Table Of ContentSolid Mechanics and Its Applications
Zengtao Chen
Cliff Butcher
Micromechanics
Modelling of
Ductile Fracture
Micromechanics Modelling of Ductile Fracture
SOLID MECHANICS AND ITS APPLICATIONS
Volume 195
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Zengtao Chen (cid:129) Cliff Butcher
Micromechanics Modelling
of Ductile Fracture
ZengtaoChen CliffButcher
DepartmentofMechanicalEngineering DepartmentofMechanical
UniversityofNewBrunswick andMechatronicsEngineering
Fredericton,NewBrunswick UniversityofWaterloo
Canada Waterloo,Ontario
Canada
ISSN0925-0042
ISBN978-94-007-6097-4 ISBN978-94-007-6098-1(eBook)
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Contents
1 IntroductiontoDuctileFractureModelling. . . . . . . . . . . . . . . . 1
1.1 RoleofMaterialDamage. . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 VoidNucleation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 ModelingVoidNucleation. . . . . . . . . . . . . . . . . . . . 3
1.1.3 VoidGrowth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.4 VoidShapeEvolution. . . . . . . . . . . . . . . . . . . . . . . 9
1.1.5 VoidCoalescence. . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Damage-BasedYieldCriteria. . . . . . . . . . . . . . . . . . . . . . . 16
1.2.1 GursonCriterion. . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3 VoidEvolutionandCoalescenceWithinClusters. . . . . . . . . 19
1.4 DamagePercolationModeling. . . . . . . . . . . . . . . . . . . . . . . 21
1.4.1 RoleoftheVoidDistribution. . . . . . . . . . . . . . . . . . 21
1.4.2 DamagePercolationModeling. . . . . . . . . . . . . . . . . 22
2 AveragingMethodsforComputationalMicromechanics. . . . . . 25
2.1 DefinationofAverageStressandStrain. . . . . . . . . . . . . . . . 25
2.2 FundamentalsofaConstitutiveModelforPlasticity. . . . . . . 26
2.3 NormalityandConvexityoftheYieldSurface. . . . . . . . . . . 27
2.4 PrincipleofVirtualWork. . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 PrincipleofMaximumPlasticWork. . . . . . . . . . . . . . . . . . 28
2.6 ExtremumTheoremsinPlasticity. . . . . . . . . . . . . . . . . . . . 28
2.6.1 UpperBoundSolution. . . . . . . . . . . . . . . . . . . . . . . 29
2.6.2 LowerBoundSolution. . . . . . . . . . . . . . . . . . . . . . . 29
2.7 Gurson’sUpperBoundSolutionforaPorous
DuctileMaterial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.7.1 VoidGrowthandNucleation. . . . . . . . . . . . . . . . . . 32
2.7.2 VoidCoalescence. . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.8 LowerBoundSolutionofSunandWang. . . . . . . . . . . . . . . 40
2.8.1 VoidGrowth,NucleationandCoalescence. . . . . . . . 41
v
vi Contents
2.9 UpperandLowerBoundApproachtoDuctileFracture
ofPorousMaterials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.9.1 ApplicationoftheDualBoundApproachtoPorous
MaterialswithVoidClusters. . . . . . . . . . . . . . . . . . 45
2.10 ApplicationoftheDualBoundApproachtoaStretchFlange
FormingProcess. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.10.1 PredictingtheLimitPunchDepth. . . . . . . . . . . . . . 49
2.10.2 DamageEvolutionDuringForming. . . . . . . . . . . . . 49
2.10.3 ComparisonofthePredictedand
MeasuredPorosity... ... .... ... .... .... ... .. 50
2.11 ApplicationoftheDualBoundApproachtoDuctileFracture
inTubeHydroforming. . . . . .. . . . . .. . . . . .. . . . .. . . . . . 53
2.11.1 ConstitutiveModeling. . . . . . . . . . . . . . . . . . . . . . 54
2.11.2 MaterialCharacterization. . . . . . . . . . . . . . . . . . . . 54
2.11.3 Finite-ElementModel. . . . . . . . . . . . . . . . . . . . . . . 55
2.11.4 MeasuringFormability. . . . . . . . . . . . . . . . . . . . . . 56
2.11.5 Results. .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . . 57
2.11.6 EvaluationoftheDualBoundApproachforTube
Hydroforming. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.12 ADualBoundApproachtoDeterminingtheVoidNucleation
ParametersinSheetMaterials. . . . . . . . . . . . . . . . . . . . . . . 63
2.12.1 ConstitutiveModelingofDuctileFracture. . . . . . . . 64
2.12.2 NotchTensileTestExperiment. . . . . . . . . . . . . . . . 66
2.12.3 Finite-ElementModel. . . . . . . . . . . . . . . . . . . . . . . 68
2.12.4 IdentificationoftheFractureStrains. . . . . . . . . . . . 68
2.12.5 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . . . . 70
3 Anisotropy. . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . .. . . . . . . . . 75
3.1 TheHill-48AnisotropicYieldCriterion. . . . . . . . . . . . . . . . 76
3.2 MaterialAnisotropyinPorousDuctileMaterials. . . . . . . . . . 77
3.3 AnApproximateUnitCellforPorousSheetMetals. . . . . . . 80
3.3.1 StressandStrainRateFieldsInsidetheUnitCell. . . 80
3.3.2 ElasticStressStateintheUnitCell. . . . . . . . . . . . . 81
3.3.3 PlasticStressStateintheUnitCell. . . . . . . . . . . . . 83
3.4 DerivationofaLowerBoundYieldCriterionforPorous
SheetMetals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.4.1 NumericalResults. . . . . . . . . . . . . . . . . . . . . . . . . 86
3.4.2 ComparisonoftheLowerBoundSolutionwith
Experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.5 DerivationofaQuasi-ExactLowerBoundAnisotropic
YieldCriterionforPorousSheetMetals. . . . . . . . . . . . . . . . 90
3.5.1 DerivationoftheFlowRuleandEquivalent
PlasticStrain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.5.2 AnalyticalDerivationoftheYieldFunction. . . . . . . 92
3.5.3 SolutionfortheMacroscopicRadialStress. . . . . . . 93
3.5.4 SolutionfortheMacroscopicThrough-Thickness
Stress. . . . . .. . . . . .. . . . . .. . . . .. . . . . .. . . . . . 94
Contents vii
3.5.5 SolutionfortheYieldFunction. . . . . . . . . . . . . . . . . . 95
3.5.6 EffectofMechanicalAnisotropyinaPorous
DuctileMaterial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.5.7 AssessmentoftheUniquenessoftheCurrentYield
Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.5.8 EvaluationoftheQuasi-ExactAnisotropicYield
Criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4 VoidGrowthtoCoalescence:UnitCellandAnalytical
Modelling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.1 VoidShapeEvolutionDuringDuctileFracture. . . . . . . . . . . . 101
4.2 Damage-BasedMaterialModelswithVoidShapeEffects. . . . 102
4.3 ModelingVoidEvolutionUsingaUnitCell. . . . . . . . . . . . . . 103
4.3.1 AnalysisofanAxisymmetricUnitCell. . . . . . . . . . . . 104
4.3.2 UnitCellBoundaryConditions. . . . . . . . . . . . . . . . . . 107
4.3.3 StressStateandMicrostructureEvolution. . . . . . . . . . 107
4.3.4 IdentificationofVoidCoalescence. . . . . . . . . . . . . . . 109
4.3.5 NumericalSolutionProcedure. . . . . . . . . . . . . . . . . . 110
4.4 UnitCellSimulationResults. . . . . . . . . . . . . . . . . . . . . . . . . 111
4.4.1 Penny-ShapedVoids:W ¼ 1/100. . . . . . . . . . . . . . . 112
o
4.4.2 OblateVoids:W ¼ 1/6. . . . . . . . . . . . . . . . . . . . . . . 114
o
4.4.3 SphericalVoids:W ¼ 1. . . . . . . . . . . . . . . . . . . . . . 115
o
4.4.4 ProlateVoids:W ¼ 6. . . . . . . . . . . . . . . . . . . . . . . . 116
o
4.4.5 SelectionofaMinimumVoidAspectRatio. . . . . . . . . 118
4.5 TheoreticalModelsforVoidGrowth,
ShapeandCoalescence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.5.1 YieldCriterion.. . . . . .. . . . . .. . . . . .. . . . . .. . . . . 119
4.5.2 VoidGrowth,ShapeEvolutionandCoalescence. . . . . 119
4.5.3 ComparisonwithUnitCellResults. . . . . . . . . . . . . . . 121
4.6 CalibrationoftheVoidEvolutionModels. . . . . . . . . . . . . . . 122
4.6.1 VoidGrowth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.6.2 VoidShapeEvolution. . . . . . . . . . . . . . . . . . . . . . . . 123
4.6.3 VoidCoalescence. . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.7 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5 Two-Dimensional(2D)DamagePercolationModeling. . . . . . . . 133
5.1 TheDamagePercolationModel. . . . . . . . . . . . . . . . . . . . . . . 134
5.1.1 ParticleFieldTessellations. . . . . . . . . . . . . . . . . . . . . 134
5.1.2 DamageEvolutionPredictions. . . . . . . . . . . . . . . . . . 137
5.2 DamagePredictions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.2.1 DamageEvolution. . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.2.2 PredictedDamageRates. . . . . . . . . . . . . . . . . . . . . . . 144
5.3 SelectionofRepresentativeVolumeElement(RVE). . . . . . . . 146
5.3.1 ParticleFieldSizes. . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.3.2 ResultsandDiscussion. . . . . . . . . . . . . . . . . . . . . . . . 148
5.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
viii Contents
6 Two-Dimensional(2D)DamagePercolation/FiniteElement
ModelingofSheetMetalForming. . . . . . . . . . . . . . . . . . . . . . . . 153
6.1 StretchFlangeExperiment. . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.2 GTN-BasedDamageModel. . . . . . . . . . . . . . . . . . . . . . . . . 158
6.3 CoupledPercolationModel–DamagePredictions. . . . . . . . . 160
6.3.1 VoidNucleation. .. . . . . .. . . . .. . . . .. . . . . .. . . . . 160
6.3.2 VoidGrowth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.3.3 VoidCoalescence. . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.3.4 Post-coalescenceTreatment. . . . . . . . . . . . . . . . . . . . 163
6.4 ParticleFieldMapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.5 CoupledModel–MeshandParticleFields. .. . . . . . . . . . . . . 165
6.5.1 FEMesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.5.2 BoundaryConditions–ToolingMotion. . . . . . . . . . . 166
6.5.3 SecondPhaseParticleFields. . .. . . . .. . . .. . . .. . . . 166
6.6 GTN-BasedFEResults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.6.1 PorosityPredictions. . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.6.2 StrainAnalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.7 CoupledFE/DamagePercolationPredictions. . . . . . . . . . . . . 174
6.7.1 DamageEvolution. . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.7.2 QuantitativeDamagePredictions. . . . . . . . . . . . . . . . 176
6.7.3 ComparisonwithMeasuredDamageLevels. . . . . . . . 177
6.7.4 FormabilityPredictions. . . . . . . . . . . . . . . . . . . . . . . 178
6.8 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7 TwoDimensional(2D)DamagePercolationwithStressState. . . 181
7.1 DevelopmentofaPhenomenologicalVoidNucleation
CriterionforPercolationModeling. . . . . . . . . . . . . . . . . . . . 182
7.1.1 ParticleSizeandAreaFractionFunctions. . . . . . . . . . 182
7.1.2 StressStateDependenceFunction. . . . . . . . . . . . . . . . 183
7.1.3 ApplicationoftheNucleationCriteriontoVarious
ParticleFields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7.2 PercolationModelingofDuctileFracture. . . . . . . . . . . . . . . . 186
7.2.1 VoidCoalescence. . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7.2.2 ProfuseCoalescenceandFailureoftheParticleField. . 187
7.3 ParticleFieldTessellations. . . . . . . . . . . . . . . . . . . . . . . . . . 188
7.4 CalibrationoftheNucleationModel. . . . . . . . . . . . . . . . . . . 188
7.4.1 ComparisonofPredictedandExperimental
FormingLimits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7.4.2 VoidNucleation. .. . . . . .. . . . .. . . . .. . . . . .. . . . . 191
7.4.3 AverageNucleationStrain. . . . . . . . . . . . . . . . . . . . . 194
7.4.4 AverageSizeofDamagedParticle. . . . . . . . . . . . . . . 195
7.5 CalibrationofaContinuum-BasedNucleationRuleUsing
thePercolationModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
7.5.1 ContinuumNucleationModel. . . . . . . . . . . . . . . . . . . 197
7.5.2 SynchronizationoftheVoidNucleationCriteria. . . . . 197
7.6 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Contents ix
8 Three-DimensionalParticleFields. . . . . . . . . . . . . . . . . . . . . . . 201
8.1 ParticleFieldGenerator. . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
8.1.1 ParticleFieldBasics. . . . . . . . . . . . . . . . . . . . . . . . . . 203
8.1.2 GenerationofRandomVariables. . . . . . . . . . . . . . . . 204
8.1.3 ObjectGeneration. . . . . . . . . . . . . . . . . . . . . . . . . . . 206
8.1.4 GenerationofObjectsWithinClusters. . . . . . . . . . . . 207
8.1.5 ParticleProperties. . . . . . . . . . . . . . . . . . . . . . . . . . . 210
8.1.6 ObjectConstraints. . . . . . . . . . . . . . . . . . . . . . . . . . . 210
8.1.7 IdentificationoftheParentElement. . . . . . . . . . . . . . 212
8.1.8 NumericalImplementation. . . . . . . . . . . . . . . . . . . . . 212
8.2 ApplicationoftheParticleFieldGeneratortoan
Al-MgAlloy. . . .. . . . . .. . . . . . .. . . . . . .. . . . . .. . . . . . . 214
8.2.1 MaterialCharacterization. . . . . . . . . . . . . . . . . . . . . . 214
8.2.2 ParametersUsedintheParticleFieldGeneration
ProcessforAA5182. . . . . . . . . . . . . . . . . . . . . . . . . . 215
8.2.3 ParticleFieldGenerationResults. . . . . . . . . . . . . . . . 216
8.2.4 ObjectDimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . 216
8.2.5 SpatialDistributionoftheObjects. . . . . . . . . . . . . . . 218
8.2.6 ParticleandVoidVolumeFractions. . . . . . . . . . . . . . 219
8.3 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
9 EstimationoftheStressStateWithinParticlesandInclusions
andaNucleationModelforParticleCracking. . . . . . . . . . . . . . 223
9.1 Particle-BasedHomogenizationTheories. . . . . . . . . . . . . . . . . . . .224
9.2 SelectionofaHomogenizationTheoryforModelling
VoidNucleation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
9.3 AParticle-BasedHomogenizationModelforaDual-Phase
CompositeSubjectedtoaPrescribedTraction. . . . . . . . . . . . 227
9.4 EffectiveModuliofaRandomly-OrientedComposite. . . . . . . . . . .227
9.5 AverageStressintheCompositeandItsConstituents. . . . . . . . . . .229
9.6 AverageStrainintheCompositeandItsConstituents. . . . . . . . . . .229
9.7 ProcedureforIntegratingaParticle-BasedHomogenization
TheoryintoanExistingDamage-Based
ConstitutiveModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
9.8 IterativeSolutionfortheEffectiveSecantModuli. . .. . . .. . . . .. .233
9.9 ApplicationoftheParticle-BasedHomogenization
SchemeintoaGurson-BasedConstitutiveModel
forDuctileFracture. . .. . . . . . . .. . . . . . .. . . . . . .. . . . . . . 234
9.10 ContinuumNucleation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .235
9.11 VoidNucleationinaParticleField. . . . . . . . . . . . . . . . . . . . . . . .236
9.12 ModelingVoidNucleationUsingPenny-ShapedVoids. . . . . . . . . .236
9.13 ANucleationModelforParticleCracking. . . . . . . . . . . . . . . . . . .238
9.13.1 StressStateandNucleation. . . . . . . . . . . . . . . . . . . . . . .240
9.14 DeterminationoftheInitialDimensions
foraNucleatedVoid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
9.15 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .243
