-
on
Ger
Carreau uid
theFWredeldinstnted
Lagrange bulk viscosity term, a purely displacement based form
of the numerical solution procedure is
eldingng a wg theeachedconve
tion of stable and suitable process parameters and effective
feed-
necessity therefor is the revelation of the physical phenomena
dur-ing the process. Hence, the demand of predictive tools in
modelingthe transactions during RFW being able to understand and
opti-mize the process is of crucial interest for industrial
applications.
The main process parameters are illustrated in Fig. 1 being
theaxial force FA, the torque T, the rotary speed n and the axial
feed sf .Therein the shaft A is the static part and shaft B is the
rotating one
this process have been identied being the heat generation,
heatfriction surfacesce [18]. Incts rst at
In 1995, Stokes modeled the molten lm of frictionpolymers by
introducing a temperature and shear dependecosity [20]. Bendzsak
tried to capture the ow regime of frictionwelds by solving the
NavierStokes equation directly [2]. Thereinthe focus has been set
on the ow traces within aluminium frictionwelds. Moal et al.
proposed a fully coupled viscoplastic model [14]for the simulation
of Inertia Friction Welding (IFW). While a Nor-ton power law was
used for the deviator stress calculation, a purelyvelocity
dependent friction law was implemented for contact forceand heat
generation prediction. In order to map the isochoric,
Corresponding author. Tel.: +49 391 67 12679; fax: +49 391 67
11379.
Comput. Methods Appl. Mech. Engrg. 265 (2013) 186194
Contents lists available at
A
.eE-mail address: [email protected] (D. Schmicker).back
control algorithms. A perseverative problem, for instance, isthe
identication of an optimal control of a position based
frictionwelding process, where for reasons of nish worked parent
parts anal axial shortening during welding has to be achieved.
Basic
conduction, plastic deformation, abrasion of theand the
diffusion of the weld partners, for instanoping simulation models
to cover all these effecan be dated back to the mid 90s.0045-7825/$
- see front matter 2013 Elsevier B.V. All rights
reserved.http://dx.doi.org/10.1016/j.cma.2013.06.007devel-tempts
weldednt vis-Moreover, neither ller material nor any inert gas
is needed forjoining providing a very efcient and uncontaminated
fuse of theweld. Benets are the high quality and symmetry of the
joint, thegood potential for process automation and the short cycle
times.In contrasts to all these virtues the main problem is the
identica-
mentioned nal shortening of the shafts lengths. In Fig. 2
qualita-tive runs of the process parameters are displayed [8]. The
threestages therein are the rubbing stage (I), the breaking stage
(II)and the forging stage (III).
Already in the 1970s the main driving physical phenomena of1.
Introduction
The process of Rotary Friction Wfusing process allowing for
combinimaterials. Due to the fact that durinmaterials melt
temperature is not rzone of the weld is smaller than atapplicable.
A penalty contact approach together with a regularized friction law
allow for an efcient deter-mination of the interface pressure and
friction forces. Themodels performance is demonstrated bymeansof a
test problem highlighting the potential of adopting it to
industrial applications later on.
2013 Elsevier B.V. All rights reserved.
(RFW) is a solid-stateide range of dissimilarwelding procedure
the[18], the heat affectedntional fusion welding.
inducing the frictional heat on the contact surface. Basically,
twokinds of drives are used in RFWmachines, either a y wheel
servesas an energy storage being referred to as Inertia Friction
Welding(IFW) or a continuously driven engine generates the power
ofwelding being known as Direct Drive Friction Welding
(DDFW).Concerning DDFW usually the rotary speed time line is given
whilethe axial force is feedback controlled in order to achieve the
above-Rotary Friction WeldingNon-Newtonian uid modeling
robust simulation of the process and the physical palpability of
the utilized material parameters. Using aspecial element
formulation on the general axisymmetric frame in conjunction with
an AugmentedA robust simulation of Direct Drive FrictiCarreau uid
constitutive model
D. Schmicker , K. Naumenko, J. StrackeljanInstitute for
Mechanics, Otto-von-Guericke University of Magdeburg, 39106
Magdeburg,
a r t i c l e i n f o
Article history:Received 14 April 2013Accepted 19 June
2013Available online 13 July 2013
Keywords:Welding simulation
a b s t r a c t
The subject of this paper isDrive FrictionWelding (DDthe
consistent and stable padopting Rotary FrictionWThe Carreau type
uid co
ature dependency is accou
Comput. Methods
journal homepage: wwwWelding with a modied
many
presentation of a holistic, fully-temperature-coupled simulation
of Direct) based on themodied Carreau uidmodel. Themainmotivation
therein isiction of suitable process parameters, which is still the
major problem inng (RFW) processes to new pairs of weld partners at
industrial applications.itutive equation is derived from a
NortonBailey law wherein the temper-for by a JohnsonCook power
approach. Its main benet is the stable and
SciVerse ScienceDirect
ppl. Mech. Engrg.
l sevier .com/locate /cma
-
s of
D. Schmicker et al. / Comput. Metho AppFig. 1. Parameterplastic
deformations, an augmented variational formulation wasused and the
elements were formulated in the r2-z space. Extensiveremeshing of
the Lagrangian mesh allowed for a complete simula-tion of the
process yielding plausible results of the ash evolutionand the
process parameters. DAlvise et al. presented simulationresults of
dissimilar weld parts [7]. The implemented friction lawhas been
adopted by Moal with the slight change, that a certaintemperature
is introduced at which the friction law turns from aclassical
Coulomb law into a thermo-dependent viscous one. In2002 Ulysse
suggested a Non-Newtonian uid law linked with aSheppardWright
viscosity approach1 for the simulation of FrictionStir Welding
(FSW) [21]. While the simulated temperature eldstend to be over
predicted, parametric studies like the inuence ofthe tools speed on
its forces have been successfully conducted withthis model. Wang
made further suggestions for estimating the heatinput at IFW, one
of which is known as Energy-Input-Based method[22]. Here, the
modeling of a frictional law is circumvented byneglecting the twist
deformation and implementing a thermody-namically consistent
heat-source being ruled by the energy decayof the y-wheel. The
disadvantage of this approach is that the heatsource is rather an
input than an output quantity of the modelwhich means that it is
not predictive. Another member of this classof model has been
pursuit by Grant [11]. Also here the process
Fig. 2. Parameter runs of the Direct Drive Friction welding
process.
1 It shall be mentioned that the SheppardWright viscosity is
technically speakinga Garofalo uid law [9], with the difference
that the material parameters therein aredened in a slight different
way.ds,parameters and properties cannot be predicted, but residual
stressesare able to be calculated which has been shown to be in
good agree-ment with measurement results. In 2010 Hamilton
published aframework for mapping the FSW process [12] wherein a
classicplasticity law incorporating elastic deformations has been
imple-mented as constitutive equation. In this context the
temperaturedependency has been taken into account by a JohnsonCook
powerapproach [13]. More recently, Chiumenti proposed a model for
thesimulation of FSW incorporating an
Arbitrary-Lagrangian-Formulation(ALE) in order to resolve the large
plastic distortions [5]. A mixedvelocitypressure formulation being
stabilized by the OrthogonalSubgrid Scale (OSS) technique accounts
for non-volumetric lockingwhile using lower order nite elements. A
classical NortonHoff law and a SheppardWright model have been
suggested asthe constitutive relation.
Basic ambition of our approach in modeling DDFW is to
increasethe robustness and efciency of the simulation, and to
express thematerial parameters in a more palpable way. We believe
that forthe sake of a robustness increase the material law needs to
incor-porate a steady transition from the Non-Newtonian solid to
theNewtonian liquid phase, which is not accounted for in prior
mod-els. Moreover, due to the lack of experimental data for the
inelasticmaterial behavior in wide strain-rate and temperature
ranges, thesimulation should predict the welding process parameters
withcoarse values of material properties. Therefore, the
constitutiveequation, the response functions of stress and
temperature as wellas material parameters have to be formulated in
an easy under-standable and accessible way.
Our constitutive model is based upon a Carreau uid law
featur-ing the solidliquid phase transition by means of a constant,
New-tonian viscosity of the materials cast. It is derived from
Nortonspower law [5,1416] and formulated in terms of the
materialsyield limit. The temperature dependency of this measure is
linkedto a JohnsonCook power law [12,13], enabling the pursued
possi-bility of assessment of the plastic ow behavior even if
hardly anyexperimental data is at hand. A distinct time integration
schemewith adaptive step size regulation, in conjunction with
remeshingprocedures of the Lagrangian formulated nite elements
ensure avery robust and smooth simulation of the DDFW process.
the RFW process.
l. Mech. Engrg. 265 (2013) 186194 187The structure of the paper
is as follows: We start with the for-mulation of the constitutive
model that reects the non-linear uidbehavior for a wide stress and
temperature range. Then we presentthe local and the weak forms of
governing equations. Spatial dis-cretization techniques including
the nite element formulationand the remeshing procedure are
discussed. After outlining the uti-lized contact approach, the time
marching scheme is depicted. Abenchmark example is introduced,
demonstrating the performanceand capacities of the model. Finally
the conclusions are drawn withregard to efciency and accuracy of
the proposed approaches.
1.1. A remark on notation
Scalars are expressed using simple non-boldfaced letters liker;
J2D and n, for instance. Tensorial expressions of higher order
than
-
3 _e0l0 _e00 1 1
Appzero are stated as boldfaced symbols like in x; er ;r and D.
Theircomponents according to a vectorial basis as well as nodal
valuesand scalars being arranged to matrices are denoted by
blackboardbold symbols as in M;v;O and Acap. Domains are marked by
calli-graphic letters like Bt or B0, for instance.
2. Constitutive modeling
2.1. Viscoplastic material model
The constitutive law is based on a NortonBailey potential ofthe
form
W r0 _e0n 1
rvMr0
n11
where r0 is the materials yield limit, _e0 the plastic strain
rate corre-sponding to the yield limit r0 and n the Norton-exponent
[15,16].The law is a classic J2-material law since the potential
only dependson the second invariant of the deviatoric part of the
stress tensor,that is to say on the von-Mises equivalent
stresses
rvM 32s s
r
3J2D
p2
Consequently, the inelastic strain rate is
D @W@r
32
_e0rvMr0
n srvM
3
where r is Cauchys stress tensor, s its deviatoric part
s devr r 13trr1 4
and 1 the second rank unit tensor. Inversion of (3) gives the
morecommon stressstrain relation
s 23r0_e0
_evM_e0
1nn
D 5
with the von-Mises equivalent strain rate
_evM 23devD devD
r6
The full stress tensor is then given by
r lB trD1 2ldevD: 7involving the abbreviation for the shear
viscosity
l _evM ;H r0H3 _e0_evM_e0
1nn
8
The spherical part of r determines the pressure state of the
materialwhich in our model is assessed by an Augmented Lagrange
methodinvolving a bulk viscosity lB. The choice of this measure is
basicallya trade-off between the strictness of keeping the volume
constraintand the condition corruption in solving the system of
equations la-ter on. The temperature dependency in Eq. (8) is now
incorporatedby linking the yield limit r0 to a JohnsonCook power
ansatz of theform
r0H r0;R 1HHRHMHR
mh i; H < HM
0; HP HM
8 e 28
acts as a trigger for remeshing corresponding to a certain
thresholdvalue e. In our simulation the threshold value was chosen
to bee 0:5, yielding reasonable results on the remeshing time
in-stances. In transferring the state variables from one mesh to
theother the element shape functions are used for
interpolation.
5. Contact modeling
The realistic simulation of the DDFW process inevitably
dependson the implementation of the contact and friction model. For
thesake of reasonable computation times a simple penalty
contactapproach has been utilized in our model, being illustrated
inFig. 5. Therein the two weld partners as well as the
self-contactof the ash with the workpiece are allowed to penetrate
eachother. Hence, the pressure p at the surface Gauss-points
isgoverned by the intersection depth s according to the
function
p 0; s < 0aps2; sP 0
29Fig. 5. Penalty contact approach.
-
The penalty term ap can be chosen in regard to the dimension
ofthe workpiece and the to be expected contact pressures.
Forinstance when saying that a penetration of 0:1 mm is
acceptableand typical contact pressures are about 60 MPa, then a
good choicefor ap is 6000 MPa=mm2. The tangential stresses szu obey
anCoulomb friction law of the form
the state variables at the proceeding time step both equations
can behandled separately according to their individual best suiting
timemarching scheme, without the necessity of implementing a
loopbetween the two subproblems (cf. [5]). Let us assume that at
acertain time t0 the conguration is given by the state variables
v0and O0. Then rstly, the mechanic subproblem is integrated
explic-itly. Therefore the nodal force balance evaluated at t0
0 fextt0;v0; _v0 fficv0; _v0 fint _v0;O0 35is solved in terms of
_v0 and the conguration at the end of the timestep t t1 is
calculated by a forward Euler differencev1 _v0 Dt v0 36involving
the time step size Dt t1 t0. Secondly, the thermody-namic
subproblem is solved implicitly by a backward Euler strategy.Thus,
with putting
_v v1 v0Dt
; _O O1 O0Dt
; v v1 and O O1 37
into (34)2 and solving the equation in terms of O1 the
temperaturesat the end of the time step are obtained. The time step
size Dt of thisprocedure is adaptively controlled by a standard
step size regula-tion algorithm being described in [19], resulting
in a very robustand stable temporal discretization scheme for the
simulation ofthe process.
7. Simulation results
In this section the model will be demonstrated in simulating
thewelding process of a forge part being illustrated in Figs. 6 and
7.Therein, the focus lies on the qualitative demonstration of
the
D. Schmicker et al. / Comput. Metho Appl. Mech. Engrg. 265
(2013) 186194 191szu gp Dvu > 0< gp Dvu 0
30
where Dvu is the relative velocity difference of the contact
inter-faces of the two workpieces and g is a temperature dependent
fric-tion coefcient. Since, Eq. (30) involves a strong singularity
at thetransition from slipping to sticking condition the
regularizedapproach
szu 2gpp arctanDvuareg
31
involving a regularization factor areg is used instead. In fact,
the sim-ulation results reveal that during the friction phase the
sticking con-dition is basically never met, hence the
regularization is more offormal nature, not really entering the
model. The heat source atthe contact interface of workpiece A and B
is then given by
rq;A aAaA aB szuDvu aABHB HA and
rq;B aBaA aB szuDvu aABHA HB32
Here, the thermal diffusity parameters
aA jc;AqAcAand aB jc;BqBcB
33
are used for portioning the frictional heat onto the two weld
part-ners following the suggestions of Chiumenti [5]. The heat ux
be-tween the two surfaces is incorporated by the heat
transmissioncoefcient aAB.
6. Temporal discretization
Since elastic wave propagation phenomena are of
subordinatenature at the process of RFW, inertia effects can be
neglected.Hence, the problem will be treated in a quasi static way.
Fictitiousforces, following from the formulation of the governing
equationusing cylindrical coordinates, are able to be accounted for
never-theless. The coupled to be solved differential equation takes
theform
f t; z; _z 0 fextt;v; _v fficv; _v fint _v;OAcap _OAconO qint
_v;O qextv; _v;O
" #
34involving the state vector z v;OT , consisting of the nodal
loca-tion and temperature values. The system is solved by dividing
prob-lem (34) into its mechanic and thermodynamic subproblem.
Since,the sensitivity of the temperatures on the nodal force
balance iscomparatively low, a decoupled treatment of the two
subproblemson the time step level is conductible.4 That is to say,
that for gaining
4 The temperatures enter the linear momentum balance only due to
the temper-ature-dependent viscosity of the viscoplastic material
law. Since for usual materialsthis dependency is of very steady
nature exhibiting no singularities (see Eq. (9) forinstance), the
impact of the extent of temperature change during one time step on
thesolution of the mechanic subproblem is rather marginal. Hence,
an isothermaltreatment of the nodal force balance is employable.
When the velocities anddisplacements at the end of the time step
can be determined only by means of the
rst sub problem at constant temperature, they can be regarded as
given whensolving the thermodynamic subproblem. Hence, at the time
step stage bothsubproblems can be treated separately.dsFig. 6.
Friction weld sample serving as a simulation example.Fig. 7.
Observed friction welding problem.
-
tentials and the performance of the model without
claimingantitative exactness of the material properties and
boundarynditions. The simulation is performed on basis of a
self-writtenite element code within MATLAB. The employed material
con-nts are comprehended in Table 1. The considered process con-ts
of only one rubbing stage at p 40N/mm2 and800 min1 for a friction
time tr 7 s. The weld partners areilar, offering the possibility of
cutting the computational effortthe half by modeling only one side.
However, one has to keep inind that this step is only admissible if
inertia forces including thetitious ones are neglected and the heat
transport coefcient asurfassumed to be independent of the rotary
speed. The presentmputation example conforms these requirements,
hence thehavior of the rotary side is mapped by a symmetry
boundary
condition. The meshing of the stationary side is performed with
anaverage element size of de 0.4 mm at the contact surface.
Theaugmented Lagrange bulk viscosity is assessed to 1:175 106
Ns/mm2 being a suited compromise between satisfying the
volumeconstraint and keeping the system of equations well
conditioned.The penalty parameter for the calculation of the
contact forces isap 1000 N/mm4. For regularization of the friction
shear stressesa factor of areg 200 mm/s is taken into account.
During the simu-lation 103 remeshing steps have been performed.
Therein, theaverage amount of elements was about 600 resulting in
approxi-mately 4000 displacement- and 1300 temperature degrees of
free-dom. In Fig. 8 the deformation shape at the end of the process
att 8 s is compared with the actual ash formation. A
qualitativeagreement between the ashes can be identied. In Fig. 9
the tem-perature eld at the end of the rubbing stage at t 7 s is
displayed.It is found out that the materials melt temperature is
not reachedat no time during the simulation. In fact, it can be
easily reasonedthat during the process of RFW the melt temperature
is very unli-kely to be reached or exceeded neither in simulation
nor in prac-tice. The explanation therefor is that in the vicinity
of the meltpoint the material is owing and not able to transmit
shear stres-ses anymore.5 Hence, once the temperature gets as high
as the meltpoint the rubbing surfaces will stick and frictional
heating will sus-pend avoiding the temperature to exceed the melt
point. Lookingcloser at the contact pressure condition in Fig. 10
it can be seen thatat the beginning the axial force is distributed
evenly across the cross-
Table 1Utilized parameters of the simulation example.
Parameter Value Parameter Value
r0;R 235 N/mm2 HM 1420 C_e0 1 s1 q 7.85 g/cm3
m 1.5 c 460 J/kgKn 5 k 50W/mKl0 108 MPas asurf 100 W/m
2K
l1 102 MPas g 0.3HR 20 C
Fig. 8. Comparison of the ash formatio
2
192 D. Schmicker et al. / Comput. Methods Appl. Mech. Engrg. 265
(2013) 1861942530350
2
4
6
8
10
12
14
16
z [m
m]
500 C
600 C
700 C
950 C
1000 C
1100 C 1200 C 1300 C poquconstasisn simtomciscobeFig. 9.
Temperatures immediateln between model and experiment.
0510150
Temperature [C]r [mm]
y after the rubbing stage at t 7 s.
-
section. However, as soon as the material starts to ow a
distinctpressure prole develops reaching its maximum in the middle
ofthe cross section at r 17 mm. Comparison of this prole with
thetemperatures over the contact surface in Fig. 9 reveals that the
max-ima of temperature and pressure do not coincide. Whereas the
pres-sure is highest near the inner diameter at r 17 mm, the
maximumtemperature is reached at r 22 mm. The unsteadiness of the
con-tour lines in Fig. 10 is due to the remeshing time instances
duringthe simulation which slightly alters the contact force
distribution.Fig. 11 displays the nal twist of the material. It is
clearly seen thatthe largest plastic deformations during the
process do not lie within
the r zplane but in tangential direction. Hence, they should
notbe neglected in modeling RFW. In Fig. 12 the power input and
outputduring the process is contrasted with each other. When the
weldstarts twisting an increasing part of heat power is generated
by plas-tic dissipation. The loss of energy due to thermal emission
and con-vection reaches its maximum at the end of the rubbing
stage. InFig. 13 the process parameter runs are shown. Since at
this examplethe process is force controlled, the left two diagrams
namely the ax-ial force and the rotary speed are input quantities,
whereas the righttwo diagrams are simulation results. The nal upset
of the welded
Fig. 10. Contact interface pressure evolution during the
process.
Fig. 11. Twist of the weld at t 8 s.
25
30Plastic dissipation work
work
Con
D. Schmicker et al. / Comput. Methods Appl. Mech. Engrg. 265
(2013) 186194 1930 1 2 3-10
-5
0
5
10
15
20
P [k
W]
Friction t
Fig. 12. Energy prol4 5 6 7
vection and radiation heat [s]
e of the process.
-
0 2 4 6 80
200400600800
n [m
in-1
]0 2 4 6 8
0
200
400
T [N
m]
ter
194 D. Schmicker et al. / Comput. Methods Appl. Mech. Engrg. 265
(2013) 186194part is 12.8 mm which is about the shortening of the
referencesample.
8. Conclusions
In the current paper a holistic model approach based on a
Car-reau type uid constitutive model for the simulation of
DirectDrive Friction Welding (DDFW) is presented. The advantage
ofthe proposed material law is the formulation in terms of the
yieldlimit such that its material parameters can be easily
identied.Having in mind that in practical applications experimental
dataare usually available for narrow stress and temperature
ranges,the current model is able to give qualitative predictions by
meansof quite fundamental parameters like the melt temperature
andthe yield limit at room temperature, for instance. Moreover,
thepaper summarizes a framework for a stable, efcient and
robustsimulation of the friction welding process, featuring a
steady tran-sition from solid to liquid material behavior when
passing the melttemperature. With the implemented time integration
and niteelement procedure, the whole process can be simulated in
lessthan one hour on a usual personal computer as the presented
testproblem reveals. As a next step it is planned to extend the
model todissimilar welds, which broadens the effort in contact
mapping.Moreover, a coupling of the model with the diffusion
differentialequation is pursued. Parallel to these model extensions
we workon a quantitative validation of the model and determination
guide-
t [s]
0 2 4 6 80
20
40
60
t [s]
F A [k
N]
Fig. 13. Process paramelines for the involved material constants
in the friction weldingprocess itself. Their results will be
communicated in a forthcomingpaper.
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A robust simulation of Direct Drive Friction Welding with a
modified Carreau fluid constitutive model1 Introduction1.1 A remark
on notation
2 Constitutive modeling2.1 Viscoplastic material model
3 Weak formulation of the governing equations3.1 The principle
of virtual power3.2 The principle of virtual temperatures
4 Spatial discretization4.1 Formulation in cylindrical
coordinates4.2 Finite element formulation4.3 Remeshing
procedure
5 Contact modeling6 Temporal discretization7 Simulation results8
ConclusionsReferences