Solid 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 Series Editors: G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchersgivingvisionandinsightinansweringthesequestionsonthesubjectof mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics;statics,kinematicsanddynamicsofrigidandelasticbodies:vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes;structuralcontrolandstability;soils,rocksandgeomechanics;fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts aremonographsdefiningthecurrentstateofthefield;othersareaccessibletofinal year undergraduates; but essentially the emphasis is on readability and clarity. For furthervolumes: http://www.springer.com/series/6557 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) DOI10.1007/978-94-007-6098-1 SpringerDordrechtHeidelbergNewYorkLondon LibraryofCongressControlNumber:2013932705 #SpringerScience+BusinessMediaDordrecht2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerpts inconnectionwithreviewsorscholarlyanalysisormaterialsuppliedspecificallyforthepurposeofbeing enteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework.Duplication ofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheCopyrightLawofthe Publisher’s location, in its current version, and permission for use must always be obtained from Springer.PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) 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