Continuum Scale Simulation of Engineering Materials

866 Pages·15.502 MB·English

by  Raabe D., Roters F. et al (editors).

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www.wiley-vch.de
Professor Dierk Raabe Dr. Franz Roters Long-Qing Chen Dr. Frédéric Barlat
received his studied Physics in is Professor of received a PhD
Ph.D. (1992) and habilitation (1997) at RWTH Braunschweig, where he got his diploma degree Materials Science and Engineering at Penn in Mechanics from the “Institut National
Aachen, Germany, in the fields of Physical in 1993. From 1994 to 1998 he was scientist at State. He received his B.S. in Ceramics from Polytechnique de Grenoble,” France, in 1984.
Metallurgy and Metal Physics. He is currently the Institute for Metal Physics and Physical Zhejiang University in China in 1982, a M.S. in The same year, he joined Alcoa Technical Center;
Director and Executive at the Max-Planck Institut Metallurgy at the RWTH Aachen. He got his Materials Science and Engineering from State Pittsburgh, Pennsylvania, USA, the research
für Eisenforschung, Düsseldorf, Germany, after PhD. degree in 1999 in the field of constitutive University of New York at Stony Brook in 1985, facility of Alcoa Inc. (formerly the Aluminum
working some time as researcher at Carnegie modelling of aluminium. From 1999 till 2000 he and a Ph.D. degree in Materials Science and Company of America). Dr. Barlat is currently a
Mellon University, USA, the High Magnetic Field was researcher at the R&D centre of VAW (today Engineering from MIT in 1990. He worked with Technology Specialist in their Materials Science
Laboratory in Tallahassee, USA, and serving as Hydro Aluminium Deutschland GmbH) in Bonn. Armen G. Khachaturyan as a postdoc at Rutgers Division. He is responsible for conceptualizing,
Senior researcher and lecturer at the Institut für Since 2000 he is senior scientist at the Max- University from 1990 to 1992. Professor Chen importing and implementing mathematical
Metallkunde und Metallphysik, RWTH Aachen, Planck-Institut für Eisenforschung in Düsseldorf, joined the Department of Materials Science models that predict the mechanical behavior of
Germany. His research fields are Computer where he is the leader of the research group and Engineering at Penn State as an assistant materials for long-term development applica-
simulation of Materials, Composites, Textures, “Theory and Simulation” in the department for professor in 1992 and was promoted to tions in the areas of metal plasticity, fracture and
Micromechanics, in which he authored more Microstructure Physics and Metal Forming. Dr. associate professor in 1998. His main research material performance. His work is used for the
than 100 papers in peer-reviewed magazines Roters published more than 30 papers in the interests include materials theory and computa- design of alloys and processes in support of
and three books. He teaches various courses on field of constitutive modelling and simulation of tional materials science. Professor Chen Alcoa's major business units, including packa-
computational materials science, materials forming. He is head of the Technical Committee received the Young Investigator Award from the ging, automotive and aerospace. Dr. Barlat is
mechanics, history of metals, and textures at “Computersimulation” of the Deutsche Office of Naval Research (ONR) in 1995, the also an invited professor at the University of
RWTH Aachen (Germany) and at Carnegie Gesellschaft für Materialkunde e.V. (DGM). research creativity award from the National Aveiro’s Center for Mechanical Technology and
Mellon University Pittsburgh (USA). His work Science Foundation (NSF) in 1999, the Wilson Automation, Portugal, where he directs activities
was already awarded with several prizes, among Award for Excellence in Research in the College on the fundamentals of plasticity and forming.
them the Adolf-Martens Award, Masing Award, of Earth and Mineral Sciences in 2000, He has actively participated in the scientific
Heisenberg Award, and the Leibniz Award. and the University Faculty Scholar Medal at committees of various international conferences,
Penn State in 2003. has been a regular reviewer in a number of
scientific journals and serves as a member of
the Advisory Board of the International Journal
of Plasticity. Dr. Barlat is published as an author
or co-author in approximately 80 papers of peer-
reviewed scientific journals and has delivered
more than 60 technical presentations at con-
ferences worldwide. In 1995, he was the honored
The Editors
recipient of the ASM Henry Marion Howe Medal
of the Material Society for the best technical
paper published in Metallurgical Transactions A.
He holds three US patents with co-inventors
from Alcoa Inc. and Kobe Steel, Ltd., Japan.
Continuum Scale Simulation of
Engineering Materials
Fundamentals – Microstructures –
Process Applications
Dierk Raabe, Franz Roters, Frederic Barlat, Long-Qing Chen (Eds.)
WILEY-VCH Verlag GmbH& Co. KGaA
April5, 2004

Contents
Preface XXI
ListofContributors XXIII
I FundamentalsandBasicMethods 1
1 ComputerSimulationofDiffusionControlledPhaseTransformations
(A.SchneiderandG.Inden) 3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 NumericalTreatmentofDiffusionControlledTransformations . . . . . . . 5
1.2.1 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 BoundaryConditions . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 CellSize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 TypicalApplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.1 LE,LENPandPEinFe-Mn-C . . . . . . . . . . . . . . . . . . . . 15
1.3.2 LE,LENPandPEinFe-Si-C . . . . . . . . . . . . . . . . . . . . . 17
1.3.3 PEinFe-Ni-C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3.4 EffectofTracesontheGrowthofGrainBoundaryCementite . . . . 21
1.3.5 ContinuousCooling . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3.6 CompetitiveGrowthofPhases: Multi-CellCalculations . . . . . . . 23
1.3.7 Gas-Metal-Reactions:Carburization . . . . . . . . . . . . . . . . . 26
1.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2 IntroductiontoPhase-fieldMethodofMicrostructureEvolution
(L.-Q.Chen) 37
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2 OriginoftheModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3 TheoreticalFundamentalsoftheMethod . . . . . . . . . . . . . . . . . . . 38
2.3.1 RepresentationofaMicrostructure . . . . . . . . . . . . . . . . . . 38
2.3.2 ThermodynamicsofMicrostructures . . . . . . . . . . . . . . . . . 40
2.3.3 TheEvolutionEquations . . . . . . . . . . . . . . . . . . . . . . . 46
2.4 AdvantagesandDisadvantagesoftheMethod . . . . . . . . . . . . . . . . 47
2.5 TypicalFieldsofApplicationsandExamples . . . . . . . . . . . . . . . . . 47
VI Contents
2.6 SummaryandOpportunities . . . . . . . . . . . . . . . . . . . . . . . . . . 48
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 Cellular,LatticeGas,andBoltzmannAutomata
(D.Raabe) 56
3.1 CellularAutomata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.1.2 FormalDescriptionandClassesofCellularAutomata . . . . . . . . 57
3.1.3 CellularAutomatainMaterialsScience . . . . . . . . . . . . . . . 59
3.1.4 RecrystallizationSimulationswithCellularAutomata . . . . . . . . 60
3.2 CellularAutomataforFluidDynamics . . . . . . . . . . . . . . . . . . . . 66
3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.2 TheHPPandFHPLatticeGasCellularAutomata . . . . . . . . . . 66
3.2.3 TheLatticeBoltzmannAutomaton . . . . . . . . . . . . . . . . . . 69
3.3 ConclusionsandOutlook . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4 TheMonteCarloMethod
(A.D.RollettandP.Manohar) 76
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2 HistoryoftheMonteCarloMethod . . . . . . . . . . . . . . . . . . . . . . 76
4.2.1 IsingandPottsModels . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.2 MetropolisAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2.3 n-foldWayAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3 DescriptionoftheMonteCarloMethodforGrainGrowth&Recrystallization 84
4.3.1 DiscretizationofMicrostructure . . . . . . . . . . . . . . . . . . . 84
4.3.2 EvolutionoftheMicrostructure . . . . . . . . . . . . . . . . . . . . 85
4.3.3 InertParticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3.4 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3.5 BoundaryConditions . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.3.6 ParallelizationoftheMonteCarloAlgorithm . . . . . . . . . . . . 88
4.4 NucleationinRecrystallization . . . . . . . . . . . . . . . . . . . . . . . . 91
4.5 InitializationofMCSimulations . . . . . . . . . . . . . . . . . . . . . . . 92
4.6 VerificationoftheMonteCarloModel . . . . . . . . . . . . . . . . . . . . 93
4.7 ScalingofSimulatedGrainSizetoPhysicalGrainSize . . . . . . . . . . . 96
4.8 RecrystallizationKineticsintheMonteCarlomodel . . . . . . . . . . . . . 97
4.9 ResultsofSimulationofRecrystallizationbyMonteCarloMethod . . . . . 98
4.9.1 AbnormalGrainGrowth . . . . . . . . . . . . . . . . . . . . . . . 98
4.9.2 StaticRecrystallization . . . . . . . . . . . . . . . . . . . . . . . . 98
4.9.3 GrainGrowthinthePresenceofParticles . . . . . . . . . . . . . . 100
4.9.4 RecrystallizationinthePresenceofParticles . . . . . . . . . . . . . 100
4.9.5 TextureDevelopment . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.9.6 Texture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.9.7 DynamicRecrystallization . . . . . . . . . . . . . . . . . . . . . . 108
4.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Contents VII
5 CrystalPlasticity
(P.R.Dawson) 114
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2 TheoreticalBackground . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2.1 MechanicalResponseofSingleCrystals . . . . . . . . . . . . . . . 114
5.2.2 LatticeOrientationDistributionsforPolycrystals . . . . . . . . . . 119
5.2.3 MechanicalResponseofPolycrystals. . . . . . . . . . . . . . . . . 121
5.3 MacroscopicCriteriaforAnisotropicStrength . . . . . . . . . . . . . . . . 123
5.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.3.2 YieldSurfacesDefinedbyExpansions . . . . . . . . . . . . . . . . 125
5.3.3 YieldSurfacesDefinedbyHyperplanes . . . . . . . . . . . . . . . 126
5.3.4 IsoparametricFlowSurface . . . . . . . . . . . . . . . . . . . . . 128
5.3.5 DirectPolycrystalPlasticityImplementation . . . . . . . . . . . . . 130
5.4 NumericalImplementations . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.4.1 BalanceLaws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.4.2 FiniteElementFormulations . . . . . . . . . . . . . . . . . . . . . 131
5.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.5.1 ApplicationtoExplosiveForming . . . . . . . . . . . . . . . . . . 133
5.5.2 ApplicationtotheLimitingDomeHeightTest . . . . . . . . . . . . 134
5.5.3 BendingofaCurvedComponent . . . . . . . . . . . . . . . . . . . 138
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6 YieldSurfacePlasticityandAnisotropy
(F.Barlat,O.Cazacu,M.Z˙yczkowski,D.Banabic,andJ.W.Yoon) 143
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.2 ClassicalPlasticityTheory . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.2.1 IsotropicYieldConditionsforPerfectPlasticity . . . . . . . . . . . 144
6.2.2 FlowRules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.2.3 SubsequentYieldSurfacesduringPlasticHardening. . . . . . . . . 148
6.2.4 AnisotropicPlasticity . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.2.5 DirectGeneralizationsofIsotropicYieldConditions. . . . . . . . . 151
6.3 MaterialStructureandPlasticAnisotropy. . . . . . . . . . . . . . . . . . . 152
6.3.1 Texture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.3.2 Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.3.3 PorosityandSecondPhases . . . . . . . . . . . . . . . . . . . . . . 158
6.4 YieldFunctionsforMetalsandAlloys . . . . . . . . . . . . . . . . . . . . 159
6.4.1 QuadraticYieldFunctions . . . . . . . . . . . . . . . . . . . . . . 159
6.4.2 Non-QuadraticYieldFunctions . . . . . . . . . . . . . . . . . . . . 160
6.4.3 YieldFunctionsinPolarCoordinates . . . . . . . . . . . . . . . . . 163
6.4.4 OtherAnisotropicYieldFunctions . . . . . . . . . . . . . . . . . . 163
6.4.5 BBC2000YieldCriterion . . . . . . . . . . . . . . . . . . . . . . . 163
6.4.6 Yld2000-2dYieldCriterion . . . . . . . . . . . . . . . . . . . . . 165
6.4.7 CB2001YieldCriterion . . . . . . . . . . . . . . . . . . . . . . . . 165
6.4.8 StrainRatePotentials . . . . . . . . . . . . . . . . . . . . . . . . . 167
VIII Contents
6.5 ApplicationtoSheetFormingandFormability . . . . . . . . . . . . . . . . 167
6.5.1 Mechanicaltesting . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.5.2 AnalysisandTreatmentoftheTestResults . . . . . . . . . . . . . . 170
6.5.3 Applicationto3103-OAluminumAlloySheetSample . . . . . . . 172
6.5.4 PlasticFlowLocalization . . . . . . . . . . . . . . . . . . . . . . . 172
6.5.5 CupDrawingSimulation . . . . . . . . . . . . . . . . . . . . . . . 173
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7 ArtificialNeuralNetworks
(E.BroeseandH.-U.Löffler) 182
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.2 BasicTerms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.3 FieldsofApplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.3.1 PatternRecognition/Classification . . . . . . . . . . . . . . . . . . 183
7.3.2 EmpiricalModeling . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.4.1 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.4.2 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7.5 TypesofArtificialNeuralNetworks. . . . . . . . . . . . . . . . . . . . . . 185
7.5.1 MultilayerPerceptron . . . . . . . . . . . . . . . . . . . . . . . . . 185
7.5.2 RadialBasisFunctionNetworks . . . . . . . . . . . . . . . . . . . 188
7.5.3 MoreNetworkTypes . . . . . . . . . . . . . . . . . . . . . . . . . 190
7.6 KindsofLearning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
7.6.1 UnsupervisedLearning . . . . . . . . . . . . . . . . . . . . . . . . 191
7.6.2 SupervisedLearning . . . . . . . . . . . . . . . . . . . . . . . . . 191
7.6.3 ReinforcementLearning . . . . . . . . . . . . . . . . . . . . . . . 191
7.6.4 BayesianLearning . . . . . . . . . . . . . . . . . . . . . . . . . . 192
7.7 ApplicationDetails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
7.7.1 NetworkTypeSelectionandConfiguration . . . . . . . . . . . . . . 192
7.7.2 InputSelection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.7.3 DataPreprocessingandInputScaling . . . . . . . . . . . . . . . . 193
7.7.4 PreventionofOverfitting . . . . . . . . . . . . . . . . . . . . . . . 193
7.7.5 OptimizationofTrainingParameters . . . . . . . . . . . . . . . . . 194
7.7.6 DiagnosticsoftheInternalState . . . . . . . . . . . . . . . . . . . 194
7.8 FutureProspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
8 MultiscaleDiscreteDislocationDynamicsPlasticity
(H.M.Zbib,M.Hiratani,andM.Shehadeh) 197
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
8.2 TheoreticalFundamentalsoftheMethod . . . . . . . . . . . . . . . . . . . 199
8.2.1 KinematicsandGeometricAspects . . . . . . . . . . . . . . . . . . 199
8.2.2 KineticsandInteractionForces . . . . . . . . . . . . . . . . . . . . 199
8.2.3 DislocationEquationofMotion . . . . . . . . . . . . . . . . . . . 200
Contents IX
8.2.4 TheDislocationStressandForceFields . . . . . . . . . . . . . . . 204
8.2.5 TheStochasticForceandCross-slip . . . . . . . . . . . . . . . . . 206
8.2.6 Modifications for Long-Range Interactions: The Super-Dislocation
Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
8.2.7 EvaluationofPlasticStrains . . . . . . . . . . . . . . . . . . . . . 209
8.2.8 TheDDNumericalSolution: AnImplicit-ExplicitIntegrationScheme209
8.3 IntegrationofDDandContinuumPlasticity . . . . . . . . . . . . . . . . . 210
8.3.1 ContinuumElasto-Viscoplasticity . . . . . . . . . . . . . . . . . . 210
8.3.2 ModificationsforFiniteDomains . . . . . . . . . . . . . . . . . . . 211
8.4 TypicalFieldsofApplicationsandExamples . . . . . . . . . . . . . . . . . 213
8.4.1 EvolutionofDislocationStructureduringMonotonicLoading . . . 214
8.4.2 DislocationCrackInteraction: HeterogeneousDeformation . . . . . 216
8.4.3 DislocationsInteractionwithShockWaves. . . . . . . . . . . . . . 219
8.5 SummaryandConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . 221
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
9 PhysicallyBasedModelsforIndustrialMaterials: WhatFor?
(Y.Brechet) 226
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
9.2 RecentTrendsinModellingMaterialsBehavior . . . . . . . . . . . . . . . 226
9.2.1 AnalyticalModels . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
9.2.2 ComputerSimulations . . . . . . . . . . . . . . . . . . . . . . . . 228
9.2.3 MaterialsModellingandMaterialsDesign: SomeExamples . . . . 230
9.2.4 SophisticatedStatisticalAnalysis . . . . . . . . . . . . . . . . . . . 231
9.3 SomeExamplesofPhysicallyBasedModelsforIndustrialMaterials . . . . 232
9.3.1 RecoveryofAluminumAlloys . . . . . . . . . . . . . . . . . . . . 232
9.3.2 CompetitionBetweenRecrystallizationandPrecipitation . . . . . . 235
9.3.3 OptimizingCastingProcessinPrecipitationHardenableAlloys . . . 238
9.4 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
II Application to Engineering Microstructures 245
10 ModelingofDendriticGrainFormationDuringSolidificationattheLevel
ofMacro-andMicrostructures
(M.Rappaz,A.Jacot,andC.A.Gandin) 247
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
10.2 Pseudo-FrontTrackingModel . . . . . . . . . . . . . . . . . . . . . . . . . 250
10.2.1 PrimaryPhaseFormation . . . . . . . . . . . . . . . . . . . . . . . 250
10.2.2 SecondaryPhasesFormation . . . . . . . . . . . . . . . . . . . . . 252
10.3 CouplingwithThermodynamicDatabases . . . . . . . . . . . . . . . . . . 253
10.3.1 PrimaryPhaseFormation . . . . . . . . . . . . . . . . . . . . . . . 253
10.3.2 SecondaryPhasesFormation . . . . . . . . . . . . . . . . . . . . . 254
X Contents
10.4 CellularAutomaton–FiniteElementModel . . . . . . . . . . . . . . . . . 254
10.4.1 NucleationLaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
10.4.2 GrowthLaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
10.4.3 CouplingofCAandFEMethods . . . . . . . . . . . . . . . . . . . 256
10.5 ResultsandDiscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
10.5.1 PFTModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
10.5.2 CAFEModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
10.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
11 Phase-FieldMethodAppliedtoStrain-dominatedMicrostructureEvolution
duringSolid-StatePhaseTransformations
(L.-Q.ChenandS.Y.Hu) 266
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
11.2 PhenomenologicalDescriptionofSolidStatePhaseTransformations . . . . 267
11.3 Phase-FieldModelofSolidStatePhaseTransformations. . . . . . . . . . . 269
11.4 ElasticEnergyofaMicrostructure . . . . . . . . . . . . . . . . . . . . . . 271
11.5 BulkMicrostructureswithPeriodicBoundaryConditions . . . . . . . . . . 271
11.6 ASingleCrystalFilmwithSurfaceandSubstrateConstraint . . . . . . . . 273
11.7 ElasticCouplingofStructuralDefectsandPhaseTransformations . . . . . . 274
11.8 Phase-FieldModelAppliedtoSolidStatePhaseTransformations . . . . . . 275
11.9 IsostructuralPhaseSeparation . . . . . . . . . . . . . . . . . . . . . . . . . 275
11.10 PrecipitationofCubicIntermetallicPrecipitatesinaCubicMatrix. . . . . . 277
11.11 StructuralTransformationsResultinginaPointGroupSymmetryReduction 279
11.12 FerroelectricPhaseTransformations . . . . . . . . . . . . . . . . . . . . . 281
11.13 PhaseTransformationinaReducedDimensions: ThinFilmsandSurfaces . 283
11.14 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
12 IrregularCellularAutomataModelingofGrainGrowth
(K.Janssens) 292
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
12.2 IrregularCellularAutomata . . . . . . . . . . . . . . . . . . . . . . . . . . 292
12.2.1 TheConcept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
12.2.2 ShapelessorPointCellularAutomata . . . . . . . . . . . . . . . . 293
12.3 IrregularShapelessCellularAutomataforGrainGrowth . . . . . . . . . . . 293
12.3.1 CurvatureDrivenGrainGrowth . . . . . . . . . . . . . . . . . . . 294
12.3.2 InthePresenceofAdditionalDrivingForces. . . . . . . . . . . . . 297
12.4 AQualitativeExample: StaticAnnealingofaColdRolledSteel . . . . . . . 299
12.4.1 TheDeformationModel . . . . . . . . . . . . . . . . . . . . . . . 299
12.4.2 TheAnnealingModel . . . . . . . . . . . . . . . . . . . . . . . . . 300
12.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302