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D0o
Mi~c0
Preface
The present compendium is the first issue of material intended as a textbook dealing withmathematical modelling and numerical simulation of thermal material processes, especiallycasting. The textbook will be used primarily as teaching material at the Department ofManufacturing Engineering, Technical University of Denmark, and mainly for course80223 Numerical modelling of thermal processing of materials (previous title: FoundryTechnology - Advanced).
The aim is to introduce technically related numerical methods in general, especially controlvolume based difference methods in such a way that the student gains an insight into theirformulation and application. An understanding of these aspects is essential before startingthe actual technological application of numerical methods or advanced theoretical studieson the subject.
The compendium is based partly on notes written previously on this subject by Associateprofessor, Ph.D. Preben Nordgaard Hansen.
To benefit from reading the compendium, knowledge of basic heat transfer, equivalent tothe level in the course 80201 Introduction to Processing Technology, is required.
Thanks are due to Jens Ole Frandsen, Ph.D. student, for careful revision of the manuscript.
Only minor corrections have been made in the 1999 version of the compendium. Inaddition, the example programs listed in the appendices are now written in theprogramming language ANSI-C, which in the future will be used as the programminglanguage in the course. Also, an extra appendix has been added dealing with how tocalculate the error function by means of Simpson-integration.
DTU, February 2000.
Jesper Hattel
Mathematical Modelling andNumerical Simulationof Casting Processes
Jesper Hattel
Thermal Processing of MaterialsDepartment of Manufacturing Engineering
Technical University of DenmarkLyngby
2000
Contents
Preface 1
Contents 2
1. Introduction 31.1 Introduction 31.2 Definition of the geometry 41.3 Governing differential equations 41.4 The discretization equations 61.5 Criteria functions 8
2. Numerical simulation of the solidification process 92.1 Introduction 92.2 Control volume based FDM 9
2.2.1 Heat conduction 102.2.2 Heat balance 112.2.3 The explicit method 152.2.4 Stability analysis 152.2.5 The implicit method 172.2.6 Formulation in polar co-ordinates 192.2.7 Multidimensional heat conduction 23
2.3 Release of latent heat 24
3. Casting of a plate in a sand mould or die 263.1 Introduction 263.2 Analytical solution 273.3 Comparison of heat quantities 293.4 Mesh generation, -the conjugate problem 31
Literature references 34
Appendix 1 36Solution of tridiagonal system, the TDMA-algorithm
Appendix 2 39Computer program for simulation of 1-D solidification
Appendix 3 43Calculation of the error function by means of Simpson-integration
1. Introduction
1.1 Introduction
There are many parameters and variables that describe and deternine the solidificationprocesses in castings. It would therefore be very costly and extremely time-consuming todesign and develop a computer program that could optimize all factors (the best process,alloy, geometry, layout, mould material, melt temperature, etc.) based purely on demandsfor the final product. When computers are used to determine optimal casting conditions, itis therefore important not to spend time on calculations that have already been made. Inother words, each time a new casted part is planned, the empirical knowledge establishedearlier should first be used. Such knowledge can usually be applied to define reasonableboundary and initial conditions for the numerical calculations.
Generally the first choice would be one of the available processes in a foundry, i.e. sandcasting, die casting, etc. A suitable alloy must then be selected. Simulation of thesolidification process can then help to optimize properties and microstructures in thevarious sections of the casting in respect of cooling conditions during the solidificationprocess.
A very important parameter for the different applications of castings is geometry anddesign. The geometry depends of course on the properties, e.g. the microstructure. There isthus relatively little freedom for the casting.
The starting point for a numerical simulation is the geometry given by the customer,modified perhaps for castability on the basis of empirical knowledge gained earlier. Thegeometry then forms the basis for the calculations that will result in the optimization of thecasting process. This optimization will include evaluations of the layout, i.e. inlet- andfeeding system, mould materials including chills, casting temperature, temperature of themould materials, porosities, reduction of thermal crack tendency, increase of feederlengths, etc.
In a foundry or steel works, parameters such as alloys, process and geometry, will bepredetermined to an even higher degree. Besides optimizing these parameters withinnarrow limits, the mathematical modelling and the numerical simulation will be relatedespecially to optimization of the casting velocity, cooling rate and layouts (i.e. castingconditions, chills, feeders etc.).
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1.2 Definition of the geometry
Starting with the engineer's description of a machine part (or other item) to be cast, atechnical drawing (paper) is entered into the computer by means of the CAD program inthe simulation system, or by a tape or diskette containing CAD drawings of the castinggeometry, that can be read directly by the CAE simulation system. Shrinkage as a result ofcooling from the solidus temperature is compensated for, and a basic geometry for thegating and feeder system is determined. A grid division of the whole geometry, includingthe mould, is then made - the so-called grid- or mesh-generation.
1.3 Governing differential equations
After the geometry has been defined and the mesh generation has been performed, the mostdemanding part of the numerical simulation follows, in respect of both the algorithmicdevelopment and the requirements for computer capacity - the actual calculation, i.e.solution of the governing equations. The most usual approach here is to solve all the basicequations, this being a prerequisite for simulating all relevant casting problems of atechnical nature. To gain an overview of the complexity of such a procedure, the governingdifferential equations are given below (in terms of tensors to save space). In the followingwe will deal only with the numerical solution of the heat conduction equation (1.6) forsimulating the solidification and cooling process.
Mould filling:
3 momentum equations
a li + Pi = ptii + p(uju,) (1.1)
where the stresses are given by
a= l- 4p + p'(u 1.j + u-. - I '5iuk.k ) + 4"5yuk.k (1.2)
The continuity equation
b = -(pu)j (1.3)
The energy equation with convection terms
pc T +u, (pc, T)., = (kT i ) ,i +011" (1.4)
Diffusion:
Fick's 2nd law=(Dc.j)ý (1.5)
4
Solidification:
The energy equation with release of latent heat
pc~t = (kT,) +Q" (1.6)
Distortions/residual stresses:
3 equilibrium equations
ai~l +p, = 0 (1.7)
6 strain-displacement relations (small strain theory)
t1 Owa+ iii) (1.8)
6 constitutive equations&~ (1.9)
In equations (1.1) - (1.6) the dots represent derivatives with respect to time, i.e.T4 = 07T/ dt , while the dots in equations (1.7) -(1.9) represent increments, i.e. & = AC.
It is clear that these calculations, in which primitive fields such as temperatures,displacements, stresses, velocities, pressure, etc. are determined, require the solution of thegoverning differential equations. As in other central parts of classical mechanics, theseequations often comprise several coupled partial differential equations with one or moredependent variables (e.g. temperature TI) and four independent variables (three spaceparameters, x, y, z and time t). All these equations express a certain balance or aconservation principle, established by specific physical quantities as dependent variables.These variables will be the temperature in the heat conduction equation, the concentrationin Fick's 2nd law, the stresses in the equilibrium equations, displacements in the Navierequations, and the velocities in the momentum equations.
The partial differential equations all represent a certain balance in a specific flux,understood here as physical quantities expressed by a potential, which influences thedependent variable (the potential). In heat conduction, the flux is thus the heat flow (theheat flux) and the potential temperature.But what in fact is such a numerical solution of one or more partial differential equations?it is important to emphasize here that a numerical solution of these equations consists of aset of numbers (values at certain points) from which the distribution of the dependentvariables can be determined. Thius a numerical solution corresponds in a way to alaboratory experiment where numerous measurement results form the basis for thedistribution of a particular dependent variable. The problem that previously was describedby a differential equation with a corresponding continuous solution, has been discretized to
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a problem where the solution as mentioned before, is a finite number of values. The aim ofnumerical solution of partial differential equations is therefore to determine the value of thedependent variable at various defined points. Thus a numerical solution of these equationswill always have these values of the dependent variable as the primary unknowns. Thenumerical method is then concerned with the setting up of algebraic equation systems,linear as well as non-linear, in the primary unknowns, and at the same time with thedevelopment of algorithms for the solution of these equation systems.
1.4 The discretization equations
The resulting equation systems in the primary unknowns are written so that for each nodalpoint in the grid there is an equation for every dependent variable. These equations areoften called discretization equations. In order to set up such equations, and in particular touse them as an approximation for the continuous solution of the original differentialequations, it is necessary to predict the way in which the dependent variable will varybetween the nodal points in the grid. Presumably this assumption seldom applies over thewhole calculation domain. On the contrary, separately defined profiles are used so that thedependent variable can be identified in a "smaller area" as a function of the values in thenodal points (often only the neighbouring nodes). Consequently, it is beneficial tosubdivide the calculation domain into subdomains, often called cells, elements or controlvolumes, so that separate profiles can be applied for each subdomain (Figure 1.1).
Higher order profiles Linear profiles
FIgure 1.1 Interpolation functions in cells or elements.
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The separate connection of interpolation functions to the subdomains means that the valuesof only a few nodal points (usually neighbouring points) appear in each discretizationequation. It can be shown that the solution of the discretization equations approaches theexact solution of the differential equation when the number of nodal points increases andgoes towards infinity. The distance between the nodal points will be reduced and gotowards zero when the number of nodal points increases, and the actual profile assumptionin the control volume will therefore lose its importance. This relation, often referred to as"the convergence theorem", is fundamental for numerical solutions of both ordinary andpartial differential equations. It should be emphasized that for a set of governingdifferential equations, there can be many different sets of discretization equations. They aretherefore not unique for the particular differential equations, but depend upon thenumerical method chosen.
Numerous numerical methods can be used, the most common being:
- Classic Finite Difference Method (FDM)
- Control-volume-based Finite Difference Method (CV-FDM)
- Finite Element Method (FEM)
All these methods use the discretization of volumes (grid generation) mentioned earlier.The methods have been used traditionally in the various areas as follows:
Mould filling:
- Primarily FDM and especially CV-FDM
- Seldom FEM
Solidification:
- Primarily FDM, CV-FDM and FEM
Distortion/residual stresses:
- Primarily FEM and some FDM (before 1970)
- Almost never CV-FDM
From a historical point of view, the numerical solution of partial differential equationsactually began already at the end of the 1800s (with very simple calculating machines andpaper and pencil !!!) in the application of classic FDM, formalized by Courant[1] in 1928.
With the advent of electronic calculating machines in the 1940s, there was a rapiddevelopment and CV-FDM was elaborated as a generalization of the classic FDM. It wasused especially for thermal calculations and flow calculations. In connection with flowcalculations, the work of Harlow & Welsh[2] from 1968 deserves mention. Here the flowequations are formulated on the staggered grid - a formulation that agrees well with theunderlying physics. The method is still the basis for more advanced flow calculations.Solidification calculations were first made with classic FDM, but as the method proved tohave serious limitations, FEM began to be used in the 1970s, and especially CV-FDM. Thework of P.N. Hansen[3J was one of the most fundamental contributions.
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As regards deformation and stress calculations, the early numerical calculations based onclassic FDM were used on one-dimensional beam problems, Bathe & Wilson[4], and two-dimensional stress problems were described by means of the biharmonic equation Griffin& Varga[5]. With the introduction of FEM in the 1950s and 1960s, this method becametotally dominant. Within the last five years, CY-POM has also been formulated forstress/strain calculations by Hattel & Hansen[6] and Cross & Baily[7]. The last mentionedfornulation resembles somewhat FEM.
1.5 Criteria functions
Considering the limitations of present-day computers, it is convenient to focus on only afew problems at a time. The number of equations to be solved is thus limited and thenecessary calculation time is reduced (CPU time). With relatively few equations in themathematical model, it is also simpler to find quick solution algorithms as well as tooptimize grid generation, i.e. obtain the desired geometrical solution with the minimumnumber of grid elements.
For each phenomenon/problem to be treated, it is often necessary to have a criteriafunction that can transform the calculated primitive fields (temperature, velocity, concen-trations, deformations, stresses, etc.) into useful and readily applicable information.
- Will the part be tight?
- Will it be sensitive to heat cracks?
- Do cold laps occur?
- Does the part maintain the prescribed geometrical tolerances?
- Does the part contain excessive residual stresses?
Such criteria functions are at present very loosely defined, but due to their key positionthere is a rapid development, and the area is thus the subject of intense research. As theprimitive fields become more accessible via improved calculation programs and increasedcomputer capacity, it will be better to evaluate the criteria functions by relating them toempirical knowledge.
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2. Numerical simulation of the solidification process
2.1 Introduction
For numerical simulation of a solidification process, it is a prerequisite, as mentioned inChapter 1, that the equations for mould filling, solidification and deformation/stress aresolved. It is often assumed, however, when describing the solidification process, that it isindependent of mould filling and deformation, so that the simulation is based only onsolution of the energy equation (1.6).
2.2 Control volume based FDM
The control volume method is easily understood because it is simple to interpretphysically. The method uses the underlying conservation principle (energy conservation),which is related to the physical process (heat conduction) that is to be analysed. This isdone together with the fundamental flux (heat flow) that describes the phenomenon,resulting in the governing balance equation (the heat conduction equation in differenceform).
Control volume based FDM has first and foremost been used for the numerical solution ofthermal and fluid mechanical problems, among them the simulation of the solidificationprocess. An obvious advantage of the method is its great flexibility in connection with theinclusion of non-constant material data; moreover, the structured calculation grid providesan extremely rapid grid generation, a factor that is of particular importance when using 3-Dmodels.
As mentioned, the method is based on the fundamental physical balance, that governs theproblem, i.e. for solidification calculations, the heat balance or energy conservation. Unlikeclassic FDM, it does not use the governing differential equation, but the physical basis forthe governing differential equation (Figure 2.1).
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CV-FDM Classical FDM
Balance on finite Balance oncontrol volume infinitesimal volume
results in a
DIFFERENTIAL EQUATION
which isdiscritized into a
DIFFERENCE EQUATION
DIFFERENCE EQUATION
Figure 2.1 Difference between classic FDM and control volume based FDM.
2.2.1 Heat conduction
In diffusive heat transfer (heat conduction), the well-known Fourier law is used as the basisfor describing the model
q=-kA46 (2.1)
where
q is the diffusive heat flow (heat flux) perpendicular through the surface of the area A, [W]
A is the area of the surface, through which the heat flow takes place, [m']
k is the heat conductivity, [W/(mK)]
T is the temperature, [K] or [°C]
x is the descriptive space parameter perpendicular to the surface [m]
Fourier's law is discretized to a form that is more compatible with numerical calculationsby replacing the partial derivative by a difference quotient
q =- , = _-k4 AT (2.2)& Ax
When rewritten, we obtain
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AT(2.3)
where the expression in the numerator is defined as the thermal resistance R,.Ar
R, = Ak [0C/W] (2.4)
Combining (2.4) with (2.3) gives
ATq=-- =, -AT=Rt~q (2.5)
Rd.
which applies not only for a thin plate, but also e.g. for a rod of length Ax, see Hattel[12].The last term in (2.5) gives a formulation identical to Ohm's law for an electric conductor,(U = "l). The thermal potential -AT corresponds here to electric potential U, the thermalresistance RP to electric resistance R,, and thermal heat flow q to electric current 1. Animportant consequence of this is that the coupling rules for electrical resistances can beapplied directly to thermal resistances. Thus for m resistances coupled in series, oneobtains for the equivalent resistance
R =>IR',h.i (2.6)
i-i
and for m resistances coupled in parallel
= E -•(2.7)Rd j-1 Rth.i
In connection with the development of numerical methods for simulation of thesolidification process, (2.6) will be used most often.
2.2.2 Heat balance
In setting up the governing difference equation, the principle of energy conservation isused. This principle is set up for the control volume under consideration (Figure 2.2) andthe difference approximations for fluxes are then inserted (the name of the method: ControlVolume-based Finite Difference Method gives the acronym CV-FDM).
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qt.
q - .Tq
z A½,q,
Figure 2.2 Heat balance on a 3-D cartesian control volume
The heat balance is now written for a control volume in general
Change of heat content pr. time unit = Sum of heat fluxes into the volume over N surfaces+ volumetric heat generation, i.e.
' qj + 0,. (2.8)i-I
Note that the choice is to calculate positively into the control volume. The left-hand side of(2.8) is given by
Vpc' - (2.9)
Now, the simplest case is chosen - a one-dimensional Cartesian control volume, Figure 2.3,i.e. N 2 in (2.8).
i-I i i+1
0 0 0
AX .1 Ax-i + i,-
Figure 2.3 Heat balance on I-D Cartesian control volumeThe heat fluxes are expressed by means of (2.5)
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Ti - Zi_S-. (2.10)
q. - - {I T"-} (2.11)
Inserting these expressions in (2.8), together with (2.9), gives
Vi (PC,)i a- Riti - I-T- i+Q (2.12)
The time derivative is discretized and the expression is rearranged as
AT T,1 - ,,, - .(pc' ) i 6 - +!L-'+". (2.13)
If the definition of thermal resistance, (2.4), is used, and it is recognized that the cross-sectional area constant is equal to A, the resistance from the centre node in a controlvolume and out to the surface, is given by
R , = 2Ak (2.14)Ak i 2Ak i
The thermal resistance between points i-i and i can then be calculated as
R -1 = -, + -4- + M- 1 (2.15)- 2Aki_, 2Ak i A
where M__,4A is the transmission resistance on the interface between the control volume i-1 and the control volume i. This resistance is used only with a transfer between differentmaterials. The size of the heat transfer resistance, depends largely on the casting process.Thus the resistance is greatest for sand casting, but much less for casting in permanentmoulds (steel dies) due to the significantly better thermal contact between the casting andthe mould. In the inner areas of a casting or mould, the value of the interface resistance iszero.
(2.15) is inserted in (2.13), and the volume V, is expressed by the space increment, Ax, (thelength of the control volume) and the area A.
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At. _ > -r ~ -r
Ax(pci-I + Axi + M_--_ Ax + Ax,., + M,•.. + (2.16)2AkI t 2Ak, A 2Ak, 2Aki., A
Which is divided by area, A.
Ax(pc AT T,_,- T T+ +I -T + Q',x At AxA-i l + A (2.17)
___H + ' +M_ ._+ _ + M•.÷ A22k,, +k, - 2ki 2ki+
The equation is written the following way
HOj AT = x, (T,-, - Tj) + nxi, , (T.+, - Ti ) + o ' (2.18)
A
where the capacity function, H10, is given by
H0j --- x, (,oc,) , / At (2.19)
and the conductivity function, Hxj, is given by
1Hxx, A,, + x (2.20)
X-I + --i + gi l-,i
2kiI 2k i +
For the solution, the governing difference equation is usually written in the form of (2.18).To express the problem in a form that can be solved numerically, the transient term ATJAtstill needs to be discretized. For discretizing the temperature in time, the followingdifference approximation is used
AT VT . - T(2.21)
At At
where T,' is the temperature at time t and Tjt"tA is the temperature at time t + At.
(2.18) is hereafter written the following way
HOj(Tj - T )= Hx (T_,-,-Tj) + Hx,+, (T,+, -T)+ Q (2.22)
A
The problem now is at what point in the interval [t ; t +At] can the right-hand side of (2.22)be written? For this purpose there are many different possibilities; two of them, the explicitand the implicit methods, will be discussed in the following.2.2.3 The explicit method
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With this method, the right-hand side of (2.22) is prescribed at the "old" time level, i.e. attime t. We obtain
HOj (Ti M -27j) = Ifx,(TA', - T') )+Hx~,(TA. - T27) + MLt (2.23)A
Presuming that the temperature at time t is known, the only unknown in 2.23 is the central(the control volume "i") temperature T, at time level t + At. This unknown temperature cantherefore be isolated directly (the result is an explicit equation in the unknown, thereforethe name of the method).
Ti,& i Hx, (T,', -j) + Hx1 +, (TA, - Ti') +OQ3 /A -(2.24)HOj
E quation (2.24) is the final descriptive difference equation that must be solved togetherwith initial and boundary conditions for a given one-dimensional heat conduction problem.
The procedure is that all temperatures for i = 1ito n (= the total number of control volumes)are initialized according to the initial conditions. The temperatures for time to = 0 are thenknown. (2.24) is then applied together with the boundary conditions in order to find thetemperatures for time t, = t, + At = At. The temperatures for time t, = At are then knownand (2.24) can then be used to determine the temperatures for time t, = 2At and so on. Thetime step need not necessarily be constant, but can change with time.
The material parameters that are contained in the weight functions are written as fortemperatures on the right-hand side at the "old" level. This is not immediately apparent.Dependency goes also in the opposite direction so that both the heat capacity and the heatconductivity are temperature-dependent. This means that the problem should actually besolved simultaneously, i.e. both temperatures and material parameters are unknowns. It canbe seen, however, that the dependence of the material parameters on the temperature in theheat conduction equation is relatively weak, and it is therefore possible to prescribe theseparameters at an explicit time level, i.e. to use the already known ("old") values. Thisreduces the problem considerably, to the explicit equation (2.24). A prerequisite for thisprocedure is of course that the time steps are kept acceptably small during the calculations
2.2.4 Stability analysis
Both numerically and mathematically, the explicit method is very simple but there is adrawback in that the calculations are not stable for all the values of the time step At.Consequently, oscillating temperature fields begin to be produced when the time stepexceeds the so-called explicit stability limit. It is a well-known phenomenon from othernumerical methods, that when the change from one condition to the next is too great (hereexpressed by the time step), the solution begins to oscillate, often becoming unstable andgoing towards either plus or minus infinity.
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For a stability analysis of the explicit method, it is necessary to rewrite (2.24) so that thecontribution of each temperature is written on the right-hand side in a separate term.
,+ H (Hxi Hx, , Hx i÷ +O,.,.1AI + .ý x ,, "'0,i +-T0 H .T., + (2.25)
H HO, HO,)- HOj HO,
For the stability analysis, the coefficient in front of the central temperature T, on the oldand on the new time level, is now considered. For all parameters fixed, other than thecentral temperature, the following reasoning applies. The case is considered where thecentral temperature at the old level is T,' and at the new level T,"'÷ .T7 is changed so thatit has the value T,', resulting in the new temperature
7'÷+. The reasoning is the following.
If T' is greater than T7' (i.e. the change has been positive on the old level), T,'`M shouldalso be greater than T`7 (i.e. the change should also be positive on the new level).Anything different would go against energy conservation. (Note that all the otherparameters are considered constant). The reasoning can be written as:
V -T/0=,ZI+& -T" M& Ž>0 (2.26a)
T -T <0:=>*4',& -T" 6 ' <0 (2.26b)
For the central temperature set equal to T,'*' , we get
nx fx,,, _ ,/A"4 T'+ 1 f- Lix, , T' +H' T, + (2.27)HO,0 HO HO, HO, HO,
and correspondingly, for the central temperature set equal to T,÷' , we get
Hx -x/,I + Hx T !T' + (2.28)= TiH + (1 HO, , ) HO, HO,
Subtracting (2.27) from (2.28) gives
T1H -& (Ti, - T7) (2.29)
As the temperature difference between left and right sides has the same sign according to(2.26), the coefficient before ,' - T,' must be positive or zero, i.e.
I_ Hx, Hx).j> 0 (2.30)HOJ HO•
To express the demand of the time step At, we rewrite as
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HO i > Hx + Hx1 , (2.31)
The relevant expressions (2.19) and (2.20) are inserted
Axi(Pc )i ? 1 + 1 (2.32)At Ax. , Ax. Ax. Ax.
2k;l.T 2ki 2ki 2kj,
and the time step is isolated
AtAX,(pc,) A 1 (2.33)
2ki-I 2k* 2k; 2ki÷I
As can be seen in (2.33), there is an upper limit for the permissible time step in an explicitcalculation. Furthermore, it will be seen that increasing the heat capacity stabilizes thecalculations (increases the permissible time step), whereas a refining of the grid, i.e.smaller Ax, increases strongly the tendency towards instability (diminishes the permissibletime step by the square of the reduction in Ax, when the transmission resistance is zero),and finally, that a large heat conductivity reduces the stability (meaning that the heat istransported quickly and "a lot happens in a short time").
2.2.5 The implicit method
The limitations that exist for the time step in the explicit method are not present in theimplicit method, which proves to be "absolutely stable", i.e. the solution goes neithertowards plus nor minus infinity. There is no guarantee, however, that nonphysicaloscillations will not arise in the solution. In such cases the numerical solution isinapplicable.
When using the implicit method for one-dimensional heat conduction, the governingdifference equation is still (2.22) but the right-hand side is written now at the "new" timelevel, i.e. at time t + At. Thus
H0, (Tj'+ - T,') = )1x( (Tt', - T/'' *) + Hx,+ (T - T1') + Q, /A (2.34)
The equation in (2.34) is implicit because the unknown central temperature T,'+' cannotnow be isolated and expressed explicitly by the known temperatures of the neighbouringpoints (these temperatures are not known either). Thus for each inner point 'T' there is anequation with 3 unknowns, T_'+&, T,.'',TZ÷'÷. The solution can only be made byformulating and solving an equation system consisting of n-2 equations formed as in (2.34)for n-2 inner nodes, and 2 further equations for the two boundary nodes (one at each end).The unknowns in (2.34) are now isolated on the left side.
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-HxJ Ti_:+(HO, +Hxi +Hxi÷,)T"M+ -Hx.,T,j," =HOT1 ' +),,,,/A (2.35)
The coefficients in front of the unknown temperatures on the left side are called A, B andC, and the right side is called D. Thus for i=2,....., n-i, we have
AivT I +B1 TI+ + C,2TI = Di (2.36)
where
Ai = -Hx,
Bi = HO, + H x, + nxix,
C, = -Hx,+,
Di = HO, T + 0,11/A
The equations are collected in an equation system written in matrix form. For the n-2 innernodes, (2.36) is used, and for node 1 and node n the equations for the formulation ofboundary conditions are used (see later).
BI C, T, DIA 2 B2 C2 T2 D2
A3 B3 C3 T3 D3
. .(2.37)
A, 2 B, 2 C- 2 T- 2 D,_A_, B._I C._, T._, D.,
A, B. T D.
As can be seen, A, B, C and D are vectors with n elements.
The boundary conditions must be formulated in equations for points 1 and n. Here onlytwo different types of boundary condition are discussed: The Dirichlet condition (knownboundary temperature, T1,,, and a special case of the Neumann condition (knownboundary flux, q,.), namely isolated boundary (boundary flux q,, = 0).The Dirichlet condition (known temperature)
r,, = T(t) (2.38)
The Neumann condition (boundary flux = 0)
T 4 ,, - Tb,,a = 0 (2.39)
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where "+" and indicate that the points are located on either side of the boundary. Ifthese temperatures are identical, heat cannot be forced through by means of conduction,according to Fouriers law (2.1). This corresponds in fact to an isolated border. Writingthese equations for points 1 and n, we obtain
The Dirichlet condition (known temperature)
i =1: T, T, T(t ) ==> Bý =1 C, =0 D,=T, (t)
i=n: T, = (t) => 4 =0 B. =1 D. =T(t)
The Neumann condition (boundary flux = 0)
i=1: q,-2 =0 => B,=1 C,=-1 D,=0
i=n: q-,• = 0 . .4,=-1 B =1 D=0
The equation system in (2.37) is called tridiagonal as only elements different from zero arefound in the main diagonal and in the bidiagonals above and below the main diagonal.Tridiagonal equation systems are solved simply by Gauss elimination with e.g. backwardsubstitution. This leads to the well-known TDMA procedure (Tri-Diagonal-Matrix-Algorithm) given in Appendix 1.
2.2.6 Formulation in polar co-ordinates
When working with a cylindrical geometry, it is often an advantage to use polar co-ordinates. In the following, the descriptive equations for diffusive heat transfer will bederived with one descriptive space parameter. In Figure 2.4 an arbitrary control volume isshown in a polar system, with radius r as the only space parameter. The distance from thecentre out to the point "i" in the control volume is termed ri.
Artj19Aaý AnL At< ,
19
Figure 2.4 Section of 1-D control volumes in polar co-ordinates.
Equation (2.13) still applies for the control volume with thickness Ar1, but the expressionsmust be changed because of the polar description. The control volumes are now cylindercaps of Ar, thickness and Az height. For the volume we obtain
Ari 2 _ (Ar + 2rAr
=Azz r, 2 + i 2+ 2ri ý,_2- - ri 2J + 2r.A
= 2Azr, Ar, (2.40)
In Figure 2.4 it will be seen that the two resistances in the control volume are now nolonger equally large, because of the cylinder geometry. The resistance from the centre tothe left surface is, Pitts & Sissom [8]:
In rer,-Ar= ,wk 2 (2.41)
2,tAzk,
while the resistance from the centre to the right surface is
r+ Sr
In 2
2z'Azk , (2.42)
In this way the resistances contained in the general balance (2.13) are found
Ar r,r,-, In
In 2 ArMA,Ri-,, =Rt +-R"i -+ R-je =- + 7 + (2.43)
2sTAzk, 2ZAzk0 2fAzr -
20
Ar In r +lIn 2 Ari+
Rijij, = R."' + R tý + = +R + 2+ + -- (2.44)2 1Azk + 2,rAzk, 2Az( +1 (2.'2)
These expressions are inserted in the balance (2.13)
2 r.Azri j(x ,Oc )i =A Ar,4 rr,-, + 2 In.
In Ar,+ ' 2 +
2nzk,, 2nfAzk, 2ffAzf>
+ T+ T+ (2.45)
r, + _= ' In r,In 2 2t,+!
r r r' 2 Mi+j+ +
2rAzk i 2,Azk,+, 2rAz(r,& . _ A- "'I2/
Just as the area A could be abbreviated in the 1-D Cartesian case, so can 2irAz here betreated likewise. Thus
AT T-1 -r 'A,(PC), At - r,., r
r,,+ 2 In r rIn -r I
?;] 2ki,_ + k, + r, -r
2
+ T+1 - T (2.46)r,+ A In r" 2zrtz
In r -2 I Ar,r, ri., 2 +Mi-*i+1
ke ki~l Ar,.. 22
The following four geometrical functions are now introduced
21
gi = In r, Ar, (2.47)r, 2
f, =In 2 (2.48)
Ar.e, = r2 (2.49)
n; = r& Ar, (2.50)
The advantage of defining these vectors is that they need to be calculated only once, at thebeginning of the programme, since they contain only geometrical information.Furthermore, they serve to make the algorithms more clear. Thus
R/-m4j = 2,rAz- R-_,• = L-- + Li +M , (2.51)ki_1 k, e,
R= i = 2f-Az* R€, = f +-+ (2.52)ki ki41 e.,
(2.46) is now written as
n(c AT. T,- -T, ~. .'A , , 2no,., (2.53)
As regards the time discretization, use of either the explicit or the implicit method will leadto similar expressions as obtained for the Cartesian case, as long as the general equation(2.53) is applied.
22
2.2.7 Multidimensional heat conduction
Extension of the numerical solution of the heat conduction equation from one-dimensionalto multidimensional is too comprehensive an area to be treated in detail so that theprogramming of the algorithms is apparent. Only the basis for such an extension ismentioned here.
The starting point for a multidimensional description is again a heat balance but now on a2-D or 3-D Cartesian grid where the central volume with side lengths Ax, and Ayj is givenby two indices (ij) in 2-D and in 3-D with the side lengths Axi, Ayj and Az, and threeindices (ij,k). In the following, the explicit algorithm is shown in 3-D.
The capacity function HO, corresponding to equation (2.19), is now
HOik = AXAyAzA (pcP)yk1/At (2.54)
and the three conductivity functions Hx, Hy and Hz, corresponding to equation (2.20), areexpressed as
Hx = AYJAzA (2.55)Uk A '+I Ax '+ Mi-i
Zkiy ,, =k ilk
Hyý=Aj AytA1 (2.56)+___ +M
2ky_• k 2kijk
= Axi 'Ay (2.57)A xkz AzyA.___ + kz•+ M# -2ko,,,_- 2koi, -
The explicit algorithm corresponding to (2.23) appears as
H 04,(T •' - 1'k) = Hx•, (T,'_, - - T'k) + Hxn,.11 (rj. - T,'k)
+ n y~i, ( TU'_ k- Tl;,) + nHy ij, II(T•]÷,'+ýk- T'k)
+ Hz ij (Ti,_ , - Ty;, )+ +z 11 k÷, (Tijk., - Tj' ) + ýM•jk (2.58)
This extension is the simplest one. As with 1-D, there is a stability requirement for theexplicit method in 3-D, but the requirement is even more restrictive. For this reason,explicit algorithms are seldom used in the numerical simulation of casting processes in3-D. Instead, the choice would be partly implicit (ADI) algorithms (Alternating DirectionImplicit) or fully implicit algorithms as e.g. SIP (Strongly ImPlicit). In FEM (Finite
23
Element Method) the well-known implicit methods such as matrix-inverting or Gausselimination would normally be used.
2.3 Release of latent heat
In the numerical simulation of the solidification process in casting, the modelling of therelease of solidification heal is very important. In principle, there are two different methodsthat can be used. In the simplest method, the solidification heat is merely modelled over thesource term, i.e. Or,. The disadvantage of this method is that it reduces drastically thestability of the calculations (this is not apparent from the stability requirement (2.33)). Onecan imagine, though, that if a large heat-producing term is prescribed in the equations,considerable changes occur in the time step, and the stability is thus reduced.
A more stable way to model the release is to add an unrealistically large heat capacity inthe solidification interval so that for each degree of cooling, considerably more heat isreleased than when applying the normal heat capacity. This addition is made in such a waythat an amount of heat is released that corresponds precisely to the solidification heatbesides from the usual "heat from cooling" in the solidification interval, when the solidustemperature is reached.
In the modelling, it is necessary to have a description of how great a fraction, f, (fractionsolid) is solidified as a function of the temperature, i.e. f,(1IJ. For temperatures above theliquidus temperature or equal to it, fs is equal to 0, i.e. everything is liquid, and attemperatures under the solidus temperature, f, is equal to 1, i.e. everything is solid (see thecurve in Figure 2.5).
Ts TL
Figure 2.5 f. (fraction woid) as (unction of temperature, here linear in the solidification interval.
In the literature, various models can be found for describing f, as a function of thetemperature in the solidification interval, among them the lever rule model, Scheil's model[9], Brody and Flemnming's model [10], and Kurz' model [11].
With a given fraction solidified, f,, (he amount of heat released [Jim'], corresponds to f,multiplied by the total solidification heat L [I/kg] multiplied by the density p [kg/in'], i.e.
QZý fS A(2.59)
24
It is assumed here that the solidification heat is uniform for the phases that are producedduring solidification. The power [W/m3] that is released at a certain point of time is foundby
C). 67, '(2.60)
rewritten as
6T, _ O (2.61)"3T W Pt
Inserting in the transient heat conduction equation gives
Pep = !:+6 ký! 9( T (2.62)
where (2.61) is applied
,iT d( t r ( 5T,• d cW ,•sc3TPC, -7 = 14 &d(ky 7 t (2.63)
The contribution from the solidification heat is now drawn over to the left side
{ Ls LfTf=..9}(k=I ýk• + Jk +9(k7 (2.64)
Thus in the solidification interval, there should be applied an adjusted ;,-value as
- C (2.65)
It should be borne in mind that ,',/dT is negative in the solidification interval (see Figure2.5) so that c, increases in the solidification interval.
Finally, it should be mentioned that the method described, whereby solidification heat isinserted in the specific heat capacity, stabilizes the calculation considerably (this can beseen also from (2.33)). The method is also the one generally used in the majority ofcommercial programs that can simulate release of latent heat as a consequence of phasetransformation.
25
3. Casting of a plate in a sand mould or die
3.1 Introduction
This chapter deals with the simplest case from a geometrical point of view, namely castinga plate-formed part in a sand mould or a die. The heat flow is presumed to be one-dimensional perpendicular to the plane of the plate. It is furthermore assumed that the plateis cooled symmetrically so that only half of the plate is considered. Even though such acase is somewhat restrictive, it has been shown that the resulting numerical calculations arein rather good agreement with experimental measurements.
The following mesh is used as shown in Figure 3.1.
Part Mould(Domain 1) (Domain 2)
Symmetry Inter- Airline face cell
I • I .......... I ° 1 ° .......... I • I • I1 2 N, N,+l N,-1 N,
Figure 3.1 1-D enmeshment in a plate casting with mouldPart :2 to N1, Mould: N,+l to N,-1.
In Appendix 2, a proposal is put forward for a construction of a computer program thatsimulates the thermal conditions for the case shown in Figure 3.1 (N, = 6 and N2 = 15),using the explicit method.
26
3.2 Analytical solution
The temperature field related to the casting of the plate in Figure 3.1 can be describedanalytically when certain conditions are met. As long as constant material data areassumed, the solidification heat is disregarded, and the two subdomains are thermallysemi-infinite, the so-called error function solution can be applied, Hattel[12]. This solutionis valid for a thermally semi-infinite body in the half space x > 0. The initial temperature isTi, and for t > 0 the surface temperature at x = 0 is increased momentarily to T., i.e.
Initial conditions T(x,O) = Tj for x -> 0
Boundary conditions T(0,t) = T, for t > 0
T(co,t) = Ti for t > 0
The governing differential equation is the one-dimensional heat conduction equation withconstant material data
d7T d' T Q"-T c 2 (3.1)kt a2 P
where a = k is the thermal diffusivity [m2/s].
PCP
With the above-mentioned initial and boundary conditions, and no heat development, i.e.= 0, the solution of (3.1) is given by Carslaw and Jaeger[13]
T(x,t)-T, =e{f x (3.2)
where eif, the error function, is defined by the following integral
2 _q f• -x J= 2 q
erf(u) m-= Je dr = erf jfe} di? (3.3)
The error function exists in tabular form in various works, but for use in a numericalmethod it is usual to perform the integration (3.3) numerically, e.g. by means of theSimpson integration
Isolating T(x,t) of (3.2) and (3.3), gives
T(x,t):= T + (T, -T,)9 xet , (3.4)
27
This solution can now be used for the specific problem (Figure 3.1). A constanttemperature of the interface will adjust itself between the cast and the mould if the thermalresistance in the interface is zero. This temperature is called the Riemann temperature.(Remember that the solidification temperature is disregarded.) The prerequisites for using(3.4) are now present. For the two domains we have the following analytical solution
Domain 1 (casting):
2 -x/,T ,Tl
T(x,t) = T, + (T -rT,)7= Je-'Ui (3.5)
Domain 2 (mould):
2T(x,t)=T +(T2 -T;) 1 Je-dr, (3.6)
where T, and T2 are the initial temperatures in domain 1 and domain 2 respectively. Notethe minus in the solution for domain 1; this is a result of the x-axis chosen.
The gradient in the interface between the casting and the mould is of particular interest. Itis found by differentiating the temperature fields (3.5) and (3.6) with respect to x, seeHattel [12]
OF - T- (3.7)
& (3.8)
A prerequisite for using the error function solution as a solution to the specific problem is,as mentioned, that the common surface temperature (Riemann temperature) is constant aslong as the analytical solution is valid i.e. as long as the two domains are thermally half-infinite (the temperature has not penetrated to the back). The Riemann temperature is nowdetermined by the following heat balance
ql'.o = qL (3.9)
Fourier's law is applied in (3.9)
-Ak1 r Ak (3.10)
The found expressions for the gradients, (3.7) and (3.8), are inserted
28
T s-T I• T - Tý-AkI 17 = -AkT (3.11)
* Ira1 t fja 2 t
-A/-fv is abbreviated, and the relation
Sk = kpc = 0 (3.12)
is applied. Thus
kPcj(T,-T,) j (T2.-T,) (3.13)
T, is isolated
kr2 = k+•kp T s (3.14)
62 = kpcP 1 2 is called the mould material's cooling capacity. Expressed by (3.14)
appears as
AT, - (3.14a)
$AMA
T. is now known and the equations (3.5) and (3.6) can be used as analytical solutions forthe temperature fields in domains 1 and 2, and a comparison can be made with numericalsolutions, bearing in mind the conditions under which the analytical solution applies(thermal half-infinity, no solidification heat, constant material data, constant T, .., etc.).
3.3 Comparison of heat quantities
Another suitable method for testing the accuracy of the numerical calculations is tocompare the analytically calculated heat quantity that is transported from domain 1 todomain 2 with the corresponding heat quantity calculated numerically.
The analytically determined heat quantity at time t is obtained by integrating the heat flux,(W), over the interface, given by the expressions (3.7) and (3.8), from time 0 to time t. Thechange in the heat content for domain 1 (the part to be cast) is equal to minus the heat thatis transported from domain 1, i.e.
A -0 - 11 -- Tt~~ o-Tt=-J) x7` =-O d J Tdt kpcl' d99A 1 A 0 Ir~ PII 1
29
=J-il T2- (3.15)
Likewise, the change in the heat content in domain 2 (the mould) is equal to the heatquantity received from domain 2, i.e.
A Fk2 1, 2' V2 (3.16)
Energy conservation gives
Q(1) An,6 Qt) A w
+ _ + 2 - = 0 (3.17)
The corresponding heat quantities calculated numerically are found by summing up theheat content changes for each control volume for each time step in domain 1 and domain 2respectively, i.e.
Q(t)( ) -T!") (3.18)A ;
:A1 - ± i (pc), (Ti' - T 1'-') (3.19)A 2 j-1 i-N,+l
where M(t) is the number of time steps at time t. For constant specific heat (specific withrespect to the volume) we obtain
= i - Tr) (3.20)
A 1 (-2
Qt) = AX (3.21)A 2 i-N,+l
Again, energy conservation demands
Q(t) + Q() + 0 (3.22)A A 2
Finally, the agreement of the numerical solution with the analytical solution can bechecked by comparing the numerical and analytical heat quantities. If there is completeagreement between the numerical and the analytical solutions, the following equations willbe fulfilled
30
Q(t)A (3.23)A, A,
Q(t) A Qt)
A 2 A - 2 (3.24)
However, there will always be a divergence between the analytical and the numericalsolutions so that in practice (3.23) and (3.24) will never be identically fulfilled. Moreover,the deviation between the two quantities can, as with all comparisons between analyticaland numerical solutions, be used as a measure of the accuracy of the numerical method (aslong as the requirements for the analytical solution are met).
3.4 Mesh generation, -the conjugate problem
The numerical solution of conjugate problems can be aided considerably by a wise choiceof calculation mesh. The problem is that the temperature on the interface between the twomaterial domains that are brought into thermal contact is unknown and not until theproblem is solved will it be known. However, the interface temperature governs thetemperature conditions in both materials and a good thermal description in the interfacedomain is therefore very important.
For the actual modelling of the casting of a 1-D plate, an analysis of the grid generation canbe made rather simply. We postulate that a good grid is obtained by selecting an identicalFourier number ( Fo = caAt/(Ax) 2 ) in each cell, i.e.
Foi_, = Fo1 = Fo1 ÷, (3.25)
This can be tested, by analysing whether the numerical solution calculates the Riemanntemperature correctly already in the first time step. For the two cells around the interface,we have
Fo, = Foe,', (3.26)
we insert
aj,, t Of' ,+ IAta 2 aN-,A (3.27)
At is abbreviated, and the expression is reduced
1 -1 (3.28)AXN.] Ax3, a,,,,
31
The numerically calculated Riemann temperature T, is now found by a heat flow balance atthe interface. The heat flow through RN, should be equal to the heat flow through R,.,,Figure 3.2.
TN I Ts TNI+l
aN t RNt +1
Figure 3.2 Temperatures and thermal resistances in cell N, and N,+l ateach side of the interface.
TS, - T T - TI RN,,T,+, +R,,+,TN,
T, =(3.29)RN, RN, , R,, + R,,
Applying now the definition of thermal resistance (2.4), gives
A ,, k + -- TN
T, kN, N1 ] (3.30)AXNý Ax"k N, + k N1,+l
kN, kN,+,l
k,, kNNl
extending with N, N, yieldsAN,. AxN,,,1
ky "+ TN_
T= N, k' AxN,+l (3.31)___+
AXN, AXN,+,
The result from (3.28) is inserted and we obtain
32
T,,, a,_, .
T,= (3.32)kN, k,,l+,
ArN &X,, a,,+,
The equation is divided by Ax,, and
+k k N,,÷
T, k, k,,,, (3.33)
_____ +
Using now the relationship from (3.12), we reach the final expression
kPCPN, TN, + ýkp c-jN.1T",, (3.34)
4FF7I,]N, + kpc71 N, I1For the first time step in the numerical calculation (3.34) and (3.14) are identical, as theindices in (3.14) corresponds to the domain numbers, and the indices in (3.34) correspondsto the cell numbers. In other words, the numerical Riemann temperature calculated alreadyin the first time step is in complete agreement with the analytical Riemann temperature.This will normally not be the case for non-uniform Fourier numbers, where there is often aconvergence of the numerical Riemann temperature towards the analytical Riemanntemperature.
The above statement supports the claim that the identical Fourier numbers in all cells willlead to a reasonable grid (even though it is only the Fourier numbers in the two cellsaround the interface that need to be identical in the above example).
33
Literature references
1. R. Courant, K. Friedrichs & H. LewyUber die Partielle Differentialgleichungen der Mathematischen PhysikMath. ann. vol. 100, (1928)
2. F.H. Harlow & J.E. WelshNumerical calculation of time-dependent viscous incompressible flow of fluid with freeSurfacePhys. Fluids, 8, pp 2182-2192, (1965)
3. P.N. HansenVannrevner i stilPh.D.-thesis, Institut for Metalliere, DTU, (1975)
4. KJ. Bathe & E.L. WilsonNumerical methods in finite element analysisPrentice-Hall, (1976)
5. D.S. Griffin & R.S. VargaNumerical solution of plane elasticity problemsSoc. Indust. Appl. Math, vol. 11, no 4, (1963)
6. J. Hattel & P.N. HansenA control volume based finite difference method for solving the equilibrium equationsin terms of displacementsAppl. Math. Modelling, vol. 19, 4, pp 210-243, (1995)
7. C. Baily et. al.Predicting the deformation of castings in moulds using a control volume approach onunstructured meshesMathematical Modelling for Materials Processing, IMA Conference Series, ClarendonPressOxford, pp 259-272, (1993)
8. D. Pitts & L. SissomHeal TransferMcGraw-Hill, (1977)
34
9. E. ScheilZeitschrift fur Metalkunde 34, 70, (1942)
10. M.C. FlemingsSolidification ProcessingMcGraw-Hill, (1974)
11. T.W. Clyne & W. KurzSolute redistribution during solidificationMet. Trans. A12, pp. 965-71, (1981)
12. J. HattelGrundlreggende varmelere for termiske materialeprocesserInstitut for Procesteknik, DTU, (1995)
13. H.S. Carslaw & J.C. JaegerConduction of heat in solidsOxford University Press, (1959)
35
APPENDIX 1
ANSI C - procedure based on the Tt3MA-algorithm for the solution ofa tridiagonal equation system.
36
FILENAME: Trisal.h
* ~Header file with prototype to the procedure Trisol*
/* Max number of equations *#define MaxNEQ 100
/* Prototype - Used to define formal parameters *extern void Trisol(int n,
double IKoefA[MaxNEQJ,double KoefB [NaxNEQ),double KoefCfMaxNEQ),double KoeEDL~axNEQj,double Result jMaxNEQJ);
FILENAME: Trisol.c
* Solving tridiagonal. system of equations (using the TDMA method) *
#include rrisol.h
void Trisol(int n,double KoefA(HaxNEQJ,double KoefB(MaxNEQ],double Koefc[MaxNEQ],double KoefD[MaXNEQ],double Result (MaxHEQ])
int i;double Beta [MaxNEQ], Gamma MaxNEQ];
Beta(0] KoefBro];Gaina[O) - Koeffo[0] / Beta[0];iff(n<1) Result[0J - Gamma[0];
else Ifor(i-1; i<n; i++) f
Beta[i] - KoefB[i] - KoefA[i) * KoefC[i-l] / Beta[i-l];Gamma~i] = (IKoefD[i] - KoefA[iJ * Garmma[i-1J) / Beta(i);
Result~n-l] - Ganuna[n-lj;for(i=(n-2); i>-1; i--)
Result~i] - Gammafij KoefC~i] -Result~i+l] / Beta~i];
37
APPENDIX 2
ANSI C - program for simulation of 1-D solidification
39
FILENAME: Exercisel.c
* *
• ID - Numerical simulation of solidification ** Exercise 1 in 80223 ** *** ***** *** ** **** ** **** * ***** ** ** ******* ***** ***** * ****** *** **** *** ** * *
P Including header files */#include <stdio.h>#include <stdlib.h>
/t Global - Constant declarations */#define Tinit cast 1320.0 /* Init. temperature of casting */#define T initmould 20.0 /* Init. temperature of mould */#define T air 20.0 /* Init. temperature of air */#define dt 0.005 /- Time step */#define maxstep 20 /* Number of steps in time loop */
/* Global - Array declarations */double dx(16], T[16], T new[16], hO[16], hx[16], r[16];int number[16];double lambda[2] [1700], rhocp[2] [1700];
/* Global - Variable declarations */long int step;int i, no, iT;double time, r67;char ch;
/- Start of main program */int main(void)
• Part I - Initializing model *** **** ** *** ** ****** *** ** ****** * ***** * *** ** * ******.* ** ** *** *** ** ** * *
/ Geometry, space increments */for(i=1; i<ll; i++) dx[i] = 0.002;dx[7j = 0.001;dx[8] - 0.001;for(i=ll; i<15; i++) dx[i] = 0.004;
/* Material numbering */for(i=l; i<7; i++) number[i] - 0; /* Casting */for(i=7; i<15; i++) number[i) - 1; /* Mould */
number[15) - 2; /* Surroundings */
/* Thermal properties - lambda */
for(i=0; i<1700; i++) Ilambda[0] [i) = 30.0; /I Casting *1lambda(l] (i) = 1.0; /* Mould *1
/* Thermal properties - rhoep */for(i=0; i<1700; i++) {
rhocp(0][i] = 5.0E6; /* Casting */rhocp[l] fi] = 2.0E6; /* Mould */
40
/* Initial conditions - temperatures */for(i=l; i<7; i++) T~i] = T init cast; /* Casting */for(i=7; i<15; i++) T(i] = T init mould; /* Mould */T(15] = Tair; /* Surroundings */
/* Resetting time */time = 0.0;
/******** * **** ****** ** *** ** ****** .*** ** * *** ***** . *** ****** ** **.* ****
* Part II - Time loop ***** *** ** *** *** **** *** ****** * ***** * *** *** *** ***** *** *** *******/
for(step=1; step<=maxstep; step++) {
/* Updating time */time += dt;
/* Weight functions //* Capacity terms */for(i=2; i<15; i++)
iT = (int) (T(i]+0.5);no = number(i];hO[i] - dxli] * rhocp[noJ [iT] / dt;
/ Resistance terms *//* Internal resistances in cells */for(i=2; i<15; i++) {
iT = (int) (Tfi]+0.5);no = number[i];rfi] - dx[i) / (2.0*lambda[no][iT]);
r[l] - r[21;
/- Resistance between mould and surroundings /r[15] = 0.1;
/* Resistance between casting and mould */r67 = 0.001;
/* Weight functions *1/* Conductivity terms */for(i-2; i<16; i++) hx[i) = 1.0 / (r(i-1]+r(i]);
/- Conductivity term at interface between mould and casting */hx[7] - 1.0 / (r(6J+r[7]+r67);
/* Calculating new temperatures, interior cellsfor(i=2; i<15; i++)
Tnew(i] = T[i] + ( hx(i] '(T~i-1]-T[i]) +hxhi+l]*(T~i+l]-ThiJ) ) / hO[i];
/* Boundary conditions *//* Centre of casting, symmetry */T_new(l] - Tnew[2];
/P Setting new temperature to old */for(i-i; i<15; i++) T[i] - Tnew[i];
41
/- Writing temperature profile to screen
printf("\n * ***** ***** TIME = %6.3f s***********\n\n", time);
printf("time step - %7.5f s\n\n",dt);printf("Cell Temperature \n");printf("---- \n");for(i-1; i<16; i++)
printf("%3i %7.2f\n", i, T[i]);printf("\nPress Enter!\n");ch=getchar);
I /* End of time loop */
* Part III - After time loop *** ** * ** ** *** *** ** * *** *** ***** ** ** ***** * *** **** *** *** **** * *** ** **** *
return 0; /* ANSI-C requires main to return integer /
/* End of main program */
42
APPENDIX 3
Calculation of the error function by means of Simpson integration.
43
The error function is defined in the following way:
erf(z) = 2 je-q'di (A3.1)
4T0
This function is not intrinsic in the programming language ANSI-C and therefore, it mustbe calculated in a separate procedure. This is done by means of the Simpson integration.
An integral is closely connected to sums and areas (with signs) since it is the lower limitfor upper-sums and at the same time the upper limit for under-sums. If the integral of afunction f(x) is going to be calculated by means of numerical integration, the principle isto divide the integration interval [a;b] into a number of sub-intervals and then approximatethe function f(x) with simple expressions in these sub-intervals. So, in numericalintegration the integral is approximated with an inexact area. The more sub-intervals, thebetter the approximation. When calculating the areas one might use either the trapeziumformula or the middle point formula, see the figures below.
Figure A3.1 Calculation of area by means of Figure A3.2 Calculation of area by means ofthe trapezium formula, the middle point formula.
Simpson's formula for calculating an integral (= area) is based on a weighting of thetrapezium formula and the middle point formula, where the trapezium formula has a weightof 1/3 and the middle point formula has a weight of 2/3. This leads to the repeatedSimpsonian formula
b hnn-
f(t)dt = - 1(a)+ f(b)+ 42' f(a + (2s- 1)h)+2 1 f(a + 2sh)
(A3.2)4b- a (4)-hl-f()
180where
b-a= --. (h is the half interval length, and n is the number of sub-intervals with
identical length).Comparing the expressions (A3.1) and (A3.2) shows that
44
a=O
b=z
f(t) = e-"
Using this in (A3.2) gives
Je"dt 1 + e + 4E e -( •l 'h9 + 2Y e ' • , (A3.3)
The last term in (A3.2) is just a small residual term, which can be discarded.
In preparation for calculating erf(z) for any given z-value, formula (A3.3) is now to be
programmed. This is done by first determining h as zn" Then one must calculate 1+ e-.
The latter result is stored in the variable "e". Further, "e" is made equal to
e = e + 4 exp(-((2i- 1)h) 2) + 2exp(-(2ih) 2)
This must be repeated for all 'i-values running from '1' to 'n -I'.
Now we only miss the last term in the first sum in (A3.3) and a final multiplication with
2 h-i and -. This gives
2 k2erl = (e+ 4 exp(- ((2 n - 1)h) 2 )).
The value of the error function erf(z) has now been determined. See the procedure on thenext page.
45
FILENAME: Erffunc.c
*
* Calculation of error function by means of Simpson integration
*.* *** * **** *t**.**** * *** * ***** ** ******fl *** **** *** **.** ** ** ***.***** ******
/* Header files */#include <math.h>
/* Constants */#define pi 3.14159265#define n 20 /* Number of intervals - influences the accuracy *1
/* Prototype */double ErrFunc(double z);
/- Function */double ErrFunc(double z)
int i;double e, h, erf;
if (z > 4)erf = 1.0;
else Ih z / (2-n);e = 1.0 + exp( -pow(z,2.0)for(i=l; i<n; i++)
e += 4.0 * exp( -pow((2.0-i-1.0)*h,2.0)+ 2.0 * exp( -pow(2.Oi*h,2.0) );
erf - (e+4 .0*exp(-pow((2.0*n-l)*h,2.0)))*h*2.0/(3.0*sqrt(pi));
return erf;
46
Preliminary draft from chapter 2
NUMERICAL MODELLING OF CASTING PROCESSES
edited by
Jesper HattelInstitute of Manufacturing Engineering
Technical University of DenmarkDK-2800 Lyngby, Denmark
Nini H. PrydsMaterials Research Department, RIS0 National Laboratory
DK-4000 Roskilde, Denmark
Preben N. HansenGiesserei Institut, Intzestrasse 5
Rheinland Westfilische Technische HochschuleD-52072 Aachen, Germany
2.2 A simple problem: Modelling temperatures in 1-D
In this section, some basic concepts of one-dimensional heat conduction analysis are presented.First the heat conduction equation is derived, and the appropriated boundary conditions inconnection with casting are mentioned. The very basic analytical solution for the semi-infinitesolid is shown, and the term thermally infinite is defined. Moreover, the application of simple one-dimensional heat balances to casting of plates is shown, resulting in e.g. Chvorinov's rule.
2.2.1 The heat conduction equation
Based on Fourier's law (2.1) together with the first law of thermodynamics (conservation ofenergy), the general heat conduction equation can be constructed.Consider an isotropic solid body with thermal conductivity k subjected to arbitrary thermalconditions on its surface (boundary). The temperature may vary with time and space and aninternal generation of heat Q,, per unit time per unit volume may be present within the body. Bothk (if the body is inhomogeneous or if the thermal conductivity is temperature dependent) and Q,may vary throughout the body and with time. Consider now the infinitesimal volume of thicknessdx shown in figure 2.1.
q, qA dx
A
x I dx
Figure 2.4 Infinitesimal volume in 1-D heat conduction analysis
We now apply the first law of thermodynamics to the volume, giving the following energybalance:
Change of enthalpy pr. time unit = Sum of heat fluxes into the volume over the two surfaces +volumetric heat generation, i.e.
-- = q1 - qx֥ + QgmV (2.23)at
The change in enthalpy per time unit is given by
2
aH aTt = Vpcr - (2.24)
Note that no distinction is made here between cp and c, the specific heats at constant pressure andconstant volume, since all mechanical effects are being neglected. Combining (2.24) with the heatbalance from (2.23) and realizing that the volume equals cross sectional area A times dx, yields
8TAdxpcP - = qý - q, + Q8,,Adx (2.25)
Dividing by Adx
q. q+,,8T A dAp T A A (2.26)
and taking the limit as dx approaches 0 gives
8p aT =_a Aq)+Qej, (2.27)Sat ar.A)
or in combination with Fourier's law, (2.1)
OT 8£ T7 +(2.28)c at +Qax &) P
This is the one-dimensional heat conduction or Fourier equation, which, in its general form inthree-dimensions, constitutes the basis of all heat conduction calculations in casting processes. Inthe case of constant material properties, which almost always has to be assumed when derivinganalytical solutions, k is placed outside the derivation in (2.28), resulting in
-- = a 'T + Q (2.29)
at x2 pcP
where a [mn/sJ is the thermal diffusivity defined by (2.22).
A special case of (2.29) deserves mention. If the temperature is not time dependent and no heat isgenerated within the body, (2.29) reduces to
82T d2T-0-- =0 orjust -=0 (2.30)ax2 dx 2 -
which is the 1-D version of the Laplace's equation; any solution of this equation is called aharmonic function.
In addition to the heat conduction it is necessary to specify appropriate initial and boundaryconditions in order to describe fully the physical problem under consideration. In doing that, an
3
understanding of the nature of the idealizations applied in the formulation of the boundaryconditions is necessary for a proper choice of the most suitable mathematical formulation of theproblem. The surface of a body (or part of it) may receive heat either through contact with anothersolid or by radiation, or through contact with a fluid.The initial conditions specify the initial temperature distribution throughout the body. In manyproblems, also in casting processes, the initial temperature might be assumed constant. During thenumerical simulation of a casting process, the initial temperature for the heat conduction (which isthe only heat transfer phenomenon taken into account in the mould) calculation of the mould willbe constant throughout the mould. The initial temperature of the melt is the superheatingtemperature, but this is the initial condition for the mould filling simulation, not the solidificationsimulation where the initial condition is composed of the temperature field immediately afterfilling. Of course, no need arises for initial conditions in the case of the steady-state problem of(2.30).
There are five principal boundary conditions, which are used in the mathematical theory of heatconduction as idealizations of the actual physical processes [1]. This classification is in essence,however, not only physically based but it is more a mathematical classification. The presentboundary condition classification is somewhat different, but it is believed to be more relevant for
-the modelling of casting processes
(1) Prescribed boundary temperature
T(P,t) = T(P,t) (2.31)
where the location P is on the surface (for the present 1-D analysis, just described by its x-value)and "'" denotes prescribed.
(2) Perfectly insulated (adiabatic) boundaryBy definition, an adiabatic boundary is one across which there is no heat flux.
ST-(P,t) = 0 (2.32)
an
where n is the outward normal to the surface of point P (in the present 1-13 analysis just the onedescribing space parameter x).
(3) Convection boundary conditionIn this case the heat flux across the bounding surface may be taken as proportional to thedifference between the surface temperature T(P,) and Tdt) of the surrounding medium, which isexpressed by Newton's convective law of cooling (2.6)
STk - (P,t) = h(Ta (t) -T(P,t)) (2.33)an
where h is termed the heat transfer coefficient and may vary with space and time in a prescribedmanner. Noted that the expression has been reduced in the sense that the minus signs on both sidesof the equation sign have been removed.It should be mentioned that an alternative way of obtaining an insulated boundary condition is toset h equal to zero in Newton's law.
(4) Radiation boundary condition
4
If the surface of the boundary is exposed to a high-temperature source, it will receive heat byradiation in accordance with an expression similar to (2.10), i.e. of the following form
k aT (p,t) = Ca(T (t) -T 4 (P,t)) (2.34)
where T(t) and T(P,t) are the absolute temperatures of the source and the surface respectively. Ingeneral radiation boundary conditions, being non-linear, render analytical solution of the resultingboundary-value problem extremely difficult.(2.34) is often rewritten in terms of the equivalent heat transfer coefficient for radiation h,, givenby (2.13), which results in an expression similar to (2.14)
aTk (P , t ) = h, (T. (t) - T(t)) (2.35)
where
h,,I= C(T.3 + T2T + T=T 2+ T 3) (2.36)
and the P and t dependency have been omitted for conveniency.The problem, however, of course is still non-linear, but if neither the temperature of the source northat of the surface varies over too wide a range during the period of interest, h,, can be takenconstant, resulting in a linearized boundary condition. This is often done in numerical calculations,where the time steps are of such short period that T. and T may be assumed constant during thetime step. Moreover, if that value is taken as that at the "old" time level, the boundary condition islinearized.A combination of the convection and the radiation boundary conditions are found at the surface ofa mould where the total effect of the heat transfer phenomena is accounted for by introducing thetotal heat transfer coefficient, h,, for natural convection and radiation
h, =h + h,, (2.37)
(5) Internal boundary (two solid bodies in contact)If the surfaces of the bodies are in perfect thermal contact, their temperatures at that surface mustbe the same. This phenomenon is well-known from the analysis of casting processes, where aconstant temperature of the interface will adjust itself between the cast and the mould if thecontact resistance in the interface R is zero. This temperature is called the Riemann temperature,see 2.2.3. The boundary condition is now formulated based on a heat balance where the heat fluxleaving one body through the contact surface must be equal to that entering the other body. Thus,for a point P on the contact surface
T 1(P,t) = T2(P,t) (2.38)
k, -'(P,t) = k 2 -(P,t) (2.39)
p tasn
where the subscripts 1 and 2 refer to the two bodies.
5
In reality, an imperfect thermal contact between the two bodies will always be present. When twosolid surfaces touch each other, because of the natural roughness of any material, actual physicalcontact takes place only at some projecting segments of the surfaces, see fig. 2.5.
Figure 2.5 Solid-to-solid contact.
Conduction takes place at these points of contact, while heat is transferred across the gaps byradiation and by conduction through the fluid, usually air, filling them. None of these modes ofheat transfer has significant predominance over the others, and thus all have to be included in thetotal heat transfer coefficient, h,,. The equality of heat fluxes must still be enforced, but adifference between the two surface temperatures, proportional to the heat flux will now exist. Theappropriate boundary conditions are therefore
ki -, -(P,t) = h. (T 2 (P, t) - T, (P, 1)) (2.40)an
k, -(P,t) = h,.(T 2(P,t) - , (P,t)) (2.41)an
This very important boundary condition is well-known phenomenon from the modelling of castingprocesses. In this case we have to specify the heat transfer coefficient between the casting andmould in order to define this internal boundary condition.A thorough description of the implementation of these boundary conditions will be given in thesection XX.
2.2.2 Analytical solutions
Obtaining analytical solutions to the general heat conduction equation with suitable initial andboundary conditions is a central topic within the field of mathematical heat transfer theory. Thisboundary-value problem is one of which has been studied extensively and many methods havebeen applied, e.g. separation-of-variables-method, Laplace-transform, conformal mapping, etc.The applications of these methods are presented numerous places elsewhere in literature, [1, 2]and will thus not be shown here. One analytical solution to the I-D transient heat conductionequation, however, deserves mention here since it is of central importance to the analyticalanalysis of casting processes, e.g. it is the basis of Chvorinov's rule.
Semi-infinite solid with fixed surface temperature
Consider the semi-infinite solid shown in figure 2.6. The initial temperature is T, and the surfacetemperature is instantaneously changed and maintained at the temperature 7, for all times greaterthan 0. Now, we seek the temperature distribution in the solid as a function of space and time. Thistemperature distribution may subsequently be used to calculate the heat flow any place in thesolid. The temperature obeys (2.29) with no internal heat generation, i.e.
8T 8•2Tat ax - (2.42)
subject to
6
boundary conditions T(O,t) - Ts for I > 0 (2.43)T(co,t)= T for t>0 (2.44)
initial condition T(x,0) =T (2.45)
This problem may be solved by [he Laplace-transform method, and the solution is
Tx,)-T, =(2.46)
where the Gaussian error function is defined by
er (u) =- 2 fe-'i drj (2.47)
TO
q.
X
x
Figure 2.6 Semi-infinite solid with fixed surface temperature
1.0
0.6
a,0.2
02 '
0.4 0.8 1.2 16 2.0x
Figure 2.7 The error function. Note that erf(u) -41 for u -w o
Note that in the definition, (2.47), q is a dummy variable and the integral is a function of its upperlimit. Suitable values of the error function for arguments from 0 to 2 may be obtained from figure2.7. More accurate values, which can be obtained from mathematical tables, are usually not
7
necessary in heat conduction problems of this type, due to larger errors in thermal properties andother conditions.When the definition of the error function is inserted in (2.46), the expression for the temperaturedistribution becomes
T(x, t)= Tý + (Tj-T,)2f J e a1 (2.48)
Of particular interest is the heat flux at any location at a specified time. This may be obtained fromFourier's law (2.1). The partial differentiation of (2.48) may be performed by Leibniz' rule,resulting in
-=OT -' )2~ &~ I (2.49)
(T7 - T) (2.50)
Substituting this gradient into Fourier's law, we get the heat flux
q _ k( T e -T x)e,/ (2.51)
We are frequently interested in the heat flux, at the surface at x = 0, which becomes
q _ kf(T,-f (2.52)
As seen, this flux clearly diminishes with time.
Penetration depth
The preceding analytical solution was obtained under the condition that the solid is thermallysemi-infinite. This condition that might seem very unlikely in the real world, covers the situationwhere the temperature at some point in the finite solid is still unaffected by the heat transfer at thesurface. In this case, the temperature field in the part of the solid that is actually affected isidentical with that in a semi-infinite body under otherwise identical circumstances. We then saythat the body acts thermally infinite.It should be noted that the analytical solution given by (2.46) states that the any point in a bodywould be affected immediately by the heat transfer at the surface, even though this point is very far
- away from the surface. In other words, the velocity of the thermal signal based on the heatconduction equation is infinitely large. This is actually the case for any diffusive equation.Experiments show, however, that this is not the case. It actually takes some time before it ispossible to register any change in temperature, due to the fact that heat conduction to some extentresembles wave-propagationVarious criterions for evaluating the penetration depth of the corresponding thermal signal havebeen given literature. For our purposes we shall define the penetration depth of the temperature asthe distance from the surface where the change in surface temperature, i.e. (T7, - T7) has beendamped to 1 %, figure 2.8.
8
T
• t'2.
!"•, •.. ttr, • • . :=-,
x 6
Figure 2.8 Penetration depth. Damping of AT at surface to 1%.
We now get:
Temperature change at x = 0 A710) = T, - T,Temperature change at x = 5 4T(0) = 0.01(T, - T0
We now express the temperature at x = 8 by (2.46) and by the definition of J. Equating these twoexpressions yields
T7+(T - T,erf( - Tt+=T +O.O(Tt-TT) (2.53)
which is reduced to
erf 0.99 (2.54)
The argument corresponding to an erf-value of 0.99 is found to 1.8, i.e.
S1.8 (2.55)
which finally results in the expression for the penetration depth
5 = 3.6Fa (2.56)
This expression is even though it is derived on the basis of the error-function solution for a semi-infinite solid often used, when an estimate of thermal penetration depth is needed.
9
2.2.3 The Riemann temperature
If two bodies of different temperature are brought together, a constant temperature of the interfacewill immeiately adjust itself between the bodies if the thermal resistance in the interface is zero,see fig. 2.9.
-T
r
x
Casting Mould
Figure 2.9 Analytical solution for two domains in contact
This temperature which is of great importance to casting processes is called the Riemanntemperature, as mentioned in the description of possible boundary conditions for the heatconduction equation. The analytical solution for the semi-infinite solid given in (2.2.2) is nowapplicable. Speaking in terms of casting processes, the present analysis will approximate thethermal conditions prior to solidification for the casting of a plate in a mould. The prerequisitesfor using (2.48) are now present. For the two domains we have the following analytical solution
Domain 1 (casting):
2 -rij4/•7
T(x,t)= T, +(T -,-- Je-'dq (2.57)
Domain 2 (mould):
T(x,t) = T, +(T2 - T,)7 e-7dq (2.58)
where T, and T, are the initial temperatures in domain 1 and domain 2 respectively. Note theminus in the solution for domain 1; this is a result of the x-axis chosen.
The gradient in the interface between the casting and the mould is of particular interest. It is foundfrom (2.52)
10
a~i' -T,-aT I =(2.59)
S T2 -7, (2.60)
axtt.a
A prerequisite for using the error function solution as a solution to the specific problem is, asmentioned, that the common surface temperature (Riemann temperature) is constant as long as theanalytical solution is valid i.e. as long as the two domains are thermally half-infinite (thetemperature has not penetrated to the back). The Riemann temperature is now determined by thefollowing heat balance
q- 0 = qL. 0 (2.61)
Fourier's law is applied
-Ak -• EL =- Ak2 T- (2.62)
The found expressions for the gradients, (2.59) and (2.60), are inserted
T -T T -T,-,Ak, =-Ak2 2 (2.63)
-A/-li is abbreviated, and the relation
k - = = ij , =if (2.64)
is applied. Thus
kp Ii(T, - T)= kPC1 (T' - T) (2.65)
Now, the Riemenn temperature, T,, is isolated
TI__ 4, 1,-TI,,+ kpc,142 (2.66)
= kpc, 12 is called the mould material's cooling capacity. Expressed by . (2.66) appears as
11
T "=AT, +A2Ti (2.67)A+A
T, is now known and the equations (2.57) and (2.58) can be used as analytical solutions for thetemperature fields in domains 1 and 2, and a comparison can be made with numerical solutions,bearing in mind the conditions under which the analytical solution applies (thermal half-infinity,no solidification heat, constant material data, constant T, .., etc.).
2.2.4 Solidification in sand moulds, Chvorinov's rule
The famous Chvorinov's rule is also based on the error-function solution given in (2.46) incombination with various appropriate assumptions. But, whereas the Riemann temperature wasbased on analytical descriptions in both casting and mould, Chvorinov's rule is derived from theanalytical solution for the mould only.Consider casting of a plate in a sand mould. Since k,.,, << k.., we can assume that the wholetemperature fall is found inside the mould, resulting in a temperature field during solidification asshown in figure 2.10.
T
sy ,, ,F ' .•
casting Mould
Figure 2.10 Casting of a metal part in a sand mould. 1-D heat conduction
The following idealized conditions are assumed
- Heat is transported in the x-axis direction perpendicular to the plane of the plate.-Thermal material data such as k and oci are constant.- All the latent heat is released at the solidus temperature, T.
With these assumptions, we can apply the analytical solution (2.46) for the description of thetemperature field in the sand mould. The heat flow into the mould is now expressed by (2.52)
q=A (T-T) (2.68)
12
where T, and T, are the solidus temperature and the initial temperature of the mould respectively.The total energy [J] that must be removed in order to solidify the melt between 0 and s with themass M is given by
Q=MAHf = AspAHf (2.69)
where AHf is the freezing enthalpy or latent heat, and A is cross-sectional area.
The amount of heat [W] produced by the solidification front advancing into the liquid can now befound from differentiating (2.57) with respect to time, i.e.
=ApAH r •(2.70)
where s is the position of the solidification front. Again, applying the concept of energyconservation results in an equality of the heat fluxes, (2.68) and (2.70)
ds ku(T , -T1)ApAH! - = A (2.71)
Separating the variables leads to the very simple ordinary differential equation
AHfp k• dt(T-T)ds = ill(2.72)
The constant
K= AHfP 'Fr"(.3K - 1~(2.73)
(T,-T,) k.u
is introduced, and the equation is now integrated
Kfds =[-dt = Ks=,F, => - =Cs2 (2.74)dt 40 o ý t 4
where C is the Chvorinov constant.
The last equation of (2.72) states that the solidification time is proportional to the thickness of thesolidified layer squared.For a solidified plate-shaped layer, the parameter s can be interpreted as the corresponding module
V(s)s = - (2.75)
A
where V(s) is the volume of the solidified material and A is the area of the cooling surface.
13
The total sotlidification time for the plate-shaped casting is found by applying the total volume of
the ha Iflplate from s = 0 Lo s = 4/2, where d is the thickness of the plate, i.e.
:,f = cm 2 (2.76)
where M is the module of the plate (= d/2).
(2.76) is the very well known Chvorinov's rule [3). It should be noted that this simple relationshipstrictly speaking is based on Qne-dimensional considerations, nevertheless, it is possible to apply itto some other multi-dimensional geometries such as cylinders and spheres, when appropriatemeasures are taken. [4]
2.2.5 Solidification in permanent moulds
A permanent mould is made out of metal, usually steel or cast iron. These materials haveconsiderable higher values of thermal conductivity, 30-50 W/mKc, compared to the one of a sandmould 0.8-1.5 W/mK. This means that the temperature distribution changes dramatically. If welook at the thermal resistances involved, we realize that the major part of the thermal resistance iscomposed of the already solidified layer and especially the mould coating which is almost alwayspresent during casting in permanent moulds. This mould coating is represented by the heat transfercoefficient h. Thus, the temperature gradients in the mould are neglected so that the temperature inthe mould will be constantly equal to its initial temperature T', as shown in figure 2.11. Moreover,it should be noted that permanent moulds normally are subject to internal water cooling, whichsupports the applicability of the temperature profile in fig.2.t 1.
T
Symme"rIme
H-x
Casting Mould
Coating
Figure 2.11 Casting of a metal part in a permanent mould. 1-D heat conduction
Now, the heat flux is expressed as the total potential difference divided by the total thermalresistance
14
=q= TT -Ti) (2.77)
as before, s is the position of the solidification front and k the thermal conductivity of the melt-metal.This heat is equal to the heat produced by the solidification front advancing into the liquid givenby (2.70), i.e.
paH ds(T -s T) (2.78)
M hA
Reducing, separating the variables and integrating yields
,, -T ) (2.79)(T, -T,)• h 2k
(2.79) gives the solidification time of a layer of thickness s. Again, the total solidification time t,for the plate is obtained by setting s equal to the half thickness of the plate. If the coatingresistance is large compared to the resistance in the already solidified layer, (2.79) is reduced to
spAR1= H f (2.80)h(T, - T)
An estimation of the validity of this simplification could be made by considering a quite extremecase of an aluminium casting of 20 mm wall thickness, i.e. s = 10 mm, h = 2000 W/(mzK) and k =200 W/(mK). In this case s/h will be equal to 5 x 10- m'K/W whereas s/(2k) will be equal to 2.5x10' m'K/W. Even in this extreme case of a thick-walled casting and a coating modelled by a quitehigh transfer coefficient, the implementation of the thermal resistance in the already solidifiedlayer only contributes with 5% to the solidification time compared to the contribution from thethermal resistance of the coating.
It should be emphasized that the expressions (2.79) and (2.80) are based on approximations of theactual physical conditions. The largest error is introduced by neglecting the heat content of thealready solidified layer. In the following we will show how this can be compensated for. Considernow figure 2.11 again. Because of the assumption of a linear temperature field, the heat that has tobe removed from the already solidified layer is given by
Q = {Asp, (T, -TO) (2.81)
For the analysis, the difference between the temperature of the melt and the interface temperaturebetween the coating and the casting, i.e. (T,-T,), is of particular interest, since T. is not known. Anexpression for this temperature difference is now obtained. For that purpose we express the heatflux in two ways. First we apply the total temperature difference together with the total thermalresistance. Secondly we apply the difference in temperature over the already solidified layertogether with its resistance. Equating those two fluxes yields
15
(TI - TO) (tI -TO (2.82)$ 1 s
-AM + A
Reducing and rearranging gives
T -T - -T (T, -Ti) (2.83)o~I+ 1+-1+/ k +Ish Bi
where Bi is the Biot-number defined as
shtBi = -- (2.84)k
Inserting (2.83) into (2.81), we get the following expression
=As{ J (2.85)
Comparing this expression with (2.69), we note that the parenthesis in the equation which istermed AHU can be interpreted as a quantity similar to the solidification heat, At4. 4/4 representsthe heat content that has to be removed from the solidified layer, so that the solidification can takeplace. The solidification time given by (2.69) will now be modified to
t' P(AH f + AH7( SZ , '(.6- I2 (2.86)
This expression can be modified further by taking into account the influence of the superheat AH1,i.e.
P(AH , ) c+ 2¾ (2.87)
where the superheat is given by
AH -; =c' (T,, - T) (2.88)
and T, denotes the pouring temperature.
This may seem to conflict with the boundary condition stating that the temperature of the meltequals the solidus temperature. But as long as (T,-T,) << T,. the approximation is still quite good.
16
The expressions found in this paragraph could also be applied for continuous casting. In this caseboth the permanent mould and the water spray is modelled by a constant temperature of T, thatinteracts with the continuous slab via the heat transfer coefficient, h.
It should be noted that all the expressions for the final solidification time are derived on the basisof the same concept involving energy conservation and separation of variables. The resultingordinary differential equations are all very simple and yield simple analytical solutions.
2.2.6 Cooling of the melt in inlet system
In connection with the mould filling it is essential to be able to estimate the cooling of the melt inthe inlet system. If this is not addressed properly, cold shuts in the inlet system itself may occurduring the filling process. Consider now the idealized inlet system shown in fig. 2.12,
Figure 2.12 Idealized inlet system. 1-D heat conduction perpendicular to the periphery.
The following parameters characterize the system
Length LPerimeter PCross sectional area AInitial temperature of surrounding mould T,Inlet surface temperature of mould T,Temperature of the entering melt T.Heat transfer coefficient melt/mould hSpecific heat of melt (ýc),Specific heat of mould (PC).&Thermal conductivity of mould k
The problem is now to find the temperature loss of the melt due to the cooling from the mouldduring the stay in the inlet system. In order to achieve this, it is necessary to solve the heatconduction equation for the above problem. This is by all means not a straightforward problem. Itis, however, possible to obtain an analytical solution in closed form by assuming that the mould isthermally semi-infinite. Then the following analytical solution for the temperature in the mould isapplicable Pitts & Sissom
T(x,t) -7,=1- erg - (hz/ .0.1Pk21( f (2.89T.- Ti k )
where X/J-
Even though (2.89) strictly speaking is applicable in the semi-infinite case only, experience showsthat it yields realistic results for the case under consideration. The surface temperature is nowfound by setting x = 0. Since Ax=O-)--0 and erA&-0)=O (2.89) is reduced to
T, Ie 2l17 erf( k (2.90)
17
The constant a = h1•/k is introduced, i.e. (2.90) becomes
TI71 - I eah'ieaf(a )) (2.91)
which is rewritten into
T.-T, e (eq(a4))t(2.92)
The cooling is now expressed by Newton's convective law of cooling, i.e.
q = Ah(T, -7T,) = LPh(T, - ]r, (2.93)
Which in combination with (2.92) yields an expression which depends only on the melttemperature and the initial mould temperature, and not the surface temperature (which is normallynot known).
qý, = LPh(T. - Ti)e2" (1 - erf(a.7)) (2.94)
The change in energy content of the melt per unit time is given by
Q,8 = AL 'cP T- (2.95)
Normally the melt is inside the inlet system in a short period of time, so that the partial derivativeof the temperature with respect to time can be approximated by difference quotient. Denoting theperiod of time t and the corresponding temperature change AT(t), we obtain
Q,,h -ALp .., AT= ,Pch AT (2.96)
The energy conservation now yields Q,,., = L.e.
•,AT L,)eh (I-ef a(ALpc -- = LP(T.-T)e 4-erj(ai.t)) (2.97)
Finally, the cooling of the melt in the time period t can be found as
AT PIT -T)e"(-erf(at)) (2.98)r Atpc,
References
18
1. H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, second edition, Clarendon
Press, Oxford, (1959)
2. B.A. Boley and J.H. Weiner, Theory of Thermal Stresses, John Wiley & Sons, (1960)
3. N. Chvorinov, Theory of Solidification of Castings, 4th Gie Berei, (27), 177-225, (1940)
4. R. Viswanath and Y. Jaluria, Knowledge Based System for the Computer Aided Design ofIngot Casting Processes, Engineering with Computers (7), 109-120, (1991)
5 D. Pitts and XX.Sissom, Heat Transfer, second edition, Shaums Outline Series, (19XX)
19
Mathematical Modelling andNumerical Simulationof Casting Processes
Jesper Hattel
Thermal Processing of MaterialsDepartment of Manufacturing Engineering
Technical University of DenmarkLyngby
2000
A 1-D analytical model for the thermallyinduced stresses in the mold surface duringdie casting
J. H. Hattel and P. N. Hansen
Labor,'tory of Thermal Prermsing of Materials. Institute of Manufacturina Engineering.Technical University of Denmark, Lyngby. Denmark
This paper presents.an analytically based method for predicting the normal stresses in a die mold surfaceexposed to a thermal load An example of application of the method is the high-pressure die casting process(HPDC), where the surface stresses in critical cases lead to cracks. Expressions for the normal stresses asa/function ofthe mechanical and thermal properties have been developed for a casting both without and witha coating. Finally, the resulting relationships are derived and evaluated, with particular emphasis on the effectof the heat transfer coefficient between the casting and the mold.
Keywords: die casting, temperature, stress, analytical solutions, coating
Introduction caused by the wrong choice of parameters for the process.The high-pressure Ji, cn;ing pro,2ss (HPDC process) This has again led to an increasing interest in optimizing
has experienced growing application. Today it is one of casting process in HPDC. A very efficient way of
the few casting processes in progress. The process is used achieving this is to use numerical calculations to simulatemore andf m casti o bigprogeress and farmoress co lex the influence of each single process parameter in themore and moIs to cast bigger and far more complex casting process.castings. Its popularity is partly due to greater awareness The literature in the field is very sparse; however,of environmental effects all over the world. Many some analytical work has been done in predictingfoundries wish to avoid the polluting sand casting thermal stresses with relation to casting processes. In theprocess and try to use permanent molds instead. paper by Weiner,' an elastoplastie analysis of a free plate• The essence of the HPDC process is that molten metal is performed. Although this does not deal with casting(mostly aluminum or zinc) is pressed into the cavity processes, the analytical solutions for the temperatureunder high pressure. The cavity is filled in a few and stress fields any of some interest, because to somehundredths of a second. After the melt has solidified, the extent they af similar to the ones derived in the presentcasting is removed from the open die and afterward the nttey ar simithe ones he pscavity surface is often sprayed with a die lubricant. The paper. Two crucial limiting assumptions, however, areclosed die is :hc: ready t9 rcceiv-c a new portion 0 r made by Weiner, namely that the plate is heated slowlymole iet Coi s thcr. -ready tn rc are used potio n re ove and that the thermal contact between the heating mediamolten metal. Cooling channels are used to remove the and the plate is infinitely good; that is, there is no thermalheat input in the dies caused by the molten metal, resistance. Neither of these conditions is present in
The theoretical background (flow, heat transfer, high-pressure die casting processes; on the contrary, thestress/strain) of the HPDC process has been a rather mold is heated rapidly by- the melt, and the thermalneglected field. Interest and development have been resistance between the mold and the melt is not zero. Inmainly on the mechanical engineering aspects, where fact, the heat transfer coefficient between the mold andmore powerful machines and more advanced measuring the casting is the most influential parameter in theequipment have been developed. This lack of insight into process. Weiner and Bole9y use the one-dimensionalthe theoretical background of the HPDC process has (I-D) analytical solutions from Weiner' to analyze theresulted in many breakdowns of molds during operation, elastoplastic stresses in a solidifying body. This work
Address reprint retsp-ts to D:, Hattzl at the Laboratory or Thermal gives one of the few exact analytical solutions forProcessing of Material, Institute of Manufacturing Engineering, thermo-elastoplastic stresses and strain in a solidifyingTechnical University of Denmark. DK-2800, Lyngby, Denmark. body. Still the heat transfer coefficient is not taken into
Received 12 August 1993; revised 28 February 1994; accepted 8 March account, as the aim of the paper is to analyze the1994 elastoplastic stresses in the solidifying body (e.g., the
550 Appl. Math. Modelling, 1994, Vol. 18, October C 1994 Butterworth-Heincmnn
Analytical method for predicting thermal stresses during die casting: J. H. Hattel and P. N. Hansen
casting) rather than to investigate the stress conditionsin the mold. However, this theory still constitutes avaluable tool for predicting the thermal stresses incasting processes, and it is very useful when it comes tothe preliminary evaluation of numerical methods forsimulating stress/strain fields in castings.
The thermally induced stresses in the surface of thepermanent mold are focused on in this paper. Themagnitude of the thermal stresses is one of the most Xinfluential parameters affecting the lifetime of the mold.One of the major problems is the cyclic thermal impacton the mold surface, resulting in heat checking andthermal fatigue. If it is possible to minimize these stresses,the lifetime of the mold can be increased considerably.
To support the nhvsical understanding of theproblem, this paper will address closed-form solutions,which are very helpful in this respect. Figure 2. Schematic drawing of thin-walled casting and mold,
The I-D model • to zero. Because the wall is not allowed to bend, e,, andIn this section a I-D model for calculating stresses in the to=owill be constant, varying not in space but in time.mold will be developed. We consider the I-D domain of All this leads to the following assumptions:
Figure 1. In order to establish the foundation of the I-D
model, many assumptions have to be made. First of all, 1. cs, = 0because the melt is not able to produce any considerable 2. a,, = .,mechanical resistance, the mold is almost free to expand 3. r, = ox. = 6y, = 0in the x-direction; that is a._ = 0. Second, because the 4. c, = s,, = constant in space. Varying in timemodel is symmetrical, the normal stresses in the y- and 5. C, = 0= S = 0z-directions are equal; that is or = ,.. Applying Hookes' generalized law,
The next assumption is not as simple or obvious asthe first two. We now consider the schematic drawing of L IsE + I -E2tATthe die and melt in Figure 2. Because the casting is often = v + - 2v 2 (1)very thin walled compared with the mold, and the moldwall is very well restrained by the rest of the mold, it we get Os, in its general formseems to be a fairly good assumption that the mold wall Eshown in Figure 2 does not bend. This means that it is 6--= - + (e• + C, + v.,)}only allowed to expand uniformly in the y- andz-directions, as illustrated in Figure 2, resulting in only EaAATthe normal strain of the strain elements being different - (2)from zero; that is, e6, = s. = &,. = 0. A consequence of
this is that all the shear stress components are set equal Employing assumptions I and 4 and reducing yields
E
(I + vXI - 2v) ((I - ,&. + 2ve, - (l + v)AT}
=0 (3)Isolating ,_ gives
Melt I +v 2v___ -- T--e-,, (4)
We now consider at, (or o,,) in its general formMoldSurface Backside -,E=-, {8,, + (6-• 1- + Z,, + at,)}
of mold of mold EaAT
YT - 2(5)
- X Employing assumption 4 and reducing, yieldsE
Figure 1. 1-D domain of melt (cating) and mold. - ( -+ vl-) -- (I + vWaAT) (6)
AppI. Math. Modelling, 1994, Vol. 18, October 551
Analytical method for predicting thermal etresses during die casting:. . H. Hattel and P. N. Hansen
e,, is isolated
ell = E dTdx (13)
Cross The integral in (14) is now multiplied and divided by pc,:Section 4-,X f= p[ ATdx (14)
Jý _ B4cXApdx
Recognizing JOL pc ATdx as the total energy per unit area,Q. transported from the surface into the mold (14), maybe written as
al = ýLQ (15)
Figure 3. Cross-section. A to B along x-axis. on which After having established equation (15), two differentequilibrium is demanded in the 1 -0 model. casting process conditions will be treated separately, one
without coating and one with coating.
Case 1: Casting without coating
Substituting c., from (4) into (6) yields Assuming that the heat flow is one-dimensional, Q mayE fl +.v 2v now be expressed as the heat flux per unit area, q,
S= (+ I- v) -.-,•vIL-- cAT - •, integrated over the time interval (Pitts and Sissom')(1-4 v~ - 2) 1 r'T.-T,-
Q = qdt=J k T.Jdt (16)+ Ell-- (I + v)atAT• (7) . Eo 7.o x//t
where x is the thermal diffusivity given by K = k/pc,, T,After some reductions (Appendix 1), we get is the surface temperature, and T, is the initial
E temperature of the die.S - v) - Note that (16) is valid only if the thermal resistancebetween the casting and the mold is zero, R = 0, meaning
The constant strain, e,, (or i,.), is determined by that the heat transfer coefficient is infinitely large, anddemanding equilibrium in the y- or z-direction for the It = co. The solution is only valid for small Fouriercross-section of the mold wall shown in Figure 3. numbers as long as the die is thermally infinite.
Equilibrium in the y-direction, assuming symmetry, The well-known expression for the depth ofyields penetration is now used to estimate when the assumption
f, of thermal semi-infinity is no longer valid.or, dx = 0 (9)
where L is the length of the I-D domain (e.g., the Setting A equal to L, and rewriting, we getthickness of the die = distance from point A to point B L- = 12.96 Kt (18)in Figure 3). Substituting or, from (8) into (9) now gives Combining this with the definition of the Fourier number
E fllv - aA T) dx = 0 (10) KE
Fo= -T (19)The integral is split up
Ei -{r - aAT) dx K
Fo-A di =2- = 0.077 (20)12.96 xr 12.96
E s,, dx - I AT dx The assumption of thermal half-infinity is thus valid for(1 - v) (1Eo Jo I Fo numbers less than 0.077 only.
= 0 (11) The time corresponding to this upper limit of the Fonumber will now be assessed, assuming the followingAssuming that ed and a at' constant, and rearranging, properties for the mold material:yields
"k = 30 w/(mK)
/•, E- A ATdx (12) pCe = 5.0 106 J/(m3 K)
562 Appl. Math. Modelling, .1994, Vol. 18, October
Analytical method for predicting thermal stresses during die casting: J. H. Harrel and P. N. Hansen
k 30w/(mK) ---K = - = .l0o m6/s
PC, 0* 10 6J(m 3K) -s
The thickness of the mold is assumed always to begreater than 0.05 m. Isolating the time from (20), we get _ -
the upper time limit:
L (0.05 m)2 3= Fo-- = 0.077 32s (21)
K 6.0.10- m 2 /s X
Realizing that the total cycle time is often 40-60 s. thedwell time is approximately 10-25 s, and the spray time2-5 s, it is obvious that the assumption of thermal infinity - --
is valid during both casting and spraying, a veryfundamentmA result. On 'he other hand the solutions - H --
presented for the stresses are only strictly valid during 0 .0. Fo . .0, a.
casting in thefirst cycle. This is due to the fact that onlyhere is the initial condition of constant temperature Figure 4. c,(Fo), case .
realistic. It is emphasized in the discussion section lateron that these results from the first cycle can be usedgenerally for the arbitrary cycle if a special procedure isapplied. But if we look at the expression, (25), we ise thetgobe
Now, substituting (16) into (17) yields maximal compression stress, which is the globalminimum, is reached for t = 0. The differentiation (27) is
= af k T, - T dt (22) thus needless.p j d (22) Figure 4 shows at, as a function of Fo, based on-- , E(25), for a tool steel with the material properties:
Performing the integration, and assuming that K= E = 2.1 * 10' Pa, a = 1.1 l10- 5 C, and v= 0.3; andk/pc,, we get the temperatures: T, = 500 °C and T = 150 'C.
a T- T Ta - We now introduce the dimensionless stress defined by, -= k 2.,/ = ý - T, 2,[;pc,L rLn
£IF /0- LE- TJ (28)in/-TIC I aE(T,. - TO
(23) (I - V)
Employing the dimensionless time, the Fourier number, Combining with (25) yields(24), is reduced to
a,, = 2a TJJ TJ '/F (24) G 1/r .-- - I (29)Figure 5 shows the dimensionless stress as a function
where Fo is given by (19). of Fo (29).Substituting (24) into (8), and rearranging, now gives
the expression for a,,
Cyr (-) [ - (25)
where AT has been replaced by T, - Ti.We are especially interested in the stress at the surface - - - - -.
at time t=OorFo= 0
-t ) 0) = E(-I -,) (26) -(1 -v)
Comparing this with the well-known formula by Danzer 06 - - - - -
et al. for the compression stress in the surface, we noticethat the result is identical. .2
Because special interest is attached to the maximalvalue of an, we will now find it from (25). Normally onewould demand that t
-a , 0 (27) "U
Figure 5. a,_(Fo). uase 1.
Appl. Math. Modelling, 1994, Vol. 18, October 553
Aitalytical method for predictirg thermal stresses during die casting: J. H. Hattel and F. N. Hansen
Case 2: Casting with coating 6fl,
The following properties are employed:AL I, -L•
Biot number, Bi =
Fourier number, Fo = Kt L 1.
The thermal effect of the coating is modelled by a heattransfer coefficient, h. The solution can be shown to be:' 0 : * a, to 12 1. 1, 20 U It 2.•6 nhx h 2K t
fT 1 -T I k T FigureG. On. tand h.
x{LC erf{4+-fl (30)k In Figure 6, Q is shown as a function of time and various
where = x/l 14 . (Again this solution is valid only for h values for a tool steel with the propertiessmall Fourier numbers, which give no limitations for this k = 30 W/(mK) pc, = 5.0. 106 J/(m'K)process; see (21).)
We are frequently interested in the temperature at Equation (37), however, can also be used in defining thex = 0, dimensionless heat, Q_
T. - T - - fA (1[Q"4--h( Q_- T =a {exp {aPt} erfc ka(Q} -= 1a
or + 22 - (38)
T - T, = (T. -- exp h2d erfe Nt~j-• K-1-• Figure 7 shows the dimensionless heat as a function
(32) of t and various a values. It should be noted, that fora = 0, Q._, is simply equal to t. This can also be seen
Identifying hA t/k as aji, (30) is rewritten as from (36), because T is constantly T, when a = 0. This
T- j = (T-_ 7;){1 - exp {a 2t} erfc jot}) (33) gives
Rearranging we get Q h(T. - rj d, =(4 - 7D d,
T.- _T = (IT_ -T,) exp (a't) erfc (aft) (34) = - TDt (39)We now employ the following relationship (Newton'slaw of cooling):
q 3
A = h(T --,) (35)
in deriving an expression for the total energy
Q=~ dt = I[T A(,- TJ d:JA Eo
- ('/' - T,) exp (a't) erfc (.af) dt (36) * ."
which is reduced to (Appendix 2)
Q1= A(T - 7D{-kiexp (ae)rc {aefi} -t) 2 41V-o S 1W IS A0 IS 10~-Ii
+ (37)Figure 7. Dimensionless heat, 4,.,.
554 Appl. Math. Modelling, 1994. Vol. 18. October
Analytical method for predicting thermal stresses during die casting: J. H. Hattel and P. N. Hansen
Combining (39) and (40), it is clear that Q..,, = t (fora = 0). As with case i, the expression for Q, that is (38).is now combined with (15) to give the strain
a (I6,= a'-'h(T - T•)c- (exp (a2t erfc (a.fi) - 1) -a __-_ -
pc, L f
(40)
066-o
Using the Biot and Fourier number, (40) is rewritten as - - -
B, = (7(T. - Till {exp (a:t) erfc {a - I) W a•n 40 O•O QM me6 10 0o7
Figure 8. a,(Fo), case 2.
Substituting (41) into (8) now gives the normal stress
E First of all it should be noted that (44) shows consistency= - T• --4T, with (25). If we set h equal to co, which corresponds to
a casting without coating. (44) becomes (25). This is{ I straightforward to see, because
U p . T,-. T., when h -. io
+ -I- ATI (42) -- 0, when h --, co
exp (024 cric (of) -- 0, when It-.Realizing that ATis equal to T. - T,, and combining (42) It f 0
with (33). we get because a
E rE(T) Figure 8 shows the a,, as a function of the Fourieror= - (T, - ) number, Fo, for the same properties as of Figure 5, except
for T2, = 500 C. As with case 1, we apply the
} }+ -dimensionless stress defined by (28) (with T. instead ofS(fall) erfc ( t - 1f + fI which combined with (45) yields
- aexp 2a,,,,.= = {exp{Bi 2Fo} erfc {Bi/l}) - 1}{1 +- a({(T. - "•){ 1 - cxp {a2t} erfc taf/t}})) 1Fo
(43) + (46)
Reducing yields which gives the curves of the dimensionless stress shown
Ea(Tý - T) - in Figure 9. It should be noted that a,4t = 0) ={(exp (a't erfc (art/) - 1) a._=(t = 0) = 0. This result is fundamentally different
(I - v) from case 1, where the maximal compression stress
I ) 2,/•o occurred exactly at t = 0 (26). It is quite obvious thatI + +-- (44) this difference is due to the time delay of the thermal
I( iiJI flj signal, which is produced by the coating. The maximalOr expressed by the Riot and Fourier numbers value of the compression stress in case 2 will be less than
in case 1, and delayed as mentioned before. Thereduction of the maximal stress is a vital parameter for
a, = (Ep {B 2 Fo) erfc(Bi.•}) - 1) the lifetime of the mold; thus it is focused uponseparately. For this, the reduction ratio is defined as
Ax Math. + ModelF (451 R 1= October 647)
Appl. Math. Modelling, 1994, Vol. 18, October f555
Analytical method for predicting thermal stresses during die casting: J. H. Hartel and P. N. Hansen
I - - _ -- 1-
In
W-2aow2 -a
_0.-I0) 0.4 a i 0 00 - C 30
Figure 9. oe,(Fo). case 2. Figure 11. Normal stress in surface plane vs. time.
as a function of the heat transfer coefficient, h (or the On the other hand, permanent molds often developBlot number). This reduction ratio is used in the cracks after some time in operation. Obviously, the molddiscussion for further investigations, has a limited lifetime; the problem is that the cracks often
begin developing considerably before the estimatedDiscussion lifetime of the mold. This phenomenon is difficult to
explain from Figure 11 alone, because the tensile stressesAs mentioned earlier in the paper, one cycle of the HPDC (which have the main responsibility for the cracks) areprocess often consists of three parts: small. In this context, creep behavior might influence the
1. Melt is pressed into the cavity and solidifies, stress field in an undesired way. This could be a2. The casting is removed, and the mold surface is relaxation of the general stress level, which in the surfaceslightly cooled by the surrounding air. is compression in mean with respect to time, resulting in
slightly coeibe surfacei ,rosunding air. hthe stress level being displaced in positive direction,3. The die surface is sprayed, resultin in eavy cooling increasing the tensile stresses in the surface.
It is well known6 that generally after a few casting cycles, Earlier in the paper it was pointed out that thea quasistationary state is reached for the thermal field in solutions for the stresses are actually valid in the firstthe die. This is seen from the temperature history in the cycle during casting only. With that in mind only thesurface (Figure 10). Typically this results in a curve for reduction ration attached to the first cycle, R(I), shouldthe normal stress in the surface plane as shown in Figure make sense. In reality this is not so. The relative nature11. (The normal stress orthogonal to the surface is close of the reduction ratio means that the systematic absoluteto zero.) Note that the quasistationary state is also errors in some sense cancel each other out when the ratioreached here within 5-10 cycles. Unless the spraying is taken. This is confirmed by a numerically based work,cools the mold surface very thoroughly, the tensile Hattel and Hansen,7 showing that R(n) is very similar tostresses as shown in Figure 11, will be small. R(I) where n is a cycle in the quasistationary state.
as
II
a0 100S -
Figure 10. Die surface temperature vs. time. Figure 12. Stress reduction ratio vs. Fo and h,.
556 Appl. Math. Modelling, 1994, Vol. 18, October
Analytical method for predicting thermal stresses during die casting: J. H. Hattel and P. N. Hansen
In Figure 12 the stress reduction ratio as a function 0.005 mof Fo is shown (compare with Figure 9). 3. Q = (630-61)oC * 2.5 * 106 J/(m'K)
2
Application + 2 (610-s70)rC.*2.6 * 106 i/(m'K)
The curves for the dimensionless heat, Figure 7, and thedimensionless stress, Figure 9, can, in a systematic way, + 0.005 m . 2.4 * 103 kg/mr ' 443 * 10g J/kgbe applied in assessing the general stress level: 2
I. Choose a tool steel, then k and pc, will be known. 0.005 mFurthermore the coating will give an assessed h value + (570-500)0C * 2.7 * l06 J/(nmK)
2. Calculate Kc and a from the expressions 2. k h1;= 3.5.106 W/mn
K= P a = k (38) now givesp% k
3. Calculate the heat that must be transported from the Q 3.5 * 106 W/m2
casting by the expression =(T, 0 - = 1000 W/(m2K)*(570-220PC
Ss =10.0Q (T-1. - T -)pc4 + j (TL - TS)PCL Using this value in Figure 7 together with a =
0.0817 s- °-I gives t = 12s.- r• 1 .)c t 6.0. 10-6 m2/s*.12 s+ 2 pL + 2 (T. - TF.)pc$ 4. Fo -.= 20.0072
With this Q value Qc...m is now calculated from (35). (O., i)'
Use the calculated values of Q_,,_ and a in Figure 7 Bi'=-hL 1000 W/(m'K) 0.1 m 3.33to find the required time, t. k 30W/(imK)
4. Using the thickness of the mold, calculate the Fouriernumber from (19) and the Biot number from the 5. a.,. is determined to be 0.24 from Figure 9.expression 6. The stress is found from (29), with T - T*,.
hL E&(T. - TDB i = -T I,, m (1 v)
5. Use the Fo and Bi numbers in Figure 9 to determine 1.1 - 10- *C * 2.1 . 10" Pa *(220-570)rCthe dimensionless stress at the time of the removal of (1-0.3)the casting.
6. Convert the dimensionless stress to real stress by (28). = -277 MpaNote that the effect of the coating on the stress has been
Example a reduction of the maximal stress to only 24% of themaximal stress value in the case of no coating. This
An example of how to use the application procedure will reduction is due to two conditions: first, the maximalnow be presented: stress, which would appear if the casting was neverT. =i 570"-C removed, is reduced. Second, the casting is removedT, = 220"C before this global value is reached, decreasing theL = 0.lto maximal stress even more. If the casting was never
removed, the effect of the coating would have been a1. For an average tool steel, k and pc, are chosen to be stress reduction of 37%. which is seen from Figure 12.
k = 30 W/(mK) pC, ý 5.0 .106 J/(m3K)
The h value from the coating will be assessed at1000 W/(m2 K). Conclusion
k 30 W/(mK) An analytically based method for predicting the thermal2. c = -- = 60.10o m'/s stresses in the high-pressure die casting process has been
pC, 5.0. 106 i/(M3'K) developed. It leads to expressions and formulas able to
a h-/• assess the stresses for the two cases of a casting with anda - without a coating. The solutions are presented in charts,
which eases the practical use considerably because the
1000 W/(minK). 6 .0 10 - ' m2/s need for calculations is limited to calculating the heatthat must be removed from the casting, Q, and a couple
30 W/(mK) of model numbers and constants. This makes the results
= 0.0817 s-0 5 very user friendly.
Appl. Math. Modelling, 1994, Vol. 18, October 657
Analytical method for predicting thermal stresses during die casting: J. H. Hanel and P. N. Hansen
Another aspect of analytical solutions is that they p density (kg/m 3)often provide increased insight into the process under R thermal resistance (K/W)consideration. The solutions obtained in this paper s thickness of casting (m)reflect the fact that controlling the HPDC process is a oa stress tensor (Mpa)coupled multiparameter problem, in which the stress a.... dimensionless stresslevel is closely connected to the difference in temperature, t time (s)but also to the dwell time and the effectiveness of the T temperature (*C)coating expressed by the h value or the Bi number. On Tl initial temperature of die (°C)the other hand, real three-dimensional HPDC problems TX liquid temperature ('C)obviously require numerical calculations if the tempera- T4,, temperature of melt ('C)ture and stress fields are the goals of the calculations. Ta,_ temperature when removing casting (*C)
T surface temperature of die *C)Nomenclature T, solid temperature ('C)
T, mean temperature of melt (*C)a internal parameter (s-°5)
a linear expansion coefficient (K- 1)Bi Biot numberCL specific heat, liquid state (J/(KgK)) Referencesc, specific heat (J/(KgK))cS specific heat, solid state (J/(KgK))CSL specific heat, solid/liquid state (J/KgK)) I Weiner, 1. H. An clastoplastic thermal stress analysis of a free plate.A5 depth of penetration (m) I Appl. Mech. 1956. 23, 397-l016,J Kronecker delta 2 Weiner, J. H. and Boley, B. A. Elasto-plastic thermal strneseýs in a
solidifying body. J. Mech. Phys. Solids It, 1963. 145-154S Young's modulus (Mpa) 3 Pitts. D. R. and Sissom, L. E. Heat Transfer. McGraw-Hill Book1 strain tensor Co., New York, 1977. p. 77
Fo Fourier number 4 Danzer, R.. Krainer, E., and Schindler, F. M. Creep behaviour ofIt heat transfer coefficient (W/(m2K)) AlSi H l10and H13 materials with regard todie life in diecasting ofk thermal conductivity (W/(mK)) brass and aluminumn. 12th International Die Casting Congress andK thermal cndifucivity (W/() Exposition No. G-T-83-012, SDCE, Minneapolis, 1983L thermal diffusivity (m2/s) 5 Pins, D. R. and Sissom, L. E. Heat Transfer. McGraw-Hill BookL length of l-D domain (mn) Co., New York, 1977, p. 78L latent heat (J/Kg) 6 Stu.m, . C., Kallien, L., Hartmann. G.. and Han,, P. N.v Poisson's ratio Modelling of advanced casting proccssta illustrated on the examplesq heat flux per unit area (W/m 2) ofhigh die casting and investment casting applications. MA TTECHQ er YO Symposiwn, Helsinki, Finland. 1990Q energy per unit area (J/ 2) 7 Hattel. J. H. and Han.men P. N. Numerical modeling of theQ,*,, dimensionless energy per unit area (J/m2) thermallyinducedstr indiecastingusinga I-Dinodcl,inpress
Appendix I
The reduction of (7) to (8):E I +XI 2v v acAT-2v I + t,,- (I +v)aAT (7)f6tCv +, 1-2 T-;"
( E f(l -v - 2v2)s, + (v + vY - (I + vXtI - v))oMATl
F(+ vXl - 2v) I+ _ -v
E 5(1 + vX - 2v)e,,, + (v + V2 - I-v'))aATl
£ 5(1 + .vXl -2 2vs,, -(1-v -v
E 5(1- vXl - 2v)E,, - (1 + vXl - 2v)aAT=e (1 + v)(l -- '2,) 1 1 -
E(l(v -l-2v) 2vg, I- +YX 2v)ccAT}
- - - aT} (8)
558 Appl. Math. Modelling. 1994, Vol. 18, October
Analytical method for predicting thermal stresses during die casting: J. H. Haetel and P. N. Hansen
Appendix 2The evaluation of the integral in (30):
E h(T, - IT) exp {a't){erfc (a..jr}} dr = h(T,, - Tit E exp {art){erfc {aJft}} dt
The integral is now calculated by partial integration. For this we need the first derivative of the complementaryerror function with respect to time. Using Leibniz's rule yields
J e r2 rf = e 2 az,-t {erfc {aj/-t, - ¾t[ Io•./ dua> -- ejFn -2~
Ib .y ', 1= 7Z J
Now, we get
J0 exp {a't)(}rfc {oajl} dt
1i 2 aexp {a2i}{l - erfa + .J -+ exp { -- exp {-a tf dt
=[Aexp {a't){l-er{aA}}i +- c 0 l Idt= Ixp {a't}{l- erf {a -)) 21ta 2
21= -iexp (a't) (1 -- cr1 ,ay/t/j 1
= 2 {exp (a ..) err -{ 1a+) al.,/
Appl. Math. Modelling, 1994, Vol. 18, October 559
4cAI C, Z
lot 2TAXAM
141
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03N
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Journal of Thermal Siresses, 24:141-192, 2001Copyright © 2001 Taylor & Francis0149-5739/01 S12.00 + .00
FINITE ELEMENT MODELING AND SIMULATION. OFWELDING PART 1: INCREASED COMPLEXITY
Lars-Erik LindgrenLuled University of Technology
andDalarna University
Luled, Sweden
Simulation of welding has advanced from the analysis of laboratory setups to realengineering applications during the last three decades. This development is outlinedand the directions for future research are summarized in this review, which consistsof three parts. This part shows that the increased complexity of the models givesa better description of the engineering applications. The important developmentof material modeling and computational efficiency are outlined in Parts 2 and 3,respectively.
The development of welding procedures is based on performing experiments [1],and a Welding Procedure Specification (WPS) is the final result. The evaluationof a welding procedure is based on joint integrity, absence of defects,microstructure, and mechanical testing, for example. Computational methodsare rarely used in the process of developing welding procedures. It is expected thatsimulations will complement the experimental procedures for obtaining a WPSsince aspects like residual stresses can then be considered when comparing differentwelding procedures. Furthermore, simulations are useful in designing the manu-facturing process as well as the manufactured component itself. Distortions areusually in focus in the first case whereas residual stresses are of interest in thelatter case.
This review concentrates on the simulation of fusion welding processes ofmetals. Theseprocesses have several common traits and are therefore similar tomodel. Publications that present finite element simulations of the mechanical effectsof welding appeared in the early 1970s, and simulations are currently only used in
Received 12 August 1999; accepted 12 April 2000.This work has been financed by NUTEK (the Swedish National Board for Industrial and Technical
Development) via the Polhem Laboratory. Its completion has also been made possible by the cooperationwith ABB Atom AB and Volvo Aero Corportion in a project about multipass welding, where this reviewwas one part.
Address correspondence to Professor L.-E. Lindgren, Department of Mechanical Engineering, LuleAUniversity of Technology, SE-971 87 LuleA, Sweden. E-mail: [email protected]
141
142 L.AE. LINDGREN
applications where safety aspects are very important, like aerospace and nuclearpower plants, or when a large economic gain can be achieved. The scope of mostsimulations has been to obtain residual stresses and corresponding deformations.There are also publications that present analyses of hot cracking and otherphenomena.
A common concern in welding simulation is to account for the interactionbetween welding process parameters, the evolution of the material microstructure,temperature, and deformation. The resultant material structure and deformationmay be needed in order to, in combination with in-service loads, predict life timeand performance of a component. Therefore, research in this field requires the col-laborative efforts of experts in welding methods, welding metallurgy, materialbehavior, computational mechanics, and crack propagation, for example. It isimportant to observe that uncertain material properties and net heat inputmake the success of simulations to a large extent dependent on experimental results[2].
The review consists of three parts. This is Part 1, "Increased Complexity," andPart 2 is "Improved Material Modeling." Part 3 is called "Efficiency andIntegration." They all outline the development of welding simulation and areseparated into sections that focus on different aspects of finite element modeling.Each section contains some recommendations based on the review and the experi-ence of the author. It is hoped that this approach will be appropriate for thosewho are entering this field of research and useful as a reference for those alreadyfamiliar with this subject.
INTRODUCTION TO PART 1
The different approaches for accounting for thermomechanical couplings and cor-responding computational strategies are given here. Initially it was only possibleto perform simulations using crude models to reduce the required computationalpower. It has been a strong development in the models, which is clearly seen inthe section "From One-dimensional to Three-dimensional Models." This has beenpossible because of increased computational power and improved computationalmethods. The latter is reviewed in Part 3. Special complications exist whensimulating multipass welding, and they are assigned a section of their own in thispart. Welding simulation is still not purely predictive. It requires some informationabout the net heat input. This is also included at the end of the article. The strategyand possible models for heat input are given there.
MODELING AND SIMULATION
Modeling is the process of preparing a computational model. It involves determiningwhat aspects of the real problem can be ignored or simplified. The deliverable of themodeling phase is usually an input file that is used in the simulation. The weldingsimulation shows the actual computing of the evolution of at least the thermal
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART I 143
and mechanical fields. However, the notation modeling and simulation are usedinterchangeably in this article. The review is primarily concerned with detailedsimulations of the welding process and not with simplified models like those wherean external force or inelastic strains replace the weld in order to obtain the residualstresses directly.
WELDING PROCESSES
Fusion welding processes, in which the metal parts are heated until they melttogether, are the scope of this review. They can be performed with or withoutthe addition of filler material. Arc welding, electron beam welding, and laserwelding belong to this category of welding processes. The differences betweenthese welding processes are expressed in terms of the distribution of heatand filler material in the thermomechanical model. Thus, the review excludes,for example, explosive welding, friction welding, and spot welding. It is con-cerned with the mechanical effects of welding and does not include more gen-eral information about welding processes and metallurgy. The reader isreferred to the classic book by Masubuchi (1980), which still is worth reading.The book by Granjon [3,4] gives an easy-to-read introduction to the metallurgyof welding, and the book by Easterling (1983) is a step toward modeling ofwelding. The excellent book by Radaj [5,6] is also recommended. It gives moreinsight into the different processes and phenomena of which one should beaware. Grong [71 also discusses modeling with the focus on welding aluminum.Furthermore, there exists a large number of books about the more practicalaspects of welding.
OTHER REVIEWS AND SOURCES OF INFORMATION
Several reviews of different sizes and with varying scopes exist that are concernedwith the mechanical effects of welding. Marcal [8] made an early summary of experi-ences from welding simulations. The reviews by Karlsson [9,10], Goldak et al.[11-14], Smith [15], and Radaj [6,16] include references to simulations performedup to 1992. The research in Japan is reviewed by Ueda et al. [17-19] and Yuriokaand Koseki [20]. The IUTAM Symposium on the Mechanical Effects of Welding'[21] summoned active researchers, and the proceedings give a good view of the stateof the art in simulation of welding at that time. Simulation software used by differentresearch groups are summarized by Dexter [22]. Chandra [23] reviews girth-buttwelds in pipes somewhat; and a more thorough review is given by Ravichandran[24], who describes the use of the finite element method for simulating weldingof pipes. The book by Karlsson [25] has some chapters devoted to weldingsimulation.
144 L.-E. LINDGREN
The current review not only updates and extends the previous reviews but it hasits own approach. The development in simulation of the mechanical effects ofwelding is outlined from the finite element modeling viewpoint. Each aspect ofmodeling and simulation has its own section that is concluded with some recommen-dations based on the review and the author's own experience.
The reader who wants to be continuously updated in the field of simulations inwelding can follow proceedings from some conferences that concentrate on thisfield. The conferences of Mathematical Modeling of Weld Phenomena [26] aredevoted to simulations and associated experiments. Trends in Welding Researchare conferences sponsored by the American Welding Institute and AmericanSociety Materials International that are equally important with a wider scope.Computer Technology in Welding, organized by The Welding Institute inCambridge, United Kingdom, also encompasses the use of FEM for weldingsimulation. The Japanese Joining and Welding Research Institute (JWRI) publishestransactions that include a section concerned with the simulation of welding. Oneissue [27] was completely devoted to the theoretical prediction of joining andwelding.
TOWARD INDUSTRIAL PRACTICE
The FEM is the most important tool used in simulating the thermomechanicalbehavior of a structure during welding. It is a general tool but may be computerdemanding. The remark by Masubuchi [3], "Hopefully, it will only be a shorttime before the computer simulation of simple joints such as the fabricationof a built-up beam and the one-pass welding of a butt weld becomes commonplaceindustrial practice" (p. 187) became more pessimistic two years later [28] whenthey concluded that "it cannot be used in everyday practice." Their conclusionwas based on the large computing time that was required in their simulations.However, this is less true now and will be even less true tomorrow. Figure 1shows the increase in the size of the computational models in welding simulationduring the last decades. The studied welding cases are not only laboratory setupsbut also real engineering structures. The required computing capacity may stillbe a problem in the case of complex three-dimensional structures. However,the major obstacle for using the simulations in industrial practice is the needfor material parameters [2] and the lack of expertise in modeling and simulation.The problems involved in simulations of welding made Marcal [8] state that"welding is perhaps the most nonlinear problem encountered in structuralmechanics." Goldak et al. [11] suggested that the difficulties experienced bythe pioneers Hibbitt and Marcal [29] discouraged others from entering this field.It is hoped that this review will show the possibilities and not only the difficultiesin modeling and simulation of welding and thereby encourage an increased use ofthe FEM in this field.
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART 1 145
IE+8
dof*nstep [214] [176]
1 E+7 [185] 4
[36] [127] [38] *t122]
1IE+6 (92] 1441
[104]1 c [37]* * [214]1
IE+5 [97] [106]110
IE46[63] 4[39]
1975 1980 1985 1990 1995 2000Year
Figure 1. Size of computational models of welding measured by degree of freedom multiplied by numberof time steps versus year of publication of work.
SIMULATION OF WELDING AS A COUPLED PROBLEM
The welding process involves many different phenomena [30,31]. In this section dif-ferent couplings are discussed, but note that this review is not concerned with spot,explosive, or friction welding. Simulations that ultimately are concerned withthe mechanical effects of welding have to compute the thermal and mechanical fields.The material behavior, due to the changing microstructure, may depend on the tem-perature and deformation histories. This requires that one consider themicrostructure evolution when modeling the material behavior. The possiblecouplings are shown in Figure 2 and explained in Table 1.
Lindgren et al. [32] outline basic equations used in thermomnechanical analysesthat account for large deformations. They are given in Appendix A of Part 3 ofthis review. Small strain approximations are often used, but one must be aware thateven moderate rotations will create spurious stresses [33]. Accounting for largedeformations is a small additional cost in this context. The argument about betterconvergence for a small deformation analysis used by Ueda et al. [34] for choosingthat formulation is irrelevant.
Most analyses are performed in two steps. The thermal analysis is followed bythe mechanical analysis (e.g., [29,35-38]). The thermal dilatation (coupling #1),which is the sum of the thermal expansion and the volume changes due to phasetransformations, drives the deformation. Some analyses only take the thermalexpansion into account, and there exist different approaches for including the effect
146 L.-E. LINDGREN
Heat flow 4 Deformation
Microstructure
Figure 2. Phenomena occurring during welding.
of the microstructure, that is, the dependency of the material properties on the his-tory of the thermomechanical process. Details about modeling the dependencyof these properties on the temperature history are given in the section entitled"Improved Mateiial Modeling" in Part 2. This dependency may be approximatedor ignored in the analysis, thereby making it possible to perform the simulationin a two-step approach according to Figure 3.
It may be necessary to update the deformation during the thermal analysis, forexample, when fixtures change the thermal boundary conditions due to thedeformation (coupling #4a). The thermal and mechanical analyses can then beperformed in a so-called staggered approach or simultaneously. The staggeredapproach has been used in most cases when the mechanical effects on the thermalfield are accounted for [39,40]. It can be performed as shown in Figure 4. The thermaldilatation is the driving force, and therefore one usually starts each time step bysolving for the temperatures, Tn"l, for time tn+ l and thereby using the geometry
Table I Couplings in Figure 2
Coupling # Explanation
Ia Thermal expansion depends on microstructure of material.lb Volume changes due to phase changes.2a Plastic material behavior depends on microstructure of material.2b Elastic material behavior depends on microstructure of material.3a Heat conductivity and heat capacity depend on microstructure of material.3b Latent heats due to phase changes.4a Deformation changes thermal boundary conditions.4b Heat due to plastic dissipation (plastic strain rate).4c Heat due to thermal strain rate.4d Heat due to elastic strain rate.5 Microstructure evolution depends on temperature.6 Microstructure evolution depends on deformation.
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART 1 147
Heat flow 3 Thermal properties
5
Deformation Thermo-mechanicalproperties
Figure 3. Performing a thermal analysis followed by a mechanical analysis is made possible by ignoringand simplifying some couplings given in Figure 2.
x". The geometry is updated in the subsequent mechanical analysis for this time step.Thus, the updating of the geometry in the thermal analysis lags one time step behind.It is possible to start with the mechanical analysis and take the thermal analysisafterward during each time step. Then the thermal load in the mechanical analysisis one time step behind. It is also possible to iterate between the thermal and mech-anical analyses in order to reduce the out-of-phase effect in either thermal or mech-anical analysis. Another approach is to solve temperatures and displacementssimultaneously. This leads to an unsymmetric system of coupled nonlinearequations. The complications involved in solving this system of equations arenot worthwhile and not necessary because the coupling is quite weak for the weldingprocesses within the scope of the review. However, Ronda and Oliver [41] used this
Thermal analysisIncrement in thermal loadGeometry x'Tn
Mechanical analysisIncrement in thermomechanical loadXn 4A i Xn+ I
Figure 4. A time step in a staggered approach for thermal and mechanical analyses.
148 L.-E. LINDGREN
approach in ABAQUS. The explicit code PASTA, used by Mahin et al. [42-44], alsosolves temperatures and deformations simultaneously, but the explicit formulationcompletely avoids the use of a coupled system of equations.
The heat generated by the deformation can be ignored in the case of welding.Boley and Weiner [45] state that it is possible to neglect the heat caused by thestrain rate for a wide class of thermoelastic problems (couplings #4c and #4d)and that it is consistent to ignore the inertia forces at the same time. Thus,the mechanical analysis can be assumed to be quasi-static. This means thatthe inertia forces can be ignored although the deformation is changing with time.The thermal strain rate (coupling #4c), part in this coupling can be accountedfor directly in the thermal analysis by modifying the heat capacity [39,40]. Argyriset al. [39,40] also showed that the heat generated by the plastic dissipation(coupling #4b) can be safely ignored. This conclusion is confirmed when the heatinput is compared to the plastic dissipated energy [46]. It is no problem to includethe heat caused by plastic dissipation since it is a small additional cost if thethermal and mechanical analyses are performed using the same software anda staggered approach.
Carmet et al. [47] performed a thermal analysis followed by a metallurgicalanalysis and finally a mechanical simulation. A fully coupled thermometallurgicalanalysis followed by a subsequent mechanical analysis was performed by Duboiset al. [48], Bergheau and Leblond [49], Devaux et al. [50], and Pasquale et al. [51].The papers by Dubois et al. [48] and Devaux et al. [50] also include a simulationof the hydrogen diffusion, which is performed after the mechanical analysis. Roelens[52-54] performed the same kind of analysis but repeated the sequence ofthermometallurgical analysis followed by a mechanical analysis for each weld pass.Thus, the effects of the deformation on the microstructure evolution and on theheat conduction (couplings 4 and 6 in Figure 2) were ignored in all these analyses.
Therm al analysis Mi rostructure
Geometry xn I Microstructure
-T +• Tnl model
L -- - - - -- i L------ ---- J
Mechanical analysis
xn b xn+l
Figure 5. A time step in a staggered approach for thermometallurgical and mechanical analyses.
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART I 149
The first coupled thermometallurgical and mechanical (TMM) analyses of weldingwere performed by Inoue and Wang [55] and Inoue [56,57], who included the plasticdissipation in the heat conduction equation, and it seems therefore that they used astaggered approach according to Figure 5. B6rjesson and Lindgren [58,59] also useda staggered approach for a TMM simulation of multipass welding. Bhadeshia [60]discusses the influence of stress on the microstructure evolution (coupling #6),but this has not yet been accounted for in welding simulation. TMM simulationsare discussed in the section "Dependency on Temperature and Microstructure"in Part 2.
Recommendations. Welding often leads to deformations that may be visible, andthus a large deformation analysis is appropriate. The uncoupled, quasi-staticapproach for computing the thermal stresses is accurate, but it is often convenientto use the staggered approach. Then the whole analysis is performed using the samesoftware, and the communication of the temperatures from the thermal analysisto the mechanical analysis via a file is avoided. This approach is necessary ifone needs to account for thermal boundary conditions that depend on thedeformation (coupling #4a). Few codes can compute the microstructure evolution.See the section "Improved Material Modeling" in Part 2 in order to understandthe implications for the material modeling.
The general considerations for finite element modeling is, of course, alsorelevant in this field. Two notes in this respect need to be emphasized here.The first concerns the consistency between the thermal and mechanical analysis[61] and the choice of element. The degree of the finite element shape functionsfor the displacements should be one order higher than for the thermal analysisdue to the fact that the temperature field directly becomes the thermal strainin the mechanical analysis. The strains are obtained as the derivatives of thedisplacement. Lindgren et al. [62] used eight-node quadratic elements in themechanical analysis and four-node bilinear elements in the thermal analysis.Usually the same elements are used in the thermal and mechanical analyses.The linear four-node element is preferred because one, in general, favors morelower-order elements than fewer higher-order elements in nonlinear problems.Friedman [63,64] uses a quadratic element, but he agrees with Hibbitt andMarcal [29] that a fine mesh with linear elements is preferable because linearquad, in two dimensions, and brick, in three dimensions, elements are the basicrecommendation in plasticity [65,66]. They perform better than linear trianglesor tetrahedrons. If triangles or tetrahedrons are used in computational plasticity,the elements should have quadratic interpolation functions for the displacementfield. Andersson [35] used higher-order triangular elements. Using a linearelement requires that the average temperature should be used to compute a con-stant thermal strain in the mechanical analysis [61] in accordance with the pre-vious discussion. The second note is concerned with plastic incompressibility.It is important to underintegrate the volumetric strain when using linearelements in order to avoid locking. Then the volumetric strain is constant inthe element.
150 L.-E. LINDGREN
FROM ONE-DIMENSIONAL TO THREE-DIMENSIONAL MODELS
Finite element models of welding must not only cover the studied domain but also befine enough to resolve details near the weld [30]. Solving the nonlinear,thermomechanical problem in a time-stepping procedure for the welding processrequires computational power. The first papers concerning the simulation of weldingreduced the computational cost of the simulations by reducing the dimension of theproblem from three to two or one. The earliest numerical predictions of residualstresses were probably those of Tall [67]. Tsuji [68] performed similar calculations.The mechanical analysis was essentially one-dimensional, although the analytic sol-ution for the temperatures was two-dimensional in the work by Tall. Thus, the longi-tudinal residual stresses were predicted as if by the collection of many, paralleluniaxial specimens. The earliest two-dimensional finite element analyses appearedin the early 1970s, by Iwaki an Masubuchi [69], Ueda and Yamakawa [70-72], Fujitaet al. [73], Hibbitt and Marcal [29], and Friedman [63]. The early simulations byFujita et al. [74] and Fujita and Nomoto [75] used only a thermoelastic materialmodel.
The typical two-dimensional finite element models for welding simulation areshown in Figure 6. Goldak et al. [11] discuss the restraints in thermal and mech-anical analyses when two-dimensional models are used. McDill et al. [76] discuss
Analysed cross-sectionPlane deformation or -----------plane strain model
a) eAnalysed cross-sectionAxisymmetric model
b)
Analysed planePlane stress model
c)
Figure 6. Two-dimensional models of welding.
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART I 151
the approximations introduced by enforcing a plane strain condition of a weldcross-section. The cross-section orthogonal to the welding direction is analyzedin the axisymmetric, plane strain, or plane deformation models. Then the heat con-duction in the welding direction is also ignored. This is discussed in Andersson [35].It is assumed in the plane stress model that everything is constant through thethickness and the stress normal to the plane is zero. This requires that the plateis thin and that out-of-plane motions are ignored. The arc moves in the planeof the model.
Most two-dimensional analyses have been under plane strain conditions(Figure 6a). That is, the weld is divided into thin slices perpendicular to themotion of the source and these slices are assumed not to interact with eachother. The use of plane strain conditions entails certain subtle consequences.Longitudinal heat flow and longitudinal displacements are assumed to be zero.While longitudinal heat flow is probably never very large [35], longitudinal inter-actions in the stress may be large. Experimental measurements of thedisplacements during welding have shown small longitudinal displacements inthe direction of motion of the source before the source has arrived, and some-what larger rearward displacements after the source has passed. Furthermore,the plane strain condition means that the net longitudinal residual stress aftercooling is not zero. It is as if the whole plate was rigidly fixed in the longitudinaldirection. The too high longitudinal restraint due to the plane strain assumptioncan be alleviated by assuming generalized plane strain [35] or plane deformation(e.g., [28,53,54,77-84]). The paper by Andersson [35], with a generalized planestrain model, is described in detail in "A Welding Simulation Revisited" in Part3. Wikander et al. [81,82] used a generalized plane deformation model and simu-lated the bending of a flat plate to a U-shaped beam and thereafter the weldingof a plate, thereby creating a hollow beam with rectangular cross-section.The strain in the longitudinal direction is constant in the generalized plane straincase, and it is a linear function of the coordinates in the case of planedeformation. The agreement between a plane deformation model and athree-dimensional model can be quite good in the case of beam-like structures[85]. However, this is not always the case. The analysis by Dike et al. [86]of a 12-in multipass weld in a 24-in plate that is restrained at its end neededa three-dimensional model to obtain a correct weld shrinkage. This is describedin the section "Increasing Complex Models." Their two-dimensional modeldid not give a weld shrinkage in agreement with measurement, whereas thethree-dimensional model did. Two-dimensional analyses have also been appliedto circumferential welds in pipes (Figure 6b) (e.g., [29,47,87-93]). Theseaxisymmetric models have a hoop strain that varies with the axial and radialcoordinates. Otherwise, they rely on assumptions similar to the plane strainor plane deformation models. Axisymmetric models for a bead on a disk havebeen used by, for example [29,55]. Hibbitt and Marcal [29] simulated the weldingof a disc shown in Figure 7. The results were compared with experiments on aplate made of HY130/150 steel. Their finite element model is shown in Figure8. The model lacked many features in material modeling, which may explainthe discrepancies between measurements and calculations; see Figure 9. The
152 L.-E. LINDGREN
.•li . disc
I• ,,,• ]_I• groove
_ . . ":5 13._/'16 "i"n
analysedcross-section
start and stop
Figure 7. Geometry studied by Hibbit and Marcal [29).
too large tensile residual stresses in the weld may be caused by neglecting thetransformation plasticity in the model [94]. See Figure 14 in Part 2 of thisreview, where this effect of the material modeling on the residual stressesdue to welding is shown. There have also been two-dimensional plane stressmodels (Figure 6c) that can be used when simulating the welding of a thin plate,for example [36,42,71,95-99]. Then the stress in the thickness direction is ignoredand the deformation is assumed to be in the plane of the plate. These modelsfollow the heat source that moves in the plane of the mesh, which requiresa large number of elements for a long weld. Therefore, the size of the smallestelement is usually larger in these models than in the previously mentionedtwo-dimensional models of the weld cross-section. The configuration studiedby Muraki is shown in Figure 10, and the corresponding finite element mesh
76.2 mm 147.6 mm
Figure 8. Finite element axisymmetric model used by Hibbit and Marcal [29].
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART I 153
Residual longitudinal stress1200
[MPa]700-
70t Computed
200
-300 Measured I I I-800 . . ..
0 50 100 150
Radial coordinate 1mm]
Figure 9. Computed and measured longitudinal stresses for the bead on disc by Hibbitt and Marcal [29].The bars denote fluctuating data from the finite element analysis.
is shown in Figure 11. The weld speed is 13.6 mm/s for the bead-on-plate case.Measured and computed strains are shown in Figure 12. Canas et al. [100]compared a plane stress model (Figure 6c) with a plane strain model (Figure6a) and found that the latter gave a wider zone with higher longitudinal residualstresses.
Simulations where two-dimensional models are used still dominate the pub-lished work. This is partly due to the fact that they give useful results and partlydue to the fact that three-dimensional simulations demand a large computingpower if a good resolution is to be obtained. Different techniques have been usedto solve three-dimensional problems without a complete three-dimensional model.Fujita et al. [95] combined plane stress models to simulate the welding of astiffener on a plate. They ignored all angular deformations and assumed that eachsection deformed only in its plane. Thus, the lower edge of the plane stress modelfor the stiffener must be straight since it is connected to a plane stress modelfor the plate also modeled as a plane stress case. Rybicki and Stonesifer [101]added stiffnesses orthogonal to a two-dimensional model of welding in orderto include three-dimensional effects. These stiffnesses were obtained from athree-dimensional model of the structure. Michaleris et al. [102] comparedtwo-dimensional and three-dimensional models of a multipass butt-welded plate.They simulated the first pass with both models. The two-dimensional modelhad a larger fusion zone, despite a reduced heat input, and a larger zone with
154 L.-E. LINDGREN
starting tab
~1ZI§ weld line
- -
clamps" -2
2218.6 mm"'"
S• - finishing tab
Figure 10. Bead-on-plate welding configuration studied by Muraki et al. [96].
weld direction weld line
Fue Fne e p /sFigure 11. Finite element plane stress model used by Muraki et al. [96).
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART 1 155
0,125[%] Measured0,075•-- - Computed
0,025
-0,025
-0,075
-0,125 .. . . .
0 50 100 150 200Time [sec]
Figure 12. Measured and computed transient transverse mechanical strains on the upper surface of theplate and 25.4 mm away from the weld centerline and 381 mm from the edge for the bead-on-plate case.From Muraki et al. [96).
high tensile longitudinal stresses than the three-dimensional model. They includedan additional dashpot in the two-dimensional model to make it match thethree-dimensional results for the first weld pass. This gave quite good agreementbetween measured and computed displacement of the butt joint's edge untilthe plate was turned and the welding continued on the other side. The opposite,moving information from a two- to a three-dimensional model, was performedby Michaleris and DeBiccari [103]. In this study they transferred welding residualstresses obtained from a two-dimensional plane deformation model to athree-dimensional model of the structure in order to study weld-induced buckling(Figure 13). The computed residual stress state in the two-dimensional model(upper part of Figure 13) was compared with measurements. Different yield stressmodels, shown in Figure 14, were tried in this comparison. The longitudinalresidual stresses on the bottom surface are shown in Figure 15. A constant tem-perature in the weld region was imposed on the three-dimensional model withorthotropic thermal expansion. Thus, it was possible to achieve different loadingin the longitudinal and transverse directions of the weld. The thermal loadwas chosen so that the stresses in the midspan of the three-dimensional modelbecame the same as in the two-dimensional welding simulation. The computedcritical buckling load of the stiffened plate agreed well with experimentalobservations.
The full three-dimensional models are based on solid models where all strainsand stress components are included. Shell elements can also be considered asthree-dimensional models because they are described with three coordinatesalthough the thickness must be thin. The shell models assume that the stress is zeroin the thickness direction and that a straight line in this direction continues to
156 L.-E. LINDGREN
Symmetry Boundary T
Condition on left edge Stiffener
// Panel
/..,•.•Support Plate
Thermal K
load withortbothropic thermalexpansion which giveloads on 3D modelthat corresponds to2D results.
F•'•<.• <>•" • -Y Weld Load-Applied to
- - Region
Figure 13. Two-dimensional model of welding and three-dimensional structural model [103].
be straight during the deformation, and therefore it is possible to analyze everythingw.r.t. a reference plane. Different assumptions about the variation of the tempera-ture over the thickness can be applied even if the total strain can only vary linearly.Typical shell and solid models are shown in Figure 16.
The first three-dimensional residual stress predictions of full welds appear tobe by Lindgren and Karlsson [104], who used shell elements when modeling athin-walled pipe. Karlsson and Josefson [37] modeled the same pipe using solidelements. The pipe is shown in Figure 17; and the shell model [104,1051, thesolid model [37], and the axisymmetric model [106] are shown in Figure 18.These models were compared in the paper by Josefson et al. [107]. These resultsand the results using the axisymmetric models by Karlsson [106] and Josefson
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART 1 157
T ('F)100 600 1100 1600 2i00 2600
600 ,
550 - .80.0
500 - _.__-a,, Bose metal 70.0450 - ....... ...... a,, Filler m etal
400 -- )0, atE, =.01 Base Metal 60.0350 V---*-a, at , =.01, Filler Metal" " 50.0 "
"• 300 ""40
250
200 30.0
150 20.0
50 10.0........................
0 10.00 150 300 450 600 750 900 1050 1200 1350 1500
T ('C)
Figure 14. Different yield strength tried in welding simulations (103].
and Karlsson [108] are shown in Figure 19. The results from the different modelswere consistent. They also agreed well with measurements except for the residualhoop stress at the surface of the weld. This discrepancy was only reduced a littlewhen transformation plasticity was accounted for [108]. It is, of course, alsopossible to question the measurements since the hole-drilling technique is appliedjust at the surface of the weld. However, the measurements are, for all threecircumferential positions, much higher than the computed results. Murty etal. [109,110] addressed this problem and obtained the wanted tensile residualhoop stress in the weld in their latter paper. The material modeling in the weldmetal was improved by estimating the phase transformations and computingthe material properties using mixture rules of the phases. Nisstr6m et al. [I 1l]made a first attempt to combine shell and solid elements in a model of a weld.Gu and Goldak [112] did the same but only for the thermal analysis. Donget al. [113] used shell elements in investigating the effect of wall thicknessand welding speed on residual stresses of a pipe. They also implemented a layeractivation/deactivation scheme in a composite shell element for simulatingmultipass welding [114]. This was used in modeling a repair weld in a pipe [115]and also for repair weld in panels [116-118]. See the section "IncreasingComplex Models," where their work is also discussed. Ortega et al. [119] useda shell model to study the buckling of a welded plate.
158 L.-E. LINDGREN
Distance from symmetry line (in.)-12 -10 -8 -6 -4 -2 0 2 4 6 8 i0
<., r- -'-r - - • 70
450 -
7
60400 0
350 Noinal 50
300 --- Variation 1
"o" 250 --- Variation 2:E 200 -..... Variation 3 30
0 150 - l Measurement 20
100 1-10lo-20
, 100 -20
-300 -200 -100 0 100 200
Distance from symmetry line (mm)
Figure 15. Longitudinal residual stress variations on bottom surface for 717-J/m weld. The computationswere performed with different yield strength [103).
Analysed surface -shell element model
a)
Analysed body -solid element model ' ----
b)
Figure 16. Three-dimensional models of welding.
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART 1 159
wel lin---,, .,
7<(b)
T~I -
5mm 10mm 15 mm(C)
Figure 18. (a) Shell model [104), (b) solid model [37), and (c) axisymmetric model [106,108) of pipe inFigure 17.
160 L.-E. LINDGREN
300(MPa] 0 Exp at 270 °
200 - * Expat 1500
100 0 Exp at 300
*- Solid (Karlsson and0 Josefson 1990)
- - - Shell (Lindgr-en andKarlsson 1988)
,Ai + tp (iosefson
and Kurlsson 1992)-200-.. Axi (Josefson and
0 Karlsson 1992)
307- Murty et.al. (1996)-300 •
0 10 20 30 40Axial coordinate (mram]
Figure 19. Measured and computed residual hoop stresses on outer surface of pipe.
Ravichandran et al. [120] used a shell element to simulate the manufacturing ofa T-beam by a one-pass fillet weld. They argue that the plane strain assumption isinvalid for this case based on results by Goldak et al. [121]. The plane strain modelis an approximation, and in many cases a good one, in the same manner as the shellelement used in their paper is. One can get useful information by completing theircoarse shell model with a fine two-dimensional mesh and assuming generalizedplane strain for this particular problem [85]. Ravichandran et al. [120] also com-pared different models of welding of a thin pipe based on shell elements andperformed simulations using a shell element. Lindgren et al. [122] madethree-dimensional models of a seam-welded pipe using solid elements. Results fromthis study are shown in the section "More Efficient Computational Methods" inPart 3.
Other early three-dimensional models were used by Goldak et al. [121], whomade an exploratory three-dimensional weld simulation of a 2.5-cm short weld.Chakravarti et al. [122,123] used a slice of three-dimensional elements in the com-putational model for a cross-section of a weld. Oddy et al. [61,94] modeled a longbead on plate using a three-dimensional model consisting of solid elements.Three-dimensional predictions of rotations and distortion in large welds were alsoperformed by McDill et al. [124], Tekriwal and Mazumder [125,126], Goldak etal. [127], and McDill et al. [76].
More three-dimensional applications followed. Ueda et al. [128] and Wang et al.[129] made three-dimensional simulations of a pipe-plate joint. They studied theeffect of the welding on the shape of holes and compared the use of instantaneousand moving heat source around the circumference. The effect of including only half
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART I 161
of the circumference was also studied. The conclusion was that a three-dimensionalmodel was recommended. Ma et al. [85] and Wu et al. [130] studied the effect ofdifferent welding procedures for a T-joint. They used a three-dimensional modelas a reference model and performed parametric studies using two-dimensionalmodels with assumed plane deformation. Choi and Mazumder [131] includedpostprocessing of the thermal field for computing primary dendrite arm spacingand grain growth in their model. Gu et al. [132] used a three-dimensional modelto investigate the out-of-plane deformation due to butt-welding of a plate wherethey account for gravity and different restraints. There have been somethree-dimensional models of multipass welding. They are described in the section"Increasing Complex Models."
Recommendations. The two-dimensional models for plane stress, planedeformation, and axisymmetric deformation are all relevant. They yield useful in-formation about residual stresses and permit the use of smaller elements nearthe weld. However, it should be noted that the effect of the two-dimensional con-straint on the results is larger for quantities like deformation and strains. Thiscan be seen in the paper by Dike et al. [86], which is described later. For example,the axisymmetric model corresponds to welding the whole circumference at once.This gives larger deformation in the axial direction than a moving heat source wherecolder regions in front of and after the arc resist this deformation better than theregion near the arc. The axisymmetric model will have the whole circumferenceat a high temperature for a short time and then larger axial deformations can occur.Two-dimensional models can also give errors in residual stresses as noted byAtteridge et al. [133] when they compare different two-dimensional models, planestrain according to Figure 6a, plane stresses as in Figure 6c, and three-dimensionalmodels (Figure 16b). See the section "More Efficient Computational Methods"in Part 3, where their three-dimensional models are explained.
The three-dimensional models are still very demanding. If possible, it would bevaluable to perform one three-dimensional simulation and then make a correspond-ing two-dimensional model [85]. Thus, it will be possible to see the influence of thekinematic constraints on the results. The two-dimensional model will be the basefor further study, like investigating the effect of different welding parameters onthe residual stresses. See also the section "Experimental Verification" in Part 3.The size of the elements and the time steps depend on the aim of the study. This,of course, is also related to how detailed a material model is used. If the overallresidual stresses are of interest, then a quite coarse mesh and simplified materialmodeling will do [106]. However, if details of the residual stresses near the weldare important, then a fine mesh and accurate material modeling is important. Thisis the case when studying hot cracking. Goldak et al. [11,121] offer some hintsfor modeling the welding process. The length of the time step is related to the sizeof the elements. Smaller elements should be accompanied by smaller time steps.Models where the arc is traveling in the mesh (Figures 6c and 16), usually havelarger elements and it is recommended that one take smaller time steps than thelength of the elements along the weld divided by the welding speed. Shorter timesteps are required at the start and finish of the weld. It is noted in Runnemalm
162 L-E. LINDGREN
and Hyun [133] that the adaptive meshing also gave smaller elements just after thewelding was finished. The mesh is usually much finer in models of the weldcross-section. One can take 100 time steps for one weld pass in these models. Theyare often combined with detailed material models where, for example,transformation plasticity may occur quite rapidly.
INCREASING COMPLEX MODELS
This section discusses more complex models like multipass welding, repair welding,cladding, hot cracking, and buckling.
Multipass welds are by far the most frequent structural application. The firsttwo-pass case, described by Ueda and Yamakawa [71,72], was a lap joint withtwo separate welds and is therefore considered as two single-pass welds. Multipassweld analyses were initially published by Ueda and coworkers [134,135], Lobitzet al. [136], Hsu [137], and Rybicki and coworkers [87-89]. Rybicki and Ueda withcoworkers have performed a large number of studies of different aspects ofmultipasswelding. They have been able to obtain quite accurate results despite rather crudemodels [I I].
Each successive weld alters the stresses and distortions caused by previouspasses. Analyzing multipass welds as a series of single-pass welds is certainly themost rigorous, albeit costly, process. Lumping successive passes together is oneway to reduce the cost. Several forms exist. Some are based on performing a com-plete thermal analysis, accounting for each weld pass and postprocessing these tem-perature histories to some kind of temperature envelope in order to simplify themechanical analysis. Some variations of the envelope technique exist. Otherssimplify the thermal and mechanical analysis in the same way by either mergingsome weld passes into larger welds or by accounting for some of the weld passes.These techniques ignore the history effects of the welding.
Ueda et al. [134,135] studied the multipass welding of plates with thicknesses of100 mm, 200 mm, and 300 mm, respectively (Figure 20). Subsequent heat treatmentwas performed on the 200-mm-thick plate in order to reduce the residual stresses.Welds in the lower groove were assumed to be laid simultaneously with correspond-ing welds in the upper groove. The cases had 7, 43, and 83 welds in each groove.These were lumped in the models to 5, 10, and 17 layers where the heat conductionis carried out for all passes, but the stress analysis is only carried out for the lastpass in each layer of welds. The configuration with 43 welds in each groove wascut into smaller specimens subjected to stress relief annealing. Plane stress conditionswere assumed in the analyzed cross-section of the weld. The experimental pieces werecut into 30-mm-long segments in order to relax the longitudinal stresses, therebyresembling the finite element model assumption of plane stress better. Good agree-ment with experiments was obtained (Figure 21) when considering the simplifiedmodeling of the influence of the phase transformations, exclusion of hardening,and other simplifications used.
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART I 163
y
2t x
S 2B 0
Figure 20. Configuration studied by Ueda et al. [134,135].
Transverse residual stress400
IMPa] l Measured - s welded
40 F-Measured
- as welded
0
Measured - 650 0C, 30°C/h I:-
-200 - Computed - 6500C, 300*C/h
Computed - as weldea\y-400 1
0 20 40 60 80 100
Distance from surface [mm]
Figure 21. Residual transverse stresses on top surface [134,135]. Heat-treated specimen was heated andcooled by 30*C/h. The maximum temperature was 6500C during this stress relaxation, and there wasno holding time at this temperature.
164 L.-E, LINDGREN
Ueda and Nakacho [136] investigated the effect of ignoring some weld passes inthe simulation of narrow gap welding. Including only every second weld pass gaveabout the same residual stresses as simulating every weld pass. However, the shrink-age at the upper part of the groove was halved in the analysis with a reduced numberof passes. This is probably due to the used heat input, which was constant per pass forall welds. Therefore, the model with fewer welds only received half the total energy.On the other hand, if the total input energy had been constant, then the residualstresses would have been more different between the two models. They found alsothat only the last weld passes had to be simulated if the residual stresses on thesurface were of primary interest.
Rybicki et al. [87-89] used a combination of lumping of welds and temperatureenvelopes to simplify the analysis. The temperature field was obtained from ananalytic solution. A series of heating steps to this envelope and a series of coolingsteps to room temperature were taken. Rybicki and Stonesifer [89] analyzed a7-pass weld by lumping it into four weld passes and a 30-pass weld into nine welds.The 7-pass weld can be seen in Figure 22, and the finite element model is shown inFigure 23. This worked quite well (Figure 24), whereas lumping of the 30-passweld into three layers did not agree with experimental data. Nair et al. [137]performed a similar, but more crude, analysis including post-weld heat treatment.They studied a pipe corresponding to the pipe analyzed by Rybicki and Stonesifer[89].
Free and Porter Goff [138] also used an envelope technique but modeled each ofthe seven weld passes individually. They used a series of heating steps to the maxi-mum temperature envelope and followed the actual transient cooling from thistemperature. They compared this with the approach of Rybicki and Stonesifer [89]using a series of steps for the cooling phase also. They found that this yielded onlyminor differences in the results. Lumping and envelope techniques are discussedand investigated in some detail by Leung et al. [139]. They concluded that it is poss-ible to introduce simplifications in the multipass welding analysis for obtainingresidual stresses. However, the transverse stress is more sensitive than the longitudi-nal stress due to the way the weld passes are lumped. Therefore, they favor a lumpingtechnique where all welds except the last in each layer are lumped together. The last
/ 7 -6
Figure 22. Seven-pass weld studied by Rybicki and Stonesifer [99).
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART I 165
I115 l mm No
8.56 mm•
R=48.59 mm
- Passes 6 & 7
-o-- Passes 4 & 5
-Passes 2 & 3
i- Root pass
Figure 23. Finite element model used by Rybicki and Sionesifer [89], where seven weld passes werelumped into four passes.
[MPal300 Left side of weld
100
100 Rig6 7 sde of we 3-100 __•
Computed-300
0 5 10 15 20 25
Axial distance from weld center [mm]
Figure 24. Measured and computed residual transverse stresses at inner surface of pipe (89).
166 L.-E. LINDGREN
weld of each layer is analyzed separately. Hong et al. [140] also evaluated theapproximations due to weld pass lumping. They tried to lump a 5-pass weld intoa 3-pass weld. They found that the lumping also affected the longitudinal stressesand, hence, introduced a weight factor to decrease the heat input and thereby alsoreduce the zone with large residual stresses so that the results compared morefavorably with the 5-pass simulation. Shim et al [141] simulated II weld passes indi-vidually and then lumped them into 6 passes, and 17 passes that were lumped into 7passes. They also found small differences between the models. Murthy et al. [109]studied a case with 20 weld passes using three different lumping models, all weldsmerged to three welds, top-layer analysis only, and a model where only the threelast passes of the top layer were included. The first two models gave approximatelythe same residual stresses on the surface but a different variation of the stressesacross the thickness.
Another technique for reducing the computational effort required in simulatingmultipass welding involves translating and superimposing the residual stress orinelastic strain field from a single pass to the other passes. It is not possible tosuperimpose the stresses directly. The stresses have corresponding nodal forcesin the finite element models. These forces can be superimposed in a separate analysisthat gives the wanted (approximate) total stress field.
Overlay weld repair consisting of several hundred weld passes was studied byChakravarti et al. [123,124]. The residual stress pattern of the multipass weldwas created by superimposing the residual stresses of individual welds and graduallybuilding the welds to patches of welds and finally combining all these patches. Thetotal distortion is determined as the elastic response of the structure to thisaccumulated stress field. The inelastic strains are called inherent strain by Uedaand coworkers [142,144]. They assume that this depends on local conditions andcan therefore be computed for fewer cases and then used in different applications(e.g., [142,144]). Yuan and Ueda [145] investigated the inherent strain for weldedT- and I-joints. The inherent strains were then used as equivalent loads in an elasticanalysis as a method for predicting the residual stresses. This superposition ofthe loads corresponding to the inherent strains can also be combined to multipasswelds. The problem of determining the relation between the inherent strain andthe welding parameters is addressed by Wang et al. [146].
There is also another complication that has to be addressed in simulatingmultipass welding [32], namely, the modeling of the addition of filler material.Two basic approaches are possible. The whole structure is included in the com-putational model in the first approach, that is, welds that have not been laidyet are present in the model. The elements, corresponding to nonlaid welds, shouldbe given material properties so that they do not affect the rest of the model. Thus,they are called quiet elements by Lindgren et al. (32] since they are present butshould not disturb the results. The elements are given normal material propertiesat the start of the weld pass. It is important then to remove all strains and stressesthat have possibly accumulated in these elements up to this point. The approachhas two advantages. It is easy to implement in most finite element codes andthe whole model can be defined initially. Variations of this method have been usedin most studies. It has been used, for example, by Rybicki et al. [87-89] and Brust
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART 1 167
and Rybicki [147], where the elements in the weld were included when the tem-perature decreased to 1 150°C. Michaleris [148] isolated the elements in the weldfrom the rest of the structure in the thermal model by using a gap conductanceelement with zero thickness. The conductance was zero until the weld was laidand thereafter it was given a very high conductivity. No interface element wasused in the mechanical model. The elements in the weld were directly connectedto the rest of the structure when the weld was laid and the stresses were zeroedinitially since the elements correspond to molten material. This was based on initialdevelopment by Tekriwal and Mazumder for ABAQUS, [149], thermal analysis,thermo-mechanical analysis [125], and later by Leung et al. [139], and Bertramand Ortega [150]. It was also used by others (e.g. [151-153], [140]). As a matterof fact, this is similar to what Hibbitt and Marcal [29] did, but they used a fluxbetween the connected nodes to bring them to the same temperature within a shorttime after the weld was laid. The use of interface elements in ABAQUS was necess-ary because ABAQUS could not prescribe the initial temperatures of nodes alongthe boundary of a weld in a general way if the element birth option was used.For example, if a single element represents the new weld and all its nodes arein connection with the surrounding material, then there is no node to which aninitial nodal temperature can be given and therefore it is not possible to give birthto an element with a given heat content. Welding analysis using ADINA [37,106]has been done using its element birth technique. The common approach is to givea heat flux to the born element in order to heat it to the wanted temperaturein a short time interval [154].
The second approach requires the restructuring of data each time the model isextended. The elements and nodes that correspond to nonlaid welds are notincluded in the finite element model until the weld is laid. They are called inactiveelements by Lindgren et al. [32]. This is a more correct approach but requiresa finite element code with the appropriate facilities and may require user interactionat each new weld pass. This approach seems to have been used by Free and PorterGoff [138]. It is less clear what Lobitz et al. [136] meant by "adding elementto the mesh." This may refer to the quite element approach. Josefson [92] usedthe inactive element approach and initially gave the element for the laid weld avery small stiffness in order to update the location of the nodes added to the modelvia the displacements of the nodes surrounding the laid weld. The paper byLindgren et al. [32] describes an implementation where all weld passes are definedat the start of the analysis and automatically included and initialized when a weldpass is laid. The cross-section of the weld is shown in Figure 25, and the finiteelement model is shown in Figure 26. A special procedure was used to movethe elements that are to be added to the model to the appropriate location withrespect to the rest of the structure that has become deformed. A smoothing pro-cedure was used to move the nodes to be added to the model to appropriatelocations before these elements are connected to the rest of the model. This processdid not give any initial strains in the elements. It is also shown in this paper that thequiet element method for adding filler material can give the same results as theinactive element method. An example of transverse stresses during welding isshown in Figure 27.
168 L.-E. LINDGREN
2.5I I I
Length= 2020 m I F III I I I
No weldII
I I 0 I III I
1800-371600
Figure 25. Cross-section of plate to be welded [321.
The quiet element technique is more practical for plane stress models [36,37]and three-dimensional models [37,125,126,155], where filler material is added inevery time step as the arc travels in the mesh. It is sometimes called the gradualelement technique in this context. Mahrenholtz et al. [156] used the quiet elementapproach for three-dimensional model of a butt-welded plate where they comparednormal welding with underwater welding. They discuss the thermometallurgicalcoupling, as they also do in Ronda et al. [157], but it is not clear if they also simu-lated the microstructure evolution. However, the inactive element approach was
2 ._5 16. -1-0
F.Fi t I I I
Figure 26. Finite element model of 28-pass weld [32).
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART I 169
Y ()0.1- _X
(MPa)0 * Vy,:4 • 300
100
0 2- -100
-300
-500
02 0 0.2 x (m)
Figure 27. Transverse stresses during multipass welding [32].
used by Gu et al. [132] for a three-dimensional model of a bead-on-plate casewhere the arc also was traveling through the mesh. They found that this approachgave a better convergence rate in the iterative solution of the finite elementequations.
Rybicki et al. [155] studied the effect of the pipe thickness on the residualstresses. Brust and Rybicki [147] and Brust and Kanninen [156] investigated thepossible benefits of backlay welding. Rybicki and McGuire [157,158] studiedinduction heating and heat sinks that are performed to create compressive stresseson the inner side of pipes to reduce the risk for intergranular stress corrosioncracking (IGSCC). An analytic solution was used for the temperature since thewelding and temperatures from measurements were the basis for an interpolatedtemperature field in the simulation of the induction heating. The use of theweld-overlay method was evaluated by Rybicki et al. [159]. The heat sink weldingtechnique was also studied in [160-163]. Bae et al. [164] investigated the mechanicalstress relieving (MSR) technique for reducing residual stresses.
Mok and Pick [80] simulated multipass welding of a T-joint and subsequentstress redistribution during loading and unloading. They used the same approachas Ueda et al. [134,135]. They compared a model where only the capping weldof each layer was simulated with a model where the residual stresses of thecapping weld was mapped to the other welds in that layer and used as initialstresses for the simulation of the capping weld. They found little differencebetween these two approaches and therefore assumed that this simplificationwas justified.
Ma et al. [85] and Kim et al. [165] investigated the effect of the welding pro-cedure on the stresses and deformations of beams that are manufactured withT-fillet welds.
170 L.-E. LINDGREN
In later work Ueda et al. [166) simulated all 14 passes of a pipe-platejoint in anaxisymmetric model. No lumping technique was used. The material modelingwas also improved using piecewise-linear, isotropic hardening and plastic strainswere removed at higher temperatures.
A large number of publications have been published based on the previouslygiven development, for example, [53,54,141,167-169]. The paper by Murty et al.[110] gives a short review of phenomena and theory and evaluates severalapplications, including some applications published by others.
Michaleris et al. [170] performed an optimization of a one-pass weld w.r.t.manufacturing and service life aspects. Michaleris [148] simulated multipasswelding and a subsequent hydrotest of the pipe. Michaleris and DeBiccari [103],described in the section "From One-dimensional to Three-dimensional Models,"studied the effect of welding residual stresses on the buckling behavior of panelswith welded stiffener. The papers by Michaleris and Sun [84] and Michaleris etal. [171] deal with thermal tensioning for the prevention of welded beambuckling.
The first three-dimensional model of multipass welding of a pipe using solidelements was presented by Li et al. [172,173]. Four weld passes, continuously laid,were simulated. Elements were activated during every time step. It is probable thatthey used the quiet element approach. It was stated that the thermal analysisrequired one month but not which computer was used. They investigated severalpipes in their experiments, but only the narrow gap welding was simulated andtested using neutron diffraction measurements. They inferred that this gave lowerresidual stresses because it had the smallest radial deformation. However, smallerdeformations need not mean smaller residual stresses even for the same geometryof the pipes. The different thermal loadings and geometry of the grooves makethis assumption unreliable without further evidence. For example, a groove withone weld across its width will give larger transverse stresses than if two weldsare used to fill its width [139]. The average value over the pipe thickness ofthe axial stress will not contribute to the radial displacement. Ortega et al. [174]simulated a three-pass circumferential weld of a pipe using the quiet elementapproach. They had a contact surface between elements for the filler materialand the base metal. Thus, they used the technique, described earlier, that manyresearchers have used when simulating multipass welds in ABAQUS. They usedcodes from Sandia Livermore Laboratories. The thermal analysis required 24 hoursand the mechanical required 48 hours on a Cray-J916. The measured and computedtemperatures agreed well. The only mechanical quantities shown were the measuredand computed axial displacements. The agreement was reasonable during the finalcooling of the pipe, but it is not possible to see any details about thesedisplacements during the welding process. Fricke et al. [175,176] also usedABAQUS but for a five-pass weld. The thermal analysis required 9 hours andthe mechanical analysis required 54 hours of computing time on a HP UNIXworkstation of model J210 with 256 MB RAM.
Dike et al. [86] used a three-dimensional model to study multipass welding withfour weld passes. A 12-in. weld was laid in the interior of a 24-in-long plate.
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART I 171
The transverse shrinkage was monitored. The thermal field, from ABAQUS, wasobtained from a two-dimensional analysis of the cross-section and swept overthe three-dimensional model giving accurate temperatures. The model for thethermal analysis is shown in Figure 28. They tried a two-dimensional model usingABAQUS and a three-dimensional model using JAC3D from Sandia National Lab-oratories in the mechanical analysis (see Figure 29). They found that thetwo-dimensional model could not predict the transverse shrinkage very well. Thiswas not only due to the restraint at the edges since a three-dimensional model witha weld laid instantaneously gave the same too low shrinkage (Figure 30). Thethree-dimensional model with a moving heat source gave a very good agreementwith experiments. They also tried to lump the last three weld passes. The transientdeformation (Figure 30) was not correct, but the residual deformation was reason-ably accurate.
Other applications that have some complexities common with multipass weldinghave also been simulated. A number of two-dimensional models have been applied tostudy cladding and weld repair. Weld repair of a pipe was analyzed by Rybicki andStonesifer [101], where 100 passes were lumped into 20 welds. Cladding of a platewas simulated by Rybicki et al. [1771 and Rosselet [178] with no individual weldsmodeled. Dupas and Moinereau [83] modeled two beads representing the first layerand one bead representing the second layer of a cladding. Ueda et al. [179] analyzed
FInterface elementsPass I
Psinterface ement
____Z . ". ý'ea S
Figure 28. Thermal finite element model using ABAQUS [86).
172 L.-E. LINDGREN
additional stiffnesses
WýSlse,[sedense mesh
Figure 29. Mechanical finite element models using ABAQUS for the two-dimensional model and JAC3Dfor the three-dimensional model (86].
Average in-plane shrinkage
[mml
0,9 3D, lumped pass 2-4,
0,7 3D
0,5-
V3D, instantaeu
-01 --- -- I--..--..'-L.- _____
0 I2 3 4
Weld pass #
Figure 30. Measured and computed average in-plane shrinkage f86].
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART I 173
only a single bead in a study of repair welding. Devaux et al. [50] studied a repairweld by a coupled thermometallurgical analysis with a subsequent mechanicaland finally a hydrogen diffusion simulation. They studied four beads on a flat surfaceand six beads in an inner corner. Junek and Ochodek [180] analyzed four beads in aninner corner with the same approach and software, except for the hydrogendiffusion. Their later paper [181] does not include any details about simulationsbut outlines their experimental and computational approach and program. They[182] compared GTAW and MMAW welding procedures for another repair weld.They found that the GTAW weld gave somewhat smaller residual stresses. Dong[183] first simulated a two-pass weld and thereafter removed elements in orderto account for the grinding out of material and the reintroduction of these elementsin the simulation of the repair weld. Finally, the planishing of the welds was simu-lated by pressing with two rigid surfaces from each side of the analyzed plate. Thisapproach was combined with a three-dimensional model for a pipe [114,115]and panels [116-118]. The pipe case is shown in Figure 31.
A two-dimensional thermal model was combined with a three-dimensionalmechanical model for simulating overlaid repair by Chakravarti et al. [123,124].They studied blocks of welds, six blocks each with 18 beads. These blocks were com-bined to different series of block welds. The three-dimensional model had only onelayer of elements in the welding direction and two beads were combined to one weldpass. The residual strain from one of the two blocks of welds, as a block had all thebeads either in the x- or y-direction, were combined to obtain the total deformationof a plate welded with different combinations of blocks. Weaved welding wasanalyzed by a complete three-dimensional model by Oddy and McDill [38] and Oddyet al. [184,185]. The difference between these two papers is the inclusion of theaddition of filler material in the model. Oddy et al. [184] compared computationalresults with measurements from repeated experiments (Figure 32) in order to checkthe variations due to uncontrolled factors influencing the experiments.
Xe R w6
Figure 31. Repair weld analyzed by Zhang et al. (1997, 1998).
174 L.-E. LINDGREN
Longitudinal Stress (MPa) Longitudinal Stress (MPa)T-Scan: Top Surface T-Scan: Top Surface
S1oo 50-100 100
-300 -3000 5 10 15 20 25 0 5 10 15 20 2!
Transverse Position (cm) Transverse Position (cm)iCmput oAvage oWeld 2 *Weld3_0Weld4]
Figure 32. Results for weaved welding from Oddy et al. [184,1851. The left diagram [1851 shows computedand measured residual stress. The right diagram [184] shows the variations in measured results usingneutron diffraction measurements.
Jonsson et al. [186] estimated the risk for hot cracking of a butt-welded steelplate. They used the increment in strain between 1400'C down to 1000°C as ameasure for the risk of cracking. An increase in this measure corresponded wellwith statistics for cracks found in real welds. Shibahara et al. [187] used an interfaceelement with a limited strength in the brittle temperature range (1200-1450'C)when studying hot cracking. Singh et al. [188] simulated the Trans-Varestraint testwhere hot cracking is investigated using a three-dimensional model. A short weldin an interior slit of a plate is laid, and an external force is applied in order tosuperimpose an additional strain. This total strain distributed over regions withdifferent temperatures is compared with the ductility of the material. Hot crackingmay occur in the region that has a temperature where the ductility is reduced. Thistest was also studied by Munier and Lefebvre [189]. Bergmann and Hilbinger [190]studied the hot cracking of a butt-welded aluminum plate. They assumed that thelarge solidification shrinkage was compensated by refeeding from the melt untila critical strain of 2% was reached. This value was obtained by fitting simulationswith experiments. Young's modulus was lowered to 0.01 MPa when the criticalstrain was reached. They obtained a good agreement between computed andmeasured locations of cracks near the edge of the plate. Yang et al. [191,192] alsostudied the prevention of hot cracking in an aluminum plate by mechanical rolling[191] and later [192] by a trailing heat sink. They also used the mechanical strainas a measure for the risk for cracks. Good agreement between experiments andsimulations were obtained for the effect of the distance between the torch andthe roll and its width and the applied force on the risk for cracking. The samewas the case in the later paper for the distance between the weld torch and theheat sink. Oddy and McDill [193] studied burnthrough in welding of pressurizedpipes using a material model with creep damage. They obtained a reasonable agree-
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART 1 175
ment with experiments when considering the simple approach they used for thiscomplex failure process. Rybicki et al. [194] used computed welding residualstresses for a two-pass weld [88] and a fracture model based on linear elasticfracture mechanics. The redistribution of the welding residual stresses due tothe crack is taken into account Brust et al. [195] transferred the computed weldingresidual stresses to another finite element model. This model with the weldingresidual stresses was combined with an analytic solution for a crack in an infinitebody. Thus, no explicit crack was needed in the mesh. These fields were combinedto give stress intensity factors for evaluating crack propagation. Thus, the analyticmodel could give the redistribution of the stresses because of the crack and itsgrowth. They analyzed a welded pipe and a pipe with a repair weld. They ignoredpossible stresses due to the in-service loads in comparison with welding residualstresses. Michaleris et al. [196] used a similar approach. They transferred allresidual fields from the welding analysis to a model with a finer mesh near a crack.Thereafter, the initial crack nodes were released, so the stresses were redistributedaccordingly and an analysis for the fracture assessment was performed. Theychecked the effect of the data transfer using the same fine mesh for the weldinganalysis as for the fracture assessment. There are many papers where the influenceof welding stresses on crack growth, for example, is studied. These residual stressesare usually obtained from simulations presented in other papers and experiments orestimated, for example, by analytic solutions. These papers are not included in thisreview.
Recommendations. Simplifying the multipass welding procedure by some kind oflumping technique must be exercised with care. All lumping and envelope techniqueschange the temperature history and will affect the transient and residual strains nearthe weld. Lumping by merging several weld passes that conserve the total heat inputis preferred. Thus, the simulation will correspond to a multipass weld but with fewerweld passes than the original.
The quiet or inactive element approaches can give the same results [32]. Strainsaccumulated during some initial phase between the time at which the element isadded to the structure and the time at which it obtained the wanted temperaturefor the new weld will not affect the results if its plastic strains are completely removedwhen this wanted temperature is reached. It does not matter if interface elements areused when connecting molten elements corresponding to the filler to the rest of themodel or if some kind of preheating of the surface is performed before the weldis laid in order to raise the temperature of the structure to the temperature ofthe laid weld. The only difference between these two methods is that there is soft,molten material pressing on the surface during the initial temperature increaseof material connected to the weld in the first case. The removal of plastic strainswill erase any possible differences between these two methods. Therefore, the choiceof method is more a matter of what is more practical for the particular finite elementcode in use.
Studies of hot cracking require more accurate models. See the recommendationsafter the discussions about the mechanical properties in the section "ImprovedMaterial Models" in Part 2.
176 L.-E. LINDGREN
IMPROVED MODELS FOR HEAT INPUT
Arc Physics, Fluid Flow, and Heat Input
The net heat input must be given to the finite element model. There is research aboutarc physics, fluid mechanics, and heat transfer to help predict the distribution of theheat input. But more developments must be made before this will be possible [1].The work by Sheng and Chen [197-199] is a step in the development of a completepredictive model for simulating welding; they incorporated a fluid-flow model inthe thermomechanical analysis. They used an interesting combination of aLagrangian reference frame, for solid material behavior, and an Eulerian frame,for the fluid flow, and mixed the results from these models according to the amountof solid and liquid material in each point (see "A Welding Simulation Revisited"in Part 3, where results from their model are shown). However, they were also forcedto apply a prescribed heat flux to their model. It is necessary to specify the heat inputas long as integrated models for arc physics, fluid flow, heat transfer, and mechanicalanalysis are not integrated into a model for welding simulation. The best approachcurrently is to measure temperatures, observe microstructure changes, and sizeof weld puddle, for example. The heat input is adjusted until good agreement withexperiments is obtained. Thus, the models are not purely predictive [12]. Goldaket al. [11,12,200] discuss different alternatives for heat input models. Goldak [201]discusses the thermal analysis of welds in some detail.
Analytic Solutions
Some papers have used analytic solutions for the temperature fields, mostlyRosenthal's solution (e.g., [87-89,96,104,147,156,157,202-204]). These solutionsmay work well for regions away from the weld. Analytic solutions based on pointsources and finite element schemes are compared by Moore et al. [205]. It canbe noted that Rybicki et al. [87,88] superimposed analytic solutions for 28 pointsources in order to match the measured temperatures. Lindgren [204] obtained quitegood agreement for the residual stresses using Rosenthal's solution for a movingheat source for a butt weld case. The use of constant properties in the analytic sol-ution leaves the question about which thermal properties to use unsolved. Datashown in "Properties for Modeling Heat Conduction" in Part 2 reveal the largevariations in thermal properties over the actual temperature range. There is nolonger any reason to use the analytic solutions because the numerical simulationof the thermal field is quite straightforward.
Finite Element Solutions with Prescribed Heat Input
Hibbitt and Marcal [29] and Andersson [35] used surface heat input and an impulseequation for the heat contributed by the addition of filler. It is also possible to dis-tribute the heat uniformly over the area, in two dimensions, or volume, in threedimensions, of elements that represent the weld. Usually some kind of ramp with
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART I 177
linearly increasing heat input for the approaching arc and constant heat input whenthe elements are melted and linearly decreasing heat input when the arc is leavingthe element are used. Shim et al. [141] showed the effect of this ramp on heatingrate and peak temperature. Nickell and Hibbitt [205] and Friedman [206] used aGaussian distribution for the surface heat input. The latter made a more detailedmodeling of the weld metal where only the deformation of the weld puddle wasconsidered. The Gaussian distribution was also used in papers by, for example,[42,44,63,125,164,198]. Goldak et al. [207] proposed a more accurate model, theso-called double ellipsoid heat source (see Figure 33). It was later extended to arbi-trary distribution functions [208]. Zhang et al. [209] implemented a heat-input modelwith arbitrary cross-section and a double ellipsoid variation over the surface. It wasused, for example, by [77,103,124,210]. The heat input is defined separately overtwo regions. One region is in front of the arc center, z > 0 (Figure 34). The otheris defined behind the arc center. The model is defined below for an arc startingat origin at = 0 and moving along the positive z-axis.
Heat input, q, in front of the arc center, is defined as
6 v3 Qe-3(@/a)Ze-3B&Ih) 2e-3((@-vO)/cI) (1)abcl it
312
and, behind the arc center, as
q=(2-fj) 6v3 Qe_3(x/a)Že_3,/h)'e_3(_vo)/c2)2 (2)abc2 3/2
2000loci ... ------ Disc shaped heat
source
1600 ------------ - Christensen
measurements
1200 - --- Double ellipsoid
800
400
0 .
0 1 2 3 4
Distance from weld [cm]
Figure 33. Peak temperatures with different models for heat input [2071.
178 L,-E, LINDGREN
yz
Figure 34. Geometry of a double ellipsoid heat source.
where a, b, cl, and c2 are the ellipsoid axes defined in Figure 34 associated with theregions over which the heat input occurs, Q is the energy input rate [W], v isthe welding velocity, andfr distributes the heat to the regions in front of and behindthe arc center. Note that if cl (2 -fr) = c2fr, then the heat input is continuous atz = 0. A plot of the heat input at the surface using the double ellipsoid heat sourceis given in Figure 35. The boundary of the heat source is defined as the region wherethe heat input has been reduced to 5% of the peak value.
Finite Element Solutions with Prescribed Temperature
The temperature field can be obtained by specifying the heat input as described aboveor by prescribing the temperature in the weld. Carmet et al. [47] and Goldak et al.[211] used prescribed temperatures. Argyris et al. [39] used a somewhat special pro-cedure for computing the temperature field because of the moving arc. They simu-lated the butt-welding of a plate. A thermal model of the plane of the plate wasused. It was assumed that the conditions were constant over the thickness. The heatloss from the lateral surfaces was accounted for by corresponding heat sinks. There-after a thermomechanical model of a cross-section of the plate was used. The tem-perature at the symmetry line (center of weld) was prescribed according toresults from the first thermal model of the plane of the plate.
Jones et al. [212,213] prescribed the temperature at the boundary of the weldpool be equal to the Tliquidus. This boundary was defined beforehand by the user.They simulated the welding of a bead on a disc. Temperatures were measuredand compared with simulations. They obtained quite good agreement with measured
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART 1 179
10-
8-
6, Heat input at upper surface of plate
4,
0i
0.5 0-1
-2
transverse direction x 0 -3 welding direction z
Figure 35. Heat input distribution using a double-ellipsoid heat source model.
temperatures. They measured the temperatures at some different locations and com-pared the results with computed values (Figure 36). The variation in temperaturesfor thermocouples at corresponding positions but from different experiments revealsvariations due to their exact locations and maybe in variation in the weldingparameters. Therefore, one can not expect better agreement between simulationsand experiments than between experiments that are performed under seeminglyidentical conditions. However, it is not difficult to achieve good accuracy for thetemperatures at some distance from the weld, especially if only a few weld passesare made. The problem of matching measured temperatures is naturally greaterthe nearer the arc the measurement is performed. A Cromel-Alumel thermocouplecan be subjected to a transient temperature up to 1200°-1300°C, and a PtRh-Ptthermocouple can take another 300'C. However, most measurements are usuallyperformed at some distance from the weld and the peak temperature falls off quicklythe farther away from the weld the measurement is made. Thus, most papers showgood agreement between computed and measured temperatures. The difference thatmay exist at higher temperatures is usually not so important. See also the sectionabout "Improved Material Modeling" in Part 2, where the problems about materialproperties at higher temperatures are discussed. The physical properties and,therefore, higher temperatures are more important when studying hot cracking,for example. Figure 37 shows that the agreement is quite good at weld 19 whenthe thermocouple is farther away from the arc but not so good during later weldpasses when the arc comes nearer the thermocouple. The results were obtainedin a first simulation where the prescribed temperature was the same for all weld
180 L.-E. LINDGREN
Sllc top
4WLi
t-, . - -Cxr, -----.-t---
... ...... . r ...4C 50.1 5"0 "Sa 250 50 4
20 zo -0 M m 0
Tim W
Figure 36. Computed and measured temperatures for the bead on disc [213].
passes. Roelens [52-54] and Lindgren et al. [32] prescribed the temperature in thecase of multipass welds. The weld metal is assigned the melting temperature duringan appropriate time interval. The latter estimate is based on dividing the lengthof the weld puddle with the welding speed. It may also be changed further whencomparing the computed temperature with experimental data. The temperaturefor a weld pass is specified during the heat input phase, as shown in Figure 38.Results from this technique for a 28-multipass weld [32] are shown in Figure 37.It shows that a reasonable agreement with experiments can be obtained withoutmuch effort. All welds were assigned Tliquidus 1520°C when they were laid, andthe same length of the weld puddle was assumed for all weld passes even if thewelding speed varied somewhat. Furthermore, the location of the cross-sectionof each weld pass was approximate. This distance from the weld to the thermocouplewill be important for later welds as they come near the thermocouple that waslocated on the top surface just at the edge of the prepared groove.
Recommendations. The technique of prescribing the temperature is easier to use asa means for heat input. However, the heating phase will not be modeled with accu-racy in this approach [1] but has to be simplified into a heating phase, usuallylinear-increasing temperature followed by a constant-temperature phase. They mustgive a correct total energy input. The cooling will thereafter be computed correctlybecause the temperature is no longer prescribed then. If the region near the arcis studied in detail, then the double ellipsoid model for prescribing the heat inputmay be better. This should then be combined with a fine mesh with about 10 elementsacross the axis of the ellipsoid area of heat input [11]. The heat input is adapted until
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART 1 181
4oc [ weld #23[°Cf -Computed
300 -.Measured w 2weldl#21
3700150190250
F 7 ru~wa lt weld #22 [1weldM#19eitpt
sa o weld #20a
17500 18500 19500 20500
Time tsecl
Figure 37. Excerpt from computed and measured temperature for a 28-multipass weld [32..
unknownM el in tem peratureMetn teeeraurerat--ure
: • prescribed
', temperature
startof wld',are has passed the node
time
Figure 38. Prescribed temperature for node associated with a weld.
182 L,-E. LINDGREN
a good fit with experimental data is obtained. This can be fusion zone, for example,but the highest accuracy can be obtained if transient temperature have beenmeasured. The temperature levels shortly after the peak temperature may be moreimportant to match than the peak itself. Then the correct total amount of net heatinput is obtained.
FUTURE RESEARCH
Finite element models of welds become more and more complex with respect togeometry, welding procedures, and the phenomena addressed. The study of residualstresses is the standard objective in finite element analysis of welds. It requires accu-rate material models. However, the study of hot cracking requires even bettermaterial models. The development of these models is given in Part 2. The solutionof multipass welds using three-dimensional models is still very demanding andrequires efficient numerical methods. This is the focus in Part 3.
The more complex the models are, the more time-consuming is the modelingprocess. The commercial mesh generating systems and finite element codes arenot especially convenient to use. Special routines are needed to support the definitionof arbitrarily weld paths. They should supply the finite element code with the in-formation of heat input as well as addition of filler material. The simulation ofindus-trial applications may have to be done without access to any information about thetemperature or size of weld puddle, for example. The only available informationcould be the welding parameters. Then it would be necessary to have models forthe net heat input related to these parameters and the joint geometry, which areused in the finite element model for the thermal analysis.
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186. M. Jonsson, L. Karlsson and L-E. Lindgren, Thermal stresses, plate motion and hot cracking inbutt-welding, Proc. of the 4th Int. Conf oJf Mechanical Behaviour of Materials, p. 273, 1983.
187. M. Shiblhara, H. Serizawa and H. Murakawda, Finite element method for hot cracking usingtemperature dependent interface element, JWRI, vol. 28, no. 1, pp. 47-53, 1999.
188. 1. Singh, W. Kroenke and M. Cola, Analytical prediction of the location of ductility dip cracking inthe Trans-Varestraint test, Proc. of 5th Int. Conf. on Residual Stresses (ICRS-5), p. 202, 1997.
189. E. MunierandS. Lefebvre, Numerical simulation of "Varestraint" test, Proc. of the 5th Int. Conf. onTrends in Welding Research, p. 712, 1998.
190. H. W. Bergmann and R. M. Hilbinger, Numerical simulation of centre line hot cracks in laser beamwelding of aluminium close to the sheet edge, in H. Cerjak (ed.), Mathematical Modelling ofWeld Phenomena 4, The Institute of Materials. p. 658, 1998.
191. Y. Yang, P. Dong. X. Tian and Z. Zhang. Prevention of welding hot cracking of high strengthaluminium alloys by mechanical rolling. Proc. of the 5th Int. Conf on Trends in WeldingResearch. p. 700, 1998.
192. Y. Yang, P. Dong, J. Zhang and X. Tian, X., A hot-cracking mitigating technique for welding highstrength aluminum alloy, Welding Journal. vol. 79, no. I. pp. 9s-17s. 2000.
193. A. S. OddyandJ. M. J. McDilI, Burnthrough prediction in pipeline welding, Int. J. Fracture, vol. 97,no. 1/4, pp. 249-261, 1999.
194. E. F. Rybicki, R. B. Stonesifer and R. J. Olson. Stress intensity factors due to residual stresses inthin-walled girth-welded pipes, ASME J. Pressure Vessel Technology, vol. 103, February, pp.66-75. 198M.
195. F. W. Brust, P. Dong, and J. Zhang, Influence of residual stresses and weld repairs on pipe fracture,Approximate Methods in the Design and Analysis of Pressure Vessels and Piping Components,PVP-vol. 347, p. 173. 1997.
196. P. Michaleris, M. Kirk, W. Mohr and T. McGaughy, Incorporation of residual stress effects intofracture mechanics assessments via the finite element method, in J.H. Underwood, B.McDonald, and M. R. Mitchell (eds), Fatigue and Fracture Mechanics: 28th Volume, STP 1321,ASTM Publication 04-013210-30, p. 499, 1996.
197. Y. Chen and I.C. Sheng, On the solid-fluid transition zone in welding analysis. ASME HTD- Vol.175/MD- Vol. 25 Heat and Mass Transfer in Solidification Processing, p. 21, 1991.
198. Y. Chen and 1. C. Sheng, Residual stress in weldment, J. of Thermal Stresses, vol. 15, no. 1, pp. 53-69,1992.
199. I. C. Sheng and Y. Chen, Modeling welding by surface heating, ASMEJI Engineering Materials andTechnology. vol. 114, no. 4, pp. 439-449, 1992.
200. J. Goldak, M. McDill, A. Oddy, R. House, X. Chi and M. Bibby, Computational heat transfer forweld mechanics, Trends in Welding Research, p. 15, 1986.
201. J. Goldak. Thermal analysis of welds, in L. Karlsson (ed.) Modeling in Welding, Hot PowderForming and Casting, Ch 2, p. 1 7, 1997.
202. J. Cafias, R. Pic6n, A. Blazquez, and i. C. Marin, A simplified numerical analysis of residual stressesin aluminium welded plates, Welding International, vol. 8. no. I, pp. 39-44, 1994.
192 L.-E. LINDGREN
203. B. L. Josefson and C. T. Karlsson. FE-calculated stresses in a multi-pass butt-welded pipe - asimplified approach, Im. J. Press. Vessel & Piping, vol. 38, pp. 227-243, 1989.
204. L-E. Lindgren, Temperature fields in simulation of butt-welding of large plates, Comm. in AppLNumer. Methods, vol. 2, pp. 155-164, 1986.
205. J. E. Moore, M.J. Bibby and J. A. Goldak, A comparison of the point source and finite elementschemes for computing weld cooling, Welding Research: The State of the Art, Proc. of 1985Int. Welding Congress injunction with ASM Materials Week'85, p. 1, 1985.
205. R.E. Nickell and H.D. H ibbitt, Thermal and mechanical analysis of welded structures, Nuclear Engand Design, vol. 32, pp. 110-120, 1975.
206. E. Friedman, Analysis ofweld puddle distortion and its effect on penetration, Welding Journal. June,pp. 161s-166s, 1978.
207. J. Goldak, A. Chakravarti and M. Bibby, A new finite element model for welding heat sources,Metallurgical Trans B, vol. 15B, June, pp. 299-305, 1984.
208. J. Goldak, M. Bibby, J. Moore, R. House and B. Patel, Computer modeling of heat flow in welds,Metallurgical Trans. B, vol. 17B, pp. 587-600, 1986.
209. J.Zhang, Y. Dong and A. Nanjundan, An enhanced heat source model for gas metal arc welding, toappear in H. Cerjak (ed.), Mathematical Modelling of Weld Phenomena 5, Institute of Materials,1999.
210. S. Das, U. Upadhya, U. Chandra, M.J. Kleinosky and M.L. Tims, Finite element modeling of asingle-pass GMA weldment, Proc. of the 6th Int. Conf Modeling of Casting. Welding andAdvanced Solidification Processes, The Minerals, Metals & Materials Society, p. 593, 1993.
211. J. Goldak, J. Zhou, V. Breiguine and F. Montoya, Thermal stress analysis of welds: From meltingpoint to room temperature, JWRI, vol. 25. no. 2, pp. 185-189, 1996.
212. B. K. Jones, A. F. Emery and J. Marburger, An analytical and experimental study of the effects ofwelding parameters in fusion welds, Welding Journal, vol. 72. no. 2, pp. 51s-59s, 1993.
213. B. K. Jones, A. F. Emery and J. Marburger, Design and analysis of a test coupon for fusion welding,ASME J. Pressure Vessel Technology, vol. 115, pp. 38-46, 1993.
214. A. Oddy. J. Goldak and M. McDill, Transformation plasticity and residual stresses in single-passrepair welds, ASME J. Pressure Vessel Technology, vol. 11, pp. 33-38, 1992.
215. J. H. Cowles, M. Blanford, A. F. Giamei and M. J. Bruskotter, Application of three dimensionalfinite element analysis to electron beam welding of a high pressure drum rotor, Proc. of the 7thInt. Con. Modeling of Casting, Welding and Advanced Solidification Processes, The Minerals,Metals & Materials Society, p. 347, 1995.
Journal of Thermal Stresses, 24:195-231, 2001 PCopyright © 2001 Taylor & Francis0149-5739/01 $12.00 + .00
FINITE ELEMENT MODELING AND SIMULATION OFWELDING. PART 2: IMPROVED MATERIAL MODELING
Lars-Erik LindgrenLu/ed University of Technology
and Dalarna University SE-971 87 Luled, Sweden
Simulation of welding has advanced from the analysis of laboratory setups to realengineering applications during the last three decades. This development is outlinedand the directions for fiaure research are summarized in this review, which consistsof three parts. The material modeling is maybe the most crucial and difficult aspectof modeling welding processes. The material behavior may be very complex forthe large temperature range considered
The development of welding procedures is based on performing experiments(Goldak et al. 1990) and a Welding Procedure Specification (WPS) is the final result.The evaluation of a welding procedure is based on joint integrity, absence of defects,microstructure, and mechanical testing, for example. Computational methods arerarely used in the process of developing welding procedures. It is expected thatsimulations will complement the experimental procedures for obtaining a WPS, sinceaspects like residual stresses can then be considered when comparing differentwelding procedures. Furthermore, simulations are also useful in designing the manu-facturing process as well as the manufactured component itself. Distortions areusually in focus in the first case, whereas residual stresses are of interest in the lattercase.
This review concentrates on the simulation of fusion welding processes ofmetals. These processes do have several common traits and are therefore similarto the model. Publications presenting finite element simulations of the mechanicaleffects of welding appeared in the early 1970s, and simulations are currently onlyused in applications where safety aspects are very important (like aerospace andnuclear power plants) or when a large economic gain can be achieved. The scope
Received 12 August 1999; accepted 12 April 2000.This work has been financed by NUTEK (the Swedish National Board for Industrial and Technical
Development) via the Polhem Laboratory. Its completion has also been made possible by the cooperationwith ABB Atom AB and Volvo Aero Corporation in a project about multipass welding where this reviewwas one part.
Address correspondence to Professor L.-E. Lindgren, Department of Mechanical Engineering. LuleAUniversity of Technology. SE-971 87 LuleA, Sweden. E-mail: [email protected]
195
196 L.-E. LINDGREN
of most simulations has been to obtain residual stresses and correspondingdeformations. There are also publications presenting analyses of hot crackingand other phenomena.
A common concern in the simulation of welding is to account for the interactionbetween welding process parameters, the evolution of the material microstructure,temperature, and deformation. The resultant material structure and deformationmay also be needed, in combination with in-service loads, to predict lifetime andperformance of a component. Therefore, research in this field requires the collab-orative efforts of experts in welding methods, welding metallurgy, material behavior,computational mechanics, and crack propagation, for example. It is important toobserve that uncertain material properties and net heat input make the successof simulations significantly dependent on experimental results (Lindgren 1996).
The review is split into three parts. Part I is "Increased Complexity"; this is part2, "Improved Material Modeling." Part 3 is called "Efficiency and Integration."They all outline the development of simulation of welding and are separated intosections where different aspects of finite element modeling are the focus. Each sec-tion contains some recommendations based on the review and the experience ofthe author. It is hoped that this approach will be appropriate for those that areentering this field of research and useful as a reference for those already familiarwith this subject.
Material modeling is important for all computational models. The materialmodel and pertinent data must represent the real material behavior with sufficientaccuracy. What is sufficient depends on the focus of the performed study. This articleshows that there are different requirements depending on the scope of the analysis. Itis also shown how the effect of phase changes has been accommodated in the models.The influence of simplifications of the material behavior has been studied in severalpapers. The simplifications are necessary due to both lack of data and numericalproblems when trying to model the actual high -temperature behavior of the material.
IMPROVED MATERIAL MODELING
Material modeling is, together with the uncertain net heat input, one of the majorproblems in welding simulation (Goldak 1989, Lindgren 1996). The thermal analysisis, in general, more straightforward than the mechanical analysis. It entails fewnumerical problems, with the exception of the large latent heat during the solid-liquid transition, and it is easier to obtain the thermal rather than the mechanicalproperties of a solid. Mcflill et a]. (1990) investigated the relative importance ofthe thermal and mechanical properties of stainless and carbon steels in weldingsimulations. Two bars of dimension 20 in x 2 in x 0.5 in were "welded" alongthe free edge. No fixture was used. One bar was made of a stainless steel andthe other of a carbon steel. The resultant radius of curvature was compared withthe computed values for the material properties of carbon steel (MS) or stainlesssteel (SS). The results are shown in Table 1. The somewhat unexpected conclusionwas that the thermal properties play a more important role than the mechanicalproperties in explaining the different behavior between these steels. This is due
IMPROVED MATERIAL MODELING 197
Table 1 Curvature for combinations of thermal and mechanical properties (McDillet al. 1990)
Thermal Mechanical Radius of CurvatureTest Properties Properties [m-I]
A MS MS 41.6B SS SS 11.9C MS SS 29.6D SS MS 16.3
Experiment MS MS 23.3Experiment SS SS 8.1
to the fact that the thermal dilatation is the driving force in the deformation and thebar is free to bend. The thermal dilatation is determined by the temperature field andis, therefore, strongly influenced by the thermal properties.
Dependency on Temperature and Microstructure
The complete thermomechanical history of a material will influence its material pro-perties. However, this can be approximated to a dependency on the current tem-perature and deformation for many materials. This simplification may be toolarge for ferritic steels where solid-state phase transformations occur that will influ-ence the thermal dilatation and the plastic behavior of the material in a way thatwill affect the residual stresses. Different approaches for modeling this dependencyare discussed. Also see the discussion about latent heats in the section "Propertiesfor Modeling Heat Conduction" and about transformation-induced plasticity inthe section "Properties for Modeling Plastic Deformation." Leblond et al. (1997)discuss the consequences of phase transformations. The influence of the stresseson the phase transformations are usually ignored and are not discussed in this article.
The simplest and most common approach is to ignore the microstructure changeand assume that the material properties depend only on temperature. The effect ofphase changes may be ignored for austenitic steels (e.g., Ueda et al. 1986, Rybickiet al. 1977, 1978, Brown and Song 1992a, b) and materials such as copper (Lindgrenet al. 1997) and Inconel (Friedman 1975, Nickell and Hibbitt 1975, Ueda et al. 1991).But phase changes have also been ignored in the case of ferritic steels (e.g., Ueda andYamakawa 1971a, Hibbitt and Marcal 1973, Mok and Pick 1990, Shim et al. 1992,Michaleris 1996, Ravichandran et al. 1997, Lindgren et al. 1999). Then, for example,the thermal dilatation is the same during heating and cooling, as in Figure Ia. Theenvelope technique and the lumping of weld passes, which are sometimes usedin simulation of multipass welding (see the section "Increasing Complex Models"in Part I), modify or even completely ignore the temperature history. In some casesthe stresses are only computed during the cooling phase (Hepworth 1980, Freeand Porter Goff 1989). Canas et al. (1996) investigated different simplifications
198 L.-E. LINDGREN
Dependency on current Et"A
Ael Ae3 T
Dependency on current Ell,& during heating andtemperature and peak cooling if no phasetemperature. change occurs
b) TI<Tp
T2 peak
p Ip
different phases whichmust be combined witha model for micro- h
structure evolution.
c)
Figure 1. Different methods for modeling dependency of thermal dilatation on phase changes in ferriticsteel.
IMPROVED MATERIAL MODELING 199
of welding analysis and found that an envelope technique for the welding of analuminum plate gave acceptable results for the longitudinal residual stresses.Surprisingly, they found only small differences if they used constant ortemperature-dependent mechanical properties. Mahin et al. (1988) simplified thecooling phase by artificially quenching the welded plate when the maximum tem-perature reached 600'C. This was done to shorten the cooling time. They usedan explicit code and could not increase the length of their time steps during coolingbut they did not perform this in their later analysis (Mahin et al. 1991) becauseit affected the history-dependent internal variables in their material model.
However, there appeared a need to account for the phase transformations forferritic steels even if their role was not fully appreciated. The effect of the phasetransformations on the thermal dilatation was included first by Ueda andYamakawa (1971b). Andersson (1978) also accounted for this effect. The effecton yield limit and Young's modulus was included by Ueda et al. (1976, 1977).Ueda et al. (1976) used one curve during heating and another curve for the materialproperties during cooling to account for the effect of the phase transformations onthe material properties, as shown in Figure 2. Phase transformations were assumedto occur instantaneously at specific temperatures. The material became fullyaustenized at 750'C during heating, and the austenite decomposed at 600'C.The latter temperature was the average of the start temperatures for the formationof bainite and martensite. The material was assumed to lose its strength rapidly justabove 750'C during heating and gained strength just at 600'C during cooling. Asimilar approach was also used by Andersson (1978) and other researchers(Josefson 1982, Jonsson et al. 1985a, Troive et al. 1989, Karlsson and Josefson1990). It is also based on given property-temperature curves, but the curves wereless idealized than those used by Ueda et al. (1976). Different curves are chosenin the analysis depending on some characteristics of the temperature history atthe considered point in the model. Usually these characteristics are the peak tem-perature and the cooling rate between 800'C and 500'C. They are the primaryparameters that determine the obtained microstructure of steels. The cooling rateis approximately the same in the entire heat-affected zone. Therefore, differentproperty-temperature curves are chosen during the cooling phase depending onthe peak temperatures (see Figure lb in the case of thermal dilatation). A single,common curve is used during the heating phase. Different curves are chosen duringcooling depending on the value of the peak temperature, TpeAk. It is also possible tointerpolate between these curves with respect to the peak temperature instead ofjust switching between them. The curves can be obtained from experiments wheretest specimens are subjected to some temperature histories typical of the processand tested at different temperatures (Lindgren et al. 1993) or compiled fromthe literature. Voss et al. (1998a, b) suggested that varying cooling rates in theheat-affected zone can be accounted for by constructing a maximum-temperaturecooling-time diagram. This diagram then gives the microstructure of the material.The latter can then be used to determine the properties of the material. They onlyapplied this to the hardness, which was compared with measurements. Theydid not use the diagram for any other material properties.
200 L.-E. LINDGREN
Yield stress Yield stress
800 _. Tp<7C5 0 C and Tc<750 0C 800 Tp.-7500C and Tc<750*C
i MPaiK- Ma etn
60000:
400 Heating 40
200
0 200 400 600 800 0 200 400 600 800Temperature I °C Temperature [*C1
Yield stress Yield stress
800 Tp<750*C and Tc.750*C 800 TpŽ-750"C and TcŽ750"C
[MPaj [MPalCoolin H eating
600Cooling
400 --- bHeating 400
200 200
0 ..... . .. .0 -- -• .0 200 400 600 800 0 200 400 600 800
Temperature rCi Temperature I*C]
Figure 2. Determination of yield stress by Ueda et al. (1976, 1977). fl is the maximum temperature inearlier heating cycles and Tc is the maximum temperature in current heating cycle of multipass weld.
Papazoglou and Masubuchi (1982) used a CCT diagram to estimate the phasechanges and the corresponding volume changes. Oddy et al. (e.g., 1989) also esti-mated the phase transformations and used a mixture rule for the austenite propertiesand the decomposition product's properties (e.g., pearlite forms austenite, whichdecomposes back to bainite). The formation/decomposition is assumed to occurlinearly with temperature within a given temperature interval. Murty et al. (1996)used a TT1 diagram to estimate the transformations that occur in a welded pipe.This information was used to predict yield stresses and volume changes due tothe phase transformations in the weld metal. This yielded an improvement inthe results when compared to the simulation of the same pipe by Lindgren andKarlsson (1988), Karlsson and Josefson (1990), and Josefson and Karlsson (1992).
IMPROVED MATERIAL MODELING 201
The most flexible way to include the effect of the temperature history is to computethe evolution of the microstructure in the material. Each phase is assignedtemperature-dependent properties, and simple mixture rules are used to obtainthe macroscopic material properties. This coupling of thermal, metallurgical,and mechanical models (TMM) was used by Inoue and Wang (1983), Wang andInoue (1983, 1985), Bergheau and Leblond (1991), Devaux et al. (1991), Inoue (1996,1998), Roelens (1994, 1995a, b), and B6rjesson and Lindgren (1998, 1999). The influ-ence of grain size and microstructure, for example, on the mechanical properties isdiscussed by Bhadeshia (1997). An example of a mixture rule is shown in FigureI c, where the thermal expansion coefficient is given one constant value for austeniteand another value for the other phases. The difference in volume between theaustenite, y, and its decomposition products is accounted for by Ct", which, in thiscase, is this difference extrapolated to 00C. The phase change may be approximatedor computed according to some model as discussed previously. The microstructuremodel used by Bbrjesson and Lindgren (1998, 2000) uses the chemical compositionas input, and its prediction can be improved by supplying data from a measuredTTT diagram (Figure 3). Verification of the microstructure model is performedby comparing with results from CCT curves. The model was applied to the samemultipass weld that was studied by Lindgren et al. (1999). Lindgren et al. (1999)assumed only temperature-dependent data, whereas Bbrjesson and Lindgren (2000)computed the material properties based on the microstructure evolution. Das etal. (1993) also computed the microstructure but used this only to account fortransformation-induced plasticity (see the discussion in the section about"Properties for Modeling Plastic Deformation"). Dufrene et al. (1996) used simplemodels for phase changes between martensite and austenite for the case ofEB-welding. This was combined with mixture rules for the material properties.However, they do not give any details of the model or the results. Ronda et al. (1995)formulates TMM models, but no simulations are given in their paper.
Myhr et al. (1997, 1998a) simulated the welding of aluminum tubes and the buttwelding of aluminum plates (Myhr et al. 1998b). They studied the importance ofaccounting for the microstructure evolution, namely, strengthening effects of hard-ening precipitates. The thermal analysis was followed by a computation of the dis-solution and recombination of precipitates and finally a mechanical analysis.The importance of accounting for the influence of precipitates on the plastic flowwas studied. See also Grong (1994, 1997) for more details.
Recommendations. Once again, the recommendation is to aim for accuracy. Themicrostructure evolution is important to include in the material modeling of ferriticsteels since their properties can change a lot due to the phase transformations.The major influence is on the thermal dilatation and the yield stress, which canbe modeled as discussed previously. It is a problem to account for phase changesin the case of multipass welding. One should have the ability to compute themicrostructure and predict the effect on the material properties, or it will be necess-ary to simplify the history dependency of the material properties to one cycle ofheating and cooling (Ueda et al. 1976) or even ignore it (Lindgren et al. 1999).It should be noted that the needed number of measured curves, such as for the
202 L-E LINDGREN
1000
800 --- ----
200
0........ ........ .. .
10 100 101 102 e 10Time [log(sec)]
Figure 3. Measured T-r7 diagram (solid lines) and computed TIT diagram (dashed lines) from B~ijessonand Lindgren (1997).
thermal dilatation of material subjected to different peak temperatures, depends onthe mesh. The intermediate curves between the heating curve and the curve corre-sponding to the highest peak temperature all are assigned to integration pointsin the heat-affected zone. Thus, it is of no use to have many curves if the modelhas just one element across this zone.
Density
Simulation of welding can usually be performed as a quasi-static analysis (see thesection "Simulation of Welding as a Coupled Problem" in part 1). The density,p, is needed even for a quasi-static analysis because it is multiplied with the heatcapacity in the thermal analysis. Handbooks may give the density as a functionof temperature. This accounts for the volume change due to the thermal dilatation.However, one must know how the density is handled by the chosen finite elementcode. A constant density may be sufficient as input if the code itself computesthe change in density when deformations are computed simultaneously with the tem-peratures (e.g., in a staggered approach). A temperature-dependent density may beused for a pure thermal analysis where no deformations are included.
IMPROVED MATERIAL MODELING20
Properties for Modeling Heat Conduction
The solution of the heat conduction equation requires heat conductivity, ;, andheat capacity, c. The density was discussed previously. These thermal propertiesare temperature dependent. They may also depend on the temperature historysince different phases may have different thermal properties. Furthermore, thelatent heats due to phase changes are needed. These depend on the phasetransformations and thereby on the temperature history and the currenttemperature. The discussion in the section "Dependency on Temperature andMicrostructure" describes different approaches on how this latter dependencycan be handled or ignored.
The dependency on temperature is completely ignored by those usingRosenthal's or other analytic solutions (see the section about "Improved Modelsfor Heat Input" in part 1). Free and Porter Goff (1989) and Dong et al. (1997) alsoassumed constant thermal properties despite the use of the finite element methodfor computing the temperatures. Their statement that large variations in thermalproperties result in very small changes of the transient temperatures is surprisingwhen considering the data in Figures 4 and 5, for example. It is also inconsistentwith the findings of McDill et al. (1990). It may be possible to obtain good residualstresses away from the heat-affected zone with this simplification (Lindgren 1986),but this is no general conclusion. Dong (1997) and Reed et al. (1997) also used con-stant thermal properties in their finite element model. Ueda and Yamakawa usedconstant thermal properties in their early analyses (1971a, 1971b) but includedthe temperature dependency in later papers (e.g. Ueda et al. 1976, 1977). Goldaket al. (1985) and Moore et al. (1985) discussed and showed the effects of using con-stant or varying thermal properties.
Most numerical models include the temperature dependency of the thermalproperties but not any effects of the phase transformations other than the associ-ated latent heats. The latent heats due to the solid-solid phase changes are oftenignored and only the heat of fusion is included in the models. The effect of thelatent heats due to the solid phase transformation is shown, for example,by Dubois et al. (1984). The magnitude of these latent heats can be estimatedfrom diagrams by Pehlke et al. (1982). They give a latent heat of about75 kJ/kg for the a-to-y phase change in steels. Murty et al. (1996) used 92 kJ/kgfor the y-to-pearlite phase change and 83 kJ/kg for the y-to-martensite phasechange. The heat of fusion, for example, is 277 UJ/kg for a 1.2% C-steel withTwl~idu, 13870 C and Thiquidu, of 148] 0 C. There is, in general, no problem inobtaining thermal properties, such as those in Figures 4 and 5 from Richter(1973). Pehlke et al. (1982) also give data for many materials. See also thereferences in Tables 2-4.
The latent heats give effective heat capacities-latent heat divided by the tem-perature interval for phase change-which are large compared to the heat capacityshown in Figure 5. This makes the heat conduction equation stiff. For example,a pure metal with a given heat input will experience a temperature increase, butsuddenly, at the melting temperature, the temperature will not increase since theheat input is consumed by the phase change. This abrupt change in the input-output
204 L.-E. LINDGREN
60 ...
50Midand low
40 ll ysteels _
30Hih alloy steels
20 ------
10
0 200 400 600 800.
Temperature [°C]
Figure 4. Heat conductivity for some steels. Data from Richter (1973).
700 f -- •
[J/kg°C]
500 Id and lowaloy steels
300High alloy steels '
100 ..0 200 400 600 800
Temperature I°C]
Figure 5. Heat capacity for some steels. Data from Richter (1973).
IMPROVED MATERIAL MODELING 206
Table 2 Reference with material properties, ferritic steels
Reference Comments
Tall (1964) No hardeningUeda and Yamakawa (1971a, b) No hardeningUeda and Nakacho (1982)Ueda et al. (1976, 1977) No hardening, power law creep during annealingUeda et al. (1988, 1991) Piecewise linear, isotropic hardeningRybicki and Stonesifer (1980)Rybicki et al. (1988)Oddy et al. (1989)Troive et al. (1989)Karlsson (1989), Karlsson and Josefson (1990)Tekriwal and Mazumder (1989, 1991a,b) Kinematic hardeningJosefson (1982) Creep druing stress reliefBac (1994)Ueda (1986, 1991)Hibbitt and Marcal (1973) Creep during stress reliefLobitz (1977)Shim et al. (1992)Brown and Song (1992ab)Ma et al. (1995)Andersson (1978)Argyris et al. (1982, 1985) Viscoplastic, Ramberg-OsgoodLindgren and Karlsson (1988)Sheng and Chen (1992), Chen and Sheng Viscoplastic, Bodner-Partom and Walker
(1991, 1992)Wikander et al. (1993, 1994)Yuan et al. (1995)Michaleris et al. (1995)Siva Prasad and Sankaranarayanan (1996)Murty et al. (1994, 1996)Michaleris and DeBiccari (1997) von Mises, kinematic hardeningRavichandran et al. (1997)Taljat et al. (1998)Hong et al. (1998b)Kim et al. (1998)
relation constitutes a stiff equation. This is the only numerical problem in the finiteelement solution of the heat conduction problem. It can be handled in different ways,including some of the following.
The enthalpy method (Andersson 1978, Jonsson et al. 1985a) can be used toreduce the aforementioned nonlinearity. Then the enthalpy, heat content, is usedas the primary unknown in the finite element solution. It is defined as
H(T) =fI pc(r) dj (1)
This is a monotonous increasing function w.r.t. the temperature and will reduce thenonlinearity when latent heats are consumed/released (see, e.g., Karlsson (1986)).
206 L.-E. LINDGREN
Table 3 References with material properties, austenitic steels
Reference Comments
Ueda et al. (1986)Ueda et al. (1991) Piecewise linear, isotropic hardeningRybicki et al. (1977, 1978, 1979, 1986)Rybicki et al. (1988)Chidiac and Mirza (1993) ViscoplasticLejeail (1997) Kinematic hardeningTroive et al. (1998)Mahin et al. (1991), Ortega et al. (1998), Rate-dependent, intern variable model with
Dike et al. (1998) combined hardening
Table 4 References with material properties, other materials
Material Reference Comments
Inconel Ueda et al. (1991) Piecewise linear, isotropc hardeningInconel Friedman (1975), Nickell and
Hibbitt (1975)Inconel Niisstr6m et al. (1989, 1992)Aluminum Canas et al. (1996) Kinematic hardeningAluminum Feng et al. (1997) Detailed model at high temperatureAluminum Dong et al. (1998a,b),
Yang et al. (1998, 2000)Aluminum Michaleris et al. (1997)Copper Lindgren et al. 1997
Another numerical approach is the fictitious heat source method (Rolph and Bathe1982) used by Murty et al. (1996). A third approach, used by Lindgren et al. (1999),is based on an effective heat capacity (Ch 5 in Lewis et al. 1996) and is easy toimplement into a finite element code; it is sometimes also called an enthalpy method.The heat capacity used in the heat conduction equation is replaced by
H n" _ Hn-I
Ceff = Tn - T 1
where the right superscripts denote time-step counters in the finite element solutionprocess.
The enthalpy method, Eq. (I), will work even for a pure metal. The othermethods work well for alloys where the melting occurs over a temperature range,but they may experience problems for pure alloys. The most straightforwardway to reduce the nonlinearity is to extend the temperature range over whichthe latent heat is released/consumed. Feng (1994) and Feng et al. (1997) used a more
IMPROVED MATERIAL MODELING 207
advanced model for the release of latent heat during solidification of aluminum.Their model gives a higher release of latent heat in the upper part of the TSnlidu,to Tliquidus interval. Otherwise it is typically assumed that the latent heat is evenlydistributed over this interval.
There is a modeling consideration of the melting behavior that sometimes has tobe taken into account in the thermal analysis. The heat in the weld pool is not onlyconducted but also convected due to the fluid flow. Many analysis imitate thisby increasing the thermal conductivity at high temperatures (e.g. Andersson 1978,Hepworth 1980, Papazoglou and Masubuchi 1982, Jonsson et al. 1985a, Leungand Pick 1986, Das et al. 1993, Michaleris and DeBiccari 1997). Ronda and Oliver(1998) and Voss et al. (1998b) used different conductivities in different directionsat varying locations in the weld pool. This was taken from Pardo and Weckman(1989).
Recommendations. There is no reason to use the analytic solutions any more sincethe numerical simulation of the thermal field is quite straightforward. It is importantand straightforward to account for the temperature dependency of the thermal pro-perties. Latent heats are also important to include. The use of effective heat capacityin handling the latent heats works well. If the chosen finite element code does nothave this capability, then it might be necessary to extend the temperature range overwhich the latent heat is distributed.
Introduction to Mechanical Properties
It is assumed that the deformation can be decomposed into a number of components.The increment in total strain is computed from the incremental displacements duringa nonlinear finite element analysis. The elastic part of the strain gives the stresses,and there are a number of inelastic strain components that can be accounted for.The decomposition is expressed in terms of strain rates as
6 + 0 + ý,j + Z + i + 0i (3)
where ýy is the total strain rate, ý' is the elastic strain rate, ?p is the plastic strain ratedue to rate-independent plasticity, '? is the viscoplastic strain rate, k, is the creepstrain rate, i" is the thermal strain rate consisting of thermal expansion and volumechanges due to phase transformations, and I'P is the transformation plasticity strainrate. The additive decomposition of the strain rate can be derived from amultiplicative decomposition of the symmetric part of the deformation gradient(Simo 1998). The book by Simo and Hughes (1997) contains detailed discussionsabout material modeling and its mathematical framework and computationalimplications. The stress increment is computed from strain increments given bystrain rates when using a hypoelastic approach. The stress increment must comefrom an objective stress rate in the case of a large deformation analysis. Thereare several possibilities w.r.t. large deformation analyses available. The basicequations are outlined in Appendix A in Part 3 of this review.
20B L.-E. LINDGREN
A welding simulation must at least account for elastic strains, thermal strains,and one more inelastic strain component in order to give residual stresses. Theplastic, viscoplastic, and creep strains are all of the same nature. They representdifferent mathematical models for the plastic deformation (See Figure 9 in the sec-tion where the varying plastic phenomena at different temperatures are discussed).The following sections deal with material properties for the thermoelastic behavior(k4 = ýý + k1), and thereafter the plastic properties are discussed that give the plasticcomponents (iý + ýýp + e) and then the modeling of transformation plasticity ( U' ).
The mechanical analysis requires much more time due to more unknowns pernode than in the thermal analysis. Furthermore, it is much more nonlinear dueto the mechanical material behavior. The mechanical properties are more difficultto obtain than the thermal properties, especially at high temperatures, and they con-tribute to the numerical problems in the solution process. This is discussed sub-sequently in more detail. Nearly all papers about welding simulations includetemperature dependencies of at least some mechanical properties (an exceptionis Cafias et al. 1994). The treatment of the dependency on the temperature historywas discussed in the section "Dependency on Temperature and Microstructure."
The high-temperature mechanical behavior is modeled in an approximate waydue to several factors: experimental data is scarce, too-soft material causes numeri-cal problems (Hepworth 1980, Leung and Pick 1986), and it is found thatapproximations introduced do not significantly influence the resultant residualstresses. Many analyses use a cut-off temperature above which no changes inthe mechanical material properties are accounted for. It serves as an upper limitof the temperature in the mechanical analysis. The meaning of using a cut-off tem-perature may vary since some studies only apply this cut-off to some properties.Ueda and Yamakawa (1971a, b) did not heat the material higher than 600'C,and Fujita et al. (1972) limited the maximum temperature to 500 0C. Ueda et al.(1985) assumed that the material did not have any stiffness above 700'C. They calledthis the mechanical rigidity recovery temperature, above which the Young's moduluswas set to zero. It is therefore likely that this corresponds to the cut-off temperature.Hepworth (1980) used 800'C as a cut-off temperature above which he did not includeany thermal dilatation. Free and Porter Goff (1989) completely ignored the tem-perature dependency of Young's modulus, neglected hardening, and used 900'Cas a cut-off temperature. Furthermore, they only followed the cooling phase ofthe compiled temperature envelope. Tekriwal and Mazumder (1991a) varied thecut-off temperature from 600'C up to the melting temperature. The residualtransverse stress was overestimated by 2 to 15% when the cut-off temperaturewas lowered.
Properties for Modeling Thermoelastic Deformation
The thermoelastic properties are used to compute the stresses. The stress-strainrelations have to be written in incremental form since the nonlinear analysis isperformed by a time-stepping procedure. Strain increments give stress incrementsaccording to Hooke's law. The stress-strain relations are usually separated into volu-metric and deviatoric parts. The first part is purely thermoelastic in the case of
IMPROVED MATERIAL MODELING 209
incompressible plasticity (see the description of plasticity later). For convenience,only the rate-independent plastic strain is included in the subsequent equations.Assuming that the thermoelastic behavior can be described by the linear Hooke'slaw gives
Ukk = 3 Ke•k = 3 K(skk - 3 6th) (4)
s =2GWe' = 2G(ej - eý) (5)
where K = E/3(l - 2v) is the temperature-dependent bulk modulus, G = E/2(l + v)is the temperature-dependent shear modulus, E is Young's modulus and v isPoisson's ratio, aO- is the stress tensor, sL = Cui - Ukkfbi1/3 is the stress deviator tensor,E is the total strain tensor, c, is the elastic strain tensor, e. = S# - Ekk&j/ 3 is the totalstrain deviator tensor, e. is the elastic strain deviator tensor, &th is the uniaxialthermal dilatation, and E,.= e, is the plastic strain tensor, which is equal to thedeviatoric plastic strain tensor for incompressible plasticity. These relations are usedto compute the thermoelastic stress in the absence of plasticity, ej = 0, but they arealso used to compute so-called trial stress in computational plasticity. The differentforms of Eqs. (4) and (5), which are needed in a nonlinear analysis, are
dKdokk = 3K(dckk -- 3d21h) + 3dKe~k = 3K(dEkk - 3di th) + "•'-Ckk (6)
dGdsij = 2Gde. + -.- sj (7)
The equations should be implemented in an incremental form that gives the exactstresses if the increment is elastic, that is, when Eqs. (4) and (5) are evaluated atthe end of an increment. This requires
Aakk = 3K(T "+')(Aekk - 3As") + AK an
T( 7 7) kk(8
AGAs¾ = 2G(Tn+')Ae', + (9)
Y G(T- ' Y
where the right superscripts denote time-step counters in the finite element solutionprocess.
The thermoelastic models tried by Argyris et al. (1982) are not in accordancewith this. Their implementation of the thermoelastic behavior is not correct evenin their so-called mixed model since all material properties in the equations corre-sponding to Eqs. (8) and (9) are evaluated at the beginning of the increment in theirpaper.
Different assumptions regarding the high-temperature elastic properties havebeen used. Friedman (1975, 1977) assumed a constant bulk modulus. This wasobtained by decreasing the Young's modulus with temperature and at the same timeincreasing Poisson's ratio toward 0.5. Young's modulus and Poisson's ratio are
210 L.-E. LINDGREN
shown in Figures 6 and 7 for lower temperatures (Richter 1973). An estimate basedon this data does not support the use ofa constant bulk modulus. It seems to decreasefor higher temperatures. Increasing Poisson's ratio and assuming incompressibilityin the liquid state have led others (e.g. Hibbitt and Marcal 1973, Andersson 1978,Josefson 1982, Jonsson et al. 1985a, Cowles et al. 1995, Troive et al. 1998) to increasePoisson's ratio toward 0.5 near the melting temperature. However, there is no reasonto have a continuous increasing bulk modulus up to infinity in the liquid state sincethere is a phase change occurring during melting and the material can, therefore,have discontinuous material properties. Leung and Pick (1986) compared an analysiswith a Poisson's ratio ranging from 0.24 at room *temperature to 0.45 at meltingtemperature with an analysis for a constant value of 0.24. They found that the resultswere almost identical, but the first analysis required 50% more computer time.Tekriwal and Mazumder (1991a) also investigated the influence of Poisson's ratio.They obtained nearly identical results, but the computer time was about the samefor all variations of Poisson's ratio. The extra computer time required in the analysisby Pick and Leung (1986) was probably due to the selected finite element implemen-tation and not due to the physical problem. The chosen values for Young's modulusis an important parameter that affects the residual stresses. If the chosen value athigh temperature is too low, then the finite element analysis will fail. Lindgrenet al. (1999) used I GPa as the lower limit for a ferritic steel.
L-E LINDGREN
[O]Paj
200 ' •Mild and low
180 ____
180160 ' High all! y steels _•• . ...
• 4o i. . . ..
0 200 400 600 800
Temperature 10C]
Figure 6. Young's modulus for some steels. Data from Richter (1973).
IMPROVED MATERIAL MODELING 211
0,33 __
0,320 ,3 1. M ild a n .d .lo w0 , 3 1 ~ ~a l l o y s t e e l s .. .0,32
0,28 iIloy steels
0 200 400 600 800
Temperature rC]
Figure 7. Poisson's ratio modulus for some steels. Data from Richter (1973).
The thermal dilatation, Eth, is the total effect of the usual thermal expansion andthe volume changes due to phase changes. The thermal dilatation is the driving forcefor the thermal stresses and is therefore an important parameter. Typical values aregiven in Figure 8 for slow heating. The influence of the volume changes due to phasetransformations on the thermal dilatation was discussed earlier in the section"Dependency on Temperature and Microstructure." Murty et al. (1996) assumedthat the transformation strain is 0.044 when 100% austenite is transformed into100% martensite and 0.007 if it is transformed into ferrite/pearlite. The solidificationshrinkage is usually ignored. The motivation may be that it will lead to plastic strainsthat are removed anyway due to this phase change. The use of a cut-off temperaturewill also exclude some thermal dilatation in the high-temperature range. Feng (1994)and Feng et al. (1997) discuss the modeling of solidification shrinkage for aluminum.Their study is concerned with solidification cracking, so they cannot resort to toolarge simplifications in the high-temperature range. Sheng and Chen (1992) andChen and Sheng (1991, 1992) also had a detailed model of the near-weld-pool regionwhere the solidification shrinkage was included.
The implementation of the computation of the thermal dilatation can be done indifferent ways. It is important to observe whether thermal dilatation is specified orthermal expansion coefficient. The latter can vary since some codes want the secantand other the tangent thermal expansion coefficient. This is due to the way theincremental thermal dilatation de th is computed. It can be computed directly byinterpolating from a given thermal dilatation curve or by the tangent thermal
212 L-E. LINDGREN
0,02 " _
0,016 _ _
0,012High alloy kteels _
Mil Iand low
0,004 allol steels
0 ___
0 200 400 600 800Temperature [0CI
Figure 8. Thermal dilatation for some steels. Data from Richter (1973).
expansion
dh = d- -dT = atdT (10)
or the secant thermal expansion
dgth =~ 2 n+[
e s - (Tn+' - Tref) - ,(T7- Tref). (11)
ABAQUS (Wilkening and Snow 1991) also computes an initial thermal strainbecause it does not assume that the initial temperature is the zero-strain state. Thus,it is important to set this reference temperature equal to the initial temperature givento the elements added to the model when a weld is laid. Hong et al. (1998a) found thedifference between using the room temperature or the melting temperature as ref-erence temperature to be small when simulating a multipass weld using ABAQUS.It is not clear if they removed all accumulated plastic strains when the materialmelted. This will remove the previous history effects at these elements. The removalof plastic strains when a material is melting is mandatory (see the next section),otherwise the chosen reference temperature will give different plastic strains, whichmay, depending on the hardening, be visible on the residual stress fields. The correctapproach is to use the zero-stress temperature of an element as the referencetemperature. This is the melting temperature for added filler material.
IMPROVED MATERIAL MODELING 213
Properties for Modeling Plastic Deformation
The mechanisms for the plastic deformation vary due to the large temperature rangeinvolved. The Ashby diagram (Figure 9) illustrates the changing plastic strain ratefor different stresses and temperatures. However, the time scale is also importantwhen determining whether rate-dependent plasticity should be included. The argu-ment for using rate-independent plasticity at high temperatures is based on theinvolved time scales. This was stated clearly by Hibbitt and Marcal (1973), whoincluded creep only when considering stress relief. The same was done by Uedaet al. (1976, 1977) and Josefson (1982). The material has a high temperature duringa relatively short time of the weld thermal cycle, and therefore the accumulatedrate-dependent plasticity is neglected. Bru et al. (1997), who studied the same prob-lem as Roelens (1995a, b), used tensile data for the strain rate of 0.1 sec - I andthe rate-independent plasticity model. Sekhar et al. (1998) show the variation ofyield stress with temperature for two different strain rates. The difference is smallfor that particular case except around 7000 to 900'C. Most studies in simulationof welding approximate the yield limit at higher temperatures and try not to makeany elaborate adjustment for expected dominant strain rates since the available dataare scarce. The plastic flow in rate-independent plasticity is determined by the con-sistency condition, which requires that the effective stress must stay on the yield
10"1 Ideal strength steady-state
(rate insensitive)
~Dislocation plasticity.dorntin,",
tont
E
aDa
10-6 (Boundary diffusi ) \(Lattice diffusion
0 Homologous temperature TiTm
Figure 9. Sketch of Ashby diagram (Ashby 1992) illustrating deformation mechanisms and differentstrain rates. Different deformation mechanisms in different temperature and stress regimes are shown.
214 L.-E, LINDGREN
surface for a plastic process. This leads to a direct relation between change of plasticstrain and change in stress. The model that has been used most widely forrate- independent plasticity is the von Mises yield criterion together with the associ-ated flow rule. Thus, the plastic strains are incompressible and are not dependenton the hydrostatic part of the stresses. The flow rule states that the plastic flowis orthogonal to the yield surface. This comes from the assumption of maximumplastic dissipation or D~rucker's postulate. The flow rule and the yield functionare given as
'i Y oai ;.(VU ii)(12)
where;.is the plastic parameter that determines the amount offlow,f = -o,(i, T)
is the yield function, &T -= - a#)(s~y -j is the effective stress, ay~ is the backstress or the center of the yield surface that accounts for the kinematic hardening,cry is the yield limit, and K is an internal variable for the isotropic part of thehardening. The consistency conditionJ f 0, is evoked to determine the plastic par-ameter A. A hardening rule is also needed to determine the amount of plastic flowand the evolution of the yield strength of the material. There are several accuratemethods for computing the stress increment for inelastic material behavior. Theeffective-stress function is a general version of the radial return method that canaccommodate different kinds of inelastic strains (Kojic and Bathe 1987, Josefsonand Lindgren 1997). See also Appendix A of Part 3.
Tall (1964) assumed the material properties to be temperature dependent but nowork hardening was accounted for. Tsuji (1967) assumed a linear decreasing yieldstress and linear, isotropic hardening. Jonsson et a]. (1985b) varied the yield limit,hardening modulus, and thermal dilatation in order to study the influence onthe residual stresses and the change in gap width in front of the moving arc. Theyalso assumed isotropic hardening. A higher yield limit and hardening modulusat low temperatures raised the residual stresses in the weld region. It also affectedthe change in gap width as the strength of the tack welds was important in the studiedcase. The thermal dilatation had only a smaller influence on the residual stresses. Thelarge restraint of the plate against in-plane motion causes the yield strength to be thelimiting, and therefore important, factor for the studied case. McDill et al. (1990)investigated the influence of the thermal and mechanical properties on the residualcurvature of a welded bar. They only combined thermal and mechanical propertiesof a stainless steel and a low carbon steel and did not vary the individual properties.For example, a larger plastically deformed region was obtained with lower yieldstrength, but the influence on the curvature was counteracted by lower residualstresses. All changes affected the residual curvature of the welded bar quite a lot.It varied from 41.6 to 11.9 In-'. The case they studied was little restrained, andtherefore it is expected that thermal dilatation together with the temperature dis-tribution is important for the deformation. The latter depends strongly on thethermal properties. This explains why they found that the thermal properties werethe most important difference between carbon and stainless steels. Bru et al. (1997)investigated the influence of the mechanical properties at high temperatures. It
IMPROVED MATERIAL MODELING 215
was a continuation of the study by Roelens (1995a,b), where microstructure evol-ution was used to compute the material properties using mixture rules. The yieldstress of the austenite was uncertain due to its dependency on grain size and its ratedependency. They used von Mises plasticity and the yield limit for a given strainrate. Furthermore, Roelens (1995a, b) used 1250'C as a cut-off temperature forstrain hardening, thermal strain, and Young's modulus. This was raised to 15000 Cby Bru et al. (1997). All these modifications did not affect the residual stresses sig-nificantly; however, they affected the residual deformation. This is probably dueto the sensitivity of the weldingconfiguration, butt-welded plates, to the deformationat high temperature during the first weld pass. This can be seen from Roelens(1995a, b). It seems that they did not have any restraints or applied gravity loadingthat would push the plates toward the welding table and the halves were only initiallyconnected with one weld.
The material near and in the weld is subjected to reversed plastic yielding duringthe cooling phase. Thus, using kinematic, isotropic, or combined hardening willaffect the stresses in this region. Andersson (1978) suspected that his use of isotropichardening was one reason for the deviations between experiments and simulations.However, this is not the case, as is shown in the section "A Welding SimulationRevisited" in Part 3. Bammann and Ortega (1993) investigated the effect ofassuming isotropic and kinematic hardening. They found that the choice of hard-ening influences the residual stresses very much in the weld metal, but further awaythe different models gave identical results. They used the rate-dependent materialmodel described later in this section. Devaux et al. (1991) found only small differ-ences in residual stresses between models with isotropic or kinematic hardening.They studied weld repair with four or six beads. No hardening has been assumedin some studies (e.g. Ueda and Yamakawa 1971a, b, Hibbitt and Marcal 1973, Uedaet al. 1976, 1977, Hsu 1977, Jonsson et al. 1985a, Karlsson 1989, Karlsson andJosefson 1990). Excluding hardening is an approximation, but it can be motivatedin some cases. Karlsson (1989) argues that most plastic strains are accumulatedabove 600'C, where the material has nearly no hardening. These plastic strainsare removed during subsequent phase transformations (see discussions later, inthe studied low alloy steel). Finally, the yield limit used at room temperaturewas estimated from hardness measurements on the welded plate and thereforealready included the effect of hardening during the final cooling. Most studiesuse linear, isotropic hardening. Friedman (1975, 1977) assumed isotropic hardeningusing a power law hardening and Ueda et al. (1991) used piecewise linear, isotropichardening. Kinematic hardening (Papazoglou and Masubuchi 1982, Wang andInoue 1985, Leung and Pick 1986, Ueda et al. 1986, Mok and Pick 1990, Tekriwaland Mazumder 1989, 1991a, b, Michaleris 1996, Michaleris and DeBiccari 1997)and combined hardening (Chakravarti 1986, 1987, Murthy et al. 1994) have beenassumed in some simulations.
The rate-dependent plasticity models use a flow potential surface that deter-mines the plastic strain rate. The effective stress can be outside this surface andthe plastic strain rate is a function of the distance to this surface. The materialbehavior is elastic if the stress state is inside the potential surface. The surface doesnot exist in the case of creep, kc, which can be considered a special case of
216 L-E. LINDGREN
viscoplasticity. A general form for the plastic flow, where it is assumed that it isnormal to the potential surface, is
o= 4 '1(a, zy, orf(, T))- (13)
where D is the flow potential and af is the flow stress of the material corresponding tothe yield limit in the case of rate-independent plasticity. Some studies like those ofArgyris et al. (1982, 1985), Inoue and Wang (1983), Wang and Inoue (1985), Chidiacand Mirza (1993), and Myhr et al. (l998a, b) used a viscoplastic material model.Wang and Inoue (1985) used a model that represents elastic and viscousdeformation, pure viscous deformation, an incompressible Newtonian fluid, andinviscid behavior. The inviscid behavior is the rate-independent plasticity discussedpreviously. Goldak et al (1996, 1997a) discuss the use of different rate dependenciesfor the plastic behavior at different temperatures and stresses. They chose to usedifferent constitutive models for different temperature regions. A linear viscousmodel is used at a homologous temperature above 0.8. The homologous temperatureis defined as the current temperature in Kelvin divided by the melting temperature inKelvin. Rate-dependent plasticity is used down to a homologous temperature of 0.5and von Mises plasticity for lower temperatures. Carmignani et al. (1999) discussthe formulation and numerical implementation of a unified model for plasticityand viscoplasticity. Mahin et al. (1988, 1991), Winters and Mahin (1991), Ortegaet al. (1992, 1998), and Dike et al. (1998) used a unified creep-plasticity modelby Bammann. They assumed
OP= Ar......rsis - ai4 - Cf(K, T)n s( - a (14)= .,ts Ln B(T) J Si." - i
where af = afo(T) + K, k = H(T)-P-recovery terms, _-t2K.P is the effectiveplastic strain rate, and &jj = h(T).i'-recovery terms. Ronda and Oliver (1998) com-pare three different viscoplastic models. The models give different results, but thereis no discussion about their applicability for the studied case and there is no com-parison with the often used rate-independent elastoplastic model. Sheng and Chen(1992) and Chenand Sheng(1991, 1992) used the Bodner-Partom viscoplastic modeland the Walker model. The latter accounts for kinematic hardening. Oddy andMcDill (1999) used a model where rate-independent plasticity and creep damagewere accounted for simultaneously in a simulation of burnthrough during weldingof pressurized pipelines.
They used the previously described von Mises plasticity together with
ýc = f(u, T)G(T) (T_ý)(15)
where co is the damage parameter, f is a function to be discussed, and n is a materialconstant. The damage parameter a), assumed to be 0 for undamaged and I at creep
IMPROVED MATERIAL MODELING 217
rupture, is determined by
C0 Y--A (16)
where A = a "(0- *) is a stress-dependent parameter combining the influence of effec-tive stress and the largest principal stress al, q and v are material parameters,and aE[0, I] is a traditional factor. The latter parameters were obtained for onetemperature and extrapolated to other temperatures by assuming that time forrupture changes with temperature in such a way that the Larson-Millar parameteris constant. This parameter is given by
L = T(m + log(t)) (17)
where T is in Kelvin and t is time. Then the material parameters in Eq. (16) can bedetermined at any temperature if data are given for one temperature by
uT = vjT 1
(CI (I + q 1))T I TC(1 + iq) - l0,,[(T IT)-
where the subscript I denotes test data at temperature T1. Constant data above 7500were assumed due to nucleation of new grains. Oddy and McDill (1999) assumed mwas 16.4. The functionf(a, 7) in Eq. (15) was used to adjust the creep rate to datafor the minimum creep rate, which occurs when w=0. They used a relation whereone additional dependency on stress, except the P" term, also was included. Thefunction had the form
A (-QRIT) • A(f(a, T)=BT+_CTe • s - (19)
and n=3 in Eq. (15).In general, the flow stress decreases with increasing temperature and increases
with increasing strain rate for a constant microstructure. An Arrhenius type ofequation, e- Q/(R ), may describe the flow stress decrease with absolute temperatureand it is roughly proportional to a power of strain rate. However, when plasticdeformation is accompanied by precipitation, phase changes, and recrystallization,for example, then the flow stress changes in a complex manner. An example ofthe flow stress for a low carbon steel is shown in Figure 10. The anomaly for higherstrain rates is due to so-called blue shortness. The flow stress for a stainless steelis given in Figure 11. The strain rate dependency can be seen. High-temperaturedata may have to be derived from tests for similar materials due to the lack of data.The influence of the material composition is often less important at higher tempera-tures (Figure 12), but not when carbide formers for added high-temperature strengthare present in the material. The influence of the yield limit at higher temperatures onthe residual stresses is also less important. This is related to the removal ofaccumulated plastic strains during phase transformations. The exclusion of therate-dependency of the plastic flow at higher temperatures can be combined with
218 L.-E. LINDGREN
3.5 se&!•.~ 055%C
200 ~ ~0.2 s~c_ •--z ......ott
0 500 1000 1500
Temperature [°CI
Figure 10. Flow stress for 0.15% C-steel evaluated at a total strain of 0.2 for different strain rates andtemperatures. Data from Suzuki (1968).
[MPa1600 ___
3(3 sei-l
S3.5 sed400
200 -
0 500 1000 1500
Temperature I°]C
Figure M1. Flow stress for austenitic stainless steel evaluated at a total strain of 0.2 for different strainrates and temperatures. Data from Suzuki (1968).
IMPROVED MATERIAL MODELING 219
using a higher yield limit than the real yield limit. This may also be beneficial for thenumerical stability of the simulations, avoiding large deformations and plasticstrains.
The influence of the phase transformations on the thermal dilatation and theyield limit was discussed earlier in the section about "Dependency on Temperatureand Microstructure." Two other aspects of the phase transformations have to bemodeled: transformation-induced plasticity (TRIP) and change in dislocationstructure. The latter is usually accounted for in the hardening model by theaccumulated effective plastic strain. The modeling of these phenomena is discussedlater in this section.
Experimental measurements of welding residual stresses in alloys with low aus-tenite decomposition temperatures showed that longitudinal stresses can be stronglyaffected by phase transformations (Nitschke-Pagel and Wohlfahrt 1992). The com-bination of an applied stress (Figure 13) and a phase change will give rise to anadditional plastic deformation, TRIP. Fischer et al. (1996, 2000) discusses the sub-ject more thoroughly in the context ofmartensitic (displacive) transformations. (Seealso Bergehau and Leblond (1991).) The transformation plasticity is usually assumedto be proportional to the deviatoric stresses in the same ways as is assumed standardplasticity models. It is given by
f f(4'- h, U., Xa- Osiv (20)
where '~Jb is the volume change when transforming from phase a to b, ka-h is therate of phase change from phase a to b, and ca is the yield limit of the weaker phase
300.. 1 .....1-[MPal
1,0 <55%C
200 L0,25%C
100 -01%
800 900 1000 1100 1200 1300
Tempemature [°C1
Figure 12. Flow stress for carbon steel at higher temperatures. Data from Suzuki (1968). The strain rate is10sec - ' and the flow stress is evaluated at e = 0.2.
220 L,-E. LINDGREN
Total strain during cooling
0,5 Martensitic
[%] transformation
0,3
0,l
-0,1 1000200 l00 Soo 0
-0,3 Application of loada3=-70MPa
-0,5 -
Temperature ['CJ
Figure 13. Effect of stress applied during martensitic transformation for uniaxial test (Cavallon et a!.1997).
a. Argyris (1982, 1985), Josefson (1985c), and Karlsson (1989) accounted for TRIPby lowering the yield limit. Denis (1984), Dubois et al. (1984), Leblond et al. (1989),Leblond (1989), Josefson and Karlsson (1992), and Oddy et al. (1989, 1992)accounted for TRIP in a more correct way. Oddy et al. (1989) showed that thismay be the explanation for the discrepancies between simulations and measurementsby Hibbitt and Marcal (1973). The effect of different material models on the residualstresses are shown in Figure 14 for conditions resembling the case studied by Hibbittand Marcal (1973). The papers by Leblond et al. (1989) and Leblond (1989) give atheoretical framework. They did not consider the effect of the stresses on the phasechanges. The paper by Oddy et al. (1992) gives a concise description of TRIP.Bammann et al. (1995) implement this effect differently by using a multiphase statevariable constitutive model. Fischer et al. (2000) give a more detailed model byincluding both the shear associated with the martensitic transformation and its vol-ume change (k'_b) to the transformation plasticity. They propose the relation
3 dq -=i? d---K kx - bs&j (21)
wher K -5(( 2 dX02where K = 5((rb)2 +jy.)/6r'., ym is the transformation shear because of themartensitic formation, ac = e4P4( - a'/o ()/(ln(c h)))is the rate of phase changefrom phase a to b, a' is the yield limit of martensite, ay is the yield limit of austenite,
IMPROVED MATERIAL MODELING 221
Residual longitudinal stress (HY 130)
800I . .Yield hysteresisIMPa40 only
600- - - including volume
change due to40 phase changeI-- . including
200 transformationl plasticity
0 : -- • -- --
'I
-200 ,;
-400 ......
0 20 40 60 80Distance from weld centre [mm]
Figure 14. Effect of transformation plasticity on welding residual stresses at top surface of weld (Oddy etal. 1989).
and qp(X) is an heuristic function. Equation (21) gives transformation plasticitybecause of an accommodation process when martensite if formed. It is calledthe "Greenwood-Johnson" effect. There is also an orientation process, the "Magee"effect. See Fischer et al. (2000) for further details about this and its relation tokinematic hardening. Further details can be found in Berveiller and Fischer (1997).Denis (1997) discusses pearlitic, that is, a diffusive transformation, in this book.An expression of the same type as in Eq. (21) is also used for this process.
Modeling the yield strength of a material that undergoes phase changes posessome questions. Usually the yield strength is determined by a virgin yield limitand a hardening contribution due to the deformation. The hardening is due tothe increase in dislocation density. The dislocation structure is represented bythe effective plastic strain in a material model. Then there is a question abouthow the dislocation density will change during phase transformations. Severalpapers have attempted to account for these effects. Friedman (1975, 1977) andPapazoglou and Masubuchi (1982) removed the accumulated plastic strains whenthe material melted and relieved the accumulated strains by multiplying the pre-viously accumulated plastic strains with a factor due to solid-state transformations.Sarrazin et al. (1997) also included recrystallization and recovery effects, whichreduced the yield strength of aluminum. It is at least appropriate to remove all
222 L,-E. LINDGREN
accumulated strains when the material melts (Lindgren et al. 1997, Brickstad andJosefson 1998). Mahin et al. (1991) included the removal of plastic strains duringmelting because they had found that this was a source of discrepancy betweensimulations and measurements in their earlier work (Mahin et al. 1988). Ortegaet al. (1992) initialized all internal state variables to zero when the material melted.They also removed the deviatoric stresses, creating a hydrostatic stress state inthe weld pool. This was also done by Dike et al. (1998). Devaux et al. (1991) arguedthat it is reasonable that the memory of previous plastic deformation disappearsfor all solid-state phase transformations in ferritic steels with perhaps the exceptionfor the martensite formation. The latter involves very small displacements of theatoms during the transformations, which they assumed did not affect the dis-locations. Brust and Dong (Brust et al. 1997, Dong et al. 1998c) introduced rateequations applied between an anneal temperature and the melting temperaturefor anneal strain. The introduction of these strains corresponds to the removalof plastic strains as they reduce the hardening. They also applied this to the elasticstrains and thereby reduced the stress. The technique can be used with the elementbirth option for multipass welding in order to remove history effects. This is alsodiscussed by Hong et al. (1998b). However, the uniaxial case discussed by themis not completely correct. The removal of plastic strains does not imply that thethermal and elastic strains should also be set to zero-the latter giving zero stressat this instant in time. Models for recovery processes at higher temperatures, likeEq. (16), exist, but physical-based models for these processes during phasetransformations are lacking. Physical-based material models are models basedon observations of the actual physical processes, phase changes, and dislocationprocesses, for example, that take place during the deformation.
There are no papers where the texture created by the directional solidification inthe weld metal is taken into account. Mahin et al. (1988, 1989) discussed this aspectw.r.t. the interpretation of the neutron diffraction measurements in the weld metal.
A list of papers where some material properties are included is given in Table 2for ferritic steels, Table 3 for austenitic steels, and Table 4 for other metals. Ifnothing else is stated in the comments, then the model is von Mises rate-independentplasticity with isotropic hardening and the associated flow rule. Most works rely onproperties compiled from the literature, and a few papers refer to experiments pre-pared especially for the welding simulations at hand. A discussion of materialmodeling with respect to obtaining residual stresses is given by Oddy and Lindgren(1997).
Recommendations. It is important to have a correct description of the materialbehavior in order to have an accurate model. The more important mechanicalproperties are Young's modulus, thermal dilatation, and parameters for the plasticbehavior. The influence of these properties at higher temperatures is less pronouncedon the residual stress fields. The material is soft and the thermal strains cause plasticstrains even if the structure is only restrained a little as the surrounding, initially coldmaterial acts as a restraint on the heat-affected zone. A cut-off temperature maytherefore be used. It may also be necessary to simplify the tangent matrix usedin the solution procedure if it is too ill-conditioned. Andersson (1978) and Jonsson
IMPROVED MATERIAL MODELING 223
et al. (1985a) used only the thermoelastic part of the stiffness matrix at hightemperatures. This reduced the convergence rate, but it may be necessary if thematrix is too ill-conditioned.
The phase transformation of ferritic steels during cooling may influence theresidual stresses quite a lot. Therefore, the cut-off temperature should at leastbe higher than austenization temperature, but it is better to use a value near themelting temperature. The influence of the material properties on stresses anddeformations is dependent on the studied configuration. A rigid structure willget larger changes in plastic strains, whereas a flexible structure will experience dif-ferent deformation due to different material properties.
The importance of including the effect of phase transformations on the mech-anical behavior for ferritic steels is also discussed in the recommendations in thesection "Dependency on Temperature and Microstructure." The effect of TRIPon residual stresses depends on the restraint of the structure and at what temperaturethe phase changes occur. If, for example, the shrinkage after the martensite forma-tion gives rise to plastic yielding before the weld has cooled completely, then theresidual stress is limited by the yield limit of the hardening material and the differ-ence in the residual stresses between an analysis that includes or ignores the volumechanges due to the phase transformation may be small.
The plastic strains generated at higher temperatures may be removed or reduceddue to phase transformation and thereby reduce the influence of high-temperatureproperties even more. It is preferable that the material is modeled as ideal plasticat this temperature; otherwise the removal of plastic strains may lower the yieldstrength, which in turn will create additional plastic strains if the stress field ison the yield surface. The plastic strains that exist during cooling will be the majorremaining effect of the properties at higher temperatures. They will in turn contrib-ute to the hardening of the material. This will also limit the influence of these plasticstrains. Thus, the use of approximate material data and ignoring the rate dependencyat higher temperatures has been found to affect the residual stresses very little (e.g.Free and Porter Goff 1989, Bru et al. 1997, Tekriwal and Mazumder 1991a). Manysteels have about the same properties when they become fully austenized at highertemperatures (Figure 12) if no special carbide formers have been added for specialhigh-temperature strength. It is appropriate to combine kinematic and isotropichardening in order to get accurate residual stresses in the weld metal, but theremay be a problem in obtaining the needed data.
Finally, it should be noted that the requirements of the material model are higherif the zone near the melt is of particular interest, for example, when hot cracking isstudied. This improved material model should, at the same time, be matched bya refined spatial and temporal discretization.
FUTURE RESEARCH
The modeling of material behavior at higher temperatures and in the presence ofphase transformations is perhaps the most crucial ingredient in successful weldingsimulations. Progress in this field is dependent on the collaborative efforts from com-
224 L-E. LINDGREN
putational thermomechanics and material science. This is a field that must be focusedon by the research community active in the field of welding simulation. The improve-ment in material modeling and increasing availability of material parameters will, incombination with the computational development, increase the industrial use ofwelding simulations.
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121. A.R. Ortega, L.A. Bertram, EA. Fuchs, K.W. Mahin and D.V. Nelson, Thermomechanical modelingof a stationary gas metal arc weld: a comparison between numerical and experimental results, Proc. ofthe 4th Int. Conf on Trends in Welding Research, p. 89, 1992.
122. A.R. Ortega, J.1. Dike, J.F. Lathrop, C.H. Cadden, D.R. Folk and J.E. Robles, Analysis andvalidation of multi-pass girth welds in pipes, Proc. of the 8th Int. Conf. Modeling of Casting,Welding and Advanced Solidification Processes, p. 779, 1998.
123. J. Dike, C. Cadden. R. Corderman, C. Schultz and M. McAninch, Finite element modeling ofmultipass GMA welds in steel plates, Proc. of the 4th Int. Conf on Trends in Welding Research,p. 57, 1995.
124. A.S. Oddy and J.M.J. McDill, Bumnthrough prediction in pipeline welding. Int. J. Fracture, vol. 97, no.1/4, pp. 249-261, 1999.
125. H. Suzuki, S. Hashizume, Y. Yabuki. Y. Ichihara. S. Nakajima and K. Kenmochi, Studies on the flowstress of metals and alloys, Report of the Institute of Industrial Science, The University of Tokyo, vol.18, no. 3. 1968.
126. T. Nitschke-Pagel and H. Wohlfahrt, Residual Stress Distributions after Weldingas aConsequenceofthe Combined Effect of Physical, Metallurgical and Mechanical Sources in: L. Karlsson, M. Jonsonand L-E. Lindgren (eds.), Proc of IUTAM Symposium Mechanical Effects of Welding, SpringerVerlag. Berlin, p. 123, 1992.
230 L-E. LINDGREN
127. N. Cavallon, L. Taleb, I.F. Jullien, F. Waeckel, Y. Wadier and L. Moche, Thermomechanicalbehaviour of a carbon manganese steel under martensitic transformation, Modelling of residualstresses, Proc. of the 5th Int. Conf. on Residual Stresses (ICRS-5), p. 239, 1997.
128. F.D. Fischer, Q-P. Sun and K. Tanaka, Transformation-induced plasticity (TRIP), Appl. Mech. Rev.,vol. 49, no. 6, pp. 317-362, 1996.
129. F.D. Fischer, G. Reisner, E. Werner, K. Tanaka, G. Cailletaud and T. Antretter, A new view ontransformation induced plasticity (TRIP), Int. J. Plasticity, vol. 16, pp. 723-748, 2000.
130. B.L. Josefson, Effects of transformation plasticity on welding residual-stress fields in thin-walled pipesand thin plates, Materials Science and Technology, Oct., pp. 904-908, 1985.
131. S. Denis, E. Gautier, A. Simon and G. Beck, Stress Phase Transformation Interactions: BasicPrinciples; Modelization and Their Role in the Calculation of Internal Stresses, InternationalSymposium on the Calculation of Internal Stresses in Heat Treatment of Metallic Materials, vol.I, p. 157, 1984.
132. J.B. Leblond, J. Devaux and J.C. Devaux, Mathematical Modelling of Transformation Plasticity inSteels 1: Case of Ideal-Plastic Phases, Int. J. Plasticity, vol. 5, pp. 551-572, 1989.
133. J.B. Leblond, Mathematical ModellingofTransformation Plasticity in Steels I: Couplingwith StrainHardening Phenomena, Int. J. Plasticity, vol. 5, pp. 573-591, 1989,
134. A.S. Oddy, J.A. Goldak and R.C. Reed, Martensite Formation, Transformation Plasticity andStresses in High Strength Steel Welds, Proc. ofthe3rdint. Conf on Trends in Welding Research, 1992.
135. A. Oddy, J. Goldak and M. McDill, Transformation plasticity and residual stresses in single-passrepair welds, ASME J Press. Vessel Technol., vol. I I, pp. 33-38, 1992.
136. D.J. Bammann, V.C. Prantil and J.F. Lathrop, A model of phase transformation plasticity, Proc. ofthe 7th Conf. Modeling of Casting, Welding and Advanced Solidification Processes, The Minerals,Metals & Materials Society, p. 275, 1995.
137. M. Berveiller and F.D. Fischer (eds), Mechanics of Solids with Phase Changes, CISM Courses andLectures No. 368, Springer, 1997.
138. S. Denis, Considering stress-phase transformation interactions in the calculation of heat treatmentresidual stresses, in M. Berveiller and F.D. Fischer (eds), Mechanics of Solids with PhaseChanges, CISM Courses and Lectures No. 368, Springer, p. 293, 1997.
139. E. Sarrazin, K. Dang Van and H. Maitournam, Modelling residual stresses in weldments ofwork-hardened aluminium alloys with microstructural effects, in H. Cerjak (ed.), MathematicalModelling of Weld Phenomena 3, The Institute of Materials, p. 652. 1997.
140. B. Brickstad and B.L. Josefson, A parametric study of residual stresses in multi-pass butt-weldedstainless steel pipes, Int. J. Press. Vessel & Piping, vol. 75, p. 11-25, 1998.
141. F.W. Brust, P. Dong, and J. Zhang, Influence of residual stresses and weld repairs on pipe fracture,Approximate Methods in the Design and Analysis of Pressure Vessels and Piping Component,PVP-vol. 347, p. 173, 1997.
142. P. Dong, J. Zhang and M.V. Li, Computational modeling of weld residual stresses and distortions -anintegrated framework and industrial applications, ASME, PVP-Vol. 373 Fatigue, Fracture, andResidual Stresses, p. 137, 1998.
143. J. K. Hong, P. Dong and C.L. Tsai, Application of plastic strain relaxation effect in numerical weldingsimulation, Proc. of the 5th Int. Conf on Trends in Welding Research, p. 999, 1998.
144. K.W. Mahin, W. Winters, J. Krafcik, T. Holden, R. Hosbon and S. MacEwen, Residual straindistributions in gas tungsten arc welds: a comparison between experimental results and weldmodel predictions, Proc. of the 2nd Int. Conf on Trends in Welding Research, p. 83, 1989.
145. A.S: Oddy and L-E. Lindgren, Mechanical modeling and residual stresses, in L. Karlsson (ed.).Modeling in Welding, Hot Powder Forming and Casting, pp. 31-55, 1997.
146. Y. Ueda and K. Nakacho, Simplifying methods for analysis of transient and residual stresses anddeformations due to multipass welding, JWRI, vol. II, no. I, pp. 95-103, 1982.
147. Y. Ueda, Y.C. Kim, K. Garatani and T: Yamakita, Mechanical characteristics of repair welds in thickplate - distribution of three dimensional welding residual stresses and plastic strains and theirproduction mechanisms, Welding International, vol. 2, no. I, pp. 33-39, 1988.
148. E.F. Rybicki and R.B. Stonesifer, An analysis for predicting weld repair residual stresses inthick-walled vessels, ASMEJ. Pressure Vessel Technology, vol. 192, August, pp. 323-331, 1980.
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149. E.F. Rybicki, J.R. Shadley, A.S. Sandhu and R.B. Stonesifer, Experimental and computationalresidual stress evaluation of a weld clad plate and machined test specimen, ASME J. EngineeringMaterials and Technology, vol. 110, October, pp. 297-304, 1988.
150. K-Y. Bae, S-J. Na and D-H. Park, A study of mechanical stress relief (MSR) treatment of residualstresses for one-pass submerged arc welding of V-grooved mild steel plate, Proc. Instn MechEngrs/Part B: Journal oj Eng. Manufacture, vol. 208, B:3, p. 217, 1994.
151. D.W. Lobitz, J.D.McClureand R.E. Nickel], Residualstresses and distortions in multi-pass welding,Proc. ofASME Winter annual meeting. Numericalmodeling ofmanufacturing processes, PVP-PB-025,p. I, 1977.
152. N-X. Ma, Y. Ueda, H. Murakawa and H. Maeda, FEM analysis of 3-D welding residual stresses andangular distortion in T-type fillet welds, JWRI, vol. 24, no. 2, pp. 115-122. 1995.
153. L. Wikander, L. Karlsson L-E. Lindgren and A.S. Oddy, Plane thermo-mechanical finite elementmodeling of welding with special reference to the material behaviour, Int Conf on Modeling andControl of Joining Processes, 1993.
154. L. Wikander, L. Karlsson, M. Nasstr6m and P. Webster, Finite element simulation and measurementsof welding residual stresses, Modelling. Simul Mater. Sci. Eng, vol. 2, pp. 845-864, 1994.
155. MG. Yuan, J. Wang, H. Murakawa and Y. Ueda, Three dimensional finite element analysis ofresidual distortions in welded structures, Proc. of the 7th Conf Modeling of Casting. Welding andAdvanced Solidification Processes, The Minerals, Metals & Materials Society, p. 257, 1995.
156. P. Michaleris, D.A. Tortorelli and C.A. Vidal, Analysis and optimization of weakly coupledthermoelastoplastic systems with application to weldment design. Int. J. Numerical Methods inEng., vol. 38, pp. 1259-1285, 1995.
157. N. Siva Prasad and T.K. Sankaranarayanan, Estimation of residual stresses in weldments usingadaptive grids, Computers & Structures, vol. 60, no. 6, pp. 1037-1045. 1996.
158. B. TaIjat, B. Radhakrishnan and T. Zacharia, Numerical analysis of GTA welding process withemphasis on post-solidification transformation effects on residual stresses, Materials Science andEngineering, vol. A246, pp. 45-54, 1998.
159. Y.C. Kim, K.H. Chang and K. Horikawa, Characteristics of out-of-plane deformation and residualstress generated by fillet welding, JWRI, vol. 27, no. I, pp. 69-74, 1998.
160. E.F. Rybicki and R.B. Stonesifer, Computation of Residual Stressesdue to Multipass Welds in PipingSystems, ASMEJ. Pressure Vessel Technology, vol. 101, pp. 149-154, 1979.
161. E.F. Rybicki, R.B. Stonesifer and W. Shack, The effect of length on weld overlay residual stresses,ASME Pressure Vessel and Piping and Computer Eng. Conference, p. 1, 1986.
162. Y. Lejeail, Simulation of a stainless steel multipass weldment, Proc. of the 5th Int. Conf. on ResidualStresses (ICRS-5), p. 484, 1997.
163. M. Ndsstr6m, P.J. Webster and J. Wang. Residual stresses and deformations due to longitudinalwelding of pipes, Proc. of the 3rd Int. Conf. on Trends, in Welding Research, p. 109, 1989.
164. M. Naisstr6m, L. Wikander, L. Karlsson, L-E. Lindgren and J. Goldak, Combined 3D and shellmodelling of welding, 1UTAM Symposium on the Mechanical Effects of Welding, p. 197, 1992.
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166. P. Dong, J.K. Hong and P. Rogers, Analysis of residual stresses in AI-Li repair welds and mitigationtechniques, Welding Journal, Nov., pp. 439s-445s, 1998.
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Journal of Thermal Stresses, 24:305-334, 2001Copyright © 2001 Taylor & Francis0149-5739/01 $12.00 + .00 ;46
FINITE ELEMENT MODELING AND SIMULATION OFWELDING. PART 3: EFFICIENCY AND INTEGRATION
Lars-Erik LindgrenLuled University of Technology
andDalarna University
Luled, Sweden
Welding simulation has advanced from the analysis of laboratory setups to real engin-eering applications during the last three decades. This development is outlined and thedirections for future research are summarized in this review, which consists of threeparts. This parts focuses on computational strategies and how they are integratedwith other methods tofacilitate the use of simulations in engineering. These develop-ments have lead to the increased application of welding simulations in industry.
The development of welding procedures is based on performing experiments [1], anda Welding Procedure Specification (WPS) is the final result. The evaluation of awelding procedure is based on joint integrity, absence of defects, microstructure,and mechanical testing, for example. Computational methods are rarely used inthe process of developing welding procedures. It is expected that simulations willcomplement the experimental procedures for obtaining a WPS since aspects likeresidual stress can then be considered when comparing different welding procedures.Furthermore, simulations are also useful in designing the manufacturing process aswell as the manufactured component itself. Distortions are usually in focus inthe first case, whereas residual stresses are of interest in the latter case.
This review concentrates on the simulation of fusion welding processes of metals.These processes do have several common traits and are therefore similar to model.Publications presenting finite element simulations of the mechanical effects of weldingappeared in the early 1970s and simulations are currently only used in applicationswhere safety aspects are very important, like aerospace and nuclear power plants,or when a large economic gain can be achieved. The scope of most simulations
Received 12 August 1999; accepted 12 April 2000.This work has been financed by NUTEK (the Swedish National Board for Industrial and Technical
Development) via the Polbem Laboratory. Its completion has also been made possible by the cooperationof ABB Atom AB and Volvo Aero Corporation in a project about multipass welding.
Address correspondence to Professor L.-E. Lindgren, Department of Mechanical Engineering, LulcAUniversity of Technology, SE-971 87 Lulef, Sweden. E-mail: [email protected]
305
306 L.-R LINDGREN
has been to obtain residual stresses and corresponding deformations. There are alsopublications presenting analyses of hot cracking and other phenomena.
A common concern in welding simulation is to account for the interaction betweenwelding process parameters, the evolution of the material microstructure,temperature, and deformation. The resultant material structure and deformationmay also be needed in order to, in combination with in-service loads, predict life timeand performance of a component. Therefore, research in this field requires thecollaborative efforts of experts in welding methods, welding metallurgy, materialbehavior, computational mechanics, and crack propagation, for example. It isimportant to observe that uncertain material properties and net heat input makethe success of simulations to a large extent dependent on experimental results(Lindgren 1996).
The review is split into three parts. They are Part l, "Increased Complexity" andPart 2, "Improved Material Modeling". This final Part 3 is called "Efficiency andIntegration". They all outline the development of simulation of welding and areseparated into sections where different aspects of finite element modeling are infocus. Each section contains some recommendations based on the review andthe experience of the author. It is hoped that this approach will be appropriatefor those that are entering this field of research and useful as a reference for thosealready familiar with this subject.
INTRODUCTION TO PART 3
The increasing accuracy of welding simulations depends on the modeling process andthe opportunity to use large models. The latter makes it possible both to reduce errordue to spatial and temporal discretization and to avoid the constraints oftwo-dimensional models. The increased size in models (Figure 1) is mainly dueto the increase in computer capacity but also reflects in improved finite elementtechniques. This part of the review shows how this has been utilized in weldingsimulations. The integration of simulations with optimization procedures andCAD systems, for example, is also important for the industrial application ofwelding simulations. Furthermore, the intensive use of experiments, which isnecessary to have confidence in models in order to apply them in design, ishighlighted. The review concludes with an investigation of one of the best earlywelding simulations that has been referenced in many later papers.
MORE EFFICIENT COMPUTATIONAL METHODS
We will focus on some of the aspects of improved finite element methods that havebeen applied to welding simulations. There is a large number of textbooks andjournals devoted to finite element techniques. The appendix gives one way offormulating the finite element method and contains references to some papersand textbooks that may be of interest. Most of the increase in the size of thecomputational models (Figure 1) can be attributed to computer development.
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART 3 307
1 E+8 .
dof*nstep [147] [149]
I E+7 [129].[26] [148] * 117]
.• (145]
I E+6 [78]
[851 c [144]0 * [146]
IE+5 [1421 [143] [117]
[141] 0[75]1 E+4
1975 1980 1985 1990 1995 2000Year
Figure 1. Size of computational models of welding measured by degrees of freedom multiplied by numberof time steps versus year of publication of work.
However, software development has also been important. More efficient and robustalgorithms for stress-computation and the solution of a system of coupled nonlinearequations, for example, have been developed. Better finite elements with reducedintegration and enhanced strain fields [2] have improved the accuracy for a givensize of the computational model. These developments are outside the scope of thisreview.
The most straightforward techniques for computational efficiency are thosementioned in the section "From One-dimensional to Three-dimensional Models"in Part 1. Another example of these techniques is given by Kassner and Wohlfahrt[3], who combined substructuring and the replacement of part of the structure withstiffnesses in order to reduce the computational effort by three to four times.
Eulerian reference frames have been used to predict residual stresses in welds [4].Later Gu et al [5, 6], Goldak and Gu [7], and Atteridge et al. [8] also used asteady-state formulation. The latter paper obtained the same results withthree-dimensional models based on Lagrangian or Eulerian meshes. However, planestrain and plane stress models (see Figures 6a, c in Part I of this review) gave tensiletransverse residual stresses in the HAZ, where the three-dimensional models gavecompressive residual stresses. Goldak and Gu [7] also computed the microstructurein the steady-state framework. The principal advantage is that using Eulerianreference frames makes the simulation orders of cheaper and faster magnitude thanLagrangian analyses and the fine mesh can be concentrated to the region of largegradients. But they are restricted to prediction of the quasi-stationary fields, thatis, stationary with respect to the reference frame moving with the heat source.
8L.-E. LINDGREN
This is valid in the middle of a long weld where the geometry is constant along thepath of the arc. The start or finish of the weld cannot be modeled. However, forthis kind of geometry it is also possible to analyze a shortened version of thegeometry in the Lagrangian reference frame; for example, the finite elementmodel of a butt-welded plate can have a reduced length in the welding direction.This analysis can be done with standard software, but it will take longer timethan the Eulerian steady-state formulation. If the residual stresses are used ina subsequent analysis of the loading of the welded plate, then it is possible toremap the residual field for this case. The interior steady-state part of the fieldsmust be stretched to be adapted to the studied geometry, and the residual stresseswill redistribute somewhat at the beginning of this analysis.
Brown et al. [9-11] describes a rezoning technique implemented using a shellprogram together with Abaqus. A traveling, dense mesh is combined with a coarsemesh. Furthermore, dynamic substructuring was added in order to increase the com-putational efficiency [12].
Adaptive meshing has been used in some studies, for Lagrangian meshes, so as toutilize the degrees of freedom in the computational model better by concentratingthem to regions where large gradients occur. Siva Prasad and Sankaranarayanan[13] used triangular constant strain elements. McDill et al. [2, 14-16] implementeda graded element that alleviates the refining/coarsening of a finite element meshconsisting of quad elements, in two dimensions, or brick elements, in threedimensions. It has been used successfully for three-dimensional simulations [17]where the computer time was reduced by 60% and with retained accuracy(Figure 2) for the electron beam welding of a copper canister. Runnemalm and Hyun[18] (Figure 3) combined this with error measures to create an automatic, adaptivemesh. They showed that it is necessary to account both for thermal and mechanicalgradients in the error measure. The mesh created by the latter error measure isnot so easy to foresee even for an experienced user. Therefore, the use of errormeasures is important.
The combination of shell and solid elements in the same model is of specialinterest in welding simulations. A thin-walled structure can be modeled with shellelements, but a more detailed resolution of the region near the weld will requiresolid elements. This was done by Gu and Goldak [19] for a thermal simulationof a weld and by Niisstr6m et al. [20] for a thermomechanical model of a weld.McDill et al. [21] developed a promising element formulation where athree-dimensional element with eight nodes and only displacements as nodalunknowns can be a solid or a shell element. It was demonstrated on a small-weldcase in their paper. The element is based on the same element as in Lindgren etal. [17] and Runnemalm and Hyun [18], making it possible to perform adaptivemeshing with a combination of solid and shell elements. It will also be possibleto determine adaptively whether an element should be treated as a solid or a shell.
The majority of analyses that have been performed have utilized implicitformulations and assumed quasi-static mechanical behavior. An explicit finite elementcode with dynamic relaxation was used by Mahin et al. [22, 23] in a fullythermomechanical coupled analysis. Artificial quenching had to be used after the platehad cooled to 600°C in order to shorten the time for cooling, which is necessary
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART 3 SW
ft a • .105 a M Pa ln •- 105 3 MP4
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Figure 2. Axial stresses (a) with remeshing at 10, 50, 100, and 200 sec and (b) without remeshing at 50 and100 see from Lindgren et al. [17].
because the explicit method cannot take large time steps and the cooling phase can bequite long. Later they [24] used an artificial increased density to facilitate longer timesteps.
Recommendations. The adaptive meshing and parallel computations are cur-rently necessary to solve three-dimensional problems with the same accuracyas in existing two-dimensional models. The Eularian approach is effectivebut less general.
EXPERIMENTAL VERIFICATION
Experiments have always been very important in the development of welding pro-cedures. The use of simulations in this field adds new tasks- for those performingthe experiments. They are needed to obtain data given to the computational models
310 L-E. LINDGREN
(a) (b)
weld position weld position
Figure 3. Adaptive meshing based on gradient in (a) thermal field and (b) gradients in thermal and mech-anical fields [18).
and to verify the computational models. Therefore, many simulations andexperiments have been performed. Nearly all simulations rely on measured transienttemperatures or the resultant microstructure in order to estimate the net heat input.This is necessary because there is no complete model that gives the net heat inputfrom given welding parameters [25]. Thus, the net heat input in the finite elementmodel changes until measured and computed temperatures agree (e.g. (26]). Manysimulations are also accompanied by residual strain measurements usinghole-drilling techniques, X-ray techniques, or neutron diffraction measurements.Transient strains have also been measured in some cases using high-temperaturestrain gauges. The transient motion of, for example, the gap in front of the arcor residual deformation has also been used to verify computational models. A listof papers with experimental verification is given in Table 1. Extensive experimentsfor verifying the computational models are presented by Ortega et al. [27], Mahinet al. [22-24], Winters and Mahin [28], and Dike et al. [29]. They used thermocouplesand measured the fusion zone and calorimetry in verifying the thermal analysis.Neutron diffraction measurements, X-ray measurements, and holographichole-drilling techniques were used to compare measured and computed residualelastic strains.
There are papers that only include experiments. They are often useful for thoseperforming the simulations. However, including them will requie a separate review.Some researchers that have performed simulations also have papers focusing onexperiments (e.g. [30-32]). The proceedings of the 5th International Conferenceon Residual Stresses [33] have sections about computing and measuring residualstresses.
Recommendations. The computational model needs at least some kind ofexperimental results in order to determine the net heat input. Using thermocouplesto measure the temperature is straightforward. It is also wise to do more experiments
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART 3 311
Table I. References with measurements
Type of Measurement- Reference
H Tall [55]TT, TS, R Tsuji [561H Hibbitt and Marcal [571TT, TD (gap width in front of arc), TS Muraki et al. [581TIT, TD (deflection of edge welded plate, TS Masubuchi [59]R Ueda and Yamakawa [60, 61], Ueda et al.
[62, 631TT', TS Lobitz [64]Tr Hsu [65]TT, H Andersson [46]TT. R, RD Rybicki et al. [66]R Fujita et al. [67]TT, R Rybicki et al. [68-71]ITT, TS Papazoglou and Masubuchi [72]
H Inoue and Wang [73]TI Dubois et al. [74]TT, H, X Argyris [75, 76]TT, TS, H, TD (changes in gap in front of arc) Jonsson et al. [26]TIT, H Wang and Inoue [771X Josefson [78]T7, X, R Leung and Pick [79]TF, TS, H, R, RD Ueda et al. [80]H(?), T7, RD (angular distortion of plate) Chakravarti [81, 82]R Rybicki et al. [83, 84]IT, H Lindgren and Karlsson [85]H, X Kussmaul and Guth [86]'IT, X, H Carmet et al. [87]TT, NDF Troive et al. [88]TI'. H, R Free and Porter Goff (89]NDF NAsstr6m et al. [901NDF Mok and Pick (911TT, NDF, fusion zone, calorimetry Mahin et al. [22-24], Winters and Mahin
[28]TS Tekriwal and Mazumder [92-94]R Ueda et al. [95]TT, infrared thermography, H Ortega et al. [27]TT', H Shim et al. [96]TT, TS, H, RD Jones et al. [97, 98]TT, TD, TS Chidiac and Mirza [99]H Cafias et al. [100, 101]TD (transverse displacement of weld edges), Feng (102]
RD (Moir& giving shear strain)TT, R Dupas and Moinreau [103]H Bae et al. (104]NDF Wikander et al. [105, 106]TD (angular distortion of T-joint) Bae and Na [107]T, X, H Roelens [108-110]
312 L.-E. LINDGREN
Table 1. References with measurements-continued
Type of Measurement- Reference
TT, X, TD (change in gap width) Dike et al. [Ill]IT, H Murty et al. [112]TIT, H, X Michaleris [113]R Wu et al. 1114]TT, RD (radial shrinkage) Yuan et al. [1151R Yuan and Ueda [116]TT, H (from Argyris et al. 1985) Siva Prasad and Sankaranarayanan [13]TT, TD Ravichandran et al. [117]NDF Runnemalm and Lin [118]TTr, H, X Sarrazin et al. [119]RD (out-of-plane motion) Gu et al. [120]TT Ravichandran et al. (121]T"", X, NDF Reed et al. [122]H Michaleris and Sun [123]TTi, H Michaleris and DeBiccari [124]TT, RD (bending of plate) Lejeail [125]TT, RD (angular distortion of flange) Troive et al. [126, 127]NDF Oddy et al. [128, 129]TT, X, RD (change in diameter) Dike et al. [29]TT, residual stresses Hong et al. [130, 131]TT', X, NDF, H, RD Stone et al. [132], see also Reed et al. [122],
Roberts et al. [36]TD (camber) Sekhar et al. [1331TT, R Lindgren et al. (1341TT, TD (axial displacements) Ortega et al. [135]TT, residual strain Munier and Lefebvre [136]X Pasquale et al. (137]TT, RD, residual stresses Kassner and Wohlfahrt [3]NDF Taljat et al. [138], Abdel-Tawab and Noor [40]cracks Yang et al. [139, 140]
*TT: Transient temperature; TD: Transient displacement; RD: residual displacement; TS: transientstrain; H: hole-drilling technique; R: relaxation technique; NDF: neutron diffraction measurements; X:X-ray measurements.
in order to verify the computational model before one uses the model to study theeffect of different changes in the design and/or the welding procedure.
INTEGRATION OF WELDING SIMULATIONS WITH DESIGNTOOLS
Many papers about finite element simulation of welding are concerned with theimprovement of the welding process, but the simulations are not used routinelyin design [34]. The results from simulations have only been used in pilot applications;the results have been evaluated and process parameters and other design variableshave been changed (e.g. [35, 36]). However, there are only a few papers where this
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART 3 313
has been integrated in a systematic approach for design. Michaleris et al. [37]performed an optimization of a one-pass weld w.r.t. manufacturing and service lifeaspects. Later [38, 39] this was also done to determine thermal tensioning, whichminimizes residual stresses. The derivation of sensitivities, variations of outputversus input, must be accurate. They are used to reduce the number of analysesin the minimization procedure. The inputs are the chosen design variables, suchas the welding procedure. The output can be residual stresses or deformations,for example. Therefore, simplified models cannot be used to compute thesensitivities. Abdel-Tawab and Noor [40] used fuzzy logic to estimate the influenceof uncertainty in data on the computed residual stresses. Sudnik et al. [41] discussedhow to estimate the different errors in the simulations, that are modeling,parametrical (e.g., uncertainty in material and process data), and numerical errors.
There is a varying need for accuracy during different stages of the design process.lsaksson and Runnemalm [42] developed a systematic approach based on simplifiedsimulations used to create a weld response matrix (WRM) used in the preliminarydesign stage of a welding procedure. The WRM should relate changes in designvariables with changes in variables that are important for the desired performance.Simplified models are practical when a large number of simulations are neededfor parametric studies. The inherent strain method [43, 44] is one way to approxi-mate the effects of the weld. Tsai et al. (1999) used this approach to find the weldingsequence which gives a minimum panel distortion with welding beam stiffeners onone side of the panel. Preliminary design decisions can be complemented by moreaccurate simulations in the final stage of design.
A WELDING SIMULATION REVISITED
The influence of different modeling and simulation aspects will be illustrated in thissection. Unfortunately, it may not always be straightforward to repeat the workof others since the description of some of the modeling aspects is lacking in theirpapers. Usually handbook data for the material properties are used and modified.The latter modification may be of primary importance for a successful analysis,and the inclusion of this information is often overlooked or excluded due to spacelimitations. The finite element model must also be shown and pertinent time-steppingand solution procedures are useful to know. These data are lacking in many papersabout finite element simulation of welding. The paper by Andersson [46] has beenchosen here as an example because it contains enough details to repeat thesimulations and good experimental results. This work is also one of the best ofthe simulations performed during the 1970s. Furthermore, there have beenspeculations about the explanations for some deviations between experimentsand simulations. These are investigated in the current study [47].
The analysis by Andersson was repeated by Oberg and "Kein~inen [48]. Theyobtained results similar to Andersson's, but there is no discussion why there aresome differences between their results and the results of Andersson. The only dif-ference was that they used a different numerical procedure for the stress calculationthan Andersson. The difference in the residual stresses may be due to differences
314 L-E. LINDGREN
in the time stepping or the finite element mesh. Andersson's configuration has alsobeen studied by Chen and Sheng [49, 50] and Sheng and Chen [51]. They usedan interesting combination of a fluid flow model and a solid model. The workpiecehad three zones: the pure solid, the pure fluid, and the solid-fluid mixture. The solidmaterial was described using the Bodner-Partom viscoplastic material model.Results from this paper are included later in this article.
Experimental Setup
Two plate halves, 25 x 500 x 2000 mm, made of a fine-grain steel, were butt-weldedtogether (Figure 4). See Table 2 for the material compositon of the plate andthe filler. The plates were first prepared by making a double V-groove andtack-welding the plates. Thereafter, the plates were stress relieved for one hourat 490°C. Chromel-alumel thermocouples for measuring temperatures weremounted in holes drilled in the plates. The plates were welded together from oneside using submerged-arc welding with three electrodes. No weld was made inthe lower groove. The welding parameters are given in Table 3.
Residual stresses were measured using strain gauge rosettes (model FRA-2-11made by SOKKY Kenkyuio Co.). Measurements were performed near a line orthog-onal to the weld in the middle of the plate (see Figure 4). They were performed on theupper and lower sides of the plates and on both sides of the weld. The latter was donein order to check if symmetry w.r.t. weld was obtained.
Finite Element Model
The cross-section of one of the plate halves (Figure 5) was analyzed assuming gen-eralized plane strain conditions. The finite element model used by Andersson isshown in Figure 6. A corresponding finite element model was used in this study[47] as shown in Figure 7. It consists of 419 four-node elements and 498 nodes.
The thermal and mechanical properties used by Andersson are shown inFigures 8 and 9. The material was described by rate-independent plasticity usingvon Mises yield condition and the associated flow rule. The heat of fusion is260 kJ/kg, and Tlidus is 1480'C and Tliquidus is 1530'C. The heat conductivity isset to 230W/mC above Thiquidus in order to imitate the stirror effect in the moltenpool. The properties used by Andersson have also been used in this study.Furthermore, kinematic hardening and transformation plasticity have been applied.The models used in Lindgren [47] are summarized in Table 4.
Comparison of Simulation and Experiments
Andersson [46] obtained very good agreement between computed and measuredtemperatures. Computed and measured residual stresses from Andersson and thecurrent study [47] are shown in Figures 10 and 11. The agreement is good, but therehave been discussions about the deviation between computed and measured residuallongitudinal stresses on the lower side of the plate (Figure 11). Andersson [46] liststhe assumption of generalized plane strain and neglecting of kinematic hardening
FINITE ELEMENT MODELING AND SIMULATION OF WELDING- PART 3 315
hold-down clamps shing tab
tac wei " -10 mm 25mm
.0
to hole drillingperformednear line
start tab
1, 000 mm •
Figure 4. Welding configuration studied by Andersson [46].
Table 2. Material composition (%)
Location C Mn Mo Si Nb
Base metal 0.13 1.57 - 0.20 0.024Weld metal 0.12 1.43 0.17 0.22 -
Table 3. Welding parameters
I U v Heat InputElectrode No. [A] (V] [mIs] (MJ/m]
I 800 30 0.025 0.9602 900 38 0.025 1.3683 900 44 0.025 1.584
316 L.-E. LINDGREN
--- cross-sectionI- weld line to be analysed
/moving electrode//t time ti t
V4
Figure 5. Welding configuration and analyzed cross-section [461.
ZY
Figure 6. Finite element model used by Andersson [461. The triangular elements have cubic basefunctions. Derivatives of displacements are also used as nodal variables, but they are not continuousacross elements near the weld.
as two possible explanations for this deviation. Goldak et al. [52] attribute this to thenonremoval of plastic strains at high temperatures. However, this was done byAndersson even if this was not stated in his paper. Kinematic hardening has beenreported to affect the residual stresses near the weld [53]. This is not confirmedin the current study using the model called kin. None of the cases studied here(Table 4) can explain the difference between measured and computed residualstresses in Figure 11. The remaining issue to investigate is whether this is an effect
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART 3 317
y
Figure 7. Finite element model used by Lindgren [47J. Quad elements with bilinear shape functions for theinterpolation of the displacement fields are used.
o E H'2 50 250 10
IkJ/keC Wt.*q .-N-You. a, I I40 200
1,5 -ar-Hardeningmodulus- weld
30 M 6
1-e--Hwnming•modulm
- hos e
05 10 502
a 0 0 00 500 10cO 150 0 500 1000 1500
Tenpermtm reC Tnpcmsn IrCI
Figure 8. Thermal and mechanical properties used by Andersson [461.
&b v
0.03 0.5 500
H- l-1 IMPaI _Yicdtimii-v• 0.4 400 weld mea
.a._Yield limit -0,02 bs metal...._,_J0.3o 300
0.2 200
0,01
0.1 100
0 0 00 500 1000 1500 0 500 100 5 I01
Tmpoatu IrCI Tern pemre Ir¢}
Figure 9. Mechanical properties used by Andersson [46].
318 L.-E. LINDGREN
Table 4. Investigated material models by Lindgren 1471
Model
stand Two different materials, one for weld and one for filler. Only temperature-dependent data.BA Same as Andersson which gives different thermal expansion during cooling.hist As BA and also different yield limit during cooling.kin As BA and with kinematic hardenng.trpl Same as hist and with transformation plasticity.
of the simplification of the welding to a two-dimensional problem. Anderssonshowed that the influence on the thermal fields is small. However, the effect ofthe plane strain or plane deformation assumptions will directly influence thedeformation, as noted by Goldak et al. [52].
Chen and Sheng [49, 50] and Sheng and Chen [51] solved the same problem withtheir combined fluid flow and solid model. Their results are compared with those ofAndersson (Figure 12), but they do not show the longitudinal stresses.
FUTURE RESEARCH
The general development of the finite element method increases its efficiency andaccuracy. This will, of course, also contribute to the field of welding simulation.
Residual stresses at upper side700
[MPa] _0 Measured longitudinal stress
500 < Measured transverse stress Andersson46
300
-100
-100 -80 -60 -40 -20 0 20 40 60 80 100Distance from weld [mm]
Figure 10. Computed and measured longitudinal stress at upper side of a butt-welded plate [46.47].
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART 3 319
Residual stresses at lower side700
[MPa] * Measured longitudinal stress
500 C Measured transverse stress Andersson46
BA stand
hist B
300 trpl kin
-100 •0 @0
3 0 0-100
-100 -80 -60 -40 -20 0 20 40 60 80 100
Distance from weld [mm]
Figure I1. Computed and measured longitudinal stress at lower side of a butt-welded plate [46,47].
100
75 o o50
25 0 0(000
25/
-50 j \-- Chen and Sheng
-75 ---- Anderssono Measured
-100 -80 -60 -40 -20 0 20 40 60 80 100Distance from weld [mm I
Figure 12. Computed and measured transverse stress at lower side of a butt-welded plate by Andersson[46] and Sheng [49. 501, and Sheng and Chen [51].
320 L-E. LINDGREN
The use of adaptive finite element codes [17] and parallel computations is a part ofthis development. It is expected that more detailed three-dimensional models willbe possible due to these developments. The increase in computational efficiency willalso pave the way for an increased use of welding simulations in combination withknowledge-based systems and optimization tools, for example, in design. This devel-opment will make welding simulations in "real time" possible. This means that simu-lation of welding does not take a longer time than the real welding. The use ofparallel computation will directly benefit the application of welding simulationsin combination with optimization methods because these use several differentmodels with varying design parameters. The different models can be solved inde-pendently and simultaneously on parallel computers with very little communicationbetween these simulations and the optimizaton software. Performing welding simu-lation of very large structures on a parallel computer will certainly also come aboutin the near future.
This three-part review outlines several developments that have made it possibleto simulate not only laboratory setups but also real enginering applications.However, several of the "Grand Challenges for 2000" given by Goldak [24, 54]are still to be met. No doubt, they will be met and new challenges will be added.The current use of computational welding mechanics in industrial applicationsmentioned in the introduction will be increased and hopefully applied whendeveloping new welding procedures.
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APPENDIX FINITE ELEMENT FORMULATION
This appendix gives an outline of one possible set of finite element formulations thatcan be used in simulation of welding. The coupling between the thermal and mech-anical fields is described in the section "Simulation of Welding as a CoupledProblem" in Part 1. The following formulations can be used in any kind ofthermomechanical coupling given in that section, but it is convenient to use thestaggered approach.
Element Formulation
It is recommended that one use a larger number of low-order elements than fewerhigh-order elements in nonlinear problems. Triangular, three-node, two-dimensional elements and tetrahedral, four-node, three-dimensional elements arenot good for problems involving plasticity [150, 151]. A so-called gradedquadrilateral element [14] is one example of a good choice of element. It is a four-to eight-node element with piecewise-linear shape functions. Thus, an edge withthree nodes can have one element on one side and two elements on the other sidewith interelement compatible fields. This formulation alleviates the creation of agraded mesh. The element integrals (e.g., element matrices) are solved by the stan-dard Gauss quadrature. The four-node element is identical to the standard four-nodeelement formulation with 2*2 gausspoints. The element integrals are evaluated intwo or four subdomains if midside nodes are added to the element. Each subdomainis then integrated using 2*2 gausspoints.
The volumetric part of the deformation is underintegrated in order to avoidlocking due to the incompressible plastic strains. The volumetric strain is constantwithin each subdomain in the numerical integration. The B-bar method [152] is usedto achieve this. The temperature used to generate the thermal loads is also taken as
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART 3 329
constant within each subdomain in order to achieve consistency between the tem-perature variation and the strain variation in the elements.
Thermal Analysis
The procedure for the thermal analysis is shown in Box 1. The finite elementsemidiscretization applied to the heat conduction equation at time
"= t + uXAt, 0 < a < 1, and a finite difference approximation of the rate of tem-perature change gives
(C]0 + At[K]0X)AT} = At{Q}0 - At[K]2{T} (A. )
where At is the increment in time, [K] ' is the heat conductivity matrix evaluated atthe temperature at time tn +", [C]' is the heat capacity matrix, {AT} is the currentestimate of the increment in temperature, and {Q]} is a vector of heat generationfor time t .
The Euler backward method, a = 1, is more stable than the more accuratemidpoint rule, a = 0.5, and is preferred by the author. The effective heat capacityis used in the heat capacity matrix in order to reduce convergence problems whenlatent heats exist [153]. The heat capacity used when computing the heat capacitymatrix is taken as
n u Hn-ICeff = -T n -, if Tn - T n- I # 0 (A.2)
where H is the latent heat.Contributions from convective and radiative boundary conditions and contri-
butions from prescribed temperatures are included in the heat generation vector,{W0 , and the heat conductivity matrix, [K]'. formulas for the matrices and vectorsare given in standard text books about the finite element method (e.g. [153]).The heat generation vector may include mechanically generated heat if this couplingis accounted for. The system of equations is modified so that prescribed temperaturesare computed during the iterative procedure, page 52 in [154]. This makes it possibleto switch a node between an unknown or a prescribed temperature withoutrecomputing the profile of the matrix. Equation (A.I) is a nonlinear system ofequations that is solved by the Newton-Raphson method. The solution procedurefor the thermal analysis is shown in Box 1.
Mechanical Analysis
The stress updating, which is crucial for an accurate numerical solution of thedeformation, is given in Boxes 2 and 3. The overall solution procedure used inthe mechanical analysis is summarized in Box 4.
The principle of virtual work applied at time t"+ ' and integrated in the currentgeometry is the starting point for the finite element discretization. The deformationis assumed to be quasi-static, that is, inertia is ignored. This gives a nonlinear system
330 L-E. LINDGREN
Box 1. Incremental solution with iterative corrections for thermal analysis
1. Initialize analysisSet time step counter n=0.Initialize temperatures {T}0={T}init.
2. Next incrementSet iteration counter i= 1.Initial estimate of temperatures for time tn+1 is set as i{T~n'l = {T~n.
3. Solve system of equationsi[Kjij{AT1 = Rt}
where
i[Kt] = (i[C]a Atoti[K] ) and .{Rj} = At(i{Q}1a-i[K]a{Tln).
4. UpdatingIncrement iteration counter i=i+l.
i+1Ii{T} = {T}n+iI{AT}
5. Check convergenceIf no convergence then go to step 3 elseincrement time step counter n=n+lGo to step 2 if more time steps should be taken.
of equations to solve. The unbalance or residual forces, {R,,}, should become zero inthe solution process
{R..} in - {F}, (A.3)
where {F}r1+,' are the external forces and {F}ri. +1 the internal forces. The internalforces are computed on the element level as
V xit+1 I = [B]n+lr{U}n+l d2 (A.4)
where {a} is the Cauchy stresses and [BJ is the same element matrix used in therelation between the velocity strains, {d}, and nodal velocities, {ii}, written as{d} = [B]+'{fiz}. It comes from the virtual strains and is evaluated at time ,n+I .
An additive decomposition of the elastic and plastic velocity strains is assumed.It can be derived from the multiplicative decomposition of the deformation gradient[155-1571. This leads to a hypoelastic stress-strain relation as increments in stressesare computed from strain increments. This implies that we have assumed the elasticstrains are small [158] if the algorithm should define an elastic material. The stress
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART 3 331
Box 2. Stress updating in reference or unrotated configuration
Step I.Compute midpoint strain increment
{fA} = [B]n+ / 2{A u}The derivative of the shape functions in [1B] is evaluated for the geometry at timetn+M 2 .The matrix is modified in order to obtain constant volumetric strain withineach subdomain for the numerical integration of element matrices and vectors.
Step 2. Compute rotation matricesUse the polar decomposition of the deformation gradient to compute rotationmatrices for geometry at beginning of the time step, midpoint geometry and theend of the time step.
Step 3. Rotate all relevant quantities back to unrotated reference
{OR}n = [VRlnTo,}n
{ERPn = [vR]T {}p
{AER) = vI R]n+l/2T {AE
where[ vR] is the rotation matrix used when the strains and stresses are stored in vector
form.Step 4. Compute stress increment and update in unrotated configuration
See Box 3 for radial return algorithm.Step 5. Rotate forward to geometry at time tn+1
n+ l = in+I n+1{opn+ = [vR In ({UR)}P
updating is a strain-driven procedure, and the midpoint strain increment is asecond-order accurate approximation [159] of the strain increment during the timestep,
AE}= {d} dt & [BJ"n+ 12 {Au} (A.5)
where {As} is the strain increment in an element, {Au) is the increment in nodaldisplacements of the considered element, and [B]f+I/ is the same matrix as inEq. (A.4) but evaluated at time tn +i/2 .
The objective Green-Naghdi stress rate is used in the hypoelastic relation. It isused since it avoids oscillatory behavior for the shear test case [160] that is due
332 L-E. LINDGREN
Box 3. Radial return method for stress updating.
1. Assume thermo-elastic trial stress increment
Compute trial stress {aRt r = {1Rl n + e[Eln + ({AER}- AS.h { })
whereC[E]n+I is Hooke's law in matrix form evaluated at temperature r + ' , A t is the
increment in thermal strain and {I}={ I I I O}T for two-dimensional stress states.
-tr 3 Jr M CCompute effective von Mises trial stress 0 R = P• , t[PM]{s tr
where
{s~ t r = [PD ]{a}tr is the deviatoric trial stress
Compute a trial yield limit YOItr = YOao(T' + I ) + H,(T+ ) pn
whereYO (T +1) is the virgin yield limit and H'(T + ) is the hardening modulus at
temperature Tn+ t , &pn is the accumulated effective plastic strain at time te.
2. Compute increment in effective plastic strain
- tr y ir
= m in( "R - a , 0)
3G(T+ I) + H'(Tn + I)where
G(T + 1) is the shear modulus at temperature To+'.3. Update stress and plastic strains
E pn+ = pn+AEp
{eR}Pn -n {ER)pn+{a}ASP
{OR}n+ = {OR) tr- [E] n+{a}Ap = {OR)tr_2G(Tn+ )(a)Ac p
where3 tr
la} -= _T is the flow direction.OiR
to the hypoelastic formulation, chapter 19 in [161]. It is defined, using tensornotation, as
6=r -- ou+ uco = RTRR T . (A.6)
FINITE ELEMENT MODELING AND SIMULATION OF WELDING. PART 3 333
Box 4. Incremental solution with iterative corrections for mechanical analysis
I. Initialize analysisSet time step counter n=0.
Initialize {U}O={U)ni t
and corresponding strains and stresses.2. Next increment
Set iteration counter i=l.
Initialize estimate of displacements ({U)n+l={U}n
and corresponding initial values for stresses and plastic strains.3. Solve system of equations
i[Kmi{AU} f Rm}
where
a elements[K.] = a.{U}i{Rm} A J BTa:] 1[BdV
iVe
= F n + I ( elements "i = Rm = F)' -{F} ext = fJ 1[j~c{}"+ 'dV)- 1 {F} I
where A denotes an assembling procedure.4. Updating
Increment iteration counter i=i+l.
i{U}n = i-I{} n +I + i I{AU}and update stresses and plastic strains.
5. Check convergenceIf no convergence then go to step 3 elseincrement time step counter n=n+ IGo to step 2 if more time steps should be taken.
where [R] is the local rotation tensor from the polar decompositon of thedeformation gradient, wo = ART, and 0 tR = RTaR is the unrotated Cauchy stresstensor.
The increment in the Green-Naghdi stress rate due to the strain increment issolved in a so-called unrotated configuration. The procedure is shown in Box 2.The radial return method is used to update the stresses, and its implementationis shown in Box 3. The logic in this box is based on rate-independent von Misesplasticity with temperature-dependent properties and the associated flow rule.The projection matrices for stress states with or without the zero normal stress (ZNS)constaint, [PM] and [pD], are from Ramm and Matzenmiller [162]. The radial returnmethod is an efficient and accurate method for stress computation [162-165], and itsextension to cases with ZNS is also given in these references. The ZNS corresponds
334 L.-E. LINDGREN
to plane stress, plate or shell formulations where it is an iterative procedure in step 2in Box 3. The accuracy of the method is especially important because errors intro-duced here cannot be compensated elsewhere in the analysis. Another methodfor stress updating is the so-called Effective-Stress-Function [166]. This is ageneralizaton of the radial return method. There are also other integration schemesavailable. See Simo and Hughes [167] for a thorough discussion of this subject.
The Newton-Raphson method is used in combination with line search [168] forsolving Eq. (A.3). The Newton-Raphson method is a second-order accurate methodto solve a nonlinear system of equations if the matrix [K,,] in Box 4 is a tangentstiffness matrix. It is connecting small changes in displacements with small changesin residual force.
IK,,f]{6U} = {6R} (A.7)
We compute the matrix on the element level as, where the dependency of externalloads on the deformation is ignored,
[k,.]{6u} = {6fl.t - {'f},, {6f},
The time derivative of Eq. (A.4) gives
[K,,,]{cIu} : {of ,.. = dBjr{U}+ [BT{6c} + [B]T{o}1 J)dfl (A.8)
The stress change is computed from the Green-Naghdi stress rate (Eq. (A.6)). Thestress terms in Eq. (A.8) and those that are present in the definition of theGreen-Naghdi stress rate are usually ignored, that is, we ignore the initial stressmatrix or geometric stiffness matrix. They are costly to compute and usually donot contribute much to the convergence rate in the solution procedure. TheGreen-Naghdi stress rate is computed from the strain increment, giving
Ik,~~1{c~u} {cBj},11] f([B[I{)} d df1{du}
We thus use only the incremental stiffness matrix
[k,,,] = jIB]TLTEI[B] dO (A.9)
where [E] = is the consistent constitutive matrix [169].L14
W I-,
I IN
-'t %*0'. l 'l. 'wI Ir
SQ
Plate Forming by Line Heating
Henrik Bisgaard Clausen
DEPARTMENT OF NAVAL ARCHITECTURE AND OFFSHORE ENGINEERINGTECHNICAL UNIVERSITY OF DENMARK - KGS. LYNGBY
APRIL 2000
Department of Naval Architecture and Offshore EngineeringTechnical University of Denmark
Studentertorvet, Building 101E, DK-2800 Kgs. Lyngby, DenmarkPhone +45 4525 1360, Telefax +45 4588 4325
e-mail [email protected], Internet http://www.ish.dtu.dkl
Published in Denmark byDepartment of Naval Architecture and Offshore Engineering
Technical University of Denmark
( H. B. Clausen 2000All rights reserved
Publication Reference Data
Clausen, H. B.Plate Forming by Line Heating.PhD Thesis.Department of Naval Architecture and Offshore Engineering,Technical University of Denmark, April, 2000.ISBN 87-89502-29-9
Preface
This thesis is submitted as a partial fulfillment of the requirements for the Danish PhDdegree. The study has been carried out at the Department of Naval Architecture and OffshoreEngineering, the Technical University of Denmark, in the period from February 1997 to April2000. Associate Professor Jan Baatrup and Professor Jorgen Juncher Jensen supervised theproject.
The study is supported financially by the Technical University of Denmark, for which I amvery grateful.
During this period of three years, I have been in contact with many people who have influ-enced my thoughts and work in many ways. Thanks to everyone! Especially Jan Baatrup,Jorgen Juncher Jensen, and Lars Fuglsang Andersen for educational and invaluable discus-sions.
Two periods of 1 and 6 months, respectively, in 1998 and 1999 were spent at the Collegeof Engineering, Seoul National University, where Professor Jong Gye Shin was so kind toinvite me and my family to stay in Seoul. These trips were partly supported by the DanishSociety for Naval Architecture and Marine Engineering (Skibsteknisk Selskab), which isgreatly acknowledged. In Seoul, I had the opportunity to work with Professor Shin and hisstudents in a laboratory concentrating specifically on aspects of automated ship production-including plate forming by line heating. Special thanks to Professor Shin and all his studentsfor their heartwarming hospitality, which made the stay a delightful experience.
Thanks also to my wife and son for their support and patience and for taking the chance oftravelling to South Korea with me.
Henrik Bisgaard ClausenApril, 2000
Executive Summary
The purpose of the present work is to examine and explain the mechanisms of the formingprocess called 'line heating' and to develop numerical tools for efficient calculation andprediction of its behaviour. The forming process consists of heating at a (steel) plate ina predetermined pattern of lines by means of e.g. a gas torch so that the plate assumes acertain, curved shape. Thus, the method is an alternative or supplement to other formingmethods such as pressing and rolling.
Today, few skilful shipwrights are capable of performing the art of line heating, as theamount of heating and the position of the lines are entirely based on experience. As thisknowledge is difficult to categorise it takes several years to learn the method, and hence theproduction method is a bottleneck impeding rapid increase of capacity. A rational methodfor the determination of heating line patterns and heating amount would be very beneficial.It can help the shipwright to determine the most efficient heating parameters, it may allowless skilled craftsmen to carry out the heat treatment, and in the long term it may lead toautomation of the process.
At the IHI shipyard in Kure (near Hiroshima in Japan) a system which automatically calcu-lates the heating parameters and carries out the heat treatment on moderately curved plateshas been developed. The only human intervention takes place when the plate is turned overby a crane. The NKK shipyard (also in Japan) has a robot too with a gas torch mounted, butas it needs off-line teaching by a shipwright it is most suitable for production of series of iden-tical curved sections. Much research is carried out in industry and at universities in the USAand South Korea to achieve automation, as the potential economic benefit is obvious. Thesavings are not directly connected with the production of the curved plates themselves, butrather with the subsequent production stages where even the smallest deviation in curvatureand cutting of the plates involves expensive corrections.
This thesis focuses on the relation between heating parameters and resulting deformations,and further a method for predicting the position of the heating lines is implemented.
The following items are dealt with.
a A finite element model is built to investigate how the resulting deformations dependon varying heating parameters. The model includes effects of temperature-dependent
111
iv Executive Swummary
mechanical and thermal properties, among these plasticity, conduction and heat loss byboth convection and radiation. A convergence test has shown that six linear elementsthrough the plate thickness and 18 perpendicular to the heating path are sufficient tomodel the process. Outside the heated zone, a much coarser mesh can be used.
* The validity of the numerical model is verified by experiments. Good agreement be-tween the simulated and the measured deformations is found.
" A method for experimental determination of the heat flux distribution from a gas torchis developed. The temperature distribution caused by a stationary torch is measuredon the bottom side of a plate, after which a finite element optimisation procedure triesto find the same temperature distribution by varying the heat flux on this numericalmodel.
" A parameter study consisting of 27 variations of the maximum temperature, the platethickness and the torch velocity is made. Hence, relations between heating parametersand deformations are obtained.
* A method for linearisation of the plastic strains from the heated zone is developed sothat they are described only by their contribution to shrinkage and bending, alongand perpendicularly to the heating path. Thus, the strain field is described by fourparameters only.
" The linearised plastic strains can be applied to an elastic analysis, which can reproducethe results of the fully elastoplastic analysis with good accuracy. The advantage is thatalready known strains (from e.g. a database as described above) can be applied to aplate of any shape, and that the deformations are computed very fast. While it takeseight to ten hours to analyse a plate heated over 20 cm, it takes but a few minutes tocarry out the elastic analysis for a plate of any configuration.
" The effect of heating at the edge of a plate is investigated. It is demonstrated that ifthe plate is heated from the edge towards the centre of the plate, the strains are almostunaffected by the edge. Thus, the simplified elastic analysis can be used to model thisparticular way of heating as well.
" Two methods-a general but rather cumbersome and a less general but simpler-are developed to interpolate in the database of relations mentioned above. Thus,a continuous function for the relation between heating parameters and deformationmeasures exists.
" A sensitivity analysis is carried out to investigate which parameters other than thosefrom the study above may influence the results. It shows that material properties andheat source modelling are important factors, which must be known before simulationsare carried out.
To find the relation between heating parameters and deflections the following approach canbe used:
Executive Summary v
1. Determine the material properties for the type of steel to which the relations shouldcorrespond, as they are determined for each type of material separately.
2. Find the flux distribution from the heat source by the method described in Chapter 6.
3. Select representative ranges of heating parameters. It is recommended to use thetemperatures 500, 600 and 700 'C for steel. However, the plate thicknesses shouldcover the range needed in the production and the torch velocities should match thepower of the available torches.
4. Divide the parameter ranges into 3x3x3 points (a total of 27 combinations) and carryout the numerical analyses for the 20 'outer' cases as shown in Figure 4.6, p. 54. Thus,the relations between heating parameters and deformations are determined for thegiven combination of steel and heat source.
5. The plastic strain is linearised by the method described in Section 3.5.
6. The linearised strains can be interpolated by Eq. (4.16) to yield a continuous distribu-
tion of the strains as functions of the heating parameters.
Thus, the relations are determined. However, the heating paths, which will produce acertain shape, remain to be determined and this can be accomplished by the rational methodproposed in Chapter 5:
7. By forcing the plate into shape by an elastic finite element analysis, the heating direc-tion can be chosen perpendicularly to the most negative principal strain direction atany point. This also provides information on how to cut the plate before forming, as agood approximation to the actual deflection of the edge is computed by the analysis.
8. Finally, the heating line positions must be converted into information on how mucheach line should be heated. This is obtained by turning the heating lines into softzones, which will absorb most of the deformations when the plate is forced into shapeby another elastic analysis. The obtained deformations can be compared to the resultsof the case study in point 6 and thus the corresponding heating parameters can befound.
The method described in 7 and 8 is not fully developed, but it provides a rational meansof determining the heating paths in a reproducible manner, which can be adjusted to theactual behaviour of the process.
vi Executive Summary
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Synopsis
FormAlet med naervwrende projekt er at undersoge og forklare mekanismerne bag den form-givningsmetode, der pA engelsk kaldes 'line heating', og at udvikle numeriske vMrktojer tileffektiv beregning og forudsigelse af dens opfcrsel. Formgivningen foregAr ved at opvaxmeen (stA1)plade efter forudbestemte linier med for eksempel en gasbroender, hvorved pladenantager en givet, krum form. Metoden er dermed et alternativ til andre metoder sAsompresning og vaIsning.
Det er i dag kun fA, dygtige skibsbyggere, som kender kunsten at formgive med metoden, davarmemrngden og placeringen af varmelinierne alene baseres pA erfaring. Da denne videner svar at kategorisere, opleves det pi vrerfter, hvor metoden allerede er i brug, at detkan tage flere Ar at oplare folk til metoden. Dermed er den en fiaskehals, der forhindrerhurtigt forogelse produktionen. En rationel metode til at bestemme varmeliniernes positionog varmemengde kan stotte skibsbyggeren i valget af varmeparametre, den kan betyde atmindre erfarne er i stand til at forestA formgivningen, og endeligt kan den pA lengere sigtfore til automation af processen.
PA IHIs vaerft i Kure (ner Hiroshima i Japan) findes allerede i dag et system, der automatiskberegner varmeparametrene og udforer behandlingen pi moderat krunmme plader ved hjwlpaf induktionsvarme - den eneste menneskelige indgriben foreg~r, nAr pladen skal vendes medkran. NKK (ligeledes i Japan) har ogsA indfort en robot med en gasbrander monteret,men den skal oplhres fra gang til gang og er siledes bedst egnet til serieproduktion. IUSA og Sydkorea forskes der i fiere universitwre og industrielle sammenhlnge ogsA intenstpA at automatisere, da man har indset, at der er store besparelser i sigte. Besparelsernekommer ikke direkte i forbindelse med produktionen af selve de krumme plader, men i allede efterfolgende produktionstrin, hvor selv den mindste fejI i pladernes krumhed og tilskwringkraver omkostningsfulde tilretninger.
Denne afhandling fokuserer pA at finde sammenhaengen mellem varme og deformationer, ogderudover er ogsA en metode til at forudsige liniernes position implementeret.
Folgende undersogelser er foretaget:
* En finite element model er opbygget til at undersoge hvordan de blivende deformationeraflhunger af wndringer i varmeparametrene. Modellen inkluderer sAvel temperatur-aflhngige mekaniske egenskaber som varmeledningsegenskaber, herunder plasticitet,
vii
viii Synopsis
varmeledning og varmetab ved bide udstrAling og konvektion. En konvergenstest harvist, at man med tilstrwkkelig nojagtighed kan modellere processen med seks lineaereelementer gennem tykkelsen af pladen og 18 elementer pA tvwrs af den deformeredezone. Uden for den opvarmede zone kan et betydeligt grovere net benyttes.
" Den numeriske model er verificeret ved hjaelp af eksperimenter. Det viser sig, at dennumeriske model giver resultater, der stemmer godt overens med de eksperimenteltbestemte deformationer.
* En metode til eksperimentelt at bestenmme fluxfordelingen mellem en varmekilde ogen plade er udviklet. Temperaturfordelingen somn folge af en stillestfende varmekildeer mAlt ph undersiden af en piade, hvorpA en optimeringsrutine finder den sammetemperaturfordeling i en numerisk model ved at variere fordelingen af varmekildensflux.
" Der er udfort et parameterstudie over 27 variationer af fremforingshastighed, maksimum-temperatur og pladetykkelse. Dette er giort for at opbygge en database over sammen-hangen mellem varmeparametre og deformationer.
* En metode til at linearisere de plastiske tojninger, sA de er beskrevet udelukkende vedderes bidrag til hhv. krympning og bojning pA langs og ph tvaers af varmeliniens retninger fundet. Hele det plastiske tojningsfelt horende til den varmebehandlede zone kansAledes beskrives med kun fire parametre.
" De lineariserede plastiske tojninger kan pifores en simpel elastisk model, som med godpraecision kan gengive resultaterne fra den fulde analyse. Fordelen er, at hvis man pAforhAnd kender de lokale deformationer (fra feks. en database som naevnt ovenfor),er det muligt meget hurtigt at fA en beregning af en vilkArlig plades deformation.Beregningerne er meget korte med denne metode - det tager Ra minutter at beregneen plade af vilkArlig storrelse, hvor det tager 8-10 timer at beregne en plade opvarmetover en 20 cms lwngde med den fulde elastoplastiske model.
" Randeffekter som folge af opvarmning i nwrheden af en kant er undersogt. Det visersig, at hvis man varmer fra kanten og ind imod centrum af pladen, er de plastisketojninger noesten, sorn hvis varmelinion havde ligget langt fra kanten. Dermed kanden simplificerede elastiske metode fra forrige punkt ogsA bruges til beregninger afopvarrmning i nwrheden af en kant.
* To metoder - en lidt omstrndelig men meget generel og en simpel men mindre generel- er udviklet til at interpolere i de lineariserede data. Dermed kan man forudsige deplastiske tojninger for varmeparametre, der ikke allerede foreligger beregninger for.
* En folsomhedsanalyse er gennemfort for at undersoge hvilke parametre, som ikke ermedtaget i ovennavnte parameterstudie, er vigtige at kendte nojagtigt. Konklusionener, at resultaterne (tojningerne) afhanger meget af materialetype og af den anvendtevarmekildes fluxfordeling.
Synopsis ix
Hvis man vil opbygge en database over sammenhzngen mellem varmeparametre og defor-mationer, kan man benytte sig af folgende fremgangsmAde:
1. Bestem materialedata horende til den type stil, som datasmttet skal here til - idetparameterstudierne foretages for et bestemt set materialedata ad gangen.
2. Bestern varmefordelingen fra kilden eksperimentelt ved hjcelp af metoden beskrevet ikapitel 6.
3. Find et reprmsentativt omride at variere varneparametrene indenfor. Det anbefalesat benytte temperaturomrAdet fra 500 til 700 'C for stAl, hvorimod pladetykkelserneskal tilpasses behovet og fremfaringshastighederne skal vwre af en storrelse, sA de tilr~dighed verende varmekilder kan levere den nadvendige effekt.
4. Inddel omridet i 3x3x3 punkter (i alt 27 kombinationer) og gennemfor numeriskeberegninger for de 20 'ekstreme' tilfmlde som angivet i figur 4.6, side 54. Dermed errelationerne mellem parametre og deformationer bestemt for en givet type materialeog varmekilde.
5. Den plastiske tojning lineariseres ved hjelp af metoden beskrevet i afsnit 3.5.
6. Resultaterne kan interpoleres ved hjaelp af lign. (4.16). Dermed har man en kontinuertfordeling af deformationerne som funktion af varmeparametrene.
Dermed er grundlaget for at oms.ette kraavede deformationer til information om den dertilhorende varmewnengde skabt. Endnu mangler en metode til at forudsige, hvor der skalvarmes, men et forslag til en rationel metode er givet i kapitel 5:
7. Ved at tvinge pladen i form med en rent elastisk finite element analyse, kan man ialle punkter pA pladen beregne retningen af varmelinierne som vrrende vinkelret pAden mest negative hovedtojningsretning. Samtidig vii denne metode angive, hvordanpladen skal skmeres ud inden formgivningen, da man f~r et godt bud p& kanternesdeformation.
8. Endelig skal varmemengden for hver varmelinie findes. Hver af linierne omdannes tilblade zoner, som vii optage mest deformation, nAr pladen tvinges i form med en nyelastisk finite element analyse. Disse deformationer sammenlignes med relationerne ipunkt 6 for at finde de tilhorende varmeparametre.
Metoden i punkt 7 og 8 er ikke f-erdigudviklet, men da den giver reproduc6rbare resultaterer den et godt udgangspunkt for en rationel metode, der kan kalibreres med den observeredeopforsel af metoden.
Contents
Preface i
Executive Summary iii
Synopsis (in Danish) vii
Contents xi
Symbols xv
1 Introduction 1
2 Literature Review 3
2.1 Production of Compound Curved Plates . ................... . 3
2.2 Types of Heat Sources .............................. 4
2.3 Finding the Heating Paths .................................... 5
2.4 Temperature Field Analyses ................................... 6
2.5 Structural Analysis ........................................ 7
2.6 Implementation of Manual Line Heating ....... .................... 7
3 Numerical Modelling 9
3.1 Modelling ........ ..................................... 9
3.1.1 Material Properties ................................... 10
xi
xii Contents
3.1.2 Heat Source .. .. .. .. .... .. .... .... .... .... .... .... .... 12
3.1.3 Conduction, Convection and Radiation. .. .. .. .. .... .... ...... 12
3.2 Meshing. .. .. .. .... .. .... .... .... .... .... .... .... .... ...... 15
3.3 Convergence Analysis. .. .. .. .. .. .... .... .... .... .... .... ...... 16
3.4 Numerical Test Oases. .. .. .. .. .. .... .... .... .... .... .... ...... 19
3.5 Linearisation of Plastic Strains .. .. .. .. .. .... .... .... .... .. ...... 21
3.6 Simplified Elastic Analyses. .. .. .. .. .. .... .... .... .... .... ...... 24
3.7 Sensitivity Analysis .. .. .. .. .. .... .... .... .... .... .... .... .... 25
3.8 Analyses of Test Case Results. .. .. .. .. .... .... .... .... .... ...... 30
3.9 Edge Effects. .. .. .. .. .... .... .... .... .. .... .... .... .... .... 41
4 Empirical Relations 47
4.1 Establishing Dimensionless Parameters .. .. .. .. .... .... .... .... .... 47
4.2 Fitting of Functions .. .. .. .. .. .... .... .... .... .. .... .... ...... 51
4.2.1 Multivariate Analysis. .. .. .. .. .. .... .... .... .... .... .... 51
4.2.2 Interpolation .. .. .. .. .... .... .... .... .... .... .... ...... 53
4.3 Prediction and Comments. .. .. .. .. .... .... .... .... .... .... .... 57
5 Heating Line Generation 59
5.1 Determining Heating Paths .. .. .. .... .... .. .... .... .... .... .... 59
5.1.1 Bending Paths .. .. .. .... .... .. .... .... .... .... .... .... 60
5.1.2 Shrinkage Path .. .. .. .. .... .... .... .... .... .... .... .... 61
5.2 Future Work. .. .. .. .. .... .... .. .... .... .... .... .... .... .... 63
5.2.1 Amount of Heating. .. .. .. .. .... .... .... .... .... .... .... 63
5.2.2 Final Check by ATFA. .. .. .. .... .... .... .... .... .... .... 64
Contents xiii
6 Experiments 65
6.1 Calibration of Termocouples .................................. 65
6.2 Heat Flux from Gas Torch ................................... 68
6.2.1 Experimental Set-up ....... ........................... 68
6.2.2 Temperature to Heat Flux Conversion ...................... 68
6.3 Validation of Structural Analysis ....... ........................ 72
6.3.1 Experimental Set-up ....... ........................... 72
6.3.2 Numerical Model and Comparison .......................... 74
7 Conclusions 77
References 79
A Differential Equations for Plastic v. K~rm~n Plates 85
A.1 Equilibrium Equations ........ .............................. 86
A.2 Strains ......... ....................................... 88
A.3 Constitutive Relations ........ .............................. 90
B Numerical Results 97
C Results of ATFA 131
D A Resistance Welding Apparatus 137
List of PhD Theses Available from the Department 139
xiv Contents
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Symbols
Roman Symbols
A Area
B•, x- and y-components of the linearised bending contribution of the plastic fieldcP Specific heatdi Dimensions in dimension analysis
E, F, G Coefficients of the first fundamental formE Young's modulus
Et Tangent moduluseT Error fraction of thermocouple No. jg GravityH Enthalpy
h Plate thicknessh Natural coordinate version of h. h E [-1; 1]hf Convective film coefficient
ij Indices
k Total number of possible interchanges between A and BL Characteristic lengthL, M, N Coefficients of the second fundamental formN-U-L Nusselt's number
Nd Number of dimensionsNp Number of dimensionless parameters
N, Number of independent dimensional setsNv Number of variables
N•,j Number of zeros in row i of CQ"(r) Heat flux from heating torch as a function of r
Qc Convective heat loss
xv
xvi Symbols
Q, Peak value of the Gaussian distributed heat flux
QR Radiative heat loss
Qwt Total heat flux from torch to plateRaL -- 9a..-TD)L Rayleigh's numberr Distance from centre of heat sourcerum Radius of the gas torch (Q"(rtrCh) = O.O1Q4.:)S-,V x- and y-components of the linearised shrinkage contribution of the plastic fieldsjj = a= j - 6 og'kk Deviatoric stress tensor
T Temperature
T Average temperature
t Natural coordinate version of T. T c [-1; 1]
TB Bulk (air) temperature
T, Plate surface temperature
t Time
U. Number of interchanges between A and B that yields equivalent sets of dimen-sionless parameters
UjR Number of prohibited interchanges between A and B due to singularity of A
Vi Variables in dimension analysis
v Velocity
V Natural coordinate version of v. f; E [-1; 1JWh = 9a- Heat per unit lengthV
w p Width of the plastic zone
x Coordinate. Aligned along the heating path
y Coordinate. Perpendicular to x and in the plane of the plate
z Coordinate. Thickness direction
Greek symbols
a Thermal expansion coefficient (of steel)ca2 Thermal expansion coefficient of air
f Thermal diffusivity
bij Kronecker's delta function
f Radiant exitance
e Strain
Ee Elastic strain
61,2 Principal elastic strainsEP Plastic strain
Symbols xvii
eP Plastic strain incrementeP Average plastic strainy' Width factor of Gaussian heat flux distribution
K1,2 Principal curvature
A Thermal conductivity (of steel)
A0 Thermal conductivity of airv Poisson's ratio. Kinematic viscosity
p Densityo Stress, Stefan-Boltzmann's constantoay Yield stressop Principal strain direction0 Number of duplicate prohibited interchanges between A and B
Matrices and vectorsVectors are written as a, matrices as B
A Square sub-matrix of H
B Sub-matrix of H6 Coefficient matrix for X
C = -D(A-'B) w
D Matrix of independent columns
D Conductivity matrixE Powers to variables to form dimensionless parametersH Dimensional matrixiDimensionless parameters
q Heat flux vector {q., qy, q•}
Powers to dimensions (6 for dimensionless parameters)v Eigenvector for principal curvature
X Multivariate parameters9Result vector for Multivariate Analysis
Operators
JAI Determinant of matrix A
Va Gradient of a = 4x2' Ox3
V.a Divergence o f a =-a + -- +-OX1 8X 2 OX3
xviii Symbols
AT Transpose of AA-' Inverse of AE,. E differentiated with respect to a
Chapter 1
Introduction
Of the many skills gathered by engineers and shipwrights in a shipyard, the ability to formcompound curved plates for the outer hull is a speciality. This production process is indis-pensable, as only the simplest barges are constructed entirely of flat plates.
Whereas a single curved plate is simply produced by rolling, the forming of double-curvedplates requires skilled labour and frequent use of heavy equipment. The double-curved platescan be shaped by applying force by means of pressing (with or without a die), peening, orforming with narrow rollers. Although these are proven techniques, there are some problems.In the case of press-forming it is difficult to predict how overbent the plate should be to obtainthe correct shape after 'spring-back', and further peening and narrow roller forming makethe plate slightly thinner in the worked areas.
Line heating is a method of forming double-curved plates by means of local heat treat-ment. It is used in parts of the shipyard industry, and much attention is paid to it in theautomotive industry (ii & Wu, 1998; Magee et aL., 1998) and within sheet metal formingtechnologies (Geiger et at., 1994). Despite its popularity the process is difficult to control,and it is generally regarded as an art performed only by the most skilled shipwrights. Themain problem is to tell in a reproducible way where and how much to heat the plate inorder to obtain a certain shape. However, it has some advantages over the above mentionedmethods, including that line heating makes the plates locally thicker, so that designers donot have to worry about the thickness of the plate with respect to approval by classificationsocieties. Further, production equipment for this method is very cheap, as it may be as sim-ple as an oxyacetylene gas torch. Of the mentioned methods, line heating seems to be bestsuitable for automation-again because of the simplicity of the equipment. A disadvantageof using oxyacetylene torches is the difficult temperature control (i.e. mixture, clearance andvelocity), and material degradation at the surface by diffusion of either excess carbon oroxygen from the combustion product.
When the plate is subjected to local heating, two things happen: The material becomessofter (lower yield limit) and at the same time it expands. The adjacent material still has its
I
2 Chapter 1. Introduction
%SFigure 1.1: Kinematics of line heating.
original strength, why the hot and soft steel will yield and make the plate slightly thicker.Upon cooling the material will regain its strength and the thermal contraction bends/shrinksthe plate. This sequence is illustrated in Figure 1.1.
The overall goal of the current work is to investigate and analyse the mechanics of the processand to find a method for predicting where and how much to heat a plate to obtain a certainshape. The thesis is organised as follows:
Literature review. An overview of past and present methods from literature is given.Chapter 2.
Numerical analysis. To understand the mechanisms and the relation between heating par-ameters and final deflections a numerical model is established, and a set of calculationswith varying heating paranmeters is carried out. Chapter 3.
Simplified elastic analysis. To save computation time a simplified numerical methodwhich can predict the response to applying already known plastic strains (from e.g. adatabase) is derived. Section 3.6.
Empirical relations. To be able to interpolate in an existing set of results (as those fromthe numerical analysis) empirical relations are derived from multivariate analysis ofthe numerical analysis. Fuirther, a simpler but less general method for interpolation issuggested. Chapter 4.
Heating paths. Determination of heating lines and heating parameters can be found ac-cording to the method outlined in Chapter 5.
Experiments. Experiments for validation of the numerical modelling and for assessmentof the heat flux from a gas torch are carried out. Chapter 6.
Chapter 2
Literature Review.
Until today, quite a few researchers have addressed the topic of line heating in the searchfor better control of the process. A range of different methods which help to understandthe mechanics has been developed, among others beam analysis approximations, equivalentforce calculations and three-dimensional finite element analyses.
That the line heating method is efficient and popular is proved by the fact that manyshipyards use it in the production. Those include (to the author's knowledge) Odense-Lind0(Denmark), Astilleros Espafioles (Spain) (de la Bellacasa, 1992; Sarabia & de la Bellacasa,1993), Fincantieri (Italy), Daewoo (South Korea), Mitsubishi, IHI (Ishiyama et al., 1999),and NKK (Kitamura et al., 1996) (all Japan), Todd Pacific Shipyard (Chirillo, 1982), AtlanticMarine Shipyards, NASSCO, and Norfolk Naval Shipyard (all USA).
In the following, a brief review is given-sorted by topic-of the methods described in theliterature.
2.1 Production of Compound Curved Plates
According to Chirillo (1982) there are a number of benefits from using line heating overpowerful plate pressing facilities, including that the process is faster, safer, and allegedlymore accurate. However, the described methods are based on experienced, manual labourand are as such trial-and-error procedures.
Table 2.1 shows the results of a questionnaire (Kitamura et al., 1996) describing the amountof time spent in seven Japanese shipyards shaping double-curved shells. On average it takes6.8 manhours (MH) in the line heating shop (hot bending) per plate to work it into thedesired shape. Prior to that the plate is typically shaped by means of rolling (cold bending),which takes 2.7 hours on average. FRom Table 2.1 it is also seen that the average manhour
3
4 Chapter 2. Literature Review
Average valueNo. of plates I MH/plate
Cold Bending 437 2.66Hot Bending 441 6.83
Total 1 878 4182MH
Table 2.1: Number of plates and manhours per plate according to Kitamura et al. (1996).
consumption is 4200. However, it is not clear whether this is per ship or per year. Anysavings of manhours in these workshops will only constitute a small fraction of the totalmanhour consumption in shipbuilding-considered alone it would not influence the overallcost of a ship. However, all the plates from this workshop must be used at the later stagesof the production, and as all errors are added up from step to step it is crucial to have agood degree of accuracy from the plate-forming shop.
Not only the 'downstream' assembly stages will benefit from improved accuracy of the curvedshells. Also in the 'upstream' production-in the plate cutting facility-proper knowledge ofthe forming process is important. Various mapping techniques emulate the actual formingprocess (Lamb, 1995; Letcher, 1993) so that the cut plates fit one another perfectly at theerection stage. Unfortunately, this does not work very well, as the mappings are typicallybased on differential geometry (which assumes evenly distributed deformation) in contrastto line heating and other forming methods which deform the plates locally. Therefore, bettercontrol of the method improves cutting.
2.2 Types of Heat Sources
The line heating process can be divided into a few categories according to the heat sources:
Gas torch is far the cheapest to buy and maintain. It is somewhat difficult to control asregards repeatability in gas amount (unless flowmeters as described in de la Bellacasa(1992) are used) and as regards keeping a constant distance to the plate.
High-frequency induction heating allows for control of the heat penetration, as it de-pends on the frequency of the induced electrical field. It is unsuitable for heating at theedges of a plate as overheating is almost inevitable. Further the equipment is ratherheavy, so it cannot replace the gas torch in manual line heating.
Laser beam is the most well defined heat source although it is very expensive. Laser isvery well suited for automation and in combination with a protection gas it reducesthe risk of oxydation of the surface.
Welding arc is a possibility but due to the low penetration of the heat, the surface is proneto melt with material degradation as a result (Kitamura et al., 1996).
2.3 Finding the Heating Paths 5
In principle, all the heat sources can be automated by mounting them on a gantry crane,which makes it possible to treat large plates.
2.3 Finding the Heating Paths
An imminent problem is to determine the locations of the heating lines. At least four differentmethods are described in the literature:
Connection of points of extreme curvature on the deflection difference surface.One method (Jang & Moon, 1998) compares the target surface with the current (ini-tially fiat) surface. The objective is to make the difference surface (target minus cur-rent) as flat as possible. This is achieved by finding the points of maximum curvatureon the difference surface and then grouping them into simulated heating lines. Byiteration, the difference surface is gradually becoming flat.
Use of elastic analysis and check with 'similarity' measure. An algorithm is devel-oped by Lee (1996), which fits heating line candidates through points of similar cur-vature on the surface. These are subjected to bending moments which emulate theheating, and the obtained shape is compared to the target shape by a square rootnorm of the differences between the points on the two surfaces. Then an optimisa-tion procedure varies the bending moments to achieve a better fit. The final bendingmoments translate into heating parameters by comparison to a database.
Use of principal strain directions from numerical elastic analyses. In Ueda et al.(1994a), the following procedure is formulated:
" Compute the strain caused by deformation from the initial configuration to thefinal one by using elastic FE analysis.
* Decompose the computed strain into in-plane and bending components and dis-play the distribution of their principal values on a graphic display.
" FRom the distribution of the in-plane strain, the region where the magnitude ofcompressive principal strain is large is chosen to be the heating zone, and theheating direction is assumed to be normal to the direction of the principal strainwith the maximum absolute value.
" Rlom the distribution of the bending strain, the region where the absolute valueof the bending strain is large is selected to be an additional heating zone, and theheating direction is assumed to be normal to the direction of the principal strainwith the maximum absolute value.
In other words, the principal bending or principal compressive strain directions inevery point determine the path. In addition, it is described how to find the amount of
6 Chapter 2. Literature Review
deformation necessary in each heating line. By making the elements at the lines foundby the above approach very soft (Young's modulus 1/1000 of normal) the strain willbe concentrated here, as it would be in the real case of line heating.
Normal strain directions determined by differential geometry. Shin & Kim (1997)use a method where differential geometry takes account of the mapping between theinitially fiat plate and the curved plate. Hence, bending and in-plane principal strainsare found. The most crucial part is to select a mapping method which resembles theline heating process as much as possible. Present work at Seoul National Universitycombines the methods of shell expansion and tracing of Nutbourne et al. (1972), Man-ning (1980), and Hinds et al. (1991). More specifically, Manning (1980) describes amethod of isometric trees to develop the surface: A tree with a long trunk with at-tached branches represents the surface, and each of the branches and the trunk aremapped onto the plane by involutes (unrolling). Hinds et al. (1991) refine this methodby replacing the involutes by traces of geodesic curvature'. The geodesic curvature canbe traced onto the plane by the method developed by Nutbourne et al. (1972), whereany line can be traced given a curvature and an arc length.
An initial shape may also be chosen as close to the target shape as possible. Such an initialshape is available by a method developed by Randrup (1997), where the cylinder shapewhich-in some sense-is closest to the double-curved shape is found. This information canbe used to roll the plate first and thus minimise the forming effort required by line heating.
2.4 Temperature Field Analyses
An early description of the total heat flux from a torch is made by Pay (1967). With acalorimeter he measured the power from a range of different burners and fuels.
Concerning the flux distribution, most references to line heating assume a model where thegas torch is Gaussian distributed as Q"(r) = Q e -7r
2 (see also Section 3.1.2). This ideacomes from Rykalin (1960) and is adopted by among others Moshaiov & Latorre (1985),Ueda et al. (1994c) and Shin et al. (1996). Yu et al. (1999) adopt a modified Gaussiandistribution, in which the heat flux outside a core radius is slightly increased to model alaser beam. Others use the solution of Rosenthal (1946) for a point source on an infiniteplate of finite thickness, for example Jang et al. (1997) or Moshaiov & Latorre (1985) in ananalytical solution. Yet others (Tomnita, et al. (1998)) calculate the heat flux by measuringthe velocity profile of the torch flame. Section 6.1 of the present thesis shows how the heatflux is evaluated from temperature measurements during heating. A simulation of heatingby high frequency induction is treated by Ogawa et al. (1994).
'Geodesic curvture is the curvature projection of auny curve, C, onto the tangent plane at points of C.
2.5 Structural Analysis 7
2.5 Structural Analysis
Various analytical and numerical analyses have been carried out to predict the response toheat treatment:
" A solution to a two-dimensional problem is found by Iwamura & Rybicki (1973). Itconsists of a finite difference implementation of a beam perpendicular to the heatingdirection.
" Jang et al. (1997) use a simplification where a circular, axisymmetric disc with springsrepresents the heated region. The spring constants are determined from the theoryof an infinite plate with a hole. The disc is then distributed along the heating pathto simulate the actual heating and the resulting strains are converted into equivalentforces by integration.
" A combination of the differential equation for a Kirchhoff plate and a membrane theorycoupled by the constitutive relations (plasticity) is derived by Moshaiov & Vorus (1987).Unfortunately, only a boundary element solution to the plastic Kirchhoff plate alonewas found and solved.
" A theory where a strip perpendicular to the heating line supported by springs representsthe heated plate is developed by Moshaiov & Shin (1991) and Shin & Moshaiov (1991).This is successfully applied to the elastic case, but when plasticity is included it isdifficult to determine the spring constants.
" Three-dimensional finite element modelling with different commercial programs is car-ried out by Lee (1999), Yu et al. (1999), Clausen (1999), Ishiyama et al. (1999), andOgawa et al. (1994). In the three former, the relation between input heat and outputdeflection is also described.
2.6 Implementation of Manual Line Heating
Interesting reports on procedures for implementation of manual line heating are availablethrough the National Shipbuilding Research Program (NSRP). The first published reportfrom Todd Pacific Shipyard (Chirillo, 1982) is a manual on how to perform line heating forforming purposes of plates and stiffeners and distorsion removal due to welding. This workis supplemented by Scully (1987) with an investigation of the possibility of using laser asheat source.
The manual line heating process is transferred to the Spanish shipyard Astilleros Espafiolesin the early 1990s, and a manual is published (de la Bellacasa, 1992) on that occasion. Astatus report is also available (Sarabia & de Ia Bellacasa, 1993).
8 Chapter 2. Literature Review
Of course, it can be avoided to use compound forming techniques, if (parts of) a ship aredesigned by means of developable spline surfaces as described by Chalfrant & Maekawa(1998). This is, however, not very interesting in the case of most ship types-and in thecontext of the current study.
Chapter 3
Numerical Modelling
Numerical modelling is a very important supplement/alternative to experiments, and inthe case of line heating, computerised calculations yield information which is otherwiseunavailable from experiments. For example, the in-plane deflections are barely measurableand no simple method can give information on plastic strains.
As a first approach to numerical modelling, a formulation based on the v. K6rmAn platewith plasticity was established. The intention was to use this in a finite difference analysis(FDA), but unfortunately the idea had to be abandoned due to problems with translatingthe intricate equations into a discrete version to be solved numerically. The derivation ofthe differential equations is, however, included in Appendix A.
Instead, a numerical model has been built by use of the finite element code ANSYS. Belowa thorough description of the model and its subsequent validation is given.
3.1 Modelling
Several assumptions and choices have been made for the formulation of the numerical models.
Kinematics. A small strain and small deflections formulation has been applied. This isjustified by the fact that the deformations induced by line heating are very small.
Constitutive formulations. The temperature field is unaffected by the structural responseso that the thermal and structural problems can be solved in sequence-with theresults of the former as input to the latter. The steel is modelled as isotropic and withplasticity taken as J2 flow theory with strain hardening, see Section 3.1.1. Cooling isimplemented as both conduction, convection and radiation.
9
10 Chapter 3. Numerical Modelling
~40.L Richter-----
>U30
.20 200 400 600 800 1000 1200 1400 10
Temperature [0C1
5 00D Pate!400Rich---------
32000 j200~ 40..................
S 000 0 40 600 800 1000
Temperature it]
Figure 3.1: Comparison of conductivity, A, and enthalpy, H, for mild steel.
Modelling. Linear brick elements are used in a fine mesh near the heated zone to capturethe temperature and stress gradients. The mesh is gradually refined towards a coarsermesh representing the surrounding plate. Only cases where the heat is applied farfrom the edges (on a sufficiently large plate) are examined as this is more general thanincluding edge effects. It is investigated later how heating near the edges affects thedeformations. The plate is considered free of residual stresses, even though fabrication,handling, cutting etc. and will all induce stresses. How severe the residual stresses areis impossible to foresee, so this effect is not included in the present modelling.
A number of cases within these assumptions axe analysed in Section 3.4.
3.1.1 Material Properties
Data on temperature-dependent material properties is generally very hard to obtain. Here,three sources are used, namely Patel (1985), Richter (1973), and Birk-Sorensen (1999), andFigures 3.1 to 3.4 show comparisons of the thermal and structural properties of the materialas a function of temperature, T.
As it is seen all the data except that for the tangent modulus is much alike, so any of itmay adequately describe the material behaviour. The Patel data set is used throughout thisthesis as it seems to be the most complete.
3.1 Modelling 1
n- 240- Pate!
200 Richte-,160 BS_-120 -- --- ----------
E 80S 40 .. ... .... ....
o 0>~ 0 200 400 600 800 1000
Temperature [0C]
300
250 ate"
0200 400 600 -800 -1000
150
100.......................................................---0------
S 0 00 20300 400 50600 70800901000Temperature ([CJ
Figure 3.3: Comparison of taungens modulus, E~, and Poisson' ramti, cr., for mild steel.
12 Chapter 3. Numerical Modelling
1.5e-05- 1.45e-05 Ic ---- .I.4e-05 ...... .. j• . ....... .. s i.c". .........1.35e-05. 1.3e-05 - -
'• 1.2e-05
.15e-050 200 400 600 800 1000
Temperature (0C]
Figure 3.4: Comparison of expansion coefficients, a, for mild steel.
3.1.2 Heat Source
Proper modelling of the heat source is important, as this is the 'driving force' of the form-ing method. As mentioned in Chapter 2 most researchers in the field employ a Gaussiandistributed, axisymmetric heat flux as a means of modelling the gas torch. This Gaussiandistribution is expressed as
Q"(r) = Q- e - 12 (3.1)
Thus, the pointwise distribution, Q", is expressed by a peak value, Q", a width factor, -,and the distance from the centre, r. The total heat input, Qwe, is calculated as the integralof the distribution:
Qwt = j Q" r dr dO = 1rOQ1, (3.2)
To make a proper choice of the torch width and to understand the term -Y better, a 'torchradius', rth, is defined as the distance where Q" is 1% of Q". Thus,
Q efe- irt oh = O.01Q •,
~(3.3)
In 100
In the following calculations, rit. is fixed at 4 cm unless otherwise stated, which meansthat -y is 2878 m- 2 .
The true distribution from a gas torch can be determined experimentally as described inSection 6.2 p. 68.
3.1.3 Conduction, Convection and Radiation
Heat conduction is governed by the first law of thermodynamics, which states that energy isconserved. Written in the form of a differential control volume formulation, neglecting heat
3.1 Modelling 13
transfer caused by mass transport and heat generation, it becomes (Kohnke, 1998)
ffTPC, -+ V 0 (3.4)
where p is density, c. is specific heat, T is temperature, t is time, V. is the divergenceoperator, and q is the heat flux vector. x is along the heating path, y is perpendicular to itand z is in the thickness direction.
Heat flux and temperature gradients are related by Fourier's law:
q--DVT (3.5)
. 0 is the conductivity matrix, as A denotes thermal conductivity, and V00 A.J
is the gradient operator. Eqs. (3.5) and (3.4) combine in the well-known form:
OT a 8 T a OT a (3.T6PCp5 = w \azw + ay AY 8) Oz (3.6)
which is the governing equation for the thermal problem.
The convection properties of the surrounding air are found from the description of freeconvection from a horizontal plate in Incropera & Dewitt (1993, pp. 460ff). Although thistheory is derived for an infinite plate with uniform temperature distribution, it is assumedthat it may describe locally heated regions as well. As shown below, the convection on theupper side does not depend on a characteristic length on the plate.
The basic assumption used in the ANSYS is that heat, Q, convected from a surface may bedescribed in the general case as Newton's postulate
Q,= hj(T. - TB)dA (3.7)
where h1 is a film coefficient which may vary with temperature, T, is the surface temperature,and TB is the bulk air temperature.
The film coefficient is given by the mean Nusselt number, NUL, relating to the Rayleighnumber, RaL, as
"NuL = 0.15Ra1a' (3.8)
Here NUL = h1 L/A0 is a non-dimensional measure of the heat convection from the surface.L is a characteristic length describing the system and A. is the thermal conductivity of air.RaL = ga.(T. -T)L
3 is a measure of the instability of the air given the physical constants anda temperature difference. g is gravity, a. is the thermal expansion coefficient of air, v is thekinematic viscosity, and f) is the thermal diffusivity.
14 Chapter 3. Numerical Modelling
5T[K] 29315 300 325 350 400 500 600 700 800 900 1000OT[0C] 20 27 52 77 127 227 327 427 527 627 727
hL [W/m K] 0 3.40 5.32 6.48 7.27 7.76 7.76 7.63 7.43 7.27 7.10
87 ......, .....................
5 " ...... ......-- ..... .. ... ...... ...... ...... ....E
... .- .. ... .
4 ..... .......i....... ......... ........ ........... ................ ..........
00 100 200 300 400 500 600 700 800
Temperature, T, [°C]
Figure 3.5: Film coefficient based on average of surface and bulk temperature, TB = 20 'C.
Solving the above equations for h1 yieldsAý go. 1{ g '(T.- TB)L 3 "
hf 0.15 •
- (015 (otT.- TB) 3 (3.9)
Hence, the convection is not influenced by the characteristic length, L, of the plate.
By use of Table A.4 'Thermophysical properties of gases at atmospheric pressure' in Incropera& Dewitt (1993) and the relation a. - -' (PA) - -- (- ), it is possible to establish the
p OT p Atable above of film coefficient versus temperature. Similarly, for the lower side of the platea relation between the Nusselt number and the Rayleigh number exists:
NUL = 0.27RaL/4 (3.10)
It is evident that the convection on the lower side does depend on a characteristic length incontrast to the upper side. However, for simplicity convection on the lower side is assumedto be equal to that of the upper side.
The heat radiated from the plate is modelled by Stefan-Boltzmann's law for radiation:
QR = j ea(T4 - TA)dA (3.11)
where A is an area, e is the radiant emissivity, and a is the Stefan-Boltzmann constant.
Here the only variable other than temperature is the emissivity, e. As suggested in Yaglaet al. (1995) the emissivity is approximately 0.5.
Precise modelling of the surface heat loss is not very crucial to the modelling, as the heatloss is quantitatively much smaller than the conduction within the steel. This is verified
3.2 Meshing 15
0.-
--8
- ----
4 4. . 1-.. . . . .. . . . . .. . .. ... ..... .... . . .. ... . ]... ... . . .. . . . . . . . . . . . ... . . .
-0 .4 .. ...... ..... . ...... ............ . ... ........................... .. ................... . . . . . . . .. ... -........ . . . . . . . . . . . . .
4 8. .. .... .. .. . .... .. .. .................... .. ..... ...... i.. ... ..................... . ......... .. .... ..
-0.9 I I _______
0 0.05 0.1 0.15 02Distmnce la]
Figure 3.6: Ratio of z- and x-direction heat flux along the heating path. The 'distance' ismeasured from the centre of the torch.
(A) (B)
Figure 3.7: Transition in the corner of the inner mesh. (A) Four elements through thethickness. (B) Six elements through the thickness.
by Figure 3.6, showing the ratio of conduction (x-direction flux) and surface heat loss (z-direction heat flux) on the heated side, along a line in the centre of the heating path (y = 0).The distance on the graph is measured from the centre of the torch. Over a large part ofthis line of evaluation, the ratio is negative, which means that the flux is directed away fromthe surface. Not until far away from the torch (as the temperature becomes more evenlydistributed), the convection becomes larger than the z-direction conduction, as can be seenfrom the small positive values of the ratio. Thus, the heat loss from the surface of the platewill not induce significant shrinkage of the outer layers.
3.2 Meshing
A mesh with as few nodes and elements as possible is built to save computation time. Thisis done by using small elements in a region around the heated zone with a transition into acoarser mesh by using only elements with six sides (not tetrahedra or wedges). The transitioncan be done as shown in Figure 3.7 where four or six elements in the thickness direction aretransformed into only one element. A sample mesh is given in Figure 3.8.
16 Chapter 3. Numerical Modelling
Figure 3.8: Sample mesh.
Generally, it should be avoided to use a single brick element without rotational degrees offreedom to model plate bending, as shear locking-the inability of the elements to representthe true response to bending moments-is inevitable. However, the linear eight-noded ele-ment is implemented in ANSYS with 'extra shapes' from Taylor et al. (1976) which reducesthis problem. With additional internal degrees of freedom which represent bending, "it mod-els pure bending exactly regardless of element aspect ratio" (Cook et al., 1989) provided thatthe element is rectangular in the bending plane. Using internal degrees of freedom makesthe element incompatible, but when the mesh is refined this will not be a problem. Further,the validity of the results calculated with this element is verified by the agreement betweennumerical results and experiments, see Chapter 6.
3.3 Convergence Analysis
It is investigated how many elements through the thickness are required to achieve a con-verged solution and whether linear or parabolic elements are most time-efficient. This is doneby comparing different configurations to a 'reference' analysis with six parabolic elementsin the thickness direction. To ensure this is actually a converged solution, a comparison isalso made with four parabolic elements in the thickness direction. All analyses are carriedout on a 10 mm plate with a target maximum temperature on the heated side of 550 0G,and a velocity of 5 mm/s. Figures 3.9 to 3.12 show plastic strain distributions along andperpendicular to the heating path, on the upper and lower side of the plate. The legendis: xyznA.Jr, xyz = component of plastic strain, ni = number of elements, type = linearor parabolic elements. The results from the 'reference' case are drawn on the graph by athicker line. The resemblance of the results indicates that the solutions in fact converge.
It is seen that four linear elements are not enough to obtain convergence (see Figures 3.9and 3.11). If six linear elements are used instead, the results match those from the (more
3.3 Convergence Analysis 17
0.0035
0.00 35 ..............t....... .
-0.0025
-0.002 ._ _ _ ......... . .......
0.01 .0 00 0.03 0...4.0.05.0.06
.0 z ..i ........ ............
-0005 0T100 .3 00 .5 00
X0.0015 -- -- --- ..............
X4/par---- --- Y4/arn*in ** Z4/pf
x4Ain,-. . Y4*¶fl- Z4Ain '- -
Figure 3.10: Comparison of plastic strains perpendicular to the heating path on the heatedmside.
18 Chapter 3. Numerical Modelling
0.004
0 .00 3........ .. ...............
-0.3
-0.15 -0.1 -0.05 0 0.05 0.1 0.15x (in]
X/ - Y6/par6p ..
Xt ''-Y4/par Z41p - -----X4i 0 - Y4flin --- A11i -- --
Figure 3.11: Comparison of plastic strains along the heating path on the bottom side.
0.008
-0.0042 ...... ..... ....
-0.004
-0.006I-0.15 -0.1 -0.05 0 0.05 0.1 0.15
x [in]
X6/par -Y6/pf ***Z* 6/pn.
X4/par ...... Y4/Pr Z4/pM - ---
Figure 3.12: Comparison of plastic strains along the heating path on the heated side.
3.4 Numerical Test Cases 19
0.008 T _
0.006 ........
0.004
0 .0 0 2 ........ ..... .... .... ... .... ..... .... . .... .... ........................ ; .......... ..........
-000 , - !.............. ....................... . .... i ................. .. .....00.002
-0.006 _____ ____
0 0.01 0.02 0.03 0.04 0.05 0.06y [in]
12Cmff 12cm--. 6cm -i 2cmnf ....-.-- 12cm ----o-.-.. cml2cm ,, f .... .... 6cm 8cm ........
12cm -- - 6cm - .8cm --------
Figure 3.13: Influence of changing the width of the mesh around the heated zone. Plasticstrains on the heated side, constant number of elements.
precise) parabolic elements. By comparison of the computer time consumption for the cases,it is obvious that linear elements should be used in favour of parabolic ones: 6/lin = 11.8CPU hours', 4/par = 68.9 CPU hours and 6/par = 128.7 CPU hours.
The influence of the width of the fine part of the mesh is examined with linear elements,and the result is shown in Figure 3.13. The legend gives the zone width and whether itis a reference solution (the solution from 6 parabolic elements shown as ref and in a thickline). The graph shows that the finely meshed area should just envelope the plastic zone,i.e. be neither too wide nor too narrow. A width of 8 cm is chosen for all later analyses.Apparently, the chosen 18 elements perpendicular to the heating direction are adequate.
3.4 Numerical Test Cases
The factors controlling the final shape of the plate include plate size, heating line position,torch speed, power of the heat source, plate thickness, and material properties. To formulaterelations from a reasonable number of simulations, the number of unknowns must be reduced.Therefore, focus is on the local strains, as they are more general than the description of the
'All computations are made on an HP Visualize C200 w/ lGB RAM, SPECfp 21.4
20 Chapter 3. Numerical Modelling
10 1 10 1915 r 2 1 20 520 3 12 21
hlrunil 15 5 14 23 10 v jrnm/sj20 6 15 2410 7 16 2515*K 8 17 26 1520 9 18 27
500 600 700 -0
Tma
Table 3.1: Numbering of test programme.
total deflection of the plate and this more or less removes the dependence on plate size.By considering only plates of identical material and neglecting edge effects, the problemis reduced to depending only on physical measures as thickness, h, torch speed, v, andamount of heat, Q. During the forming process, the quantities h and v are easily measured,but how much heat is actually absorbed by the plate is difficult to say. Meanwhile, themaximum temperature is very important as there must be some proportionality between thetemperature and the local deflection. The surface temperature can be measured and is afunction of the other parameters, Q, v, and h.
For these reasons, only the maximum temperature measured on the heated side, T.. theplate thickness, h, and the torch velocity, v, are considered in the simulation programme.For the simulations the heat input that produces a certain maximum temperature is foundby manual iteration. Once these three parameters have been set, the temperature field wilbe fully governed, as thermal gradients, heat penetration depth, and heat input dependdirectly upon those parameters. Thus, to obtain a certain temperature, the heat input mustbe varied depending on the thickness and the torch velocity. An error of the heat modellingis that the torch width, r~0 mj, is always the same, no matter the heat input. For a gas torchthis is physically impossible, whereas induction heating or a laser beam are more likely tohave this characteristic.
The simulation programme has to cover a representative range of the possible parametervalues, so three thicknesses (10, 15 and 20 mmn), three velocities (5, 10 and 15 ') and threemaximum temperatures (500, 600 and 700 00, reference temperature at 20 00) are chosen.
I M
A B x
Symmetry rA
Figure 3.14: Schematic presentation of a symmetrical model.
3.5 Linearisation of Plastic Strains 21
Thus 27 (33) simulations are to be carried out. For details and numbering, see Table 3.1.All simulations are made on a 1 by 1 m plate with symmetry conditions at the centre lineand heat is applied along a 20 cm path in the x-direction in the centre of the plate (frompoint A to B in Figure 3.14). As mentioned earlier, a torch radius, rt•,,, of 4 cm is used. Amodel as shown in Figure 3.8 consists of 5100 elements and 6400 nodes and solves in about500 iterations or eight hours. The results of the 27 test cases and a thorough discussion isgiven in Section 3.8.
Although the line heating process is used and approved by the American Bureau of Shippingup to 900 'C, lower temperatures may be sufficient "... because efficient bending was alreadybeing performed with lower temperatures" (quoted from Chirillo (1982, p. 6)). Embrittle-ment is also more likely to occur at higher temperatures due to the formation of martensite.Further, the constitutive model (bilinear plasticity law) is unable to model temperaturesabove the phase transformation temperature of low carbon steel (723 0C). Among others,the thermal expansion coefficient, the yield limit and Young's modulus differ in heating andcooling near the phase transformation temperature (Ueda et al., 1985). Using temperatureshigher than 700 'C in the numerical test programme is therefore not recommendable, eventhough it may work fine in reality.
3.5 Linearisation of Plastic Strains
To facilitate the analysis of the vast amounts of data from the three-dimensional simulations,some simplifications must be made. The goal is to describe the local deformation field withas few parameters as possible, leading to the following assumptions:
" The only area of interest is the vicinity of the heat affected zone (HAZ) where theplastic strain field is analysed, as shown in Figure 3.15.
" The plastic strain does not vary along the heating path, so effects of starting and stop-ping are not considered. Hence, only a section, labelled D in Figure 3.15, perpendicularto the centrepoint of the heating path (x = 0) is analysed.
D
Figure 3.15: Definition of zone of interest.
22 Chapter 3. Numerical Modelling
Z S x,y Bxy
h
Original . -7•
Figure 3.16: Linearisation of strains.
* It is further assumed that the plastic strain field can be described by four parametersonly, namely shrinkage, S, and bending, B, with contributions to the longitudinal (x)and the transverse (y) directions, respectively. Thus, the strain distribution is assumedto be linear in the thickness (z) direction.
The first step towards simplification of the strains is to find an average strain distribution,P,•(x), which applied uniformly to D, will yield the same result as the original strain field.
The average &,, is found by integration over the width of the plastic zone in every layerof nodes in the z-direction. This yields what may be called a 'plastic displacement' or theamount of deformation caused by all the elements in the zone, due to the plastic strainsalone. Therefore, the average strain contribution, tP, can be found by dividing the 'plasticdisplacement' by the width of the HAZ, wP:
ep(= - [ e EPY(y, z)dy (3.12)ýY(Z =wP Yo
The resulting strain distribution, igP,(z), is then linearised by means of the least squaresmethod as shown in Figure 3.16 to yield the acquired bending and shrinkage contribution,13, and S=,,, to the plastic strain. To make a simplified analysis equivalent to the fullyelasto-plastic analysis, this strain must-of course-be applied to an area of the width, wp.
A BC DE F
G HI J
Figure 3.17: Result representation.
3.5 Linearisation of Plastic Strains 23
1 -.. .. ..... • -• g -.• --.., ....... . ..... .... .. ... ...- .
.. ... r " • - , I - - T T ..... ... .... ..... ......... ...... .... ./ .. ." 1 T .. ..................... ... ....
'• • ' • , ,. i • • .oo ,• I -.. --.-.-. ---...... --....... ......:= ..... . .. . ... .....0 . " . 01...01 aa .O.•L... .. 3 0_.._. . 0 0,3 ..... ..... . W ý 1 ....... 4 0= 0 1 0 1
... 1. 0 -- -- -.o s .....
-a " 55 . ............-
,~~ ~ ": ..... ..... o4W-""15 ' - . .
... .. ..... ........ ..... .. ...... . ........... .... .. .. ... ... ........ ...... . .
-0.03 • 0030 .M0 0.01 0.015 O0M 0.05 0.03 0.035 0.0 -(.15 -al 4).05 0 0.05 at 0.15
o Ix 45 . ..... ..- .....-- .•. .- . . .. -* -00 4 .... .• ........ -} -...... ----------,.W ..... ......................................... ...... 0• ........ ............ •.. ..... ............ i.. ........ ..... '::-..
-a--.W.---5. - - ,-- - .- -.. ..o oo . .. .. . . .. .... ..... . ... . .i .. ........ ..........0.3
0.005 0901 0.015 002 0.M2 0.03 0,035 0,0 4-.15 -al -am. 0 0.0 0.1 Q.15
y, o ....... ... .................. ....:t.............o~~~~~m2~~~~ -. ....... ...... ..... ...... .. ... .....•S
O w '; IT" ------.. .... -" .....- .. .--.. .....--.. .. ..- .......... ..... ... .......... ,...... ....
oR) I MI t' ... .. .. .. ."> - • t.. ... . ..J... • ::::::::::::::: ::::::::::::::: :::
0.01 .............. .I..
4W.03 4W,02 4).00 0 Owl0 0.002 0,003 0.004 0.5 0 10 M0 M0 Q 5() 6W 70
0 0
4W•15 Aw.l0 4.005 0 0.()05 -0MO5 -.awl -a000 0 a0.00
Figure 3.18: Test case 4. Final/plastic strain in x-, y-, and z-directions for x-, yl-, andz-components. Linearised data compared to averaged plastic strain, tfP,.
24 Chapter 3. Numerical Modelling
The plastic strain distribution and the linearised strains from the 27 analyses are presentedin Appendix B, where the plastic strain is shown as in Figure 3.17 and as an example inFigure 3.18. Above the line (A through F) the graphs show how the plastic strain x-, y-, andz-components are distributed along and perpendicularly to the heating path on the top andbottom sides of the plate. Thus, pane A holds e., evaluated perpendicularly to the centrepoint of the heating path, while B holds the same property, evaluated along the heatingpath. Panes C, D, E, and F are the same except for EP and e, respectively. The G paneshows the eP,•,, distribution through the thickness in the centre of the heating line. Pane Hshows the maximum temperature at each time step. Finally, the I and J panes show howthe averaged plastic strains are linearised. The dash lines are the original plastic data, whilethe thick straight lines represent the linearisation in either {Sý, B.I or {S,, By}.
3.6 Simplified Elastic Analyses
The assumption as regards the test cases is that the deformation from the line heating processis local. Therefore, an equivalent analysis can be carried out, if the linearised plastic strainsare applied to a plate in an elastic analysis. Thus, a plate of any shape can be analysedusing the plastic strains according to the heating parameters as input.
Unfortunately, ANSYS is unable to use the plastic strains directly as input in the initialstate. Instead an 'artificial temperature field analysis' (ATFA) can be used, which combinesartificial thermal expansion coefficients and temperatures to simulate the linearised plasticstrains. The artificial material properties are applied to a region of the same size as theplastic zone shown in Figure 3.15 and the region outside is assigned a zero thermal expansioncoefficient. The 'plastic' region is divided into two elements in the thickness direction eachwith separate orthotropic thermal expansion coefficients, as shown in Figure 3.19.
In the following it is assumed that the elements are equally thick and subject to a constanttemperature, T. Then the strain at the bottom, e,4, and at the top, e4, must be equal to thelinearised strain at those z-coordinates.
T T (3.13)
and
c4ý a' T = S= + B. a' S_,9 + B. 3.4T (3.14)
4 _ TAt the interface between the two elements, the strains will be averaged at load vector as-sembly, resulting in a linear distribution from top to bottom. The same derivation is validfor the y-direction.
ATFA can be applied to a FE model with relatively large elements and still yield preciseresults. One thing is, however, crucial to correct modelling: Consider the boundary region
3.7 Sensitivity Analysis 25
E4
Figure 3.19: Strains and artificial properties for two elements in the thickness direction.
between the zones with artificial expansion coefficients and the surrounding area with zero
expansion coefficient. The nodes at this interface are assigned the average thermal expansionfrom the element in the two zones, and the elements connected to those nodes receive anincorrect expansion. If the elements have independent, coinciding nodes at the interface,and the nodes are connected by stiff ends (constraints), the elements at the interface will be
thermally independent but still provide the true kinematics.
To validate the results from this elastic method, comparison to the 27 elasto-plastic analyses
is made. Figures C. 1 to C.4 in Appendix C present a comparison of the z-direction deflectionat the symmetry (heated) line and at the far edge of the plate parallel to the heating path.It is seen that good agreement is found between the two types of analysis, and consideringthe calculation time saved (6 hours versus 30 seconds) this is very encouraging.
3.7 Sensitivity Analysis
A number of simulations are made to examine the sensitivity of the model to variations ininput parameters related to previous assumptions. Information is provided on how variationsin material properties, torch width and plate size influence the linearised strains with caseNo. 14 as a reference. The following simulations are made:
* Two simulations inlvestigate changes in the yield limit (sensi and sens2). In the former,
all data for the yield limit is raised by 10%. In the latter, the room temperature
yield limit is changed from 250 MPa to 400 MPa leaving all other untouched. Linearinterpolation is used for temperatures between 20 00 and 400 00, where the yield limitis still 210 MPa.
26 Chapter 3. Numerical Modelling
Yield limit Yield limit Hardening rule Radius Plate sizeSemsi Sens2 Sens3 Sens4 Senas
S. 11%r -71o 10% -30% 5%B2 3% 85%T. 21% o :v
S, 07b - 16%67-o% 8H,& Wo -2 11 -24o
Table 3.2: Influence of change in parameters on residual plastic strains.
0 -- 0No.I4-N.-1
00' . - . . 0.'.....................0"..............
0 0.
4~ (06 -0,4 -0(0QU 0 -01 ýCO -0 OW6 -0 M3 0
00
0.0 ..-... .......... ............................... .................
0.014 -. v
-O b4 -0&0 0 -0.ý 4MU 4"3 0
Figure 3.21: Comparison of averaged plastic strains, E% , for sens4 n es versus case No.14
3.7 Sensitivity Analysis 27
" Test with a narrower torch ('sens3'). A torch radius, rm, of 3 cm is used instead ofthe usual 4 cm. The maximum temperature is kept constant.
" Test of the influence of using kinematic hardening instead of isotropic hardening('sens4').
* One test investigates the influence of the plate size on the local plastic deformation('sens5'). A plate size of 1.5 m by 0.75 m is used with symmetric conditions.
It is seen from Table 3.2 that the bending in the longitudinal x-direction, Bx, is highlyinfluenced by the yield limit. However, S., S., and B. are much less sensitive to thosechanges. Figure 3.20 shows how the averaged (not linearised) plastic strains are changedwith the increase in yield stress. The change is mainly on the heated side, where the x-components of the plastic strains are increased compared to the reference case, so thatbending becomes more predominant.
The results have proved insensitive to the use of either isotropic or kinematic hardening.
A narrower torch will produce less deflection in general as the required heat input is smallerto attain the temperature of 600 0C. However, the x-direction bending, B,, is larger. SeeFigure 3.21, where the heated zone is narrower and the heat penetration is smaller for anarrower torch, which again implies that the ratio of shrinkage to bending can be controlledby the choice of heat source. A broader torch will give more shrinkage at the expense ofbending-most notably in the x-direction, provided that the maximum temperature is thesame. As the results are sensitive to changes in torch radius, direct measurements of thetemperature distribution on a heated plate are used to find the heat flux distribution inChapter 6.
Finally, sens 5 shows that the results are for practical matters independent of the plate size.
Figure 3.22: Temperature profiles. The maximum temperature is 500 'C and 50 'C betweeneach contour.
28 Chapter 3. Numerical Modelling
:2190.:092m-108.09
- 130 09-- .900E, D8
-- 100.08:s
liE. 0910 O+ 09
:2308,0D92708+09
(B)
Figure 3.23: Stress az-components, a.. (A) Intermedihate and (B) final step.
-. 2902:09- 250. 09230 09
MR_ 17008.09
- 930.8 09
:3002. 0.87 008+8-
190E+09.__230s:09
270E,09
(B) RO
Figure 3.24: Stress y-components, as, (A) Intermediate and (B) final step.
3.7 Sensitivity Analysis 29
m -. 0317-.010073
__- .000376-0 22
--. 00371
- .002227
' .00370
(B)MIM
Figure 3.26: Plastic strain, y-components, EP.. (A) Intermediate and (B) final step.
30 Chapter 3. Numerical Modelling
3.8 Analyses of Test Case Results
General Mechanics
A general description of the stress and strain development for test case No. 9 is given in thefollowing. Thus, T.,, = 500 'C, h = 20mmur, and v = 15 mmn/s.
Figure 3.22 shows the temperature profiles from the test case. It is seen that the temperaturesdo not propagate very far into the surrounding material-the contour for 50 'C is well insidethe 4 cm boundary of the finely meshed area. Of course, the heat will eventually disperse,but at much lower temperatures, which suggests that the heating paths do not interactthrough the temperature fields but rather because of the stresses associated with the localdeflections of neighbouring paths.
When heated, the area immediately under the torch will be subjected to large compressivestresses, a. Figure 3.23 shows a., at an intermediate load step and after cooling and likewiseFigure 3.24 shows ar.. The torch position in the intermediate state is denoted by an orangepatch and the torch is moving from right to left. As regards the a.,, a large area in front of thetorch is in compression, but it gradually decreases to zero further away from the torch. Thea,, also shows compression in an area in front of and below the torch, but due to y-directionsymmetry there is additionally an area with tensile stresses, which becomes permanent onthe bottom side of the plate. After cooling, a., is mostly tensile, whereas a,, is tensile on thesurfaces and compressive in the centre. It should also be noted that the tensile stresses arelarger in the z-direction than in the y-direction, as this information is used later to explainthe development of plastic strains.
The plastic strains, eP, are shown in Figures 3.25 and 3.26 in the intermediate and finalstates. Here, the final state is purely compressive strains-except just in front of the torch,where the y-component plastic strain is slightly positive (shown in grey). To analyse howthe strains are developed, a plot of plastic strains versus temperature is shown in Figures3.27 and 3.28 for nodes in the thickness direction in the centre of the heating line. The timebetween all the points except for the last 13 (during cooling) is a quarter of a second. Asthe penetration of the plastic zone is low, only strains from the top four nodes are shown.
R~om Figure 3.27 it is seen that Er in the top layer is initiated at about 100 TO and that itrapidly increases to its maximum value at 500 00. When the torch passes by, the materialcools down and the plastic strains are constant for some time, while the contraction of thesteel changes the stress state from compressive to tensile. This corresponds to the green areabehind the torch in Figures 3.23(A) and 3.24(A). When the top layer reaches tensile yielding,the material is still very much expanded and there is a large temperature difference betweenthe top and bottom sides of the plate. As the temperature is still 350 'C, there is excessiveplastic straining, which is reduced during cooling. For the second layer the behaviour isthe same, but with a higher temperature for the onset of plastic yielding because of smaller
3.8 Analyses of Test Case Results 31
0.001-
0~~~ ~ ~~~~~~~~ .--...... ....--- .. . .: ........ ....
-0 .0 0 2 .-..... . .. . ... ..... ..... ...
-. .
-0.0040 50 100 150 200 250 300 350 400 450 500
T,, [o1c
Figure 3.27: Plastic strains as functions of temperature in nodes 1 to 4 in the thicknessdirection. zr-components.
2 ----0.• ..... .i ... .. .1....... .....•.......... .... ... .. ...........-3
-0 .0 0 2 ............ ....... . ....... .. L -.... ., .... . .-0003 ...........
-0.0040 50 100 150 200 250 300 350 400 450 500
T.x [OCi
Figure 3.28: Plastic strains as functions of temperature in nodes I to 4 in the thicknessdirection. y-components.
32 Chapter 3. Numerical Modelling
Tm41 v h Case No. S. B. S. Bu[°C] [mins] [mm] ______
10 1 -5T.1.0 -o 4.49.10 -4.96.10q- 3.59.0 -
5 15 2 .510- .5- - 78.20 3 18.1 - .55.10- -1.3.1 2 80.10ý-10 4 -22.O 7.78.10- - S- 3.99.0
500 10 15 5 - .13.0 - 1 -10- 2.. - .89.20 6 -1.4. - 2. 33.10 - f -f1.0 - -210 7 -3.49.10 - 1.00 -1.99.10 - 375-o
15 15 8 -1.73.10-- 2.27.10 - 4 -1.3210 - 4 2.50. 10- 4
20 9 l.30.1- 4 2.25.10 - -10Y.1 - 1.96.110 10 -6.801 2.32.0 - 846.1O 43410
5 15 11 -4.54. - 1.61 10ý- --4.28.o0- 5.m89.1 -
20 12 279.0 278 - 4281 10 5910 13 5.22. 563.10 - -505 5.41.10 -
600 10 15 14 -:3.10 - 2 08.10 465.10-
20 15 -1.961 2.75.10 -19310 - w3.36.110 16 -4.44.10 7.88- - -3.56. -o 5.38.
15 15 17 -2.52.10 - 4 2.30.10 - 4 -2.10.10 - 4 3.97 10- 4
20 18 -:.69-10- 2.66.10 - --. 59.10 - 4 2.85 10-s10 19 70 -7.0 1- 3 75. -1.28.1 - 4.8-1 0
5 15 20 -5.621 - 1.18i.T - -6.T42.10 7-48.10-120 21 -3.-10- 27510- -4.00 - 65181010 22 5.83 10- 4 .75. 7 -. 29. - .14
700 10 15 23 05 64910 -
20 24 260 2 8610-8 27.1 0- 4.mrr770
10 25 Tho-. -6.03 o0- 85.1-15 15 26 -390 10
- 3.14.10 - -31 6 51.10-20 27 -42 70 10 .81 10- 4 2 -73 110
Table 3.3: Linearised plastic strain.
temperature gradients. Moreover, this layer is not reduced nearly as much as the top layerbecause the temperature for tensile yielding is lower (150 0C).
Figure 3.28 gives eP. Apart from beginning with a small tensile plastic strain, the develop-ment of e is the same in compression as described above, although the maximum plasticstrain is not as large. However, at the last stages of cooling eP increases. This is caused bythe difference in x- and y-components of the stresses. According to J2 plasticity theory, thesign of the plastic strain increment iP is determined from the product s5iski~kL, where sijare the stress deviators and &kl are the stress increments. Even though ac and a. are bothtensile, a. is so much larger that 6, is positive at the expense of P. This means that ebecomes numerically larger than e$.
The data necessary for making plots as in Figures 3.27 and 3.28 is only available for testcase No. 9 to limit the storage space needed. However, another eight test cases have beenmade for use in Chapter 4, and the temperature/strain plots are given in Figures 3.29 and3.30 for comparison of results related to different input parameters (shown in Table 3.4).
The linearised, averaged data for the 27 test cases is given in Table 3.3, but even with these
3.8 Analyses of Test Case Results 33
x-components y-components
--- -- ... • ::.. ..... 1: 2........0 -.. . ............. 0 ...... = . ..... .. .. .........
•0 0 .. ..* ........ ..... .p. . .. ,.... . ... .• L ........ ........ *.........,.......... ........4 ........
6 -- 6 -
... ...... .......... .......... _ . ....... ...... .............. •~ ~ ~ ~. .... ...................L, .... ".
" ~ ~ ~ ~ ~ ~ ... .. ...................:
0. 03 . - 2.. ... . ........ 4• 03 .... . ......
4. . ... 4)(0 ....... .. ..... .... ... ....... ... .... ......
0 Im0 2M0 (M 0050 0 1w0 2w0 Y]0 (M 50
0 • 2 -... ----. -...... •- .. ....4 -o - 4 -- -
°" ~ ~ .. ..... - •ii "iA I M -... 7 ........ i. .......................................... ., ............
4 W300 .......
0 1'M m• m0 IM 0 0 0 1•0 m0 M),40
T=- lrc T-, I•c)
.0 0 1..... ... . 3 . ... ý0 .. ..... . .... .2 -------. .4 ...... -- 3 .. .•).C•2~ ...... ....- ..•.-L ... .... . -0.0......0..
4 W 3o .... .... ...... .... ........ o. • ...... .... ................. 4
•.• ~ ~ ~ ~ ~ ..... ...... i .. ........ ........ . ........... ..........
0 Im0 2w0 Yk] 4m0 0 6 0 I00 M0) X)I) m0 0
T• r°c) T-• 1,C)
Figure 3.29: Plastic strains as functions of temperature in nodes 1 to 7 in the thickness
direction. Node 1 is on the heated side.
34 Chapter 3. Numerical Modelling
x-components y-compoflefts
] . .. . . .
0 .... .... .
40).W2 - 4.W
4)........ ......................... 54
4).W)5 - - ---- - .- 4 )
o I '(2W )3() lW5W (37W 0 IWX 2W 3(0W 5W WI 7WT- 1C) T- ICc
2 ------------ --
4- M I .- ....-----------------------
404W4
o IM 2W) 3W 4W 5W6 6(0 70 0 IM 2W) 3W 4W 5W)6( 7W)T- Cc] T foci
0 12
6() - -6
4(04-
-().W .... .... .4 W 3
.........-
0 I0)2W)3W0l)4()5( W)7W) 0 0) 2W)3W 4W 5W0)607W)T_ Cci T- MC
o~ - 6- I .
7 7
4W3~~~ - ...- 4 . ....
4M)
0 IM 2W 3C() 4 5W0 6W 7W 0 1) 20) 3W 4()( 50) 6()( 7()(T, Cc) T- Cc)
Figure 3.30: Plastic strains as functions of temperature in nodes 1 to 7 in the thicknessdirection. Node I s on the heated side.
3.8 Analyses of Test Case Results 35
h [mm]10 15 20
0.0 0 . .... . . .... ............ 000 6 0{ )
5 0.0
...................... .:,. .) .0. 4).(30
-0.001 2.~8I4.
X00 60 0) 607 00.ý 0.0••T000
00 -0 0 0.73
152-o 4".50 60 7050 70W 70
0.040,000006 • [•-i• 0.000• 6ý ..... ....... .... 0.032"03"04-..-• --:'--- -- •
1o0O ............. : ......ý) 0 2,M n .. ..•....... ..... 0 ..... ......... .... ....... ............ .. 0 . ..... .. . .. . ....
Figure 3.31: Tendencies with temperature, Tan a C]. Legend: e
simplifications, the amount of data is slightly overwhelming so to assess the tendency of thedata, the variation with T,., h, and v is shown in Figures 3.31, 3.32, and 3.33, respectively.Figure 3.31 presents the variation in temperature, and almost without exception the plastic
strains become numerically larger with increasing temperature-which is not very surprisingas higher plastic strains induced by higher temperatures will increase the strain hardeningof the material. Thus upon cooling, the material is harder the higher the temperature, onthe assumption used in the modelling that the steel is not annealed by the heat treatment.
Figure 3.32 shows the variation with velocity. With increasing velocity, the shrinkage con-tribution, S',,, decreases and the bending contribution is rather constant or even increasing,
7.5 1 5
v [mtn/s] 12.5 -2 6 12.5 h [nun]7.5l3 7
125 4 8 17.5
Tm ro
Table 3.4: Heating parameters for additional test cases.
36 Chapter 3. Numerical Modelling
h (mm]10 15 20
O(ý3 0i "25 U' :0 . .... ... . . .... . . I " o oo.. . ... ..
O.Wa5 0 : -: :: - 0
-900 3 •. .................- -5 ... ...... " .. ...
01 1E
0.0006000060."05
0.00 ............... ................ 0.004i0ý0 ...............
ý " ..... .............. 0............•.ooo4~ ~ ~ ..... .2 ...... iiiZ 1 112
500 00-OL* -ft
5 10-- 15 5 10 15
Z 0 0,IS70 0)0
4M154
Figure 3.32; Tendencies with velocity, v, [,mm/s]. Legend: SH
which means that bending becomes dominant at higher velocities. At high velocities, theheat is not conducted so far into the plate so the heated side mostly contracts with bendingof the plate as a result. At extremely slow torch movement with full heat penetration, thestrains will-ideally--only consist of shrinkage. At medium penetration depth the materialon the bottom side will act as a hinge, resulting in bending.
It is seen from Figure 3.33 that the shrinkage, S,,,, is decreasing with increasing thickness.It is also observed that the bending z-cornponent is increasing with the thickness. Again,this relates to the heat penetration as explained above. However, the bending y-componentis mostly decreasing with increasing thickness, which at first may seem very strange. Toexplain this, the data behind the lower, right graph is analysed further. It corresponds totest cases 25, 26, and to 27, from which the averaged and linearised data is presented inFigures 8.25 to 8.27 (Appendix B). For a given temperature and velocity, 9P is almostconstant on the heated surface, but with thicker plates the value in the centre of the platedecreases. This corresponds to larger bending, B~, and smaller shrinkage, S.. In the y-direction, all the strains are generally decreasing so that bending, B., and shrinkage, S.,,decrease.
3.8 Analyses of Test Case Results 37
v [mm]5 10 15
O00( __0.00 0.0002...2 .• + : T -" ' ...... 0 " .1: . . ....... •
IXJ ----- -- I
500 0.04
_
0 000/ ............. t .......... 400 0 0
410004000.3• a 0002
-0.0004 4)000360G6I • . ... 0.0005 -0.0004
-60 oooo
4)0008 I •.~060 .•010 15 20 to 15 200.001 000021I 0.0008t
0.00(5 0.0004 0ý0004
7 0 0 ,oo0< 5 0, m • 0c .. .. ... . ." . . .. . .. . .-0.0015 40.0003 ""41.00011
Figure 3.33: Tendencies with thickness, h [mm]. Legend:
Comparison of x- and y-components of Shrinkage and Bending.
With the numerical data at hand, it is interesting to know how it can be compared quantita-tively. For this purpose, it is relevant to know whether bending or shrinkage is predominantand how the magnitudes of the deflections can be compared in the tested temperature range.
To facilitate the interpretation, the data is compared as ratios of y- and x-componentsand as ratios of bending and shrinkage magnitudes, respectively. Firstly, Table 3.5 showsthe relation between the longitudinal and the transversal strains, and one can see that thebending perpendicular to the heating path is larger than that along the heating path.
Further, the ratio of y- and z-components in shrinkage, SI/S,, is not far from unity, so thatthe instances of shrinkage along and perpendicular to the heating path are comparable inmagnitude. However, the tendency is that the ratio increases with rising temperature, T,,,=,and decreases with rising velocity, v. Thus, in cases where shrinkage perpendicular to thepath is mostly wanted (which is likely to be the general case), higher temperatures shouldbe preferred. The dependence of the thickness, h, is rather ambiguous.
The ratio of the y- and z-components in the bending, BY/BX, has the same tendencies,apart from the dependence of h where the ratio is clearly inversely proportional. In general,
38 Chapter 3. Numerical Modelling
Tmnr = 500 °C" [nn/s] h [mm- Case Sn/Sr Average By/B. Average
10 1 0.87 7.995 15 2 0.75 0.84 1.92 3.67
20 3 0.89 1.1010 4 0.64 5.13
10 15 5 0.72 0.73 0.76 1.34 2.47 2.6420 6 0.84 0.9510 7 0.57 3.40
15 15 8 0.76 0.71 1.10 1.7920 9 0.80 0.87
Tm0: =600 -C" [ra/s h [mm) Case SISý Average IB[IB Average
5 10 1.24 18.705 15 11 0.94 1.06 3.65 8.00
20 12 1.01 1.6510 13 0.97 9.62
10 15 14 0.84 0.93 0.95 2.23 4.36 5.2520 15 0.98 1.2210 16 0.81 7.40
15 15 17 0.83 0.86 1.73 3.4020 18 0.94 1.07
Tm0: =700 °Cv [mn/s] h [mm] Case Sf/Sj Average 11I/BJ Average
10 19 1.74 130.395 15 20 1.14 1.33 6.33 46.32
20 21 1.11 2.2510 22 1.25 12.93
10 15 23 0.98 1.10 1.13 3.56 6.05 18.39
20 24 1.06 1.6710 25 0.98 5.03
15 15 26 0.91 0.97 2.08 2.8020 27 1.01 1.29
Table 3.5: Ratio between x- and y-components.
transversal bending is much larger than bending along the heating line. Again, high temper-atures have to be preferred to achieve a high ratio between the y- and x-components and,further, low velocities to improve the ratio can be chosen. However, the latter will reducethe absolute amount of bending and increase shrinkage.
Comparison of Bending and Shrinkage
Next, comparison between bending and shrinkage (Table 3.6) shows that their ratio, B:,n/S,t,,is highest for low temperatures and high velocities. If the required behaviour is to have ahigh ratio between bending and shrinkage, a low temperature should be chosen. But if-atthe same time-a high ratio of y- and x-components is required, this implies high tempera-
3.8 Analyses of Test Case Results 39
Tm0: =500 0Cv [mnn/s] h imm] Case BZ/S, Average B,/S, Average
10 1 -0.08 -0.725 15 2 -0.63 -.70 -1.60 -1.35
20 3 -1.39 -1.7210 4 -0.18 -1.47
10 15 5 -1.02 -0.94 -0.92 -1.89 -1.73 -1.6620 6 -1.63 -1.8410 7 -0.32 -1.88
15 15 8 -1.31 -1.12 -1.90 -1.8920 9 -1.73 -1.89
Týr =600 00v [mm/s] r s h 1ýn Case Br/Sr Average BI/Sd Average
10 10 -0.03 -0.515 15 11 -0.36 -0.46 -1.38 -1.17
20 12 -0.99 -1.6310 13 -0.11 -1.07
10 15 14 -0.67 -0.73 -0.69 -1.78 -1.53 -1.4720 15 -1.40 -1.7410 16 -0.16 -1.46
15 15 17 -0.91 -0.88 -1.89 -1.7220 18 -1.57 -1.80
Tm,_ =700 0Cv [mm/s) hIrt] Case B/ A e /S A B /Sv Average
10 19 -0.01 -0.385 15 20 -0.21 -0.33 -1.16 -1.03
20 21 -0.76 -1.5410 22 -0.08 -0.84
10 15 23 -0.45 -0.54 -0.56 -1.64 -1.40 -1.3620 24 -1.10 -1.7310 25 -0.25 -1.30
15 15 26 -0.81 -0.82 -1.84 -1.6520 27 -1.41 -1.80
Table 3.6: Ratio between bending and shrinkage. Test cases and averages.
tures. Therefore, either of the two must be chosen. The B/S ratios are less sensitive thanthe y/x ratios, so the latter are the governing factor.
Calculations with Very Slow Movement
A sufficiently low heat source velocity will produce a 'pure' shrinkage state in the heatedregion. Table 3.6 shows that only thin plates subject to low velocities (cases 1, 10, and19) have really low ratios of bending to shrinkage, and this it is only really true for thex-components. In the y-direction it seems that even lower velocities are needed, especiallyfor the thicker plates. Therefore, more simulations are carried out to see if 'pure shrinkage'is attainable for thicker plates.
40 Chapter 3. Numerical Modelling
v mm/s h fmm] Case S 1B. Sy B10 28 -9.16 .10- - -2.79 .10- -2.21 .10 - 3.19 .10- 4
2 15 29 -8.37 10 - 258 100 1 39 lO 6.40 10w-
20 30 -6.. 1 0- 1.67.10 -8.65.0 - 8.47.1-1 201 31 848.10 7 680 08- -- 0_T
0.5 20 32 -9l35 .10- 4r b6.0W5 .10- ' -1.7 3.-- --- T - 4
vTinm/s' h lin Case B,/S, BU/SU10 28 0.03 -0.14
2 15 29 -0.03 -0.4620 30 -0.25 -0.98
1 20 31 -0.09 -0.570.5 20 32 -0.06 -0.42
Table 3.7: Test cases with low torch velocity.
The additional simulations along with linearised results are given in Table 3.7. See alsoFigures B.28 to B.32, p. 125ff. Not surprisingly, the 10 mm plate is still very close to 'pureshrinkage' at v = 2 mm/s. However, thicker plates still show a good deal of bending, espe-cially in the y-component, and even at the unrealistically low speed of 0.5 mm/s bendingis still rather conspicuous. Either the fact must be accepted that bending will always bepresent or methods like double sided simultaneous heating or high-frequency induction heat-ing must be used. The latter is able to generate heat in the material as well as on the surface.Further, changing the frequency of the induced current will control the heat penetration tosome degree.
Power Consumption
As mentioned earlier, the maximum temperature, T.,, is chosen as a parameter instead ofthe power transferred from the torch to the plate, Q,,, mainly to ease the control during theforming process. Another reason is that the results from the ANSYS simulations based onpower rather than temperature are not reproducible. Depending on time stepping, meshingand solution options, one specified power can lead to various temperature distributions,whereas a specified maximum temperature leads to consistent results.
However, it is interesting to know approximately how powerful the torch must be to performshaping, especially at high velocities. Table 3.8 shows the approximate power input for eachof the simulated test cases along with energy per unit length of heating, Wh = J. Thepower output from the heat source must of course be higher, as the figures in the tableindicate what must be put into the plate (efficiency coefficient being ignored).
Using fast torch movement is least energy consuming, on the assumption that the totalheating length is the same no matter what the chosen torch velocity is. However, thereis not really a choice, as the temperature and the velocity are governed by the necessarydeformation at the heating line. There may be a limitation in the heat output of the availabletorches which hinders the combination of high velocities and high temperatures.
3.9 Edge Effects 41
T• v h Cas Qtý 1wh
[0G[ [mam/s [m1mi [W] [MJi/r]10 1 4367 0.87
5 -75- 2-T 500- 1.00
-in- 9 g-~ -- ru500 10 15 5 7665 .693
10 m 8050 0.515 15 -9 -77W- 0.51
20 -7-795-0 0.5210 10 5200 1.04
5 15 1T 600w 1.202T- -r2- _Wf-r T--r r3 75 0.79
600 10 M5 1r- 820 0.82
20 155- 8400- 0.84-"15 15 17r 920T 0.r1
10 19 6150 1.235 15 20 6900 1.38
20 21 -T6760 1.35-10 122 8900 0.-89
700 10 15 23 9575 0.9
7T-r -Tr r15 15 25 r15 720W 11900 0.79
10 28 2450 1.232 15 29 2900 1.45
700 20 30 1150_ 1.-1 2 3 2400 24
05 2U 32 M80 .6
Table 3.8: Power consumption.
3.9 Edge Effects
Until now, all the strains have been derived from cases where heating is applied far fromthe edges. Although this represents a large part of the potentially heated area, heating isvery commonly applied near the edges when plates of overall positive Gaussian curvature(concave) are formed.Therefore, it is analysed how heating to and from an edge influencesthe strain distribution. Simulations based on the same parameters as case No. 14 with thetorch leaving and entering the plate at right angles and 45', respectively, are carried out.
Figure 3.35 shows the x-components of the plastic strain near the edge of a plate. Onlyhalf the plate (divided at y = 0) is shown to give a view of the strains in the interior of theplate. In the top of the figure, the result of heating towards the edge is presented, and thereis clearly a large influence from the edge, which is partly connected with the increase intemperature when the edge is crossed, see Figure 3.34. Further, the area which supports thestresses in front of the heat source is gradually reduced as the edge is approached. Therefore,
42 Chapter 3. Numerical Modelling
the final state at the edge will consist of large compressive x-direction plastic strains, eP,and large tensile y-direction plastic strains, eP. The development of cP and 6 at the edgeis depicted in Figures 3.40 and 3.41. Even before the heat reaches the edge the steel yields,resulting in the mentioned compressive and tensile strains.
When heating takes place from the edge towards the centre the strains are almost unaffectedby the edge effect.
When heating takes place at an angle of 450 the results are much as above: The strain fieldis disturbed compared to that of heating in the centre of a plate, and when heating takesplace from the edge inwards, the strain field is almost normal. See figures 3.37 and 3.38.Figure 3.39 shows the plate divided in the centre of the heating line (y = 0).
The conclusion is that to minimise the effect of heating at an edge, the heating should becarried out from the edge towards the centre of the plate. Thus, the artificial temperaturefield analysis (ATFA) can be used with the plastic strains derived from the centre of theplate and reasonable results can still be obtained. Neglecting the edge effect corresponds tothe assumption of neglecting the effect of starting and stopping in Section 3.5.
6 0 0 ........ ." -- " -- -r ......... . . . . . .. .. . .. .. . . .. . .. . .. .
,o ....... ..... .F- T - - - -........
0 to 20 30 40 50 60 70 80Lo sMp
Figure 3.34: Temperature as function of load steps.
3.9 Edge Effects 43
(A) M-007
m- 00155
M-00120-. 00102-. 00004___ .__ 00067
-.___ 0003
-.___ 0001
(B) 004
Figure 3.35: Plastic strain az-components when (A) leaving and (B) entering the plate atright angles.
(A)- 008
_______00t52-- 000,,
-. 000415
(B)008
Figure 3.36: Plastic strain y-components when (A) leaving and (B) entering the plate atright angles.
44 Chapter .3. Numerical Modelling
(A)
-. 0007,
-. 00030
.n0005
000D45
(B)
Figure 3.37: Plastic strain cr-components when leaving and entering the plate at 45".
3.9 Edge Effects 45
(A)
I -'.0021
-. 00180
-. 00144
-. 00109.00073
-. 00030
-.l 00002
.00033
.00019
(B) 7
Figure 3.38: Plastic strain y-components when leaving and entering the plate at 45' .
(A) _00250-. 00215
-. 00180- 00109
I-.00073
m -00038
-. 00002
(B) =.00M
Immom.00069
Figure 3.39: Plastic strain y-components when leaving and entering the plate at 45. Platedivided at y = 0.
46 Chapter 3. Numerical Modelling
0
-0.0004 .... 3
ti4
-0006 - ....30.40 50 60 70
-0.0014 .-.....
0.0015
-0.002
0 100 200 300 400 500 600 700T,.,ý FOCI
Figure 3.41: Temperature versus plastic strain, cP, at the edge.
Chapter 4
Empirical Relations
The subject of this chapter is how functions for relations between heating parameters andlinearised plastic strains, S., Bý, S., and B., can be found. Firstly, independent variablesare established from dimensional analysis, and secondly the coefficients to the independentvariables are determined by means of multivariate analysis.
4.1 Establishing Dimensionless Parameters
A rational way of finding all independent sets of dimensionless parameters which may beimportant to a physical problem can be found in Szirtes (1997). With manipulation ofa matrix (the so-called dimensional set) the dimensionless parameters can be determined.Below derivation of the set and description of its application to the problem are given.
Firstly, the physical properties for expression of the dimensionless parameters must be cho-sen. A natural choice is the parameters mentioned in Section 3.4, namely T,., h, andv. However, those are not alone sufficient to form dimensionless parameters, so additionalones must be included: density, p [kg/M 3], conductivity, A [W/(Km)], yield limit, a. [Pa],and specific heat, cp [J/(Kkg)]. Secondly, a system of independent dimensions is neededas a descriptive basis for all the variables. Here, length [m], time [s], temperature [K] andmass [kg] aptly describe all variables. The relation between the eight variables and the fourdimensions in a monomial power form is
V1 1 VE2 VC3 VC4¢ VSCS V•fG V7 V98' = dql' a•2 e• 4 q412= 4 (4.1)
with cj being the sought powers to the variables, Vj, and qj the powers to the dimensions, di.If dimensionless parameters are sought, qi equals 0. Each variable, Vj, can be expressed interms of the dimensions as
h h-jh , (4.2)Vi = d1 d2 d3 d4
(4
47
48 Chapter 4. Empirical Relations
Insertion of this in (4.1) and taking note of the fact that the powers to the dimensions onboth sides of (4.1) must be the same yield
h = qi (4.3)
or
H E = = 0 for dimensionless parameters (4.4)4.8 8.1 4×1 4×1
The relation between variables and dimensions is expressed in H, which is called the dimen-sional matrix. For example, the dimension of density, p, is [kg/m 3], which is representedby a -3 and a 1 in the p-column. For use in (4.7), H is divided in to submatrix B and therightmost square submatrix A:
k S. c p A a T. h vK -1 -1 1m 2 -3 1 -1 1 1s -2 -3 -2 -1kg1 1 1
B A (4.5)
Likewise, the vector c is divided into two parts, the last as long as the number of thedimensions and the first (here called J) holding what is left. Since the system of equationsis underdetermined, d can be chosen arbitrarily. Obviously,
d=T Ji I4 4 (4.6)4ý1I 4x4 4x4
and therefore (4.3) can be rewritten as
[ A] 6 = {g} (4.7)8x8 8Xl 831
By premultiplying with the inverse of the term in the square brackets the following is obtained
=[1 0{,7dB- A]- to6
= [- - {01 (4.8)8X8 8Xl
This is, however, just one of the possible solutions to the underdetermined system of equa-tions. Different choice of J gives three more:
E [AI- B A-'] [DOT ] = [-A-RBDT] (4.9)8x8 8x4 8x4
4.1 Establishing Dimensionless Parameters 49
Variables
Hmatrix A matrix
Dimensionless
variables H Dmatrix Cmatrix
Figure 4.1: Definition of the dimensional set.
The rows in DT can be chosen arbitrarily, but they have to be linearly independent. Now,the powers of the variables to form dimensionless parameters are found in C. For example, adimensionless parameter, ir,, is given by the powers in the first column of C. If ET is writtenunder the dimensional matrix and a new matrix C = -D(A-IB)T is defined, a structurecalled the dimensional set is derived as outlined in Figure 4.1.
Selecting D to be the identity matrix and evaluating C yield the following dimensional setfor the present problem:
S C % p ,A av Tn h vK- -1 -1 1
m 2 -3 1 -1 1 1s -2 -3 -2 -1kg 1 1 1ýr 1 0 0 0 072 1 0 1 0 -273 1 -1 0 0 274 1 -1 1 -1 -1 (4.10)
All empty spaces in the matrix denote a 0, but as they are needed for countinq, in C they areincluded there. Horizontal reading in D and C, reveals that 7r, = S&, 72 = 2 , 7ra3 = a-•,
and 7r4 = , This is just one of many possible dimensionless sets. By interchangingthe order of the columns in the set or by selecting D differently, an infinite number ofdimensionless sets can be found-however, just a few of those sets will be dimensionallyindependent of each other. Matrix A must be non-singular at all times and further, onlyinterchanging columns between the matrices A and B will yield distinct sets-althoughthis is merely a necessary and not sufficient requirement for obtaining independent sets of
50 Chapter 4. Empirical Relations
iri. Thus, for a fixed D the number of distinct sets, Ns, is calculated as (Szirtes, 1997,Eq. (10.20)):
Ns = k - (UR - V) - vc (4.11)
The terms in (4.11) are
k Nd) is the total number of possible interchanges between A and B. Nv is the numberk= Nd
of variables and Nd is the number of dimensions. Here, Nv = 8 and Nd = 4, so k = 70.
UR = -- ( 1 _ll) ( i) is the number of prohibited interchanges due to possible sin-
j=l (
gularity of A. Nzi is the number of zeroes in any row, i, in C and Np is the numberof dimensionless variables, fr. The value of UR must be summed over all the rows inD which contain zeroes. As Np = 4, Nz., = 4, Nz,2 = 2, Nz,3 = 2, and Nz,4 = 0,U = 35 + 5 + 5 = 45.
V represents the number of duplicate prohibited interchanges. If for example two columnsin A are interchanged with two columns in B and each of these interchanges resultin a singular A, this will count as two occurrences of Un. However, A is singularby just one of the interchanges, so that it must only count once-this is corrected by09. It is determined by counting zeroes: If two rows have two or more zeroes in thesame columns this counts as one occurrence of 0. In this case, the first and the thirdcolumn in row one and three share zeroes, which implies one instance of 0. Likewise,the second and the third column in row one and three share zeroes, which is anotherinstance of t9. Altogether, V = 2.
U, is the number of interchanges which yield equivalent sets. To reduce the size of theproblem, an example with three rows is given: If C is as shown below, U. = n1 + n2 +n3 + n1n2 + nln3 + n2na + n1 n2n3, where ni counts the elements in row i which are theonly non-zero elements in its own column. For example,
C 1005001C = 10480001
L 900032]
has n1 = 1 ('5' is the only non-zero element in column 4), n2 = 2 ('4' and '8' in columns2 and 3), and n3 = 2 (columns 5 and 6). In the current set, n, = 0, n2 = 0, n3 = 0,and n 4 = 1, which means that U, = 1.
Altogether, Ns = 70 - (45 - 2) - 1 = 26.
Performing the rather exhausting manipulation of the dimensional set to find all the setsresults in a large number of apparently different dimensionless variables. However, any two
4.2 Fitting of Functions 51
fl•pl Tv. T,..~c fl ! A.u
T, A2 Tc= A2 A T...2 ,
1- -A- __ 1 _h,,vT~
TMo tmn 6'a.vma P rý BL,,x'-J hv pch
Table 4.1: Table of parameters based on dimensionless variables.
dimensionless variables are dimensionally dependent if one is equal to the other raised toany power. Thus, for example I and JS are equivalent properties. Further, as the presentproblem always concerns the same torch and the same type of material the variables relatingto heat conduction and stress analysis can be neglected. This leaves the mixtures of thethree heating parameters which-from a point of view of dimensional analysis-must begoverning for the forming process. Eleven independent parameters consisting of T,., h,and v are identified from seventeen dimensionless parameters as shown in Table 4.1. Ifrelations between the heating parameters and the linearised plastic strains exist, they mustbe composed of combinations of those terms.
The variables, along with T., h, and v, are raised to various powers (positive and negative),and the correlation between the variables and the linearised plastic strains is calculatednumerically (by means of (Lee, 1998)). All in all about 100 variables are tested for correlationwith the linearised plastic strains, and the variables with the numerically largest correlationare chosen for further treatment by multivariate analysis.
4.2 Fitting of Functions
4.2.1 Multivariate Analysis
Consider the situation where results (e.g. S.) from n sets of analyses or experiments mustbe fitted with a function of p independent variables. The results in the vector, 9,, can beexpressed as a function of some dimensionless variables, X, with coefficients, b, and a fittingresidual, 9, as
Y2 = 1 X22 X23 ... X 2p + e2 (4.12)
Yn1 I [ell!
For example, y, contains the results from case No. 1 (the value of either S., Bý, S, or B,),XI1 contain the independent variables from Table 4.1 evaluated by the heating parameters,
52 Chapter 4. Empirical Relations
S.b X Corr.
77,87.10 - -0,78
' -3,37.10-6 , -0,74
2,58.10 - 13 --0,54
01 h " 1 , l-2,36.10- s n -0,90
1,07.10 - '3 -0,88
Figure 4.2: Graph and coeffcients for S•.
B®b X Corr.= -3,71.10-4
S-74,2-31i0--T- ý -0,88
0 - u 1,49 - -0 --,41
040)20 -i,01"I0-1 • -0,82
1 2 3 4 5 6 7 0 9 2o1t10-3141516 L71819,28523 S755"10-0 (Tm.. - 300)7v- 0,31
3,8-310 -I Tn, 1 h2 0,85
Figure 4.3: Graph and coefficients for B.
Sw,"4 b X Corr.
0-2-,02"10-
d2'-0 - 1 -0,87
w*VA 2,52"10 - ' Tn,0 1 v 0,238,39.10- 7 N -0,84
-2,78.10 -= T -0,60
Figure 4.4: Graph and coefficients for S•.
4.2 Fitting of Functions 53
By
b X Corn.2,31.10 - 3 1 -1,22.10 - ' Týma. 0,81.2,G6 10- -F 0,62
o ______ 8410-ý T r 064*MVA-7,3210 Tl t 0,75
•z - WAm• -6,98.10 -5 Tar 0,81
1,34.10_' - 0,4910 S2168-10
- NY 0,29
-1731.10- l
!F 0,47
5,43.10 - Tm2 v 0,50
52,6 • -0,68-7,29._10_- T_ T= 0,81
Figure 4.5: Graph and coefficients for S,.
Tmax, h, and v, according to case No. 1. The coefficients, b, can then be found from theordinary least squares method:
(XrX)b = XrP (4.13)
If (xrX)- l exists, b is expressed as
b = (xrx)-ixrg (4.14)
The terms from Table 4.1 are tested for correlation with the linearised strains, and byaltering the terms and coefficients of the fitting polynomials, functions with as few terms aspossible are found. By inspecting the fit quality, an assessment of whether the dependenceon the variables is too strong or too weak is made. Thus, a term like (Tine - 300) 2 inFigure 4.3 seems appropriate, and even though some of the terms in the polynomials havesmall correlation values, they prove to be essential to obtain a proper fit. Figures 4.2 to 4.5show the polynomials, X, and their coefficients, b, for each of S1, Br, S, and B. alongwith bar charts which evaluate the fitted polynomials and the linearised strains. As seen,the fitted polynomials yield a fairly precise representation of the data from cases 1 to 27,which indicates that the dimensionless parameters found in Section 4.1 are governing for thephysical problem.
4.2.2 Interpolation
To spare the effort of performing a dimensional analysis followed by multivariate analysisanother method is available. Instead of fitting a polynomial to the data set, parabolicinterpolation through the data points is applicable.
54 Chapter 4. Empirical Relations
p
//"20 0
I s6
19 24
n 25 x2 2
o 7 L 1
p 21 z 16
n~ x 5 x 2
q 4 a 18r 8 b 12
Table 4.2: Corresponding element nodes and test cases.
The data set comes from 3x3x3 variations of the heating parameters, which means that there
are three results in each 'direction'. Therefore, a parabolic interpolation between the datacan be used, and for simplicity the shape function of a twenty-noded element is adopted.
With reference to Figure 4.6 the s-direction corresponds to values of velocity, v, the t-direction corresponds to thicknesses, hi, and the r-direction corresponds to different temper-atures, T,,,.4 . The 'element nodes' are numbered 'i~j, k,. .. , z,a,b' and each of the test casesare numbered from 1 to 27. It should be noted that some of the results are discarded, asthere are only 20 'nodes' for results. Those are Nos. 5, 11, 13, 14, 15, 17, and 23. The restof the cases correspond to the node numbering as shown in Table 4.2 and Figure 4.6.
The benefit of using this method is that the interpolation is exact at the points where datais given (as opposed to MVA). Only 20 nodes (results) are used, which is both a benefitand a drawback, as fewer simulations are required to obtain data for interpolations, at thecost of precision. Further, the method is less general than MVA as it cannot use more (or
4.2 Fitting of Functions 55
2 3 4 5 6 7 8 9 1 11 12 B3 14 15 16 17 18 19 M 21 2 23 24 25 26 27
Figure 4.7: Comparison of FEM results and interpolation fit for S,.
ý"35 MO.Mv4o,•30
0."25 N Dor i
O.in10
I2 3 4 5 6 7 8 9 10 11 12 13 14 t5 16 17 19 19 M0 21 M 23 2M Z52 27
Figure 4.8: Comparison of FEM results and interpolation fit for B..4M23
4•,006,
2 3 4 5 6 7 8 9 10 1112 13O141 1617 1 19 2122 23 24 23 262
Figure 4.9: Comparison of FEM results and interpolation fit for SB.
I2 3 4 3 8 9 101 12 13 14 35 16 17 18 14 W 21 n2 23 24 23 2W 27
Figure 4.10: Comparison of FEM results and interpolation fit for BS.
56 Chapter 4. Empirical Relations
less) than three data points in each 'direction'. Eq. (4.15) gives the interpolation functionin general terms, where r, s, and t are natural coordinates in the range [-1;1] and uij,...,b arethe values of either S., Bý, S., or B. in each of the test cases.
uiýýIatd ý (ui(i - s)(1 - t)(1 - r)(-s - t - r - 2)
+uj(1 + s)(1 -t)(1- r)(s- t- r- 2)
+ uk(1 + s)(1 + t)(i - r)(s + t - r-- 2)
+ uj(1 - s)(1 + t)(1 - r)(-s + t - r-- 2)
+ u.(1 - s)(1 - t)(i + r)(-s - t + r- 2)
+"u (1 + s)(1 - t)(1 + r)(s - t + r- 2)
+ u0 (1 + s)(1 + t)(1 + r)(s + t + r- 2)
+ u,(1 - s)(1 + t)(1 + r)(-s + t + r - 2))/8 (4.15)
+ (uq(1 - s2)(1 - t)(1 - r) + Ur(1 + s)(1 - t2 )(1 - r)
+ u,(1- s2)(l + t)(I - r) + u,(1 - s)(1 - t2)(1 - r)
+ u.(1 - s2)(1 - t)(1 + r) + u.4(1 + s)(1- t2)(1 + r)+÷u.-(1 s2)(1 +t)(1 +r) + u.(1- s)(l t2 )(1 +r)
+ u(t - s)(1 - t)(1- r2) + uý(1 + s)(1- t)(1 - r2 )
+ u 0 (1 + s)(1 + t)(1 - r) + Ub(1 - s)(1 + t)(1 -r))/4
Written in terms of case numbers, heating parameters, and S. as an example this becomes
Szwrn~ tm = (S. 1 (1 - O)(1 - h)(l - t:.)(-v - h - Tm: - 2)
+ Sý,7 (1 + O)(1 - h)(1 - tma)(O - h - t, - 2)
+ Sx,9 (1 + 0)(1 + h)(1 - +mhx)(O + h - Tm0: - 2)
+ SI,3 (1 - O)(1 + h)(1 - Tro)(- + h - tma - 2)
+ Sý,25 (1 - O,)(1 - h)(1 + t,,,0 )(-O - h + t.,,a, - 2)" Sý,5'z(1 + •3)(1 - h)(1 + Ta)•-- h + •az- 2)
+ sý,27 (1 +Ob)(1 +h)(1 +tmx(+ h+t x 2) (4.16)+ S., 21 (1 - V)(1 + h)(1 + +,.,)(- + A + Th + - 2)
+ Sý,6 (1 - i2)(1 + h)(1 - t7 ý,) + Sý, 2(1 - iO)(1 _ h2) ( 1 - ".)+ S.,22(1 - f2)(1 - h)(1 + t,.) + S., 26 (1 + 0(1 - h)(+ + )
+ Sý,24 (1 - f02)(1 + h)(i + T,,ý.a) + Sý,20(1 - 0)(1 - h2)(1 + t,ý)
+ So(1 - 0)(1 - (1 - t2) + S.,,6 (1 + 3)(1 - h)(1 - t2)
+ S.,18 (1 + 00)(1 + h)(i - ?$,.) + Sý,12(1 - 0)(1 + h)(1 - t2))/4
4.3 Prediction and Comments 57
where T h,. , ii, and 0) are natural coordinate versions of T, , h, and v:
=T, - ,-500 2 1 Tmaxc[-1;1]200h-0.010.2- E[-1;1]
0.010
v - 0.00503= 0.0 2-1 V)e[-1;1]0. 0 _10
The results of this interpolation are shown in Figures 4.7 to 4.10. By comparison of MVA andparabolic interpolation it is seen that the latter reproduces the fitted data more precisely.In the next section, it is investigated whether one of the methods predicts the results for yetunknown heating parameters better than the other.
4.3 Prediction and Comments
Additional cases are carried out to test the ability of the polynomials to predict-by inter-polation-the plastic strains for yet uncalculated heating parameters. The extra cases arethe same as in Table 3.4, which is shown again in Table 4.3 for convenience.
Figures 4.11 and 4.12 show the predictions along with actual calculations. Generally, the fitis acceptable, except for the value predicted by MVA for case No. 5, which is 54% too large.
It may be concluded that the variation with input parameters is sufficiently consistent(smooth) so that interpolation in a database is possible.
Looking at the results used for linearisation in Appendix B, reveals that some of the caseshave not reached a steady level in the centre of the heating path where the strains areevaluated. As a verification of this, the strains from the individual cases are discussed onthe basis of the graphs in the appendix. As shown in Figure 3.15, the strains are evaluatedin the centre of the heating path (x=O), so that it is crucial that the strains have in factreached the quasi-steady state in that position. According to the convergence tests (figures3.11 and 3.12), this should be the case. However, when the longitudinal distribution ofstrains in Figures B.1 to B.27 (the upper right pane) is considered this is sometimes not thecase. Especially, cases No. 10 and 19 show a large variation along the heated line, and as
7.5 1 5v[mmn/s] 12.5 2 6 12.5 h [mn]
75 3 7125 4 8 17.5
550 650
Table 4.3: Heating parameters for additional test cases.
58 Chapter 4. Empirical Relations
-3'02-04 222"-04
-4520F04 2,52-04
OMVA OMVA-. 52-04 OW, ] ,04 O•
-3 6 7 8o ] 2 3 4 6 7
(A) (B)
Figure 4.11: Comparison of predictions for (A) S. and (B) Bý.
-'52-Ct 8.02-04
•,OE*¢ MVA O2E) MVA
"2-E04 2OE.Ct-ID•
dg ,O4520 04
-tE-Ct.OEO.OENNI 0.0E+08
3 4 3 6 7 H I 2 3 4 3 6 7 8
(A) (B)
Figure 4.12: Comparison of predictions for (A) S. and (B) By.
both cases have low velocity it suggests that velocity is the cause of the problems. This isfurther supported by the lack of a quasi-steady state in cases 28 to 32, which all have verylow velocities. At low velocities, the heat will accumulate in front of the torch and lead to aslowly increasing (i.e. not constant) temperature.
Another possible cause of the deviation is that the process of linearisation is questionable.When the plastic strain penetration is small-which is the case at high velocities and ofthick plates-the plastic strain distribution is poorly represented by a straight line as seenin e.g. Figure B.6. However, in Section 3.6 the linearised strains are applied to elasticanalyses which yield deformation in good accordance with those from the plastic analyses,so this seems not to be a severe problem.
In conclusion, slightly longer heating paths must be employed in future simulations if reliabledata should be deducted, and results from linearisation of low velocity cases should be usedwith care.
Chapter 5
Heating Line Generation
In the previous chapters, it is investigated how the response to heating in a certain positioncan be simulated, both by use of direct elasto-plastic and simplified elastic analyses. However,calculating the result of heating is of little use if it is not known where to heat the plate.Therefore, a scheme for the prediction of heating patterns and parameters is proposed in thefollowing.
A straightforward method is to use an optimisation procedure, in which the artificial tem-perature field analysis (ATFA) from Section 3.6 is employed to vary the position and thestrength of the heating lines. However, this proves to converge poorly, other methods aresuggested in the following.
Two methods for predicting the heating patterns are investigated. One is based on informa-tion on the principal bending directions of the target surface by use of differential geometry,and the other is based on the principal membrane strain directions from 'mapping' the tar-get surface onto a flat surface. The two methods ame not mutually exclusive, as commonpractice in manual line heating is to obtain bending first (by fast source movement) followedby shrinkage (by slower movement).
5.1 Determining Heating Paths
As suggested by Ueda et al. (1994a), the heating paths may be determined from informationon the principal compressive strain direction: Heating along a line will often result in morebending in a direction perpendicular to the heating path than along the heating path, seeTable 3.5 p. 38.
There is a mixture of both bending and shrinkage in a heated region, but when the heatingpaths axe traced, a choice must be made between principal directions from either bending or
59
60 Chapter 5. Heating Line Generation
shrinkage separately, as the information comes from two different methods which can hardlybe combined. Instead they can be applied in succession, so that the lines for bending aredetermined first followed by of the shrinkage lines.
5.1.1 Bending Paths
If the target surface is given as a function (e.g. a spline surface), the principal bendingvectors, 01,2, can be calculated from the eigenvalue problem stated in standard differentialgeometry literature (e.g. Lipschutz (1969)):
[M N] _[ F GF]-=0 (5.1)
where n is the eigenvalue (principal curvature), and {E, F, G} and {L, M, N} are thecoefficients of the first and second fundamental forms, I and II, respectively.
Once the eigenvalues (principal curvatures) have been established, the corresponding eigen-vectors (principal curvature directions) can be calculated from
[ M I N]_ KI [E F]3] [vi] = [0] (5.2)
where n, is chosen to be the numerically largest principal curvature and v is the correspondingeigenvector. As this system of equations is singular (the determinant of the coefficient matrixis equal to zero (5.1)) is reduces to
(L - KiE)vl + (M - niF)v2 = 0 or (5.3)(M- IjF
Vl=S (- -IE 1 , sE1Z (5.4)
If, for instance, the surface, x, is given as a function of x and y as x = xe1 + yE2 + f(x, y)e3then the coefficients of the first and second fundamental form are simply
E= 1 + (f )2
F = f,. f,y
G = 1 + (f,y )2
L = f.,/1 + (fy) 2 + (f,) 2 (5.5)
M = f'ý
V1 + (f,' )2 + (ft )2N = f•
5.1 Determining Heating Paths 61
where ,• denotes differentiation with respect to x.
When the vector field for the principal directions of curvature has been found, it must betraced into heating lines. As mentioned, heating produces more bending perpendicularly tothe heating direction than along it, and therefore the bending paths can be traced perpen-dicularly to the field. It can be carried out as outlined in the next section, with the additionthat the sign of the principal curvature (positive or negative) suggests whether one or theother side should be heated.
5.1.2 Shrinkage Path
According to Table 3.5 of SV/SX ratios, the largest shrinkage direction is not necessarilyperpendicular to the heating path. On the contrary, the heating path must be along theprincipal membrane direction at low temperatures and perpendicular to it at high temper-atures. The principle of tracing the lines is, however, the same and therefore only the hightemperature (perpendicular) case is dealt with here.
As mentioned in Chapter 2, there are various methods for mapping the curved plate onto aplane. Despite the local nature of the deformations induced by line heating, not knowing theposition of the lines in advance implies the use of averaging (not local) mapping methods.Differential geometry can be used, provided that a function for the target surface is known,or elastic finite element analysis can be applied as follows.
To compute the in-plane strains, a model of a flat plate can be forced into shape by means ofspecified z-direction (out-of-plane) deflections, which means that the nodes will move onlyin this direction and not in the x- or y-direction. However, for deflections larger than themagnitude of the plate thickness, this results in too large membrane strains, so that this non-linear geometric effect must be included. This will-on the other hand-introduce deflectionsin the x- and y- directions of the nodes. Thus, the specified z-defiections become slightlywrong as the fixed z-deflection is now specified for slightly altered x- and y-coordinates. Thisdilemma can be solved by reversing the process. If the forced deformation is applied to anoriginally curved plate, the plate is guaranteed to become perfectly flat. Even though thereis an in-plane deflection the required z-deflection of a node does not change-the value muststill be zero. It should be noted, that the found membrane strains are now the negative ofthe ones required.
FRom the strain field of the elastic analysis the principal direction, 0s, and the principalstrains, C1,2 , can be computed from the x- and y components of the strain field, E., C., andE.Y. For convenience, el is the most negative (compressive) of the two. As information on zcomponents of the strain is not needed and this is assumed to be zero the principal direction
62 Chapter 5. Heating Line Generation
is found from the plane strain formulation:
OP, 1 a&ctan(Exu) (5.6)
El,2 = 6 + Ey±ý(O e ) E (5.7)
Eq. (5.6) may be the direction of either of the two perpendicular principal directions in theplane. As the direction for E, is needed, El and E2 can be compared with (5.8) to find theprincipal strains based on the just found angle, 0r, and thus establish which of the strainsbelongs to the direction:
'te ((e. + s,) + (e. - 6,) cos 20p, + E., sin 20p,) (5.8)
If the found Et,,, is equal to e, the angle, 0,, used in (5.8) was in fact the wanted. Otherwiseit belongs to the other principal strain, and " may be added due to the nature of 'arctan'2used in (5.6).
This evaluation of the principal strains and angles is made in all nodes of the finite elementmodel and an addition of 90' is made to find the heating line directions field. A visuali-sation of the directions and shrinkage magnitudes calculated by the shrinkage method fora parabolic plate ( 2
= a (X2 + y2)) is presented in Figure 5.1. The length of the arrowsrepresents the strain magnitude, and a threshold is applied so that only the areas with the30% largest membrane strains are designated for heating.
Before tracing the strain directions into actual heating paths, a requirement for the distancebetween the heating lines must be determined. Since a heating line can only produce acertain amount of bending or shrinkage the distance can be based on an inverse proportion-ality between required strain and distance. If much shrinkage or bending is required, thedistance between the lines should be small. This can be expressed in the following way: Thedisplacement from the heating line (that is shrinkage strain, S., multiplied by zone width,wP) should be equal to the elastic analysis strain, El, multiplied by an unknown width, 4J,which is the distance between the lines of interest:
4L = Lu (5.9)6i
Finally, the heating lines can be traced. Figure 5.2 shows the vector field converted intoheating lines as calculated by ANSYS with threshold and distance criterion applied. This isthe cause of the pattern of alternating short and long heating lines as the distance shouldincrease towards the middle of the plate. The edges of the plate are not perfectly straight,which suggests how the plate should be cut prior to heat treatment. The heating patternagrees well with e.g. Ueda et al. (1994b, p. 246).
Even though this method has not yet been verified by experiments, it is a first step towardsa rational way to determine the position of the heating lines, and as such it is not importantwhether it is perfectly correct, as it can be amended according to further investigations.This is certainly an interesting area to be dealt with in the future.
5.2 Future Work 63
Figure 5. 1: Heating line directions and magnitudes.
Figure 5.2: Heating pattern on parabolic plate as calculated by ANSYS.
5.2 Future Work
Besides the refinement of the above methods, a few thing axe needed to accomplish the taskof automatically deriving line heating information.
5.2.1 Amount of Heating
A method is still lacking for finding the velocity and temperature corresponding to theheating paths above. This can be done by a soft zone method suggested in Ueda et al.(1994a):
The zones at the heating lines are assigned a Young's modulus thousand times smallerthan the surrounding material in an elastic numerical model. Thus, the strains will be
64 Chapter 5. Heating Line Generation
concentrated in the soft zones and yield information on how large strains each heating linemust produce. Subsequently, a comparison to a database (from experiments or numericalanalyses) will reveal which parameters produce the strains in the soft zones.
Again, this is a mere suggestion as it needs further development. However, it seems feasibleas the method of Ueda et al. is used (with modifications) by the IHI shipyard in Japan.
5.2.2 Final Check by ATFA
Once the heating parameters (temperature and velocity) have been determined for each ofthe heating line candidates, it can be checked by means of the combination of artificialtemperature field analysis (ATFA as described in Section 3.6) and the data from Table 3.3p. 32, whether the target shape was obtained to a satisfactory degree of accuracy. If not, ameans of reiteration of the the above procedure must be devised. Otherwise, the data canbe sent to immediate processing in the workshop.
Chapter 6
Experiments
A number of experiments are carried out to establish a basis for validation of the assumedheat flux distribution and of the deformations calculated numerically. All experiments aredone at Seoul National University, where a numerically controlled gantry crane equippedwith an oxyacetylene torch can move about in a given pattern. Along with the torch are alsoa contact-type measuring probe, a water cooling system and a magnetic distance controller.The torch is a type for gas cutting (<YS> 3051 No. 7). A photograph of the machineis shown in Figure 6.1. Temperatures are measured with thermo couples connected to aHewlett-Packard 34970A data acquisition unit with a 16-channel scanner for thermocouples.The thermocouples are created by attaching thermocouple extension cable to the plate bymeans of resistance welding (i.e. by discharging a large electrical capacitor). For details, seeAppendix D.
6.1 Calibration of Termocouples
First, it must be ensured that the thermocouples are calibrated after the crude assembly,where the welding process may introduce impurities and thus cause the thermocouples to beinaccurate. A plate is prepared with 14 thermocouples arranged along an 11 cm long line onthe bottom side of a plate. To produce an even temperature as a basis of the calibration, theplate is insulated with mineral wool and then heated approximately 30 cm. from the therniocouples. A photograph of the insulated plate is shown in Figure 6.2. The plate is heated forhalf an hour so it reaches a temperature of 200 'C. Then the plate-still being insulated-isallowed to cool down. Figure 6.3 shows the time history of the temperatures, and duringcooling they are quite the same. After almost 2 hours the insulation is removed from theupper side. Figure 6.4 shows the error, efT, at each channel, j, as a function of the averagetemperature, T, over all the channels.
2' _ 7-T(6.1)
65
66 Chapter 6. Experiments
Figure 6.1: Heating and measuring machine, iCALM, at Seoul National University.
Figure 6.2: Plate insulation for calibration. The thermocouples are under the piece of mineralwool to the left on the bottom side and the cables.
6.1 Calibration of Termocouples 67
..0 .......... -__ _ _ 2 ------- -
3.180 ...4........... ..........
5
........ .. .. .. . .7........ .... .~~I40 --
4 IU20
*2~~ - -2-
3 .................
40~ ... ... .. .. ~ ) 8 0 0 2
~~~Figure 6.4 : Temperature pretg eito as a function of avedrange temperatureo1nou
betwee the dots
68 Chapter 6. Experiments
where T1 is the temperature measured in the Jth channel. As seen in Figure 6.4, the temper-ature error is smaller than ±2% half an hour after the heating has ceased. This error mayvery well be due to temperature differences in the plate despite the insulation.
This leads to the conclusion that the same error (if any) is made when all thermocouplesare welded. Although a reference temperature is not measured the relative differences aresmall, so it is most likely that the measurement of' temperatures is precise. According to thespecifications for the acquisition unit, the instrument error alone is smaller than 1 TC.
6.2 Heat Flux from Gas Torch
6.2.1 Experimental Set-up
The first set of experiments is designed to evaluate the true heat flux distribution from agas torch. The idea is to heat a plate with a stationary torch and measure the temperatureprofile at a given time, after which the profile is compared to FE simulations to establishthe corresponding heat flux distribution. The same plate as during calibration is used. It is11.5 mm thick and the thermocouples are positioned at the distances from the centre of thetorch shown in Table 6.1 (two extra channels are added since the calibration). Again, thethermocouples are on the opposite side of the plate, as the wires would otherwise melt.
A total of 12 experiments is carried out, in which the torch clearance is varied between 40,50, 70, and 100 rmm with three repetitions each. The temperature profiles for a temperatureincrease of 400 'C are shown in Figure 6.5. The curves represent the rise in temperaturefrom the initial state (which varies due to the cooling period between the experiments) tothe time, when the maximum temperature is increased by precisely 400 'C. This time is onaverage 15.1, 16.3, 28.3, and 74.6 s, respectively. One of the results from the experimentswith a torch clearance of 70 mm is discarded, as it does not fit the other two results.
6.2.2 Temperature to Heat Flux Conversion
To find the heat flux distribution which resulted in the measured temperatures, an optimi-sation procedure is carried out on a FE temperature analysis. As indicated in Figure 6.6 theheat flux profile on the upper side is varied to fit the measured temperatures on the lowerside. This is done with 14 heat fluxes as design variables, the temperature error at each
TC No. 1 12 3 4_ 15 65 7 T 811 9 110 1111 12 113 114 15 16Pos. [nnI1 0 15 10 1u 5 20 2 30 40 150 160 170 180 190 1110 140 170
Table 6.1: Position of thermocouples from the torch centre.
6.2 Heat Flux from Gas Torch 69
350 - Ae
250 ...... .....
E 200 ..I ...... .....
E:504 40 mm2 I0.... . ..........
50......
00 20 40 60 80 100 120 140 160 190
Distance. r Im=1
400
2 Am
S200~ ......
1010 50 mm
5-0 20 40 60 8010M3120 1401IN 18400
N O - ........... 2 2....j250 -
1150. .... mm.50 1 r
50
00 20 40 60 80 100 120 140 160 180
Distanc. r [ns1
Fiue654TmeaueWrfls
70 Chapter 6. Experiments
I %
Figure 6.6: Optimnising heat flux distribution to fit measured temperatures.
.. 350 - .......B 30oponI -----.......
Z250 .. .t 0 ....... ... .......
50-
0 .2 ON OW O A 0,12 0 0.01 0.02 0.03 004 0,05DitNg. . r [ml Distase. r [m]
(A) (B)
Figure 6.7: Clearance 40 mm. (A) Measured and optimised temperature profiles. (B)Optimised heat flux distribution and equivalent Gaussian distribution.
350 M..- 25+0 ~s2Iý .. ... - ----505...... ....... 1........ .......
0 0.02 0.04 0.06 010W 0.1 0.32 0 0.01 0.02 0.03 0.04 0,05
(A) (B)Fgr6.:Clearance 5m.(AMesrdadoptimised temperature profiles. (B)
Optimised heat flux distribution and equivalent Gaussian distribution.
6.2 Heat Flux from Gas Torch 71
Q.,
I Q3i 2
rro r2 r3 r4
Figure 6.9: Definitions for integrated heat flux.
(.10' *
Hoot flux
500 -
200-
0 0O0 I]0 20 20 msDistance fron co~et
Figure 6.10: Resulting heat flux from measurement of combustion gas velocity field. Datafrom Tomita et al. (1998).
of the measuring points as state variables, and a suitable norm of the errors as objectivefunction.
The optimisation procedure leads to a temperature distribution for a torch clearance of40 mm, which agrees well with the measured data, as shown in Figure 6.7(A). The flux itselfis shown in Figure 6.7(B) along with a graph of a Gaussian distribution with a total heatpower, Qý, of 3000 W and a torch width, rm, (as defined by (3.3) p. 12) of 4 cm. Figures6.8(A) and (B) show the same results, but for a 50 mm clearance. The drop in the heat fluxin the centre of the torch can be interpreted as a stagnation point of the torch jet flow, as theheat transfer between a fluid and a solid relates to the velocity of the fluid. A similar resultis found by Tomita et al. (1998), who measured the velocity field from a much smaller torchand converted it into heat flux by means of computational fluid dynamics, see Figure 6.10.
The optimised flux distribution can be integrated, on the assumption that it is sectionally
72 Chapter 6. Experiments
linear (see Figure 6.9), to determine the total power absorbed by the plate.
Qas, = Z- QJ = zf*J- •- r [5f~-1\-, + • - rdrdo2Q" -- ;I r'Q 1
= 2n _7: - r-1 I r 2 -rr j dr+Qff l.. i rddri (6.2)S r j - -r 3 _ 1 - 1 r -
IIrZ{J-1 (2r - 3rfrji + rj_,)/3 + Qi_1(r,2-rj)
where Qj is the total heat power in section j, rj is the jth distance from the torch centreand Q' is the individual heat flux from the optimisation.
By use of (6.2), the effective heat power for a clearance of 40 mm is calculated to be 2850 Wand similarly for the clearance of 50 mm to be 2830 W. It is not possible to find the heatdistribution from clearances of 70 and 100 mm, as the heating times are too long. Thus,the relation between heat flux distribution and temperature distribution is too diffuse forthe optimisation procedure to converge. Using a shorter heating time (and hence lowertemperatures) should ensure convergence for 70 and 100 mm clearances as well.
6.3 Validation of Structural Analysis
Another set of experiments is designed to evaluate the correctness of the numerically deriveddeflections in Chapter 3.
6.3.1 Experimental Set-up
Heating is performed on seven plates of the dimensions 600x600 mm in two parallel lines at1/3 and 2/3 of the width of the plate, see Figure 6.11. The thicknesses and torch velocitiesare in Table 6.2. The torch is kept at a distance of 40 mm from the plates so that the true
Plate h [mm] v jmm/sj1 12.5 3.02 12.5 3.03 9.7 3.04 13.0 4.05 12.8 5.06 13.0 4.07 13.0 4.0
Table 6.2: Forming experiments numbering.
6.3 Validation of Structural Analysis 73
600O
Figure 6.11: Experiment layout.
600
* . . * * . . * 550
* * * *. * * * * .450
* * * * *, * * • 350
600* * * * * * a * 250
* * * * * * *, * 150
* * * * * * * * 50
50 150 250 350 450 550200 400
Figure 6.12: Measuring points. All figures in [mm].
(A) (B)
Figure 6.13: Measured deflections from (A) experiment 2 and (B) experiment 3. Not toscale!
74 Chapter 6. Experiments
Position 50 -150 -200 250 350 400 450 55050 2.102 --0.112 -1.082 -1.144 -1.132 --1.007 -0.075 1.978-150 -1.977 --0.224 --1.218 --1.082 -1.256 --1.094 0.099 1.779-250 -1.816 -0.323 --1.356 --1.53 -1.343 -1.554 -0.448 1.542350 -1.455 -0.796 --1.915 --1.902 -2.188 -1.989 -- 1.02 0.883450 -0.721 -1.753 -2.673 --2.935 -2.897 -2.836 --2.127 0.062550 -0.112 -2.45 -3.744 -4.104 -4.253 -3.756 -3.519 -1.306
Table 6.3: Deflections from plate No. .3. All figures in [mm].
Figure 6.14: Finite element model of experiments.
heat flux distribution is known. Before and after forming a plate, it is measured with thecontact-type probe, so that relative deflections at each point are available. Measurementsare carried out at the points indicated in Figure 6.12.
Unfortunately, only experiment No. 3 (the thinnest plate) yields deflections large enough tobe measured by the contact probe. All the other plates deform less than the accuracy ofthe measuring unit--compare Figures 6.13(A) and (B). The deformation of the plate fromexperiment 3 is shown in Table 6.3.
6.3.2 Numerical Model and Comparison
A model similar to experiment No. 3 is made in ANSYS. It consists of 28740 elements and34334 nodes (see Figure 6.14) and is thus very large compared to the test cases in Chapter 3.It takes about a week to solve the problem on the same computer as the test cases.
6.3 Validation of Structural Analysis 75
4500 SOO.
35)35 - - --- --350-
NOe -
- ISO ....
0-
(A) (B)
Figure 6.15: Measured and simulated temperature profiles.
0.6
I 0.5
I 0.4
0. .1 (1
* " 1 0.2
I I 0.1
0 0.1 0.2 03 04 05 0.6
0.00375 ---U0.3--- -
0.0225 ... ..0,001
0.0075----0
Figure 6.16: Contour plots of numerical simulation results and experimental data.
76 Chapter 6. Experiments
Temperatures are measured on the bottom side during the experiments, and a comparisonbetween those and simulated temperatures is given in Figure 6.15. The (A) graph shows thetemperature as a function of time at a fixed point on the bottom side-both measured andsimulated. The (B) graph shows the temperature profile perpendicular to the heating pathat the time of maximumn temperature on the bottom side. There is good agreement betweenthe measured and simulated temperatures, although the simulated plate cools a little tooquickly (either due to surface heat loss or conduction). The agreement between simulatedand measured deflection described below means that the difference in cooling rate does notinfluence the results. According to the simulation, the maximum temperature on the heatedside, T,.~, is 635 T.
The deformations calculated by an elasto-plastic analysis prove mainly to be governed bypure bending about the heating lines, as this geometrically linear analysis does not accountfor the influence of the membrane stresses on the out-of-plane deflections. Including non-linear geometric effects allows for this coupling between membrane and bending modes, butincluding the non-linear effects directly in the elasto-plastic problem will make the probleminsurmountable for the available computers. To solve this problem, the plastic strains areapplied to an artificial temperature field analysis (as described in Section 3.6), which isso much simpler that the non-linear effects can be handled easily. The strain distributionfrom the elasto-plastic analysis and the linearised strains are presented in Figure B.33 in theAppendix.
For comparison of the two surfaces from experiment 3 and the numerical analysis, a contourplot is made in Figure 6.16. It shows the plate from above, with contours representing theout-of-plane deflections of the plate-each contour is 0.75 mmn apart. The two deformationsurfaces axe aligned (rotated and translated) to account for rigid body motions. The mea-sured deformations are the short lines (as the measuring grid does not extend to the edgesof the plate) and the numerical results are the long lines. It is seen that the results from theexperiment and the numerical analyses are very similar indeed. On the left half of the plot,the contours are nicely aligned. On the right half, the contours do not fit as well due to atwist of the plate in the experiment and stronger bending along the heating line. However,the results are all in all very encouraging.
Though only one deformation experiment was successful, it demonstrates that the numericalmodel is capable of simulating the actual behaviour of a steel plate being heated.
Chapter 7
Conclusions
A number of investigations are carried out during the research on line heating. The conclu-sions reached are as described below.
" A numerical finite element model for the investigation of the behaviour of a platesubjected to local heating is built on the following assumptions/restrictions:
- The material is isotropic and without residual stresses
- Plasticity is modelled by an isotropic hardening law
- Heat loss by convection and radiation are included
- The temperatures are lower than 700 'C (i.e. no phase transition of the steel).
" A parameter study is carried out with this model to obtain relations between heatingparameters and resulting strains. The results of the parameters study are furtheranalysed to find a physical explanation of the mechanisms of the process.
" The strain field caused by the heat treatment can be linearised to consist of bendingand shrinkage contributions along and perpendicularly to the heating path. Hence, thestrain field can be described by four parameters only.
" Empirical relations between the heating parameters and the linearised strains are de-rived. This gives continuous functions which can be used to predict the plastic strainsfor heating parameters not yet calculated. The prediction is fairly accurate as seen bycomparing with direct calculations.
" A method for simplified elastic analysis which yields results equivalent to the fullyelastoplastic analysis is developed. This can be used for fast prediction of the defor-mations of a plate of any shape subjected to given plastic strains.
" A sensitivity analysis shows that the plastic strains are very sensitive to changes inmaterials properties and torch modelling.
77
78 Chapter 7. Conclusions
" The effects of heating near an edge are investigated. The conclusion is that the plasticstrains are almost unaffected by the edge if heating is carried out from the edge towardsthe interior of the plate.
" A rational method for predicting the position of the heating lines is proposed andimplemented. As yet, the method is very simple but being rational it provides a basisfor further development/refinement so that it may take account for residual stresses inthe plate and anisotropy.
* A method of 'soft zones' which can convert the information on heating line positionsinto heating parameters is suggested.
* Experiments to evaluate the validity of the numerical simulations and to find the trueheat flux distribution from a gas torch awe carried out. The findings are that the nu-merical simulations-despite the assumptions- accurately find the same deformationsas measured during the experiments. Further, it turns out to be possible to establishthe flux distribution from measured temperatures on a plate.
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Clausen, H. B., Three-Dimensional Numerical Simulation of Plate Forming by Line Heating,in Chryssostomidis, C. & Johansson, K. (editors), Proceedings of the 10th InternationalConference on Computer Applications in Shipbuilding (ICCAS '99), pp. 387-398, Vol. 2,MIT Press, Cambridge, Massachusetts, USA, 1999.
Clausen, H. B., Numerical Methods for Plate Forming by Line Heating, Ship TechnologyResearch, 47(3):102-109, 2000.
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Geiger, M., Arnet, H. & Vollertsen, F., Laser Forming, in Geiger, M. & Vollertsen, F. (ed-itors), Laser Assisted Net Shape Engineering, pp. 81-92, University Erlangen-Nuremberg,Germany, 1994.
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Iwamura, Y. & Rybicki, E. F., A Transient Elastic-Plastic Thermal Stress Analysis of FlameBending, Journal of Engineering for Industry, pp. 163-171, 1973.
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Ji, Z. & Wu, S., FEM simulation of the temperature field during the laser forming of sheetmetal, Journal of Materials Processing Technology, 78:89-95, 1998.
Kawakami, M. & Shiojiri, H., Thermo-Elastic-Plastic Plate Analysis with a Plate/ShellElement Using ADINAT/ADINA, Computers and Structures, 21(1/2):165-177, 1985.
Kitamura, N., Sakai, Y. & Murayama, H., 3-Dimensional Line Heating System For CurvedShell Plate for Shipbuilding, Technical report, NKK Tsu Laboratories, 1 Kokan-cho Ku-mozu, Tsu, 514-03 Japan, 1996.
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Letcher, Jr., J. S., Lofting and Fabrication of Compound-Curved Plates, Journal of ShipResearch, 37(2):166-175, 1993.
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84 References
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Appendix A
Differential Equations for Plasticv. Ka'rmain Plates
In this chapter the differential equations of a plate with elastic-plastic materials propertiesare derived. The kinematics of the plate is based on the v. Krxm.n plate theory, and thebehaviour of the material is based on the J 2 flow theory with isotropic hardening. The goalis to develop an efficient finite difference formulation to be used for the simulation of lineheating.
Consider an infinitely small patch of a thin plate as depicted in Figure A.l. The patchsubjected to forces and moments must be in equilibrium, which is ensured by means of forceand moment projections. The following derivation is based on the v. K~rman plate theoryas described in Timoshenko & Woinowsky-Krieger (1959) and Pedersen & Jensen (1983). Inthe following, u. denote in-plane deflections and w denotes out-of-plane deflection. Elastic-plastic constitutive relations are from Tvergaard (1997) and partly from Tvergaard (1985).
Nn-IŽlfl0., +Orr
Figure A.1: Definition of forces and moments on an infinitasimafly small patch.
85
86 Appendix A. Differential Equations for Plastic v. Kirmnn Plates
A.1 Equilibrium Equations
First all forces as defined in Figure A.1 axe projected onto the main axes, XZ, x 2 and X3.As regards projection onto the X3-axis, the deflection of the patch has to be considered, asdeflections are assumed to be relatively large.
x1 ON'--dxzdx2 + !-dx2dxl = 0 (A.1)8Xl OX2
ON2 2 ON1 2X2 "g--dx2dxi + N1---dxldx2 = 0 (A.2)
OX2 OXl
X3 : '- "dx1 dX2 + ON23 dx 2dxl + qdxldx2axl OX 2
- -Ow /'+ Ni + ON,2 '\Ow O2 Wd\
- Nn2Odx1+ ""22+ONddX2 xl]+ 02 dx1 (A.3)5Xld2 OX 5 X2Owa 1d2 O 2 d,5-X-l~Nd+ 2 diX -T
H x2 -1
"N2 -Yw x+ (N•,l±O•-•1dx 2 )Kf Ow 02w (E2 \
Ow { ON12 "N {Ow 02w N- N 12 O--dx2 + -+ -dxI i- + W-dxl| dx2 =0
5X2 O12x x / X2 OXX2 /Eq. (A.1) yields
ON11 ON21OaN + x2= 0 or N01 0 = 0 (A.4)
Eq. (A.2) yields
ON22 + =Nn 0 or N 0 2,0 = 0 (A.5)
012 ax1
Eq. (A.3) yields
ON 13 N23 +Oxj
ON,, Ow 02w ON1 1 O2w+ y-xl Ox - + Ni x---l2 + y dxi-'yi
ON 22 Ow 02w ON22 02w2 5 + N2- +.T2 2
ON 2 1 Ow a 2w ON 21 0 2w- -+ N2a---'T-• +i-'x -""'•-Z 03X1 X•1 OIX2 O2 OXlOX2
ON 12 Ow 2w ON12 92w+-y- 5X2 + N 12 T0 - + --- dxl 0 1 0 " =0X oX2 X ozlOX2
A.1 Equilibrium Equations 87
Neglecting higher order terms gives
ON13 ON23 +q+2W N 1l 02 ON22 Ow 8 2w ON21 Ow+ +q + + N, ++ N22- 2 +'Xl x2 ~OX1+Nla x20 ý Ox 2 Ox1
02w 0+/N12 Ow 02w+N 2 10 X- 2- I +N ioX 2 =0
N.3 ,. +q + N.aw,0a +N.#,. w,O = 0 (A.6)
Secondly, all moments acting on the patch are taken about the main axes:
Xl W • axa2 +-2•dx 2dXia*l + N23 dx1& 4 ' . -dX 2 dx 1 4 0 (A.7)ax: OOld 012 OX2
x 2 : aM1 - dX2dN1 - 19 lI1 dxglX 2 - N13 dx24K 2- ON21 dxidxý- = 0 (A.8)
x 3 : 2Nn2dx2 dý + ON•ld ld x -2N' - Ox 2 '2 0 (A.9)
Eqs. (A.7), (A.8), and (A.9) leads to
OM 12 + OM 22 + ON23 OX2 =0ýXl I NX 2 ÷ FX2 2
OM21 + OM1 + N ON 13 OX =0x2 Ox1 axl 2
ON 12 OIl _ ON 12 Ox2 0
8x, 2 ax 2 2
Again, neglecting higher order terms gives
aM12 Oaf 22 ON293 X2 = 0
X - + N2 OX2 249M 2 1 +- 9M H l + N 1 3 + a N 3 a I = 0
Ox2 Ozx Ox, 2
MaP,a +NY3 = 0 (A.10)
and
N12 = N 2 1 (A.11)
88 Appendix A. Differential Equations for Plastic v. K&rmin Plates
Eqs. (A.4), (A.5), and (A.11) inserted in (A.6) yield
ON13 O2N23 ONl Ow 02w ON 22 Ow O2wOx' ON + q + ON" aw + Nn1ir2 + (-9-N--- + N22"-i'X_2ax X x 5x, X X2 8X2 f2
ONll Ow 0 2w ON22 0w 02w
x x+ N 12o1 0X2 0 2 0+XNl2 0 XlX2
0N 13 O9N 23 0 2w 02 w 02wOX1 +- 7 -+q+Nnii--2+N2"-+- +2NI1 ""-2 - =0ax XX
1 'I ý 2 ZOXlJ2
or
N. 3 ,. +q + N.0 w,.0 = 0 (A.12)
By differentiation of each of the two equations represented by (A. 10) by x, and x2, respec-tively, and addition of them M. P,.af +N03,0 = 0 is obtained. Insertion of this in (A.12) leadsto
M =,g,., = N.,9w,t• +q (A.13)
which is the general equation of equilibrium of a plate with relatively large deflections, i.e. thev. KdrmAn equations. To use this in the case of plasticity, the incremental form is needed,which gives (by differentiation with a time-like parameter):
Aý.O,.o = NA0 w,.0 +N.0'tb,.g +4 (A.14)
where * denotes differentiation with respect to the parameter.
A.2 Strains
As the plate is subjected to large deformations, the strain measure f = must be discussedin detail. On the assumption that the strain may be divided into three contributing parts,respectively, from bending E,#, from membrane deflection e,$, and from temperature e.T#the following is obtained
eC0 = e6p + e.0 + eaT (A.15)
orom standard plate theory using the Kirchhoff assumption, the following relation betweencurvature, w,,#, and strain is obtained:
6. 0 = -X3W,a,3 (A. 16)
Regarding the membrane contribution to the strain the strip of a plate deformed as inFigure A.2 can be considered. Its original length dxl is elongated to a new length dl, which
A.2 Strains 89
WD, X3
dxlOw
Figure A.2: Deformation of a strip.
may be described in terms of the two perpendicular sides of a triangle. Thus, di is wvritten
dl • dxl + •'•xdXI) + •'•xldXJ)
= X I +x ý u-' 2 +2Oq q
1'2
+ +-+
dx, _ + 2"
(A. 17)
Expansion by Taylor series gives
Sl dx?{(1+0l• 2 2 +2 + (ZW 2'
I 2 a2 i-, T -
1 [ M -1) o•)
_dy1~ +2' +~
Exasinb Taylo serie gives
Neglecting second order terms except 8wlOxl leads to
( On x 1 ( Ow 2
k Ox, 2 Ox1 }
90 Appendix A. Differential Equations for Plastic v. Kgrmgn Plates
which for ell yields
dx 1 +w 2) x
ell d 1
dz1- a, +½Itl 2
l (A. 19a)
In the same manner the strain in the x2-direction gives
e2= 2,, +w,• (A.19b)
As shown in (Timnoshenko & Woinowsky-Krieger, 1959, p. 385), the shear strain due tobending and shearing is
e12= 2 (u 1 , 2 +u 2 ,1 +w,l W, 2 ) (A.19c)
Thus, (A.19a), (A.19b) and (A.19c) may be written as
e.0 = ½ (u1,9 +uo,a +w, w,p ) (A.20)
In general the strain due to temperature, e. 0 , may be expressed as
e4 = &p&AT
where
6.fl is Kronecker's delta,E is the linear modulus of elasticity (Young's modulus),
& is the linear coefficient of thermal expansion,AT is the difference between a reference temperature and the heated condition,v is Poisson's ratio
All in all the strain in (A.15) may be expressed as
Ea = ½ (u.,# +up,0 +w, wn, 0 ) - x 3w,0 0 +±0 &AT (A.21)
On incremental form this equation is
iap = ½ (ila,0 +u•0, , +tia WO +W,a X36,) -- xa, +6a. ('AT + &T) (A.22)
A.3 Constitutive Relations
A relationship between strain and stress in the material has to be established. Using J2 flowtheory for the plane stress case gives
a.0 = I,0ýý6 (A.23)
A.3 Constitutive Relations 91
where
denotes plane stress variables (except for 6),
,ý.O are incremental stresses,L,,3.6 are plane stress elastic-plastic relations between strains and stresses,if are incremental strains
(A.24)
The elastic-plastic relation, La9.5, may be expressed as the sum of the elastic and the plasticcontribution. Thus,
where
E£°A = 1 - V2 {10 - v) (6e,606 + 656oi) + Vuofldd}
is Hooke's law for plane stress[u LL----E- = for fhgi• _ 0
0 for rh~oi•o < 0
EE E-1
+ E 1-2vEt 3T £oa3z
Lijkt = Cijkt - ILmijmkl
mig = V/1 m
Iik + V 2 2v
0 kksis = 6-0 -4
a• = V/5sijs1 (assuming a v. Mises yield surface) (A25)
Et is the tangent stiffness
92 Appendix A. Differential Equations for Plastic v. K6nrmgn Plates
Insertion of the strain definition (A.22) and the constitutive relations (A.23) in the equilib-rium equations (A.14) still remains. This is done by using the following relations:
h
N•0 = o0 dX3 (A.26)
simply integrating over the current stress state.h
N 0 0 =I: L0 ,.3-, 6 c76dx3
-1 (L00,6 - PrAtihmY) .46 dx3
- x3?b, 6 + &6, (AAT + &t) dx3
h h
- :e.~sgý07 3T 6d3 + e.6 .0, 5/6fd 4X3Z + 6763ýýj: ;f1 0 l
h
h
which leads to
N+•(hZ0 ,6 - A(7 6) - b,76 (A.27)h
where A-=i) J-1 g fnr6 4&xjdx 3 (A.28)
h J (c0 OT6 - Mrn0 ofl 7 6 ) (676 e 6 + 6T
- t 6L,007 6z3d 3 + J &yý1~6P 323dX33 - 675C 001 6X3 dx
+ 6,6A7 4.firi4 6X3dzs - I 74fh.0 7h,6Xsdz + fI iýT6fitiioiityx 3dX3-T h! ~
A.3 Constitutive Relations 93
K
(2 ) T (1)
=tb"6,s T2 o076) + ) M 0 01 6 (A.29)
For insertion in (A.14), k.0 has to be differentiated with respect to op.
Although the Poisson ratio, v, for mild steels varies approximately 50% in the temperaturerange from 0 to 7000C, it is chosen to keep v constant. This ensures that only E, E,, andso, vary through the scope of the plate, and thus the integral of Mk0 differentiated by ao isas follows:
kpfl= ul,0 g7 6 (72~h W316Y) + tb75 +1 E04.(•,as +.,rs) M01 7A 6(1) + (, + _ M) AM ,O) (A.30)
From (A.30) it is obvious that Cj5,a and , must be found. The former is simplyfound by differentiation:
SE'a ,!1-
£o'lw = 1 -V {2(i - V) (6,6,6 + 66-o) + v3 5 676} (A.31)
The latter is found by rearranging part of the integrand of M 0 )76 by means of MAPLE V1:
kc2 kc33( 'E'-- 1) +) (1o ++ • 533s.03 S3 +f " M
/ NS1 + I5
2(1 + v)2 (E' 1)- (,2 1 + 3s 2(E1) Dkc5
(A.32)
~Ewhere v= and E* -
1-2v TI*
'Computer program which derives algebraic solutions from user input
94 Appendix A. Differential Equations for Plastic v. Kiirm6.n Plates
This may be differentiated, with the following result:
A
A
fI(ki,.a rk 2k3 + k, (k2,aojk3i + k2 k3 ,0 4 )) k4k5 - klk 2ks (k 4 ,.flk5 + k4 k5,aoI__ kk) 2 x3dX3
(A.33)
where
k, E - 1
=E,a/I Et - EEt,a/I
kz2 =Es 0 0 - s_____1 + v*
E SIO - S33 *) 6 a0 as C03 E 'Jf
(I1+vj)(1+ V) + v
Az2,.P = E,a4 (sO 833 (92 ~ (saa T~3 (,-)2(1 + v.)(1 + u)J0)/ + \S.'0--m (1 + r)(1 + ii)
k 3 =E s6- s3 (+V T1+ 6,6~
(3. =(,t ~ S3 V)2 6 ) + E ~S-y6a/3 533,a. (1 (V*
Az, 4 taI - (I + vf)l + v) *(d -s V)(1 + V
k4 =E' 2v- 1
k 5. ,a$ 5
Et - Et.3=k,
k5= 21 +9 - 32 (E*- 1)Az eI+ V 2(1 + v) (E* +
1 + 9* (6_ 33. (E* - 1) + 3s23ki,,0 ) 3s 2(E' - 1)l.Az0=
2aa Is + V 2(1 + v) (E' + +v3 2(1 + v) (E* + 2Lj3 i) 2
It is also neccessary to derive ý,,1 and 6,4
S'a0 used in (A.33). These are
6, 6 7.0 = (ik-Yaas +idI+tb'aD 7 W6d+7hi)7W'a006 +W~aaj tb,6 +w, zl,005 (A.34)= 1 t0 AT + ATtas +&,, 0 gT + &,o(A.35)
A.3 Constitutive Relations 95
Inserting (A.26), (A.27), and (A.33) along with the definitions in (A.25), (A.28), and (A.31)inthe equilibrium equations (A.14) yields a fourth order differential equation in u., w, andT, which is to be solved numerically.
batyhd 3 ( + tb,.j 3(6.06,.tr4(2)
+ ( 76,of +6T,,p) M(o) + (•6 + &5) •M ,o
(h 0 6 - 6a.y) - wd,6 W, 0 + [ aadXs] tb, +4
(A.36)
As defined earlier, the strain components &-6, -7TJ are
1.e.= I% (i,6 +i,6,7 +wr W,6 +W,• tw, ) (A.37)
• = &,& (AT + &i) (A.38)
By use of the summation rule for the index notation, (A.36) represents one equation only,although it is very long indeed. The terms in (A.34)-(A.38) marked with dots are the onlyvariables to solve for in each iteration of the solution scheme. Since dotted terms are notmultiplied, the equation is linear in each increment.
The solution of the equation can be reached by means of the finite difference method bydiscretisation of the above systems of equations. This is, however, not done in favour of theease of using a ready-made finite element code like ANSYS.
96 Appendix A. Differential Equations for Plastic v. Krmr.n Plates
This page is intentionally left blank.
Appendix B
Numerical Results
For a description of the panes in each figure, the figures below are referred to. Above the line(A to F) the graphs show how the plastic strain x-, y-, and z-components are distributedalong and perpendicularly to the heating path on the top and bottom sides of the plate.Thus, pane A holds eP, evaluated perpendicularly to the centre of the heating path, whileB holds the same property, evaluated along the middle of the heating path. Panes C, D, E,and F are the same but for Eý and cP, respectively. The G pane shows the P distributionthrough the thickness in the very centre of the heating line. Pane H shows the maximumtemperature 'measured' for each time step. Finally, the I and J panes show how the averagedplastic strains are linearised. The dash lines are the original plastic data, while the thickstraight line represents the linearisation in either {S., B,} or {Sý, Bv}.
Case No.10 1 10 19 B1 2 11 20 50 3 12 21 C D
10 4 13 22hmm 15 5 14 23 1 [0nun/sl E F
20 6 15 2410 7 16 2515 8 17 26 15GH20 9 18 275oo oo• W I J
S5ý00 600 71 OWTmaz
97
98 Appendix B. Numerical Results
...... ...... n ... .............
(M ,- V
4W:2lS -.... 1 ... .0..&031~~42 -002 ~ I___
0 ems 0.01 0.015 0.0 0.025 0.03 0.035 0.0 A015 4.1 -005 0 0.05 0a1 015y [mlx1m
-0~ IX- N y
0.W35 0.030 . 00,005OM OW500303 ON 0 434 5 41 A5 0 00 1 01
0.033 1- 0x33 --
004150.35-0.(3X35
03 0... 5 ... 0.0. 0.02 0025 03 -003 --. 040.00 5 - 0. -0050-.0-0--,
0032 .( ... ... - --
0 0 t4
003003-03-.0320.031 0 0.015 0.W = 0-003 0,30.035 0 30 20 3105 40 50 60. 70.1
0 550 _ _1M ---- SI4
v=5~~~~ tm/fT=O ']
99
-0 ,a-
-0030 .' 0 . ..... ... . ...
.. o .... 3 ...... I-0~4 - ±T
ý . ] .. ...-- .... ... ...
U- t
0 Oa)5 0.02 0.025 0.02 0.025 0.03 0.035 0.041 -0.2 4005 0 .05 0.12 .1
a"53 0."5 02
----- ---- T- 4 W
0..2 ... ..........
0.32 OM~ Ib 0.10M0050.3005O.0 .15- 41 . 0 005 01 01
O WV 0. - MP 1 0.W 3 2- -w -- -- - ----O.W5 --"------.- aM3~-i
M03 I ------ 50 - .-
0.03 -------
+ LM0.2 -- ý -- -
0.014 ---- -- -----
AM03 -003 W0.3 0 0,032 0032 0.03 0.03 0 20 20 30 40 w0 70
Lod te
0 0
& s~I
0.021 -I 0.02-.0.032 -0.02 40."5 0 0.ý5 -0.01 -0032 400" 0 0.M5
Figure B.2: Test case 2. Final plastic strain in x-, Y-, and z-directions for x-, y-, andz-components. Linearised data compared to averaged plastic strain, tn,,. h=l15 [mm],v=5 [zmm/s], T.=500 CC].
100 Appendix B. Numerical Results
0 0
,()M02 ..... W4." --
4 ()W6 --- - ...... ........ 4 (6 - -
-0 42 I I V 4 ." 8..... .... .. ........ ... ... ..
0. .0 ...) 0.. .........0045 00J .01 00.025 00.035 00........00 0 00-.... ..... -0.... ..2-0...........4
0.11 015.
0."2 ~0."52. . .0 ý--X0 -4)- -- ------
4(.~ -04-...C.....
y Wl I m
0.045 . .04350.03 Zmp
0.2 MS 0 O2502
V .042 r0,12 .. -...0.015 I M1
V M Ow ...... V
00" 0.4 .2 005 00 .2 .0 0...........
0.4 0.1 - -0 W5 00.3 0.045O, -0 -0.2 -0.05 0 0.03 0.2 0.25
0 350
T tw
0045-.........................................................
00. ........... .0.0....
0015 -..... ....... .... 0,150 -
v=5ff-DWl.Z~lD3 [m /0 100 M M 4(C5)].7
101
00 --0.0.04-Wn
-0.(M38
....... ... ....... .3M 2 - ...4M4.0334 - .. . .. ..
-0(12 wrz. -4W16 -
ý.0(04 -0(020 .M3 0.01 0.015 0.0 0.025 0.03 0.35 0.0 -415 -01 -005 0 a0.0 0.1 0.15
y Wm m
0.03350."05
-O M00 4-" ....... ....4 01 .0. ----- Z
4032 - -- 0(--..........4 2 -
-0020 0105 0.01 0.015 0,02 0,025 0.03 0.035 0.0 A1,5 -0.1 -0.0 0 0,05 0.3 .1
y (.l x [M]
01045 0.035- ___
0.0335........................................0..3
0(02M 5 -.. ......... ...------ --0.02 0(02
0.03) -- -4 - -.3 -. ------ I--------1
0 .500 O5 D0 .50.03 0.035 0.0 4015 -0 -005 0 0.05 0.1 0.15
I NoC'- 0(4i 350---i-.-- -------0. I~ I -7 j------------ I M
-003 -002-.03 0 0,033 0.02 0.033 0.03 0,035 0 10 w '0 40 5 w6 70
0 1 B, 10 B,
MW5 O.W5 ........0..5...
0.01 0.0'4W035 -0.w] AM00 0 0.M0 -0135 403] AM0 0 .0
Figure BA: Test case 4. Final plastic strain in x-, y-, and z-directions for x-, V-, andz-components. Linearised data compared to averaged plastic strain, Rn.I. h=l10 [mm],
102 Appendix B. Numerical Results
0 0
...........0..4
-. 03 ..... 4 -0.03)2
-0.03) -0.03)6A14
0 0035 00) 0.015 0.02 0.05 0.03 035 0,04 .0. 15 0.' 4005 0 0.05 0.1 0.15
U -.e~ .... 0
...... ..........
0 0.05 0,01 00)i5 00U2 0025 0.03 0.035 0,4 -0.25 A0) 405 0 0,05 a.1 0)15
0.=35 0=3 5......... .... ....
0.032 1 032-
0.30 00 05 - .".. ..'..
0 0 - -- - - - - - - - - -0 0,035 0.01 0.0)5 0.02 0.025 0.03 0.035 0.04 ý.1 01 005w 0 0.05 0)1 0)15
y jal )m
0 550
O' Mj - .150...
0.0" ..-.......
0.0)6
0.1 0
0 B T 0 B
0.05 ...... 1.. ... . . - 0.035 ----' -------- ..
0.015 0.0)5-0.35 40.w) 4"S~ 0 0.ý5 4=035 -0.03 4M835 0 0.ý5
Figure B.5: Test case 5. Final plastic strain in x-, y-, and z-directions for x-, y-, andz-comnponents. Linearised data compared to averaged plastic strain, tP,. h=15 [mm],v=lO [mm/s], T.=500 f'C].
103
4M2 -a"2 7
-awl3 -awl1
0 0035 0.01 0.015 002 0.0 0.03 0,035 004 -015 -O -05 0 0.05 0.1 0.15
0 0."32
-. "3 - 0. - Y"
0.03 ....... 0....33
0-0.06 -0.5 03 -05
y I-)
0am3 ---- -- - ----
0.02-0 .M343 0.01 3 0.0152a O 30.03 4 0W 30 215 3a 0 40 050. 60.1
0.035--.1
x-- 40,05 ~ - _
m 'I
A~2 .w .. .... .. ~ i-
0.015 --
03.............
-0035 -- 0.-- 3 -0= 0 =031 0.3 0"
Figure BA6 Test case 6. Final plastic strain in x-, y-, and z-directions for x-, y-, andz-components. Linearised data compared to averaged plastic strain, iP, h=20 [mm],v=IO [mm/s], T...=0 0' M0C.
104 Appendix B. Numerical Results
4 1J '•. ............. ......
4Wo .- <-.-- ...... ... 012 .... . ..... ...... ._::. .............
01 .. ........ o4 ..... + .. l ' L .......t .....Z .o, ....... ...... ...-...................
0 ,00W5 001 0,015 0.02 0.025 0.03 0,035 0,N• -0.15 401 ý.05 0 0,05 0.1 0.15
y 1m-1Im
0.(• 5 Ow-l•++ , , ,000
0A "0 0 ... .. . ... ... -.... g - - .....
...... . i .........~~ ~~~..... + _ _ . : : .•i .i .... .. , Z - .+ .00 ... ....... ................ .....
A. M l . ....i " T....... .• ...... .. . . ...... ..... .... P ... .... ... :.. ... + . ... ... -...... ..4oM 5 .m ........... A M.. .... !....... ...... ..... .L.....
0 .005 0.01 0015 002 0.025 0.03 0035 0.0 4) 15 ý.1 "0.05 0 0.05 0.1 0.15
y [m] x Im)
OM35 I ' I 0.ý -
0.0025 .. .. T" " " -'" -' -1...................0.2 .............. .. ............
...... .... .. . .. ... .... .OM 5~~~O.0 ...................................
O M : O wl, .......... .. .... ..........
0 O.5 0.10 U"15 O.02 0.M2 0.03 0,035 0.0 4O,15 4.1 A)05 0 0.05 0.1 0.15
y 1m-1Im
0o• • .. . . . . . . . .
- W ... .... .L+., + 4ML ..... ......... . ....... ........ ........... ~ ~ ~~ ..... ...... .... ......
L 0 . .. . . ......L .......... .... .... ....7
0 0 15 ...... . .. .. .• ..
......0 ... L ..... .... .. .- ...- . ..i.. . .. ..... .. ....
• o oow o o , . .. .......... ..F ...... ........... +• ....... ... ..... ..... ......
+.01 +.W]01 4.03 M2 5 W 0. 00005 4.0 10 M10 M 0. M5 70 W ,W IM
FiueB7:Ts a ep W.Fnlpatcsri nayadzdrcn o - - n0-o p n ns 0is rie Baa c m ae o a ea e ls i tan •• ~ O [a ]
.... [ m/ ,T . = 0 ........... OW ..........
105
.............
.1 .0 4 . .....
-0.32 --- W1 - - --- ---10 0.035 0.01 0.015 0.02 0.025 0.03 0.035 0.04 .. 15 Al2 4005 0 0.05 0.1 0.15
0------
00324Y; ----W- - -001-0.2 ...... -..... . -026 --- -
0......~. . ..
........ .. t 02 - .0.... .......
0.032 -I .0,.....3W 2 ... ... .
0ý 5. - ..... . ýW 4.. .. ....
0.033- 0,012.3 0.015 10.03205OM 0.034 0N '01 20 0 Ow 5 0 1 0 .10
O.W~i M I
Fi ur ---t5 ---- --------- --j-
0 .M 0.013 U1 0.W ) W5 0.3.035O 1 40014M 0 O A 5B.S:~~~~~~~ Tetcs 1Fnlpatc tani ---1n -drcin o - - n
z-opnns Lie5sddtMoprdt vrge lsi tan x.h1 m]v=15 [m m /s- T15O 45 C]...... .......... ......... ... ..
106 Appendix B. Numerical Results
0 * 0-. 03 ...-... 4 ..... 2
II T.0 ~ ... ...... -
0 .005 0.01 00)5 0OM 0025 0.03 0U35 0.04 4.15 40. 0.05 0 0.05 0.A 0.15
y (-0 X [M]
-0 OKY)
-004"2 -00
0.00 .,, 0.. .. ..... ---
Vim I VIn)f
0ý .... .. ~ .500.000-
--
~ -o- -
--- -----... ... ........ ......... ............ ......... .
-0.020.00 0k 0.015 ). 0.0025 0.0030.004 0 15 20I 30 40 .50 60. 30.
0.25 7--
V 0.0015 .000 .../3 0 . 0.. 3 V 0.015 -00 T0 5
v=1 [mm/s] 0,.0.O=5O 00 / 0].02 .3005O 5 1ý0 .50101
107
0 0fj5- T
.awl . .. .. ------ -~p-4= - - ------ ---...4
4W16 + 3 +. J0 05.5 0,01 0015 OM O 50,03 0,035 ON-a.15 -al -0.05 0 0.0 0.1 01
yaml xOni
..... .. . .. .
.. ~~~~ ~~ ~ ~ ~ .. ..... .. ....... ........
.005 ....... ..... - - ---- -------..... 1.0 0.05 0.01 0.015 0.0 0.02 0.03 0.035 0.0 -f.15 -a.1 -05 0 0.0 0,1' .1
y [m-1Im
0.... ... ...
005 ----- aW --I.05
0a0l awl 0..2.05.0...
0 0,5-~-
0 00 .1M1 OM Q 5OW O35 0.04 -. 15 -0. -0.0 0 0M0 0a 0&15
...........
- 0b]5 0 .~.-- I
0.0
z-omoens inarde daacmae oaergdpatcsrin 4,.h1 m]
v=5___ [mm] T.=600_ f ]
108 Appendix B. Numerical Results
0704(1X5 -. ...... 4C 4AU)2 - .. ..... W
IM ....... .. ....
-0036 4=8-1~
0 0.03 0.01 0015 002 0025 0.03 0.035 00M 415 -01 -005 0 0.05 0.1 0.15y [ml-Im
-0."5A 5 t ... ...
-0V 1 V
003 0"100,85 OM -0 0,03 - .O500 .3 . 15 ýI 4 5 0 00 1 0
..) .0...0. ...
V 0.03 . -.... 0033
O M- .. . .. 1
0 .5 .1 .15 O ,05 0,3 0.3-0WA15 401 -005 0 0.05 0 I 015y [m.x[m
0 7
0(11.......
0.012
0.014.6. ... .....
40"3 -00 4 "-.3 0 0.032 0.03 0.03 0.03 0 10 20 30w 0 ( 70
0 0
.. .... .0.01 -....... ...... 001 ........... --
09015 0.0I5 _____-0,0315 031 -0jý5 0 0(45 -0.035 -0.03 4".30 0 0,M5
Figure B.11: Test case 11. Final plastic strain in x-, y-, and z-directions for x-, y-, andz-components. Linearised data compared to averaged plastic strain, iP.,. h=15 [mm],v=5 [mm/s], T,.~=600 "C].
109
.0.0f 0,4- - -
.. .. ... ...... I
... .0 6 ... ]. ....
-0.03... .. .. ... 14 -
-00315 -, -
0 0.M3 003 0.015 0,02 0.05 0.0 0.035 0.04t 401 403 005 0 0.05 0.3 0.5
y [m.x[m
0."35 0.0345-0.3 ----- yt 003
0....... .... 0.....- -
0.03 5 . ..... .....- .1I
0 0.03 0.03 0.035 0.0 0.025 0.0 0.035 004 A0l5 41 -005 0 005 0.I 0.15[m x. [ml
0.035 1 T 10.02 -M
0.35--------- --J i 0035 ---I
0033 5 -0,033 0.01 5 .W 0 0ý 0.03 00 5 -0.0415 4003 40 0 005 0. 0.135
Fiue .2:Tstcs 12. Fia(lsi tri nxyan -ie]os o > - nz-opnns iersddt oprdt aeae lsi tan x.h2 m]
v=5 mm/s, T~6OO C]
110 Appendix B. Numerical Results
0 * ,0
-00 3 --- -- L T . .....
.... ... .... ... ...
0 .035 0.01 O.tto 0.02 0.025 0.03 0.3 .4 lt 41 -,5 0 0. 0. 0.AS
0.0351 0.4 015-00
0 -- ý---*--- - L
- -uXL -'-~ -V -0.000....
V -0.0320 005 00 . .2 ...2 0.0 0..3 004 -000 25 -0! 005 0 05-. 45-0 , 0....... 0,......
00 60416 .- .... ...... ... ....
., 3 ....0.......A003 -0.0034 -.- --..... ... .. I
V -0.003 V 40.033 - I .....
ýU0.032 . 03 --
0 0.03 0.01 0.015 0.02 0.025 0.03 0.035 004 41.15 .0.1 -0.05 0 0.05 0.! 0.25
- 005 0005 ..... m!..
-------
0oa 0e
0 7 00
2 03 -t.......
0.0 0.05
0,0! 0.01 ____40,035 -.0,01 40005 0 0.005 -0.0015 40.00! O.M3 0 0.0005
Figure B.i3: Test case 13. Final plastic strain in x-, y-, and z-directions for x-, y-, andz-components. Linearised data compared to averaged plastic strain, Rx, h=l1[nn]v=10 [mm/s], T.~=600 PC]. V
....4 ......... -
. ........ .......
0 0.005 0.01 0,015 0.02 0.025 0.03 0.035 0.04 -A.15 -0.1 .0.05 0 0.05 0.1 0.15y [m-1'D
0000-0.0005-
'4-M
...... 0. 31 ..... ......
0 Oteji 0,01 0.015 0,02 0025 0.0 0,035 0.0 -0 fl,, -0.1 -0.05 0 0,05 01 01
000,1 0.155
00(345 0.0035 ... ..40.034 - ' 0.01 34 .....-...... ..... .. ...... ...
0.0335 .... 0...3 .. ..
0M2. .. ... 4-----3-.on0.03 5 ---- --- . 0.0015 .- . ... .
0 031 IM 0.1005 . I 0.030.3 W-.15 -01 .. 00.0 . 1
00
0. -- ---- 6.............. .
T0."3--
0.01 7--. T 300.012- jI0.014 I I0.0160 J'
-0.03 -0.02 -0.001 0 0,03 0.002 (0.033 0.00,1 0,035 0 to M0 30 Q0 50 60 70
0 0 N"
0.005 -. . . . ... ,,0035- ŽS
0.01 - .... 00. .. .. - 0.0
0.015 0.0350.031 -0.001 400015 0 0iD5 0.0315 .003 001).5 0 0.005
Figure B.14: Test case 14. Final plastic strain in x-, y-, and z-directions for x-, y-, andz-components. Linearised data compared to averaged plastic strain, jP,, h=1 5 [win],-v=10 [mra/s], T.,=600 j"C].
112 Appendix B. Numerical Results
0 ... 0-O-- -0t2- --- ----- I
... .......... .... .. ...
I tr s ... .- . .. . .. . ...
------- -431 ----- ......
.004)M6 -050 0.055 0.01 0.015 00 0025 003 0.035 0.0 40.15 41 I -05 0 0.05 0.1 0.15
y Tm)Kim
0."5 0."5
.. ...3 I.....
0,5 5 0 )M5 . . .. ................
........ . L.... -L -- 02
0 00)5 001 0.015 0.02 0.025 0.03 0.035 0.0 -0.1 40.1 -0,05 0 0.05 0.1 0.1$
0.0535 , , . 00350(3)3 Z n.- ----
0= 0.053 -
0.0 25 -. 2 ........ --- .0.........-
V V
0.015 0.0515 - .. .....
0"05 0."0...)1........
0
-0.2 4052 0.0)1 0.1 0(302 0.0252 0.053 0.0354 0.05 04 10 3`05 0 050 0. 30.
Al.-- .......
00 0I0-N
0.02 000,0515 W 0.051 -0.OR] 5 0 3 0. .tW5 0, 105) w05 Q.~ 0 700
Figur B.5 etcs 5 ia lsi taninxyadzdrcin o > - nz-opnns Lierie daacmae0oaeae latcsrii h2 m]
v .lO [n..../..., .,,4 = O .C.......---- .5..........
113
-0.025 -- -05*32-
4 ) MI .. ]. ..... .. - 0 . 0 2 5 64M2-0.028 -
4W.016 -0.0I0 0=05 0.01 0.015 0.02 0.02 0.03 0.035 0,04 A015 4.1 -0.5 0 0.0 at1 0.15
y[m
0"5~~4*3 -------.w
-OMM ý'E-0.03 -----awl-0142
...................... ... ... .
-0.023 ....... .m .......... I----0/XIS 0 ...........0 0. 0 0,02... . . 02 0,0 005 00....-.1 -. 0...0 .1 01
002'~~~.. ....-w4~00' ~ ....*.
0.. 0....a I0 005 00 001 0.0 0.05 0.3 0.35 004 015 .. .. 0...0.0 0 ..1 0.5
....... [.........
0.022--'~<~ ~I I -~-k.0I0
40240.02 0.02 1 0 .01 0.0210.05 ) 0.0230.2045035 0(N -. 10 20 l -a050 60 70 80 0,10
OW Oep5
t 0 ... .......
If----25--
aw ... . .... LW......
0 00 .m0.021 0 .01 55 0,0 0*1*5 0.03205 O 1 1 405 0 5 0. 0.15
Figur B.x Tetcs[6Mia patcsri nx- - n -ietin o - -0-opnns Lierie dat copae to avrgdpatcsrIn hl m]
v=15... [mm.] T.n....0.. -]
114 Appendix B. Numerical Results
0 0
F- T- . ..
.J ........ .-......
o 0,00 0.01 0005 0.02 0025 0.03 0.035 004 -015 -01 -005 0 0.05 0.1 0.15
0- -
-05 -
0 0."0 0.01 0.015 0,02 0.25 0.03 0.035 0.0 4015 0.1 4005 0 0.0 0.A 0.15
0.004 0.000.035 0.05-
0.W 3 .. .. ... ..-.
V 0.002V 0.002.
0=015 -4 001 000 .'00500 .2 0 050407
O.500 00500 OM00 005Oý-0.15 01 0.05 0 0.05 0.1 0.15[m K [m
0 1 60___0.0 2 .. ......0.00- .0..
..........
0.014 t -- 1.. ....... ...
0.016 0
~O04003020l0 0.000.020.33000.13M 5000 0 10 2M 30 40 9) w 70
0 Bý
0.01 . . . 00)-. ...... 7..7...,. ........ ... ..
0.015 0.05
Figure B.17: Test case 17. Final plastic strain in x-, y-, and z-directions for x-, y-, andz-components. Linearised data compared to averaged plastic strain, tx,,. h=15 [rmmT,v=15 [rmm/s], T.=600 ['C].
115
0 0
4W4.'W7 ., .t ....... ..... ....- ...
0 .M5 0.00,0 o1 n O.M~ O 5o.W 0.035 0.0 4i0 13. 40. 0.05 0 0,05 01 01I 11-12200 .00.5
4 ".28 .t .... , -~ ~ ----- - ------- -. . ... ......- 01.4 26
0 0035 0.0'~~ ~... 0 .0 0.0 0.02 0. 0 0.3..4-02 01 005 0 00
0031 . .02! - --
0 .5 0.1005 M 005-03003 . .25 -0. 0.05 0 0.05 0.2 0.15
0.025 1 sw
0.015 . W122
IM . .... . ...... .. ----- _. ------
0.0~ 04023402202201 00210022.M0.030,0.035 0ý 410 20 3 405 OM 0. 70.
0.2
0.01 - - 1----- -- .....
I I
0.02 ----- ------ -.1 ----~- ---.-.. .
0.02 0,2______40215 4022 -OwS 0 OrnM 40225 0.02 4"l5 0 OwlS
Figure B.18: Test case 18. Final plastic strain in x-, y-, and z-directions for x-, y-, andz-cornponents. Linearised data compared to averaged plastic strain, tP,. h=20 [mm],v=15 [mm/s], T,.=600 fO].
116 Appendix B. Numerical Results
004"2 ~ ~ - ...
0( . ..........0........
4034-4. 4W23
0 .W 0010.15 O. .05 .0'0.52NA5 41. 4)05 0 0.05 0.1 0.15
0.03 - I'St
4 W0 2 -. 0,032 -----....... . .. .-.....
10
.0038.0060403 0.03 2001 O .03400.038 0,0 .0 )2 O 0.10 20 305 40 050 601 01
A 005 0 W 05 4... . ............. .......
M I .... .... w
0450.10 015 5 0. 0XW5 '0.035 '0.035 ONAl .)A 5 00. 01 0.135
y 1-1 [.
Figur B.1: Tet cas 19.Fina platic srainin r, y- and.-dir.t.os.fo x-, - n0-opnns LW2rie daacmae- oaergdpatcsrinyn.hI m]
0= [m %] T '7O I[I II
117
o ------ 0........A w l2 ..... .....2 .. ... .........
.0~ ..... -] .jn-- .A....-4 1
-()Mo 0t6 ' 1 4M1 -.
-0044 -4M6u -.
0O 1 -0AA --)2 T.
0 0.045 0.01 0.015 0.02 0.021 0.03 0.05 0.04 -015 401 -05 0 0.05 0.2 .1
[ml [wl
0.7 0--~------ , W
0.... ... ..
0.0.()M .
o 0045 0.02 0.05 0.02 0,02 0.03 0.035 004 105 42 _()05 0 005 021 0.15
0 ." 4 ..
v 1 T0.."
----- +
0.0I T---~r 7 'O --
2 . ... 0 . ............ 0 .0 .04 0..0) .. .. . 0 0.0 0. .. 0 20 30. 4Ow60l
Fiur -. . . .. 0. ; i ~- --0 . W5 U -0.04 IM 0010.3 O 5 N1
B.20:~~~ ~~~ Tetcs 0 ia patcsri na - n -ietin o - - n0-opnns L7eWsddt oprd oaeae lsi trit, = m]
v=5 2 [m/ 6M70P]
118 Appendix B. Numerical Results
0 -,0
4=OX2 - --P--
-M .... .. ..-.V-00M84)8-
W I2 - 1 .... .....-----
4W(16 .0.014 --0.02W5 0.01 0,015 002 0025 003 0035 004 4.15 -0. 0.05 0 005 0.1 0.15
0.I80)5 --- . . . . . .G M0 .......
-0005.. .. ....
a w - ....... ... ... .. _ ...
4 1-0. 1 5 . .... . ............. .... . 1 .....
-0 (025 -- - -A-O. 2
o O.M2 0.02 O0025 002 0025 0.03 0.035 0,0 -0.15 -Cl 005 0 0.05 0,1 0.25y, [ml X Iro]
0.W)5 0,025-
0.005 ..... 00235-.0.M2 .... 0..... ....... 23 -.
V 0.0225 V 0 6w5
0.=25 .' ... 0,005 --M0.2 0,2 .. .........-.... ......
0."5 . .0... ......
0 -5 00 D500 .2 .3 0 .04 -0.25 -0 .-05 0 005 0.1 0.15
0~ 1 70
00..... .- ... ----- -
0025 .* .k.........0.0 . . ... .
0.02 -00 4 -. 0 00)2 0,02 0.02 0028 0 10 20 30 Q M 60 70
0BB
0.025 .1.. . ......0.015~. . . -
0.025 0.02 1 _-0.2( 002 ,S 0 0ff0 40=5 -0022 4M 0 0.M5
Figure B.21: Test case 21. Final plastic strain in x-, v-, and z-directions for x-, y-, andz-components. Linearised data compared to averaged plastic strain, e?,. hi=20 [mm],v=5 [mm/s], T.=700 P'C].
119
0 0-
............
V 4M8 ...... V -,25--**
4W216 -010250 0.005 0.01 U.15 0.02 0.05 003 0.035 0.04 -. 15 401 -0.05 0 0,05 0.1 0.15
00 -,. ---
-0........0...... - -( ) - ---.....
005-4095 T I -T-
0 .025 0.01 0"15 0.02 0,02 0.03 0,035 0.04 0.1 401 -0.0 0 0.05 0.3 0.1$
0.07 0.02006 -006 ---------- ---- -----
0,025 . .0..........25 . . . .....
0.03I... .... . .... ... ------- 0' --
0 O. .1 0.015 0.02 0.02 0.03 0.035 0.0 -0.15 -0I 00 0 0.05 0.1 0.15
0 7M---2-- K ----r - ---
10............
-0ý402-004-0.0 0 0.M20.02 0.060) 0.01 0 30 20 30 40 M0 w V0
0~ 0 T___
0.01 _1 0 ____4W 0.043 4"S 0 0(00) 40335 0.031 0.(05 0 0.W)
Figure B.22: Test case 22. Final plastic strain in x-, y-, and z-directions for x-, y-, andz-components. Linearised data compared to averaged plastic strain, eb,,. h=10 [mm],V=10 [mm/s], T.~=700 P'C].
120 Appendix B. Numerical Results
0 *0 1
V 4 M. 18 .......... ......- .V .......
il 1 4W514 ..... ..40514 ... .0......
4W 8)6 -4)A 80.05= 0.0' 0.015 002 0025 0.03 0.035 0.04 ' 0.15 -0 I -05 0 0.0 0.1 0.15
y 1-1 m
01025 -- x. . Y12p 0"l -
-0(800
4U-00525
-0.0 W3
.........4. -0 354--------
0 0.055 0.01 0.015 002 0025 0.03 0.035 004 -0.s -01 005 0 0.05 0.1 0)5
0.06 -
0.05 0.5... .....
0.M5 f.*V 0.3 0, 0..03 0.1-M00,0.3 O3 W1
00
0.052 0.N5- ----0.055~~ ~~ 0.0-- 7.M .2 005 .3 005 00 -05 -' 05 0 00 0I 01
[ ...... ......
0 - 1..... ......
0,0126- .1 ...A 0.01458-..... . .......
0.016-
006 0
0.055 0.05
4M5 4I 45 0 OMlS -010) -0.05 -O.l 0 OblS
Figure B.23: Test case 23. Final plastic strain in x-, y-, and z-directions for x-, y-, andz-components. Linearised data compared to averaged plastic strain, iP. h=15 [mm],v=IO [mm/sL, T.= 700 PC]. .
121
0 0-0022 r -0.21'
- j 4W016--1OMI -0.0210 0,025 0.01 0.015 0,02 0,025 0.0 0.035 0.0 -. 15 -0.1 -. 05 0 0.05 0.1 01
y In-1im
0.05 0"
....... . ...........
0 0.025 0.01 0.025 0,02 0.05 0.03 0.035 0.0 -0.13 -. -0.05 0 0.05 0.1 01
lol (.1l 01
0.025 OM- ------- O M.4.. ----- 0-3 -- ------- ---0.023 -tC4 0,23 , r
0 . W34 5 - -- -- - .. 3 5 ~ ~ -.. -.. .-.. ....O M
0. 2 0,2 0.0 0.... 0.0 0.02 0.3 03..4.01 0 .. 5 .5 02 0
N.M
0.025."
-0020030020.021 01 0 .020.22 M23.04=0506 0. 201 2 .0 3 40 0.0 0. 701
0.0-................ .L.... .. -ii0.0 ...- - 0.0...25... ..0.015 -.------* ---------- 0015 .
0.02- ._ __ . 002. .
-002 -0.03 - ---- 00-.0 0.0225 0.1 ,5 0 03
Figure B.24: Test case 24. Final plastic strain in x-, y-, and z-directions for x-, y-, andz-components. Linearised data compared to averaged plastic strain, etn,. h=20 [mm],v=10 [mm/sL, T.= 700 rC].
122 Appendix B. Numerical Results
-00 4- 0."51 -
-0 31 -- V-- - 0.0 -. . .........
-020 0.035 0.01' .1 . OM 00 .15 O5M 4-0 A .05 0 0.05 01 0.95
..... ... ..
4WIYAM
0~00 O03 0.01 0.015 002 OM2 003 0.035 0.041 1 4 00 01 01
y (ml.[m
00M 0'"30" A 8 - ----
0.. ...........0.........OW W7
0.2 0. 03 0.3 0.0- . .. . 005 0. 005 0..0.
0.03 .. . .. .
0 0I02L0.08 OM 0.1005 O .2 .3 005 . 1 . 0 I01 0
0-0
2M KZs-
0.09403( 4M03 -00305 ýý O 0 . 000 001 0.01 1 -.M3W5 70 W 0W I
v=15 5 fmm/s T4="O 0CC].A 45 0
123
0 0
4 W 25 O I L-
0 . .1O05O .025 0.03 0.035 0.04 415 04 4.05 0 0.0 0.1 0.15
4 4-0 04 ------.
0.02 1) 7 .- ... ..
ý.WIJ ------- - M- .....
4M3.023-
0 0,0 5 0.0 0,015 0.02 0,05 0.03 0,035 004 4015 *01 005 0 0.05 0.1 01
00.
.a 0 .-. . .... .....-- --
0.045..........
0.0IV OW301
4021 5 0.01 0.015 ON M 0.0 .3 0X5 4-0415 .0.01 40 5 0 0.0X01 .5
Figur B.26 Tes cae2.Fnlpatcstani.-1- n -ircin o - - n0-opnns LMerie dat copae tavrgdpsictai. .= . .m]
v=1 [ins] .......,=0 WOC]
124 Appendix B. Numerical Results
0 - -- - -
-.--0-- - -------- - - ,
V IM
A(25 * .L.... ......I 4(M2 -
0 .0)5 0.02 0.015 0.02 0.025 0.03 0.05 0.0 -).15 .1).1 .(.05 0 0.05 0.2 0,25y (M] X rn!
~ ~ -0 -. - ----- -- X~
04")2 . .... W
............... ........ ...... .I. .
0 0.05 0,02 0.015 0.0 0.025 0.03 00D35 004 -0.25 -02 -005 0 0,05 021 0215
0.0) ý'*
0.02 0100
0.02 0
-0.0)4.0)0.012( 0.1 0.(0l020.0)30.0.03.U3521ON6 0.20 40 304050& 0 05 OA 0.120
7W I
o o n - ........ . 'I ...... ...... --- ...
0.02 -0.0-0.0)25 UM 000 -. 01.MOM5 0 0.w5" 0 10)2 W 30 4()2 5()~ W 0 W 0W I5
v=15 Imis TI7O rcl
125
-0 .0 -- - .0..... - ----
-------- ---- I ----- -----
V -0.-- ------ ----- .4W002 -
-----2 -- --- - ----- 'I .00 2 ......
....... ..... ............
4).W 7 d 0020 OM02 0.0 0.015 0.0 0.025 0.03 0.035 0.0 -0.5 401 0.5 0.05 0.1 A
[ml 4m ........ , "
0.........
..*... ... . 0
0 OW05 0.02 0.015 0.02 0.025 0.03 0.035 004 -0 I5 0A -005 0 0.05 0,1 0.15y 1m] ~mn
0,012 0.012.
V~~~~--- -~ --- -. ~ , ~ -~I 0'" m ... .......
0 0.02 0.01 0.015 0.0 0.025 0.03 0.035 0.04 A015 -01 405 0 0.0 0.1 0.25
.. ...... -- --- ......
...........................
-0.102 40M5 0 025 &001 0.025 0 20 Q0 W0 W0 2 220
0 0 B
.. 0.2 ...... ..... .. ....... OM05 .
0.01 0.021 --0.022 -0.0 4"S5 0 0.ý5 40D0 4025 402W0.25 4022 40200 0 00fl5
Figure 8.28: Slow case 1. Final plastic strain in x-, V-, and z-directions for x-, y-, andz-components. Linearised data compared to averaged plastic strain, tn,,. h=I10 (inn],v=2 [mm/si, T.n~,=700 PC].
126 Appendix B. Numerical Results
........... -011 2 -
0M.0)00
-O0-
tP -0,003 _0.....-.
-0,007 0 008
0 .0005 0.01 0.015 0.02 0.025 0.03 0.035 0.0 -0.15 -0.1 -0.05 0 0.05 0.1 0.15y [ml x [.In
0.0,900491
000 ......... 00046 -
00453000024 -ýO
000014 -.
000045 0.0) 00)5 0.02 0.025 0.03 0,035 0040.I s 15 1 00 0.05 0r1 0)5
.0042 ..... ...0044-
0..... ..
0.016 0-04-0604-020 0,0420,00440.006 011 0.0) 0 20 40 W N0 I04 [20
0 I10*
0.045 0.045- ........... .....
0.015 0.i100 0 05-0.00)5 -0004) 4000*5 0 0.0005 -00025 -0.002 -0.0015 -0.014) 0005 0 00005
Figure B.29: Slow case 2. Final plastic strain in x-, y-, and z-directions for x-, y-, andz-coxnponents. Linearised data compared to averaged plastic strain, ti,,,. h=15 [mm],v=2 [mm/s]1 T..= 700 P0C].
127
0 0=~5
----~ ---- ...
4M)3 -- - 1.,0 / .05...... .0 h ........... VS 402I-- -
.0 2 64 M3 5 ................. .....
*00212 -- *
-0014 .......-... 4M2 2-
0 fk1"5 0.0' 0.015 0.02 0.025 0.03 0U35 0.0 415 -0,1 4005 0 0.05 0.1 0.15
[wl K [m
0.----- 0.2 -- -- - -
Ybs
40 n2 .-... ----- 0....2 -.......
-0023 43033 -
4"05 -0.5 O .
0 OWS 0.01 002 0025 OM .03 0.035 0045 4 A b 01 01
OM . ..... 025 ----- -
0w:4
VV
0,2
0 0,025 0.01 0.035 0.02 0.025 0.03 0.035 0,04 4.15 -01 4M0 0 0.05 0.1 015
0.015 -.... .
0,01554 -------
0.02 -0 L0.6-004ý AM02 0 OM2 0.02 0036 0."8 0 W 40 6 W 0 IM3
0025- --- -- - 0025 -- - - ----- --
O 0.0 0.01 -
0.015 ... I. ..... ..... ... .. .. ..... 0.015 -0.02 0.02
4W.015 AM02 4"fl5 0 0.0)5 4.02 -0.025 -0.02 -0.) 0 0.ý)5
Figure B.30: Slow case 3. Final plastic strain in x-, y-, and z-directions for x-, y-, andz-coxnponents. Linearised data compared to averaged plastic strain, iP,. h=20 [mm],v=2 [minls], T.= 700 f C].my
128 Appendix B. Numerical Results
AW.88*150
0 0035! 0.01 0.015 002 0025 0.03 0035 004 415 -01 -005 0 00M 0.1 015y [mlx1m
0 0Owl
0,0M24W.33 -
-0,0T w3X)V V
...3 .......
0 00 0.035 0.01 0.015 0.02 0.025 003 0.035 004 -0.15 -0.8 -0.05 0 0.05 0.1 0.85
4W II 0M0.03 0.: -----
0.00.
0.0 8 -.. L .... ... 0.
0:-. W'SWýL ..... 01
0
4038 -003O 0 0(015 OMO M003 0.N5 -035 -0102.005 -003 -0( 0. 04815
Figur B.31 Slwcs[.Fnlpatc tani mM-]n -drcin o m - n0-opnns Lierie dtcoprdoaeagdlsictaIn ,.h20[u,v= [7ms] T- .7.....C]
129
0.0ý5
........- 0~xJ . ..... -- ---- ----- --
4W-0.02 ........
00 4 00' O 0.015 0.02 0.02 0.03 0.035 0.04 -0.15 41 .0.05 0 0,05 0.1 0.15y. .m. .. ....... ml ... ..
-0.) -- w - .c.....*----
VL
0 .0) 0.01 0.015 0.02 0.025 0.03 0.035 0.0 4.15 -41 -0.05 0 0.05 0.1 IS1y [ml'I"
0." 0.0
.~~ .... 'M0--4 ----- - -- .......... -- --- --
0--- -- 3 -- ~ - - -
0,
A 4
0.0204.40)-(04ý4.02 0 0ý=0420.44 O6,04 0,01 0 20 Q W0 W '04 I20
0 B, 0 0-
0.1.....r 0.01-11
0.0415 -1-.01........................... 0,025 1 ... ... .... ....
0.02 00.AW5 4I 4=5 0 0,.) -04=5 0,042 4M4 40405 a
Figure B.32: Slow case 5. Final plastic strain in xn-, y-, and z-directions for xn-, y-, andz-components. Linearised data compared to averaged plastic strain, tx.,. h=20 [mm],v=O-5 [rnm/s], T.,,~=700 P C].
130 Appendix B. Numerical Results
4l.W 2 . -. . ..... .. ... oI ......--
...t ....0.. . . .......
I0~ ...0 . ....
.......... 05
~~003I 1 -003
-ow0........1.2 - ...... ......2---50.035 ~ ~ ~ ~ Ow 0.0 ..0. 0.0 0.02 0.03 0.3 ..0 0..4 0. 0. . , .4 0.5. .
-0 O 5001005 .23W500 005 1N - 0 01.003 OA 053.
-0."3 -0."
O.W7~~ -0......036- .
V .036. .. .... VI~ 0036'
0.035 - oM I. ..... . .. ..
0... ..... L L - l 0032..
0 __0
0 0035 0.01 0015 0.02 0O .02 00 0.035 0.0 0,045 0 01 .2 .3 0.4 0.5 0.6
0 r T 8M
0032 ........ y --x.-
0037 ...... ....
4M08-06 4W04 002 0 032 0 .00400M6003 0 5 0 30 w 350 2W3 25 303
0j 0 B
0.034 0.035
0,021 0.024M.035 41033t 4W005 0 M0.5 -0,025 -0.03 4M.05 4M]03 40Wf5 0 0.005
Figure B.33: Experiment. Final plastic strain in x-, y-, and z-directions for x-, y-, andz-components. Linearised data compared to averaged plastic strain, ePY h9.7 [rum],v=3 [mm/s], T..,0 =635 ["C].
Appendix C
Results of ATFA
This appendix gives the results of validating the artificial temperature field analysis. Foreach of the 27 test cases in Chapter 3 the linearised plastic strains are applied to an elasticanalysis. The following graphs show the out-of-plane deflections at the symmetry line (theheating path) and at the edge parallel to the heating path.
The plates appear to be heated from below (so that it curves downwards), and the curvesshowing most curvature are the ones at the heating line.
131
132 Appendix C. Results of ATFA
00 0
Cm _________- ~ S at,________
4- o .1--
*~~t t,~ii_ 06.2I _
o _m i_-ij K
- 4Ito
't Ito_ _ _ __ _ __ _ _
0.6gure CA: Coprsno.0stp cadpreeatcaayi.2 0es 02s 0. tho6h8
133
- -- - !.o 1
_ _ 2_ I I
____'_ _I 'I , _-.__.. i I I ;•-- -,-06I --2 2 0 00 .6 44 .02 02 0
01.I-t - -! 1
•iue ,: opaio I~lsolsi m ueeatco~ss e•cs hog •
134 Appendix C. Results of ATFA
A 100 _ I --____ o
oI-I
® i i I -] '____ ...o____•;
cal _1-- i.//0\ ---
-01 ....... -..... _• _ -....... I-- o .-.. -.-i-,. , - I- -_....04 -' __ _ _ I .r-- = - .---1-. I ,_ I-
24
Figure.3 C s of I __ -st h- -.-.. .......... . ___
* .
Figure 0.3: Comparison of elastoplastic and pure elastic analysis. Test case 17 through 24.
135
I N
1 U- . 4 G
Figure 0.4: Comparison of elastoplastic and pure elastic analysis. Test case 25 through 27.
136 Appendix C. Results of ATFA
This page is intentionally left blank.
Appendix D
A Resistance Welding Apparatus
Thermocouples are made of K type extension cable (2x0.25 mm type K from Thermocoax).The ends of the cable are welded onto the plate by resistance welding.
The welder is made of a 36 mF capacitor, C1, and a large tyristor. To protect the tyristor,the peak current is restricted by a coil, L, with two windings, see the chart in Figure D.l.Further, the circuit consists of a small capacitor, C2 , to trigger the tyristor, a charging anda discharging resistance, R, and Rf, and a rectifier.
Figure D.2 shows thermocouples welded onto a plate.
A Hewlett-Packard 34970A data acquisition unit with a 34902A 16 channel reed multiplexerfor thermocouples is used to gather temperature versus time data.
iS, R, R,ýW.Id d
T Iz
Figure D.I: Electrical diagram for thermocouple welding apparatus.
137
138 Appendix D. A Resistance Welding Apparatus
Figure D.2: Thermocouples welded onto a plate.
PhD ThesesDepartment of Naval Architecture and Offshore Engineering
Technical University of Denmark . Kgs. Lyngby
1961 Str0m-Tejsen, J.Damage Stability Calculations on the Computer DASK.
1963 Silovic, V.A Five Hole Spherical Pilot Tube for three Dimensional Wake Measurements.
1964 Chomchuenchit, V.Determination of the Weight Distribution of Ship Models.
1965 Chislett, M.S.A Planar Motion Mechanism.
1965 Nicordhanon, P.A Phase Changer in the HyA Planar Motion Mechanism and Calculation of PhaseAngle.
1966 Jensen, B.Anvendelse af statistiske metoder til kontrol af forskellige eksisterende tilnwermelses-formler og udarbejdelse af nye til bestemmelse af skibes tonnage og stabilitet.
1968 Aage, C.Eksperimentel og beregningsmessig bestemmelse af vindkr~after pi skibe.
1972 Prytz, K.Datamatorienterede studier af planende bddes fremdrivningsforhold.
1977 Hee, J.M.Store sideportes indfiydelse pA langskibs styrke.
1977 Madsen, N.F.Vibrations in Ships.
1978 Andersen, P.Bolgeinducerede bevwgelser og belastninger for skib pA lmegt vand.
1978 R6meling, J.U.Buling af afstivede pladepaneler.
1978 S0rensen, H.H.Sammenkobling af rotations-symmetriske og generelle tre-dimensionale konstruk-tioner i elementmetode-beregninger.
1980 Fabian, 0.Elastic-Plastic Collapse of Long Tubes under Combined Bending and Pressure Load.
139
140 List of PhD Theses Available from the Department
1980 Petersen, M.J.Ship Collisions.
1981 Gong, J.A Rational Approach to Automatic Design of Ship Sections.
1982 Nielsen, K.Bolgeenergimaskiner.
1984 Nielsen, N.J.R.Structural Optimization of Ship Structures.
1984 Liebst, J.Torsion of Container Ships.
1985 Gjersoe-Fog, N.Mathematical Definition of Ship Hull Surfaces using B-splines.
1985 Jensen, P.S.Stationxre skibsbolger.
1986 Nedergaard, H.Collapse of Offshore Platforms.
1986 Yan, J.-Q.3-D Analysis of Pipelines during Laying.
1987 Holt-Madsen, A.A Quadratic Theory for the Fatigue Life Estimation of Offshore Structures.
1989 Andersen, S.V.Numerical Treatment of the Design-Analysis Problem of Ship Propellers using VortexLattice Methods.
1989 Rasmussen, J.Structural Design of Sandwich Structures.
1990 Baatrup, J.Structural Analysis of Marine Structures.
1990 Wedel-Heinen, J.Vibration Analysis of Imperfect Elements in Marine Structures.
1991 Almlund, J.Life Cycle Model for Offshore Installations for Use in Prospect Evaluation.
1991 Back-Pedersen, A.Analysis of Slender Marine Structures.
List of PhD Theses Available from the Department 141
1992 Bendiksen, E.Hull Girder Collapse.
1992 Petersen, J.B.Non-Linear Strip Theories for Ship Response in Waves.
1992 Schalck, S.Ship Design Using B-spline Patches.
1993 Kierkegaard, H.Ship Collisions with Icebergs.
1994 Pedersen, B.A Free-Surface Analysis of a Two-Dimensional Moving Surface-Piercing Body.
1994 Hansen, P.F.Reliability Analysis of a Midship Section.
1994 Michelsen, J.A Flee-Form Geometric Modelling Approach with Ship Design Applications.
1995 Hansen, A.M.Reliability Methods for the Longitudinal Strength of Ships.
1995 Branner, K.Capacity and Lifetime of Foam Core Sandwich Structures.
1995 Schack, C.Skrogudvikling af hurtiggiende f--rger med henblik pA sodygtighed og lav modstand.
1997 Simonsen, B.C.Mechanics of Ship Grounding.
1997 Olesen, N.A.Turbulent Flow past Ship Hulls.
1997 Riber, H.J.Response Analysis of Dynamically Loaded Composite Panels.
1998 Andersen, M.R.Fatigue Crack Initiation and Growth in Ship Structures.
1998 Nielsen, L.P.Structural Capacity of the Hull Girder.
1999 Zhang, S.The Mechanics of Ship Collisions.
1999 Birk-Sorensen, M.Simulation of Welding Distorsions of Ship Sections.
142 List of PhD Theses Available from the Department
1999 Jensen, K.Analysis and Documentation of Ancient Ships.
2000 Wang, Z.Hydroelastic Analysis of High Speed Ships.
2000 Petersen, T.Wave Load Prediction-a Design Tool.
2000 Banke, L.Flexible Pipe End Fitting.
2000 Simonsen, C.D.Rudder, Propeller and Hull Interaction by RANS.
2000 Clausen, H.B.Plate Forming by Line Heating.
'04 Ila, my6c)"
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Residual Stresses and Deformations in SteelStructures
Lars Fugisang Andersen
DEPARTMENT OF NAVAL ARCHITECTURE AND OFFSHORE ENGINEERINGTECHNICAL UNIVERSITY OF DENMARK - KGS. LYNGBY
DECEMBER 2000
Department of Naval Architecture and Offshore EngineeringTechnical University of Denmark
Studentertorvet, Building 101E, DK-2800 Kgs. Lyngby, DenmarkPhone +45 4525 1360, Telefax +45 4588 4325
e-mail [email protected], Internet http://www.ish.dtu.dkl
Published in Denmark byDepartment of Naval Architecture and Offshore Engineering
Technical University of Denmark
(© L. F. Andersen 2000All rights reserved
Publication Reference Data
Andersen, L. F.Residual Stresses and Deformations in Steel Structures.PhD thesis.Department of Naval Architecture and Offshore Engineering,Technical University of Denmark, December, 2000.ISBN 87-89502-46-9Keywords: Numerical welding simulation, dynamic meshing,
welding response prediction, experimental residualstress evaluation.
Preface
This thesis is submitted as a partial fulfilment of the requirements for the Danish PhD degree.The work has been performed in collaboration with the Department of Naval Architectureand Offshore Engineering, the Technical University of Denmark, and Odense Steel ShipyardLtd. during the period of June 1997 to February 2001, including six months' leave of absence.The study has been supervised by Professor Dr. Techn. Jorgen Juncher Jensen, whose helpand encouragement have been highly appreciated.
The study was financially supported by the Danish Academy of Technical Science (ATV)and Odense Steel Shipyard Ltd. The support is greatly acknowledged.
Special thanks to Lars-Erik Lindgren at LuleA University of Technology for inspiring discus-sions and a pleasant stay. Also thanks to my co-supervisors Lars Malcolm Pedersen, AnnBettina Richelsen and Henning Kierkegaard for their interest.
Thanks to my colleagues at ISH, friends and family for invaluable help and support. Specialthanks to Linda for her support and understanding during the entire study.
Executive Summary
The objective of the present thesis is to contribute to the understanding of process-relateddistortions and stresses in steel structures. The following main aspects are covered:
1. Production strategy emphasising distortion minimisation
2. Efficient numerical simulation of welding
3. Experimental method for residual stress evaluation
The work concerning production strategy primarily serves to put welding simulation andresidual stress evaluation into an overall perspective and to describe the factors motivatingthe present work. However, instead of merely describing the problems caused by geomet-rical distortions, a new production strategy is proposed with the emphasis on the effect ofdistortion minimisation. The intention is merely to establish a basis for discussion of thetechnological paradigm of a shipyard and its relation to competitive power.
The objective of the second part of the thesis is to enable welding response prediction inlarge-scale industrial welding applications. This objective is achieved by the developmentof a dedicated 'welding simulation package' using a commercial software as platform for thesoftware development. Initially, a basic model is established allowing the essential weldingmechanics to be captured. Dynamic activation of fillet elements, dynamic coupling of parts,contact modelling and tack weld modelling are identified as essential factors for accurateprediction of the distortion field. On this basis, the computational efficiency is subsequentlyincreased by the implementation of a graded element and the development of an efficientdynamic mesh refinement scheme. An investigation of the mesh density in dynamic meshingshows that good predictions of the qualitative distortion pattern may be obtained withfew elements if the mesh density is appropriately distributed. The experienced increase inefficiency is considerable and allows not only models to be computed within shorter time butalso to simulate welding applications which were previously far beyond the computationalcapability.
For further increase of the computational efficiency, a template combining several techniquessuch as dynamic meshing and substructuring is developed. The combination of local solid
iii
iv Executive Summary
models and a global shell model makes it possible to represent properly the boundary condi-tions for the weld region and to calculate the distortions in a large-scale structures by takinginto account variations in structural stiffness, welding sequence, tack weld positioning, weld-ing parameters etc. The basic capability needed for a welding response prediction tool isthus established.
Figure 1: Predicted shape of a fully welded assembly (scale 15).
The objective of the third part of the thesis is to enable residual stress evaluation in steelplates in order to allow the potential influence of residual stress on process-related distortionsto be evaluated. A short review of existing experimental stress measuring methods shows thatnone of these are suitable for the purpose, and a modified incremental hole-drilling techniqueis therefore developed. An analysis of the hole-drilling configuration reveals that considerableimprovements may be obtained by optimising the hole-drilling configuration to plates of finitethickness. This results in a set of guidelines improving the stress calculation accuracy, whichis supported by an extensive error analysis. As the incremental hole-drilling technique doesnot allow the residual stresses to be resolved through the entire plate thickness, it is modifiedto include two coupled measurements which solve the problem.
Synopsis
FormAlet med nwrvwrende studium er at bidrage til forstAelsen a! procesrelaterede deforma-tioner og sp~endinger i st~lkonstruktioner. Arbejdet er inddelt i folgende hovedomrAder:
1. Produktionsstrategi fokuseret pA deformatiousminimering
2. Beregningseffektiv sirnulering af svejsedeformiationer
3. Eksperimentel metodeudvikling for evaluering a! residualspaendinger
Arbejdet omhandlende produktionsstrategi, saetter svejsesirnuilering og residualspaendings-mAling ind i et overordnet perspektiv og beskriver de faktorer, der motiverer dette studium.Derudlover foreslAs en ny produktionsstrategi med fokus pA minimering a! deformationer.1-ensigten er at skabe et grundlag for diskussion af et skibsvarfts tekuologiske paradigme ogdets relation til konkurrenceevue.
FormAlet med anden del af studiet er at gore det muligt at forudsige svejsedeformationerog spamdinger i relativt store indlustrielle svejseapplikationer. Dette m~l nAs gennern etab-lering 31 en dedlikeret 'svejsesirmuleringspakke', der udarbejdes med et kommercielt softwaresom platform for programudviklingen. Indledningsvist etableres en grundmodel, der tilladermodellering af de va~sentlige svejseeffekter. Dynamisk aktivering af tilsatsmateriale, dy-namisk kobling af de enkelte dele under svejsning og kontaktinodellering identificeres somvwsentlige faktorer for bestemnmelse 31 deformationsmonsteret. Baseret pA denne grund-model foroges beregningseffektiviteten herefter ved implementering af graduerede elementerog udvikllug 31 dynamiske netgenereringsalgoritmier. En analyse a! netfinheden viser, atdeformationsnwnsteret med god nojagtighed kan forudsiges ved anvendeise 31 relativt fA ele-menter, sAfrerut disse fordeles hensigtsmiessigt. En v--sentlig effektivisering opnis herved ogtillader ikke kun hurtigere beregning men ogsA sixunlering af svejseapplikationer, der tidligerevar uoverkomnmelige.
For yderligere at 0ge beregningseffektiviteten udvikies en metode, der kombinerer forskel-lige teknikker som f~eks. dynamisk netgenerering og superelementer. Korubinationen a!en global skal-element model og lokale 'solid'-modeller tillader passende modellering 31randbetingelserne for svejseomrAdet saint deformationsberegning i sammensatte konstruk-tioner under hensyntagen til variationer i strukturel stivhed, svejseraekkefolge, placering a!
v
vA Synopsis
haeftesomme, svejseparametre niv. Den grundI~ggende funktionalitet som er nodvendig foret generelt svejsesimuleringsvaerktoj er hermed etableret.
Formn~let med tredje del af rapporten er at evalucre residualspaendningstilstanden i st~lpladerfor herigennemn at kunne estimere deres potentielle indflydelse pA procesrelaterede deforma-tioner. En hurtig gennemngang af eksisterende eksperimentelle metoder til residualspa~nd-ingsrnuling viser, at disse ikke er umniddelbart egnede til formAlet, og en modificeret inkre-mentel hulboringsmetode udvikies som folge heraf. En analyse af hulboringskonfigurationenafslorer, at viesentlige forbedringer kan opuAs ved optimering af konfigurationen til pladeraf endelig tykkelse. Dette resulterer i et s.Tt retningslinier, der minimerer usikkerhedenispa~hdcingsberegningen, hvilket komnmer til udtryk i den efterfolgende og omnfattende fe-
jianalyse. Idet inkremientel hulboring ikke tillader residualspmndingsm~ling gennem helepladens tykkelse, modificeres denne til at omnfatte to koblede mAlinger, hvorved problemetloses.
Contents
Preface i
Executive Summary iii
Synopsis (in Danish) v
Contents vii
Symbols xiii
I Introduction and Motivation 1
1 Introduction 3
1.1 Objectives and Background ...... ........................... 3
1.2 Structure of the Thesis ..................................... 4
2 Production Strategy Emphasising Distortion Minimisation 5
2.1 Background .............. .................................. 5
2.2 Definition of Strategy and Technology ............................ 6
2.3 Problem Definition ....... ................................ 6
2.4 Proposed Technological Strategy ....... ........................ 9
2.4.1 A Common Technological Trajectory of Shipbuilding ............. 9
2.4.2 Radical Approach .................................... 10
vii
viii Contents
2.4.3 Short- and Long-Term Aspects ......................... . 14
2.4.4 Gains and Critical Issues ............................... 14
2.5 Selected Subjects for the Present Thess ......................... 15
II Efficient Modelling and Simulation of Welding 17
3 Scope of Modelling and Simulation 19
3.1 Focus in the Field of Welding Research .................... 19
3.2 Target Application ...... .......................... 21
3.3 Intermediate and Final Objective ........................ 23
4 Modelling and Simulation of Welding 25
4.1 Integration in Commercial Codes ....... ........................ 27
4.2 Geometrical Aspects ........ ............................... 27
4.3 M aterial M odelling ......... ........ ............ ... 28
4.4 Numerical Aspects ........ ................................ 31
4.5 Finite Elements and Mesh Grading ............................. 33
4.6 Thermal Modelling and Simulation ............................. 37
4.6.1 Physics of Arc Welding ................................. 37
4.6.2 Heat Source Modelling ................................. 38
4.6.3 Thermal Model ........ .............................. 42
4.7 Mechanical Modelling and Simulation ...... ..................... 45
4.7.1 Dynamic Coupling and Contact Modelling ................... 45
4.7.2 Simulation Results .................................. 48
Contents ix
5 Dynamic Meshing 51
5.1 Computational Efficiency .................................... 51
5.2 Mesh Refinement Scheme .................................... 53
5.3 Data Mapping ........................................... 55
5.4 Mesh Density in Dynamic Meshing ...... ....................... 58
5.5 Double-Sided Fillet Welding ....... ........................... 64
5.6 Obtained Simulation Efficiency ....... ......................... 67
5.7 Effect of Tack Weld Positioning ....... ......................... 70
5.8 Experimental Evaluation .................................... 71
5.9 Summary of Chapter 5 ........ .............................. 73
6 The Local/Global Template 75
6.1 Concept of the Local/Global Template ........................... 76
6.1.1 Model Geometry ........ ............................. 79
6.1.2 Combining Solids and Shells ....... ...................... 79
6.1.3 Utilisation of Substructures ...... ....................... 80
6.1.4 Updating of the Global Model ............................ 81
6.1.5 Extraction of Boundary Conditions ........................ 83
6.1.6 Capability of the L/G Template ........................... 84
6.2 Distortion Prediction in Large Assemblies ...... ................... 85
7 Improved Material Modelling 93
7.1 Microstructure ......... .................................. 94
7.2 Material Model 2 ........ ................................. 98
7.3 Material Model 3 ........ ................................. 101
7.4 Transformation Induced Plasticity .............................. 105
7.5 Influence of Material Models and TRIP .......................... 107
x Contents
8 Conclusions and Recommendations for Part II 113
8.1 Conclusions ......... .................................... 113
8.2 Recommendations for Future Work ...... ....................... 115
III Experimental Method for Residual Stress Evaluation 117
9 Residual Stress in Steel Plates 119
9.1 Influence of Residual Stress ................................ .119
9.2 General Approaches to Residual Stress Evaluation ................... 120
9-3 Categorisation of Steel Plates ....... .......................... 124
9.4 O bjective . .. .. .. .. . .... . ... . ... .. .. .. .. .. .. .. . 124
10 Selection of Method for Modification 125
10.1 Requirements ........ ................................... 125
10.2 Selection of Experimental Method .............................. 126
10.3 Basic Hole-Drilling ......................... .... ... 127
10.4 Incremental Hole-Drilling ............................. 130
11 The Integral Method 133
12 Modified Hole-Drilling Technique 139
12.1 Investigation of Hole-Drilling Configuration ........................ 139
12.1.1 Characteristics of Conventional Incremental Hole-Drilling ....... .139
12.1.2 Hole-Drilling Configuration for Plates of Finite Thickness ....... .140
12.1.3 Summary of Investigation ....... ........................ 144
12.1.4 Proposed Hole-Drilling Configuration ....................... 145
12.2 Calculation of Correlation Constants ...... ...................... 145
Contents xi
12.2.1 Finite Element Model ....... .......................... 146
12.2.2 Strain Integration over Strain Gauge ....................... 148
12.3 Stress Measurement through the Plate Thickness ................... 152
12.4 Example of Experimental Measurement ...... .................... 155
12.5 Milling Stresses ........ .................................. 157
12.5.1 Experimental Investigation of Milling Stresses ................. 161
12.6 Experimental Precautions ....... ............................ 162
12.7 Error Calculation ........ ................................. 163
12.7.1 Strain Errors ........ ............................... 165
12.7.2 Hole Depth Errors ....... ............................ 167
12.7.3 Hole Radius Errors ................................... 168
12.7.4 Correlation Constant Derivation Errors ..................... 169
12.7.5 Material Parameter Errors .............................. 170
12.7.6 Other Error Sources ....... ........................... 171
12.7.7 Overall Accuracy in Stress Calculation ..................... 171
13 Example of Experimental Investigation 175
13.1 Plates and Plate Manufacturing Processes ........................ 175
13.2 Experimental Procedure ....... ............................. 176
13.3 Results ............................................... 178
14 Conclusions of Part I1 183
Bibliography 185
A Details of Stress Error Calculation Analysis 191
A.1 Details of Strain Errors ........ ............................. 191
A.2 Details of Hole Depth Errors .................................. 195
B Representative Depth 201
List of PhD Theses Available from the Department 203
xii Contents
This page is intentionally left blank.
Symbols
Abbreviations
CCT Continuous cooling transformationCCW Counter-clockwiseDOF Degree of freedomFE Finite elementFEA Finite element analysisFEM Finite element modelFZ Fusion zoneHAZ Heat affected zoneHSS High-speed steelLSP Least square projectionL/C Local/globalMAG Metal active gasTRIP Transformation-induced plasticityTTT Time-temperature-transformationWPS Welding process specification
Symbols in Part II
Roman Symbols
A1 Highest temperature in the phase transformation temperature rangeA3 Lowest temperature in the phase transformation temperature rangeA3-,1 Phase transformation temperature rangeC Elastic stress-strain matrixE Young's modulusEt Tangent modulusH Hardening modulusI Amperage[K) Global stiffness matrix[M] Diagonal matrix
xiii
xiv Symbols
M# Material model #M#T Material model # with TRIP includedNj Shape functionQ Effective energy input rate{Q} Global load vector[Q] Matrix of shape functionsT TemperatureATs/ 5 Cooling time between 800 and 5000CTXak Austenitising peak temperatureU VoltageV VolumeY Yield stress of weaker phasea,, a2 , b, c Descriptive parameters of heat sourceff, fr Fractions of heat deposited in the rear and the front of the heat sourceft,. Volume fraction of austenite transformedh Combined heat transfer coefficienthý Convective heat transfer coefficient{q} Global displacement vectorq;, q, Power density in the rear and the front of the heat sources Deviatoric stresst Timev Welding speedx, y, z Cartesian coordinates
Greek symbols
a Yield surface shift stressf, y, t, Parameters of the Avrami equationC. C Accumulated ineffective strain while the element is dead'd Elastic strain1P Plastic strainEt h Thermal strainftot Total strainftrp Transformation-induced straintEr Emissivity77 Heat source efficiencyu Poisson's ratioý, 77, ( Normalised coordinatesa Stress(7b.1 Stefan-Boltzmann constanta•e Effective stressa, Effective stress corrected for the shift stressaer Trial stressgy Yield stress
Symbols xv
7 Heat source position at time zero, (A* Field quantities of old and new mesh, respectively
Xý Phase fraction already transformed in calculationA X Transforming phase fraction
Symbols in Part III
Roman Symbols
A, B Calibration constantsCe Covariance matrix of vector {u}CUV Cross covariance matrix of vectors {u} and {v}E Young's modulusF Nodal circumferential forceL9 Length of strain gauge[N] Matrix which relates measured strain to cartesian stressP, Q, T Stress variables used for decoupling of equationsR. Hole radiusRa,r The hole radius where the 'near edge' becomes dominant for a given R,Rý Radius of mean strain gauge circleTP, Plate thicknessa, b Correlation constantsc, d Correlation constants used for the second half of the plate thicknessf Function representing the qualitative variation of E, with hole depthh Hole depthh9 Half width of strain gaugeh.x The maximum hole depth at which stress can be measured
Subscript; number of increments drilledSubscript; increment in which the stress is acting
p, q, t Strain variables used for decoupling of equationsr, 0 Radii and angles related to gauge geometrys Standard deviationu Nodal displacementz Intermediate depth between surface and hole depth h
Greek symbols
a Angle measured CCW from gauge 1 to the direction of am..)3 Angle measured CCW from the direction of a,, to Er
f 1, E2, E3 Relaxed surface strain at gauge 1, 2 and 3Er Relaxed radial surface strainfý Average longitudinal strainv Poisson's ratio
xvi Symbols
o-•q Equibiaxial stress component6ý,• Maximum principal stressesUmin Minimum principal stressesU.0o.r Nominal equibiaxial stress6sh Pure shear stress component713 Shear stress
Part I
Introduction and Motivation
1
Chapter 1
Introduction
1.1 Objectives and Background
The objective of the present thesis is to contribute to the understanding of process-relateddistortions and stresses in steel structures with the emphasis on production methods, weldingmechanics and methods for residual stress measuring.
The importance of process-related stresses and deformations is exemplified by describing theimpact of geometrical distortions on shipbuilding. Instead of merely describing the problemscaused by geometrical distortions, a new production strategy is proposed emphasising theeffect of distortion minimisation.
The problem of process-related geometrical distortions has, however, relevance to industriesother than shipbuilding - especially considering welding. The welding technology, being themain joining technique used in industry, has significant influence on geometrical inaccuraciesand there is a general desire to integrate welding response prediction in preliminary designtools.
Methods for prediction of mechanical welding response are usually subdivided into empir-ically based methods, numerical methods or combinations. Empirical methods suffer fromthe amount of empirical data required and the lack of flexibility in predicting the responsefor new problems. The numerical methods provide the flexibility but are compromised bythe computational complexity, the lack of material data and the demands for computationalcapacity.
Research within the area of numerical welding simulation has been highly focused on weldquality, leading to a successive refinement of methods for modelling complex thermal, struc-tural and metallurgical effects occurring in the weld zone of small components. Little re-search has been directed towards methods for distortion prediction in true-scale components
3
4 Chapter 1. Introduction
in which detailed information about weld quality is exchanged for knowledge of macrode-formations in the full structure. Even though encouraging results have been reached, a lotof research is still needed before the mechanical welding response can be predicted for mostindustrial applications, and even further progress is required before the methodologies areefficient enough to meet the demands of preliminary design tools.
The scope of the present work is primarily to establish methodologies for welding responseprediction and to increase the computational efficiency of numerical welding simulation.Moreover, templates for prediction of structural response in large structures should be de-veloped based on localised modelling. Hence, emphasis is put on predicting the qualitativeresponse of large structures in contrast to a detailed analysis of a localised problem.
The last problem treated in the present work deals with experimental methods for residualstress evaluation in steel plates. Residual stresses which originate from the plate manufac-turing process are believed to influence the residual distortions resulting from welding andcutting operations. As a first approach the stress levels and distributions should be estimatedroughly in order to evaluate the error of excluding initial residual stresses from the analysis ofwelding and cutting distortions. For this purpose an experimental stress measuring methodis developed for evaluation of residual stresses in medium thick steel plates.
1.2 Structure of the Thesis
The contents of the thesis are presented in three parts composed as follows.
The primary aim of Part I is to establish a basis for discussion of the technological paradigmof a shipyard and its relation to competitive power. A conventional technological strategyis discussed and compared to a production strategy emphasising distortion minimisation.Secondly, Part I puts the objectives of Part II and III into an overall perspective.
The scope of Part II is to establish a model allowing the essential welding mechanics to becaptured in fillet welding. Subsequently, different techniques are employed to increase thecomputational efficiency to a level where large-scale industrial applications may be inves-tigated in terms of welding deformations, taking welding sequence, tack weld positioningetc. into account.
Part III deals with experimental methods for residual stress measurement in large, medium-thick steel plates. An experimental method is developed for evaluation and categorisationof steel plates as regards residual stress.
Chapter 2
Production Strategy EmphasisingDistortion Minimisation
The aim of the present chapter is to discuss the technological strategy of a common shipyardwith regard to future development of technology and competitive power. With the focus onproduction technology, a technological strategy which emphasises distortion minimisation isproposed and compared to a conventional strategy.
2.1 Background
In the 1970s and 80s European shipyards virtually lost the leading position of the market.However, over the past 10 years Europe has regained a relatively strong position based uponthe niche of specialised ships. Many European yards were shut down in the 1970s and80s and in retrospective, it is recognised that these were the yards which did not receivegovernment subsidies or the yards which continued at a traditional technological level wheregood craftsmanship was the only competitive factor.
The shipbuilding market is extremely competitive, and the industry itself estimates todaya surplus production capacity of approx. 30%. In the market of large cargo carriers, veryfew innovative product technologies have led to marked differentiation. In other words,Korean or Japanese yards are capable of delivering products at the same technological levelas European yards and vice versa. Hence, product quality is hardly a competitive parameter,especially as the use of recognised classification societies ensures a uniform minimum qualitylevel. As a consequence, the industry may be regarded as matured and key technologiesare production technologies rather than product technologies. The conclusion for Europeanyards with regard to the market situation and competitiveness is simple. It is necessary toimprove production efficiency significantly in order to survive at a market with importantexternal influential factors as fluctuating currencies, taxes and government subsidies.
5
6 Chapter 2. Production Strategy Emphasising Distortion Minimisation
2.2 Definition of Strategy and Technology
The terms 'strategy' and 'technology' are often misused and are therefore defined below.
Technological strategy is the understanding and creation of action patterns which leadto the desired goal through technology (Stacey [64]).
Interpreted in this way, strategy is not only the stated strategy, but also the actual patternsof actions and the reasons behind them. Creation of technology is defined as:
Creation of technology covers new processes but also the combination of new as well asexisting processes through knowledge.
That is, technology is both processes and patterns.
2.3 Problem Definition
As ship production involves many tasks, various problems could be focused on with significantimpact on the competitiveness of a yard. Before the proposed technological strategy isoutlined, it is therefore appropriate to discuss the problems to be solved.
In the imlplementation of new technology, value and non-value adding processes in regard tonew technology should be considered. It is a well known fact that more than 70% of the ship-building costs is related to the non-value adding processes of moving materials, equipment,information and people around. When new technology is introduced in the production it isseldom aimed at these non-value adding processes. On the contrary, the focus is onl reducingthe smaller percentage of time where actual value is added to the ship, e.g. through cutting,joining, fitting and painting. In order to increase the production efficiency by e.g. automa-tion, it is necessary to perform a thorough analysis of the consequences of the automatedprocesses. The reduced process time should be compared to the time used for e.g. extrahandling and fitting. Very often the analysis reveals that savings on value adding processesare cancelled by an associated increase in the non-value adding processes, as the processchain becomes more complex in order to facilitate the new technology.
The structure of a contaliner vessel contains thousands of elements, which are produced andassembled in a complex pattern. Today, ships are built by the block assembly productionmethod (Figure 2.1), according to which the grand-blocks are assembled of blocks, assemblies,sub-assemblies and elements produced in various manufacturing areas. The grand-blocksare subsequently transported to the dock for final assembly. This method facilitates parallelproduction, where many related tasks are performed simultaneously.
2.3 Problem Definition 7
Figure 2.1: The block assembly production method.
As a result of the block assembly production method, ship production is closely relatedto logistics. Logistics are a key task in the improvement of the production efficiency as it
addresses the non-value adding processes (the 70%). The primary task in logistics is to
optimise the capacity usage of existing production facilities, based on the elements definedin the structural breakdown. The structural breakdown is, however, a compromise between
the consideration for production facilities and factors related to the operational performance
of the product, such as strength, cargo capacity and classification rules. Automation enters
the task of logistics as it influences the structural breakdown in terms of facilitation for
automated production, but more important, it may introduce new production facilities with
radical consequence for logistics. Thus, logistics at a shipyard involve many factors, especially
if the production facilities are considered to be variables. A serious obstacle in optimisinglogistics is the large degree of unpredictability in production time. A careful estimate ofthe unpredictability of some of the key activities is ± 30%, which necessitates operationwith large buffer times. The main reasons for the severe degree of unpredictability in theproduction time are
* Geometrical distortion of structural elements due to thermo mechanical processing,
e.g. cutting and welding.
* Poor set-up of assemblies prior to welding, resulting in geometrical deviation.
* Insufficient quality in the performed processes, e.g. welding defects and bad cutting ofthe edges etc.
As the most significant factor influencing production time, geometrical distortions are con-
sidered in the following. The first obstacle, is that distortions generally are accepted as
unavoidable in common shipbuilding practice and only severe distortions are noticed in the
production. Hence, it is difficult to establish a reference of acceptable levels of distortions.
8 Chapter 2. Production Strategy Emphasising Distortion Minimisation
Secondly, distortions at one level of the production have extensive consequences at subse-quent levels as the geometrical deviations are accumulated through the process chain. Bythe common productions methods, it is frequently necessary to bring down the accumulateddistortions to an acceptable level as illustrated in Figure 2.2. The accumulation as well as thedifferent sources of distortions complicate the identification of the exact source of distortionat a late stage of the process chain.
/ -
/ACOfld
St• In ww.pvc•Jcfn •u
Figure 2.2: Reduction of accumulated distortion during the production process.
Straightening and adjusting distorted elements within the steel structures are one of themost time-consuming tasks in the production, but it is extremely difficult to estimate thepercentage of labour related to these problems. A cautious estimate of labour used forstraightening and adjustment of distorted elements is 30% of the total labour.
The geometrical distortions caused by thermal processing constitute a severe problem asthey cannot be estimated with sufficient accuracy and as a pattern of distortions throughoutthe process chain cannot be recognised. Equally important are the distortions introduced bya poor set-up of assemblies, which concerns the problem of inaccurate geometrical measuringand insufficient erection procedures.
From the above definition of problems the following conclusion is drawn. The main issuesto be addressed in the proposal of a new production strategy are the unpredictability inproduction time and the rationalisation of both the non-value and the value adding pro-cesses. As unpredictability in production time is related mainly to geometrical distortion,the production strategy to be developed should emphasise distortion minimisation.
2.4 Proposed Technological Strategy 9
2.4 Proposed Technological Strategy
In this section a technological strategy is proposed with short- and long-term aspects andwith the emphasis on production technology.
Different approaches can be adopted to improve production efficiency. The existing techno-logical trajectory within the paradigm' of ship building can be followed or the technologicalframes of production, logistics and process can be combined to create a paradigm shift withnew technological trajectories. First, a conventional shipbuilding strategy is discussed.
2.4.1 A Common Technological Trajectory of Shipbuilding
Since the 1980s, the key technology has been considered to be automation. Shipyards havebeen investing heavily in automation in order to increase production efficiency. The au-tomation of production has affected many areas of production in a positive manner. Anexample is the automated welding processes, which increase the efficiency of an operator upto five times compared to manual welding. The structural breakdown of ships in elementsdesigned for automation has furthermore proved more efficient, especially as automation wasthe indirect reason for leaving sequential production flow in favour of parallel production ofrelated elements.
Production facilities at a shipyard represent enormous investments. Characteristic of thetechnological paradigm of a conventional shipyard is therefore that production facilities andgeneral logistics are considered as fixed parameters constituting the frames from which newtechnology should emerge.
New production technologies are in general applied incrementally at all stages in the produc-tion. Technology for improving and measuring quality is applied whenever an unacceptablelevel is reached at some stage in the process chain. The process in question is analysed andthe problem is corrected or counteractions are taken. The later in the process chain theproblem emerges, the greater the risk of not being able to identify the problem source andcounteractions as repair or erection are likely to be taken. The problem source might neverbe identified if it is not recognizee as a problem at the stage where it is introduced, e.g. ifit is within the specified tolerance and hence accepted on an erroneous basis. The aboveapproach will unavoidably be a slow incremental and very research-intensive process.
Now, consider the conventional application of automation technology where a specific taskis brought into focus. As an example, a repetitive welding task on an assembly is identifiedfor automation. In order to implement the new time saving technology the assembly isdecomposed to facilitate automation. Thus, a non-value adding process of extra handling
'A technological paradigm is a model or pattern of solution of selected technological problems based onselected principles. A technological trajectory is defined as the pattern of normal solving activity within aparadigm.
10 Chapter 2. Production Strategy Emphasising Distortion Minimisation
and fitting is created and furthermore the resulting welding deformation is increased asthe structural stiffness of the panels is reduced. All in all the new technology might be anon-value adding process.
Roughly speaking, the common technological trajectory is incremental application of tech-nology all over the process chain, solving a problem or rationalising a task within a narrowperspective. The conventional approach is commonly accepted as it is based on existingproduction facilities and incremental improvements are generally obtained. In this contextit appears to be cost-saving.
2.4.2 Radical Approach
The radical approach is somewhat more conceptual as several interconnected problems areaddressed, constituting a significant potential for improved competitiveness. The strategyis primarily based on reduced length of the process chain, automation and 'the principle ofgeometrical restraint' related to welding deformations.
Welding deformations and stresses occur due to the thermal cycle experienced by the materialnear the weld, while it is restrained by the surrounding material. The principle of geometricalrestraint is simply defined as
A high degree of geometrical restraint in welding results in high stresses and smalldeformations, while an unrestrained weld produces less stresses but larger deformations.
In consequence, a simple panel will distort more than a large rigid assembly if both areset-up, tack welded and welded in one run. In this context distortions should be observedat a global level, as deformations locally at a less restrained part of a large assembly will beof the same magnitude as the local distortion of a simple panel. However, the main point isthat distortions in a large rigid assembly tend to be less global compared to a simple flexiblepanel and therefore less critical.
If the 'block assembly production method' illustrated in Figure 2.1 is considered, it followsfrom the principle of geometrical restraint that distortions are pronounced at the first stagesof the process chain where the structural rigidity of the simple panels is low. Hence, a largeassembly composed of prewelded globally distorted panels will suffer from inadequate fit dueto globally deformed simple panels.
As local distortions in a simple panel compared to those in a large rigid assembly are believedto be of the same magnitude, if follows directly that accumulated distortions are reduced byshortening the process chain.
2.4 Proposed Technological Strategy 1
Figure 2.3: The affect of shortening the length of the process chain.
The principle of geometrical restraint is applied by
" ensuring sufficient structural rigidity at the first stages of the process chain as globaldistortions of simple flexible panels axe reduced
* Shortening of the process chain as less distortions are accumulated
The ideal manufacturing process as regards distortions is therefore to cut plates with high-precision laser cutting, assemble them with high accuracy into grand-blocks and subsequentlyweld them by extremely flexible robotics in one run. If the flexible robotics at the same timewere capable of cutting, fitting, NDT, cleaning and painting by changing tools, grand blockscould essentially be finished at the same station. However, this is not a feasible process forseveral reasons:
" The yard layout should be altered drastically and a large number of stations withflexible robots are necessary to cover the required capacity
* The flexibility of the robots should be extremely large and such technology will prob-ably not be available for many years
To provide a more feasible technological strategy, the concept should be adjusted to a con-ventional shipyard using different technologies for short- and long-term strategies.
The desired goal is to reduce the problems described at the end of Section 2.3, i.e. mainlyto reduce rework and unpredictability in the production time by reducing the geometricaldistortion. The action patterns leading to the desired goal are outlined below.
A feasible reduction of the process chain is to limit the number of processes at the beginningof the process chain. Thus, both critical global distortions are reduced and less distortionsare accumulated, i.e. the utilisation of the principle of geometrical restraint is maximised.In more detail the plates should be cut, set up and tack welded to form assemblies with
12 Chapter 2. Production Strategy Emphasising Distortion Minimisation
sufficient rigidity before the first weld is applied. Subsequently, the normal procedure isfollowed.
The technologies to be developed or emphasised in order to effectuate the above are
" Redesign of structural breakdown
" Flexible snake-like robotics
" Design tool for distortion analysis
" Distortion minimising processes
A short description of these technologies is given in the following list.
Structural breakdown: The simple panels should be replaced by structurally rigid sub-assemblies designed to accommodate access of flexible robots. This might involverestructuring of all succeeding assemblies. Simple methods for calculating assemblystiffness (degree of geometrical restraint) should be developed for use in the redesignof structural breakdown.
Flexible robotics: Flexible robotics must be purchased or developed. Telescope roboticsto weld joints accessible from above combined with snake-like robotics for joints to beaccessed horizontally or through narrow passages are necessary to provide the requiredflexibility. A detail of a snake robot is illustrated below.
Figure 2.4: Detail of a flexible snake robot.
2.4 Proposed Technological Strategy 13
Design tool for distortion analysis: A design tool capable of estimating process-relateddeformations is necessary to understand the mechanics of welding deformations and torecognise deformation patterns throughout the process chain. The design tool shouldenable analysis of the relation between geometrical restraint and welding deformationand thus provide the minimum requirements to structural rigidity of the low-level sub-assemblies. Even more important, the tool should enable analysis of optimal weldingsequences, not as regards process time but as regards distortion.
To provide the necessary flexibility with respect to problems that may be analysed,a numerical tool is preferable compared to a tool based on empirical data, especiallyconsidering the amount of empirical data which will be required.
Distortion minimising processes and methods: The laser technology applied to weld-ing and cutting is a distortion minimising process as it applies a minimum of heat tothe material. Gutting deformations are not only caused by the thermal processing butalso by imprecise gantry systems and by relaxation of residual stresses in the steelplates originating from the plate manufacturing process. The laser cutting technologyon precise gantry system should therefore be combined with investigation of the effectof residual stresses. The assembling of elements should be carried out with high preci-sion in order not to contribute to the accumulation of distortions. Using the laser forposition marking simultaneously with laser cutting will enhance the accuracy. As thelaser is capable of cutting, marking and welding at the same station the logistics couldbe further improved. However, laser welding of subassemblies requires extremely largeflexibility of the lasers and the robotics.
An extremely important technology for distortion-related production quality is mea-suring methods. Relatively easy and accurate methods for geometrical measuring arerequired in order to monitor distortion related quality and will facilitate the localisa-tion of the problem sources. Even more important, an easy measuring method willdrastically improve the accuracy of the assembly set-up. Technologies as 3-fl vision,laser scanning and the well known theodolite systems should be investigated.
Other technologies for set-up and assembly positioning should be investigated. Con-cepts for optimal positioning of assemblies using supports or semni-automatic roboticsmight be considered, especially if access by flexible robotics from beneath the assemblyis decided upon. Moreover, the possibilities of using a kind of jigsaw puzzle method(tacks and cut-outs) for set-up should be investigated.
The above represents only some of the possible technologies which can be used to obtain thedesired goal. Moreover, the above technologies should be divided into base, key and pacingtechnologies to place the correct share of resources on the development of each technology[19].
14 Chapter 2. Production Strategy Emphasising Distortion Minimisation
2.4.3 Short- and Long-Term Aspects
The technologies are set in perspective to time by describing three stages in the technologicalstrategy. Common to the short- and long-term strategy is the redesign of structural elementsto shorten the beginning of the process chain.
Stage 1: Structural redesign is performed using simple methods for calculation of geo-metrical restraint. Laser cutting and laser marking are combined with conventionalwelding methods with low heat input. Only weld lines accessible from above will bewelded by robotics whereas the rest is performed manually.
Stage 2: Structural redesign is done using design tools to determine structural rigidity aswell as optimal welding sequence. Laser cutting, laser marking, 3-D measuring, set-uipand tack welding are performed at one station to reduce distortions before the firstweld. All joints are welded by telescope and snake robotics using high-performancewelding with low heat input (e.g. twin arc).
Stage 3: The capability of the flexible robotics is increased allowing all production pro-cesses to take place at the same station. The cutting and the welding are performed byYAG lasers. A quality monitoring system based on automatic geometrical measuringand NDT robotics is established.
As all invention and innovation are complex non-linear iterative processes the time neededfor development of these technologies cannot be estimated. However, it is emphasised thatthe short term technological strategy only requires laser cutting, laser marking and structuralredesign in order to shorten the process chain.
2.4.4 Gains and Critical Issues
In the proposed technological production strategy technologies related to logistics, geometri-cal distortions, design and automation are combined to form a new technological trajectory.Compared to a conventional 'incremental' strategy a synergy is obtained by addressing allthese issues in one step. The effect of fewer stages at the beginning of the process chainillustrates this synergy, as both geometrical distortions and logistics are reduced as well asvalue adding processes can be highly automated without increasing the non-value addingprocesses.
It should be emphasised that distortions aire not eliminated but only reduced according to'the principle of geometrical restraint'. Hence, the potential gains depend on the obtainablereduction.
PRom the alteration in process flow alone, i.e. independently of the distortion reduction, thefollowing benefits are apparent:
2.5 Selected Subjects for the Present Thesis 15
" The shortening of the process chain by performing all processes at one or two stationsreduce the non-value adding processes and enhance the logistics.
" The value adding processes become highly automated at the same time as the non-valueadding processes are reduced.
Less geometrical distortion provides the following gains
" Less accumulated distortions throughout the process chain
" Reduced unpredictability in production time resulting in improved logistics
* Improved product quality
From the technological portfolio included in the proposed strategy several benefits are ex-pected :
" Improved understanding of process-related deformations by the design tools
" Improved quality monitoring by measuring systems
The above benefits lead to easier identification of sources of geometrical distortion.
It is clear that considerable investments in production facilities will be necessary alongwith the employment of the proposed technological strategy. If the proposed strategy iscompared to the conventional with respect to investments in production facilities only, theproposed strategy will presumably turn out to be rather expensive. However, the analysismust be performed in a broader perspective and benefits resulting from the improvementof operational production efficiency and competitiveness must be included in the analysis.Obtaining the same benefits through the conventional strategy is not believed to be realisticdue to the characteristics outlined in Section 2.4.1. Hence, the central analysis is ratherto compare the potential gains of the proposed technological strategy with the requiredinvestments to implement it and the risks connected with the development of technologies.A key issue in the above analysis is the obtainable distortion reduction, which should beestimated.
2.5 Selected Subjects for the Present Thesis
The first subject chosen for investigation is related to the understanding of process-relateddeformations by the use of design tools. Such design tools do not exist and neither do thebasis on which they should be developed. As the primary source of geometrical distortion
16 Chapter 2. Production Strategy Emphasising Distortion Minimisation
is welding deformations, it has been chosen to investigate methods for welding responseprediction in plate structures, which is the subject of Part II.
The second subject dealt with concerns the 'raw material' in ship production i.e. the steelplates. All attempts to understand or predict process-related distortions are based on theassumption of uniform raw material. High levels and large variation of residual stress in thesteel plates will therefore invalidate or at least introduce large errors in the investigations.Currently, there is no suitable method for analysing residual stress in steel plates and theirsignificance is therefore difficult to evaluate. In Part III an experimental method is developedfor this purpose.
Part II
Efficient Modelling and Simulation ofWelding
17
Chapter 3
Scope of Modelling and Simulation
The savings potential related to reduction of welding distortions and the need for a weldingresponse prediction tool was identified in Chapter 2. The desired capability was describedas follows:
A design tool capable of estimating process-related deformations is necessaryto understand the mechanics of welding deformations and to recognise patternsthroughout the process chain. The design tool should enable analysis of therelation between geometrical restraint, welding method and welding deformationand thus provide the minimum requirements to the structural rigidity of thelow-level subassemblies. As a result, the tool should enable analysis of optimalwelding sequences.
If speed and easy use are added to the desired capability, the optimal design tool for distortionminimisation is outlined. If these requirements are compared with the state of the art innumerical welding response simulation it soon becomes apparent that such design tools arefar-fetched. The shortcomings in welding response prediction are well illustrated by Goldak[23], who in 1990 stated ten great challenges for welding research for year 2000. Almost allof these still hold true, including the first challenge of stress computation in a 10 in weld ofhigh-strength steel.
3.1 Focus in the Field of Welding Research
Historically, electric arc welding appeared in the late 19th century shortly after electricalpower became available. Failure of welded bridges in Europe in the 1930s and the AmericanLiberty Ships in World War II initiated research in welding mechanics. The welding research
19
20 Chapter 3. Scope of Modelling and Simulation
carried out since then is vast and a general review is out of the present scope. Several reviewsare, however, available such as the extensive reviews by Radaj [49], Goldak et al. [26] andthe very recent one by Lindgren [36].
Numerical modelling and simulation of welding are a difficult and challenging problem due tothe complex mechanisms involved. The wide range of problems concerned can be generalisedinto the fields shown in Figure 3.1.
Heat Flow +(Fluidyamc
Figure 3.1: Coupled fields in welding analysis.
The fields are strongly interrelated and couple in almost every possible manner. Estab-lishment of a model accounting for all the physical effects and their couplings would bean incomprehensibly large and complex task. Hence, welding research is characterised bychoice of a focal area for thorough analysis and use of suitable assumptions. Thus, the 'art'of welding research is to choose simplifications without invalidating the research focus.
The objective of the present work is the assessment of distortion information for analysisof manufacturing related problems. It is far more common to address the design relatedproblems in which stresses are sought for evaluation of the strength, stability, fatigue andcorrosion properties of a structural detail. Although deformations and stresses are physicallyclosely related, they are in general quite distinct in a modelling sense as the assumptions tobe made depend on the focus. This leads us to the inherent problem of welding which is thediscretisation of the domain.
Thermal, metallurgical and mechanical effects require the weld zone to be resolved downto a fraction of a millimetre, while the far field region can be coarsely meshed. Hence,the variation in mesh density might be several orders of magnitude when common weldingproblems are modelled. The transient nature of welding requires furthermore the problem tobe discretised in time and every part of the weld line regions to be densely meshed at somestage. As welding distortion problems tend to be truly 3-dimensional, these simulationsset excessive computational demands. As a result the research within numerical weldingsimulation has been highly focused on local weld effects, leading to a successive refinementof methods for modelling stresses and complex thermal and metallurgical effects which occur
3.2 Target Application 21
in the weld zone of small components. Little research has been directed towards methodsfor distortion prediction in true scale components, in which detailed information about theweld zone is exchanged for knowledge about macrodeformations in the full structure.
The latter research field is essential to the present work and the scope of modelling andsimulation may be defined by the task of
establishment of computational efficient numerical methods for welding distor-tions prediction in large-scale structures.
This field is not only essential for analysis of manufacturing related problems but also fordesign-related analysis as the global restraint on the weld zone may dominate the local strainand stress fields.
The first challenge is to reduce the complexity of the problem to the extent where at leastqualitative information about distortions can be preserved. For this purpose it must be keptin mind that the strains in the local weld region may be more sensitive than stresses withregard to detailed phenomena as e.g. transformation induced plasticity (TRIP). This is dueto the stress being zeroed at high temperatures and to the stress limitation imposed by theyield stress, whereas the material, by contrast, is highly deformable at elevated temperatures.The region of interest in distortion prediction is, however, seldom the weld zone itself anda relatively coarse discretisation of the domain may provide a reasonable prediction of theaverage deformation important to the global deformation field.
The second challenge is to reduce the computational demands imposed by the problem ofdiscretisation. A few feasible solutions have been described in the literature involving e.g.dynamic or adaptive redefinition of mesh topology [54], special elements for mesh grading[40] or substructuring [10]. The common characteristic is, however, that these techniqueshave only been investigated in relation to extremely simple geometries and applications ase.g. edge heating or bead on plate.
For both the challenges the application itself imposes a set of requirements on the modellingof welding distortions. To identify these requirements a target application is described inthe following.
3.2 Target Application
The requirements imposed on the modelling by the application are defined by the character-istics of the structure, the process and the manufacturing. Such characteristics are structuraldimensions, clamping, material, weld sequence etc.
22 Chapter 3. Scope of Modelling and Simulation
Figure 3.2: Example of structural complexity in ship construction.
With regard to geometrical dimensions and structural complexity, shipbuilding is presumablythe worst case scenario. By relating the ship structure shown in Figure 3.2 to the dimensionof the localised weld zone, the inherent problem of discretisation becomes clear. Fortunately,full analysis of such structures is not necessary in order to benefit from welding distortionsimulation. In Chapter 2 the first few stages of the 'block assembly production method' wereidentified as the major source for accumulated distortions throughout the process chain. Thelow structural rigidity allows the simple stiffened panels to distort globally, which causes asevere misfit in the subsequent assembling of simple panels into subassemblies. Low levelsubassemblies and simple stiffened panels are hence defined as the elements of primaryinterest, which, fortunately, reduces the computational demands.
Figure 3.3: Low-level subassembly.
3.3 Intermediate and Final Objective 23
The simple stiffened panel chosen for welding distortion analysis in the present work is shownin Figure 3.3. Even though the panel appears to be simple in a shipbuilding context, itscomplexity is far beyond the 'state-of-the-art' capability of welding simulation. The chosenL-shaped panel is interesting as regards manufacturing as it is known to cause problems.The structural stiffness of the panel is low both in the in-plane and the out-of-plane direc-tion. The low in-plane stiffness is caused by the L-shape and the cut-outs, which not onlydeteriorate the structural stiffness, but also increase the demands for accuracy as the slitsmust accommodate the longitudinal stiffeners on the adjacent panels.
The panel has a total weld length of 8.1 mn and is welded by either MAC (C02) or coveredelectrodes. The use of covered electrodes is known to yield large deformations and is thereforechosen for the analysis.
Plate thickness 10 mmMaterial mild steel (0.14%G)A-measure 5 mmWelding speed 5 mm/sHeat source efficiency 0.75Voltage 34 VAmperage 265 AElectrode ESAB OK FEMAX 33.81
Table 3.1: Welding parameter specification.
Prior to the production of a structural element, a manufacturing specification has been made.This specification includes the welding process specification (WPS), which basically providesthe requirements to weld quality and dimensions. The manufacturing specification doesnot contain information about weld sequence, tack weld positioning or additional supports.These parameters are determined on the grounds of shop floor experience or determined bythe robot motion planner, whose basic concern is minimisation of the inefficient travellingtime.
The modelling and simulation requirements imposed by the application itself are defined bygeometry, support, tack weld positions, welding sequence and the WPS. These parametersare essential for the prediction of welding distortions and should be carefully considered.The WPS defines the standard parameters to be accounted for in modelling as e.g. weldgeometry, heat input and weld speed. The additional characteristics such as global geometry,tack welds and support define the geometrical restraint of the weld zone.
3.3 Intermediate and Final Objective
The ultimate objective is to enable numerical analysis of welding distortions in structuralelements of the size and complexity defined by the target application. This objective is
24 Chapter 3. Scope of Modelling and Simulation
broken down into the intermediate objectives described below:
a Find pertinent parameters and establish a basic model accounting for the essentialwelding mechanics
* Increase the computational efficiency by special elements and dynamic redefinition ofmesh topology
e Increase the computational efficiency, exploiting that the response of a large domainis purely linear elastic
* Calculate the welding response of the target application by taking into account theeffect of tack weld positioning and welding sequence
The main issues are the accuracy and the computational efficiency to be achieved by theabove, as this will determine the potential of the tools developed with regard to futureapplication in the industry.
Chapter 4
Modelling and Simulation of Welding
In Chapter 3 the focal area was defined as numerical methods for computational efficientdistortion prediction in large-scale structures. To enhance the computational efficiency, lesssignificant phenomena should be omitted from the modelling and simulation. However, thelevel of detail to be included should at least enable the qualitative deformation field to bepredicted and accordingly, assumptions must be evaluated in this context.
Welding research can be described by the fields of microstructure, heat flow, mechanicsand fluid dynamics as previously shown in Figure 3.1. Detailed weld pool phenomena may,however, be regarded as a separate research field due to the level of refinement and as tem-perature fields and weld bead geometry cannot yet be predicted and subsequently accountedfor in a macroscopic context [49]. Until then, simplified heat source models are employedfor estimation of thermal loading. The implementation of simplified heat sources will bedescribed in Section 4.6.2.
By disregarding the weld pool physics, the fields and the couplings commonly considered'nay be described by Figure 4.1 and Table 4.1.
Microstructure
// 3 4
25
26 Chapter 4. Modelling and Simulation of Welding
Coupling Description Strength1 Temperature-dependent microstructural properties Strong
2. Thermal properties depend on microstructure Medium2 b Latent heat in phase change Medium
3 Stress state affects phase change Weak4. Mechanical properties depend on microstructure Strong4 b Phase changes yield volumetric strain Strong5 Heat generated by deformation Weak
6 Thermal expansion Strong
Table 4.1: Description of couplings between fields.
The microstructural stress dependency (coupling 3) is, however, only of interest in an ex-tremely detailed analysis and the deformation heat (coupling 5) can safely be ignored [33].
Welding simulations emphasising the mechanical aspects of welding need as a minimum tocalculate the heat flow and the resulting mechanical response. The field of microstructurecan be accounted for by using more or less sophisticated techniques for material modelling.Direct calculation of microstructural evolution is most sophisticated. A less general butsimpler method is indirect incorporation of microstructural aspects in the material descrip-tion exploiting the microstructural dependency on temperature and deformation. The lattertechnique is suitable for the present application and its implementation will be discussed inSection 4.3.
Different assumptions can be made when the transient aspects of the couplings are mod-elled. The fully coupled thermal-metallurgical-mechanical analysis, in which all equationsare solved simultaneously, is the most comprehensive and is rarely seen. It is more commonto apply a fully coupled thermal-metallurgical analysis followed by a mechanical analysis f8],which may be implemented in a staggered manner. If the calculation of microstructure isomitted, the thermal and metallurgical analyses are coupled by the use of a sequential orstaggered approach.
The geometry of the part to be welded and the objective of global distortion prediction im-pose a set of requirements on the modelling. These requirements are the topic of Section 4.2.
With regard to the numerical formulation, thermo-elasto-plastic finite element analysis hasbecome the standard. In the present work a small strain implicit analysis disregardinggeometrical non-linearities is applied. This and other numerical aspects are discussed inSection 4.4.
4.1 Integration in Commercial Codes 27
4.1 Integration in Commercial Codes
The choice between a commercial and an in-house code is always a topic of discussion inresearch projects. An in-house code is often highly specialised but full source code accessprovides the flexibility needed. On the other hand, considerable effort is usually requiredto establish features of little value for the research focus. In contrast, commercial codes areextensive with respect to pre- and postprocessing facilities and allow a variety of problemsto be investigated within different fields of physics. This comprehensiveness, however, limitsthe depth of analysis within each field and thus also the research potential. Some commercialcodes allow users to integrate their own coding to obtain the flexibility often required in aresearch project. This appears at first to be extremely beneficial as a lot of functionalityas pre- and postprocessors are ready at hand. However, it should be noted that the sourcecode access and the documentation in general are very limited because of rights protection.In consequence, integration of user functionality in commercial codes is to some extentcomparable with the task of guessing the contents of a black box by trial and error.
In the present project user coding has been integrated in the commercial code ANSYS,where the integration is facilitated by a set of subroutines made available for user call. Thepresent user coding comprises the structural elements, element activation/deactivation func-tionality, material nmodels, plasticity formulation, transformation induced plasticity, meshingalgorithms, data mapping and history tracking algorithms for dynamic remeshing etc.
4.2 Geometrical Aspects
The large computational requirements of three-dimensional welding simulation have led re-searchers to simplify the geometry by restraints. Especially 2-D models of sections transverseto the welding direction have been thoroughly investigated on the assumption of plane strain,generalised plane strain or generalised plane deformation. Depending on the research focus,the use of restraints to simplify geometry may invalidate the results and the effect needscareful consideration.
For the prediction of global deformations it is essential to have a proper 3-D representationof the structure. Uncoupling of the local weld region and the surrounding global structurewould lead to unacceptable results, as the elastic interaction between the global structureand the weld zone may be large enough to dominate the residual distortion and stress state[91. The elastic coupling is three-dimensional and the structural restraint of the weld linesvaries significantly due to the geometrical complexity of the global structure. In addition,temperature changes and previous welding in the global structure produce stresses whichcannot be predicted by modelling the local region alone. Furthermore, the joining of partsby welding itself changes the structural stiffness in a transient manner. Consequently, weldingdistortions in such structures can only be predicted by use of three-dimensional FE models.
28 Chapter 4. Modelling and Simulation of Welding
The 3-Dl dimensionality of the problem is likewise illustrated by the effect of tack welds.The individual parts of the panel are allowed to move relatively to each other in welding,restrained only by contact and tack welds as no other fixtures are present. As the weld mate-rial is deposited, the parts are locked relatively to each other in the distorted configuration,and hence the positioning and the length of the tack welds influence the final deformationpattern. To capture the effect of tack welds it is not only necessary to include the tack weldsin a 3-D representation of the geometry, the contact between parts must also be included aswell as the dynamic addition of filler material by the use of an element 'birth' technique.
Another aspect of the geometry is the kinematic boundary conditions of the panels. Panelsare generally welded on a production plane and the initial support therefore depends on theflatness of both the plane and the panel. As the panel distorts in welding the support willshift, which causes a change in moments and forces resulting from gravity. As the support isdefined by contact, which is computationally demanding in numerical simulation, it has notbeen considered worthwhile to include this detail in the model. As a result, gravity is alsoneglected as artificial fixation of the model in space together with gravity forces will causeunphysical stresses.
Only by modelling the welding mechanics as described above it will be possible to predictthe effect of welding sequence. Unfortunately, these requirements also imply very largecomputational demands which cannot be met by standard FEM techniques.
4.3 Material Modelling
Evolution of nmicrostructure can either be calculated or modelled indirectly through theniicrostructural dependency on the thermal and mechanical history. The indirect method isnot as flexible as the direct calculation, which allows the material properties at a discretepoint to be calculated from thermal history, current phase fractions, material propertiesof each constituent and deformation history. Both methods are, however, based upon thematerial data available and, in the case of direct microstructural calculation, also on TTTor COT diagrams. Accurate material data in the high temperature range is in general hardto obtain and combined with the assumptions made in numerical modelling, the calculationof material properties becomes at best a reasonable approximation.
The present approach is the indirect method, using material data obtained from literature.The material considered is a carbon-manganese steel with the chemical composition givenin Table 4.2. The thermal properties have been taken from Lindgren and Karlsson [38) andare shown in Figure 4.2.
C 0. 14 S i 0.346 Mn 1.42 P 0.23S 0.007 Or 0.026 Cu 0.016 Ni 0.012
Table 4.2: Chemical composition of the structural steel (percent).
4.3 Material Modelling 29
1". 40
~c12M - - - --- -----
35 F
----- --U
220
10
0 00 2W 4M3 6 8 I" 120 140 160
Temperatmec ['CI
Figure 4.2: Thermal conductivity and thermal capacity.
The latent heat in the solid phase transformation and in the solid-liquid transformation hasbeen included in the heat capacity. The latent heat of 260 kJ/kg between Tjjd,,=1480 GC
and T!jijdý=1530°C may cause numerical instability due to the steep gradient in heat ca-pacity which, however, can be overcome by using an enthalpy formulation. Latent heat canbe regarded as energy accumulated in the weld pool while heating takes place. This heatsink effect and the later energy release affect the size of the weld pool and smearing thelatent heat over a wider temperature range should be avoided. The stir effect caused byfluid flow in the weld pool has been modelled by application of a thermal conductivity of230W/mOC at temperatures above Tjjidý, as suggested by Andersson [1].
In the mechanical analysis the material is modelled as thermo-elasto-plastic with temperature-dependent material properties. Phase transformation is modelled through the thermal di-latation, which includes not only the thermal expansion but also the phase transformationstrain associated with the solid phase change. Thermal dilatation can therefore be consideredas the driving force in mechanical analysis.
The effect of microstructure is described in a simplified manner by using the peak tem-perature Ta.k and the cooling time between 8000C and 5000 C, Atg/,. In Section 4.6.3 itwill be shown that Atsl 5 may be assumed to be constant for the weld region. The peaktemperature varies, however, considerably and determines the grain size of austenite, whichaffects the relation between phase transformation and cooling time [68]. The constant cool-ing time and the austenitising peak temperature become therefore the primary parametersfor determination of phase transition temperatures and resulting phase constituents.
The mechanical material properties have been adopted from Jonsson et al. [32], who investi-gated a fined-grained steel of the same chemical composition. It should be noted that Atsl 5reported here is 22 sec., which is approximately twice the size of At8/ 5 to be calculated in
30 Chapter 4. Modelling and Simulation of Welding
Section 4.6.3 for the fillet welds of the target application. Poisson's ratio u and Young'smodulus E are shown as functions of temperature in Figure 4.3.
2W ......223 4M 0
8(0.2
2 75 IVO
0L 000 IIW 1<0'. 1 "
Temperatur I°CO
Figure 4.3: Young's Modulus, yield stress and Poisson's ratio versus temperature.
The thermal dilatation including phase transformation strain is shown in Figure 4.4. Theupper curve is followed during heating, whereas the curve followed upon cooling depends onthe peak temperature reached whereby a hysteresis effect is seen in the thermal cycle. Forpeak temperatures different from those listed, linear interpolation is used.
16Heawing.+ T < 70WC-
,4 •Z'__• 7, 'awc .....9 IOP°C .....
1.2 • 1300°G -
O.2
OA
06
0.2 , ,
200 3W) 4W 5X0 6 7W0 8W0 0OTempra.t rci
Figure 4.4: Thermal dilatation as a function of temperature and peak temperature.
In the present work, kinematic hardening is applied using the peak temperature-dependenthardening moduli shown in Figure 4.5. The effect of phase transformation and high temper-atures should be considered when hardening is accounted for. As the material is deformed
4.4 Numerical Aspects 31
T Tk4iSO
----------_------o ----- --
24
0 2W 0) 8WTemperature [iCi
Figure 4.5: Hardening modulus, H.
an increasing number of dislocations cause the material to harden. The number of dislo-cations is, however, reduced in phase transformation and at high temperatures by thermalrestitution, and it is reasonable to believe that the deformation history of the material isreset at high temperatures. These mechanisms are, however, not fully understood [36] andthe temperature range used for resetting is to some extent a matter of belief. In the presentwork, the deformation history is reset linearly in the temperature range 900C to 13000C.
At the end of the present PhD thesis material modelling is investigated in considerably moredetail. This work, described in Chapter 7, includes experimental evaluation of microstructurethroughout the HAZ and improved material models accounting for transformation-inducedplasticity (TRIP).
4.4 Numerical Aspects
In the present work the thermal and mechanical analysis have been sequentially coupled.The formulation used is implicit and the mechanical analysis is assumed to be quasistatic.The non-linear problem is solved by an iterative incremental Newton-Raphson scheme and adirect elimination solver. No kinematic non-linearities have been considered i.e. small strainsand displacements are assumed. The general formulation of the thermal and mechanical FEAis the standard and can be found in e.g. Cook et al. [15].
The material is modelled as thermo-elasto-plastic with temperature dependent material prop-erties. Plasticity is assumed to be rate-independent and is modelled by using the von Misescriterion, the associated flow rule and kinematic hardening. The plasticity formulation ap-plied in the general Newton-Raphson scheme at the structural level is described below.
32 Chapter 4. Modelling and Simulation of Welding
by subtracting the yield surface shift stress a from the deviatoric stress s
f 1 = {s} - {a} (4.1)
the yield function becomes
f = & -0 a = V3/2{I)[M]{f}) - (4.2)
where 6, is the effective stress corrected by the shift stress, a the temperature dependentyield stress and [M] a diagonal matrix with diagonal [111222]. By solving the global systemof equations by the general Newton-Raphson procedure, a set of incremental displacementsis provided. Calculation of the total strain gives the trial stress to be used in the plasticstrain updating:
{P"r} = [C] {"I . - f 6"). _ { E}n-I) - {} (4.3)
where [C] is the temperature dependent elastic stress-strain matrix and where cot, e" and ePare the total, thermal and plastic strains, respectively. If the effective trial stress 3I' exceedsthe yield stress ac, the associated flow rule allows the plastic strain increment to be calcu-lated. Here the radial return algorithm [13] is used, which ensures fulfilment of f = 0 at theend of the increment:
{AP} = AAfatr} (4.4)
where
{atr} = 9 -- {ýtr} (4.5)
A+=Pa0 (4.6){&r})T[C]{afT} + F
and E and E, are the elastic and the tangent modulus, respectively. By means of the plasticincrement, which is purely deviatoric due to plastic incompressibility, the plastic strains areupdated and the stress is calculated:
{EP}n = {E'}- I + {AEP} (4.7)
= [C] ({61*1}.- { -th}. _ {Ep}n) (4.8)
The yield surface shift stress is updated as [2):
= 2{E {AEP (4.9){a}.= {a._wt3(E - E,)
After calculation of the above parameters, the consistent tangent stiffness matrix can beestablished, which is needed in the calculation of the element stiffness matrix.
4.5 Finite Elements and Mesh Grading 33
The material properties of Section 4.3 are incorporated through aC, v, E and E,. In weldingsimulation only a limited number of elements are exposed to high temperatures, causing,soft' spots in the global stiffness matrix. This may cause numerical instability, especiallyat the unrestrained elements dynamically activated to simulate the laying of the fillet. Toovercome this problem a mechanical cut-off temperature can be applied, meaning that themechanical properties of the cut-off temperature is applied for all higher temperatures. Inthe present analysis a temperature of 10000C has yielded satisfactory numerical stability.The structural stiffness at this temperature is insignificant and will therefore not affect theaccuracy of the solution.
Numerical instability may occur for several reasons and is usually experienced as convergenceproblems, large hydrostatic stresses or excessively large deformations. Elastic incompress-ibility occurs as Poisson's ratio approaches 0.5 for the liquid state, but numerical locking iseasily prevented by using a mechanical cut-off temperature. A second source of numericalproblems concerns the consistency between the displacement strain field and the thermalstrain field in mechanical analysis as discussed by Oddy et al. t48]. If the same order ofshape functions is used for the thermal and the mechanical analyses the displacement strainfield is one order lower than the displacement and the temperature field. In thermal loadingthe temperature field directly becomes the thermal strain in the mechanical analysis andinconsistency between the thermal strain and the displacement strain field is created. Inthe present work linear shape functions are used and the inconsistency problem is solvedby assuming constant temperature within each element in the mechanical analysis. Plasticincompressibility also tends to cause numerical problems in fully integrated elements. Thishas been solved by using the B-method for reduced integration as described in Hughes [31}.Hereby the dilatational terms of the strain-displacement matrix B are replaced by termsaveraging the dilatational strain within an element. The approach has proved very efficientand eliminates numerical locking and spurious variation in hydrostatic stresses.
4.5 Finite Elements and Mesh Grading
In welding simulation the choice of finite elements has a significant influence on both ac-curacy and computational cost. For most applications the linear hexahedron is preferableto both the linear and the quadratic tetrahedron [14]. Mesh refinement is usually dividedinto h-refinement, p-refinement or a combination. In h-refinement the polynomial degreeof the shape functions is preserved whereas the element density is changed and vice versafor p-refinement. Szabo [65] showed that h-refinement is superior to p-refinement in non-smooth stress fields as experienced in welding, and in consequence linear hexahedrons andh-refinement are chosen for welding.
The basic problem in welding is the discretisation of a domain holding a localised regionwith large gradients in field variables. Thermal, microstructural and mechanical effectsrequire the weld zone to be resolved down to a fraction of a millimetre while the far field
34 Chapter 4. Modelling and Simulation of Welding
region can be coarsely meshed. The variation in required mesh density might therefore beseveral orders of magnitude in the modelling of a common welding problem. Mesh gradingin three dimensions using standard elements commonly results in distorted element shapesand inefficient grading as illustrated in Figure 4.6. 3-D grading with regular undistorted
Figure 4.6: Example of 3-D mesh grading using standard methods.
elements as shown in Figure 4.7 can be obtained by use of constraint equations to coupleDOFs or by a transition element as the graded element developed by McDill et al. [40]. Thisgrading technique is extremely efficient as it allows for large variation in mesh density withina short distance and the number of elements is therefore minimised. The use of identical
Figure 4.7: Improved 3-D mesh grading.
meshes for the thermal and the structural analyses facilitates the thermal loading of themechanical model by transfer of temperatures. Requirements to mesh density are decided
4.5 Finite Elements and Mesh Grading 35
upon evaluating gradients of field variables in the thermal and the mechanical models. Theserequirements are formulated by a set of meshing parameters describing the basic elementsize, the maximum levels of refinement allowed, the position of the heat source and theadditional scaling parameters. These parameters are defined in an input file which is usedfor automatic mesh generation.
The thermal element applied is a standard 8-noded hexahedron with linear shape functionsand birth and death functionality. The mesh grading in the thermal model is accomplishedby constraint equations to couple DOFs at interelement mismatch. The use of constraintequations restrains the DOFs located in mid-edge or mid-face positions of the larger elements,which results in a more 'rigid' mesh in these locations. In practice the restraining of theseDOFs has insignificant effect on the calculation accuracy, especially considering the accuracyof the subsequent mechanical analysis where temperatures are averaged within each elementfor thermal loading. The constraint equations are applied automatically upon meshing inthe developed meshing routine.
The structural element applied is a version of the graded element developed by McDillet al. [40]. The 8-26 noded element illustrated in Figure 4.8 is an isoparametric hexahedronsuited for mesh grading. The element is similar to the familiar 8-noded linear brick but theconstraints associated with mesh grading are embedded in the shape functions to ensureinterelement compatibility. Nodes 1-8 are mandatory vertex nodes whereas nodes 9-20 and21-26 are optional mid-edge and mid-face nodes. The optional nodes can be included in any
3
129
20 17is0
21 10 2
16 13
Figure 4.8: Node numbering for graded element.
desired combination. The shape functions listed in Table 4.3 must be calculated sequentiallyfrom p26 and down, and if a node is absent the matching shape function is zeroed. In thismanner only nodes present will be accounted for in the subsequent shape functions.
36 Chapter 4. Modelling and Simulation of Welding
Node P(ý, , ()
26 1(1 - Il1)(1 - 171I)(1 - C)25 ½(1 - KI1)( - I)D(' + 0)24 !(1 - 1ýi)(I - 7)(1 - (1)
23 '(1 - If1)(1 + q)(1 - I()
22 '(1 - ý)(1 - j)(1- I[)
21 1(1 +.)(1 - In)(1 - (1)20 4(1 + U)(1 - 71)(1 - 0l) - ½(P21 + P 24 )19 4(1 - 0)(0 - 7)(1 - IC) - 2(Pn2 + P24 )
18 4(1 - 0)(1 + 7)(1 - (I) - 2(P22 + P 23 )17 1(1 + f)(1 + 7)(1 - I() - I(P21 + P23 )
16 4(1 + )(1 - 1i77)(1 - 0) - R(P21 + P26 )15 4(1 - ]l(l - 7)(1 -)0 - 12 (P2 4 + P2)14 4(1 - 0)(l - 171)(1 - C)- 2(P2 + P26 )
13 4(1 - IEI)(1 + 0)0 - ) - 2(P23 + P26)12 1(1-+ C)(1 - l]l)(1+C)- + (P2 + P25 )
11 4(1 - 10)(1 - 11)(1 + C) - 2(P24- + P25)10 4(1 - V)(1 - 11)(1+ C) - 1(P22 + P25)9 4(1 - l)(1 + 1)(1+C)- + (P23 + P25 )8 81(1+ o(1-- 7)(1 -o)-½2(PH, + P16+P20) -14(P21 +P24,+P26)7 8(1 - -)0(1 - 77)(1 - 1) - (P1 4 + P1 5 + P29) - 41(P22 + P24 + P26)
6 8(1-E)(l1+i)(1-0 )-1(P 1 I-3 +P 14 + PIS)- 4(P22 +P 23 +P 2 6)
5 1(1+V)(1-+-i7)(1-C)-½(Pj3 +P 16 +P1 7 )-4(P 2 1 +P 23 -+-P 26 )
4 +(2l',)(1- )(1+C)-½(Pl, +P 12 ±+P2 0)-I(P 2 1 +P2 4 +P 25 )3 i(1 - 0(1 -7)(1 + '(P) +P 11 +F 19 ) - 12 1(1-0(1 +7-)(1 +C-(P 9 + PIO + PIS) - (P2 2 + P23 + P0)
1 A1C + 4 P1 2 +P 1 I) - 4(P 21 + P23 + P25 )
Table 4.3: Shape functions for graded element.
Depending on the number of optional nodes and their positions, the element is subdividedinto a number of linear subdomains varying from 1 to 8. The combination shown in Fig-ure 4.9 results in 4 linear subdomains. As the boundaries of each subdomain are linearthe compatibility between the graded element and the adjacent linear elements is ensured.Each subdomain is independently integrated by a 2x2x2 Gaussian quadrature scheme forthe deviatoric strains, whereas the dilatational strains are underintegrated by the previouslydiscussed B-approach. To obtain consistency between the thermal loading and the thermalstrain, constant temperature is assumed in each subdomain.
The variable number of nodes and the interelement compatibility make the graded element
4.6 Thermal Modelling and Simulation 37
Figure 4.9: Example of linear subdomains in grading.
extremely efficient in mesh grading algorithms. The mesh grading algorithm developedautomatically identifies mid-edge and mid-face locations and adds the necessary node to thedefinition of the graded elements.
4.6 Thermal Modelling and Simulation
In this section thermal modelling and simulation are considered with the emphasis on heatsource modelling, and simulation results for fillet welding are presented. Before introducingthe heat source models, the general physics of arc welding is shortly described to illustratethe complex processes in this localised region.
4.6.1 Physics of Arc Welding
Electric arc is the most common heat source in welding. The general process is describedby an electric field between the positive anode and the negative cathode surrounded by anionisation gas. On the metal there is a thin layer of surface electrons which are acceleratedin the electric field towards the anode. These electrons collide with the atoms in the gas,causing impact ionisation where the atoms are decomposed into electrons and positive ionswhich cause further ionisation. The current of electrically charged particles in the arc andthe temperature are interrelated as high temperatures increase ionisation and the increasedionisation causes the temperature rise due to the released energy. To obtain welding condi-tions the temperature or the current must initially be brought up to a certain level, whichis done by igniting the arc. Arc ignition is accomplished by e.g. the short-circuit currentwhich occurs as the anode and the cathode are brought into brief contact. The short-circuitcurrent shortly increases the temperature and the current and subsequently the arc can bemaintained in the electric field existing under normal welding conditions. The arc is sur-rounded by a magnetic field directing the charged particles towards the centre of the arc,causing the arc to localise in spots on the anode and the cathode. When the electricallycharged particles impact on the anode and the cathode, the anode and cathode spots are
38 Chapter 4. Modelling and Simulation of Welding
heated to high temperatures (approx. 3000-40000C) whereby both the electrode and theworkpiece melt. Due to several forces as e.g. the suction force of the plasma flow, dropletsof the electrode material are deposited on the workpiece.
The base metal is fused almost instantaneously at the arc spot and a weld pool is formedas the fusion zone expands. The plasma pressure forms a crater in the fused material whichflows towards the rear of the weld pool where it solidifies. As some of the fused material ispushed upwards a weld bead is formed. Arc welding is commonly classified in surfacing arcand immersing arc according to the penetration depth of the fusion zone. For a given heatinput the characteristic shape of the fusion zone is dependent on the ratio between am~perageand voltage. As the amperage ratio is increased the plasma pressure becomes larger causingincreased penetration depth and decreased width of the fusion zone. The heat transportin the weld pool is mainly convective due to the fluid motion. The flow characteristics inthe weld pool depend on several phenomena as e.g. plasma pressure, electromagnetic forcesand buoyancy forces. The temperature-dependent surface tension force is, however, usuallydecisive for the flow characteristics in the weld pool. By using different additives as e.g.Sulphur, the surface tension can be altered to give a high vertical fluid motion which furtherenhances deep penetration welds.
A thorough description of welding processes is found in the book by Radaj [49].
4.6.2 Heat Source Modelling
As it is indicated above accurate modelling of the weld pool dynamics is an extremelycomplex and yet unsolved task, which involves numerous physical effects and couplings.Fortunately, the complexity of weld pool modelling is no hindrance of modelling the macro-scopic effects of welding. Generally, the multiphysics of the weld pool is decoupled from themechanics of welding, either by prescription of the isotherm at the liquidus boundary or thepower density distribution in the region of the weld pool. The general problem is thereforeto estimate the shape of the weld pool and to choose a heat source description enabling thegeneral aspects of the heat source to be captured.
As the shape of the weld pool is believed to be the dominant factor for e.g. angular defor-mations in welding, distinction between heat source models should be made with respect totheir capability to model correctly the shape of the weld pool. Several methods are avail-able for heat source modelling varying from the classical analytical solution by Rosenthal[52] to direct prescription of the liquidus boundary obtained by experiments [27]. The heatsource description used in the present work is the double ellipsoidal power density distri-bution presented by Goldak et al. [25]. The 'double ellipsoid' sketched in Figure 4.10 is avolumetric heat source, which is well suited for modelling weld pools where heat distributionis dominated by fluid flow.
4.6 Thermal Modelling and Simulation 39
Figure 4.10: The double ellipsoidal power density distribution.
The arrangement of the two ellipsoids, approximately of the shape and the size of the weldpool, allows for different gradients in the heat flux in the front and the rear of the source.The ellipsoids are described by the semi-axes a,, a2 , b and c and the following equations:
6y/, z, 0 'afQ e-3l be- zlce - 31x+)v(r
- l/al 1 (4.10)ql~x~y~zat =abcjr y/
q,(x, y, z, t) = fbcfQ -3y2 /b2e3ý/S
ff + , = 2 (4.12)Q = 7UI (4.13)
where
qj, q, : power density in front and rear ellipsoidsff, f, fractions of heat deposited in the front and the rear
v, t, r welding speed, time and heat source x-position at t=OQ : effective energy input rate
2l, U, I : heat source efficiency, voltage and amperage
Variation of the semi-axes and the heat deposit fractions allows the 'double ellipsoid' to befitted to a given heat source.
40 Chapter 4. Modelling and Simulation of Welding
Figure 4.11: X-section of fillet weld.
hrom the polished X-section shown in Figure 4.11 the boundary of the fusion zone (FZ)and the heat affected zone (HAZ) can be estimated, which allows respectively the liquidusisotherm (ý 1530 0C) and the isotherm at the outer HAZ (; 800 0C) to be identified.The heat source efficiency can be estimated from literature and the net energy rate canbe calculated by Eq. (4.13). Thus, the information available for fitting the heat sourceparameters is the estimated isotherms in the plane transverse to the welding direction andthe net heat input. By the use of a finite element model for temperature calculation thesemi-axes are adjusted so that the calculated isotherms becomes similar to those obtainedby evaluation of the X-section. In the FE code the volumetric heat flux is applied as nodalvalues, and the Gaussian distribution is discretised according to the variation allowed by thethermal finite element employed. If a coarse mesh and elements with linear shape functionsare used, a considerable error might be introduced in the discretisation, which should beaccounted for when the heat input is adjusted.
The above procedure has proved to be sufficiently accurate for most applications, but ad-ditional information about isotherms in and around the weld pool may be estimated fromexperiments using infrared temperature measurements and thermocouples. Such experi-ments combined with a more flexible power density distribution [24] may be necessary foraccurate metallurgical analysis.
In the following an example of application is given. A fillet weld was performed by use of acovered electrode mounted in a guiding device. The specifications are listed in Table 4.4.
4.6 Thermal Modelling and Simulation 41
Plate thickness 10 mmMaterial mild steel (0.14%C)A-measure 5 mmWelding speed 5 mm/sHeat source efficiency 0.80Voltage 34 VAmperage 265 AElectrode ESAB OK FEMAX 33.81
Table 4.4: Welding parameter specification.
Based on the polished X-section shown in Figure 4.11 a microstructural evaluation wascarried out (see Section 7.1) and the extent of the fusion zone and the HAZ was estimated.By means of 3-D FE model for iterative solutions, the liquidus and the outer HAZ isothermsshown in Figure 4.12 were fitted to those observed in the microstructure. The descriptiveheat source parameters are given in Table 4.5.
Temperature [C]A A= 800B = 1530
.2mm
Figure 4.12: Calculated solidus and outer HAZ isotherm.
a, = 0.0040 m a2 = 0.025 mb = 0.0625 m c = 0.050 mff =0.2 f, =1.8
Table 4.5: Descriptive parameters for heat source.
In the thermal model the pressure-dependent thermal contact between the base plate andthe web is not modelled, which causes the discontinuity over the boundary to be slightly
42 Chapter 4. Modelling and Simulation of Welding
larger than the true one (see Figure 4.12). Still, the flexibility of the double ellipsoid hasproved sufficient for the present application since it allows a reasonable fit of the calculatedisotherms to those observed. The longitudinal extent of the weld pool is difficult to eval-uate without experimental verification but the calculated length of approx. 26 mm seems,however, reasonable.
4.6.3 Thermal Model
By application of the thermal solids and the grading by constraint equation as discussedin Section 4.5 a basic model for analysis is established. The model for single-sided weldingshown in Figure 4.13 has the dimensions given in Table 4.6.
Figure 4.13: 3-D thermal FE model (single sided welding).
Length Width Height Thickness A-measureBase plate 200 100 - 10Web 200 - 45 10Fillets 200 - 5Tacks 20 5
Table 4.6: Dimensions of basis model [mm}.
The number of DOFs for the single sided welding model is 12720 whereas the number forthe double-sided adds up to 20794. Two tack welds are positioned at the ends behind the
4.6 Thermal Modelling and Simulation 43
web and are the only connection between the web and the base plate at the start of thesimulation. The weld deposit is modelled by activation of elements immediately behind thecentre of the heat source. When activated, an element adopts the nodal temperatures fromsurrounding elements already active. The front nodes do not share nodes with activatedelements and are therefore set to ambient temperature. Elements activated at the start of atime increment have consequently been given an amount of energy before the volumetric heatflux becomes effective within the element. This error and the error caused by the inactivefillet elements in the front ellipsoid are accounted for by adjusting the net heat input. Theuse of the front ellipsoid is desirable for smoothening of the thermal gradients in the web andthe base plate in front of the heat source. Furthermore, it enables tack welds to be properlyheated as the heat source passes without element activation.
The thin elements positioned at the outer edges of the thermal model are infinite boundaryelements, which represent the heat conduction across the boundary to an exterior domainnot included in the model. If these elements are not included, the cooling time to ambienttemperature will be unrealistic long. A description of the infinite boundary element can befound in ANSYS User's Manual [2].
Additional boundary conditions included in the thermal modelling are convection and radi-ation. Convection and radiation are combined into the following heat transfer coefficient:
£Eyo&0 4((T + 273)' - (Ta,..h + 273)') + hý, (4.14)T
where
ht : combined heat transfer coefficient
Izý convective heat transfer coefficient
tern emissivityumI Stefan-Boltzmann constant
T.b:ambient temperature
The temperature-dependent emissivity and the convective heat transfer coefficient are adoptedfrom Brown and Song [0]. In practice, however, the thermal conductivity is strongly domi-nant and radiation and convection can be omitted from the analysis without significant lossof accuracy.
An example of calculated isotherms is given in Figure 4.14 where the heat source approachesthe end of the weld line. The peak temperature is calculated to be just above 19000C.The thermal gradients are seen to be very steep in front of the heat source as the torchtravels faster than the heat propagates. Further down the weld line, the distance betweenthe isotherms is increased as the cooling rate decreases. The primary results from thethermal model are the isotherms for evaluation of the descriptive heat source parameters(Section 4.6.2) and the general temperature field for loading of the mechanical model. Fur-thermore, the cooling time At,8 /5 must be evaluated for use in material modelling. In Fig-ure 4.15 temperature has been plotted against time and distance. The distance is measured
44 Chapter 4. Modelling and Simulation of Welding
- 102- 30851
51412.S 92'= 11331340
o 1752
Figure 4.14: 3-D thermal FE model.
from the centre of the fillet at an xy-cross-section and vertically down into the base plate(see Figure 4.14). The contours for 5000C and 800 0C have been plotted on the base, whichallows AtB/ 5 to be evaluated. The assumption of a constant At8 i 5 in the fusion zone and theHAZ is reasonable as only limited variation with the distance from the fillet centre is seen.A value of 11 secs. has been chosen as an approximation of Atls/ for the entire region.
Tempctr [tCi °C....
5W100
-y -.... ----- .....
Q00
F r . Te u fro file cenre
Figure 4.15: Temperature versus time and distance from fillet centre.
4.7 Mechanical Modelling and Simulation 45
4.7 Mechanical Modelling and Simulation
Based on the graded element, a mechanical model with a topology equivalent to that ofthe thermal model has been established. Considerable effort has been placed to allow formovement of the web relative to the base plate. The technique employed to capture thisphenomenon and its effect on the deformation pattern aie described below. Furthermore,results of stresses and deformations obtained from the basic model are presented.
4.7.1 Dynamic Coupling and Contact Modelling
The web and the base plate are allowed to move relatively to each other in welding, restrainedat the beginning only by contact and tack welds. As the filler elements are activated, theparts axe locked relatively to each other in the distorted configuration. The modelling of thisphenomenon involves the dynamic activation of fillet elements, dynamic coupling of partsand modelling of contact and gap.
The dynamic activation of fillet elements in the mechanical model is based on average ele-ment temperature. Fillet elements behind the xy-plane containing the maximum nodal tem-perature are activated at temperatures below T.0u1d.,. The temperature-dependent elementactivation enhances numerical stability as the number of 'soft' high-temperature elements isreduced.
Figre .16 Tepertur-dpenentactvaton f sruturlflerturelmns
Elemntsareactvatd stessandstrin ree An ncrmenal heral srai fomultio habeenappied whch alow th thrmalstrin o b zered y smpl deltin th thrmastrin ccuulaed.Resttig o th reainng tran cmpoent reuirs secil atenioin asmal dforatio anlyss. s th toal tran isobtine frm noal ispaceent
46 Chapter 4. Modelling and Simulation of Welding
relative to the original undistorted configuration, resetting can only be done by introducinga strain component which holds the strain accumulated in the element while it is dead.The total strain is calculated by subtracting the accumulated dead strain, which yields acorrected total strain from which trial strain, plastic strain and elastic strain are calculated.The use of an updated Lagrangian large strain formulation would more correctly solve thisproblem as the element would be completely redefined in its new distorted configuration bydeleting all accumulated strains. The error introduced by the present approach is, however,believed to be small.
A small gap between the web arid the base plate has been included in the model to makeit possible for the web and the base plate to move independently. Laying the first filletthe web will bend out towards the heat source due to the thermal gradient through thethickness. The filler material will dynamically couple the web and the base plate in thedistorted geometrical configuration. The coupling is obtained through constraint equations,applied and updated incrementally at the end of each time step. Nodes of inactive filletelements are restrained relatively to the web. In the activation, nodal positions of the filletelements are found by interpolation between the updated nodal positions of surroundingnodes at the web and the base plate. Subsequently, constraint equations axe computed andapplied solving the inter-element incompatibility. Thus, through activation of fillet elementsand constraint equations, the web and the base plate are fixed relatively to each other in thedistorted configuration. Considerable improvement of numerical stability is found by usingconstraint equations to control the free nodes at the front of the fillet. These elements arenumerically soft and even a small residual force tends to cause highly distorted elements.
Thermal expansion will set up contact forces between web and base plate. Contact problemsare highly non-linear and solving them requires significant computer resources. In addition,the transient regions of contact are generally unknown, which makes it difficult to reducethe number of contact elements needed in the analysis.
The contact problem is solved by ANSYS surface-to-surface contact modelling. A contactpair consists of two surfaces made of contact and target elements, respectively. The gapbetween web and base plate allows the contact and target elements to be positioned withoutinitial contact. The 4-noded contact and target elements deform with the underlying solidelements whereby the contact is modelled as flexible-to-flexible. The contact elements areconstrained against penetration into the target surface at the Gaussian points, which are usedas penetration detection points. Contact is defined by a normal contact stiffness factor and apenetration range allowing the contact force to be increased gradually with penetration. Thealgorithm used for contact is the augmented Lagrangian method where an iterative series ofpenalty updates is used to find the Lagrange multipliers.
The modelling of friction constitutes a problem as the temperature-dependent friction co-efficient is practically unknown. The assumption of frictionless contact was compared to atemperature-independent friction coefficient of 0.3. The only significant difference observedwas a considerably longer computation time in the case of friction. In consequence fric-tionless contact has been assumed for the present application so that the parts are free toslide.
4.7 Mechanical Modelling and Simulation 47
Contact Pressure [MPa]Heat source Z-position 0 2
25
PaR 75100
125r- 150
r--- 200300
Figure 4.17: Contact pressure.
Figure 4.17 illustrates the contact pressure of the contact elements at the bottom of the web,plotted from the heat source and approx. 120 mm in the direction opposite to welding. Thehigh pressures are found just outside the high-temperature/soft-material region located alittle behind the heat source. Further behind the heat source, the contact region is shifted tothe opposite side of the web, partly due to the moment induced by the high-pressure regionand partly due to the high-temperature/soft-material region close to the fillet. Both thepressure regions are travelling in a transient manner at some distance of the heat source.
The contact elements are not suited for grading and the incompatibility causes the contactpressure to be wrongly distributed to nodes at element faces containing mid-edge nodes. Asmid-edge nodes only attain forces from two of the three surrounding contact elements, thesetend to penetrate a little more than intended, which causes small contact pressure peaksin these positions. The influence on the computed results is, however, insignificant whichwas found by comparing calculated element stresses to modelling results where the contactsurfaces had been refined to obtain regular non-graded elements.
For a given web height, the curvature of the web is primarily a result of the thermal straingradient through the thickness. Thus, the amplitude of the deformation becomes highlydependent on the distance between tack welds, allowing the maximum amplitude to beestimated as a function of the free length between tack welds.
The displacements are illustrated by x-displacements in Figure 4.18(a). At first, the relativedisplacement between the web and the base plate appears to be relatively small but, nev-ertheless, it has significant influence on the qualitative deformation pattern. This influenceis evaluated in Figure 4.18, in which the model including dynamic coupling and contactis compared to a model where the web has been fixed to the base plate, by coupling theinterface nodes in the initial position. In the latter model the web is seen to be affectedby the unphysical restraint as the qualitative deformation is changed merely to angular de-formation, which increases continuously towards the end of the weld line. Clearly, this isnot physically correct and in consequence, modelling of dynamic coupling and contact isnecessary to predict the qualitative deformation patterns.
48 Chapter 4. Modelling and Simulation of Welding
Figuep418: x-displce]ts20 -. 207-1486 -. 155
-a~a -. 104-. 260 -. ,052
F 217 .141F 278 42c6
(a) Contact and dynanic coupling included (b) Fixed web
Figure 4.18: x-displacements.
4.7.2 Simulation Results
Dynamic coupling and contact modelling have been included in the basic model. As theprocess of tack welding is not simulated, tack welds are initially assumed to be stress-freeand undeformed. The model, tack welds and structural boundary conditions axe shown inFigure 4.19.
Figure 4.19: Structural boundary conditions and tack welds (note display rotation).
4.7 Mechanical Modelling and Simulation 49
X-dispj=j Y-displ=l- 208 -. 1 a-:148 -.077 ,-.868
.03 91- 6. .14
003 5 :255
96 363.156 .472217278339 .797
(a) x-displacernents (b) y-displacernents
Z-di.p[=] S,...-329 -351-.283 -263-23 8 -175-.193 87-.147 a-.102
818-.056 11- 011 2640355 3520805 440
y
(c) z-displacernents (d) Longitudinal strm
Sý..(..]350233-217-15083, 7
50.7, 83250
(e) Transverse strem
Figure 4.20: Simulation results.
50 Chapter 4. Modelling and Simulation of Welding
The simulation results are presented in Figure 4.20 as deformations in the x-, y- and z-directions. Stresses are given as longitudinal stress and stress transverse to the weldingdirection. The direction of transverse stress has been aligned with the free face of the filletin the xy-plane.
The deformations of the web and the base plate appear to be rather small measured inabsolute terms, but the present angular deformation of the base plate will in fact cause ay-displacernent of more than 20 mm per metre, i.e. approx. two times the plate thickness.When the results are evaluated, it should, however, be kept in mind that the structuralstiffness of the present T-profile is relatively low compared to common assemblies and hencethe predicted deformations will lie in the high range.
A general characteristic of the stresses is that they tend to reach a steady state at somedistance of the tack welds. This is best seen in Figure 4.20(d) which shows the longitudinalstress. Little variation is found in the longitudinal stresses in the fusion zone and the innerHAZ. The stress state in this particular region is, however, very sensitive to the materialmodel applied. In Chapter 7, other and more sophisticated material models will be presentedwhich are seen to produce compression stresses in the fusion zone and high tensile stressesin the inner HAZ. Hence, a detailed discussion of stresses in this region is postponed toChapter 7.
The model for single-sided welding simulation consists of 38160 DOFs and the computationaleffort required to solve the problem is approx. 23 hrs1 . In general the computation time is noteasily estimated from the model size. For a direct solver the computation time is roughlyproportional to the bandwidth squared times the number of DOFs, times the number oftime steps. As the bandwidth does not only depend on model size but also on the transientstate of contact, the bandwidth is not a readily applicable measure for computation time.The computation time needed for simulation of even small welding applications, such asdouble-sided welding of a 500 mm long T-profile, will, however, be unacceptable and thesimulation of large industrial welding applications by the above technique seems thereforeto be far-fetched.
The main issues of the present chapter are in summary: A basic model has been established.Different physical phenomena have been evaluated by computation or through literature andthose believed to be important to a qualitative prediction of welding deformations have beenincluded with plausible results. Dynamic activation of fillet elements, dynamic coupling ofparts, contact modelling and tack weld positioning were identified as prime factors withregard to the qualitative deformation field. Experimental verification is, however, neededto give a quantitative evaluation of the predicted distortions. Finally, it is recognised thatthe basic model is of limited practical value in terms of distortion prediction due to itsgeometrical size and the computational effort required.
'HP J5000 workstation, 2GB RAM
Chapter 5
Dynamic Meshing
The objective of the present chapter is to increase the computational efficiency in weldingsimulation. The graded element introduced in the previous chapter has already attributedto increased efficiency and its superior mesh grading characteristics establish the basis forthe dynamic meshing algorithms to be described below. In dynamic meshing it is utilisedthat the thermal and mechanical activities are localised in the region around the heat source,and the basic task is therefore to provide a dense mesh only where needed and thus reducethe number of DOFs.
5.1 Computational Efficiency
As previously mentioned, the computational demands of 3-D welding simulation have forcedresearchers to focus on detailed investigation of welding phenomena in the local weld region,usually employing a 2-D model. The first 3-D model was presented in the late 80s byLindgren and Karlsson [38] based on shell elements. 3-D simulations has since appearedfrequently in the literature but it should be noted that these are seldom concerned withgeometries more complex than edge heating, bead-on-plate or butt welding. To the author'sknowledge the basic model, presented in the previous chapter, is the first physically realisticmodel for simulation of fillet welding and in this context it is hardly 'basic'.
The main obstacle in 3-D welding analysis is the computation time. The development in thisresearch field can mainly be attributed to the rapid evolution of computer power but also tothe computational methods developed to increase efficiency. Such methods involve automaticFEA, substructuring, dynamic meshing, adaptive meshing and code parallelisation.
Automatic FEA regards problem analysis based on user input supplied at higher level ofabstraction than that generally required for problem description in FEA. Automatic FEA
51
52 Chapter 5. Dynamic Meshing
includes in the present work model generation, dynamic remeshing and the actual computa-tion using process and geometry based input. The degree of automation is, however, limitedto analysis with predefined heat sources, materials and geometry types.
The basic idea of dynamic and adaptive meshing is to refine the mesh only where needed toobtain a sufficient solution accuracy with the least possible computational effort. Weldingsimulation is the ideal application for dynamic and adaptive meshing due to the localisedload travelling with time in a large and only slightly changing domain.
Dynamic and adaptive meshing involves in the mechanical context dynamic redefinition ofthe mesh topology in quasistatic non-linear finite element with history-dependent materials.Dynamic meshing and adaptive meshing are only distinct in the way the mesh refinementis decided upon. In dynamic meshing a predefined mesh refinement scheme is used whereasmesh refinement in adaptive meshing in general relies on error estimators.
Rezoning was first applied in welding in 1993 by Brown and Song [10] in a 2-D edge-heatingapplication. Later in 1997 Lindgren et al. [37] applied dynamic meshing for a corresponding3-D simulation. The latest contribution known by the author is Runnemnalm and Hyun [54]who applied an adaptive mesh scheme for 3-D bead-on-plate simulation.
In adaptive meshing the general objective is to obtain a mesh which yields a solution with anequidistributed error below a predescribed limit compared to the exact solution. As the exactsolution is commonly unknown, different methods exist to obtain an error estimate. Themost common error estimator is presumably the posteriori Zienkiewicz-Zu error estimator, inwhich a post-processed higher-order solution, usually obtained by least square approximationover the global domain, is used instead of the exact solution [39, 71]. By dividing the localelement error with the global error a relative error norm is established and can be used fordetermining which elements to refine or coarsen in an adaptive scheme for mesh refinement.Runnemnalm and Hyun [54] compared adaptive meshing based on the thermal flux to thatbased on effective stress. As the results were very different, they concluded that a combinederror norm based on the maximum of the two should be applied. However, this approachis likely to result in a dense mesh all along the path behind the heat source due to residualstress gradients. In consequence, the computational cost increases significantly as the heatsource travels with time.
The above adaptive approach is not suitable for the present work. The scope is predictionof the overall displacement field, and if this can be maintained in coarsening along witha reasonable representation of the elastic stiffness, compromises can be made with regardto the stress and strain description in low-temperature regions. Hence, the Zienkiewicz-Zu error estimators based on mechanical field derivatives are not suitable for the presentpurpose and will cause a mesh too dense in the lower temperature range. Instead a dynamicmeshing scheme was decided upon, combined with a coarsening procedure which emphasisesthe retaining of nodal displacements.
Dynamic meshing involves managing of mesh topology, interelemnent compatibility, datastructures relating new and old meshes. and data mapping of field variables.
5.2 Mesh Refinement Scheme 53
5.2 Mesh Refinement Scheme
The dynamic meshing capability developed for the fillet welding application is based on theposition of the heat source and the predefined weld path. In consequence the mesh topologyis predetermined and in contrast to adaptive meshing, the mesh needs therefore not to beulpdated in each increment but can be refined in advance to accommodate a given lengthof the weld path. Both the mesh updating and the extra DOFs needed to accommodateadditional weld length are time-consuming in simulation: Thus, computational efficiencycan be further enhanced by choosing a reasonable relation between mesh updating andincluded weld length.
The mesh refinement scheme is based on a normalised geometrical parameter. The basicnorm applied to an element i is adopted from McDill et al. [42]:
E(i) = nint( o(At 51log(2)
In the present implementation r- is the distance between the element centroid and a predefinedpath, hi is the longest diagonal of the element, c; is a scaling parameter for the degree ofrefinement and nint is the nearest integer function. Thus, the norm is simply based on theratio between the 'size' of the element and its distance from a refinement path. A negativevalue of E(i) indicates the need for coarsening, a positive the need for refinement whereaszero leaves the element unchanged.
A rule-based mesh management algorithm has been developed for the fillet welding ap-plication. Initially, the dimensions of fillet, web and base plate are defined along with theapproximate dimensions of the largest element allowed. On this basis, a coarse and ungradedroot mesh is established. In the mesh management algorithm the distance r has been takento be the perpendicular distance between element centroid and a refinement path, which isbasically a line representing the region which is desired to have a high mesh density. By useof refinement paths related to each refinement level, flexibility is added to the mesh defini-tion as e.g. the 'tail' of refined elements behind the heat source can be controlled for eachlevel. Starting at the initial mesh, i.e. at refinement level 0, only procedures for refinementneed to be considered. A lower bound of element size is defined by choosing the maximumrefinement level allowed. Looping through the mesh, elements with a positive refinementnorm are refined one level. This procedure is repeated until the maximum refinement levelis reached. Binary grading has been used as a mesh grading restriction and as a result, anelement can as a maximum join 4 elements at a face and is refined if this number is exceeded.
In the definition of a new and updated mesh, rule-based refinement and coarsening proce-dures can be applied to the preceding mesh as proposed by McDill et al. 142]. Alternatively,the root mesh can be used as refinement basis each time the mesh topology is changed sothat only a refinement procedure is required to coarsen and refine the mesh. This techniquehas been applied here as it offers a simpler implementation and a gain in computational
54 Chapter 5. Dynamic Meshing
Figure 5.1: Examples of meshes in dynamic meshing.
efficiency. In Figure 5.1, three out of twelve meshes used for a simulation are shown, repre-senting start, intermediate and final calculation. As the mesh is coarsened at some distancebehind the heat source it is not allowed to coarsen to the maximum degree in the weldregion. This restriction was implemented to provide the flexibility necessary to capture thedeformations resulting from the welding.
Once the topology of a mesh has been defined, redundant nodes must be eliminated. In thethermal model constraint equations must be derived to obtain interelement compatibility.As regards the mechanical model, the elements to be graded must be identified and theproper nodes added to their definition. To speed up the process spatially addressable searchtrees has been defined for node search.
A consistent data structure is essential to relate a new and updated mesh to a precedingmesh. A data array structure in two planes is used for mesh refinement data. One planeis used for current mesh data whereas the other holds data for the preceding mesh. Whena new element first enters the model it is assigned a unique element number, which is usedas key in the data array. The information saved in the mesh data array comprises the levelof refinement, parent number, graded face key, children and geometrical part affiliation.When the mesh is updated, the history of each element is easily recovered by comparisonof the information of the preceding and the current mesh refinement data. The use of datastructures allows for efficient mapping of history-dependent field variables between the oldand the new mesh.
5.3 Data Mapping 55
5.3 Data Mapping
As the mesh topology is changed, the positions and the numbers of nodes and Gaussianintegration points are altered. In history-dependent problems, the field variables must bemapped from the old to the new mesh in order to establish the initial condition for the nextiteration. Different problems must be addressed dependent on whether the quantities arenodal values or Gauss point values and whether the mesh is refined or coarsened. Firther, thereplacement of a graded with a non-graded element and vice versa, corresponds to coarseningand refinement, respectively.
Data mapping must be considered carefully as errors accumulated in mesh updating sig-nificantly may affect the final results. In the following, field variables of the old mesh aredenoted by 0* and those of the new by 4. Both interpolation and least square projection(LSP) techniques are applied in data mapping.
Data mapping by interpolation is used to find the quantity of a point from nodal values. Apoint quantity of the new mesh is found by interpolation of the nodal quantities of the oldmesh. The element in the old mesh containing the point is located and its shape functionsare used as interpolation functions. The shape functions Nj must be calculated on the basisof the normalised coordinates of the point with respect to the element. The normalisedcoordinates (4, 27, () are unknown and must be calculated from the cartesian coordinates byinverse isoparametric mapping. In the case of e.g. non-rectangular fillet elements Eq. (5.2)defines a non-linear system of equations which is solved using an iterative numerical solutiontechnique. In the case of rectangular elements, Eq. (5.2) defines a linear system of equationswhich can be solved directly.
n
t= •N j (4,7,() t;, t=x,y,z (5.2)j=l
Once the normalised coordinates and the shape functions are calculated, the nodal quantityof the new mesh is easily found by Eq. (5.2) substituting t and t* with the field quantities 4and 0'.
The mapping of Gaussian integration point values is somewhat more complex. The mappingis carried out using a least square method for projecting the integration point values of theold mesh to the nodal positions. FRom these 'artificial' nodal values the values of the newGauss points are subsequently calculated by interpolation. Standard least square collocationhas been applied, yielding the following matrix equation to be solved:
[Q]T[Q]{4'} = IQ]'{O"1 (5.3)
on represents the 'artificial' nodal values to be estimated, 09 represents field variables at theGauss points of the old mesh and component Qij of (Q] is the shape function for node j,evaluated at the isoparametric coordinates of the Gauss point i.
56 Chapter 5. Dynamic Meshing
In the thermal analysis initial nodal temperatures must be prescribed for the new mesh. Inmesh refinement, temperatures of added nodes are obtained by interpolation in the temper-atures of the old mesh. In coarsening, the temperatures of the common nodes are copiedwhereas the redundant nodal values of the old mesh are simply dropped. The mapping oftemperatures is not subjected to any significant inaccuracy as refinement and coarseninggenerally occurs in domains with small thermal gradients.
Data mapping of the mechanical quantities are, however, not so simple. The data needed toestablish the initial conditions for the first iteration using the updated mesh comprises:
* Displacements
" Peak temperature
" Dilatational strain
" Plastic strain
* Yield surface shift strain
" Accumulated ineffective strain
In mesh refinement the added nodal quantities are obtained by interpolation in the nodes ofthe old mesh. The mapping of Gauss points quantities in mesh refinement is a little morecomplex. The number of Gauss points in the old element to be coarsened varies between8 and 64 depending on the number of linear subdomains, whereas the number of nodesin the old element lies in the range 8 to 20 as at least three faces are non-graded. TheLSP technique is used to minimise the error in projecting the Gauss point quantities to thenodes. As this is done in the old mesh the establishment of [Q] is straightforward as theisoparametric coordinates are ready at hand. Inverse isoparametric mapping is, however,needed as the Gauss point values of the new mesh are calculated by interpolation in the'artificial' nodal values of the old mesh.
A good accuracy is usually obtained in refinement as the increased number of node andintegration points easily represents the variation of field variables in the domain. The mainsource of inaccuracy in refinement is related to the changed stiffness of the cluster of elementsreplacing the elements to be refined. Coarsening is subjected to somewhat larger errors. Notonly is the stiffness of the domain altered but the field variables need also to be representedby a reduced number of variables.
Mapping of Gauss point data in coarsening is managed by the LSP method. All values ofthe integration points belonging to the cluster of elements to be replaced are projected ontothe nodes of the new large element. Hence, inverse isoparametric mapping is required toestablish the [Q}-matrix. Subsequently, the Gauss point data is found by interpolation ofthe 'artificial' nodal values.
5.3 Data Mapping 57
The mapping of nodal values in coarsening can be carried out by different techniques. Ifthe LSP method is applied, the domain must include both the element to be coarsenedand the surrounding elements, as shared nodes must be assigned equivalent values [411.Alternatively, the values of nodes common to the old and the new mesh can be retainedand the values of redundant nodes can be dropped. Errors are introduced by either of thetechniques, primarily due to the different techniques used to obtain the nodal values andthe integration point values. As previously discussed, the Gauss point values are obtainedby the LSP method, taking into account the integration points of the cluster of elementsto be replaced. If the method used to obtain the nodal values is not based on the valuesof the same domain a mismatch is created. This mismatch is most significant if values ofredundant nodes are simply dropped but it is, however, also present if a larger domain isincluded. In the present Work the node dropping technique has been applied in combinationwith a corrective technique which emphasises the preservation of nodal displacements. Theelastic strain is defined as
ed = ftq _ -l _ -P CfU VU ' (5.4)
where
4 elastic strainftt total strain
c thermal strain
ef'j plastic strain
eqf accumulated ineffective strain while inactive
'3 was introduced in Section 4.7.1 and represents the ineffective strain accumulated in anelement while dead. In the first iteration using the new mesh, the total strain is calculatedfrom the nodal displacements and the trial strain is calculated in equivalence to the elasticstrain in Eq. (5.4). The mapping error becomes visible in the calculation of the trial strainas the total strain has been obtained by the node dropping technique, whereas the otherstrain components have been obtained by the LSP technique. The method for correctionexploits that the total strain is a part of the element saved base variables, so that it can befound by the LSP technique and compared to the total strain derived from nodal displace-ments. By subtraction of the differently derived total strain components, a corrective strainis established which is included in c?,. The resulting gain is that the nodal displacementshave been preserved and that the trial strain calculated in the first increment is found fromvalues based on the LSP technique representing the coarsened domain. What has not beenaccounted for is the stiffness change attributed to the replacement of a cluster of elementsby a larger less flexible element, which results in a slight change in the element load vector.
Figure 5.2 compares the von Mises stresses before and after mesh updating for equivalentload steps. It should be noted that the displayed stresses are integration point values copiedto the nodes by a heuristic method. Furthermore, the postprocessor does not include mid-edge and mid-face values in the plotting which causes a discrepancy in these positions asonly values for the small elements at a 2:1 joint are included in the plot averaging.
58 Chapter 5. Dynamic Meshing
397
(a) Before coarsening (b) After coarsening
Figure 5.2: Von Mises stress before and after mesh updating.
However, if the plot errors are disregarded very good agreement is obtained between stressfields. This is further supported by the L2-norm used for convergence checking, which issmall in the calculation used for mesh updating. However, if the contact surface is redefinedin areas of primary contact, the contact pressure may be altered and cause a somewhathigher convergence norm in the first iteration. This is, however, not visible if the resultingvalues of the field variables are compared.
The data structure previously discussed relates the old mesh to the new and provides theinformation necessary for efficient data mapping between meshes. However, in some casesthis data structure is not available for data mapping. In simulation of double-sided filletwelding, it is desirable to 'reset' the mesh data after the first fillet has been laid and definean altered root mesh for simulation of the second. Elements as those of the second fillet,which did not exist in the first model, are defined as inactive elements which do not needany initial data. Before data transfer is initiated, the relationship between the new and oldmesh must be established as it e.g. is of relevance to know the clusters of elements whichhave been merged by the coarsening. This relation is established by finding the elements inthe old mesh which contains the new nodes and integration points. For this purpose, themethod used in Brown and Song [1O] has been adopted to locate a spatial point in a mesh.
The above technique for data mapping is far more general than the mapping method basedon mesh history but requires a considerably larger computational effort.
5.4 Mesh Density in Dynamic Meshing
The mesh density required in welding simulation is decisive for the computational efficiencyand was shortly addressed in the discussion of adaptive meshing versus dynamic meshing.
5.4 Mesh Density in Dynamic Meshing 59
Mesh density requirements in dynamic meshing are investigated by evaluation of the fieldvariables in both the thermal and the mechanical model. It is, however, clear that a com-promise between resolution and computation time must be made and that the mesh densityfor the present work have been chosen with the emphasis on computational efficiency.
The first issue to consider is the basic dimensions of the finest elements and the size of timeincrements. A series of computations was made to find a set of dimensions where furtherrefinement did not cause considerable changes in the field variables in the region surroundingthe fillet. Specifically large gradients occur at start and stop and in the regions around tackwelds. Observed from a co-moving frame an approximate steady state is reached betweentack welds. In this 'steady state' region the requirements to discretisation in the weldingdirection are reduced and the length of the elements can be extended. The models presentedso far employ elements of constant length along the weld line but in the models to beintroduced, the element length in the intermediate region grows towards the double lengthof those in the start and the stop region. Thus, the computational efficiency is significantlyincreased for long models with few tack welds. As regards the time stepping procedure, itis common practice to use the element length divided by the weld speed as time increment.This reduces the dependency between heat source position and mesh topology as the discretevalues of power density obtained by the double elliptic heat source do not change seen froma co-moving frame.
The next issue to consider is the level of mesh refinement needed in the different parts ofthe model. Temperatures and thermal gradients change rapidly in the immediate vicinity ofthe heat source but the gradients diminish quickly with distance as previously illustrated inFigure 4.15. As regards the mechanical field, the large thermal gradients produce correspond-ing mechanical changes in heating but at high temperatures and in cooling, the similaritybetween the thermal and the mechanical fields disappears for two reasons. At high tempera-tures the material stiffness is reduced significantly and the gradients in the mechanical fieldvariables disappear. In cooling, the thermal gradients diminish quickly, whereas volumetricexpansion and rapid change in mechanical properties follow from the phase transformation.
The mesh density in front of the heat source has been decided upon by evaluating themechanical gradients as these impose higher demands than the thermal gradients. Thelargest gradients occur in the HAZ region as illustrated by Figure 5.3, which shows theequivalent plastic strain in the plane transverse to the welding direction. In a similar fashion,the von Mises stress increases rapidly from almost zero to approx. 200 MPa in the region.The mesh density has simply been chosen to accommodate this region of high gradients,either by the finest elements or by the graded elements facing the finely meshed region.
The mechanical effects in cooling strongly influence the mesh density needed in the regiontravelling behind the heat source. Two phenomena in cooling are important to describe themechanical activity. First, the material regains its strength as the temperature is decreased.Secondly, volumetric expansion and rapid change in material properties follow from the phasetransformation. Basically, mesh coarsening can be conducted with little loss of accuracy ifthe domain has similar mechanical properties and small gradients in field variables. If the
60 Chapter 5. Dynamic Meshing
-Z~z-- .02777.
Figure 5.3: Equivalent plastic strain in the plane transverse to the welding direction.
field gradients do not develop in time, coarsening can be argued only to result in a lessgood representation of stresses and strains and in a slight stiffness alteration, whereas nodaldisplacements can be retained with reasonable accuracy. Fuirther mesh coarsening is thereforerestricted by the transient change in mechanical gradients occurring in the domain behindthe heat source, which is related primarily to the occurrence of phase transformation andthus to the phase transformation temperature range. A reasonable basis for determination ofthe length of the densely meshed 'tail' travelling behind the heat source would therefore bethe A3 -temperature which represents the lower temperature bound of phase transformation.Presently, the minimum A3 is 4000C but later, in Chapter 7, other material models willbe investigated with A, as low as 2550 C. As the cooling rates decrease significantly whenthe temperature drops, a low A3-temperature will result in a long tail of fine elementstravelling behind the heat source. This is apparent analysing the temperature range shownin Figure 5.4.
The above approach offers a reasonable increase in computational efficiency based on thecurrent material model because of the relatively high value of A3. However, the gain isnot significant and if a lower A3 -temperature is considered, the gain quickly turns into zerofor the 280 mm long T-profile shown in Figure 5.4. Therefore, it is highly interesting toinvestigate the loss of accuracy associated with a shortening of the tail of the most refinedelements. The applied material model uses the peak temperature of a material point as thedecisive parameter for the temperature range of phase transformation. Decreasing the taillength to increase the computational efficiency will cause an averaging of peak temperatureand consequently an averaging of transformation strain. This will mainly affect the HAZelements adjacent to the fillet, since this particular region experiences the largest differencesin peak temperatures. In Figure 5.4, the highest level of refined elements is coarsened onelevel at about 5250C and further one level at approx. 45000 with the exception of the HAZelements which are maintained. In consequence, coarsening is carried out before the A3-temperature of 4000C is reached. The loss of accuracy has been evaluated in a comparison
5.4 Mesh Density in Dynamic Meshing 61
TF rature
236272308344380
4 88152
Figure 5.4: Slow cooling in the lower temperature range.
of the dynamically meshed model to a model with fully refined elements along the weldline. To further evaluate the dependency on mesh density, a coarse model has been includedin the evaluation for comparison. The coarse model is identical to the full model but themaximum element refinement has been decreased one level. The stress and displacementfields are shown in Figures 5.5 and 5.6 for evaluation.
(a) Full model (b) Dynamic (c) Coarse
Figure 5.5: Von Mises stress [MPa].
62 Chapter 5. Dynamic Meshing
(a) Bil model, x-disp. (b) Dynamic, x-disp. (c) Comre, x-disp.
*. C000000
(d) Full model, y-disp. (e) flynamnic, y-disp. (f) Coarse, y-disp.
(g) Fill model, z-disp. (h) Dynamic, z-disp. (i) Coare, z-disp.
Figure 5.6: Displacements [mm].
5.4 Mesh Density in Dynamic Meshing 63
The finely meshed dynamic model shown in Figure 5.5 yields stress values within 10% ofthose of the full model. This is satisfactory as the applied coarsening procedure emphasisesthe retainment of displacements on account of accurate stress mapping. As a general problemit should also be noted that the stress values are Gauss point values and that the displayedvalues in differently meshed regions are therefore related to different locations.
Based on visual inspection of the displacement fields it can be concluded that dynamicmeshing provides a very good approximation to the deformations predicted by the full model.Hence, it is validated that coarsening can be carried out before the phase transformationhas finished without the introduction of significant displacement errors.
As regards the coarsely meshed model quite different displacement and stress fields are pre-dicted. The deviations in x-displacements are the most significant as the web is seen togo into a different deformation mode with maximum displacements at the top of the web.This deviation is independent of the structural stiffness of the surrounding structure as thecharacteristic deformations are preserved for both the full and the coarse model when thegeometrical dimensions are increased. Three other issues may, however, be related to thedeviation. First, the heat source modelling deviates due to the difference in mesh discreti-sation and in the coarse model the net heat input must be corrected by the discretisationerror. Secondly, the thermal gradient through the web is essential to the bending momentintroduced. Since the temperature is averaged on element basis, four elements through thethickness result in a rather poor representation of the thermal strain. Thirdly, especiallythe bending stiffness of a coarsely meshed structure is larger than that of a finely meshed.All three factors cause deviations but proper discretisation of the region surrounding theheat source seems to be most essential. This is indicated as the dynamically refined modelpredicts the displacements with good accuracy in spite of a stiffness only slightly differentfrom the coarsely meshed model.
The conclusion of the above analysis is highly interesting with respect to computationalefficiency. Without the capability of dynamic mesh refinement it is necessary to employ acomputationally demanding model with a dense mesh along the weld path. This is requiredas a coarser but faster model fails to predict even the correct qualitative deformation pat-tern. The dynamic model allows both the quantitative and qualitative deformations to be
calculated with good accuracy and significantly reduced computational effort.
The next problem to be analysed is the degree of mesh refinement needed in dynamic meshingin order to obtain a reasonable prediction of the distortions. Compared to the first meshrefinement scheme the extent of the finely meshed region is reduced to decrease the numberof DOFs and, hence, increase the computational efficiency. A series of calculations was made
to find a mesh refinement allowing the distortions to be calculated quickly with reasonableaccuracy. The mesh shown in Figure 5.7 provides a suitable compromise between accuracyand speed. An additional step towards increased efficiency was made in the time steppingprocedure. An increment corresponding to the length of an element is common practice butthe use of double sized calculation steps had almost no visible impact on the results, evenin the centre region with long elements. The actions taken and the assumptions made so
64 Chapter 5. Dynamic Meshing
Figure 5.7: Mesh used for computationally demanding applications.
far to increase the computational efficiency are considerable and should supposedly influencethe displacement and stress fields significantly. Comparing the calculated displacement andstress fields shown in Figure 5.8 to those of the full model, it is, however, concluded thatthis has been accomplished without seriously affecting the accuracy of the results.
It is, however, important to recognise that the above approach does not offer detailed mod-elling of the plastic straining which may occur, if the yield stress is exceeded towards theend of the thermal cycle or if plastic straining occurs outside the regions exposed to highthermal loads. Increased mesh density in such regions may be included by use of adaptivemeshing but, again, the computational demands will increase dramatically.
No matter whether dynamic or adaptive meshing is applied, the effect of mesh densityon structural stiffness (especially bending stiffness) and thus displacements is difficult toevaluate. Normally, a repetitive mesh refinement is used until asymptotic displacementconvergence is seen and based on this, the mesh density is decided upon. Two problems ariseby the use of this approach for the present application. First, the computational capacitydoes not allow the problem to be refined sufficiently. Secondly, the changing mesh densityconstantly redefines the problem to be investigated. In the models presented, two linear solidelements are used through the thickness as a minimum in the far field region. This resultswithout any discussion in a structure which is too stiff in bending and must be describedas an absolute minimum if dominant parasitic shear should be avoided. As further meshrefinement is undesirable in respect to computational efficiency, the application of bubblefunctions to the graded element should be considered to improve the bending behaviour.
5.5 Double-Sided Fillet Welding
Simulation of double-sided fillet welding has become computationally attainable by the useof dynamic meshing. The requirements to mesh density are increased as the residual stresses
5.5 Double-Sided Fillet Welding 65
(a) x-disp. [mm] (b) y-disp. [mm]
I 0l 0m 0B O a0 MmODOOMMOD
(c) z-disp. [mrm] (d) Von Mises stress [MPaJ
Figure 5.8: Results predicted by dynamic model with reduced refinement.
and deformations from the first fillet are included as initial conditions. The elements in theregion of the first fillet are not coarsened to the lowest refinement level as a higher density isrequired to capture properly the residual stress and deformation. Further, the temperature inthis region rises sufficiently to cause considerable stress relaxation. All together the numberof elements is increased considerably.
In the simulation of the second fillet the web has already been locked to the base plate bythe constraint equations of the first fillet, and the relative displacement between the partsremains therefore unchanged in simulation of the second. The filler elements are 'born'as the thermal expansion is at its maximum and locking the second fillet in this positionmodels properly the restraint which follows upon cooling. The mesh used for simulation isbased on the mesh refinement scheme of the dynamic model initially introduced. Figure 5.9
66 Chapter 5. Dynamic Meshing
shows von Mises stresses and mesh topology viewed from the side of the second fillet andthe first fillet, respectively. The stress state which follows upon welding of the first filletwas shown previously in Figure 5.5(b), and Figure 5.9(b) therefore illustrates a considerablestress relaxation in the first fillet caused by the welding of the second.
000000000 WOMllDMOfODO
(a) View of second fillet (b) View of first fillet (stress relaxation)
Figure 5.9: Von Mises stress in simulation of second fillet [MPa].
(a) x-disp. (b) y-disp. (c) z-disp.
Figure 5.10: Displacements in simulation of second fillet [mm].
Figure 5.10 shows the final displacements for the T-profile. The effect of laying the secondfillet can be evaluated on comparison of the above results to those of the dynamic model inFigure 5.6.
5.6 Obtained Simulation Efficiency 67
5.6 Obtained Simulation Efficiency
At this point the simulation efficiency has been increased significantly compared to the basicmodel initially investigated. The efficiency obtained through graded elements and dynamicmeshing has the drawback of a decrease in simulation accuracy, which, however, appears tobe quite acceptable with regard to distortion prediction. Besides the decrease in computationtime, the increased efficiency also allows for simulation of welding applications previouslyunattainable.
The developed mesh refinement and data mapping algorithms are quite efficient. The tasksinvolved in the procedure are shortly listed:
" Extract Gauss point data and nodal data from model
* Save model and results
" Generate a new and updated model
" Map data onto model
" Initialise elements in the new model
The time needed for the above procedure depends on the model size. For the model shownin Figure 5.1 the time elapsed before solving is continued in the updated model is less than90 secs.
The computational saving obtained by dynamic meshing is not pronounced for small modelsbut the saving increases dramatically as the length of the model is extended. In contrast,the computational requirements becomes extensive as a fully refined model is extended inthe longitudinal direction and as a result rather limited models, measured in welding length,can be dealt with. In consequence the comparison between a dynamically meshed and afully refined model is not straightforward since direct comparison in computation time canonly be made for small models where the advantage of dynamic remneshing has not yet beenfully put into effect.
In general the increase in efficiency is not easily quantified. As previously mentioned, the useof a direct solver gives a computation time roughly proportional to the bandwidth squaredtimes the number of DOFs, times the number of time steps. In dynamic meshing thebandwidth changes in a transient manner due to the constantly changing contact and thevarying number of DOFs. In consequence a detail like a differently positioned tack weld mayinfluence the state of contact and alter the computation time considerably.
In order to provide a quantification of the obtained computational efficiency a few examplesof computation time will be given. The workstation available was an HP J5000 with 2Gb
68 Chapter 5. Dynamic Meshing
RAM. The computation time needed for each of the four models used in the investigationof dynamic modelling is listed in Table 5.1.
Model DOFs Computation timeFully refined along weld path 37593 32 hrs.Dynamic 10236-21723 13 hrs.Dynamic (reduced refinement zone) 8667-14442 3 hrs.
Coarse mesh 14931 3 hrs.
Table 5.1: Computation time for 280 mm T-profile, one-sided welding.
In the comparison it should be noted that the time stepping was changed for the dynamicmodel with reduced refinement zone. Further, this model is seen to have fewer DOFs thanthe coarsely meshed model, which is due to the reduced refinement at all refinement levels.The computation time includes both the thermal and the mechanical analyses of one-sidedwelding.
For industrial applications a weld path of 280mm is of little practical value. To study thecomputational capability of the dynamic model with reduced refinement zone a considerablylarger application was investigated. Double-sided welding of a 1690mm T-profile, tack weldedin five positions, was simulated. The total length of the weld path was 3380mm, whichsurpasses most industrial welding applications, except those commonly found at a shipyard.Figure 5.6 shows the deformed geometry by the use of a scaling factor of 30.
The computation time used for the above model is given in Table 5.2.
Model Computation timeThermal, first side 10 hrs.Mechanical, first side 31 hrs.Thermal, second side 15 hrs.
Mechanical, second side 64 hrs.Total 120 hrs.
Table 5.2: Computation time for 1690 mm T-profile, double-sided welding.
Comparing the computation time with the capability of the workstation and consideringthe length of the simulated weld path, the above example clearly illustrates that numericalsimulation can be applied to the investigation of welding applications in industry.
5.6 Obtained Simulation Efficiency 69
Figure 5.11: Deformed geometry of large model (scaling factor 30).
70 Chapter 5. Dynamic Meshing
5.7 Effect of Tack Weld Positioning
The positioning of tack welds influences significantly the deformation pattern. This is illus-trated by the large T-profile presented in the previous section and the change of tack weldpositions.
Q),
13)
(4) J
11 (3)
(a) (b)
Figure 5.12: Deformation mode depending on tack weld positions (scaling factor 30).
Altogether 5 tack welds were applied to each T-profile, positioned at the weld line as indicatedby the numbers in Figure 5.12. In welding, the first fillet is laid on the right side of the web,starting at the far end.
It is interesting to note that the deformation mode of especially the web is highly dependenton the tack weld positioning. In fact, the modes are directly the opposite of each otherwhereby it corresponds to the difference in tack weld positioning. In the simulation it wasfound that the positions of the first few tacks were decisive for the deformation mode as theinfluence of the tack welds is not sufficient to cause large alterations once the mode has beeninitiated.
5.8 Experimental Evaluation 71
5.8 Experimental Evaluation
A rough quantitative evaluation of the predicted deformations will be presented in this sec-tion. Experimental evaluation of the simple T-profile shown in Figure 5.13 was carried out byBirk-Sorensen [7]. The hatched edge was restrained between to rigid plates, bolted togetherto form a cramp device. This boundary condition is not well suited for FEA as it involvesrestraint based on unspecified contact. It has, however, been modelled on the assumptionof fully restrained out-of-plane displacements and unrestrained in-plane displacements. The
C
Figure 5.13: Simple T-profile, plate thickness: 12mm.
welding parameters for the present specification are somewhat different from those of thetarget application outlined in Section 3.2. The dimensions of the T-profile are given in Fig-ure 5.13 and the WPS in Table 5.3. The longitudinal and the transverse deformations were
Plate thickness 12 mmMaterial mild steelSpecified A-measure 3.5 mmWelding speed 7 mm/sVoltage 28 VAmperage 280 AProcess MAG FCW
Table 5.3: Welding parameter specification
measured by a large micrometre screw gauge between the locations A-B and C-D, respec-tively, as indicated in Figure 5.13. The out-of-plane deflection was measured by the use of adial indicator at location E. The measured values are presented in Table 5.4. Even thoughthe utility value is compromised by the small number of experiments, a good estimate ofthe magnitude are provided by the investigation. The mean values are used for evaluationof the numerical model.
72 Chapter 5. Dynamic Meshing
Specimen Fillet no. Longitudinal Transverse Deflection1 1st 0.17 -0.05 1.80
2nd 0.02 -0.16 3.802 1st - - 2.16
2nd 0.12 -0.10 4.903 1st - -
2nd 0.24 -0.30 5.504 1st 0.29 -0.16 2.36
2nd 0.22 -0.18 5.035 1st 0.26 -0.23 2.04
2nd 0.10 -0.36 4.406 1st 0.22 0.06 2.73
2nd 0.10 -0.15 5.03Average 1st 0.24 -0.10 2.22
2nd 0.13 -0.21 4.78Std. variation 1st 0.05 0.13 0.35
2nd 0.08 0.10 0.59
Table 5.4: Measured deformations [mm] (Birk-Sorensen [7]).
The numerical model was established by changing the dimensions and welding parametersin the input file. Further, a series of iterative thermal simulations was carried out to esti-mate the descriptive heat source parameters based on a polished x-section. The dynamicupdating was carried out every 8th time step resulting in 13 models for each fillet weld.The y-displacements resulting from simulation of the first and the second fillet are shown inFigure 5.14.
Y-disp. 1-]
S 0.086
1 •.045
2.314
__ 3 544 2164 .50
(a) First fillet (b) Second fillet
Figure 5.14: Y-displacements
5.9 Summary of Chapter 5 73
The longitudinal and the transverse displacements are strongly dependent on the exactlocation of measuring points. Depending on the y-coordinate of the measuring points, thetransverse displacement varies from -0.11 to 0.07 mm in the case of the first fillet. Likewisethe longitudinal displacement varies from -0.02 to 0.17 mm from the base plate to the web.In consequence, neither good nor bad agreement between the numerical and the measuredvalues can be concluded from the experiments available.
The measuring of deflection is, however, better defined and less dependent on the exactposition of the measuring point. The predicted deflections are 2.44 mam for the first filletand 4.65 mm for the second. Comparing these values to the average deflection values listedin Table 5.4, the deviations are less than 10% and 3%, respectively.
The above evaluation shows good agreement for the deflections. Further, the order of mag-nitude was verified for all the predicted distortions. The longitudinal and the transversestrains could unfortunately not be evaluated and proper verification will require experimentsdesigned specifically for the application.
5.9 Summary of Chapter 5
In the present chapter the computational efficiency of welding simulation has been increasedconsiderably. Based on the mesh grading characteristics of the graded element, a dynamicmesh refinement scheme has been developed with the emphasis on displacements in the datamapping algorithms. By investigation of the mesh density required in dynamic meshing, itwas found that good predictions of the qualitative distortion pattern could be obtained, evenif the mesh refinement was decreased considerably to increase the computational efficiency.
By application of dynamic mesh refinement, the welding of the second fillet was shown tocause considerable stress relaxation in the fillet already laid. Further, the effect of tackweld positions was seen to be significant as a change in positioning resulted in very differentdistortion modes of the web.
The use of dynamic mesh refinement allows simulation of welding applications which waspreviously far beyond the computational capability. The welding simulation of a T-profilecontaining more than 3.3 metres of welding illustrates clearly that numerical simulation maybe applied to large industrial problems.
74 Chapter 5. Dynamic Meshing
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Chapter 6
The Local/Global Template
The efficiency obtained by graded elements and moving mesh algorithms is inadequate whenthe objective is simulation of welding in large structures such as ship sections. The essentialproblem of moving mesh refinement is that the structure should be discretised at leastto the level where geometry, stresses and deformation can be represented with adequateaccuracy at ambient temperature. More precisely, unwelded regions should be discretised torepresent geometry and restraints properly. The welded regions should also hold the stressesand deformations accumulated in welding and, hence, further discretisation is required. As aresult the number of DOFs necessary to represent a given structure at the coarsest refinementlevel very quickly increases beyond what is computationally attainable as the size of thestructure grows.
In the present context a template is defined by a bundle of techniques, combined to enablewelding simulation in large structures. Several templates of varying complexity have beenproposed in literature. Most templates are based on heuristic or idealised models for calcu-lation of the welding response locally at the weld, which are subsequently mapped on to theglobal structure. Recent examples can be found in the papers by Tsai et al. [66] and Seoand Jang [55], who employed an experimentally based inherent strain method and a highlyidealised analytical method, respectively. The work carried out by Brown and Song [10], whoapplied dynamic meshing and dynamic substructuring to a 2-D edge heating application, ismore interesting.
Generally, templates are always based on further approximations compared to the full simu-lation. The essence is how these approximations are chosen and how they affect the accuracyof the prediction. To predict the qualitative distortions of an assembly rather extensive re-quirements are imposed on the template to be developed. The requirements previouslyidentified for the 3-D fillet welding application are still relevant, but in addition it is nownecessary to include the boundary conditions formed by the surrounding structure and toinclude the stress and deformation caused by preceding welding at the assembly. Thesefactors are essential to a design tool capable of predicting the welding response as a result of
75
76 Chapter 6. The Local/Global Template
welding sequence. A template denoted the local/global template (L/G template) has beendeveloped for the purpose.
6.1 Concept of the Local/Global Template
The target application introduced in Section 3.2 is chosen for analysis. The subassemblyis a part of a transverse web frame in a container vessel and is particulary interesting as ithas caused significant problems in production due to welding deformations. A global modelrepresenting the subassembly is purely composed by linear elastic shell elements. Comparedto solids, the use of shell elements significantly reduces the number of DOFs and improves thebending behaviour. The element applied is a 4 noded quadrilateral 3-D shell of the discreteKirchhoff type without shear deflection capability [2, 15]. The global model is illustrated inFigure 6.1.
Figure 6.1: The global model.
As the global shell model is not suited for welding simulation, a local solid model is built foreach weld line using the dynamic models with reduced refinement introduced in Section 5.4.The idea behind the L/G template is to link the local models to the global model and thusenable a proper representation of both the boundary conditions for the local models and
6.1 Concept of the Local/Global Template 77
the accumulated distortions in the global model. Moreover, it is exploited that a large partof the structure can be modelled by substructures due to its linear elastic response at lowtemperatures. Figure 6.2 shows the first few steps in the L/G template.
(a) (b)
(c) (d)
(e)
Figure 6.2: Steps in L/G temnpate.
78 Chapter 6. The Local/Global Template
The steps illustrated can be described as follows:
(a) The global shell model, also shown in Figure 6.1, and the local solid model of the firstweld line are generated.
(b) The shell elements represented by the local model are removed and the boundary nodesare identified. Substructures are generated to represent the structure of the globalmodel and these are coupled to the local model.
(c) Stress and displacement fields are found by welding simulation.
(d) Nodal positions in the centre plane of the local model are used to update the displace-
ments in the global model.
(e) The next local model and the surrounding substructures are generated. Based on thedisplacements found in (d), the loads acting on the boundary of the local model arecalculated and included in the formulation of the substructures.
() Stress and deformation are calculated as in (c) and the cycle (c)-*+(e) is continued untilall weld lines have been included.
The tasks defining the L/G template are therefore:
" Generation of geometrically consistent local and global models
* Substructure generation including thermal and structural loading
" Coupling of shell and solid elements
" Method for updating accumulated distortions in global model
* Extraction of boundary conditions from the global to the local model
The local models are not been included in the above list as they have already been attendedto in the previous chapters.
The above discussion concerns only the mechanical modelling as it is far more complex thanthe thermal. For smaller assemblies it is sufficiently efficient to skip the thermal substructuregeneration and include all shell elements of the far field domain in the local thermal analysis.As the material at a weld line is assumed to cool to ambient temperature before the nextfillet is laid, there is no interaction between the thermal models. Due to the simplicity of thethermal modelling, the point of focus of the subsequent discussion will be the mechanicalaspects of the modelling.
6.1 Concept of the Local/Global Template 79
6.1.1 Model Geometry
To facilitate the exchange of data between the local solid models and the global shell model,consistent geometry must be ensured. The shells in the global model corresponding to thelocal solid models are from now on denoted the equivalent shells. These axe based on thecentre planes of the local models and have a mesh density corresponding to the coarsestrefinement level of the solids.
The stiffness of the structure increases as the structure in welding is gradually changed frombeing attached only by tack welds to being fully joined at the weld lines. The change instiffness is included in the global model and thus also in the substructures by modelling thetack welds as well as the joints after welding. The gap occurring in the global shell modelbetween the centre planes of the web and the base plate, causes a problem when the pastsmust be joined. The gap is unphysical as it is only related to the modelling of plate thicknessby shell elements. A connecting shell element is applied to join the web and the base plateand the properties of the shells constituting the joint are modified. A T-joint, fillet welded onboth sides, is very rigid in bending transversely to the weld line, whereas the axial stiffnessis only changed by the added stiffness of the fillets. This can be approximated by decreasingthe elastic modulus and increasing the thickness, which leaves the axial stiffness unchangedwhereas the bending stiffness is increased. A series of FE analyses was made to find a setof properties for the shell joint, providing a reasonable approximation to the behaviour of ajoint modelled by solid elements.
6.1.2 Combining Solids and Shells
The mixing of solids and shells is advantageous in welding simulation. Shells are superiorto solids in the far field region as they improve the bending behaviour and as there are norestraints on the aspect ratio, whereby the number of elements can be dramatically reduced.However, in the non-linear region with large gradients through the thickness, solids arerequired to obtain an adequate resolution.
The shell and solid elements have different node arrangements and different degrees of free-dom and in order to combine them, the compatibility problem must be solved. Figure 6.3(a)illustrates the DO~s to be coupled in the combination of solids and shells. The nodes inposition 1-3 belong to the solid elements and have only translational DO~s. The node inposition 4 belongs to the shell and has 3 rotational DO~s in addition to the translationalDOE.
Generally, constraint equations are used to relate the DO~s of the shell to those of the solids.However, the use of constraint equations does not allow the solids to expand thermally in thethrough-thickness direction, which cause spurious stresses. Secondly, it is computationallymore efficient to include the solid-shell transition arrangements in the substructures to begenerated from the shells, so that definition and computation only need to be performed
80 Chapter 6. The Local/Global Template
4)1
3) 6
(a) (b)
Figure 6.3: Connecting solids to shells
once. This is not possible by use of constraint equations as all relevant nodes must bepresent at the time of generation.
A different approach based on beam elements has been applied to solve the above problems.As shown in Figure 6.3(b), two beam elements are connected perpendicularly to the nodeof the shell in position 4. The beams are used to obtain large bending stiffness and lowaxial stiffness, which is achieved by defining a large moment of inertia and a small elasticmodulus. The beams, having 6 DOFs at each node, will rotate and translate with the nodesof the shells. The connection to the solids is now established coupling the translationalDOFs of nodes located in equivalent positions. The low axial stiffness of the beams allowsthe solids to expand thermally in the through-thickness direction and the bending stiffnessallows rotation about the in-plane axes to be transferred. The only restraint imposed on thesolids is the confinement of shear deformation as the beams force the nodes of the solids toremain perpendicular to the centre plane in bending. Further, the in-plane rotational DOFsof the shells are not specified in the transition.
The use of beams facilitates the coupling between solids and substructures made of shells.The beams are simply included in the substructuring and only the translational DOFs aredefined as independent master DOFs.
6.1.3 Utilisation of Substructures
Substructuring is a technique of matrix reduction used to reduce the system matrices to asmaller set of independent variables. Thus, a substructure is simply a collection of elementsacting as one element defined by a reduced set of DOFs. Below, the reduced set of DOFs isdenoted m for master DOFs and those to be eliminated are denoted by s for slave DOFs.The standard system of equations
[K]{q} = {Q) (6.1)
6.1 Concept of the Local/Global Template 81
is rearranged into
[K,,,] [K {q,,.} fQ.11 (6.2)[[f.] [K..] {qI- {Q.)
Solving for {q,}
{q,) = [K.,]'{Q.}-jKs]'[K.,]{qm} (6.3)
and substituting into the first equation of (6.2) yield
[[K.m] - [K,,][K8 5]1-[K.m]] {qmq} = {Q,,}-[K,,][Ks,,]-{Q,} (6.4)
where the large square bracket term is the substructure stiffness matrix and the right handterm is the equivalent force vector.
The entire far field region clear of the non-linear weld line regions is in general suited forsubstructuring. If the thermally affected part of the far field region is substructured in thethermal analysis it should later be expanded to provide the thermal load on the equivalentstructural substructure.
Due to the matrix reduction all elements are required to have constant linear elastic prop-erties. Loads acting on slave DOFs must be applied at the time of substructure generationand awe transformed into the equivalent load vector acting on the master DOFs.
When the global shell model has been substructured the bandwidth of the stiffness matrixshould be considered. After the global system of equations are set up, the bandwidth of theproblem is reduced by renumbering the nodes based on nodal connectivity through elements.If the local model is encircled by a single substructure a large bandwidth may be causedby inappropriate connectivity. Dividing the global model into a number of substructuresmay significantly reduce the bandwidth and thus the computation time. The use of multiplesubstructures is illustrated in Figure 6.2(e).
The subassembly must be spatially fixed for FE analysis by definition of boundary conditionspreventing rigid body translation and rotation. As the local models change for simulationof each weld line the boundary conditions are defined on the global shell model and henceincluded in the substructuring. For the present analysis the subassembly was fixed at thebottom of the base plate at the crossing of the webs. This later turned out to be impracticalfor comparison of deformations between differently welded assemblies as even small rotationsat the fix point cause considerable rotation of the entire subassembly. Hence, restraints atthe outer boundaries were applied in the comparison.
6.1.4 Updating of the Global Model
To find the welding response of the global structure and to generate updated substructuresfor the next local analysis, the global shell model must be updated by using the displacement
82 Chapter 6. The Local/Global Template
fields calculated in the local solid models. As the local solid models are not included in theglobal model the updating procedure is considerably complicated. The task is to apply thecalculated displacements to the global shell model by a procedure which will subsequentlyallow the deformed global model to respond linear elastically in loading.
The centre plane deformations of the solid model are extracted and imposed on the equivalentshells in the global model as shown in Figure 6.4.
(a) Local solid model (b) Equivalent shells
Figure 6.4: Updating displacements of equivalent shells in the global model.
When the displacement field is imposed by displacement restraints, an inelastic deformationfield can be obtained by updating the geometrical configuration of the equivalent shells. Thiswill yield the global displacement field, but as the displacement restraints are removed anelastic spring back will occur. The elastic spring back is significant and must be accountedfor in the updating, which can be done in either the local or the global model. Both methodshave drawbacks being estimations in comparison with inclusion of the local solid model inthe global model, which, however, is computationally infeasible.
The adjustment of nodal displacements to account for the spring back can be done in thelocal solid model by disregarding the boundary conditions supplied by the substructure andperforming an equilibrium calculation. This procedure requires the stiffness of the local shellsto be equivalent to that of the solids in order to yield the same spring back. This cannot beobtained for several reasons. Two solid elements through the thickness in the outer regionof the local solid model make the region too stiff in bending. The shell elements are betterin bending but are on the other hand incapable of shear deflection. Even if bubble functionswere included in the graded elements and shell elements with shear deflection capability wereapplied, the different bending and deflection behaviour would not allow the stiffness of thelocal solid model to be properly represented by the equivalent shells.
To overcome the problem of non-equivalent stiffness, the spring back may be accounted for in
6.1 Concept of the Local/Global Template 83
the global model by application of nodal spring back forces on the boundary of the equivalentshells. The procedure may be described by the following steps:
Step 1 Extract displacements from the centre planes of the local model and apply them tothe equivalent shells by translational displacement restraints. Perform equilibriumcalculation to find global deformation field.
Step 2 All DO~s (rotational + translational) at the boundary of the equivalent shells arelocked in the just calculated positions. The equivalent shells are killed and an equi-librium calculation is performed to make them stress-free in the updated geometricalconfiguration. The nodal reaction forces at the boundary of the equivalent shellsare extracted.
Step 3 Reactivate the equivalent shells and apply nodal forces corresponding to the reactionforces at the boundary. Perform an equilibrium calculation to obtain the updatedglobal shell model.
If additional weld lines have been simulated, the above procedure is repeated based on thedeformed geometry of the global model. This implies of course that the deformed globalmodel has been used to obtain the boundary conditions for the local models. Thus, theoutlined approach allows the accumulated distortions to be calculated. Information on thestress state is, however, not available but may be extracted from the local solid models ifrelevant.
Non-linear geometrical effects were included in the updating to test the influence. However,no significant effect was registered.
6.1.5 Extraction of Boundary Conditions
For simulation of the first weld line, the global model is only used to generate the substruc-tures representing the stiffness of the surrounding geometry. For the subsequent weld linesit is, however, necessary to take pre-stressing and pre-distortion into account. After the ge-ometry of the global model has been updated, the boundary nodes for the next local modelare identified. By use of restraints, the translational and rotational DO~s for these nodesaxe prescribed to zero in order to match the boundary of the undistorted local model to beused for simulation. An equilibrium calculation is performed, and the nodal reaction forcesat the restrained boundary are obtained. The nodal forces, corresponding to reaction forces,are subsequently applied as initial loads in the substructure generation. The pre-distortionand pre-stressing of the local solid model are thus produced by the loading included in theattached substructure. The pre-distortion of the local model has been tested against thedistortions of the equivalent shells in the global model taken after the last update. Thedisplacements were in very good agreement, which is seen by comparing the y-displacements
84 Chapter 6. The Local/Global Template
Y-dlspl. [r].-- 0.192-m -0.115
- -0.037aim 0.040- 0.117_--- 0.194m• 0.271
r--- 0.3480.4250.502
A(a) Equivalent shells (b) Local solid model
Figure 6.5: Comparison of y-displacements between equivalent shells and local solid model.
of the equivalent shells to those of the local solid model in Figure 6.5. As regards the stressstate, only small deviations are seen near the fillets which are caused by the differences ingeometry. It is consequently concluded that the different structural stiffnesses of the localsolid model and the equivalent shells do not cause significant errors in the pre-loading oflocal models. This is presumably due to the homogeneous stress and displacement fieldcharacterising the domains not yet welded.
6.1.6 Capability of the L/G Template
The L/G template provides a method for simulation of welding in large assemblies. It isutilised that the weld lines already welded and cooled may be represented by a linear elasticdomain on which displacements and reaction forces are imposed. Further, shells and solidshave been combined and applied in the domains where they each have the best characteristics.
Combining the methods for substructuring, updating and extraction of boundary conditionsinto a template, dramatically increases the computational efficiency as the alternative is amodel comprising solid modelling of all weld lines combined by shells.
The solid models described in Chapter 5 fulfil the requirements previously identified forwelding simulations. These comprise computational efficiency, full thermo-elasto-plastic 3-Dmodelling, dynamic adding of filler material, contact and relative displacements betweenparts as well as the effect of tack welds and their positioning. Combined with the template,the structural stiffness of the subassembly and the accumulated distortion and stress are alsoaccounted for. In conclusion, it should now be possible to predict the welding deformationswhich arises from different welding sequences.
6.2 Distortion Prediction in Large Assemblies 85
There are, however, also drawbacks related to the L/O template. First, it is presupposedthat the welds are cooled to ambient temperature before the next weld is initiated. Secondly,stresses cannot be represented in the weld line regions by the global models as these elementshave been made stress free in the updating procedure. Thirdly, and possibly most important,the modelling complexity related to the implementation of the template is rather large. Thedevelopment of dedicated subroutines did, however, facilitate the use of the template andmade it relatively easy to modify e.g. the welding sequence. But these disadvantages areinsignificant compared to the capability offered by the template, especially as there is noalternative available which enables welding simulation in large assemblies.
6.2 Distortion Prediction in Large Assemblies
The L/G template has been applied to predict the welding distortions from two arbitrarywelding sequences, which are shown in Figure 6.6.
3
A
(a) Sequence 1 (b) Sequence 2
Figure 6.6: Welding sequences investigated.
The boundary conditions were changed to facilitate displacement comparison between mod-els, using the locations A, B and C for restraining. All three points were fixed in theout-of-plane direction. In addition, B was fixed in the z-direction and A was fully clamped.
It was chosen to weld the two sides of a web immediately after each other as the numberof updates in the simulation thereby is reduced. Figures 6.7 to 6.8 show the out-of-planedisplacements for weld sequence 1 as a result of weld lines completed. The figures explainby example the necessity of representing the variation of structural stiffness along the weldlines as it clearly influences the angular deformation. The deformed shapes are shown witha scaling factor of 15.
86 Chapter 6. The Local/Global Template
Y-diap (MI
_6000-4,333
-01 6067
2 3 334 .000
(a) Weld lines 1-2
Y-disp [=I•• 9.3303
• 4ý333_2.657
0 6612.3334.000
(b) Weld lines 1-4
Figure 6.7: y-displacements as a result of weld lines 1-4 (scaling factor 15).
6.2 Distortion Prediction in Large Assemblies 87
y-disp (=]m-11.000
=3 -1.667-6.000-4.333-2.667-1,06..
2.33634.000
(a) Weld lines 1-6
09.3303
=3 -,667_• -6.00 0-4.333
0-2667
C• 2.3334.000
(b) Weld lines 1-8
Figure 6.8: y-displacements as a result of weld lines 1-8 (scaling factor 15).
88 Chapter 6. The Local/Global Template
(a) x-displ-pements
(b) z-displacements
Figure 6.9: Final x- and z-displacements using weld sequence 1 (scaling factor 60.)
Figure 6.9 illustrates the in-plane deformations of the base plate. The deformed shape isshown with a scaling factor of 60. The subassembly is seen to contract along the weld linescausing the ends of the L-shaped panel to move towards each other. In-plane deformationsas those predicted, are difficult to correct due to the relatively large in-plane stiffness ofthe subassembly compared to the out-of-plane stiffness. This will cause the subassemblyto distort in the out-of-plane direction when in-plane correction forces are applied. Hence,in-plane distortions are critical when the subassembly is subsequently to be assembled withother parts.
6.2 Distortion Prediction in Large Assemblies 89
The effect of welding sequence is illustrated by a comparison of the final y-displacementsof weld sequence 1 to those resulting from weld sequence 2 as shown in Figure 6.10. Thedifferences observed are not only related to magnitude but also to the qualitative deformationpattern and it may be concluded that weld sequence 2 results in less deformation.
A skilled welder may presumably pick out a weld sequence close to the optimal sequencein terms of distortions. The significant issue, however, is to combine practical experiencewith numerical prediction as it allows detailed analysis of numerous effects which cannot begrasped from practical experience only.
The above results illustrate that the L/C template can account for variations in structuralstiffness and, more important, for the welding sequence in a real size subassembly. Thus,the basic capability needed for the welding response prediction tool outlined in Chapter 3has been established. Unfortunately, no experiments were made to validate the results. Itis, however, safe to conclude that the predicted level of magnitude is the same as the levelobserved for the application in production.
As regards the computational efficiency the L/G template takes advantage of numeroustechniques:
" Graded elements
" Dynamic meshing
* Increased element length in 'steady state' regions
* Combination of solids and shells
" Substructuring of far field domains
" Substructuring of welded regions
A significant increase is obtained by the above techniques but seen in relation to the targetapplication, holding more than 8 m of welding, it is still a time-consuming task to computethe welding distortions. The time used for e.g. weld sequence 2 adds up to approximately290 hrs. on an HP J5000 workstation. This includes thermal and mechanical computation,generation of substructures, computation of substructure loads, global model updating, ex-traction of boundary conditions as well as model preparation time. R~om a simulation pointof view this is efficient considering the size of the subassembly and, especially, if the capabil-ity of the workstation is taken into consideration. Even a low-cost multi processor PC andthe use of code parallisation will make it possible to reduce the computation time to lessthan 150 hrs. and the use of powerful workstations will therefore reduce computation timeconsiderably.
90 Chapter 6. The Local/Global Template
-11:0300
I .9.333a-2 _7 0667
I 1 -6 0002 .333
11.000
(a) Weld sequence I
y-disp 1-11.1100_67
I-_91333
-600
I 2. 667
ii,000
y2
(b) Weld sequence 2
Figure 6.10: Final y-displacements resulting f'rom weld sequences 2 and 1.
6.2 Distortion Prediction in Large Assemblies 91
Computational efficiency was a requirement initially defined for the welding response predic-tion tool outlined in Chapter 3. From an industrial point of view, 150 hrs. may be acceptablefor the investigation of a single well defined weld sequence. It is, however, necessary to em-ploy an iterative procedure for the investigation of optimal welding sequence and powerfulworkstations are therefore required.
The subassembly investigated is extremely small in a shipbuilding context. To investigatee.g. the effect of increased structural stiffness at subassembly level on the distortions accu-mulated, much larger assemblies must be investigated. This is not feasible considering thetime required by the L/G template.
Fortunately, other industries do not impose the same requirements as shipbuilding. Heavyvehicle construction or chassis construction for trains and trucks offers the ideal applicationfor the L/G template. If e.g. a front frame of a wheel loader is considered, it containsapproximately 10 m of welding in heavy steel plates and is manufactured in large series.The size of the frame is computationally manageable and the large number of frames to beproduced legitimise the time required by the L/G template to provide detailed informationfor use in e.g. the design of fixtures.
92 Chapter 6. The Local/Global Template
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Chapter 7
Improved Material Modelling
The objective of the present chapter is to improve material modelling and to evaluate thematerial model applied in the preceding chapters.
Prior to the discussion of material models, an experimental investigation of microstructureis performed to enable some verification of the phase constituents predicted by the materialmodels. Further, the investigation nicely presents the large microstructural changes in theHAZ and the fusion zone.
Material modelling was discussed previously in Section 4.3. The temperature-dependentmaterial model used an indirect approach to account for the microstructural evolution, whichstrongly influences the material properties. In the present chapter a similar model is basedon a different set of material data, which improves the modelling of microstructural evolutionduring phase transformation.
Another but quite different material model is subsequently presented. This model deviatesfrom the previous model, as all parameters are obtained through CCT diagrams, coolingcurves and material data for each phase constituent.
The implementation and the effect of transformation induced plasticity (TRIP) are alsodescribed. TRIP is a phenomenon which may significantly influence the mechanical effectof the solid-solid phase transformation. Transformation plasticity is an irreversible strainoccurring in phase transformation, which acts to relieve the macroscopic deviatoric stressduring phase transformation.
After the discussion of microstructure, the material models and the implementation of TRIPare described. Finally, the influence of material modelling is evaluated and compared to theinitial material model, denoted material model 1.
93
94 Chapter 7. Improved Material Modelling
7.1 Microstructure
The discussion of microstructural evolution in the HAZ and the FZ is based on fillet weldingperformed with the welding parameters previously given in Table 4.4. The steel is a carbon-maganese steel with a chemical composition of 0.14% C, 0.37% Si and 1.42% Mn. Thecomposition of the filler material is 0.08% C, 0.5% Si and 0.8% Mn.
The weld region is commonly subdivided as shown in Figure 7.1. The outer region is de-scribed by a tempered zone where the temperature is too low for austenisation. The partlyaustenitised zone with some retained ferrite comes next and is followed by a fine-grainedzone in which the austenitic decomposition is controlled by diffusion processes, which resultsmainly in ferritic and pearlitic microstructure. The next zone is called the coarse-grained orthe underbead zone, characterised by austenitic grain growth yielding bainite and marten-site. A partly fused zone is seen between the HAZ and the fusion zone, which contains amix of the base and the filler material.
MixedCoarse Grained
f-
Fiine GrainedT/Teme •erecd,
:FZ iBa se Metal
Figure 7.1: Zones of the HAZ.
By means of an electron microscope a series of photos in scale 1x200 was taken through theHAZ. The followed path starts beneath the HAZ in the horizontal base plate and continuesvertically upwards into the FZ as shown in Figure 7.2. The microstuctural evolution isillustrated in Figure 7.3, which should be read left-right.
7.1 Microstructure 95
Figure 7.2: Polished X-section of fillet weld.
96 Chapter 7. Improved Material Modelling
-650x10 In
A
B-
C->
D->>
Figure 7.3: Microstructure through the HAZ (WOO0).
7.1 Microstructure 97
The annotation A- to E in Figure 7.3 are used to explain the evolution of microstructure inthe HAZ. (A) designates the tempered base material, which is a matrix of primary ferriteand pearlite. The region between (B) and (C) is the partly austenitised zone, characterisedby temperatures between A1 and A3. At A, the pearlite starts to austenitise, whereas theferrite grains are left unchanged. As the temperature rises the carbide starts to diffuse intothe adjacent ferrite grains causing these to transform into austenite. In cooling the formedaustenite is transformed into a fine grained structure of ferrite and pearlite between the non-austenitised ferrite grains. In the region from (C) to (D) the ferrite-pearlite matrix has beenfully austenitised, resulting in a fine grained ferrite-pearlite microstructure upon cooling. Inthe region (D) to (E) the temperature exceeds the critical grain growth temperature T,9 ofapprox. 12000C. The transformation of austenite into ferrite and pearlite is controlled bydiffusion and nucleation processes, which primarily start at the austenite grain boundariesand moves inwards. As the austenitic grains become larger, the cooling time is not adequatefor the diffusion controlled transformation to reach the centre of the austenite grains andbainite and eventually martensite starts to form. At the end of the zone the microstructure isseen to be fully bainitic-martensitic and at (E) the structure changes abruptly going throughthe mixed zone into the fusion zone.
The cooling gradients in the fusion zone are similar to those of the coarse-grained HAZ butthe resulting microstructure is different. This is partly due to the lower carbon content in thefiller material but can mainly be attributed to the presence of ferrite oxidic inclusion, whichserves as nucleation site for ferrite. Thus, in contrast to the HAZ, ferrite forms not only at thegrain boundaries of the austenite but also within the austenite grains and, consequently theferrite transformation is promoted and can occur at higher cooling rates. The microstructureof the FZ contains a variety of ferritic/pearlitic microstructures, e.g. allotriomorphic ferrite,Widmannstitten ferrite and accicular ferrite. A discussion of the transformation mechanicscan be found in the book by Hrivfrik [30] and in the paper by Bhadeshia [5].
By the present experimental evaluation of microstructure it is not possible to estimate theamount of each phase constituent in a certain location. The presence of bainite and marten-site was, however, verified and further the microstructure was seen to change almost abruptlygoing from the coarse-grained HAZ into the fusion zone. This is not remarkable in a mi-crostructural context, but in welding simulation the differentiation is rarely seen, as themechanical properties of the fusion zone are commonly assumed to be identical to those ofthe outer HAZ. In steels containing more than 1-2% of carbon, the mechanical properties forbainite and martensite in the coarse-grained HAZ will deviate considerably from the ferriticmicrostructure of the fusion zone and a considerable error might be introduced.
The above evaluation of microstruture provides in addition estimations of the isothermsexperienced in the HAZ. These isotherms have been used for fitting of the descriptive heatsource parameters.
98 Chapter 7. Improved Material Modelling
7.2 Material Model 2
Material model M2 is basically the same as M1 with the exception of the material data onwhich it is based and the modelling of yield stress. The primary reason for introducing asecond material model is to improve the modelling of phase transformation during cooling.
Phase transformation is accompanied by a transformation strain and large changes in the me-chanical properties. The mechanical effect of phase transformation is increased as the phasetransition temperature is lowered and as the temperature range in which the transformationoccurs is reduced. At low temperatures the material has regained its structural stiffnessand the mechanical effect of the volume expansion is accordingly much larger compared tothat of the austenite --* ferite/pearlite transformation occurring at a higher temperature.Moreover, the transformation strain is counteracted by the thermal contraction and a nar-row phase transition temperature range therefore causes a larger dilatational strain duringtransformation.
The phase transition temperature range and the austenitic decomposition products dependon the chemical content of the steel, the cooling rate and the austenitising temperature. Thiscan be seen from CCT diagrams of e.g. Seyffarth [56], which show that even small changesin these parameters may result in quite different material properties. In the temperaturerange A, - A3 austenite is formed in heating. In cooling the austenite is decomposed into thephase constituents ferrite, pearlite, bainite and martensite. A high cooling rate enhances theformation of bainite and martesite with mechanical properties which deviate considerablyfrom those of ferrite and pearlite. In consequence, the increased material stiffness will addto the effect of phase transformation at low temperatures.
In the search for mechanical material properties, good agreement between experimental andmodelling conditions with regard to the above parameters should therefore be strived for.As previously described, the effect of microstructure is modelled in a simplified manner byuse of the austenitising peak temperature T,ak and the cooling time At8 /5 . In Section 4.6.3it was shown that Ats/5 may be assumed to be constant. The peak temperature varies,however, considerably and determines the grain size of the austenite, which significantlyaffects the relation between phase transformation and cooling time [68]. The microstructuraldependency on these parameters allows therefore the phase transition temperatures and theresulting phase constituents to be indirectly included, if the description of material propertiesis representative for At8 /5 and the actual T, 0 k.
The mechanical material properties for M2 have been adopted from Karlsson and Josefson[34], who gathered experimental data from several sources. The cooling rate in simulationwas stated to be At 8 /5 = 6 - 9 secs. and the material composition reported here is similar tothat of the fillet welds in the target application. The gathering of data from several sourcesdoes not support consistency between data types, but it should be noted that the importantdilatational strain was gathered from one source whereby valuable information about phasetransformation temperature ranges was obtained as a function of austenisation temperature.
7.2 Material Model 2 99
This information appears to have been combined with yield stress values to establish therelation between yield stress, temperature and austenisation temperature.
Poisson's ratio v and Young's modulus E are shown as functions of temperature in Figure 7.4.
250 ' - 0.5
0.45
2W 0.4
- 0.35
t 5o 50 0.3 .
8W 0.2
0,15
50 0.1
0.05
000 2W .8M I"SW l
Temperature [°(CI
Figure 7.4: Young's modulus and Poisson's ratio versus temperature.
The thermal dilatation including phase transformation strain is shown in Figure 7.5 and theyield stress in Figure 7.6. In both cases the curve shown for Trak = 810 0C is followed inheating, whereas the curve followed upon cooling depends on the peak temperature reached.For peak temperatures different from the listed, linear interpolation is used.
1.6 -T-=950
14 810
1.2
0.8
0.4
0.2
00 2W M 0 0 I"
TFlmpcnfture adlp
Figure 7.5: Thermal dilatation as a function of temperature and peak temperature.
100 Chapter 7. Improved Material Modelling
Fl--
'ý- 50-500TW 1050.....
Tý 1950 ------
20
0
-- 4 0 T200 40O ........
Tempertatre 'I°C
Figure 7.6: Yield stress as a function of temperature and peak temperature,
Karlsson and Josefson [34] assume ideal plasticity and hence no dependency on the de-formation history. In the present work kinematic hardening is applied and the hardeningmodulus shown in Figure 7.7 has been adopted from Andersson [1], who investigates a simi-lar structural steel. The dislocation hardening is, however, reduced in phase transformationand at high temperatures by thermal restitution. This is accounted for by resetting of thedeformation history in the temperature range 900GC--1300GC.
T 1150i = 8 10 --....
24
12
00 200 4M 6W 800 I
Tempure rC]
Figure 7.7: Hardening modulus, H.
Based on the thermal model discussed in Section 4.6.3, At8 /5 was approximated to 11 secs.for the fillets of the target application. In Ml the cooling rate was twice as small sinceAt8 / 5 was specified to 22 secs. for the thermal dilatation. In M2 At8 / 5 was approx. 6 - 9
7.3 Material Model 3 101
secs. and offers therefore a better resemblance with the calculated cooling rate. The highcooling rate will result in large fractions of bainite and martensite, which is supported bythe experimental evaluation of microstructure. As peak temperature-dependent yield stressis also included in M2, it is reasonable to conclude that M2 offers improved accuracy in themodelling of phase transformation and cooling effects.
7.3 Material Model 3
In the present section an alternative material model is established. In contrast to the previousmodels, the present is based on microstructural evaluation, calculated cooling rates, CCTdiagrams and material parameters for the individual phase constituents.
The most significant changes of material properties in the HAZ are found in the fine- andthe coarse-grained zones. These zones are distinguished by the occurrence of austenitic graingrowth, which retards the -y -* a transformation and consequently lowers the critical coolingrates for martensite and bainite formation. Prediction of the resulting microstructure in theseregions therefore requires CCT diagrams obtained at different austenisation temperatures.The CCT diagrams shown in Figures 7.8 and 7.9 are obtained for a carbon-manganesesteel with a chemical composition of 0.17% C, 0.51% Si and 1.39% Mn which is slightlydifferent to that of the specimen used for microstructural investigation. The austenitisingtemperatures are 9000C and 13000C, respectively, and the effect of austenitic grain growthis clearly illustrated.
tO000
Austenltzed at 900 C'
- C2
700 - -
A' 0600 -
Figre .8 CC iga_ 18 aseiieda40 c)
'00300 IT 1 1200
0.1 1 to, to, to' to' 10,
Time[' s]
Figure 7.8: CCT diagram [18] (austenitised at 900 OC).
102 Chapter 7. Improved Material Modelling
900 A jstenitized at 1300 C'
- - - - - - -- - --. -. AC,
A F600
500 I 3 1 I0' I 30'
4o0 L -L
00A 10,' If
z 101 10, t01
Tine Cs]
Figure 7.9: COT diagram [18] (austenitised at 1300 0C).
The shown CCT diagrams are based on the cooling time from 850 to 5000C. A cooling curvefor the coarse grained zone has been calculated by thermal FEA and is plotted in the CCTdiagram in Figure 7.9. Similarly, a cooling curve for a material point experiencing a peaktemperature of 9000C has been plotted in Figure 7.8. The cooling curves are very similarand confirm the assumption of a common Ats/5 for the HAZ. As implied by the name,CCT diagrams are based on continuous cooling and the formed phase fraction indicatedat the end of each domain is strictly not valid. The cooling profiles are, however, similarto those of continuous cooling and the fractions indicated can be taken to be very goodapproximations. On the basis of this assumption, the decomposition products and the phasetransition temperatures have been estimated and listed in Table 7.1. A1 has been taken asMf.9o%, defined as the temperature where 90% of the martensite has been formed.
Tak Ferrite Pearlite Bainite Martensite A, A,
1300 0C - - 60% 40% 5600C 330 0C
900 0C 51% 2% 28% 19% 7030C 330 0C
< 800 00 80% 20% - - 8000C 750 °C
Table 7.1: Estimated phase fractions and phase transition temperatures.
The phase transformation kinetics in the fusion zone is characterised by ferrite oxidic inclu-sions promoting the formation of ferrite at high cooling rates. The resulting microstructureof the fusion zone is a variety of ferrite-pearlite microstructures which differ considerablyfrom the bainite-martensite in the coarse-grained HAZ in terms of mechanical properties.
7.3 Material Model 3 103
The CCT diagrams shown in Figures 7.8 and 7.9 are therefore only representative of theHAZ as they do not account for the ferrite oxidic inclusions, which shifts the ferrite trans-formation curve to the left. As no data was found in the literature to support the choice oftransformation temperature range or material properties for the weld metal, a transforma-tion range similar to that of the coarse-grained HAZ is assumed and material data similar tothat of ferrite/pearlite is chosen. The properties of the transition zone between the FZ andthe HAZ will be a mixture of the respective microstructures and a temperature transitionrange between 1450 and 16000C is assumed.
Final phase fractions and transition temperatures can now be estimated for a given peaktemperature by linear interpolation between the values given in Table 7.1. As the tempera-ture is assumed constant within each linear subdomain in an element, the material propertiesare likewise assumed to be constant. In the estimation of the material properties at a giventemperature T, it is initially checked whether or not transformation occurs. During trans-formations the present phase fractions should be estimated. Along any constant cooling ratecurve the volume fraction of austenite transformed can be assumed to follow the Avramiequation 162]:
f, = 1 -exp(-TP t") (7.1)
where ft, is the fraction, t, the cooling time and -f and fi are constants depending onthe cooling rate only. By means of the parameters obtained from the CCT diagrams, theconstants -y and f can be calculated and Eq. (7.1) can subsequently be used for calculation ofthe austenite fraction which has been transformed. This approach models the transformationkinetics more convincingly as the transformation starts slowly in the beginning of the phasetransitions temperature range, picks up speed in the medium range and finishes slowlytowards the end.
When the austenite fraction has been calculated, the fractions of the remaining phase con-stituents must be evaluated. The austenite transformation tends to start by formation offerrite, then pearlite is formed, which is followed by bainite and finally martensite. As thefinal fractions are known, these fractions are added in the above sequence until the fractionof decomposed austenite is reached. In consequence, martensite will, as an example, notappear until the three preceding phase constituents have been formed.
From the phase fractions, it is now possible to estimate the mechanical properties. Theproperties for the individual phase constituents shown in Figures 7.10 to 7.12 are simplymultiplied by the phase fractions to yield the average values. This is an approximation asplastic flow will start in the weaker phase below the yield stress predicted by the linear mix-ture rule, whereas fully plastic flow will occur at stress values above. The linear mixture rulehas, however, proved to be a good first approximation [631. The yield stress for bainite andmartensite has been adopted from Sjbstrbm [63], who specifies the properties as a functionof carbon content. The remaining properties are adopted from Bdrjesson [8], who extendsSjdstr6m's material properties to a wider temperature range and modifies the hardeningmoduli.
104 Chapter 7. Improved Material Modelling
SOOA nn
'73.
05
Figure 7.10: Yadeield stresus versus temperature.
7.4 Transformation Induced Plasticity 105
The purpose of the above material model is primarily to illustrate an alternative concept formaterial modelling and thus describe additional aspects of the material behaviour. Directapplication would require further knowledge of several aspects. The assumptions madeabout the filler material concerning phase transformation temperature range and materialproperties are quite uncertain. Final phase fractions and phase transformation temperatureranges for the base material are based on only two CCT diagrams. Moreover, the materialproperties for each phase constituent are associated with large uncertainty as e.g. bainite andmartensite become highly unstable at elevated temperatures and tend to harden excessivelyin deformation, which complicates the experimental assessment of properties. Altogether,use of material model 2 is preferable as it is based on experimentally obtained values andfewer assumptions are accordingly made.
7.4 Transformation Induced Plasticity
Transformation-induced plasticity (TRIP) is a phenomenon which may cause additionalplastic deformation of a multiphase material during the time of solid phase transformation.Transformation plasticity is an irreversible strain occurring in the weaker phase constituentat macroscopic stress levels far less than the yield stress. The phenomenon can be attributedto two mechanisms denoted the Greenwood-Johnson mechanism and the Magee mechanism,respectively.
The Greenwood-Johnson mechanism is related to the volume change in phase transformationand to the fact that phase transformation occurs heterogeneously. When a small volume ofmaterial transforms and changes volume, the restraint of the surrounding region causesstresses large enough for local plastic deformation. Due to their random distribution, theseplastic micro-strains do not cause macroscopic plastic straining and only the volume changeis observed at the macroscopic level. A macroscopic deviatoric stress will, however, alignthe microscopic plastic strains and consequently cause a macroscopic plastic strain in theweaker phase constituents, i.e. a transformation-induced plastic strain.
The Magee mechanism is related to shape change in the martensitic formation. The forma-tion of martensite occurs in specific crystallographic orientations which tend to be random,observed at the macroscopic level. The presence of a macroscopic stress will favour somedirections over others as these result in reduced internal energy of the material. Thus, anadditional plastic strain may be produced in the formation of martensite.
Several models for transformation plasticity have been proposed, which vary from artificiallowering of the yield stress to extensive models working at the subgrain level. Leblond et al.[35j showed that the plastic strain could be decomposed into the classical plastic strain andthe transformation-induced plastic strain, yielding a total strain composed as
,¶9 _ e+ th PI +rp (7.2)i - j -- j + j
106 Chapter 7. Improved Material Modelling
The model applied here to estimation of etrp is that of Greenwood and Johnson 128] combinedwith a scaling factor for partial transformation as used by e.g. Oddy et al. [47]. The usedequation is on incremental form written as
AIP = 5 Vi 15 (2 -2x - Ax)Ax (7.3)
Avis the specific volume change, Y' is the yield stress of the weaker phase, Lii is the deviatoricstress, AX is the transforming fraction and y,. is the fraction already transformed. Eq. (7.3)shows that transformation plasticity causes a strain which acts to relieve the macroscopicdleviatoric stress during phase transformation. In the incremental formulation the deviatoricstress is assumed to be constant during the increment. This tends to overpredict the TRIPstrain if the increments are too large since the rate of e,,, in reality will decrease during theincrement when the deviatoric stress is relieved. Good approximations to integration of thestrain over the increment is, however, easily achieved by subincremuentation. The decreasein temperature during an increment is used to determine the number of subincrements.
Depending on the degree of structural restraint and the phase transition temperature range,the influence of TRIP may be significant. The effect is illustrated by a Satoh test with a peaktemperature of 13000 C. A Satoh test is simply a bar restrained in the longitudinal directionbeing exposed to a thermal cycle. Based on the material model 2, the stress developmentwith and without transformation plasticity has been computed and is shown in Figure 7.13.The effect of TRIP is stress reduction during phase transformation, resulting in higher finalstresses. For decreasing peak temperatures the effect declines as the phase transformationtemperature range shifts towards higher temperatures, where the lower yield stress decreasesthe effect of phase change and thus the effect of TRIP. If the material goes into yielding atthe end of the thermal cycle, the effect will furthermore be decreased by the upper stressbound. The Satoh test serves well as an illustration of the effect of TRIP but the complexityis significantly increased in welding simulation where adjacent material with varying thermalhistory interact.
To the author's knowledge the influence of TRIP on welding deformations has not beendiscussed in the literature. The effect of outer restraints is straightforward as rigid restrainingyields less deformation and higher stresses so that transformation plasticity is increased.In addition to outer restraining, a material volume going through phase transformation isrestrained and stressed by the surrounding material. The effect of TRIP becomes to someextent self-regulating as TRIP reduces stress behind the phase transition front, whereby theeffect of transformation plasticity is reduced in subsequent transformation.
The present model is a rather simple approximation to the effect of TRIP and far morecomprehensive models can be found. The present implementation is, however, believed tobe a significant improvement in the general material modelling.
7.5 Influence of Material Models and TRIP 107
cT - - -
(a) TRPnticldd()TRPicue
Figur 7.3 feto2RI nsrs nsmuaino ao)es T~=30 )
7. Iflene f atril odlsan TI
The mst exensiv probem reated o wel insiuaonsthlckfex rmnalaeil
data. Reibeeprmntleauto1f aeilpoetisi ifclt o peesv
and ~ ~ ~ veyepniets.I3steeoerrl oeadpbihdee oerrl.Mtra
dat is thrfr feesdadcmie rmsvrlsucst utteproeo h
(a)c o TRIP ha eno includted (bfTIPinlue
7.5 dinflurence oftee tematerial Modelsishotyumase.M and TRIPeesen
Thelcamost bextenieproblenMI reated to wedingsmtion i2iclds the ylac ofes exerimdental maeradautaReisable texperimuenta evaluatin ofep oesals material prpetesidadffc lt compeheonsivetrandiver maexpnive dtak. It isteraeforeerarl donaeand pulihed maevenl more rarelycin Matriadtisthefilrmaefreiftnaeue and comine from several sourceios to suitdewerthaed prose ofather einesiatioan, whihor tossomeetetinaiates wtlaguneaithes cosistencyrof the datafet.o TIn, ith fol-welowiong itscl hnmeon should bereor kep incmindethat the modelsig prsete atbs rodapoiatinreuto the mat feriable be avio r.a oe per ob ,wihsol ecmae oM
Beow the differentsmatplerinall model andiou the ptefetoTr IPaeeauaesntrm.fsrs
ahd dformaltion. staeistogydpnntnthe material models aredeotdliM2dM and ao T isapeddifhtherdeffet of tRhP haclsiben aoune fRP oilsr. t h eednyalmoeshv enapid
The dinfference bewenfh material modelsin is motdshortintly summarisedo loangtdin2ale stessen
108 Chapter 7. Improved Material Modelling
This is seen in Figure 7.14 where the longitudinal stress is shown in the centre cross-sectionof a 3-D model with fully refined weld path.
In the comparison of the stresses resulting from M1 with those of M2, extremely largedeviations are seen in the HAZ and the FZ. In some locations the difference is more than 600MPa as the stress is altered from high-tensile stresses to high-compressive. This difference ismainly related to the modelling of phase transformation, as the transformation temperaturerange (A,,,) and its dependency on austenisation temperature varies significantly betweenM1 and M2. In M2 A 3 -1 goes from 806-7470C to 388-2530C as the peak temperature isincreased. The lower range in M1 is 560-460 0 C and hence it follows that large differencesin material modelling are experienced in the HAZ where all peak temperatures are present.By use of M2, the high levels of compressive stress in the I-AZ and the fusion zone thereforearise as the surrounding material has regained most of its structural stiffness, when thematerial of the FZ and the HAZ expands in transformation. The effect of expansion is,however, overestimated by M2 as the effect of TRIP has not been accounted for. TRIPhas been included in M2T and is seen to reduce the compressive stresses to more moderatelevels in the HAZ and the fusion zone whereas the stress state outside the HAZ is basicallyunaffected. In the comparison of the results of M1 with those of M12T it is interesting tonotice that the stress state at some distance of the HAZ becomes quite similar and that thedeviation to some extent may be regarded as a local phenomenon.
Due to lack of experimental data, the mechanical properties and A,,, were approximated inM3 for the weld metal. If the stress state resulting from M3 is considered, an abrupt changeis observed from the HAZ into the FZ. Small changes in the approximated data are foundto influence the stress state significantly in this particular region and M3 should thereforenot be employed on the basis of the material data currently available. The experiencedsensitivity further emphasises the importance of proper material data estimation.
7.5 Influence of Material Models and TRIP 109
z-stss [MPal-275- 194
mo 114
47128m208
289___369450
(a) Ml
(b) M2 (c) M2T
(d) M3 (e) M3T
Figure 7.14: Longitudinal stress dependency on material model and TRIP
110 Chapter 7. Improved Material Modelling
As regards the transverse stresses, similar but more moderate deviations are experiencedwhen the material models are compared.
As prediction of welding distortions is the stated objective of the present work it is evenmore interesting to evaluate the effect of material modelling with regard to deformations.The distortions resulting from application of M1 and M2T are shown in Figure 7.15. Thex-displacements of the web are seen to change considerably as M2T is applied instead of M1.The relative displacement between the bottom of the web and the base plate is essentiallythe same for both models (see also Figure 7.16). Thus, the deviation is mainly related to anincreased angular deformation, causing the web to tilt. Otherwise a reasonable agreementbetween the displacements is observed, especially if those of the base plate are compared.
The influence of TRIP on displacements mainly concerns magnitude as the displacementsin general are reduced approx. 10% when TRIP is accounted for. This means that thedeviation between material model 1 and 2 is reduced by the inclusion of TRIP.
FRom the above evaluation and comparison of material models it may be concluded thatthe stresses in the HAZ and the fusion zone depend significantly on the material modelapplied. This calls for experimental verification which, however, is a significant problem.Unlike deformation stress is a quantity which can only be measured indirectly and accord-ingly uninterpreted reference values do not exist. In the HAZ, where the microstructurevaries significantly, it is not yet possible to establish reliable reference values and accord-ingly verification of the numerical models is not possible.
For the purpose of the present work it is, however, essential that distortions are far lessdependent on material modelling, which enhances successful prediction of welding distortionsby numerical simulation. Still, the difference between M1 and M2T is too significant as theincreased angular deformation of the web not only causes a quantitative but also a qualitativechange in the deformation pattern of the web. All simulations should consequently be redoneusing M12T in order to obtain an improved prediction. Fortunately, the application of M2Tdoes not pose any problems and is straightforward. It must, however, be emphasised that thematerial modelling is associated with large uncertainties and that experimental verificationof the predicted displacement fields is highly recommendable.
7.5 Influence of Material Models and TRIP
Am=Ails4016
OW7
0 189.Nl0.3930.4%0ý8O.M
y
(a) x-displmement, M1 (b) x-displacenient, M2T
43ý
in0. 136OMO.M70.4120.5ý8U13
y
(c) y-displacement, M I (d) y-displwement, M2T
zAi". 1.14no-0.233
=119,6,4.122
MAW6=4W9=14.012OW5
0 W2
y
(e) z-displacement, MI (f) z-displ;ixenient, M2T
Figure 7.15; Comparison of displacements of MI to M2T
112 Chapter 7. Improved Material Modelling
-0.220An0016-0.087- AP8
___ 029___0.393
0700
(a) x-displacement, MI (b) x-dispiacement, M2T
Figure 7.16: Comparison of x-displacements of MI to M2T at centre cross section.
Chapter 8
Conclusions and Recommendationsfor Part II
8.1 Conclusions
The objective of the present part of the thesis was to enable welding response prediction inlarge-scale industrial applications. This objective has been achieved as summarised below.
Based on the commercial software ANSYS, considerable alterations and extensions have beenmade to enable physically realistic simulation of large-scale welding. Most of the work hasbeen implemented as Fortran routines facilitating speed and flexibility. Some of the mainalterations and extensions are
" Databases, data structures and data exchange
" Torch modelling
" Graded element
" Material modelling
* Plasticity formulation (TRIP)
" Automatic FEA (geometry- and process-based input)
" Dynamic Meshing:
- Meshing algorithms
- Mapping algorithms
- Dynamic coupling
113
114 Chapter 8. Conclusions and Recommendations for Part II
- Dynamic activation
e Local/Global template:
- Meshing routines
- Data exchange between local solids and and equivalent shells
- Shell updating procedure
The use of a commercial software as platform for the software development has been offundamental importance to the present work as a lot of functionality such as postprocessorsand solvers was hereby ready at hand. By the extensions and alterations, a dedicatedwelding simulation package has been developed, providing the functionality necessary forwelding distortion prediction in relatively large plate structures.
A basic model was established, allowing the essential welding mechanics to be modelled.Dynamic activation of fillet elements, dynamic coupling of parts, contact modelling and tackweld positioning were identified as essential factors as regards the qualitative distortion field.
To extend the functionality of the basic models, the computational efficiency was subse-quently increased. Based on the mesh grading characteristics of the graded element, anefficient dynamic mesh refinement scheme was developed with the emphasis on displace-ments in the data mapping algorithms. In the investigation of the mesh density required indynamic meshing, it was found that good predictions of the qualitative distortion patterncould be obtained, even if the mesh refinement was decreased considerably, to increase thecomputational efficiency. The obtained increase in efficiency was considerable and allowednot only models to be computed within shorter time, it also allowed for simulation of weldingapplications that previously were far beyond the computational capability. The investigationof tack weld positioning using a T-profile containing more than 3.3 metres of welding clearlyillustrates that numerical simulation may be applied to relatively large industrial problems.
An investigation of material modelling revealed that stresses in the region around the HAZand the fusion zone were highly dependent on the material data applied. Fortunately, dis-tortions were far less dependent on the material data which enhances successful prediction ofwelding distortions by numerical simulation. Still, material data dedicated to the applicationbeing investigated is necessary for successful prediction.
As the computational efficiency obtained by dynamic meshing did not allow the specifiedtarget application to be analysed, a template combining several techniques such as dynamicmeshing and substructuring was developed. The combination of local solid models and aglobal shell model allowed the boundary conditions for the weld region to be properly repre-sented and the distortions in a real-size subassembly to be calculated by taking into accountvariations in structural stiffness, weld sequence, tack weld positioning, welding parametersetc. Thus, the basic capability needed for the welding response prediction tool outlined inChapter 3 has been established.
8.2 Recommendations for Future Work 115
8.2 Recommendations for ]Future Work
To verify properly the welding response predicted in the presented work, it will be neces-sary to perform dedicated measurements of welding distortions with a rather high accuracy.Further, material data obtained for the specific application to be analysed would enable animproved evaluation of the modelling accuracy as it is not possible at present to estimatethe inaccuracy related to the material data and that related to the modelling.
As regards the numerical simulation, quite a few improvements and analyses may be carriedout. The maximum plastic strain experienced in a common welding simulation is approx-imately 6%, which calls for a large strain formulation. It would therefore be highly inter-esting to implement a large strain formulation in order to evaluate the error introduced bythe current small strain formulation. This may be done by implementation of a finite strainalgorithm based on Green-Naghdi stress and centred strain.
The solid elements applied in the local models tend to be too stiff in bending. The inclusionof bubble functions in the element formulation is straightforward and would improve thebending capability.
A solid graded shell element developed by McDill et al. [43] is well suited for application in theglobal model. Implementation of this element would solve nicely the incompatibility problemrelated to the solid-shell transition, which is currently solved by use of beam elements. Theproblem of different stiffuesses of the local solid models and equivalent shells is also relatedto the combination of solids and shells. As they are interchanged in the L/G template anerror is introduced. In the current implementation the error is reduced by the choice ofupdating procedure but further investigation to obtain an improved stiffness resemblancewould be beneficial.
Computational efficiency may be increased further in several ways. Code parallelisationand faster computers are to be expected but as regards modelling techniques the authorhas few suggestions. The numerous factors influencing welding distortions and the complexdeformation modes produced do not support the use of templates in an attempt to expandthe results from small models to large and complex structures without actually simulatingthe major part of the weld length. An extension to the L/G template would be the abilityto divide a weld path into short coupled models, still allowing the welding to be performedin one continuous run. The author has tested several concepts to solve the weld line divisionproblem with little success. A template offering a simple solution to the problem wouldincrease considerably the computational efficiency.
116 Chapter8. Conclusions and Recommendations for Part lI
Part III
Experimental Method for ResidualStress Evaluation
117
Chapter 9
Residual Stress in Steel Plates
The problems caused by process-related geometrical distortions in shipbuilding were de-scribed in Section 2.3. Distortions accumulate throughout the process chain, so that com-pensation rework is needed in order to maintain an acceptable distortion level. Geometricaldistortions are primarily caused by thermomnechanical processing and poor set-up of assem-blies prior to welding. An additional source of distortion may be initial residual stress in theunprocessed steel plates. Residual stress in the steel plates is a result of the plate manufac-turing process, which involves numerous processes depending on the product type and thesteel manufacturer. In investigation of the problems related to residual stress in steel plates,three main issues have been addressed:
1. Does residual stress affect geometrical distortions in heavy steel construction
2. How can residual stress be evaluated
3. Can the residual stress state in steel plates be categorised for general use
The above list is a three-step ladder: If there is no potential effect of residual stress, there isno reason for investigation and if residual stresses cannot be measured, there is no generalway to evaluate and categorise the steel products.
9.1 Influence of Residual Stress
The effect of residual stress in processing is difficult to evaluate as it is indirect. A steelplate containing high levels of residual stress is in self-equilibrium and causes no distortionrelated problems until the state of equilibrium is disturbed by processing, such as cuttingand welding. If the process itself causes geometrical distortions, it is difficult to quantify
119
120 Chapter 9. Residual Stress in Steel Plates
the deformations which may be attributed to the residual stress alone. Several phenomenaexperienced in production indicate, however, that residual stress is a factor to be accountedfor. An example is the cutting of plates into strips by laser. The distortions resulting fromidentical processing of plates with similar dimensions and material composition may varysignificantly depending on the steel manufacturer. This indicates that the manufacturinghistory influences the cutting distortions.
Hence, based on production experience, residual stress is known to influence geometricaldistortions and accordingly an investigation is needed. This leads to the next issue concerningresidual stress evaluation, where either a numerical or an experimental approach can beadopted.
9.2 General Approaches to Residual Stress Evaluation
The numerical approach offers a significant advantage as it allows the entire stress fieldof the steel plate to be estimated whereas experimental measurement is confined to stressevaluation in discrete locations. To take advantage of this it is, however, required thatnumerical simulation enables the residual stress to be predicted with sufficient accuracy.This issue is addressed below where the possible sources of residual stress are first identified.The discussion is confined to hot rolled steel plates as these constitute the majority of thesteel plates used in shipbuilding.
Processes commonly available at the steelworks for manufacturing of hot rolled steel platesare listed in Table 9.1.
Slab heating
Descaling
Hot rolling
Hot levelling*
Cooling (forced or air)
Normalising*
Cold levelling*
Edge cutting*
Table 9.1: Plate manufacturing processes.
Whether or not the processes marked by an asterisk are applied in the plate manufacturingprocess, is highly dependent on material, product type and steel manufacturer. Likewise,
9.2 General Approaches to Residual Stress Evaluation 121
the application of each process type varies significantly depending on common practice andthe production equipment available. As a result it will be necessary to choose one producttype from one steelworks for numerical simulation.
The next problem is to single out the processes affecting the final residual stress state. It isimportant to distinguish between stresses and deformations as plastic deformation and hightemperatures do not provide a unique relation between stress and strain. Below some of themanufacturing processes are described with respect to residual stress and deformation.
Slab heating furnace: The slabs are heated to approx. 1200 00. Uneven heating of the slabcombined with low yield stress may cause plastic deformation. Stresses and deforma-tions at this stage are, however, insignificant seen in relation to the extensive materialdeformation occurring in the subsequent rolling. The uneven temperature distributionin the slab will, however, influence the rolling.
Hot rolling: Rolling is a pronounced dynamic process involving extensive plastic deforma-tion at high strain rates and high temperatures. Many factors affect the plate flatnessin rolling. Plate flatness and transverse roll gap variation (crown) are interrelated.Zero crown is preferable in regard to flatness but a small positive crown (thicker atthe centre) is necessary to keep the plates on track. Constant crown is difficult toobtain as the rolling conditions vary heavily during and between passes. Generally, thecrown setting is based on experience but due to variations in temperature, materialparameters and plate dimensions the crown might not be appropriate. Long-edge orlong-centre shaped plates are consequently manufactured and if the length mismatchbetween centre and edge is sufficiently high, buckling may occur. In addition to edgeand centre waves, other deformation patterns such as longitudinal and transverse wavesmay occur in rolling. These may be attributed to roll friction effects, irregular coolingetc. Hence, a considerable number of parameters influence the plate flatness althoughthe overall manufacturing conditions appear to be constant.
The residual stress level after the last pass has proved difficult to predict by simulation.Edberg and Mantyla [17] showed that the residual stress state was highly dependenton the material model applied and concluded that considerable improvement of thematerial modelling was necessary for reliable prediction. Further, it will be necessaryto account for microstructural effects during rolling.
Hot rolling is carried out at temperatures above recrystallisation temperature. If highlevels of residual stresses are present after the last pass, these will to some degree berelaxed by recrystallisation and recovery. The level of stress relaxation is, however,difficult to predict, especially the relaxation during phase transformation.
Cooling: The material temperature subsequent to rolling or normalisation is commonlyabove phase transformation temperature. Thermal strain gradients and phase trans-formation are the primary factors influencing residual stress and deformation producedin cooling. Temperature variations may result in thermal strain gradients large enough
122 Chapter 9. Residual Stress in Steel Plates
to yield plastic deformation. The effect of phase transformation is related to the vol-umetric expansion of the transformed austenite. If the transformation occurs inhomo-geneously in the plate, the expansion will cause plastic deformation and consequentlyresidual stress. Although stress influences the initiation of phase transformation theeffect is secondary compared to temperature. Hence, thermal gradients are the primefactor in analyses of both thermal strain gradients and phase transformation effects.
The heterogeneous temperature state is a direct consequence of surface cooling. Thesize of the thermal gradients are, however, dependent on the cooling media used as e.g.forced water cooling produce gradients far higher than those resulting from cooling inair at ambient temperature. Additional irregularities in temperature may be attributedto variation in convection and in air cooling also to variation in radiation and to localheat transfer at the cooling bed supports.
Normalising: In normalisation the steel plates are commonly reheated in a 'walking beam'furnace to above the reaustenitisation temperature (ý 9300 C) in order to achieve afine-grained austenite. All stresses are assumed to be relaxed in the austenitisationprocess whereas deformations remain unchanged or might even be increased in thetransport by the 'walking beams' through the furnace.
Cold and hot levelling: A leveller consists of a series of vertically overlapping rolls throughwhich the plate is drawn and hence plastically deformed in alternating bending. Theabsolute bending diminishes from entry to exit and the final bending is adjusted sothat the plate becomes flat after the elastic spring back. The symmetrical stress stateresulting from bending is superimposed on the pre-existing stresses. In alternatingbending, pre-tensiled material will exhibit larger plastic deformation in tension andless plastic deformation in compression compared to non-stressed material. Thus, al-ternating bending results in elongation of a pre-tensiled material. By the same principlepre-compressed material is shortened in the process and the overall result is improvedflatness.
At the leveller entry the deformed plate is forced into plane whereby the correspondingstresses are superimposed on the residual stress state which yields the above stressstate denoted the pre-existing stresses.
Above, the effect of each process was discussed with respect to stress and deformation.To single out the primary processes in terms of final residual stress and deformation itis, however, required that the effect is traced all the way down to the final state of theproduct. The conclusion drawn with regard to distortions is that residual deformations area result of all processes from slab heating to cold levelling. As an example, a cold spot ina heated slab may result in poor flatness in rolling which affect the deformation field in allsubsequent processes. As regards residual stress, a conclusion is not easily drawn. Residualstress appears mainly to be a result of cooling and cold levelling. It is, however, not evidentto what degree stresses produced in hot rolling and hot levelling are retained during phasetransformation and cooling, especially as hot levelling in some cases is performed when phase
9.2 General Approaches to Residual Stress Evaluation 123
transformation has already begun. Yet, if normalisation is applied, all stresses are relaxedduring the austenisation and cooling and cold levelling may be singled out for analysis.With respect to residual stress and residual deformation it should, however, be noted thatstress and deformation are recoupled in the levelling process as the plate is forced into plane.Thus, if cold levelling was applied in the plate manufacturing process, the task of numericalsimulation would be tremendously complex as all processes should be included.
The above discussion does not provide an adequate basis for proper evaluation of the pos-sibilities offered by numerical calculation with regard to residual stress prediction. It is,however, evident that products to which cold levelling has been applied impose excessive de-mands on simulation. For all other products, the cooling process appears to be the rationalprocess to be singled out for simulation.
The alternative to numerical calculation is experimental evaluation. Techniques which can beapplied to experimental stress evaluation comprise diffraction techniques, mechanical stressrelaxation techniques, ultra sonic techniques etc. None of the available techniques provide,however, a cheap and reliable method for residual stress measurement through the thicknessof a steel plate. The choice of the experimental approach therefore involves development ofan experimental technique for the purpose.
The accuracy in residual stress prediction by the experimental and the numerical methodsis difficult to evaluate. Stress is unlike deformation a quantity which can only be measuredindirectly and accordingly uninterpreted reference values do not exist. In general large varia-tions are expected in the stress predicted by numerical simulation. The results depend on thematerial model applied, the effects included and the material data available. As an example,the exact modelling of transformation induced plasticity (TRIP) will significantly affect theoutcome. The inaccuracy in experimental evaluation concerns primarily the assumptionsmade as the stress is related to a measurable quantity and to the uncertainty in measuringwhich, however, can be approximated by stress calculation error analysis.
Compared to the numerical method, the experimental approach does not offer detailed in-formation on the residual stress field but only information on the stresses in a small volumeof material. On the other hand an experimental method has the advantage of being corn-pletely independent of the plate manufacturing history as it can be applied to any plate inthe production. This is essential for the choice of approach. The main issue of categorisationof steel plates with respect to residual stress is evaluation of differences in residual stressbetween plates manufactured under similar conditions. This is only possible by an experi-mental method as the effect of all factors, including the unknown ones, is directly included.Subsequently, if residual stress similarity is supported by the experimental investigation andthe stresses are of a significant magnitude, numerical analysis would serve to provide addi-tional information on the residual stress distribution. This is, however, beyond the presentscope and accordingly only the experimental approach has been chosen for stress evaluation.
124 Chapter 9. Residual Stress in Steel Plates
9.3 Categorisation of Steel Plates
Previously, the manufacturing processes were discussed in relation to residual stress anddeformation. On this basis, the possibility of categorising steel plates according to theirresidual stress may be discussed. Deformations and stresses introduced in the initial pro-cesses will only slightly or not at all contribute to the final residual stress state. Residualstress and deformation are, however, recoupled if cold levelling is applied at the end of themanufacturing process. Production experience shows that deformations in rolling are influ-enced by many factors and that deformations may occur in a so far unpredictable patterntinder manufacturing conditions which appear to be identical. Hence, as unpredictable de-formations are transformed into residual stress by cold levelling, it is very unlikely that coldlevelled steel plates can be categorised according to residual stress.
In general, the cold levelling process is only applied if waves or other deformations areexperienced in the production. This occurs primarily in the rolling of thin to medium thickplates and in practice the majority of steel plates are not cold levelled. For the purposeof categorisation, it is therefore essential to know whether or not cold levelling has beenapplied.
Thus, for the majority of steel plates, the cooling process is believed to be the prime factorinfluencing the final residual stress state. For a steel plate of a given type and from agiven steel manufacturer, rather small deviations are expected in the cooling process, whichsupports the possibility of categorisation. In the categorisation it is, however, essential toverify that the deviations in residual stress are small for a given product.
9.4 Objective
The present objective is limited to the development of an experimental method for estimationof residual stress in medium thick steel plates. The method must provide sufficient informa-tion on the residual stress state to allow the effect on cutting and welding distortions to beevaluated. This evaluation is, however, beyond the present scope just as the categorisationof steel plates with respect to residual stress.
Chapter 10
Selection of Method for Modification
The requirements for the experimental method to be developed are specified in the following.Existing methods for residual stress measurement are discussed with regard to the formulatedrequirements and a method is chosen for modification.
10.1 Requirements
The level of detail required in the residual stress description depends on the application. Thepresent objective is to provide sufficient information on the residual stresses for evaluationof the effect on cutting and welding distortions and the requirements are specified on thisbackground.
Welding and cutting deformations are influenced by residual stress in different ways. Residualstresses are released in thermal processing but also mechanically in cutting at the free edgeswhich are formed. The size and the shape of the thermally affected zone, in which theresidual stresses are released, depend on several factors such as welding technique, type ofweld, material and plate thickness. For thermal as well as mechanical cutting, a descriptionof the mean in-plane stress variation is in general sufficient. The stress variation throughthe plate thickness will only result in local deformations at the edges, whereas the overalldistortion is determined by the mean stress. This is, however, not adequate in the caseof fillet welding. Here, residual stresses are inhomogeneously released through the platethickness due to the large temperature gradients and may result in angular deformation.The level of detail required for general welding applications is therefore both the in-planeand the through-thickness stress variation. The plane stress condition is in general a goodassumption for the steel plates to be investigated and the through-thickness variation mayaccordingly be described by the variation of in-plane stresses with depth.
125
126 Chapter 10. Selection of Method for Modification
Besides the requirements to the residual stress description, the measuring method shouldfacilitate
" Residual stress measuring through the thickness of steel plates more than 8 mm thick
" Residual stress measuring in steel plates as large as 4 m wide and 16 m long
" Cheap and reliable residual stress measuring
To the author's knowledge no existing method complies with the above requirements.
10.2 Selection of Experimental Method
Residual stresses are defined as the stresses existing in a body at homogeneous temperatureand with no external loads acting on the body. In the plate manufacturing process residualstresses arise due to inhomogeneous plastic deformation of the material during the process.According to their nature residual stresses can be determined as follows:
" From macroscopic elastic strains released while material is removed from parts loadedby residual stresses. This is the principle behind mechanical methods.
" F'rom lattice strain distribution. This is the basis of diffraction methods.
" From their effect on physical properties. This is the basis of magnetic and ultrasonicmethods.
The mechanical methods are all of destructive character, whereas the others are generallynon-destructive. The most widely applied methods are at present the mechanical straingauge methods and the diffraction methods.
Mechanical methods: A large variety of mechanical methods is available but few arecapable of measuring stresses in the interior of the material. The methods for incrementalhole-drilling and sectioning are of interest to the present application. Incremental hole-drilling has been investigated by several authors [22, 46, 50, 57] but has until now beendedicated to stress measurements at small depths only. However, the method seems to havethe potential to resolve stresses at larger depths.
Among several sectioning techniques, Ueda [67J has successfully employed the 'inherent strainmethod', which requires laboratory conditions and highly specialised mechanical cuttingdevices designed for low heat generation in the cutting process. The sectioning techniqueshave proved valuable but costly and extremely time consuming.
The problem of material removal without introducing machining stresses is common to allmechanical methods. Several advanced material removal techniques have been developed forthis purpose [11, 21].
10.3 Basic Hole-Drilling 127
Diffraction techniques: The two diffraction methods available are X-ray and neutrondiffraction. X-rays penetrate typically less than 20 inm into the material and the methodis therefore restricted to determination of residual stresses at the immediate surface of thespecimen. Neutron diffraction offers excellent possibilities of determining residual stresscomponents in the interior of a material. However, the technique requires a thermal neu-tron reactor whereby the experimental expenditure becomes extremely high. Secondly, thespecimen size is restricted whereby stress measurement in large steel plates is only possibleby combination of the neutron diffraction with a sectioning method. Diffraction techniquesmeasure the lattice strain distribution resulting from both the microscopic and macroscopicresidual stresses. As regards geometrical distortions in the production only macroscopicresidual stresses are of interest and the microscopic stresses need to be filtered out in the in-terpretation, which has proved difficult in many applications. As neutron diffraction requiresa relatively large material volume for measuring, it becomes less sensitive to the influence ofmicrostresses but heavily texturised material may still cause problems.
Magnetic and ultrasonic methods: Magnetic methods are based on the coherence be-tween residual stresses and the electromagnetic signal generated when a ferromagnetic ma-terial is subjected to a magnetic field. Ultrasonic measurements involve measurement of thespeed of sound in the material which is related to residual stress. Both methods are promis-ing but literature documenting the capability in terms of stress resolution and accuracy isstill scarce.
The method chosen for further investigation is the incremental hole-drilling method. Thearguments are
" Low experimental expenses
* Applicable to steel plates of large dimensions
* On site measuring possible
* Promising potential for adequate stress resolution in the interior of the material
* Promising potential for sufficient stress calculation accuracy
Before the introduction of incremental hole-drilling, basic hole-drilling is briefly described.
10.3 Basic Hole-Drilling
The basic principle of hole-drilling is measurement of elastic strain relaxation while stressedmaterial is removed. A new and unique stress state is reached and by inverse calculation,the stresses present before drilling can be calculated. Standard hole-drilling is a well known
128 Chapter 10. Selection of Method for Modification
method, frequently used for measuring local surface stresses in various isotropic elastic mate-rials. A general description of the standard method is given in the ASTM E 837-95 standard[4] and in general a lot of literature is found on the subject. In the ASTM standard holedrilling is briefly described as
The hole-drilling strain-gauge method measures residual stresses near the surfaceof a material. It involves attaching strain gauges to the surface, drilling a holein the vicinity of the gauges and measuring the relieved strain. The measuredstrains are then related to relieved principal stresses through a series of equations.
Standard strain gauge rosettes have been developed for the purpose of hole-drilling and asthere is a general desire to minimise the damage to the specimen tested, the rosettes arerelatively small. A typical 00/450/900 rosette is shown in Figure 10.1.
Figure 10.1: Standard O/45°/90 hole-drilling rosette
Counter-clockwise (CCW) numbering of the gauges is chosen as standard in the present work.The gauges are radially orientated, placed equidistantly from the centre. The directions ofthe gauges 1 and 3 are mutually perpendicular and gauge 2 is placed at one of the bisectors.Other configurations of the rosette can be used, but the 00/450/90' configuration abovesimplifies the equations for residual stress calculation.
In principle the experimental procedure is simple. The rosette is cemented onto the testspecimen, a hole is drilled in the centre of the rosette, either through the plate or to a depthexceeding approx. 40% of the mean diameter of the strain gauge circle, as the surface strainsare assumed to be fully released at this depth. Finally, readings of relaxed strains are made.
The theoretical solution of radial strain relaxation is derived as follows. Consider the princi-pal stresses in the plane parallel to the surface, the relieved radial surface strains are relatedto the principal stresses as
10.3 Basic Hole-Drilling 129
Er(/3) = K(3)a.. + K(03 + 900)ummj (10.1)
where
K(,3) are calibration constants0
,ao,•,6,i are maximum and minimum principal stresses, respectively)3 is the angle measured CCW from the direction of a,,, to C.
The constant K(Q3 ) is an even function and can be represented by
K(Ol) = ZAncos 2n)3 (10.2)
As a good approximation only the first two terms are retained [50]:
K(0) = A + B cos 2f3 (10.3)
and subsequently the radial surface strain is written as
El = (U,• + amUi)A + (ama: - ami,)B cos 2/3 (10.4)
The constants A and B still need to be determined. For a hole through an infinite plate ananalytical solution for the stress state was obtained by G. Kirsch in 1898 and A and B areeasily calculated [51]. For a blind hole, not passing through the entire thickness, no closedform analytical solution is available and the constants are either obtained experimentally orby numerical methods [4, 29].
After determination of the calibration constants three unknowns remain, namely 6rea., 0 .mi.and a. The angle a is measured CCW from gauge 1 to the direction of the maximum principalstress and determines the reference direction of /3 (see Figure 10.1). The three equationsnecessary are obtained by measuring the relieved radial surface strains Cj, £2 and e3 at theequally numbered gauges. The following solution is obtained by using the 00/450/900 rosette:
£3 + El " (CI - E3)2 + (2E2 - El - C3)2 (10.5)
Uina:,Uiin = 4A 4B
130 Chapter 10. Selection of Method for Modification
or = 1arctan [2C2 CI C3 ] (10.6)
where oa should be positioned in the correct quadrant. The calibration constants A and Bdepend on the material properties but are approximately made independent by applicationof the constants
2E
yielding
C. + V a K ý. 2 )6 + 1 b 2ý. )~ cos 2 ,3 (10.8)
The correlation coefficients a and b relate respectively the mean biaxial stress and the shearstress to the released radial strain at the gauge.
The standard hole-drilling method is confined to measurement of surface stresses and is basedon important assumptions which must be evaluated on application [4, 29]. The assumptionabout residual stresses being uniform with depth, measured from the specimen surface, ismost important. Strain measurement errors can be minimised by use of strain averaging,described in references [29, 601 and at the same time the uniformity of stresses may beevaluated.
10.4 Incremental Hole-Drilling
By application of incremental hole-drilling it is possible to obtain the residual stresses as afunction of depth measured from the specimen surface. Plane stress condition is assumed andthe residual stress variation is therefore described by the variation of the in-plane stressesthrough depth. The calibration constants are derived by numerical methods, whereby Sig-nificant flexibility is obtained as any geometrical shape of a specimen can be investigated.
The experimental procedure in incremental hole-drilling is similar to the procedure for tra-ditional hole-drilling. However, the hole is drilled in predetermined depth increments andreadings of released strain are logged at each increment.
Different stress calculation procedures for incremental hole-drilling have been developed.However, only the integral method [6, 22, 46, 57] and the power series method take thegeometrical effect, explained in the following, into account.
10.4 Incremental Hole-Drilling 131
The relaxed strain, measured at the surface subsequent to the drilling of an increment, ispartly due to the stresses released in the increment and partly due to the previously releasedstresses which yields a modified response as the geometry is changed. The two contributionsare illustrated in Figure 10.2.
+-
Figure 10.2: Geometrical contribution to relieved strain in drilling of the second increment.
The power series method and the integral method are briefly discussed below.
The power series method: In the power series method proposed by Shajer [57] it isassumed that the residual stress variation can be expressed by power series. An arbitrarystress field can be divided into an equibiaxial and a pure shear stress component. Considerthe equibiaxial component given as
ab(h) = bo + b1h + b2h2 +... (10.9)
in which It is the hole depth and the b-coefficients are the unknown constants. The surfacestrain response at hole depth h is approximated by
Eb(h) = boA 0(h) + b1A1(h) + b2A2(h) +... (10.10)
where Ai(h) is the strain response at the surface which results from a unit stress field cor-responding to the i'th term in the power series stress polynomial. The A.(h) constants arecalculated by application of finite elements. By use of the equibiaxial part of the incremen-tally released strain and Eq. (10.10), the b-coefficients can be determined by a least squareanalysis and subsequently the equibiaxial component of the residual stress variation can becalculated by Eq. (10.9). By a similar calculation of the shear stress component the arbitrarystress field can be determined. Unfortunately, only the first two terms of the polynomial haveproven applicable, as the third order term causes significant stress calculation inaccuracy.When only the constant and the linear term are included, the method becomes very stableas the strain data is filtered in the least square analysis but only linearly varying stress fieldsare adequately described by the method.
As there is no indication that residual stresses in hot rolled steel plates are well representedby a linear variation through the thickness, the power series method is rejected for furtheranalysis. Later, in Chapter 13, it will in fact become apparent that the residual stressvariation in hot rolled, normalised and cold levelled steel plates does not vary linearly.
132 Chapter 10. Selection of Method for Modification
The integral method: The integral method, which is discussed in detail in the followingchapter, is well suited for description of non-linear variations in stress fields. However, thestress calculation inaccuracy increases quickly with the number of hole depth incrementsused in the calculation. Therefore, fine spatial stress resolution through the depth causespoor stress calculation accuracy and a compromise must be made.
The integral method has been chosen in the present work as it offers flexibility as regardsthe stress variation and as it has potential for improvement.
Chapter 11
The Integral Method
The first steps of developments of the integral method were made by Bijak-Zochowski [6]followed by several other authors, e.g. Shajer [57], Flamman and Manning [22] and Niku-Lari [46]. In the following the integral method is described.
Center line
Coge
inc. 12 Z3h
4MI
Figure 11.1: Definition of hole depth h and stress depth z.
As a hole is drilled to a certain depth h, the residual stresses released at every intermediatedepth z, between the surface and the depth h, contribute to the relaxed strain measured atthe surface. The relaxed strain contribution from the residual stress released at the depthz depends on the actual hole depth h. This was previously described as the geometricaleffect (Figure 10.2) and is taken into account by the integral method. As an arbitrary stressdistribution can be decomposed into an equibiaxial and a pure shear stress component, therelation between radial strain and released residual stress can be written as
= L+V jh a(z, h) ocq(z) dz + I fo b(z, h) og,(z) cos 20(z) dz (11.1)
where
co,(z) = aa,,x(z) + Umnin(Z)
,.,(z) = a,,n=(z) - ami,(z)
133
134 Chapter I. The Integral Method
The function a(z, h) is the radial strain response at the gauge, corresponding to a unitequibiaxial stress acting at the depth z in a hole of depth h. Similarly, b(z, h) is the radialstrain at f = 0, corresponding to a unit pure shear stress, acting at the depth z in a hole ofdepth h. As the angle )3 is measured CCW from the direction of o..z, which is dependenton z, 3 itself becomes a function of z.
The derivation of a(z, h) and b(z, h) is cumbersome but can be done numerically by e.g. FEanalysis. However, in order to determine ma,(z), amin(z) and a(z), simultaneous solution ofthe integral Eq. (11.1) in three unique angular positions is required. Moreover, the releasedstrain cr(h) cannot be logged continuously due to temporary stresses in the drilling process.
The above stress calculation procedure is extremely difficult and Eq. (11.1) is thereforediscretised to facilitate drilling of the hole in n successive increments:
1 + V (11.2)(c,)i 1 - Z.aq (a.mz + umi)j (11.2)
j=1
1 =
+ 2E Zbis (a,,w - umin)j cos 2)3j 1 < j :5 i < nij=1
It should be noted that index i refers to the number of increments drilled, whereas j refersto the increment in which the stress is acting. In matrix notation Eq. (11.2) becomes
{Er} = [- [a] {(o. + amin} + 1 [b] ((ama: - mi,) cos 2/3} (11.3)
The principal stresses a6.j and a,mnj are the uniformly distributed stresses acting withinincrement j, which are equivalent to the stress actually present in terms of released strainat the gauge. Hence, the discrete form given by Eq. (11.3) is an approximation as uniformstress is assumed in each increment.
The correlation matrices [a} and [b] are lower triangular matrices containing the componentsaij and bij, respectively. The correlation constant aij relates an equibiaxial stress acting inincrement j, in a hole i increments deep, to the strain response at the gauge. Figure 11.2illustrates the load cases used for derivation of aij components in a hole three incrementsdeep.
135
all
021 a2 2
a31 a32 a33
Figure 11.2: Application of unit stress loads for derivation of aij
The bij components are similarly the correlation coefficients between pure shear stress andthe strain response at the gauge. Hence, [a] represents the axisymmetric and [b} the non-axisymmetric part of the loading.
The equivalent uniform stresses within the depth increments can be calculated by solvingEq. (11.3) on the basis of three readings of released strain in unique angular positions for eachincrement. However, as the equation is non-linear, solution can be facilitated by linearisation.The following method is based on a 00/450/900 CCW positioning of the strain gauges asshown in Figure 10.1.
By definition of the following set of strain and stress variables it is possible to decoupled theequations which must otherwise be solved simultaneously. The strain variables are
p(h) = (cj(h)+f3(h))/2
q(h) = (ei(h)-e3(h))/2 (11.4)
t(h) = (2E2(h)-Ej(h)-C3(h))/2
and the stress variables are
P(z) = (ai(z) + 93(z))/2
Q(z) = (aI(z) - 03(z))/2 (11.5)
T(z) = rlW(Z)
136 Chapter 11. The Integral Method
where
z is the current depth from surface
h is the hole depthP is the equi-biaxial normal stress componentQ and T are the max. normal and shear stress component in pure shear stress statep, q and t are the strain components corresponding to P, Q and T0l and G3 are the normal stress in the directions defined by the gauges (see Figure 10.1)T13 is the shear stress componente# is the strain measured at gauge #
By use of these strain and stress variables, Eq. (11.3) can be written as
+ VE-[a]{P} = {p} (11.6)
Efb]{Q} = {q} (11.7)
E[b]{TI = {t} (11.8)
where
(P}, {Q} and {T} contain the stress variables Pj, Qj and Tj for each increment{p}, {q} and {t) contain the strain variables pi,qi and ti for a hole i increments deep
The cartesian stress components for each increment can be retrieved as
01(z) = P(z)+Q(z)
03(Z) = P(z)-Q(z) (11.9)713 (z) = T(z)
and the principal stresses and the angle a defining the maximum principal stress direction(Figure 10.1) are given as
Umaz(Z),Omin(Z) = P(z)± Q(z)±T(Z)2 (11.10)
a~z) = 1/2 arctan(fŽ())(1.1Q(z)
The angle a is positioned in the correct quadrant by the following scheme, in which thecalculated arctan value is denoted y = arctan , ,1 11 Jý
137
Numerator \ Denominator El63 >0 El-C3 <0
2E2 -C1 -E3 >0 o 1/2,y a =1/2(1800o-y)2f2- El- E3 <0 =1/2(3600o-y) a =1/2(1800 +-y)
Table 11.1: Corrected value of a.
To solve the decoupled Eqs. (11.6) to (11.8), it is necessary first to obtain the correlationmatrices ta] and [b). In Section 12.2 a method using finite elements for derivation of thecorrelation constants is described.
The number of stress calculation increments should be chosen on the basis of the stresscalculation accuracy obtainable in the last increment. Considering the matrix Eq. (11.6), arelative error in {p} may cause a significantly greater relative error in {P). The maximumrelative error is described by the condition number N of the matrix, which is calculatedas the norm relative to the determinant. For a lower triangular matrix like [a) and [b),the determinant is the product of the diagonal elements, whereby a small diagonal elementincreases N and thus decreases the stress calculation accuracy. With evenly spaced incre-ments, the size of the diagonal elements will decrease with depth due to increased distancebetween released residual stress and the strain gauge. Moreover, an increasing number ofincrements will reduce the size of the diagonal elements as less residual stress is releasedin each increment. Consequently, the number of increments should be limited to maintaingood stress calculation accuracy. A commonly applied number is 3 to 5 increments fromthe surface to maximum stress calculation depth. Stress calculation errors are discussed indetail in Section 12.7.
138 Chapter 11. The Integral Method
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Chapter 12
Modified Hole-Drilling Technique
Until now mainly the work of other authors has been discussed. In the following a modifiedincremental hole-drilling technique for plates of finite thickness will be introduced, allowingthe residual stresses to be measured through the thickness of the plate. Further, additionaladvantages compared to "conventional" incremental hole-drilling will be presented.
12.1 Investigation of Hole-Drilling Configuration
St. Venant's principle can be used to show that the effect of a local self-equilibrating load willdecrease quickly with distance. Accordingly, the hole-drilling method is in general not welladapted for measuring of stress acting remote from the surface as the correlation betweenreleased surface strain and relaxed residual stress will decrease significantly with depth. Ifreleased surface strain is plotted against hole depth, it will increase quickly, reach a maximumand then start to decrease. The hole depth at which the maximum is reached is denoted themaximum strain release depth or the maximum stress resolution depth.
As the application of interest involves residual stress measurement at relatively large depths,it is relevant to minimise the problem of decreasing strain sensitivity. To this end, thehole-drilling configuration is analysed in detail for plates of finite thickness.
12.1.1 Characteristics of Conventional Incremental Hole-Drilling
Authors investigating incremental hole-drilling have so far limited their research to hole-drilling configurations based on small hole radii R. and small mean gauge circle radii R..Investigated values of R. and R. are usually less than 2 mm and 3 min, respectively, andeven though non-dimensional scaling is possible, the benefits resulting from larger dimensions
139
140 Chapter 12. Modified Hole-Drilling Technique
have not been examined. There are several reasons for the choice of small dimensions. Thesmaller radius, the more local measurements, which is necessary when stresses are measuredin specimens with large in-plane stress gradients, e.g. in measurements of welding stresses.Moreover, the smaller radius, the less specimen damage.
In conventional hole-drilling configurations, the specimen dimensions are considered to beinfinite compared to the hole. On this assumption the following characteristics of hole-drillingcan be summarised from the work done by several authors, notably Shajer [57-59].
The maximum theoretical stress resolution depth is approx. 0.5 R., as the released equib-iaxial strain component at this depth has zero tangent, which results in a singularity [5]The practical stress resolution depth is, however, only 30 to 40 percent of R. as the strainrelaxation gradient hereafter becomes very small. Using the conventional R4, values of 2-3mm, the maximum resolution depth is approx. 1 mm, which does not fulfil the specifiedrequirements of the current application.
The maximum strain release depth depends on the mean gauge radius R,, whereas the holeradius R. influences the magnitude of the released strain. Hence, the shape of the 'relaxedstrain - hole depth' curve is qualitatively determined by R, and quantitatively by 1R..Moreover, the principle of dimensional similitude can be applied due to the linearity of theproblem. That is when all dimensions in the hole-drilling configuration are scaled uniformly,the released strain plotted against non-dimensional hole-depth will remain constant.
As it will be shown below, these characteristics are highly dependent on the assumption ofinfinite specimen dimensions.
12.1.2 Hole-Drilling Configuration for Plates of Finite Thickness
In measurements of residual stresses in steel plates, large in-plane stress gradients are notto be expected. This allows for larger dimensions in the hole-drilling configurations, whichwill prove advantageous to maximum stress resolution depth, stress calculation accuracy andusable milling methods.
In Section 12.1.1 above, it was indicated that stress calculation depth is simply increasedby increasing R.,. The influence of hole-drilling dimensions on the assumption of infinitedimensions needs, however, not to be valid as concerns plates of finite thickness and hencean analysis is carried out.
The analysis is performed by calculating relaxed surface strain as a function of hole depthby axisymmetric FE, using an equibiaxial loading of 1 MPa. As only linear elastic strainsare involved, the principle of dimensional similitude is valid and the results are made non-dimensional by the plate thickness T,,.
Figure 12.1 shows the effect of increasing 1R,, if a constant &1R/., ratio and a constant platethickness are maintained. The maximum stress resolution depth obtained by increasing R,
12.1 Investigation of Hole-Drilling Configuration 141
0 RRRm /T0 6 =,T, T035R./R Ok= 1.0 ---- ----
=3.0 --l.Oeq5 =3.5 " '~-,.- 6
-2.0 "-6
-2.5-
-3. -
0 0.2 0.4 0.6 0.8 1h T,
Figure 12.1: Qualitative change in E, by varying PR.
is approximately 75% of the plate thickness T.1, as the strain relaxation gradient becomeszero at this depth. Thus, the finite thickness limits the maximum stress calculation depth.
The gradient of released strain with hole depth is important as a small gradient near themaximum strain release depth decreases the stress calculation accuracy in the last incre-ments. As R, is increased the shape of the 'strain-depth' curve goes towards a 'limitingshape', which provides the best overall stress calculation accuracy as well as the maximumstress resolution depth. It should be noted that the 'limiting shape' of the stress-strainrelation provides an improved accuracy for hole depths larger than approximately 50% ofthe plate thickness. To determine the R. range, facilitating good overall stress calculationaccuracy and stress resolution depth, a maximum stress resolution depth not less than 70%of T, has been defined as a criterion. In Figure 12.2 the maximum stress resolution depthhas been plotted against R. and by using the chosen criterion, R. should fulfil PRI/TP > 1.2to ensure high stress resolution depth and good accuracy.
0.8
0.70.65
~0.60355
0.50.45
0.40.350.3 L
04 0.6 0.8 I 1.2 1.4 1.6 1.8 2R I/T,•
Figure 12.2: Max. stress resolution depth versus R1.
142 Chapter 12. Modified Hole-Drilling Technique
In the above calculation of max. resolution depth versus R., a very small hole radius &4 wasemployed as the correlation between Ra and R. thus is suppressed. The effect of increasingR. and keeping R. constant is, however, shown in Figure 12.3. In the lower range ofR4/1R, a qualitative resemblance between the curves is apparent by visual inspection. Theresemblance is confirmed by Figure 12.4, where c, of Figure 12.3 has been multiplied by afactor chosen to be (R/IRa)2 .
0.0e.6 RTp 0.30• •0.35 --- - ....
0.40
0.50• o , 0.55 o
-2.0c-6 " ."% °'a .. • 0.65 .....
-3.0e-6 00 40 0-----. 8. .0 .35,. 0.85
.. 0.90.4.0-6 05.95
-5.0c-6 " " """ "
.2-.Oe.6 \ ... •
-6.0e60.00 0.20 0.40 0.60 0.80 1.00
h/Tpi
Figure 12.3: Strain relaxation versus hole depth by varying R..
Ra/Rý= 0.30 -0.0e-6 8 0.35 ---......
c, ~~~0.40 ++"..•, ~~0.45 -o-
0.50 - -
,• 2.0e-6 0.55 -e~~0.60 -- -~~~0.65 -'"..
_E 0.70 "-' ...4.0e-6 0.80 .....
••0.80 --°----
-8.0e-6 R /p ::
0.00 0.20 0.40 0.60 0.80 1.00
Figure 12.4: Qualitative change in e, by varying PR..
12.1 Investigation of Hole-Drilling Configuration 143
As only limited variation of Er(Ra/R,) 2 is seen in the lower range of RIR,., er may herebe approximated by
C nt & Wh (1)(12.1)
where
0non, is the nominal equibiaxial stressf is a function representing the qualitative variation
A deviation is seen in Figure 12.4 as &1R./ goes towards unity. This effect is denoted the'near edge' effect as it occurs when the strain gauge is located near the edge of the hole. The'near edge' effect results in an undesirable qualitative change of c,, characterised by largestrain release at small hole depths and small or negative strain release as the depth is furtherincreased. To evaluate the 'near edge' effect the function f is analysed by comparison ofcr(R.m/R) 2 with a reference chosen to be E,(R•/R•) 2 at /•R/, = 0.3. If the deviationsare larger than 10% the 'near edge' effect is defined as dominant. For the analysis, er hasbeen calculated by variation of R. in discrete steps for selected values of R, and T,. Usingthe chosen criteria, the critical, maximum hole radii &,, may be estimated. The results aregiven as critical clearance between the strain gauge centre and the hole edge, (R,-R.,,)/Tp,and are shown in Figure 12.5.
0.55
0.50 ......**
0.45•--0.40.'
035
': 0.300.25
0.20 * -* o0.15 T=40mm
0.1-00.0 0.5 1.0 1.5 2.0 2.5
Rm iTp
Figure 12.5: &,,. given as (R, - R,,•)Ti versus Rm/Tj.
Again, the finite plate thickness is clearly observed to influence the results. The criticalclearance becomes constant at RmIT, ;z4 1.2, which is the same value providing a stressresolution depth of 70% (Figure 12.2). For RI/Tt < 1.2 the critical clearance increaseslinearly resulting in an almost constant critical RO/R. ratio. For R.,,/Tj > 1.2 the criticalclearance becomes constant, which allows the R/Rm ratio to be increased with increasingR,. This facilitates the stress calculation accuracy as the amplitude of the released strainthus is increased without impairing the qualitative shape of the 'released strain - hole depth'
144 Chapter 12. Modified Hole-Drilling Technique
curve. Hence, for R,,/Tl > 1.2, the constant critical clearance of (R. - Ra)/Tp1 i 1/2makes it possible to approximate the released strain as
._ h 2, - >1.2 A BL C<1p T, \R.,/ Tp, Rý 11 (12.2)
where a polynomial form of f(h/Tp,) may be obtained by using e.g. a least square method.
12.1.3 Summary of Investigation
The analysis of hole-drilling configurations for plates of finite thickness subjected to equibi-axial stress can be summarised as
" Qualitatively, the 'strain release - hole depth' relation is influenced by finite platethickness and a 'near edge' effect.
" The finite plate thickness T,1 limits the maximum theoretical stress calculation depthto approximately 75 % of T,,. A near to optimal qualitative shape of er as concernsstress calculation accuracy is reached for n > 1.2.
The 'near edge' effect is prevented by choosing - < 1 - ý2 for RP/T > 1.2.
" The above restrictions makes it possible to approximate the 'released strain - holedepth' relation by Eq. (12.2).
According to Eq. (12.2), Ra/P, should be chosen to be as large as possible within the abovelimits to increase further the stress calculation accuracy. Using the incremental hole-drillingmethod the in-plane stress is, however, assumed to be uniform across the diameter of thehole, which must be taken into account when R. is chosen. After R, has been determined,R. should be chosen to fulfil R. < Rm - Td/2, in which the length of the strain gaugesmust be taken into account. As local surface stresses will be introduced in the milling ofthe hole, some distance between the hole edge and the strain gauge is desirable to minimisetheir influence.
Using the above guidelines, an optimised 'released strain - hole depth' relation is benefittedfrom, so that the stress calculation accuracy is increased for final hole depths larger thanapproximately 50% of the plate thickness. The analysis reveals, however, that the hole-drilling technique investigated is incapable of measuring stresses through the thickness of aplate. Section 12.3 details a method to address this issue.
The validity of dimensional similitude adds a generic aspect to the modified hole-drillingmethod, as the above analysis is valid for any uniform scaling of the hole-drilling configura-tion.
12.2 Calculation of Correlation Constants 145
12.1.4 Proposed Hole-Drilling Configuration
The above analysis is used to determine a hole-drilling configuration for the derivation ofcalibration constants which will later be employed in experimental investigation. As theconfiguration analysis only yields lower bounds for R., hole-drilling dimensions are onlylimited by the assumption of uniform in-plane stresses in the material to be removed. Notto violate this assumption, dimensions have been chosen just large enough to provide areasonable clearance between the hole edge and the strain gauge to minimise the effect ofmilling stress.
Hole-drilling configurationPlate thickness, TI 101mmMean gauge circle radius, R, 22mmHole radius, RP 15mmP./Rý 0.68Max. hole depth 5.0 mmStress calculation increments 4Gauge type HBM LYllGauge length, L9 3 mm
Table 12.1: Configuration for experimental investigation.
The configuration in Table 12.1 provides a drilling clearance of 5.5 mm (Rm - Ra - Lg2),which is believed to be sufficient to exclude the influence of milling stresses. The problem ofmilling stresses is discussed in Section 12.5.
A hole depth of half the plate thickness has been chosen for reasons which will becomeapparent in Section 12.3.
12.2 Calculation of Correlation Constants
Below, the procedures used to derive the correlation constants are described. Consider thedecoupled Eqs. (11.6) - (11.8):
1+vE+[a]{P} = {p}
S-[b]{Q) = {ql
1.k[bJ{T} = ft)
146 Chapter 12. Modified Hole-Drilling Technique
In order solve these equations, it is necessary to derive the correlation matrices [a] and [b],which are 4 x 4 lower triangular matrices, as four stress calculation increments are used inthe chosen hole-drilling configuration:a2 OaO2 0o0] [o-[2 2 1(23
[~ 1 0 0 0 611 0 0 0~[]=a 22 0 0[b] : = 12.3)
a3 1 a32 a33 0 , 631l
b32 33 0
a41 a 42 a43 a44 1b41 b42 b43 b44
The matrix component aij relates an equibiaxial stress acting in increment j in a hole iincrements deep to the relaxation strain at the gauge. Similarly, bij relates a pure shearstress to the relaxation strain. Bijak-Zochowski [6) determined the correlation constantsby experimental investigation whereas Shajer [57] was the first to apply FE analysis to thederivation. In the present paper the correlation constants are derived by FEM.
12.2.1 Finite Element Model
By numerical calculation of the correlation constants significant flexibility is added to thehole-drilling method as any specimen can be investigated without restrictions on geometricalshape.
Residual stresses can be considered to be elastically 'locked-in' stresses, formed by inhomo-geneous plastic deformation in the case of hot rolled steel plates. Hence, relaxation strainswill be elastic strains only and the correlation constants can be calculated by using linearelastic axisymmetric finite elements. In Figure 12.6 the principle of superposition is appliedto illustration of the loading used to calculate the correlation between relaxation strain andresidual stress.
A B C
Figure 12.6: Relaxation strains can be derived by loading of inner circumference.
The stress state prior to drilling (A) is superpositioned to the redistribution stresses (B) inorder to obtain the resulting stress state after drilling (C). It is seen that the redistributionstresses (B) can be calculated by loading the inner circumference of the drilled hole bystresses equivalent but opposite to those present before drilling. Hence, by loading the innercircumference of the hole, the correlation between residual stress and strain relaxed due tothe hole can be calculated.
12.2 Calculation of Correlation Constants 147
The finite elements used for calculation of the correlation constants are semi-analytical axi-symmetric solids to which non-axisymmetric loading is applied [2]. The nodal displacementsare given as Fourier series.
Nuý(x,z,O) = Z u.(x, z) cos(n6) (12.4)
n=ON
u,(x,z,O) = ,u(x,z)cos(nO) (12.5)n=O
Nuo(x,z,O) = E uon(x,z) sin(nO) (12.6)
n=O
The nodal circumferential forces are also expanded into Fourier series with the same harmonicvariation. It is assumed that the residual stresses are acting in a plane parallel to the surface.As previously described, any plane stress state can be decomposed into an equibiaxial anda pure shear stress field. The equibiaxial stress field corresponding to aij is given as
F.(x,z,O) = Feo(x,z) (12.7)
whereas the pure shear stress corresponding to bhj is given as
F.(X,z,O) = F 2(x,z)cos(20) (12.8)Fo(x,z,O) = Fa2(X,z)sin(20) (12.9)
( Y
(a) Radial variation (b) Tangential variation
Figure 12.7: Harmonic loads super-positioned for pure shear stress.
Due to the orthogonality of the harmonic functions, displacements can be calculated individ-ually for each load component. The resulting displacements are calculated by superposition.Hence, aij components are calculated as the average strain in the gauge area when Fro,
148 Chapter 12. Modified Hole-Drilling Technique
corresponding to a unit stress, is applied to the j'th increment in a hole i increments deep.The calculation of bij components is similar, but the nodal circle forces F,2(x, z) cos(26)and F0 2(x, z) sin(20), each corresponding to a unit stress, are applied individually and bij istherefore given as the sum of relaxed strain.
In Figure 12.8 a detail of the FE model used for calculation is shown.
Figure 12.8: Detail of FE model. Nodal loads for calculation of a32 are applied.
Variation in hole depth is modelled by assigning a very small elastic modulus to the elementsto be removed. The elements used are quadrilateral linear revolution solids. No further im-provements in calculation accuracy were obtained by use of finer mesh or quadratic elements.
12.2.2 Strain Integration over Strain Gauge
As the strain gauge is assumed to be linear, the measured strain is the average strain inthe area covered by the gauge. Furthermore, it is assumed that the gauge has no transversesensitivity and consequently only longitudinal strain is measured.
When the correlation constants ai1 and bij are calculated, the end point displacements ofthe strain gauge are used for calculation of the average strain released in the strain gaugearea. This is an approximation, as the strain is assumed to vary linearly over the length ofthe strain gauge.
Axisymmetric finite elements are used in the calculation of correlation constants. In equibi-axial loading the strain is independent of the circumferential co-ordinate. However, in pureshear loading the radial and the tangential displacements vary with cos(29) and sin(29),respectively. Figure 12.9 shows the gauge geometry and defines the geometrical parametersused in the following. The parameter h. denotes the half width of the strain gauge.
12.2 Calculation of Correlation Constants 149
1. 2.
Rn r2.h,
Gauge
y
rb
Figure 12.9: Strain gauge geometry.
The calculated longitudinal displacement is written as
UZI = Ur, cos(0,) - up, sin(01) (12.10)
u = u, cos(9 2) - uo, sin(02) (12.11)
and the average longitudinal strain as
= jh, UX2Z - UýI(12.12)
Correlation constants aij : In the calculation of aij the loading is equibiaxial.
u,(r,O) = u,(r) (12.13)
uo(r,O) = 0 (12.14)
The radial displacement is approximated by a polynomial:
ur(r) z cr'+c -.,r"-'+...+cjr+co (12.15)
150 Chapter 12. Modified Hole-Drilling Technique
which is found by least square analysis of discrete values of radial displacements betweenthe radii r. and rb. The polynomial approximation is good for even low values of n (n ; 4).The radii rj(0 1 ) and r2 (02 ) are expressed as
r,(s) = (12.16)cos(6j)
r2 c(s2 ) (12.17)cos(6 2 )
By insertion of the above in Eq. (12.15), the longitudinal displacements of the strain gaugeend points can be written as
Ta
u., ( u ( )) cos(61 ) (12.18)
ux2 c U,( ) eos(6 2 ) (12.19)cos(0 2)~
The average longitudinal strain for equibiaxial stress on the basis of end point displacementsis now obtained as
fh, Ux2-U.dy
11 f T) T r__ ra )hL K fUr( (6 (9 dO2 - f ur( r. r. c 01 ) (12.20)
Correlation constants bij : In the calculation of bij the loading is pure shear loading:
ur(r,6) = ur(r)cos(20) (12.21)ug(r,O) = uo(r)sin(20) (12.22)
The radial and the tangential displacements are approximated by polynomials:
U,(r) z dnr -+ d,_r -1 + ... + dr + do (12.23)u6 (r) , fnr' +f• 1r
- 1 +...+fir+fo (12.24)
which are found by least square analysis of calculated displacements between r. and rb at0 = 0' and 450 C, respectively. Good approximations are again obtained for low values of n(n > 4).
12.2 Calculation of Correlation Constants 151
The longitudinal end point displacements then become
U. a(r cos(91 )cos(291 ) - To0r sin(9i)sin(291) (12.25)cos(lo sl cos(c2)u___u_( _ Tbu ( ) sin(92 )sin(292 ) (12.26)
U. 2 - Ul ( 0 )r cos(02 )cos(20,) - uc(r ) sin(02)sin(202) (12.26)
After some rearrangement the average longitudinal strain due to a pure shear stress can befound as
z h UI L9 UZIdy
atan( Žt•b)
h J ( -- Tb cos(292) do2cosU( 2 ) cos(9 2)
0atan( • )
f 2 uo( Tb rbsin(02)tan(0 2) dO2J cos(0 2 )
0
ra TaJur( r9)) cos(2l) do1
0
atan(ýh)
+ f2 uO( r,) r. sin(9i) tan (01) ,(1.7J2u(cos(01 ) di 1.70
Using the FE model and the above strain integration method, the correlation matrices havebeen derived for the chosen hole-drilling configuration:{-0.129 0.000 0.000 0.000 1
[a) = -0.169 -0.120 0.000 0.000 (12.28)-0.198 -0.149 -0.095 0.000-0.212 -0.162 -0.110 -o.o63JE-0.178 0.000 0.000 0.000 1
[ -0.234 -0.171 0.000 0.000 (12.29)-0.272 -0.215 -0.140 0.000-0.297 -0.242 -0.173 -0.1031
These correlation matrices apply to any hole-drilling configuration fulfilling the requirementsfor dimensional similitude.
152 Chapter 12. Modified Hole-Drilling Technique
In general improved strain integration accuracy may be obtained by the following alternativemethod. Strain components are calculated by finite elements in several discrete locations inthe gauge area and the longitudinal strain components are found by transformation. In aleast square analysis a 'best' surface is fitted through the data points by polynomial functions.The average strain is calculated by integration of the polynomial function over the gaugedivided by the area. In the chosen hole-drilling configuration, insignificant improvement incalculation accuracy is, however, obtained as the strain variation is small in the gauge area.This is due to the small gauge dimensions compared to the dimensions of R. and Rm..
If standard hole-drilling rosettes are considered, the relative size of the gauges compared toR. and R, is in general large. In Figure 12.10 the variation in longitudinal strain relaxationis shown in the gauge area for a unit shear stress (b33). The relaxation strain is given as thevertical distance between the plotted surface and the zero strain plane.
Strain
0-0.Se -6
1I.. e-6
Figure 12.10: Example of longitudinal strain variation in the area of the strain gauge.
Due to the non-linear variation in the longitudinal direction of the strain gauge, the as-sumption of linear variation made in the end point method should not be applied to rosetteconfigurations with relatively large gauges as in most standard rosettes. Here, the bestsurface fit method should be applied.
12.3 Stress Measurement through the Plate Thickness
The analysis in Section 12.1.2 showed that the effect of finite plate thickness was a theoreticalmaximum stress calculation depth of approx. 70% of the thickness. As the intention is toresolve stresses through the entire thickness of the plates it is necessary to develop furthermodifications to the presented hole-drilling technique.
When a hole has been drilled to the centre of the plate and the stresses have been estimated infour increments, it is still necessary to find the stresses in the remaining half of the thickness.
12.3 Stress Measurement through the Plate Thickness 153
It is not feasible to place strain gauges at the bottom of the hole and drill a second holewithin the first. This is again due to the effect of finite thickness on strain relaxation. Inorder to resolve the stress through the thickness an infinite number of holes within eachother using this method should in theory be drilled.
Another method is proposed. A second hole is drilled from the opposite side of the plate.A small drill causing insignificant strain relaxation is drilled through the plate, so that it ispossible to align the second hole at the centre of the first. New strain gauges are positionedon the second side and another four increments are drilled in order to reach the Centre andobtain a hole through the plate.
Center line
First Side GaCge 1
Seon Sie
measured strai values.
Thelin tequationd equival fen t to Eq. e (11.6)drtonwe hedcupe q. 1.)
Then becutom es uvln oE.(1
P,F P2A +pO + CI 1 0 0 0 C15 c16 C17 CIS8 PP2 +o PO E- C21 C22 0 0 C25 C26 C27 C28 A (12.30)P3 +PO I 3 C32 C33 0 C35 C36 C37 C38l P6A4 +7) PO C41 C42 C43 C44 C45 C46 C47 C48. P7
1P8
154 Chapter 12. Modified Hole-Drilling Technique
where Po is the initial strain relaxed while drilling the first hole. The strains relaxed whiledrilling the second are denoted Pi to p4. P, to P4 are the unknown stress variables to becalculated whereas P5 to P8 are the already known stress variables related to the first halfof the hole. The calibration constants which relates the strains released in the second holeto the stresses calculated for the first are cij,j > 4.
Elimination of the first row of Eq. (12.30) and rearrangement yield
rPi [Ci 5- C05 C16- COS C17-C07 CISa-c l C POSE 2 c 25 -c 05 C26 c06 c 27 - c0 7 c 28 -c 0 8 J P6
1+v 1p3| 53- Co5 C3 6 - C06 C37 -C07 C38 -c C08 P7t4 J -C4 5 O C46s - C0 6 c 4 7 - c 0 7 C4s - Cos P8
[C'1 0 0 01 P '
c21 c22 0 0 P2 (12.31)C31 C32 C33 0 P3C41 C42 C43 44 I P343
For simplicity the above is written as
EW {p- [Co] {PO} = [C]{P} (12.32)
The equations giving {Q} and {T} are found in a similar way and the decoupled equationsfor the modified hole-drilling are
EI• p - [Co0 {Po} = [Cl{P} (12.33)
E {q} - [Do]{Qo} = [D]{Q} (12.34)E It} - [Do]{TI} = [D){T) (12.35)
All correlation matrices are derived by finite elements as previously described. The unknownstress variables can be found by inverting the calibration constant matrices. Subsequently,the principal stresses or the cartesian stresses are calculated by Eqs. (11.9)-(11.11).
The correlation constants have been derived for the present hole-drilling configuration andare given by Eq. (12.36). The correlation constants representing the geometrical effect ofthe pre-released stresses are clearly important, although some are very small. The constantsare valid for any other configuration in which all dimensions, including plate thickness, areuniformly scaled.
12.4 Example of Experimental Measurement 155
E 0 0 0 C05 C06 c07 coalC,1 0 0 0 c1 5 c 16 C17 C1 8
c 21 C22 0 0 C25 C26 C2 7 C28
C3 1 C32 CM 0 C35 C36 C3 7 C3 8
C41 C42 C43 C4 4 C45 C46 C4 7 C48J
-143 0 0 0 7 28 49 72-189 -134 0 0 0 27 54 84 x 10- 3 (12.36)-217 -161 -102 0 -11 21 52 86-229 -173 -116 -65 -23 12 46 81
0 0 0 0 do5 d06 do7 dosld 1 0 0 0 d1 5 d16 d17 d18d2 l d22 0 0 d25 d26 d 27 d28 =
[d3 d32 d33 0 d35 d36 d37 d38
d 41 d42 d 43 d 44 d 45 d 46 d 47 d 48 J
0 0 0 0 -17 19 47 72-191 0 0 0 -30 9 39 65-254 -193 0 0 -46 -6 26 53 X 10-3 (12.37)-296 -241 -169 0 -64 -24 8 36-324 -270 -204 -141 -089 -48 -14 15
The present method allows for residual stress measurement through the thickness of theplate.
12.4 Example of Experimental Measurement
An example of an experimental residual stress measurement is given for the first half of theplate thickness. The hole-drilling configuration (Table 12.1) from the analysis performed inSection 12.1.2 is applied. All parameters used for the experimental investigation are listedin Table 12.2.
156 Chapter 12. Modified Hole-Drilling Technique
Thickness of steel plate, Tt 10mmIn-plane specimen dimensions 400 x 400 mmMean gauge circle radius, R.m 22 mmHole radius, R. 15 mmMax. hole depth 5.0 mmStress calculation increments 4Uniform increment size 1.25 mmGauge type HBM LY11Tool face type HSSYoung's modulus 211 GPaPoisson's ratio 0.28
Table 12.2: Parameters used in experimental investigation.
The milling procedure used for experimental investigation is described later in Section 12.5.The following strains.were recorded:
[:-36.21 -35.77 -28.531
{augli} {gauge2} {g }] = I 5431107.5 -68.41 (12.38)-84.37 -81.08 -60.21i [la6] (12.38)-86.01 -81.87 -56.56]
Using the calculation procedure described in Chapter 11, the variation of cartesian stresseswith depth has been calculated and is shown in Figure 12.12. The corresponding variationof principal stresses with depth is shown in Figure 12.13.
-I m-raO4-
40
A.i
Figure 12.12: Derived cartesian stresses from strain measurements.
12.5 Milling Stresses 157
40 76
10 74
-208
0 1 2 3 4 5 0 1 2 3 4 5flpth belo .,ui [mm) Dcph blw srace-, m
Figure 12.13: Derived principal stresses and angle to a,.., as measured from gauge 1.
Representing the stress variation with depth, the values are found by interpolation betweenstresses acting at discrete depths. Half the increment depth is commonly used, but asdescribed by St. Venant's principle, the location will in reality be closer to the surface.In Appendix B it is shown that half the increment depth may be applied to the presenthole-drilling configuration.
From the above example of stress analysis it is not possible to evaluate the accuracy of themethod. Stress calculation accuracy is the subject of the following sections.
12.5 Milling Stresses
When the hole-drilling method is applied it is important to take precautions not to intro-duce significant machining stresses in the drilling of increments. Moreover, the incrementalmethod requires a fiat-bottomed hole in order to establish an exact measure of incrementdepth. An issue which has received much attention from authors investigating hole-drillingin general is the employment of advanced drilling techniques to eliminate drilling stress.Prevailing methods investigated are air-abrasive drilling and high-speed drilling which arecompared to conventional milling. Especially, the high-speed drilling has proved valuable asonly low stresses are introduced and a well defined hole geometry is obtained. Air abrasivedrilling also introduces low-level stresses but the hole geometry is generally difficult to con-trol. Conventional milling has proved to produce milling stresses at a higher level but welldefined geometry is obtained when a fixed bench milling machine is used. However, the onlydrilling method applicable to the proposed hole-drilling configuration is conventional milling,as equipment for high-speed and air-abrasive drilling is not available for the required holedimensions. For that reason machining stresses due to conventional milling are investigated.
Quite different levels of machining stresses have been reported, varying from compressiveto tensile stresses. In articles by e.g. Niku-Lari et al. [461 and Mordfin [44] the method isreported to produce low-level stresses only, but others as Bynum [11] and Flaman [21] reportstresses which far exceed the acceptable level. A Round Robin test, i.e. an investigation for
158 Chapter 12. Modified Hole-Drilling Technique
the purpose of comparison, has been performed by SEM [70] and shows divergence amongthe results obtained by the different investigators. As will become apparent in the followingdiscussion, the divergence is likely to be a result of different drilling techniques and hole-drilling configurations.
In the machining operation the material is deformed plastically and, moreover, high tem-peratures and gradients are present. This causes uneven plastic deformation in the surfacelayer, which results in large residual stress levels and gradients in the immediate vicinity ofthe machined surface. The stress state resulting from machining can be described by themechanical, the thermal and the metallurgical contributions, which are strongly intercon-nected.
The mechanical contribution represents the plastic deformation of the surface during cutting,resulting in high-compressive stresses at the surface and low tensile stresses in the bulk ofthe material after processing. The effect mainly depends on the size of the yield zone, whichagain is dependent on the geometry of the tool nose, on the yield stress reduction due totemperature etc.
The thermal contribution is defined by the plastic compressive strain, caused by thermalexpansion and reduced Young's modulus at elevated temperatures, resulting in tensile surfacestresses after cooling. The higher the temperature the higher the tensile stresses.
In cutting, the thermal contribution will normally dominate the mechanical effect [3] and,hence, the resulting surface stresses will be predominantly tensile. However, at high machin-ing temperatures the microstructure of the surface layer is changed. The metallurgical effectis mainly related to the martensitic formation caused by phase change and rapid cooling. Inthe transformation from austenite to martensite the material volume expands considerably,which yields large compressive stresses at the surface as the expansion is restrained by thebulk material. Thus, if the machining temperatures exceed the phase transition tempera-ture (approx. 730 0C for steel) the resulting surface stresses from the mechanical, thermaland metallurgical contributions tend to be-compressive.
The different effects described above will result in different stress variations with depth belowthe machined surface. Thus, the resulting stress state may be complex, varying between high-tensile and high-compressive stresses. Therefore, in a characterisation of surface stresses aseither compressive or tensile, it should be interpreted as the predominant stress state in thefirst few tenths of a millimetre;
The complex dependency of machining tool and stress variation is illustrated by Figure12.14 adopted from [20]. The residual stress variation resulting from surface milling of a4340 steel with a hard metal tool face is measured by use of x-ray diffraction and etching.The predominant surface stresses are compressive but the immediate surface stresses are seento become more and more tensile with increasing tool wear (wearland) and the compressivestresses are increased considerably. Consequently, machining stresses may affect hole-drillingvery differently, even when the same tool type is applied.
12.5 Milling Stresses 159
OePh below milled sutlc,,0 am042 aml 0010 0612 001a a0ns 00cl
-=.1 04 r .. ..ndI l l- 0.2 mm wuar rand
- - 0.4 mm ,
u -$ -ai
-- 1000
0 0.0 010 01s 0.10 o2n 0m am5 son 0
oelth beWow milled tusro. mm
Cinloloy 310 (C46) ma~tic
PeTopherl clea.nce .............. r, 0"
min) . ...................................... (1 )
Dliph I oun. . . . ..n -....). .0.(0.040)W id .h t.. i.......................................... 5c5, (In.' 0)
Fool Wtogshh~O.Jooi o ) I Il ..... I.................. I........... ............. 0. USa lm
Figure 12.14: Residual stress resulting from surface milling (adopted from [20]).
Cutting tool faces made of high-speed steel (HSS) and of hard metal as ceramics or carbidesare discussed below. Hard metal is more brittle than HSS and in consequence the hard metaltool nose is blunter in order to be wear-resistant. The yield zone arising by use of a hardmetal tool are therefore more pronounced than the yield zone arising from an HSS tool. Thisis illustrated in Figure 12.15.
aHard Metal HSS
Figure 12.15: Yield zones arising from using a hard-metal tool and an HSS tool.
The size of the yield zone is a measure for the wear of the machining. Moreover, HSS toolsare designed for a temperature of approx. 600 OC at the cutting edge, whereas hard metal
160 Chapter 12. Modified Hole-Drilling Technique
tool faces exceed a temperature of 1000 0G. This further indicates that phase transformationoccurs by use of hard metal tools under optimal cutting conditions but not by use of HSStools. In general, hard metal tools therefore tend to give compressive surface stresses of anabsolute value larger than the tensile surface stresses which HSS tools tend to give.
Rtom the above discussion it is concluded that the choice of cutting tool strongly affects themachining stresses. The type of cutting tool is, however, an information usually not givenin articles on milling stresses in connection with hole-drilling. Moreover, it is concluded thatHSS tool faces in general are preferable to hole-drilling in low-alloy steels.
In the following, the relation between hole-drilling configuration, machining stresses andstress calculation accuracy is discussed in order to justify the use of conventional drilling inthe proposed hole-drilling method.
As previously described, large machining stresses are introduced in the immediate surfacelayer which results in oppositely balancing stresses in the bulk material. These stresseswill decrease with the distance from the machined surface in accordance with St. Venant'sprinciple and their influence on the strain gauge readings will depend on the drilling clear-ance, defined as the distance from the hole edge to the beginning of the strain gauge(R,. - &- L,/2). A small drilling clearance makes the hole-drilling method more sen-sitive to machining stresses. A relatively large drilling clearance decreases, however, thestress measuring sensitivity, which is also undesirable. The latter problem was discussed inSection 12.1.2.
As regards the radial stress variation with the distance from the machined surface, it isexpected to be independent of the hole size if the machining conditions are maintained.Hence, the influence of machining stresses on strain readings will decrease significantly if thehole-drilling configuration is enlarged uniformly, and if the Ra/R,. ratio is kept constant,the stress measuring sensitivity is retained. Nawwar et al. [45) experimentally investigatedthe relation between the drilling clearance and the measured drilling stress in aluminium.They found that the drilling stress decreased exponentially with increasing drilling clearance,and at clearances above 4 min no drilling stress was recorded. In the present hole-drillingconfiguration the drilling clearance is 5.5 mmn and the material of interest is mild steel.According to the findings of Nawwar et al. [45] and the fact that mild steel in general is lesssensitive to machining than aluminium [21], the chosen drilling clearance should be sufficientto exclude the effect of machining stresses.
In conclusion, the presented hole-drilling method has an additional advantage compared totraditional hole-drilling, as the large rosette dimensions make it possible to apply conven-tional milling. In Section 12.5.1 this is verified experimentally.
12.5 Milling Stresses 161
12.5.1 Experimental Investigation of Milling Stresses
In order to investigate the stresses introduced in milling, it is essential to use a stress-free specimen. Stress relaxation can be obtained by thermal annealing, which is a time-temperature-related phenomenon. For low-alloy steels, residual macrostress is almost fullyrelieved at the surface by annealing at 600 OG for one hour. The annealing must be fol-lowed by furnace cooling to room temperature in order to prevent the occurrence of coolingstress [12, 69]. Residual stress in the interior of the material is relieved more slowly and thetime needed for full relaxation may be longer. The annealing mechanism can be explained asfollows. The residual stresses correspond to elastic strains formed in uneven plastic deformna-tion. When sufficient thermal energy is applied to the material, dislocation slip transformsthe elastic strains into microplastic strains so that the residual stresses diminish. It is im-portant to keep the annealing temperature well below the phase transformation temperatureas microstructural change is undesired. According to Vohringer [691, a temperature of 6000Cis suitable for most low-alloy steels. The test specimens used in the present investigationare therefore annealed at 6000C for 3 hours after which the specimens are left to cool inthe furnace. It should, however, be noted that the thermal annealing will only reduce theresidual stresses but not entirely eliminate them.
The annealed specimen was investigated by the hole-drilling procedure and a fixed benchmilling machine. The stresses to be measured were introduced in the fixture of the specimenand reference values were measured at the surface by a standard rosette strain gauge. Asmall hole was drilled at the centre and the increments were subsequently milled using aturning head with low radial in-feed, an HSS tool and large amounts of cutting fluids. Veryaccurate hole geometry was obtained by this procedure. The results of the investigation werepreviously shown for all increments in Section 12.4. For investigation of the milling stress,the reference stress measured at the surface will be compared to the stress calculated hyincremental hole-drilling. To exclude the effect of milling stress in hole-drilling these valuesshould be similar. The obtained values are listed in Table 12.3.
Measured surface stress Stresses found by hole-drilling1st increment 2nd increment
a.= 49.7 MPa a, = 47.5 MPa a,,. 0 = 31.8 MPa0
min = 40.1 MPa 0 min, = 35.4 MPa 0mP, = 21.8 MPa
a =72.70 a = 69.2 0 a = 75.70
Table 12.3: Stress measured at surface and stress calculated by hole-drilling.
162 Chapter 12. Modified Hole-Drilling Technique
A second measurement, performed equivalently to the above on another specimen, yieldedthe following results:
Measured surface stress Stresses found by hole-drilling1st increment 2nd increment
=ý 12.3 MPa a...: = 13.9MPa 0.X, = 8.6 MPa=ýi -10.1 MPa 0 min = -9.3MPa 0 'nin = -3.3 MPa
a =1340 a = 1320 a = 1330
Table 12.4: Stress measured at surface and stress calculated by hole-drilling.
Other error sources than milling stress influence the stress calculation. These compriseresidual stresses still present in the specimen, strain measurement errors and geometricalerrors. Comparison of the measured surface stresses with the stresses calculated in thefirst increment by the hole-drilling method shows, however, good agreement. It should benoted that the representative depth for the stresses calculated for the first increment isapproximately 0.6 mm beneath the surface and consequently the values cannot be compareddirectly without taking the stress gradient into consideration.
A number of error sources influence the calculated results and it is not possible to separatetheir individual contributions. Still, the experimental investigation indicates that millingstresses can be disregarded by use of the chosen hole-drilling configuration and the experi-mental investigation therefore supports the findings of Section 12.5.
12.6 Experimental Precautions
If an end mill is used, the variation of cutting speed in the radial direction results in non-optimal milling conditions towards the centre. In order to minimise the effect, a small holewhich does not influence the residual stress state, is drilled through the plate to remove thecore. Subsequently, a turning head with radial in-feed is used to produce the final hole. Inthis way, optimal cutting conditions can be maintained to a higher degree. In addition, anaccurate hole radius can be obtained by calibration of the turning head.
The experimental procedure for investigation of residual stresses in large plates is not trivialto set up. If a large-size milling machine with a large positioning table is not available,it is necessary to use a rigid transportable milling machine. If the plate to be investi-gated is uneven and unequally supported, stresses might be introduced by the weight ofthe machinery or non-optimal milling conditions might be caused by the lack of support.Gravity-introduced stresses are, however, easily checked by recording the strain readingsbefore and after positioning of the milling machine.
12.7 Error Calculation 163
12.7 Error Calculation
Appropriate stress error calculation is necessary to evaluate the accuracy of the experimentalresults. As indicated by Shajer and Altus [61], the stress calculation accuracy obtainable inincremental hole-drilling is extremely poor in the interior of the material if small standardrosette configurations are used. In the present section the results of an extensive stresserror calculation analysis will be presented for the proposed hole-drilling configuration. Thedetailed analysis has been placed in Appendix A.
Consider again the decoupled equations given in Chapter 11:
1+v[]P p (12.39)
[b()= {q} (12.40)
~[b]{T} = (t) (12.41)
The stress calculation errors can be divided into two categories, which are errors related tothe left and the right side of the above equations, respectively.
" Right side errors:
- Strain measurement errors
- Milling stress
" Left side errors:
- Hole depth measurement errors
- Hole diameter measurement errors
- Hole eccentricity errors.
- Material parameter measurement errors
- Correlation constant derivation errors
Experimental investigation as described in Section 12.5 may be used for a rough estimateof the total error contribution from all the above error sources. These contributions can,however, not be separated and consequently the significance of each error source cannot beevaluated.
The errors caused by the listed error sources are estimated by investigation of the relationbetween each error source and the basic matrix equations. By considering Eq. (12.39) theright-side errors may be written as
1 + V[af P = {p-4-6p) (12.42)E a P
164 Chapter 12. Modified Hole-Drilling Technique
By inversion of both Eq. (12.39) and Eq. (12.42) the following equation is obtained bysubtraction:
{6P) [a]--Ia'f&p} (12.43)
Hence, stress calculation errors resulting from right-side errors may exist independently ofresidual stresses.
Apart from material parameter errors, the left-side errors affect the correlation constants ase.g. a variation in hole depth causes a change in the correlation constants. Left-side errorsare written as
1+ V[ 6]fP P = fp} (12.44)
By inversion of Eqs. (12.39) and (12.44) the following is obtained by subtraction:
fsp} = ([a+ &J-1> - [a]-') {p} (12.45)
Hence, left-side errors cause stress calculation errors relative to the residual stress present.
All the error types listed are assumed to be independent of each other and, moreover, nor-mally distributed with a zero mean. Independence among errors is an assumption as e.g. theleft-side errors are related to the calculated stress and therefore dependent on the right sideerrors. This dependency is, however, believed to be insignificant and therefore neglected. Inaddition, the relation between geometrical errors and correlation constants must be consid-ered. Hole depth errors are e.g. not linearly related to the correlation constants and thereforeneither to the cartesian stresses. Once more this effect is explained by St. Venant's principleand an error resulting in a hole too shallow will thus affect the stresses more than an errorresulting in a deeper hole. The assumption of a linear relationship is, however, adequate forsmall depth errors arid the assumption is therefore made.
The number of chosen stress calculation increments is essential to the stress calculationaccuracy that may be obtained in each increment. In general it is necessary to make acompromise between good stress calculation accuracy and good spatial stress resolution.The stress error calculation analysis forms an improved basis for choosing the number ofincrements. The present number is four.
The error analysis comprises both the cartesian and the principal stress components. In theanalysis of e.g. hole depth errors and in the transformation from cartesian to principal stress,the matrix equations are non-linear and the standard deviations have been calculated by an
12.7 Error Calculation 165
approximate technique based on linearisation by second order Taylor expansion as describedby Ditlevsen [16] (see Appendix A).
As regards the notation, E[x] denotes the mean and s. the standard deviation for thestochastic variable x. The cross covariance matrix of the stochastic vectors {u} and {v}is denoted C,, and defined by the covariance of the elements in the matrix formed by thedyadic product of the vectors:
C" = Cov [{u}, {v T ] (12.46)
so that the covariance matrix of {(u becomes C• or just Cý. Characteristic of the uncer-tainty modelling is that matrices are contracted to vector form to facilitate the calculation.
In the evaluation of all error types, the interval of confidence has been chosen to be 90%and the correct value is thus assumed to be within ±1.64 standard deviations from the valuemeasured.
12.7.1 Strain Errors
Strain errors include instrumentation errors, thermal strains and strains introduced in ma-chining. The standard deviation for the strain is assumed to be
s, = 2 x 10- ' (12.47)
The calculation of the covariance matrix for the cartesian stress is facilitated by rearrange-ment of the matrix Eqs. (11.6)-(11.8) into one equation which relates the measured strainsto the cartesian stress:
{a} = [N] {E} (12.48)
where
W' = {{o [TGA3 171 3} )}
{clT 116 = ' {{EIT 21T,{411T1)
The covariance matrix then follows as
Ca = [N] C, [N]T (12.49)
where the covariance matrix C, is given as a diagonal matrix containing only the varianceof the measured strain as all measurements are assumed to be independent with the same
166 Chapter 12. Mlodified Hole-Drilling Technique
standard deviation. It should be noted that this is an approximation as e.g. errors relatedto the strain gauge amplifier will tend to give fully correlated measurements. The standarddeviation of the cartesian stresses may now be calculated by Eq. (12.49), taking the squareroot of the diagonal elements in Ca.
The error bounds have been calculated for the experimental investigation of fixture stressdescribed in Section 12.5. The results are shown in Figure 12.16.
0 0 11
Figure 12.16: Cartesian stresses pl otted with error bounds for errors in strain measurements.
The error bounds are seen to increase with depth as the correlation between residual stressrelaxation and surface strain decreases.
Residual stresses measured by hole-drilling are usually given by the principal stresses andthe angle of the maximum principal stress measured from a reference direction. Thesecomponents are calculated from the stress variables P, Q and T by Eqs. (11.10)-(11.11). Therelations have been linearised for estimation of error bounds as described in Appendix A.The results are shown in Figure 12.17.
1 1 4
Figure 12.17: Principal stresses plotted with error bounds for errors in measured strain.
It is seen that the direction of the maximum principal stress is extremely sensitive.
12.7 Error Calculation 167
12.7.2 Hole Depth Errors
Hole depth errors affect the components of the calibration constant matrices [a] and [b]. Inorder to calculate the standard stress deviation due to depth errors it is necessary to makethe following assumptions:
" Depth errors are non-linearly related to the calibration constants as described bySt. Venant's principle. Since the errors, however, are small the relationship is assumedto be linear.
" Milling depth errors Ah are assumed to be independent with the same standard devi-ation and zero mean.
A standard deviation Sh of 0.04 mm is assumed for the hole depth. The standard deviationsfor the components of [a] and [b] are denoted s', and s", respectively. These are conservativelyestimated as the depth is decreased in each increment which results in larger errors than acorresponding depth increase. A hole depth decrease within the i'th increment decreases allthe elements of the i'th row in the calibration matrix and consequently the row elementsare fully correlated. As the depth errors are independent there is no correlation betweenelements in different rows. Thus, the elements ai1 of [a] are normally distributed stochasticvariables described by
E[aij] = dij (12.50)Cov[aki, amn] = Paa,,,a•,sS (12.51)
= 1 for k= maa - e (12.52)
10 else
bij is described equivalently whereas the covariance between ai. and bij components is de-scribed by
Cov[ak•,bmn] = Pw,bks•,S'ln (12.53)= 1 for k= m
1 0 else (12.54)
The s' and At components are derived by decreasing the depth of one increment at atime. The corresponding aij and bij components are calculated as previously described andsubtraction of the original correlation constants yields the desired standard deviations.
If the equation
1E+ta- (12.55)
168 Chapter 12. Modified Hole-Drilling Technique
is considered, the covariance matrix of the inverse matrix [a] - ' must be derived in order tocalculate the standard deviation of {P}, and similar derivations must be performed withrespect to {Q} and {T}. This involves linearisation of the cross covariance for multivariablefunctions, which is done in Appendix A. The error bounds calculated for the cartesian andthe principal stress components are shown in Figure 12.18 and Figure 12.19, respectively.
1an - e pt - - r o a dt rr
0 _ 201 .0 1 20
0 I 2 2l 4 0 [ 2 2 4 5
Figure 12.18: Cartesian stresses plotted with error bounds related to hole depth errors.
40 40
Figure 12.19: Principal stresses plotted with error bounds related to depth errors.
12.7.;3 Hole Radius Errors
Hole radius errors are left-side errors and affect the correlation matrices [a] and [b]. Inthe error analysis the radius error is assumed to be constant with depth i.e., it does notvary between increments. Through FEM analysis, hole radius changes are seen to be non-linearly related to the correlation constants aij and bij. Errors resulting in a large holeradius affect the correlation constants slightly more than errors giving a smaller radius. Thecorrelation constant errors resulting from radius errors are, however, extremely small. Onthe assumption of a standard deviation Of sm~d = 0.02ram the relative correlation constanterrors have been estimated by FEM. As a positive radius error is applied, the relative errorsare conservatively estimated. The maximum relative error, denoted c,, is
c, = 0.00228 (12.56)
12.7 Error Calculation 169
Consequently, small errors in hole radius have no visible effect on the stress calculation. Asthe absolute size of radius errors is unchanged when a hole-drilling configuration is enlarged,the effect on stress calculation errors is decreased. Hence, an additional benefit of the chosenhole-drilling configuration has been identified.
12.7.4 Correlation Constant Derivation Errors
The correlation constants ai and bij are derived by axisymmetric finite elements with non-axisymmetric loading and strain integration over the strain gauge area based on a polynomialfound by the least square method. In this procedure errors are introduced both as modellingerrors and numerical errors. For error analysis the same relative error is assumed for all thecorrelation constants. The choice is between fully correlated or fully independent correlationconstant errors. As correlation constants are calculated by the same method, fully correlatederrors are assumed in the present analysis.
Consider once again
1-v[a{P} = {p} (12.57)
Introduction of the correlation constant error yields
-+ (+ ci ) [a]{P+6P) = {p} (12.58)
By inversion of Eqs. (12.57) and (12.58) the following is obtained by subtraction:
(6P}= ( n j"A)E[a] -fp)
H (12.59)
Hence, the covariance of {P}, {Q), {T} and any combination can be derived from
Cxvr = +Cc,, (12.60)
by substitution of X and Y and thus the variance of the cartesian and the principal stresscomponents can be derived. By use of a relative error c,, of 2% the error bounds for cartesianstresses have been calculated and are shown in Figure 12.20.
170 Chapter 12. Modified Hole-Drilling Technique
40 (40
cm.,- 02
Fiur 1200 Catsa40esspotdwt ro bud eae ocreaincntn
calclto e0rrors.
Th0 reutn 10ts teserr r se ob ml.A hsisas h aeo h rn
FIghure 1220 Cartesitan streassesmpliotte wit erryorbonsrelated ersisthnon-correlvation cosantcralculationc errors.tl ietderr ilices hesrs aito ewenrmns
The resultingthertesian stressherrorsare seem oba sallslhsi.as h cs fth rn
cipal5 strss error bonsteraesutsr a rre otrshon h sue ro f2 sdfiutt
Only Young's modulus is investigated with respect to stress calculation errors. Errors inYoung's modulus are represented by a scalar in the matrix equations:
CE{PI = {SP} (12.61)
CE is the relative error in Young's modulus. The covariance of the stress variables is givenas
Ex X{} (12.62)
By substitution of X and Y' with P, Q and T, the cartesian and the principal stress compo-nents may be derived. The contribution to stress calculation errors from modulus errors isshown in Figure 12.21 where a 2% error have been used. The contribution is clearly seen tobe insignificant.
12.7 Error Calculation 171
40 4040
1276 0te ro ore
Fniguorean 122:Catsa stressaclainesrpotted withecetict errorbons reateas sued to Youg' mdlso.
12.7.6 Otheral ErrrcSuracenStrs aclto
Anothepreeigscinthdfert error sourceis hole ecetrctehcni o analysed in the presen work eshaje[61]d htasdargud thvatitheseccentricitywerrorsfshouldse 2a -e 3qtiesthoed Holeer raiusherorslaberaie htthey causee errors ofopoie sigtthegl gaugtes of the rosette. As the oe exprrosmereaeunipmpotnt toaistres calcultion exerreeors smal ineetricityror.se are als assumedeciton bellso.
mnathie prceingh sedctin the asmdifferen derror sourcesrhavevbeetanalysed.on thmbasis ofesti
applied a value of 0.0 1mm in his analysis. The standard deviation of measured strain shouldonly be decreased if high-precision equipment and very skilled technicians are available.
In summary, the strain measurement errors and the hole depth errors are seen to be mostimportant to the stress error bounds calculated.
The total error bounds are obtained on the assumption of independent and normally dis-tributed errors, which results in independent normally distributed stress calculation errors.As previously mentioned, the left- and the right-side errors are strictly not independent asthe left-side errors are relative to the stress present. Furthermore, non-linear equations havebeen linearised, and thus it is assumed that the resulting stress calculation errors are nor-mally distributed, which they are not necessarily. On these assumptions the total standarddeviation for a stress component is calculated as
S, = F s (12.63)i=i
172 Chapter 12. Modified Hole-Drilling Technique
The assumed errors used for the error analysis are summarised in Table 12.5. The parametersin the experimental hole-drilling analysis are given in Table 12.2.
Std. strain error 2 x 1-Std. hole depth error 0.04 mmStd. radius error 0.02 murStd. Young's modulus error 2%Std. calibration constant error 2%Interval of confidence 90%
Table 12.5: Std. errors.
The total stress calculation errors are given in cartesian and principal stress components inFigures 12.22-12.23.
-40404
X 4.
o I 2 2 4 S0 I 2 2 4 002 2 4 &
Figure 12.22: Cartesipan residual stressecmonns plotted with error bounds for independent creacorltion between all uncertain variables.
of.. the maximu prnia stres is sent-epoleemnd
12.7 Error Calculation 173
In the hole-drilling configuration the number of stress calculation increments was chosen to befour. If three or five increments were chosen instead, the stress calculation accuracy would,respectively, be improved and reduced. Hence, a compromise between stress calculationaccuracy and stress resolution is made by choosing the number of increments. In his choiceof a number of stress calculation increments for an application the investigator should alsotake the characteristics of the error types into account. The strain measurement errorsresult in stress calculation errors independent of stress, whereas the stress calculation errorsrelated to hole depth errors are relative to the stress. Consequently, the relative size of atotal stress calculation error increases with decreasing residual stress level. Hence, a numberof three stress calculation increments should be considered in measurements of low-levelstresses as the strain error contribution might otherwise compromise the usefulness of themeasurements.
Compared to the incremental hole-drilling methods investigated by other authors, the hole-drilling method developed in the present work offers improved stress calculation accuracy.The improvements follow mainly from the enlarged and optimised hole-drilling configuration,which results in improved correlation between released residual stress and relaxed surfacestrain with depth. Moreover, the size of positioning errors, like hole radius and eccentricityerrors, is independent of the dimensions used in the hole-drilling configuration. Thus, thestress calculation accuracy is improved by use of the proposed hole-drilling configuration.
174 Chapter 12. Modified Hole-Drilling Technique
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Chapter 13
Example of ExperimentalInvestigation
A modified hole-drilling technique has been applied to 10 mm hot rolled, normalised andcold levelled steel plates. The experiments were carried out before finalising the investigationof the modified hole-drilling technique and, in consequence, the experimental investigationhas not been optimally performed. Below, the applied experimental procedure is describedand evaluated against the procedure outlined in Chapter 12. Further, the level and thereproducibility of residual stresses are evaluated.
13.1 Plates and Plate Manufacturing Processes
The investigated plates had the following characteristics:
Plate characteristicsSteel quality AH36, mild steelYoung's modulus 211 GPaPoisson's ratio 0.28Yield stress ý410 MPaLxWxT, 10 x 3m x 0.010mm
Table 13.1: Plate characteristics.
The processes applied to the plate manufacturing are listed sequentially in Table 13.2.
175
176 Chapter 13. Example of Experimental Investigation
Plate manufacturing processes1. Slab heating2. Descaling3. Rolling4. Hot levelling5. Air cooling6. Normalisation7. Air cooling8. Cold levelling9. Edge cutting
Table 13.2: Sequentially listed plate manufacturing processes.
The investigated plates were followed through the production line. Process time, time be-tween processes and temperatures were recorded in order to evaluate differences. Moreover,the levelling parameters were kept constant in order to ensure similar manufacturing con-ditions. Very small deviations in both time and temperature were recorded and hence theoverall manufacturing conditions should facilitate uniform manufacturing.
13.2 Experimental Procedure
The most significant difference between the modified hole-drilling technique described inChapter 12 and the present application is that the measurements were only carried outhalf-way through the thickness of the steel plate, as the necessary theory had not yet beendeveloped. Other differences and factors influencing the accuracy of the measurements arediscussed below.
In the milling procedure different sizes of end mills were successively used until a diameterof 30 mmr was obtained. The diameter increase for the last milling was only 1 mm, whichfacilitated good surface finish and accuracy due to little torque in the milling. However, theprocedure of changing end mills made it difficult to establish an exact reference of depthand an additional error source is therefore introduced. This may be prevented by use of aturning head which will also allow for accurate adjustment to the desired diameter.
The steel plates investigated were uneven and characterised by edge waves. The poor flatnessresulted in undesired stresses introduced by the weight of the milling machine and by inad-equate support in the milling process. When measurements were performed at the corners,the milling machine was positioned next to the plate. But at the centre it was necessaryto move the milling machine around on the plate to find a good position where no or onlyvery low strains were recorded at the gauges. A gap between the ground and the plateresulted in inadequate support during milling and consequently in non-optimal milling con-ditions. Hence, of the five plates produced for the purpose, only three were chosen because
13.2 Experimental Procedure 177
of inadequate support in the areas to be investigated. Measurements performed on the firstplate were omitted from the analysis due to strain measurement fluctuations caused by poorelectric grounding of the strain gauge amplifier.
The experimental procedure applied and the conditions under which the experiments werecarried out have without doubt increased the stress calculation errors. The error sourcesaffected by the above are mainly
" Hole depth errors
* Hole diameter errors
" Strain measurement errors
The standard errors used in the calculation of error bounds in the succeeding analysis aregiven in Table 13.3 and are seen to be somewhat larger than those applied in the erroranalysis in Section 12.7.
Std. strain error 3peStd. hole depth error 0.08 mmStd. radius error 0.04 mmStd. Young's modulus error 2%Std. correlation constant error 2%Interval of confidence 90%
Table 13.3: Std. errors.
The diameter errors can be neglected according to the discussion in Section 12.7.3. The holedepth errors and the strain errors are, however, the major contributors to the total stresscalculation errors and the standard error increase will seriously affect the error bounds.
2
Figure 13.1: Numbering of measurement positions.
Measurements were performed on two plates, five measurements on each. The numberingof positions is shown in Figure 13.1, where the corner measurements are positioned 350 mmfrom the edges. The two plates are identified by their production ID-numbers, 8921 and8925.
178 Chapter 13. Example of Experimental Investigation
13.3 Results
The procedure described in Chapter 12 for residual stress calculation and error analysis isapplied to both plates. The results are given in cartesian stress components according tothe co-ordinate system in Figure 13.1.
Results, plate 8921
" 2 1 1 1
Figure 13.2: Derived cartesian stresses, position C1, plate 8921.
1 0 P p
2 .10 .10 . 10
-40 40
SO .313 -00
1 I g u 4 Derivd 1 0 pl2t3 483
Fio. ~gureo 13.3: Derive breian stesses, positonC2 plate l 8921. .~~oli
Figure 13.3: Derived cartesian stresses, position C2, plate 8921.
13.3 Results 179
S0 0
I ~j : jo
o2 1 4 50 i 2 4 5 0 i 2 3Lftthb -h -r. n-I I 18 ýrI-de- l.s
Figure 13.5: Derived cartesian stresses, position C4, plate 8921.
40 00 ,,0
!-10 Z10
1 2 1 1 1 2 1 1 Q 2 3 4 5
Figure 13.7: Derived cartesian stresses, position CENR, plate 8921.
40 4 4
-71 2 0 0 1
Figure 13.6: Derived cartesian stresses, position CENR, plate 8921.
180 Chapter 13. Example of Experimental Investigation
Figure 13.8: Derived cartesian stresses, position C2, plate 8925.
40 40 40
P t 1 2
I *I I• I I-
Figure 13.10: Derived cartesian1 stresses, position C4, plate 8925.
o 1 2 4 50 I 2 3 4 0 0 1 4 5
Figure 13.18: Derived cartesian stresses, position C2, plate 8925.
0,._ C ,_- " ,' _-
.0 4 .4
Fiue1.17eie atsa tesepsto ETE lt 95
13.3 Results 181
The residual stress levels arising from production of normalised and cold levelled mild steelplates are seen to be low or moderate. F'urther, the stress calculation uncertainty boundsare seen to compromise the calculated results as the possible error range in the 3rd and the4th increment is much larger than the stress values calculated.
If the reproducibility of residual stresses in plate manufacturing is considered, there is nogeneral agreement between stresses measured in the same position at the two plates. Thisconclusion is also valid when the uncertainty bounds are taken into consideration. Thestresses are also seen to vary considerably between corners of the same plate, which indi-cates a non-symmetrical stress state. The above is illustrated by Figure 13.12, showing thecartesian stress component a, for the plates.
30 77
10 2
.2. . -.....
-3 -0
(a) Plate 8921 (b) Plate 8925
Figure 13.12: Comparison of stress components u, in corner positions.
It is clear that a decisive conclusion about the reproducibility of residual stresses in the man-ufacturing of normalised and cold levelled steel plates cannot be based on the investigationof two plates. The results, however, support the discussion in Chapter 9 of the possibilityof categorising steel plates with respect to residual stress. Here it was concluded that dis-similar residual stresses in cold levelled steel plates were likely since unpredictable residualdeformations are transformed into residual stress in cold levelling.
In the stress error analysis a modified set of standard errors was applied due to the deteri-orated experimental conditions. By use of the experimental procedure described in Section12.5.1 on plates with a better flatness, the standard errors assumed in Table 12.5 are reason-able and might even be reduced depending on the equipment available. By comparison of
182 Chapter 13. Example of Experimental Investigation
Figure 13.13 to Figure 13.14 the error bounds are observed to be reduced by almost a factortwo if the reduced standard errors are employed.
40 40 0
. 10 1 01 .Io . .
20 -o -20
-40 -4 i I0 -40o 2 3 4 5 2 4 0 I 2 4 1
Figure 13.13: Applied standard errors (position C3, plate 8921).
,o" ,0 0. '00 o 0
.10 • ~.10; ]
400
-0 2 3 4 0 t 4 0 1 2 4I).pi g Ný. col- f...] lioshý boo o.~m ohb. ,0o 1_1.
Figure 13.14: Standard errors from Table 12.5 (position C3, plate 8921).
An additional reduction of calculation errors can be obtained by reducing the number ofstress calculation increments but it should be noted that the improved accuracy is obtainedat the expense of stress resolution through the plate thickness. Thus, if the experimentsabove were performed in accordance with the procedure outlined in Chapter 12 and thenumber of increments were reduced to three, it would then be possible not only to evaluatethe stress level but also to some extent the variation.
The general conclusion for the experiments is that only low level residual stresses are foundin the hot rolled, normalised and cold levelled steel plate and that these are unlikely to haveany significant influence on cutting and welding distortions.
Chapter 14
Conclusions of Part III1
The objective of the present part of the thesis was primarily to enable residual stress eval-uation through the thickness of large, medium thick steel plates. This objective has beenachieved as sumnmarised below.
A short review of different stress measuring methods showed that none of the existingmethods fulfilled the requirements specified. A modified incremental hole-drilling techniquehas therefore been developed for the purpose.
An analysis of the hole-drilling configuration revealed that considerable improvements couldbe obtained by optimisation of the configuration to plates of finite thickness and a set ofguidelines was established. By application of these guidelines an optimised 'released strain -
hole depth' relation is benefitted from and the stress calculation accuracy is increased. Theanalysis also showed that the theoretical maximum stress resolution depth is approximately75% of the plate thickness, whereas the stress resolution depth obtainable in practice isabout 50-60% of the plate thickness. To this end, an extension of the incremental hole-drilling method has been proposed, allowing the stresses to be resolved through the entireplate thickness.
For evaluation of the method an extensive stress calculation error analysis was performed.The analysis revealed that strain measurement errors and hole depth errors are the dominanterror types anid, consequently, these should be attended to when the method is applied.However, the error analysis also showed that low-level stresses can be compromised by thecalculation errors in the deep interior of the plate. For such applications, the number ofstress calculation increments should be reduced to improve the attainable accuracy.
A discussion of residual stress in connection with plate manufacturing processes indicatedthat cold levelled steel plates would be difficult to categorise according to residual stress. Thisis supported by the experimental findings in Chapter 13 as no general agreement betweenthe cold levelled plates was observed anid as the stress state within each plate appeared tobe non-symmetric. The stress levels measured in the hot rolled, air cooled and cold levelled
183
184 Chapter 14. Conclusions of Part III
steel plates are, however, small and are unlikely to have any significant influence on cuttingand welding deformations.
Fortunately, the major part of the steel plates used in the production has not been coldlevelled. In this case, the primary process influencing the final stress state is believed tobe the cooling process. The small deviations expected concerning the cooling process for agiven product type manufactured at a given steelworks support the possibility of categori-sation. The experimental method developed in the present work allows these products tobe investigated and the influence of residual stress on cutting and welding distortions to beevaluated.
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Appendix A
Details of Stress Error CalculationAnalysis
In the present appendix further details are given on the stress error calculation analysispresented in Section 12.7. The subject is mainly analysis of strain errors and hole deptherrors described in Sections 12.7.1 and 12.7.2, respectively. The following two sections maydirectly be substituted for the above-mentioned and should be read in their context.
A. 1 Details of Strain Errors
The strain errors include instrumentation errors, thermal strains and strains introduced inmachining.
The standard strain deviation is assumed to be
s,= 2 10Q' (A.l)
The standard deviations of the stress variables { PI, { Q} and (T) are calculated in thefollowing. As a first step the matrix equations (11.6)-(11.8) are rearranged. Eq. (11.4) iswritten as
fi p) = $ qcoH{E1J Jt)~ (A.2)
191
192 Appendix A. Details of Stress Error. Calculation Analysis
where the subvector components are the incremental values and the submatrices of the order4 x 16 contain the coefficients correlating the measured strains to the strain variables {p}, {q}and {t}. The matrix equations (11.6)-(11.8) are now written as{{P} 1 fJ{eJ)
{Q} = [Al] { E, } (A.3){T}) {E3J
where
EF, [A]-'[p~q} 1[M] = E [B]-i [q.,1[E [B]-' It.111
Eq. (A.3) expresses the stress variables directly as a function of the measured strains. Thisfacilitates calculation of the covariance matrix for the vector containing P, Q and T L16],which is
Cp-Q = [M] CE [M]T (A.4)
where
{PQT}T = {{pI T, {Q} T , {T} T }
{e)T = {{f,}, {e 2 }T, {f 3 }T}
The covariance of a vector is given as the covariance of the elements in the matrix formed bythe dyadic product of the vector by itself. The covariance matrix CG is given as a diagonalmatrix which contains only the assumed variance of measured strain as all strain measure-ments are assumed to be independent with the same standard deviation. The covariancematrix CPQT will be used later in this section for calculation of the standard deviations ofthe principal stresses.
The covariance matrix of the cartesian stress components is obtained by the same principleas C-P•. The cartesian stresses are expressed in terms of P, Q and T by the relations givenin Eq. (11.9) which results in the matrix equation
{631 = [N] J{E } (A.5){T13}) fF{3})
where
FrE [A]- '[pf] , ' [B]-' [qJ1][N] = /f [Afl' [qj] -1 E [B]-,' [t•,] (A.6)
E [B-' [t 1 ] I
A.1 Details of Strain Errors 193
The covariance matrix then follows as
Ca = [N] C [gN]T (A.7)
The standard deviation of the cartesian stresses is now found by taking the square root ofthe diagonal element of Ca.
The standard deviation has been calculated for the experimental investigation of fixturestress in Section 12.5. The results are shown in Figure A.1.
I 1 1 4 1 2 1 1 4
Figure A.I: Cartesian stresses plotted with error bounds for errors in strain measurements.
The error bounds increase with depth as a consequence of the decreasing correlation between
residual stress and strain relaxation at the surface.
Residual stresses measured by hole-drilling axe usually given by the principal stresses andthe angle of the maximum principal stress measured from a reference direction. Thesecomponents are calculated from the stress variables P, Q and T by means of Eq. (11.10)-(11.11). The relations are non-linear and the method already introduced for calculationof standard deviations is therefore not valid. Instead, an approximate technique based onTaylor expansion will be applied to linearise the expressions (Ditlevsen [16]). A differentiablefunction f of the uncertain quantity X is approximated by
1ii
f(X) 'ý f(E[X]) + f (E[X])(X - E[X]) + ýf (E[X])Var[,X] (A.8)
The variance of f(X) is then
I ar[f(X)] :z f(E[X])'Var[X] (A.9)
Generalisation of Eq. (A.9) to fpoctions of more uncertain quantities results in the followingapproximation to the variance:
194 Appendix A. Details of Stress Error Calculation Analysis
7n n
Var[f(XI, X 2 ,.- •X,)] ~ ffCov•[X, Xj]j=1 j=I
{f'} T Cg{f'} (A.10)
where
U) T = (j, " ,,f,)fj : derivative of f with respect to XK at the mean (E[X1], E[X 2],... , E[X,])
The linearisation technique is applied to the angle and to the principal stresses, which arecalculated from the stress variables P, Q and T by means of Eqs. (11.10)-(11.11). Thestandard deviations are calculated for each increment and the values of P, Q and T aretherefore related to an increment in the following. If the maximum principal stress aU, isconsidered:
U., = P + V (A.11)
{f'} in Eq. (A.10) becomes
\(- ii{,.)= nazQ., ? VEIQ?+EI(Tl2 (A.12)I J[ 7,i nl/ /
7 •E[Q]ý+E[T1,
Cx in Eq. (A.10) becomes the covariance matrix Cp--, which is obtained from Eq. (A.4)by taking out the elements related to the increment of interest. The standard deviation ofthe maximum principal stress in an increment is retrieved by using Eq. (A.10), which yields
s,... = moPQT {O" I (A.13)
The standard deviation of the minimum principal stress is calculated equivalently.
Next, the angle a measured from the reference direction towards the direction of maximumprincipal stress is concerned:
a = 1/2 arctan ( T (A.14)
A.2 Details of Hole Depth Errors 195
As it is only a function of Q and T, {f'} becomes
If' = I 'Q 2(E[TL+EIQI') (A.15)
The Cw are obtained by selecting the relevant elements of CTvr in Eq. (A.4) and thestandard deviation is calculated as
S' = {r'}ITCQT{a'1 (A.16)
Principal stress error calculation is again carried out by use of the fixture stress example inSection 12.5, yielding the results shown in Figure A.2.
a 2 40
Figure A.2: Principal stresses plotted with error bounds for errors in measured strain.
In comparison with the calculated cartesian stress errors, especially the direction of themaximum principal stress is seen to be extremely sensitive.
A.2 Details of Hole Depth Errors
Hole depth errors affect the components of the calibration constant matrices [a] and [b]. Inorder to calculate the standard stress deviation due to depth errors it is necessary to makethe following assumptions.
•Depth errors are non-linearly related to the calibration constants as described bySt. Venant's principle. As the with err, are small the relationship is assumedto be linear.
196 Appendix A. Details of Stress Error Calculation Analysis
e Milling depth errors Ah are assumed to be independent with the same standard devi-ation and zero mean.
A standard deviation sh of 0.04 mm is assumed for the hole depth. The standard deviationsfor the components of [a] and [b] are denoted sa? and sý , respectively. These are conservativelyestimated as the depth is decreased in each increment which results in larger errors than acorresponding depth increase. A hole depth decrease within the i'th increment decreases allthe elements of the i'th row in the calibration matrix and consequently the row elementsare fully correlated. As the depth errors are independent there is no correlation betweenelements in different rows. Thus, the elements aij of [a} are normally distributed stochasticvariables described by
Efaij] = "ij (A.17)Cov[akb,am.] = P-k,,nS• k1Sn (A.18)
1 for k = mPa~ki,a., = 0 else (A.19)
bij is described equivalently whereas the covariance between aij and bij components is de-scribed by
Cov[akz,bmn] = Pab_,,b.Sb (A.20)1 fork=i
al, = 0 else (A.21)
The s* and sb. components are derived by decreasing the depth of one increment at atime. The corresponding aij and bij components are calculated as previously described andsubtraction of the original correlation constants yields the standard deviations given below:
{ 5 a}T = 1 5711, 4 1, 41, 4 1,S 72, ... .8 a (A .22 )
= -{ 79,15,32,28,0,53,25,22,0,0,45,17,0,0,0,28} x 10- 4
{Sb}T = ,sbi ,,sb ,,4,,S12,..-,S4 } (A.23)
= -{82,19,10,6,0,73,9,7,0,0,58,9,0,0,0,41} x 10-4
If the following equation
a = {P} (A.24)
A.2 Details of Hole Depth Errors 197
is considered, the covariance matrix Cý of the matrix [a] must be established in order tocalculate the standard deviation of {P}. The matrix [a] is contracted to a vector {a}:
{a}T = {all, a21, a3l, a4 h, a12 ,. . , a44) (A.25)
Hence, Ca is a 16 x 16 matrix. Ca is calculated using Eqs. (A.22) and (A.18). To facilitatelater matrix operations, the order of C, should not be reduced. The next step is to derivethe covariance matrix of {a] - '. For simplicity the [a] - ' is denoted [u]:
-a0 0 0O•l02
[a] =a02 2 a22 (A.26)010-2232-33 a33
-022033a4 +•02t033a24,+ -22, 304s --02ý .2.4 -- 42+a a 3 -. 43-
011d22033a44 022033244 033044 44
[u] is likewise contracted to a vector denoted {ul
{u} T = {uI , U21, U31, U41, U12,..., u44) (A.27)
To calculate Cý a linearised expression for the covariance of two functions of n variables isneeded. In general notation this is given by
i=1 j=l
{/f'TCg{g } (A.28)
where
f,i and g~j are the derivatives of f and g, respectively, with respect to Xi at the mean{XJ} = (E[XI], E[X2],..., E[X.1).
Thus, in order to calculate the covariance between the individual uij components, the partialderivatives of the {ul with respect to the components of {a} at the mean {J} have beenfound. A matrix with the partial derivatives of {u} with respect to {a} is now established.
U1 1 ,ll U 1I,a2 • . .. U11a441
- (A.29)lu. n44U21,4
.U44,,mH ... ... 1-4
198 Appendix A. Details of Stress Error Calculation Analysis
C, is then given as
C, = [U,,]C 4 [ua]T (A.30)
C,, which is of the order 16 x 16, is divided in submatrices of 4 x 4 to facilitate the calculationof the covariance of {P}. By dividing [u] in column vectors
[u] = [{u1}, {u2}, {U3}, {U4)] (A.31)
C, can be written as
[CUIh CUIU2 CU-M- C91•.
Cr3w1 C 4, 2 Ccr 3 C,, 4
Each submatrix is a 4 x 4 covariance matrix for the respective column vectors. FromEq. (A.24) the covariance matrix Cp of the vector P may now be obtained as
CP = ( ({pErT .Cý.*{p}) (A.33)
where *-multiplication is analogous to row-column multiplication, submatrices are treatedas elements. Hence, the result of Eq. (A.33) is a 4 x 4 matrix (see Ditlevsen [16]).
Next the full covariance matrix Cpyr-- is established. The vector {PQT} is defined as
{PQT}T = { {p}T, {Q}T, {T}T} (A.34)
so that Cp- may be written as
= COP CO C'P (A.35)C7 [CqP CPO C1 J
The first submatrix in Eq. (A.35) has already been calculated as Cp in Eq. (A.33). Thevectors {P), {Q} and {T) may be expressed as
A.2 Details of Hole Depth Errors 199
{P} = E/(1 + v) [u]{p} (A.36){Q} = E [v]{q} (A.37)
{T} = E (v]{t} (A.38)
where
(u] = [al-' and [v] = [bj - '
By application of the same principle as in the calculation of Cp and the symmetry of CTrthe remaining components can be derived as
C O = E 2 {q}T*Co* {q} (A.39)
C T = E2 {tIT*C0 *{t} (A.40)
SCO. = E2 {qlTWC_*{t} (A.41)
CP 0 - 1+- v { *}T C.,0* {q} (A.42)
E 2
Cpt - 1+ v { _}T, C., 0 * (t) (A.43)
where {u} and (v} are the contracted vectors of [u] and [v], respectively. C, is derivedequivalently to C, whereas the cross covariance matrix Cf,, for the matrices (u] and [v] isgiven as
C.ý = [u,.]C•[vb]T (A.44)
Cob is derived similarly to C. by Eq. (A.20). The matrix [v,b]T with the partial derivativesof {v} with respect to (b) is obtained in the same manner as Eq. (A.29).
When the covariance matrix CP-Q-T has been established the cartesian stress components aswell as the principal stress components may be analysed. The covariance of the cartesianstress components is derived by
Ce, = l(Cp+CO+- CpO+COp) (A.45)
C,- = -(Cp + CO - CpO - Cop) (A.46)Cn, = CT (A.47)
in which the variance of the cartesian stress components is found in the diagonal. Applicationof the calculation method to the example of fixture stresses yields the results shown inFigure A.3.
200 Appendix A. Details of Stress Error Calculation Analysis
1 1
[2,0th bl. wlu '%- o~I D hbek. .nttce [ni :l NpbbIo, . .wc f-Irn
Figure A.3: Cartesian stresses plotted with error bounds related to hole depth errors
The variance of the principal stress components is calculated by the linearisation techniquefor the relation between principal components and the stress variables P, Q and T, whichresults in the Eqs. A.16 and A.13 given in Section A.1. In these equations the derived CFis used (Eq. (A.35)). By application of the stress error calculation method to the exampleof fixture stress, the following error bounds are calculated and presented in Figure A.4.
,60 ,0 110
1 4
Figure A.A: Principal stresses plotted with error bounds related to depth errors.
The remaining error sources are treated with sufficient detail in Section 12.7 and are thereforenot presented here.
Appendix B
Representative Depth
In the calculation of residual stresses acting in an increment, uniform stress with depth isassumed. Representing the stress variation with depth, interpolation between incrementalstress values is performed. In the interpolation it is necessary to use a relative representativedepth for each increment defined as z, = h/TI. Half of the increment depth is commonlyused but as described by St. Venant's principle, the representative depth is closer to thesurface.
Based on the stress calculated for a specific increment, the relative representative depth canbe approximated by the FE model previously applied to calculation of calibration constants.In the incremental method the following approximation is used for the depth increment Ah:
Thus, Pi is not the average stress in the increment but the stress equivalent to the actualstresses acting in terms of strain relaxation measured at the gauge. For calculation of therepresentative depth of Pi, the loading used for calculation of aii is divided into n equalparts. Hence, the unit stress is applied in n successive steps and the response at the straingauge is calculated by using the hole depth of the full increment at all times. As the strainresponse is given by
n
a,, a (B. 1)k=1
the relative representative depth za may be calculated as
Ef 2k- akZ. k=1 ý- it S2
r (B.2)
aii
201
202 Appendix B. Representative Depth
Similarly, z' is calculated on the basis of bij. The relative representative depth for theprincipal residual stress components in an increment is given as
- P - - (B.3)p + V 2_T
As z' and Z4 will have very similar values, the dependency of the stress state is limited. Inkeeping with St. Venant's principle, the relative representative depth measured from theincrement start will decrease with the hole depth. This effect is, however, dependent on thehole-drilling configuration as described in Section 12.1.2. If the chosen configuration is usedonly a modest deviation from z, = 0.5 is expected, which is related to the limited variationin strain relaxation with depth. This is supported by the results shown in Table B.1
Representative stress depthsincrement 1 z' = 0.473 zb = 0.475
increment 2 z' = 0.461 zr = 0.460increment 3 z' = 0.453 z' = 0.450increment 4 Z4 = 0.443 z = 0.443
Table B.I: Representative depths in the selected hole-drilling configuration.
PhD ThesesDepartment of Naval Architecture and Offshore Engineering
Technical University of Denmark • Kgs. Lyngby
1961 Strom-Tejsen, J.Damage Stability Calculations on the Computer DASK.
1963 Silovic, V.A Five Hole Spherical Pilot Tube for three Dimensional Wake Measurements.
1964 Chomchuenchit, V.Determination of the Weight Distribution of Ship Models.
1965 Chislett, M.S.A Planar Motion Mechanism.
1965 Nicordhanon, P.A Phase Changer in the HyA Planar Motion Mechanism and Calculation of PhaseAngle.
1966 Jensen, B.Anvendelse af statistiske metoder til kontrol af forskellige eksisterende tilnxrrmelses-formler og udarbejdelse af nye til bestemmelse af skibes tonnage og stabilitet.
1968 Aage, C.Eksperimentel og beregningsmwssig bestemmelse af vindkrmfter pi skibe.
1972 Prytz, K.Datamnatorienterede studier at planende bAdes fremdrivningsforhold.
1977 Hee, J.M.Store sideportes indflydelse pA lan gskibs styrke.
1977 Madsen, N.F.Vibrations in Ships.
1978 Andersen, P.Bolgeinducerede beviegelser og belastninger for skib pi lwgt vand.
1978 R'6meling, J.U.Buling at afstivede pladepaneler.
1978 Sorensen, H.H.Sammenkobling at rotations-symmetriske og generelle tre-diznensionale konstruk-tioner i elementmetode-beregninger.
1980 Fabian, 0.Elastic-Plastic Collapse of Long Tubes under Combined Bending and Pressure Load.
203
204 List of PhD Theses Available from the Department
1980 Petersen, M.J.Ship Collisions.
1981 Gong, J.A Rational Approach to Automatic Design of Ship Sections.
1982 Nielsen, K.Bolgeenergimaskiner.
1984 Nielsen, N.J.R.Structural Optimization of Ship Structures.
1984 Liebst, J.Torsion of Container Ships.
1985 Gjers0e-Fog, N.Mathematical Definition of Ship Hull Surfaces using B-splines.
1985 Jensen, P.S.Stationare skibsbolger.
1986 Nedergaard, H.Collapse of Offshore Platforms.
1986 Yan, J.-Q.3-D Analysis of Pipelines during Laying.
1987 Holt-Madsen, A.A Quadratic Theory for the Fatigue Life Estimation of Offshore Structures.
1989 Andersen, SX.Numerical Treatment of the Design-Analysis Problem of Ship Propellers using VortexLattice Methods.
1989 Rasmussen, J.Structural Design of Sandwich Structures.
1990 Baatrup, J.Structural Analysis of Marine Structures.
1990 Wedel-Heinen, J.Vibration Analysis of Imperfect Elements in Marine Structures.
1991 Almlund, J.Life Cycle Model for Offshore Installations for Use in Prospect Evaluation.
1991 Back-Pedersen, A.Analysis of Slender Marine Structures.
List of PhD Theses Available from the Department 205
1992 Bendiksen, E.Hull Girder Collapse.
1992 Petersen, J.B.Non-Linear Strip Theories for Ship Response in Waves.
1992 Schalck, S.Ship Design Using B-spline Patches.
1993 Kierkegaard, H.Ship Collisions with Icebergs.
1994 Pedersen, B.A Free-Surface Analysis of a Two-Dimensional Moving Surface-Piercing Body.
1994 Hansen, P.F.Reliability Analysis of a Midship Section.
1994 Michelsen, J.A Free-Form Geometric Modelling Approach with Ship Design Applications.
1995 Hansen, A.M.Reliability Methods for the Longitudinal Strength of Ships.
1995 Branner, K.Capacity and Lifetime of Foam Core Sandwich Structures.
1995 Schack, C.Skrogudvikling af hurtiggdende fwrger med henblik pA sodygtighed og lav modstand.
1997 Simonsen, B.C.Mechanics of Ship Grounding.
1997 Olesen, N.A.Thrbulent Flow past Ship Hulls.
1997 Riber, H.J.Response Analysis of Dynamically Loaded Composite Panels.
1998 Andersen, M.R.Fatigue Crack Initiation and Growth in Ship Structures.
1998 Nielsen, L.P.Structural Capacity of the Hull Girder.
1999 Zhang, S.The Mechanics of Ship Collisions.
1999 Birk-S0rensen, M.
Simulation of Welding Distortions of Ship Sections.
206 List of PhD Theses Available from the Department
1999 Jensen, K.Analysis and Documentation of Ancient Ships.
2000 Wang, Z.Hydroelastic Analysis of High-Speed Ships.
2000 Petersen, T.Wave Load Prediction-a Design Tool.
2000 Banke, L.Flexible Pipe End Fitting.
2000 Simonsen, C.D.Rudder, Propeller and Hull Interaction by RANS.
2000 Clausen, H.B.Plate Forming by Line Heating.
2000 Krishnaswamy, P.Flow Modelling for Partially Cavitating Hydrofoils.
2000 Andersen, L.F.Residual Stresses and Defornations in Steel Structures.
.19,
I.1
46)
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