Analysis and Computation of Microstructure in Finite Plasticity PDF
Preview Analysis and Computation of Microstructure in Finite Plasticity
Lecture Notes in Applied and Computational Mechanics 78 Sergio Conti Klaus Hackl Editors Analysis and Computation of Microstructure in Finite Plasticity Lecture Notes in Applied and Computational Mechanics Volume 78 Serieseditors FriedrichPfeiffer,TechnischeUniversitätMünchen,Garching,Germany e-mail:[email protected] PeterWriggers,UniversitätHannover,Hannover,Germany e-mail:[email protected] AboutthisSeries Thisseriesaimstoreportnewdevelopmentsinappliedandcomputationalmechan- ics- quickly,informallyandata highlevel.Thisincludesthe fieldsoffluid,solid andstructuralmechanics,dynamicsandcontrol,andrelateddisciplines.Theapplied methodscanbeofanalytical,numericalandcomputationalnature. Moreinformationaboutthisseriesathttp://www.springer.com/series/4623 · Sergio Conti Klaus Hackl Editors Analysis and Computation of Microstructure in Finite Plasticity ABC Editors SergioConti KlausHackl Inst.fürAngewandteMathematik LehrstuhlfürMechanik-Materialtheorie UniversitätBonn Ruhr-UniversitätBochum Bonn Bochum Germany Germany ISSN1613-7736 ISSN1860-0816 (electronic) LectureNotesinAppliedandComputationalMechanics ISBN978-3-319-18241-4 ISBN978-3-319-18242-1 (eBook) DOI10.1007/978-3-319-18242-1 LibraryofCongressControlNumber:2015937885 SpringerChamHeidelbergNewYorkDordrechtLondon (cid:2)c SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia (www.springer.com) Preface ThepresentvolumepresentsresearchcarriedoutwithintheresearchgroupFOR797 “Analysisandcomputationofmicrostructureinfinite plasticity” (MICROPLAST), financed by the German Science Foundation (DFG). The research group was es- tablishedin 2007andcame to an endin 2015.Duringthisperiodmorethatthirty researcherscontributedactivelytothisendeavor,organizedineightsub-projects. The aim of MICROPLAST was to understandand modeldislocation based mi- crostructureformationand evolutionin materials, anddeveloptools to computeit effectively.This is importantbecause plasticity determinesa numberof industrial and natural phenomena, ranging from the deformation of the earth crust to form- ingofmetals.Mostplasticprocessesarestronglyinfluencedbytheformationand evolution of dislocation patterns and other plastic microstructures. Classical plas- ticity modelswereable to make usefulpredictionswithoutaccountingforthe mi- crostructure,andfocusingonaphenomenologicalunderstandingofthemacroscopic material response. These models have, however, strong limitations, concerning in particular the transferabilityof the results, the applicability to large deformations, tosmallsamples,andthetreatmentofageingandfatigue. The foundationof MICROPLAST was inspired both by new mathematical and modeling developments and by new experimental techniques. Around the turn of the millenniumnovelvariationalconceptsbecameavailableallowingto formulate finiteplasticityandassociatedmicrostructureformationwithinarigorousandpow- erfulmathematicalframework.The systematic variationalformulationof the evo- lutionofinternalvariablesviadissipationpotentialsanddissipationdistancesmade itpossibletoapplytheconceptsofrelaxationtheoryandΓ-convergence,thatwere originallydevelopedforstaticproblems,totheevolutionofinelasticmaterials. Atthesame timeexperimentaltechniqueshadadvancedto astagewhereitbe- came possibleto performin situ measurementsof dislocationmicrostructuresand their evolution, thus giving valuable information for the development of the cor- respondingmodels.Forthisreason,MICROPLAST wasestablishedasaninterdis- ciplinary cooperation of scientists from the fields of mathematics, mechanics and materialssciences,workingcloselytogetheronacommongoal. VI Preface This volume contains reports on the various achievements reached within MICROPLAST. These range from experimental evidence on the mechanisms of microstructureformationto micromechanicaland multiscale modelsof these pro- cesses.Theyincludederivationsofrelaxedenvelopesofnon-convexenergiesusing noveldevelopmentsofvariationalcalculus,aswellasfullmathematicalanalysisof themodelsestablished,andthedevelopmentofsuitableandfastnumericalschemes. Itisourhopethatwesucceededingivinganinsightintoanewdevelopingfield, andthatthereaderswillfindthematerialinthisvolumeinterestingandbeneficial fortheirresearch. Thecontributionsinthisvolumehaveundergoneaninternalpeer-review.Weare convincedthatthisprocessledtosignificantimprovementsinthepaperscontained inthisbook. Finally,wewouldliketothanktheGermanScienceFoundationfortheirgenerous supportandthealwaysverypleasantandprofessionalcooperation. April2015 SergioConti KlausHackl Contents 1 NumericalAlgorithmsfortheSimulationofFinitePlasticitywith Microstructures ............................................. 1 CarstenCarstensen,DietmarGallistl,BorisKrämer 1.1 Introduction............................................. 1 1.2 PreliminariesandNotation ................................ 3 1.3 ConvergentAdaptiveFiniteElementMethodfortheTwo-Well ProbleminElasticity ..................................... 4 1.3.1 ReviewoftheModelProblem ...................... 5 1.3.2 AdaptiveAlgorithm .............................. 6 1.3.3 ConvergencefortheDeformationGradient ........... 9 1.4 GuaranteedLowerEnergyBoundsfortheTwo-WellProblem ... 10 1.4.1 NonconformingFEMandDiscreteEnergy Functional...................................... 10 1.4.2 LowerEnergyBounds ............................ 12 1.4.3 GuaranteedErrorControlforthePseudo-stress........ 15 1.4.4 NumericalExperiments ........................... 16 1.5 DiscontinuousGalerkinMethodfor DegenerateConvex MinimizationProblems ................................... 17 1.5.1 OptimalDesignBenchmark........................ 18 1.5.2 DiscontinuousGalerkinMethods ................... 20 1.5.3 LiftingOperatorR ............................... 20 1.5.4 ConnectionwiththeNonconformingMethod ......... 21 1.5.5 AdaptiveFiniteElementMethod.................... 22 1.5.6 ComputationalExperiments........................ 22 1.5.7 Γ-shapedDomain ................................ 23 1.5.8 SlitDomain ..................................... 24 1.6 ConclusionsandOutlook.................................. 25 VIII Contents 2 VariationalModelingofSlip:FromCrystalPlasticitytoGeological Strata...................................................... 31 SergioConti,GeorgDolzmann,CarolinKreisbeck 2.1 Introduction............................................. 31 2.2 ExperimentalObservationofSlipMicrostructuresinNature .... 33 2.2.1 ChevronFoldsinRocks ........................... 34 2.2.2 KinkBandsinStacksofPaperunderCompression..... 34 2.2.3 SimpleLaminatesinShearExperimentsinCrystal Plasticity........................................ 36 2.3 TheHunt-Peletier-WadeeModelforKinkBands .............. 37 2.4 VariationalModelingofMicrostructure...................... 38 2.5 ModelsinCrystalPlasticitywithOneActiveSlipSystem ...... 41 2.5.1 VariationalFormulationofCrystalPlasticity.......... 42 2.5.2 RelaxationResultsinCrystalPlasticitywithOne SlipSystem ..................................... 44 2.5.3 HeuristicOriginoftheLaminates................... 46 2.5.4 RelationtoKinkBandsinRocks.................... 49 2.5.5 ElasticApproximation ............................ 51 2.5.6 Higher-OrderRegularizations ...................... 52 2.6 BeyondOneSlip-System.................................. 53 2.6.1 TwoSlipSystemsinaPlane ....................... 53 2.6.2 ThreeSlipSystemsinaPlane ...................... 54 3 Rate-IndependentversusViscousEvolutionofLaminate MicrostructuresinFiniteCrystalPlasticity ...................... 63 ChristinaGünther,DennisM.Kochmann,KlausHackl 3.1 Introduction............................................. 63 3.2 VariationalModelingofMicrostructures..................... 64 3.3 SingleSlipCrystalPlasticity............................... 67 3.4 PartialAnalyticalRelaxationviaLamination ................. 67 3.5 Rate-IndependentEvolution ............................... 70 3.5.1 EvolutionEquations .............................. 70 3.5.2 LaminateRotation................................ 71 3.5.3 LaminateInitiation ............................... 72 3.5.4 NumericalScheme ............................... 72 3.6 SimulationofRotatingLaminates .......................... 73 3.7 ViscousEvolution........................................ 75 3.7.1 EvolutionEquations .............................. 76 3.7.2 LaminateRotation................................ 77 3.7.3 LaminateInitiation ............................... 77 3.8 Comparison of the Laminate Evolution for the Rate- IndependentCaseandtheViscosityLimit.................... 78 3.9 ConclusionandDiscussion ................................ 85 Contents IX 4 VariationalGradientPlasticity:Local-GlobalUpdates, RegularizationandLaminateMicrostructuresinSingleCrystals .... 89 SteffenMauthe,ChristianMiehe 4.1 Introduction............................................. 90 4.2 AMultifieldFormulationofGradientCrystalPlasticity ........ 93 4.2.1 IntroductionofLong-RangeFieldVariables .......... 93 4.2.2 IntroductionofShort-RangeFieldVariables .......... 96 4.2.3 EnergyStorage,DissipationPotentialandLoad Functionals...................................... 99 4.2.4 Rate-TypeVariationalPrincipleandEulerEquations ... 102 4.2.5 ExplicitFormoftheMicro-forceBalanceEquations ... 103 4.3 AlgorithmicFormulationofGradientCrystalPlasticity......... 103 4.3.1 Time-DiscreteFieldVariablesinIncrementalSetting... 103 4.3.2 Update Algorithmsfor the Short-RangeField Variables........................................ 104 4.3.3 Time-DiscreteIncrementalVariationalPrinciple....... 105 4.3.4 Space-Time-Discrete IncrementalVariational Principle........................................ 106 4.4 Example1:AnalysisofanF.C.C.CrystalGrainAggregate...... 108 4.4.1 SlipSystemsandEulerAngles ..................... 108 4.4.2 Voronoi-TessellatedUnitCellunderShear............ 109 4.5 Example2:LaminateMicrostructureinSingleCrystals ........ 110 4.5.1 DoubleSlipSystems.............................. 111 4.5.2 ImplicationsofSamePlaneDoubleSlip ............. 112 4.5.3 LaminateDeformationMicrostructureinSingle CrystalCopper .................................. 114 4.6 Conclusion.............................................. 118 5 VariationalApproachesandMethodsforDissipativeMaterial ModelswithMultipleScales ................................... 125 AlexanderMielke 5.1 Introduction............................................. 125 5.2 VariationalFormulationsforEvolution ...................... 127 5.2.1 GeneralizedGradientSystems andthe Energy- DissipationPrinciple.............................. 128 5.2.2 Rate-IndependentSystemsandEnergeticSolutions .... 132 5.3 EvolutionaryΓ-Convergence .............................. 134 5.3.1 pE-convergenceforGeneralizedGradientSystems..... 134 5.3.2 pE-convergenceforRate-IndependentSystems........ 137 5.4 JustificationofRate-IndependentModels .................... 138 5.4.1 WigglyEnergiesGiveRisetoRate-Independent Friction ........................................ 139 5.4.2 1DElastoplasticityasLimitofaChainofBistable Springs ......................................... 141
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