-
A Coupled Multiscale Model of Texture Evolution andPlastic
Anisotropy
J. Gawad∗,†, A. Van Bael∗∗,‡, S. K. Yerra∗∗, G. Samaey∗, P. Van
Houtte∗∗ andD. Roose∗
∗Department of Computer Science, Katholieke Universiteit Leuven,
Belgium†University of Science and Technology AGH, Poland
∗∗Department of Metallurgy and Materials Engineering, Katholieke
Universiteit Leuven, Belgium‡Limburg Catholic University College
(KHLim), Belgium
Abstract. In this paper we present a multiscale model of a
plastic deformation process in which the anisotropy of
plasticproperties is related to the evolution of the
crystallographic texture. The model spans several length scales
from the macro-scopic deformation of the workpiece to the
microscale interactions between individual grains in a
polycrystalline material. Themacroscopic behaviour of the material
is described by means of a Finite Element (FE) model. Plastic
anisotropy is taken intoaccount in a constitutive law, based on the
concept of a plastic potential in strain rate space. The
coefficients of a sixth-orderFacet equation are determined using
the Taylor theory, provided that the current crystallographic
texture at a given FE inte-gration point is known. Texture
evolution in the FE integration points is predicted by an ALAMEL
micromechanical model.Mutual interactions between coarse and fine
scale are inherent in the physics of the deformation process. These
dependenciesare taken into account by full bidirectional coupling
in the model. Therefore, the plastic deformation influences the
crystallo-graphic texture and the evolution of the texture induces
anisotropy of the macroscopic deformation. The presented
approachenables an adaptive texture and yield surface update scheme
with respect to the local plastic deformation in the FE
integrationpoints. Additionally, the computational cost related to
the updates of the constitutive law is reduced by application of
parallelcomputing techniques. Suitability of on-demand computing
for this computational problem is discussed. The
parallelisationstrategy addresses both distributed memory and
shared memory architectures. The cup drawing process has been
simulatedusing the multiscale model outlined above. The discussion
of results includes the analysis of the planar anisotropy in the
cupand the influence of complex deformation path on texture
development. Evolution of texture at selected material points
isassessed as well.
Keywords: multiscale modelling, texture evolution, multilevel
model, sheet forming, finite elementPACS: 02.70.-c, 46.15.-x,
46.90.+s, 62.20.F-, 81.40.Lm
INTRODUCTION
The crystallographic texture has a significant influence on the
mechanical and physical properties of sheet productsas it causes
plastic anisotropy. Obviously, this should be taken into account
during the simulation of sheet formingprocesses with the Finite
Element (FE) method. The evolution of the texture occurring in the
deformed material shouldbe reproduced in the plastic anisotropy of
the material. An appropriate and reliable constitutive model for
the materialshould involve the updating of the constitutive law and
must correspond to the quantitatively accurate prediction ofthe
texture. This is a challenging task, since phenomena at multiple
scales are involved in the physics of the process.These
requirements make the entire task complex and computationally
costly. Moreover, one has to decide whetherthe evolution of
material properties in the FE simulation should either occur
continuously (i.e. in every time step) orwhether it can be deferred
to selected update temporal points.
The current paper exploits several concepts proposed previously
by Van Houtte and co-workers. For the sake ofclarity, some of the
ideas are briefly recapitulated in the paper. For a detailed
description, the reader is referred to theoriginal papers: the
Facet method is comprehensively discussed in [1] and the Advanced
LAMEL (ALAMEL) modelis presented in [2].
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MULTISCALE MODEL
The macro-scale part of the model presented in this paper
follows the idea of the Finite Element method with ananalytical
macroscopic yield locus. There are many successful examples in the
literature that an accurate analyticalexpression can describe the
plastic anisotropy of the polycrystalline. Such yield locus model
can be identified byevaluating a micro-scale model, e.g. one based
on the Full Constraint (FC) Taylor assumption [3], a
self-consistentmodel (e.g. [4]) or a more advanced one, such as the
ALAMEL model used in this paper. Thus, a strong connectionbetween
microscopic phenomena and macroscopic material response is
introduced. This link persists even if themicroscale properties of
the material are assumed to remain constant. However, such
formulation does not permita reliable description at the crystal
level. Another, completely different approach discards entirely the
analyticalyield locus expression and makes use of the
Representative Volume Element (RVE) formulation, owing to
thehomogenisation theory.
We take the aforementioned issues into account in a new
multiscale model of a deformation process presented in thepaper. In
this model, the phenomena occurring at the multiple length scales
are mutually dependent. The anisotropyof plastic deformation
contingents upon the evolution of crystallographic texture and vice
versa. The FE mechanicalmodel of cup drawing process is considered
as a macroscopic description of the deformed workpiece. The
analyticalconstitutive law describing the plastic anisotropy, based
on the concept of a plastic potential in strain rate space,
isimplemented into the commercial FE code ABAQUS/Explicit.
The plastic anisotropy of textured polycrystalline material can
be described by means of a plastic potential in strainrate space.
Recently, Van Houtte et al. [1] proposed the Facet method that
utilises a homogeneous polynomial todescribe the plastic potential.
For materials without strain rate sensitivity, the deviatoric
stress tensor S derived fromthe plastic flow stress can be
calculated for a given strain rate tensor d as:
S =∂ψ(d)
∂d, (1)
In the Facet method, the plastic potential ψ(d) in the strain
rate space is expressed using the function:
ψ(D) = [Gn(D)]1n , (2)
where Gn(D) is a homogeneous polynomial of degree n and n is an
even natural number greater than 1. The Facetmethod exploits the
property of the strain rate tensor d that it can be converted into
a vector D in a five-dimensionalspace. Consequently, the Facet
equation is given by:
Gn(D) =K
∑κ=1
λκ (Sκ,pDp)n , (3)
where D is a five-dimensional strain rate vector, Dp, p = 1 . .
.5 are the components of the strain rate vector, λκ and
thecomponents of 5D deviatoric stress vectors Sκ,p, p = 1 . . .5,κ
= 1 . . .K are parameters. Einstein summation conventionis used for
the index p. Provided that λκ ≥ 0 for all κ and n is a positive
even number, it can be proven that theequipotential surface as
defined by Equation (2) and Equation (3) is always convex [1].
The identification of the parameters in Equation (3) requires
numerous evaluations of the stress vectors for thecorresponding
nearly equidistant strain rate modes. This can be done by means of
the FC Taylor or a more advancedmicromechanical model. Previous
experience [1] shows that an acceptable approximation of the
equipotential yieldsurface requires a degree n≥ 6 and at least 402
strain modes. The amount of strain modes evaluations can be
reducedby a factor of two if the stress differential effect need
not be accounted for.
For the purpose of the parameter identification, it is suitable
to rephrase the Equation (3) in matrix form. Theparameters λκ can
be determined by solving the following system of equations:
[W ]{λ} ∼= {C} , (4)
where the vector C selects the equipotential surface and the
components of the matrix Wi j = (Di ·S j)n are identifiedas the
homogeneous n-th degree polynomials of the plastic work dissipated
per unit volume in function of themacroscopic plastic strain rate
D. The 5D deviatoric stress vectors S j correspond with the D j
strain modes for whichthe micromechanical model calculates the
plastic work. The system (4) is solved using the Non-Negative Least
Squares(NNLS) algorithm [5], which ensures that all components of λ
are positive or zero.
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Algorithm 1 The pseudo-code of the texture updating algorithm
implemented in ABAQUS user subroutineRequire: D is calculated for
every integration point
1: for all integration points do2: P← P+Ddt3: if criterion for
updating is satisfied then4: Subject the texture to deformation P .
run the ALAMEL5: P← 06: for all strain modes do7: calculate stress
tensor S . run the FC Taylor model8: end for9: Store the checkpoint
data
10: Calculate λ coefficients by solving system (4) . identify
the Facet model11: Commit the updated constitutive law12: end if13:
end for
Texture evolution in the FE integration points is predicted by
the ALAMEL micromechanical model [2]. TheALAMEL belongs to the
class of statistical models for the plastic deformation of
polycrystalline materials. Themodel follows the Taylor assumption
that plastic deformation must be homogeneous for thin regions
situated at bothsides of the grain boundary. However, one of the
advantages of the ALAMEL over the classic FC Taylor approachis
related to the stress equilibrium conditions imposed at the grain
boundaries in the simulated microstructure. Itwas shown in [2] that
the qualitative predictions of the texture are more accurate than
those obtained using the FCTaylor model. The ALAMEL represents the
state of the crystallographic texture by means of a discrete form
of theorientation distribution function (ODF). The crystallographic
orientations are expressed using three Euler angles inBunge
notation [6]. Triclinic class of sample symmetry must be imposed to
the original ODF because the initialorthorhombic symmetry is
expected to disappear due to the texture updating.
In this work, we assume that the texture may evolve
independently in every FE integration point. This evolution issplit
into intervals between the updating events. An overview of the
updating method is presented in Algorithm 1. Inorder to decide
whether such event has to be activated, the history of deformation
is being tracked. The state of thematerial is described by means of
the accumulated plastic strain:
Pi =∫ ti
ti−1Ddt, (5)
where i is the ordinal number of the updating event, D is the
plastic strain rate as calculated by the FE in the
integrationpoint, ti−1 is the time of the previous updating event.
The time intervals |ti− ti−1| can be imposed either explicitlyor
determined on-line using a more elaborated criterion. In this work
we utilise the following predicate function todetermine whether the
texture-related material properties should be updated:
f (Pi) ={
1 : ||Pi|| ≥ Pcr0 : otherwise , (6)
where Pcr is a parameter. The criterion is defined as satisfied
if the predicate (6) results in value 1. The activationof the
criterion (6) is considered as a necessary condition for
modification of the local texture representation. Theaccumulated
plastic deformation Pi is passed to the ALAMEL model, which applies
the appropriate lattice rotationsto the crystals belonging to the
virtual microstructure. Afterwards, the updated texture data are
used as an inputfor the reconstruction of the plastic potential
(2). This step requires series of stress tensor calculations for
theprescribed strain modes. The Taylor theory was used [3],
assuming the identical critical resolved shear stresses on24
{110}+{112} slip systems. Provided that the stress components are
evaluated, the system (4) is solved. As aresult, the updated yield
surface is affected by the modification introduced by the texture
prediction model.
It is worth noting that the mechanical properties of the
material remain constant between the successive updates.However,
the time steps used by the FE are very short, which is intrinsic to
the explicit time integration solvers.
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Computational issues
Keeping in mind that the complexity of the problem translates
into the time demanded for computations, we analysedthe predominant
components of the execution time for our model. For a reasonable
value of the parameter Pcr chosen as0.1, the number of updating
steps Nu was appraised approximately as Nu≈ 6000. This reckoning
was made by countingthe virtual activations of the criterion (6)
observed in the FE simulation of the cup drawing process performed
for theconstant material properties. If the computer program
implementing our multiscale model is executed on a single CPU,the
estimation of the time necessary for the computations is given by:
Ts = NuTu+TFE = 6000 ·10[min]+1620[min]≈43[days], where Tu is the
amount of time for a single texture and constitutive model update,
TFE is the time spent onthe FE calculations.
Algorithm 1 exhibits several opportunities for parallelisation.
The iterations of the top-level loop (line 1 in Al-gorithm 1) can
be executed independently of each other. In the current study the
workload related to that loop isdistributed among the compute nodes
of the high performance cluster (HPC) facility. Our implementation
takes intoaccount a varying problem size, since the number of
updating criterion activations may vary from one time step
toanother. Instead of requesting a constant pool of the CPUs from
the HPC queuing system, the code starts the workerprograms on
demand, with respect to the task size and the current availability
of the idle CPUs in the cluster. Therefore,the actual number of
processors that are put at our disposal may vary from several to
hundreds. For this reason it is dif-ficult to estimate the speedup
of the parallel algorithm in terms of the classic definition.
Nevertheless, the assessmentof the overall efficiency can be done a
posteriori.
One can notice that the evaluation of the micromechanical model
for the set of strain modes (line 6 in Algorithm 1)can be performed
without any communication between the processors. The multicast
communication is required onlyprior to the loop 6, because every
processor involved in the task must have the current texture
available. The schemeoutlined above is suitable for parallelisation
on either multi-core (Symmetrical Multi-Processor, SMP) or
distributedmemory architecture. In the current implementation the
former choice is made. Thus, the execution of the loop 1
indistributed among the nodes of the cluster, and the internal loop
6 is parallelised on the multi-processor nodes.
In view of the fact that the two-level parallelisation is
implemented in the code, it is possible to tune Algorithm 1
forexecution in the HPC environment. For instance, the criterion
for texture updating may include an additional part whichdepends on
the global FE time increment number. In order to improve
concurrency of the block 3–12 in Algorithm 1,one can decide that
the criterion is fulfilled only in the time increments that are
multiple of N, where N is a parameterof the algorithm. The
distributed checkpointing operation (line 9 in Algorithm 1)
allocates the I/O operations over theparallel filesystems. Step 6
in Algorithm 1 can benefit from availability of modern multi-core
CPUs, as it shows nearlylinear speedup. The aforementioned
improvements are utilised to improve the performance of the
model.
RESULTS
In this work the cup drawing is considered, which is a simple
and typical example of sheet forming process. Theinvestigated
material is a low carbon DC01 steel. The model outlined in the
previous section has been implementedin the FE code
Abaqus/Explicit. The sample is discretised using the reduced
integration elements C3D8R. A localcoordinate system is attached to
every element in the FE mesh. Initially, the coordinate system is
common for theentire FE mesh, but the local coordinate systems
co-rotate with the material during the simulation. Coulomb
frictionwith the coefficient µ = 0.05 is imposed at the contact
surfaces. The Young modulus is assumed 210 GPa and Poissonratio
0.3. A Swift-type hardening law is used, with the following values
of coefficients: K = 659 MPa, n = 0.399 andε0 = 2.96 ·10−2. The
coefficients were corrected following the method given in [7]. The
Facet expression of degree 6was used, the parameters of the
Equation (3) were determined using 201 strain modes.
The geometry of a calculated intermediate cup is presented in
Fig. 1. It can be seen that part of the elements haveundergone at
least one updating of the material properties. The updates take
place in the zones that correspond to thelocalization of the
largest accumulated plastic strain. Forasmuch as the
element-by-element updating approach is used,the amount of
constitutive law updates is moderate at the early stages of
deformation, as is demonstrated in Fig. 1.The final geometry of the
cup is shown in Fig. 2. It can be seen in that figure that two ears
appear for 0◦ and 90◦ to therolling direction (RD).
The calculations were performed on the HPC cluster. The parallel
machine consists of 256 nodes, which are mainlydual-core and
quad-core AMD Opteron systems. The design of our code exploits the
fact that the PBS/Maui job systempromotes relatively short,
parallel jobs. The simulation of the process required 136034 FE
time increments to reachthe total simulation time 5 · 10−3s. The
actual number of the texture and yield surface updates as counted
during the
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FIGURE 1. The partially deformed sheet during the cup drawing.
The finite elements where the update of material propertieshas
occurred are marked gray. Grayscale denotes the number of the
updating events in the element. The update of texture
andconstitutive law is limited to the zones corresponding to the
largest accumulated plastic strain
FIGURE 2. The final cup geometry obtained from the multiscale
model. The points A, B and C denote the locations of
texturemeasurements. The local reference frames for the
experimental textures are indicated with x1, x2 and x3 axes
calculation was Nu = 5939. Measured execution time of parallel
computations was only Tp = 2265[min]≈ 1.57[days].This lead to a
posteriori estimation of the speedup, S = TsTp ≈ 27.
In order to verify the results of the numerical model, an
experimental study was additionally performed. The analysisincluded
the initial and final state of the material. The measurements of
the texture in a cold-rolled DC01 steel sheet(thickness 0.5mm)
provided an orientation distribution function (ODF) which was
utilized as the initial input data. Thetexture in the analysed
sheet exhibits a strong γ-fibre, the maximal ODF value was 19.04,
texture index was 7.108. Thediscretisation of the ODF consists of
5000 equally-weighted grains. Such a discretisation is considered
as sufficientlyrepresentative for the lattice orientations in low
carbon steel grades [1].
The X-ray measurements of the final texture were taken from the
fully deformed experimental cup. The locationsof the sampling
points are indicated in Fig. 2. The points A, B and C are selected
to be placed close to the cup rim at90◦, 45◦ and 0◦ to the rolling
direction, respectively.
The evolution of texture in the point A is shown in Fig. 3. In
order to facilitate the evaluation of the results, allthe
calculated textures are rotated to the coordinate systems used in
the experiment. Therefore, in all cases the radialdirection of the
cup is aligned with the vertical x1 axis of the experimental
sample. To illustrate the evolution of thetexture, the sections of
ODFs are presented with the φ1 Euler angle ranging from 0 to 360◦
and φ2 = 45◦. Accordingto the update criterion (6), the plastic
strain increments of uniform norm ||Pi||= Pcr are applied, Pcr =
0.1. Figure 3ashows the state of the initial texture, while Fig. 3b
presents the state of the texture after deformation P1, Fig.
3cpresents the deformation texture after the second update with
deformation P2 and so on. The φ2 = 45◦ section ofthe ODF determined
for the experimental cup in the point A is presented in Fig. 3j.
The texture indices are alsolisted. The strongly decreased texture
index corresponds to the broadening of the original texture. The
index increasescontinuously after all subsequent updates due to
increased intensities for particular orientations along the
original γ-fibre. A remarkable congruence between the experimental
and the ALAMEL-predicted texture can be noticed in that
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a)
b)i = 1
c)i = 2
d)i = 3
e)i = 4
f)i = 5
g)i = 6
h)i = 7
i)i = 8
j)
Texture Index (TI)a) b) c) d) e)
7.11 3.44 3.77 4.28 4.93
f) g) h) i) j)5.66 6.27 6.92 7.54 7.71
FIGURE 3. φ2 = 45◦ sections of the ODF for: a) initial
experimental texture, b) – i) evolution of the texture in the point
A (90◦ toRD) predicted by the ALAMEL model for the successive
deformations Pi (||Pi||= 0.1), j) – texture measured in the
correspondingpoint of the experimental cup. Texture index is given
for every texture a) – j)
figure. This is particularly relevant for the final deformation
texture, see Figures 3 i and 3 j. When comparing the maintexture
component, one has to remark that the intensity of the texture is
very similar in both cases. Additionally, thisis confirmed by the
texture index of the experimental texture which coincides with the
ALAMEL result. It shows asubstantial improvement to the previous
findings (e.g. [8, 9]), where the angular position of the texture
componentswas correctly identified by means of FC Taylor model, but
the intensity of the texture was overestimated.
The agreement between the yield locus calculated from the
measured textures and the simulated one can be visuallyinspected
using a π-plane section of stress or strain rate space. The
following relations are satisfied for the π-plane:d11 + d22 + d33 =
0 and S11 + S22 + S33 = 0 for strain rate and stress space,
respectively. Figure 4 shows the π-planesections of yield locus in
stress space and equipotential surface in stain rate space,
calculated for the point A. Asseen in that figure, the shape of the
equipotential surface calculated using the Taylor model on the
basis of the textureevolution predicted with the ALAMEL is fairly
consistent with the result obtained for the real deformation
texture.The convexity of the yield surface is also reproduced by
the model.
Successive deformation of the material leads to a new
deformation texture, as it can be seen in Fig. 5. The φ2 =
45◦sections of the ODF for point B reveal only small discrepancy
between prediction and measurement. The final texturein point B is
approximately orthorhombic and strongly resembles the final texture
in point C. The texture in point A isnearly identical to the one in
point C. It is noticeable that the texture indices of the predicted
ODF in the three pointssignificantly correspond to the experimental
ones.
The plastic anisotropy of the flat samples is commonly
characterised by the q-values, i.e. the ratio of the plastic
strain
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FIGURE 4. π-plane section of a) yield locus in stress space and
b) equipotential surface in stain rate space. The sections
arecalculated for the texture data in point A, as denoted in Fig.
2: plain line – initial texture, open symbols – final texture
predictionshown in Fig. 3h, filled symbols – experimentally
determined deformation texture shown in Fig. 3i
a)
b)
c)
d)
FIGURE 5. Comparison of the final deformation texture: a), c)
are predicted by the ALAMEL in point B (45◦ to RD) and C (0◦to RD),
respectively, b), d) were determined experimentally in the
corresponding points. Texture index: a) TI=8.363, b) TI=8.265,c)
TI=7.541, d) TI=7.443
in the width direction to the one in the longitudinal direction.
The relation between Lankford r-value and q-value isexpressed by q
= r/(r + 1). The subsequent changes of the crystallographic texture
induce the accommodation ofplastic anisotropy as shown in Fig. 6.
For better comparison with results calculated for the experimental
texture,the q-values are presented as a function of the angle to
the sample radial direction, i.e. in the experimental
samplereference frame. The filled symbols in Fig. 6 denote the
results obtained for the experimental textures following themethod
described in [10]. A smooth evolution of the planar anisotropy
towards a new stable configuration is observed.
a) b) c)
FIGURE 6. Evolution of q-values in consecutive update steps
(open symbols). The initial q-values are denoted with plain
line,filled symbols denote q-values calculated for the final
texture measured on the experimental cup. In the plot for the point
A the linesare calculated using the intermediate textures b) – i)
as shown in Fig. 3, respectively. Number of plastic strain
increments Pi of thenorm Pcr is denoted by the index i
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Interestingly, in the point A and C the anisotropy after the
first texture update seems to be substantially lower than inthe
initial material. The subsequent updates lead to an increase of
q-value variation which finally ranges from 0.52to 0.8 in point A
and 0.48 to 0.81 in point C, respectively. This is not the case for
the points B, where the tendencytowards augmented planar anisotropy
is instantaneously obtained. The agreement between q-values
obtained fromthe measured textures after deformation and the
simulated textures is satisfactory, although some discrepancies
areobserved for the point C.
CONCLUSIONS
A new multi-scale approach to crystallographic texture
prediction in a complicated metal forming process was pre-sented in
the paper. The concept of analytical constitutive law describing
the plastic anisotropy in the FE was applied inthe model. A
state-of-the-art Facet method for plastic potential in strain rate
space was implemented into the FE code.Owing to the successive
reconstructions of the yield locus, coupled to the evolution of the
crystallographic texture,the mutual interactions between micro- and
macro-scale are taken into account. Therefore, the plastic
deformationinfluences the texture, while the evolution of the
texture induces the changes in the anisotropy of macroscopic
defor-mation. The presented approach enables the adaptive texture
and yield surface update schema with respect to the localplastic
deformation in the FE integration points. Despite the fact that the
model does not predict the true continuousevolution of the
microstructural and mechanical properties, the quantitative
forecast of the micro- and macro-scalematerial behaviour is
reasonably accurate.
The results of this study confirm the capability of the ALAMEL
model to predict the evolution of the crystallo-graphic texture in
steels. In comparison to the previous attempts (e.g. [8, 9]) in
which the FC Taylor theory was used,the quantitative accuracy of
texture intensities is improved.
The limitations of the model due to its computational complexity
are discussed in the paper. To overcome theserestrictions,
parallelisation techniques have been implemented in the code. A
satisfactory speedup of the computationson a HPC cluster compared
to a single processor calculations is achieved.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the financial support from
the project IDO/08/09, funded by K.U.Leuven, andfrom the Belgian
Network DYSCO (Dynamical Systems, Control, and Optimization),
funded by the InteruniversityAttraction Poles Programme, initiated
by the Belgian State, Science Policy Office. Albert Van Bael
acknowledges thefinancial support from the Interuniversity
Attraction Poles Program from the Belgian State through the Belgian
SciencePolicy agency, contract IAP6/24. The authors are grateful to
P. Eyckens for his supervision of texture measurements onthe DC01
steel samples. Research is conducted utilizing high performance
computational resources provided by theUniversity of Leuven,
http://ludit.kuleuven.be/hpc. The scientific responsibility rests
with the authors.
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Welcome ScreenProceedingsTitle PageISBN/Copyright
InformationPrefaceContentsPLENARY LECTURESOlgierd (Olek) Cecil
Zienkiewicz (1921-2009): A Biographical TributeThe Finite Element
Method in Industrial Forming Processes. A Brief Historical
PerspectiveModeling, Testing and Numerical Simulation on Hot
Forming of HSSAdvanced Numerical Methods for F. E. Simulation of
Metal Forming ProcessesIntegration of Numerical Methods and
Analytic Approaches for Effective Solution of Metal Forming
ProblemsLightweight Steel Solutions for Automotive
IndustryPredicting Shear Failure of Dual-Phase SteelsMaterial
Models for Accurate Simulation of Sheet Metal Forming and
Springback
MINI SYMPOSIAADVANCES IN MODELING AND EXPERIMENTS FOR MATERIALS
FROMING IN HONOR OF THE 60TH BIRTHDAY OF PROF. DONG-YOL YANGA
Computational Model for the Numerical Simulation of FSW
ProcessesEffect of Superimposed Hydrostatic Pressure on Bendability
of Sheet MetalsProposal and Use of a Void Model for the Simulation
of Ductile Fracture Behavior in Material FormingAn Elastoplastic
Finite Element Modeling Coupled with Orientation Image Based
Micromechanical ApproachNumerical and Experimental Investigations
on Deformation Behavior of Aluminum 5754 Sheet Alloy under Warm
Hydroforming ConditionsCalculation of Recrystallization Textures
Using Slip Systems Activated during Deformation of MetalsClassical
and Thermodynamically Consistent Nonlocal Formulations for Ductile
Damage: Comparison of ApproachesNew Advances in Numerical
Simulation of Stationary Processes Using Arbitrary Lagrangian
Eulerian Formalism Application to Roll Forming Process3-D and
Anisotropic Effects on the Prediction of Burst in Aluminum Tube
HydroformingFinite Element Investigation of Hole-Expansion
Formability of Dual-Phase Steels Using RVE ApproachNumerical
Simulation of Time-Dependent Spring-Back Behavior for Aluminum
Alloy 6022-T4 SheetModeling of Directional Hardening Based on
Nonassociated Flow for Sheet FormingCup-Drawing Behavior of
High-Strength Steel Sheets Containing Different Volume Fractions of
MartensitePrediction of Residual Stresses and Distortion in
Quenched Extruded Shapes Using Generalized Plane Strain
TheoryEffect of Asymmetric Rolling on Plastic Anisotropy of Low
Carbon Steels during Simple Shear TestsManufacture of Mould with a
High Energy Efficiency Using Rapid Manufacturing
ProcessFully-Integrated Numerical Analysis of Microinjection
Molding with Localized Induction HeatingAnalysis of a Micropattern
Forming on the Thin Sheet Metal for Electronic DeviceDevelopment of
Alloy and Superalloy Large Shafts by Friction Welding
ProcessMaterial Modeling and Springback Prediction of Ultra Thin
Austenitic Stainless Steel SheetAn Investigation on the Accuracy of
Numerical Simulations for Single Point Incremental Forming with
Continuum ElementsImplementation of an Evolving Nonquadratic
Anisotropic Behavior for the Closed Packed Materials
ADVANCED ELEMENT TECHNOLOGY AND MESHLESS METHODS FOR LARGE
DEFORMATION PLASTICITYNumerical and Experimental Analysis of the
Dynamic Behavior of Aluminum-Composition Cork Sandwich
BeamsContinuous-Discontinuous Model for Ductile FracturePrediction
of Forming Limits Based on a Coupled Approach between Anisotropic
Damage and Necking ModelsA New Axi-Symmetric Element for Thin
Walled Structures
GENERAL TOPICSA Simple Modeling of Asymmetric RollingA
Methodology to Determine the Friction Coefficient in Flexforming
(Fluidcell Forming) ProcessFinite Element Analysis of
Layer-Integrated Steel Sheets Undergoing BendingMould Filling
Analysis of Aluminium Extrusion DiesAnalysis of Multistep Forming
Process of Metallic Bipolar Plate for MCFC Using Various Shapes of
PreformsForming Apparatus to Investigate the Effect of Temperature
on the Superplastic Behavior of AlloysProcess Modeling in Cold
Forging Considering the Process-Tool-Machine InteractionsMetal
Forming Process Effect Evaluation on Structural Behavior of an
Aeronautic PanelSheet Hydroforming Process Numerical Model
Improvement through Experimental Results AnalysisSheet Hydroforming
Pre-Bulging Numerical Model ImprovementFormability, Flow and Heat
Transfer Simulation of Hot Press Forming B-Pillar Part and
ToolsNumerical Design of Drawbeads for Advanced High Strength Steel
SheetsAnisotropic Properties of Stainless Steel-Clad Aluminum
SheetSheet Forming Simulations of Automotive Parts Using Different
Yield FunctionsHydroforming of Flanged Tubular PartEffect of the
Yield Stress and r-value Distribution on the Earing Profile of Cup
Drawing with Yld2000-2d Yield FunctionEffect of Dummy Block Shape
on Deformation of Oxide Film of Billet in Copper ExtrusionOn the
Die Face Design for Stamping an Automotive Engine HoodThe Design
and Performance Evaluation of Hydroformed Tubular Torsion Beam
AxleCoining as a Microforming ProcessLocal Bifurcation and
Instability Theory Applied to Formability AnalysisAnalysis of
Earing in Deep Drawn CupsUsage of Similarity in Incremental Bulk
Forming for Speed Up the FE-SimulationFast Numerical Simulation of
Sheet Metal Forming Using the Program QuickFormA New FE Modeling
Method for Isothermal Local Loading Process of Large-scale Complex
Titanium Alloy Components Based on DEFORM-3DModelling the
Thermomechanical Behavior of Magnesium Alloys during Indirect
ExtrusionNumerical Analysis of AHSS Fracture in a Stretch-bending
TestPrediction of Shear-induced Crack Initiation in AHSS Deep
Drawing Operation with a Phenomenological Fracture ModelFailure
Prediction in Fine Blanking Processes with Stress Limit ModelLocal
Interpolation for Tools Surface DescriptionA 3D Contact Smoothing
Method Based on Quasi-C Interpolation and Normal Voting-Application
to 3D Forging and RollingA Study of the Development of a New Type
of Bulb Bracket for Offshore Structures Using Suitable Casting
SteelEffect of Thickness Stress in Stretch-BendingCASTING, MOLDING,
QUENCHING, AND MICROFORMINGSimulation and Experiment on Direct
Continuous Casting Process of Lead Frame Copper AlloyWarpage
Simulation and the Experimental Verification of an L-Plate Sand
Mold Casting by Using the Thermoelastoplastic FEM CodeMicropowder
Injection Moulding of 316L Stainless Steel Feedstock and Numerical
Simulation of the Sintering StageAn Optimization Study of Hot
Stamping OperationA Streamline-Upwind Model for Filling Front
Advection in Powder Injection MouldingEffect of Viscosity on the
Microformability of Bulk Amorphous Alloy in Supercooled Liquid
RegionProcess of Equiaxed Grains of RE-Al-Alloy under Slope
VibrationExperimental Study of Local Microforming for Bi-HTSA
Numerical Model of the Temperature Field of the Cast and Solidified
Ceramic MaterialNumerical Optimization of the Method of Cooling of
a Massive Casting of Ductile Cast-ironSoftware Analytical
Instrument for Assessment of the Process of Casting Slabs
THERMAL MECHANICAL PROCESSINGNumerical Simulations on Warm
Forming of Stainless Steel with TRIP-EffectPlate Rolling Modeling
at Mill 5000 of OJSC "Magnitogorsk Iron and Steel" for Analysis and
Optimization of Temperature RatesFinite Element Analysis on Spread
for In-plane Roll-bending ProcessFEM Analysis of Defects and
Microstructure Evolution during Hot Working of Specialty
AlloysAtomistic Numerical Simulation on Nanoupsetting Process of
Copper BrickElastoplasticity Behavior of Type 5000 and 6000
Aluminum Alloy Sheets and Its Constitutive ModelingSimulation of
7050 Wrought Aluminum Alloy Wheel Die Forging and Its Defects
Analysis Based on DEFORMModeling Asymmetric Rolling Process of Mg
AlloysExperimental Analysis and Numerical Simulation of the Flow
Behavior of Thin Polymer Films during Hot EmbossingA Study on
Temperature Distribution and Curved Structure for Thick Plate by
Single-pass Induction Heating
MATERIALS, JOINING, AND POWDER FORMINGEffect of Isothermal
Reheating at Different Holding Times on the Microstructure of
Al-Mg2Si in-situ Cast CompositeSelf-pierce Riveting of Three
Aluminium Alloy and Mild Steel SheetsA Homogenization Approach to
the Yield Strength of Spherical Powder CompactsAnisotropic
Constitutive Model and FE Simulation of the Sintering Process of
Slip Cast Traditional PorcelainNumerical Modelling of
Thermal-electrical Phenomena in Spark Plasma SinteringStrength and
Efficiency during Lap Joining Molding of GMT-SheetA Study on the
Application of Submerged Arc Welding for Thin Plate of A-Grade 3.2
Thickness Steel in Ship Structure
TRANSITION FROM RESEARCH TO PRACTICEAn Analytic Model for the
Prediction of the Bar Temperature in a Roughing MillA New Model for
the Prediction of the Steepness Profile across the Strip in Flat
RollingAccuracy Analysis of Anisotropic Yield Functions Based on
the Root-mean Square ErrorFinite Element Analysis of Cross-wedge
Rolling Process
MODELING METHODS, INELASTIC MATERIAL MODELS, AND MULTISCALE
MODELINGRapid Parallel Calculation of Shell Element Modeling Used
on GPUPolymer Modeling in Wall Ironing Simulations of a PET-Steel
LaminateA Coupled Multiscale Model of Texture Evolution and Plastic
AnisotropyDiscrete Element Method, a Tool to Investigate Complex
Material Behaviour in Material FormingInvestigation of Ti6A14V
Orthogonal Cutting Numerical Simulations Using Different Material
ModelsFinite Element Modeling of Ring Rolling ProcessesCrystal
Plasticity Finite Element Analysis of Loading-Unloading Behavior in
Magnesium Alloy SheetThermomechanical Forming of Al-Mg-Si Alloys:
Modeling and ExperimentsPrediction of Gas Leak Tightness of
Superplastically Formed ProductsA Crystalline Plasticity Finite
Element Method for Simulation of the Plastic Deformation of AZ31
Magnesium AlloysMono and Multiobjective Optimization Techniques
Applied to a Large Range of Industrial Test Cases Using Metamodel
Assisted Evolutionary AlgorithmsA Quantized Crystal Plasticity
Finite Element Model for Nanocrystalline Metals: Connecting
Atomistic Simulations and Experiments
MATERIAL CHARACTERIZATION, PROCESS AND PRODUCT DESIGNOn the
Determination of Flow Stress Using Bulge Test and Mechanical
MeasurementAsymmetric ColdAVarm Rolling Simulation by Crystal
Plasticity Multiscale Finite Element Analysis Based on
Crystallographic HomogenizationAnalysis of Material Flow in Screw
Extrusion of AluminumA Method of Springback Prediction and Tool
Shape Compensation for Multicurvature Sheet Metal BendingEvaluation
of Die Chilling Effects during Forging of Nimonic-80A
SuperalloyAnalysis on the Cause of Twisting Defects Occurred in
Sheet Panel Formed through Local Embossing and Edge
BendingSimulation and Knowledge Based Process Planning through the
Use of MetamodelsThe Effect of Skin Passing on the Material
Behavior of Metal Strip in Pure Bending and TensionSimulation and
Analysis of Hot Forging Process for Industrial Locking Gear
ElevatorsApplication of a New Semiempirical Model for Forming Limit
Prediction of Sheet Material Including Superposed Loads of Bending
and ShearingReduction of Springback of Sheet Metals by
BottomingProcess Simulation of Aluminum Sheet Metal Deep Drawing at
Elevated TemperaturesAn Explicit Approach for Strain Gradient
Plasticity FormulationsFE Simulation of Cross Roll Straightening: A
Strain Tensor Field ApproachInvestigation on Mechanical Properties
of Boron Steel for Variation of Quenching Temperature and Its Hot
Press Forming SimulationInvestigations on the Predictability of
Coining Stainless Steel AISI 410Process Design by FEM Simulation
for Shape Ring Rolling of Large-sized RingSimulations of Forming
Limit Diagrams for the Aluminum Sheet Alloy 5754CCMechanical
Properties and Microstructure of AA1050 after ECAE (Equal Channel
Angular Extrusion)Computer-assisted Rheo-forging Processing of A356
Aluminum AlloysStraight Tube Hydroforming of Dual Phase (DP780)
Steel Tubes with End-FeedOn Springback Prediction with Special
Reference to Constitutive ModelingWear Behavior of Al-Mg2Si Cast
in-situ Composite: Effect of Mg2Si Different Volume FractionsThe
Effect of Li Additions on Wear Properties of Al-Mg2Si Cast in-situ
CompositesInfluence of Surface Effect on Nickel Microdeep Drawing
ProcessTemperature Dependent Constitutive Modeling for Magnesium
Alloy SheetPrediction of Microstructure in High-strength Ductile
Forging PartsAn Analytical Comparison of Single and Bilayered Tube
Hydroforming Systems Using Finite Element MethodFinite Element
Analysis and Die Design of Nonspecific Engineering Structure of
Aluminum Alloy during ExtrusionEffects of Al-5Ti-lB Master Alloy on
the Microstructural Evaluation of a Highly Alloyed Aluminum Alloy
Produced by SIMA ProcessEffects of Boron on Microstructure and
Tensile Properties of AIMg2Si Metal Matrix Composite
OPTIMIZATION METHODS AND NUMERICAL METHODSOptimization Design of
the Die of Synchronization Rolling Axle on Cross-wedge Rolling with
MultiwedgeOrthogonal Metal Cutting Simulation Using Advanced
Constitutive Equations with Damage and Fully Adaptive Numerical
ProcedureAn Investigation into the Optimization of Loading Path in
T-Shape of Tube HydroformingMPS-based LS-SVR Metamodeling Technique
for Sheet Forming OptimizationAn Efficient Triangular Shell Element
Based on Edge-based Smoothing Technique for Sheet Metal Forming
SimulationAn FE Based On-line Model for the Prediction of Work Roll
Thermal Profile in Hot Strip RollingOptimum Blank Design of
Thick-plate Metal Forming without Blank HolderPhenomenological
Microstructure Simulation of Incremental Bulk Metal Forming Using a
Multimesh MethodGenetic Algorithm for Design and Manufacture
Optimization Based on Numerical Simulations Applied to Aeronautic
Composite PartsStatistical Analysis of Magnetic Abrasive Finishing
(MAF) on Surface RoughnessAn ALE Based FE Formulation for the 3D
Numerical Simulation of Fineblanking ProcessesInvestigation on
Sintering Mechanism of Nanoscale Tungsten Powder Based on Atomistic
SimulationRheological Models of Blood: Sensitivity Analysis and
Benchmark SimulationsEffect of Imposing Temperature Gradient in
Stretch Forming Process for Ferritic Stainless Steel SheetsMesh
Generation Based on Virtual GeometryThe Influence of Geometrical
Parameters on the Incremental Forming Process for Knee Implants
Analyzed by Numerical SimulationNumerical Modeling of Hot Press
Forming Process of Boron Steel TubeExplicit Simulation of Roll
Forming Process with EAS Solid-shell Elements
HYBRID, INVERSE, AND ADAPTIVE METHODSMicropatterning of a
Bipolar Plate Using Direct Laser Melting ProcessInverse Analysis in
Hydroforming of a Refrigerator Door Handle Using MOGAAutomated Bead
Design Considering Production ConstraintsSpringback Compensation
Based on FDM-DTF Method
DAMAGE EVOLUTION, FAILURE, FORMABILITY, AND FRACTUREPredicting
Ductility and Failure Modes of TRIP Steels under Different Loading
ConditionsHole Expansion Simulations of TWIP Steel Sheet
SampleResearch on Fracture of Aluminum Foil in Microscale Laser
Peen FormingA Study on Critical Thinning in Thin-walled Tube
Bending of Al-Alloy 5052O via Coupled Ductile Fracture
CriteriaDuctile Fracture of TRIP780 Sheets under Multiaxial
LoadingA Study on Fracture Locus of Stl2 Steel and Implementation
Ductile Damage CriteriaTransverse Crack Modeling of Continuously
Casted Slabs through Finite Element Method in Roughing RoUing at
Wide Strip MillPrediction of the Localized Necking under
Nonproportional Strain Paths by M-K Theory and FE AnalysesStudy on
Edge Crack Propagation during Cold Rolling of Thin Strip by FEMThe
Nucleation and Growth of Microdefects in Hot Compression Process3D
FEM Simulation of Rolling Load Working on Piercer Plug Mannesmann
Piercing ProcessApplication of a Dislocation Based Model for
Interstitial Free (IF) Steels to Typical Stamping Simulations
Author IndexHelpSearchExit
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