Micromechanical Modeling Strategies Toward High-Fidelity Digital Twins of Polycrystals

ARENBERGDOCTORALSCHOOLFacultyofEngineeringScienceMicromechanicalModelingStrategiesTowardHigh-FidelityDigitalTwinsofPolycrystalsNikhilPrabhuDissertationpresentedinpartialfulfillmentoftherequirementsforthedegreeofDoctorofEngineeringScience(PhD):MaterialsEngineeringOctober2025Supervisor:Prof.Dr.-Ing.M.DiehlCo-supervisor:Prof.Dr.-Ing.M.Seefeldt MicromechanicalModelingStrategiesTowardHigh-FidelityDigitalTwinsofPolycrystalsNikhilPRABHUExaminationcommittee:Em.prof.dr.ir.J.DeSchutter,chairProf.Dr.-Ing.M.Diehl,supervisorProf.Dr.-Ing.M.Seefeldt,co-supervisorProf.dr.ir.A.VanBaelProf.dr.ir.Y.SwolfsProf.dr.ir.N.MoelansProf.Dr.-Ing.A.Hartmaier(Ruhr-UniversitätBochum)DissertationpresentedinpartialfulfillmentoftherequirementsforthedegreeofDoctorofEngi-neeringScience(PhD):MaterialsEngineeringOctober2025 ©2025KULeuven--FacultyofEngineeringScienceUitgegevenineigenbeheer,NikhilPrabhu,Celestijnenlaan200Abox2402,B-3001Leuven(Belgium)Allerechtenvoorbehouden.Nietsuitdezeuitgavemagwordenvermenigvuldigden/ofopenbaargemaaktwordendoormiddelvandruk,fotokopie,microfilm,elektronischofopwelkeanderewijzeookzondervoorafgaandeschriftelijketoestemmingvandeuitgever.Allrightsreserved.Nopartofthepublicationmaybereproducedinanyformbyprint,photoprint,microfilm,electronicoranyothermeanswithoutwrittenpermissionfromthepublisher. Preface Everychoicewemake,nomatterhowsmall,changesourpathinlife.Over time,thesesmallshiftsaccumulateandcarryustoplaceswecouldneverhave predictedatthestart.It’sabitlikethosestoriesofparallelworlds,whereonetinydecisionchangeseverythingthatfollows.IsometimeswonderhowdifferentlythingsmighthaveturnedouthadIbegunmyPhDatanother universityacrosstheDutch-speakingBelgianborderinsteadofatKULeuven, almostfiveyearsago. Lookingback,Iamdeeplygratefulforthedecisionsthatbroughtmehere,to thispointofcompletingthehighestacademicdegreeinmyfield.Thisjourney wouldnothavebeenpossiblewithouttheconstantsupportofmanypeople whoseinfluencehasshapedmeintothepersonIamtoday.Itrulybelievethat weare,inmanyways,theaverageofthepeoplewesurroundourselveswith, andIhavebeenfortunatetobesurroundedbytheverybest. Anobviousmentionamongthemismypromotor,MartinDiehl,whobeganhisprofessorialcareeratKULeuvenjustafewmonthsbeforeIstartedmy PhD.BeingequallynewtoLeuvenatthattimehelpedusbondaswenavigated theintricaciesofuniversitybureaucracyandtheadditionalchallengesbrought bythepandemic.Youweremorethanjustaformalsupervisor.Youwereamentorwhoguidednotonlytheworkbutalsothepersondoingit.Ineverycollaboration,Iwasimpressedbyhowdeeplyinvolvedyouwere,eveninthe smallestdetails,andyoualwaysmademefeellikeaco-authorratherthanjust astudent.ThisissomethingIwillcarrywithmeintomyownfuturecareer, whenIhavethechancetoguideothersatthestartoftheirjourney.Thankyou sincerelyfortheskills,insights,andconfidenceIhavegainedthroughworking closelywithyou. Iwouldalsoliketothankmyco-promoter,MarcSeefeldt,forthemanydiscussionswehavehadovertheyearsandforhisinsightfulanddetailedfeedbackonthiswork.Mysinceregratitudealsogoestothemembersofmy i iiPREFACE examinationcommitteefortheirsupportandcommitment,fromquicklyfindingasuitabledateforthedefensetocarefullyevaluatingthisthesisandengaginginacademicallyenrichingdiscussionsduringtheexamination.Twoofthechapters presentedinthisthesisalsoresultedfromcollaborationswiththeDepartment ofMaterialsEngineeringatKULeuven.IwouldthereforeliketothankPushkar andAkshayforinitiatingthesecollaborationsandturningthemintotangible outcomesthroughthejointpaperswearecurrentlypublishing. Afterwritinganentirethesisoncrystalplasticitymodelingthatrunsover200pages,itwouldbeunfairnottodedicateatleastapagetothankand acknowledgethepeoplewhowerenotofficiallyassociatedwiththisworkbut whoplayedamajorroleinkeepingmesaneinmypersonallife—somethingthatundoubtedlyaffectsprofessionalperformanceaswell.First,myNUMA colleagues,whograduallybecamefriendsthrougheverythingfromthewonderful NUMAhikestobeachvolleyballtournamentsandthesillyyetengaginglunchtableconversations:Giovanni,Ignace,Javier,Kobe,Pieter,Sanaz,Stijn,Tom, Toon,Vince,Wouter,andEdgar.Morethanfouryearscouldeasilyhavebeenfilledwithmanypost-lunchdrowsyafternoons,butthankstomyoffice mates—Evert,Leonie,Marlies,Nick,andYiqing—Istayedenergizedthrough ourperiodic(andsometimesunnecessarilylong)discussions.Ahugethanks alsogoestotheofficialpartyplanningcommitteeofNUMA,whomadesurewe hadafter-workfunactivitiesand,mostimportantly,thefundingtomakethem possible.Finally,IamverythankfultotheDutchlanguageexpertsinmyclose circle,TomandIgnace,whosefeedbackwasessentialforachievinganatural translationoftheabstractintoDutch. Iwouldalsoliketothankmynon-NUMAfriends,especiallyNishanthand Anjana,forthemanyweekendsspentexploringBelgium,tryingnewrestaurants, and,mostimportantly,keepingKannadaaliveinourconversationssothatI neverfelttoofarfrommyhometown.AnothercircleoffriendsIamgratefulforcamethroughahobbythatkeptmefitduringthistime—volleyball,myfavorite sportaftercricket(ofcourse).Thiswasalsoanamazingyearformyvolleyball aspirations,aswealmostwontournamentswithbothmynon-NUMAteamand myNUMAteam—awinningmomentIwillalwayscherishand,hopefully,the beginningofmanymoretocome. Finally,comingtothemostimportantpeopleinmypersonallife—thecoreof mysupportsystem.AanuandAmma,myfirstgurus,whonotonlytaughtme lettersandalphabetsbutalsolifelessonsthroughoutmychildhoodthatmade methepersonIamtoday.Withoutyourtrust,support,andencouragement, noneofthiswouldhavebeenpossible.AsImentionedinthebeginningabout thesmalleventsshapingthebranchesofparallelpossibilities,oneofthemost impactfuldecisionsthatgovernedthispathwasyourdecisiontosupportmein relocatingabroad,allthewaytoSweden,topursuemymaster’s.Whilethis PREFACEiii wasabigdecisionfinancially,youcushionedmefromallthechallengesthat camewithit.Thelove,support,andtrustyouhaveshownfrommymaster’s daystonowareunparalleled,andItrulyoweeverybitofmysuccesstoyou. Youhavesetthebarveryhighforparenting,andIcanonlyhopetobehalfas goodarolemodeltomychildrenasyouhavebeentome. Tomybrother,Nitishanna—fromgrowingupandplayingtogethertostanding shouldertoshoulderthroughsharedresponsibilitiesanddecisions—thankyou foryourconstantsupport.IwishyouandVarshahonniallthebestforthenew andexcitingphaseoflifeahead. Manythanksalsotomyin-laws,RavindramamandRoopamai,andtheirsupportivefamily,forstandingbymeandtrustingmethroughallthe uncertaintiesandimplicationsthisPhDbroughttomypersonallife.Iwould alsoliketotakethisopportunitytorememberthosewhoarenolongerwith us—AjjaandGanapathiBappa—whoseblessingscontinuetoguideme. Andfinally,mybetterhalf,Lakshmi,orasIfondlycallherKittu.Youentered myliferightwhenIwasabouttomoveabroadformymaster’s,andreturningtotheanalogyofbranchingpaths,yourarrivaliswithoutdoubtthemostimportant eventalongthisjourney.Yourconstantsupport,comfort,andbeliefinme haveliftedmethroughmanydifficultmoments,andIfeeltrulyluckytohave yourcompanionship.AlthoughmuchofmyPhDwasspentinalong-distance relationship,IamverygladthatwegottoshareitsfinalphasetogetherwhileI waswritingthisthesis.Youhavebeenthelargeandsignificantjigsawpiecein thepuzzleofmylife,providingstabilityandclaritywheneverthingsseemeduncertain.Yoursupportmeansalottome,andIamexcitedtocontinue navigatingthefuturetogetherwithyoubymyside. IamsureIhavemissedmentioningmanynamesofpeoplewhohaveplayed importantrolesatdifferentpointsalongthisjourney.Iwouldliketoexpressmy gratitudetoallofthemaswellformakingthisamemorablechapterofmylife. PopularizedAbstract Sciencefictionoftenimaginesengineersdesigningfuturisticmachinesentirely inthedigitalworld,wherevirtualdesignsaretestedandperfectedbeforebeingsentdirectlytoproduction.Intherealworld,progresstowardthisvisionismade throughtheconceptofdigitaltwin—avirtualcopyofaphysicalcomponent,keptinsyncwithitsrealcounterpartthroughaflowofinformation.Atthe coreofsuchtwinsarecomputationalmodelswhosepredictionsaregroundedin mathematicalformulations,experimentalobservations,orboth.Mostcurrent digitaltwinsrelyonmodelsthatcaptureonlythevisible,macroscalefeatures ofacomponent.Yetthestrengthandreliabilityofcrystallinematerialsaredeterminednotonlybywhatisvisibleonthesurfacebutalsobyunderlying featuresthatarenotimmediatelyapparent:fromtheinteractionsofmicroscopic crystals,orgrains,tothearrangementofatomsthemselves. Tomakedigitaltwinsmorereliable,itisessentialtozoomclosertothegrain scale—themesoscale—wherethecollectivebehaviorofmanyinteractinggrains determineshowthelargercomponentresponds.Inthisresearch,mesoscale simulationswererefinedtoenhancetheirreliabilityandpredictivepower.The workfocusedoncrystalplasticitymodeling,whichdescribeshowcrystalline materialsdeform,andwasinformedbyadvancedexperimentaldata.Thesedatarichexperimentsrevealedmaterialbehaviornotonlyatthelevelofpolycrystal aggregatescontainingoverthousandsofgrainsbutalsoatthelevelofindividual grains.Byintegratingthisinformation,themodelsbecamemorephysically groundedandmorereliableevenwhenappliedbeyondtheconditionsoftheir initialcalibration. Therefinedmodelsweretestedinscenariosinspiredbyexperimentstoexamine whetherfaintcorrelationsobservedinmeasurementstrulyreflectedcause-andeffectrelationships.Thisanalysisrevealedunderlyingmechanismsthatcould explainwhycertaintrendsemergeinexperimentaldata.Atthesametime,the studiesalsoemphasizedtheneedforrobustandreliableprocessingofdatasets fromcomplexexperiments,astheseproceduresoftenrequirematerial-specific v xiiBEKNOPTESAMENVATTING gedomineerderegimeaccuraatvoorspellen,hunvoorspellendekwaliteitafneemt zodraplastischemechanismenactiefworden,waarmeebeperkingeninde constitutievebeschrijvingenduidelijkwerden. Omditprobleemaantepakken,leiddehetopnemenvanaanvullendeexperimen-teledataindemodelinitialisatietotaanzienlijkeverbeteringeninvoorspellende capaciteiten.Ditwerdgerealiseerddooreennieuweiteratievestrategieteontwikkelenomkorrelsteinitialiserenmetexperimenteelgemetenelastischerek,terwijlcompatibiliteits-enevenwichtsvoorwaardenbehoudenbleven.Dezeaanpakresulteerdeinduidelijkverbeterdemesoschaalvoorspellingenvanelastischerekbijmeerderestadiavanmacroscopischebelasting,metde grootsteimpactinhetelastisch-gedomineerderegime,waarmeehetbelangwordtaangetoondvanhetvolledigbenuttenvanbeschikbareexperimenteleinformatie bijdeconstructievandigitaletweelingen. Dederdevraagbelichteengroottekortvanfenomenologischemodellen, namelijkhunverminderdebetrouwbaarheidbuitenhetgekalibreerdedatagebiedvergelekenmetfysisch-gebaseerdebenaderingen,ondankshunbekendeefficiëntie eneenvoud.Eenmogelijkeverbeteringwerdgevondenindeparameterisatie vandeverhardingsmodulimatrixdiedeinteractiestussenglijvlakkenbepaalt, welketraditioneelsteuntopvereenvoudigingendiealdecenniateruggaan.Deze werdherziendooreenfysisch-geïnformeerdeparameterisatieintevoerendie zowelvervormingsgedragalsmateriaaltoestandconsistenterkonreproduceren buitenhetkalibratiegebied,inclusiefbelastingstrajectwisselingen.Metdeze verfijningbehoudenfenomenologischemodellenhunvoordelen,terwijlzijnauwer aansluitenbijfysisch-gebaseerdevoorspellingenovereenbrederscalaaan belastingscondities. Devierdevraagbreiddehetonderzoekuitnaarmaterialenmethogereanisotropie,inhetbijzonderdehexagonaaldichtgestapeldestructurenvanTi-6Al4V,waarzowelfenomenologischealsfysisch-gebaseerdemodellenuitdagingen ondervindenbijbetrouwbareparameterkalibratie.Dezemoeilijkheidissterker bijeerstgenoemde,omdathetgebrekaanfysischeonderbouwingvande parametersertoeleidtdatzevaakfungerenalsafstemmingsvariabelen,waardoor hununiekeidentificatiewordtbemoeilijkt.Omditaantepakken,werdeenmeerstapsoptimalisatiekaderontwikkeld,gebruikmakendvanuitgebreidein situhoge-energieröntgendiffractiegegevensverzameldtijdenstrekproeven.Door zowelmacro-alsmesoscopischeresponsenindedoelfunctieoptenemen, kwamendegekalibreerdeparametersnietalleenovereenmetliteratuurwaarden, maarvertoondenzijookconsistentevoorspellendeprestatiesindaaropvolgende validatiestudies.Naastdekalibratieleverdedezestudienieuweinzichtenin glijactiviteit,korrelheroriëntatieentextuurevolutietijdensvervorming,waarmee hetcrucialebelangvangoedgeïdentificeerdeparametersvooreennauwkeurige modelleringvanhexagonaaldichtgestapeldesystemenenuiteindelijkvoor BEKNOPTESAMENVATTINGxiiibetrouwbaredigitaletweelingenwerdbenadrukt. Totslotwerdendegekalibreerdeparameterstoegepastineenstructuur-eigenschapkoppeling,waarinmicrostructurelekenmerkenwerdenverbondenmetrestspanningsstatenineenwarmgewalsteTi-6Al-4V-legeringnaeenslijpschijf-snijbewerking.Gelokaliseerdeuit-het-vlakrestspanningsmetingen,verkregenviadefocusionenbundelincombinatiemetdigitalebeeldcorrelatienabijhetsnijoppervlak,toondenonverwachthogetrek-endrukkrachten,metsubtielecorrelatiesmetgelaagdemacro-getextureerdegebieden.Omcausaliteittetoetsenendebepalendeconditiesteidentificeren,werden kristalplasticiteitssimulatiesuitgevoerdoprepresentatievemicrostructurenonder vereenvoudigdebelasting.Destudiebevestigdedatdeoriëntatievandemicrogetextureerdegebieden—diezowelelastischealsplastischeresponsenstuurten wordtbeïnvloeddoornabuurinteracties—combinatorischderestspanningsstaat beïnvloedt.Destudiebevestigdekwalitatieveen,inaanzienlijkemate,kwantitatieveovereenstemmingmethetexperiment,enlietbovendienzien hoebetrouwbareparametersdigitaletweeling-raamwerkeninstaatstellenom microstructuurzinvoltekoppelenaanmacroscopischeprestaties. Samenvattendtoontditproefschriftaandatdevijfonderzoeksgebiedennietalleenhunrespectievelijkevragenmetvertrouwenbeantwoorden,maargezamenlijkookdemesoschaal-modelleerstrategieënverderbrengeninde richtingvanhooggetrouwedigitaletweelingenvanpolykristallijnematerialen. Tegelijkertijdblijktdathetverbeterpotentieelopdezeschaalenormis,endatdit werkslechtshetbeginvormtvandevelemogelijkhedenvoorverdereverfijning eninnovatie. ListofAbbreviationsBCCbody-centeredcubicCDDcontinuumdislocationdynamicsCPcrystalplasticityCRSScriticalresolvedshearstressDAMASKDüsseldorfAdvancedMaterialSimulationKitDDDdiscretedislocationdynamicsDICdigitalimagecorrelationEBSDelectronbackscatterdiffractionFCfullconstraintsFCCface-centeredcubicFEMfiniteelementmethodFFTfastFouriertransformFIBfocusedionbeamHCPhexagonalclose-packedHEDMhigh-energyX-raydiffractionmicroscopyHEXRDhigh-energyX-raydiffractionIPFinversepolefigureLPBFlaserpowderbedfusionMDmoleculardynamicsMECmodelingefficiencycoefficientMTRmicro-texturedregionxv xviLISTOFABBREVIATIONSNDnormaldirectionRDrollingdirectionRVErepresentativevolumeelementSEMscanningelectronmicroscopyTDtransversedirectionTEMtransmissionelectronmicroscopy ListofSymbols𝐾bulkmodulus𝑀Taylorfactor𝐅deformationgradienttensor𝛾shearstrain𝜇shearmodulus𝜈Poissonratio𝜏resolvedshearstress𝐈identitytensor𝐋velocitygradienttensor𝐌Mandelstresstensor𝐑rotationtensor𝐒secondPiola–Kirchhoffstresstensor𝐔rightstretchtensor𝐕leftstretchtensor𝝈Cauchystresstensor𝜺Green–Lagrangestraintensor𝐧slipplanenormal𝐬slipdirection𝜉slipresistancexvii ContentsPopularizedAbstractvAbstractviiBeknoptesamenvattingxiListofAbbreviationsxviListofSymbolsxviiContentsxixListofFiguresxxiiiListofTablesxxvii1Introduction11.1DigitalTwinsintheWorldofMaterials.............11.2StrategiestoModelMechanicalBehaviorofCrystals......51.3KnowledgeGapatMesoscale...................81.4CrystalPlasticity..........................101.4.1FromSingleCrystaltoPolycrystals...........111.5ObjectiveoftheThesis.......................141.6OutlineoftheThesis........................162Background172.1ContinuumMechanics.......................172.1.1FiniteStrainKinematics..................182.1.2MeasuresofStress.....................212.2SimulationFramework.......................232.2.1Decompositionof𝐅....................23xix xxCONTENTS2.2.2TimeIntegration......................252.3ConstitutiveModeling.......................272.3.1DislocationDensity-basedModel.............282.3.2PhenomenologicalModel.................322.3.3IsotropicPlasticitywithDilatation............332.4ExperimentalCharacterizationTechniques............342.4.1FundamentalsofX-RayDiffraction............362.4.2High-EnergyX-rayDiffraction..............372.4.3High-EnergyX-rayDiffractionMicroscopy........383ComparisonofFull-FieldCrystalPlasticitySimulationstoHigh-EnergyDiffractionMicroscopyExperiments413.1Introduction.............................423.2SimulationSetup..........................433.2.1MicrostructureModel...................443.2.2ConstitutiveModelsandParameters...........443.2.3BoundaryConditions...................483.2.4InitialElasticStrain....................493.3ResultsandDiscussion.......................513.3.1IncorporationofInitialElasticStrain..........523.3.2OverallPredictionQuality.................533.3.3MonotonyoftheElasticStrainEvolution........573.3.4StressEvolution......................623.3.5VolumetricStrain.....................643.4ConclusionsandOutlook.....................654IncorporationofPhysics-basedStrengtheningCoefficientsintoPhenomenologicalCrystalPlasticityModels694.1Introduction.............................704.2Dislocation–DislocationInteractionsinFCC...........744.3ParametrizationsoftheHardeningModuliMatrix.......794.4MaterialParametersCalibration.................804.5ModelPerformanceAssessment..................814.5.1MicrostructuresandTextures...............824.5.2CalibrationLoadCases..................824.6MeasurementofCorrelation....................834.7EvaluationofPredictiveCapabilities...............844.8ResultsandDiscussion.......................854.8.1HardeningBehavior....................864.8.2DeformationBehavior...................894.8.3PredictiveCapabilitiesofthePhenomenologicalModels904.9SynthesisandInterpretationofResults..............934.10ConclusionsandOutlook.....................95 CONTENTSxxi5DeterminationofCrystalPlasticityParametersfromSynchrotronDiffractionExperiments975.1Introduction.............................985.2Background.............................1015.2.1Ti-6Al-4V..........................1015.2.2AdditiveManufacturingofTi-6Al-4V..........1025.2.3PhasesinAMTi-6Al-4V.................1025.2.4MechanicalCharacterizationofthePhases........1035.3ExperimentalData.........................1045.4InverseModeling..........................1075.4.1SimulationSetup......................1075.4.2ComputationofLatticeStrains..............1095.4.3MultilevelMaterialParametersCalibration.......1105.4.4FormulationofLossFunctions..............1125.4.5InitialGuessValuesandConstraints...........1135.5ResultsandDiscussion.......................1155.5.1ElasticConstants......................1165.5.2PlasticConstitutiveParameters..............1185.5.3EvolutionofIntensities..................1275.5.4GrainReorientation....................1295.5.5DominantSlipModes...................1355.6ConclusionsandOutlook.....................1376Microstructure–MechanicalPropertyLinkageStudyonTi-6Al-4VAlloy1396.1Introduction.............................1406.2Experiment.............................1436.2.1Background.........................1446.2.2Observations........................1446.3GuidingInsightsfortheComputationalStudy..........1476.4SimulationSetup..........................1486.4.1AppliedBoundaryConditions...............1486.4.2Microstructures.......................1496.5ResultsandDiscussion.......................1506.5.1Test-Case1:Bicrystal...................1506.5.2Test-Case2:PolycrystalcontainingMTRs........1576.5.3GrainOrientation,SlipActivity,andResidualStress..1626.6ConclusionsandOutlook.....................1667ConclusionsandOutlook169AChallengeGrainsandTheirNeighborhoods179 Chapter1Introduction Imaginearoveronadistantplanetpreparingtotraversetreacherousterrain,a criticalcomponentwithinanuclearreactorenduringextremetemperatures,orastructuralelementsubmergeddeepintheoceanunderimmensepressure.Insuch high-stakesanddynamicenvironments,decisionsabouttheperformanceand safetyofthesecomponentsnolongerrelysolelyonsimulationsperformedduring theirdesignormanufacture.Instead,theydependonreal-timemonitoringthroughsensorsandpredictiveinsightsthroughcomputationalmodels.At present,thereal-timeaspectinvolvingmicromechanicalbehaviorstillfeelslike afuturisticidea,largelybecauseofboththeslowpaceofsimulationsandthe difficultyofnon-destructivelyextractingbulkmicroscaledataevenfromstate-of-the-artcharacterizationtechniques.However,withadvancesincharacterization andcomputation,thispossibilityismovingclosertorealityjustastheconcept ofdigitaltwins[1,2],whichwasonceseenasanascentandambitiousvision, nowhasrapidlymaturedformacroscalecomponentsoverthepasttwodecades. Whetherreal-timeornot,acriticalelementofanyhigh-fidelitydigitaltwinis theunderlyingcomputationalmodels,whosecontinueddevelopmentisessential forrealizingthistransformativeshift.1.1DigitalTwinsintheWorldofMaterials Intheworldofproductlifecyclemanagement,adigitaltwincanbeunderstood asavirtualrepresentationofaphysicalproductwithabi-directionaldataconnectionbetweentherealproductanditsvirtualcounterpart[2,3].Thephysicalproductcouldwellbeasinglecomponentoracomplexsystemof 1 2INTRODUCTION components,forexample,aturbineengine[4,5,6,7].Recentadvancesinsensortechnologiesenablereliable,real-timemonitoringofthephysicalsystemsduringoperation,althoughstilllimitedtothecomponentscale[8].Thecollecteddataarethentransmittedthroughaninformationchannelto replicateenvironmentalconditionsonthevirtualcounterpart.Thisenablesthe coexistenceofphysicalanddigitalsystems,wherethedigitaltwinoperatesunder sensor-drivenconditionstosupportproactiveandreactivedecisionmaking[9, 10]. Whilethedefinitionofdigitaltwinsvarieswiththefieldofapplication[11],a commonprerequisiteremainscentraltotheconcept:accuratestructuraland functionalreplicationofthephysicalcounterpart.Inpractice,atypicaldigital twinofasystemofcomponentsfocusesprimarilyoncapturingmacroscale geometricfeaturesandcomponent-wiseperformancemetrics,whichencapsulate structuralandfunctionalcharacteristics,respectively[3,10].Theperformance metricofacomponentmadeofapolycrystallinematerialisoftenderivedfrom computer-aidedengineeringtoolsthathomogenizethematerialstate,reducedordermodeling,orfromdata-driventechniques[3,12].However,theextensive knowledgeaccumulatedoverthepastcenturyonpolycrystallinebehaviorhas shedlightonwhysuchsimplificationsmayleadtomisrepresentationsata foundationallevel. Itiswellestablishedthatthebehaviorofamaterialatthelengthscaleofacomponentdependsonunderlyingmechanismsatsmallerlengthscales.Fig.1.1schematicallyillustrateshowtheperformanceofacomponentinamechanicalsystemdependsnotonlyonitsmacrostructurebutalsoonpropertiesderivedfromitsmicro-andnanostructures.Furthermore,asthe materialrespondstoappliedloading,concomitantchangesoccurinitsmicroandnanostructures,whicharetypicallyomittedinsimplifiedstructuraland functionalrepresentations. S.R.Kalidindietal.[3]attributethisissuetothetraditionallysiloednature ofproductdevelopment,wheremechanicalormanufacturingengineersarecommonlyinvolved,oftenwithoutleveragingtheprocess-structure-property linkageinsightsprovidedbythefieldofmaterialsscienceandengineering.The authorsalsoenvisionsignificantimprovementsindigitaltwindevelopmentthroughtheintegrationofmaterialsscienceexpertise.Theirapproachgoes beyondderivingfunctionalresponsefromperformancemetricsalone,instead treatingthematerialasacomplexmultiphysics,multiscalesystem,whosestructureandpropertiesevolveovertime.Theyproposedamathematical frameworkthatemploysstatisticaltoolssuchasn-pointspatialcorrelationsand principalcomponentanalysisformaterialstructurerepresentation,alongwith aframeworkcenteredaroundtwo-stepBayesianapproachtoenablemultiscale functionalpredictions.Whiledata-drivenapproachesenhancecomputational DIGITALTWINSINTHEWORLDOFMATERIALS3 Figure1.1:Aschematicdepictionofmultiscalefeaturesofamaterialssystem ofacomponent,whicheventuallydetermineitsresponsetotheexternalloading. Thebidirectionalarrowsindicatescalebridgingthroughhomogenizationor localization.FigurereproducedfromS.R.Kalidindietal.[3] 4INTRODUCTION efficiency,theystillrelyfundamentallyonphysics-basedinsights,eitherfrom costlyexperimentsorfromsimulationtechniquesthatfaithfullycapturethe underlyingphysicalbehavior.Inaddition,theavailabilityofreliablesimulation techniquesconsiderablyreducestheneedforextensiveexperimentalefforts. TheapproachproposedbyS.R.Kalidindietal.[3]advocatesastochastic frameworkfordigitaltwinsofmaterialssystems,mainlybecauseitfocuseson macroscalecomponentscomposedofalargenumberofgrains.Atthisscale,nondestructive3Dcharacterizationofstructureandpropertiesisoftenimpractical. Furthermore,thematerialsexhibitnon-deterministicbehavior,meaningthat evenunderthesameprocessingconditions,theresultingmicrostructuremaynotbeidentical.However,inapplicationsthatdonotconcernbulkpolycrystals,for instance,ametalinterconnectonanintegratedcircuit[13,14],thesizeofthe componentisusuallyintheorderofthesizeofindividualgrains.Insuchcases, adeterministicdigitaltwinmaybemoreappropriate,asthestructureand propertiescanbeexplicitlycharacterizedandincorporated.Here,ahigh-fidelity simulationtechniquethatfaithfullycapturestheobservedphysicalmechanisms isessential. Digitalthreads[15,16,17]areanotherimportantconceptinproductlifecycle management,thattightlycoupleswithdevelopmentofhigh-fidelitydigitaltwins.Inthecontextofpolycrystallinecomponents,adigitalthreadcanbeviewedasa unifieddataframeworkthatlinksinformationgatheredacrossallphasesofthe component’slifecycle,fromconceptanddesigntomanufacturing,operation, andeventualretirement[15].Itenablesthecontinuousgenerationandupdating ofdigitaltwinsbyprovidingcomprehensiveaccesstoextensiveexperimental andoperationaldata. Theaccumulationofdetailedinformationacrosstheentirelifecycleofacomponentofferssignificantpotentialforthedevelopmentofhigh-qualitydigitaltwins.Additionally,supportingthisdevelopment,advancesinnondestructivematerialcharacterizationtechniquesnowenabletheextractionof datanotonlyfromthesurface,butalsofromthebulkofthematerial[18].For example,suchmethodscannowprovideexplicit3Dmicrostructures,residual stressstates,andotherbulkcharacteristics[19,20].Animportantchallenge thatarisesishowsuchextensivedatacanbeefficientlyutilizedtoenablethe developmentofhigh-fidelitydigitaltwins.Amajorsteptowardthisgoalis theadaptationofcontemporarymathematicalframeworks,whichcomprisea myriadofcomputationalmodelingtechniques,toincorporatestrategiesthat effectivelyleveragetheextensivebodyofdatanowavailable. STRATEGIESTOMODELMECHANICALBEHAVIOROFCRYSTALS51.2 StrategiestoModelMechanicalBehaviorof Crystals Figure1.1illustratedhowmacroscopicbehaviorofapolycrystallinecomponent isgovernedbyfeaturesatsmallerlengthscales.Inthelowerrightcornerofthefigure,theexternalenvironmentalfactorsthatinflictchangesinthe materialstateofengineeringcomponentsarealsolisted.Giventhevastarray ofpotentialphysicalphenomena,thefocusofthisthesisisnarroweddownto modelingelastoplasticbehaviorwhenanexternalmechanicalloadisapplied. Incrystallinematerials,giventhetemperatureisbelow ≈1/3 ofmeltingpoint andstrainrateisbelow ≈105 s −1 ,plasticdeformationisdominatedbymotion of1Dlinedefectscalleddislocations[21,22].Forthisreason,itistypicallythe predominantfeatureofphysics-basedmodelsformulatedforcrystalplasticity [23,24]. Dislocationsareone-dimensionallinedefectsinanorderedlatticeofacrystal,oftenobservedasline-liketracesonthesurface[21].Dependingonthelength scalethatisconsidered,differentfeaturesofthedislocationsinacrystalcanbeusedinplasticitymodeling.Thelengthscalemayrangefromthesizeofseveral atoms( ≈10−10 m)tothesizeofindustriallyrelevantengineeringcomponents ( ≈101 m).Modelingplasticityincrystallinematerialsthereforerendersa challengeofcoveringawiderangeoflengthscales.Amyriadoftechniquesexist tosimulateplasticityandthemostrelevantonesarediscussedinthefollowing sectionsinincreasingorderoftheirlengthscaleofapplicability.AbInitioCalculations Asone-dimensionaldefects,thedislocationshavebothlong-rangeandshort-rangeimplicationsinthelattice.Thelong-rangestressfieldinducedbythedislocationareconvenientlymodeledwithintheframeworkofdislocationelasticitytheory[22].Theshort-rangestressfieldatthedislocationcore,however,istooimportanttobemodeledbyelasticitytheoryanddemandsanexplicitelectronicdescriptionoftheatomicbonding[25].Toaddressthis necessity,theabinitiocalculationsbasedondensityfunctionaltheoryadoptsa quantummechanicsapproachtomodelatomicbondingandinteractions[26,27].Thecalculationsarebasedonatomicnumberandotherfundamentalquantities andthereforeareconsideredtobeveryaccurate,havefewestofempiricalparameters,andareobjectivewithrespectivetotheenvironment[25].Theyprovidevaluableinsightsondislocationmobility,cross-slip,andshort-range interactionswithotherdefectswhichcanbeutilizedinsimulationsathigher scale.Despitetheiraccuracyinrealisticallyrepresentingtheelectronicstructure 6INTRODUCTION throughaquantum-mechanicalframework,thesemethodsaretypicallylimited tomaterialvolumescontainingonlyafewhundredatomsduetotheirimmense computationalcost.AtomisticMethodswithEmpiricalPotential Thehighcomputationalcostsofabinitiocalculationscanbeovercomeby adoptingtheapproachofclassicalmechanicsinsteadofquantummechanics.In thisapproach,ratherthanexplicitlyaccountingfortheelectronicstructuresofmaterials,theatomicinteractionsaredescribedbyempiricalinteratomic potentials[28,29].Differentstrategiesareadoptedtoformulatefunctionalformsoftheinteratomicpotentialsthatarebasedontheory,intuition,andfitstoboth experimentalandabinitiodata[28].Moreover,recentadvancesinmachinelearningalgorithmshaveledtothecomputationallyefficientreconstructionofpotentialenergysurfaces[30,31].Thesesimplifications,incomparisonto abinitiocalculations,reducethecomputationalcostsandallowsimulationof materialvolumescontaininguptomillionsofatoms[23].Inthecontextoftime andlengthscalesofdislocations,adislocationwithalengthof 100 Burgersvectorscanbemodeledovertimescalesofnanoseconds[23].Importantly, thechoiceofinteratomicmodelsshouldbemadecarefully,ensuringthatthey adequatelycapturethephysicsofinterest,astheyformthefoundationofMD simulations.DiscreteDislocationDynamics Asacrystallinematerialdeformsplastically,thedislocationsmoveandinteract (formstructures,multiply,orgetannihilated)withotherdislocations.During interaction,thedislocationsformcellstructuresthataretypicallyontheorder of 1µm ,givenamoderatetohighdegreeofplasticdeformation[32].Ifthiswere tobesimulatedusingatechniquethatexplicitlymodelsindividualatoms,it wouldrequiremorethan 1010 atoms,which—giventhecurrentstate-of-the-art atomistictechniques—wouldbecomputationallychallenging[33,34].Toaddressthisissue,astepfurtherinthelengthscalecomesdiscretedislocationdynamicssimulations(DDD)[35,36],wherethematerialisnolongerdescribedbyatoms butsolelybydislocations,whosemovementandinteractionsgovernplasticbehaviorinamaterial.InDDDsimulations,thedislocationsaremodeledascurvedlinesin3Dspace,andtheirinteractionswithotherdislocationsare explicitlycaptured[37].Thisallowssimulationvolumesontheorderoftensof micrometersovertimescalesofmilliseconds.Theaveragedislocationdensity inmetalsistypicallyaround 1010to1015 meterspercubicmeterofvolume, STRATEGIESTOMODELMECHANICALBEHAVIOROFCRYSTALS7 andthistechniquetrackstheevolutionofeachoneofthem.Asthedislocation densityincreaseswiththeappliedstrainduetomultiplicationevents,the applicabilityofthistechniquebecomeslimitedbythemagnitudeofdeformation [38].Furthermore,itrequiresverysmalltimestepsintheintegrationscheme toaccuratelyresolvedislocationmotionandinteractions,particularlyamong closelyspaceddislocations[34].ContinuumDislocationDynamics Thedensity-likedescriptionofdislocationsincontinuumdislocationdynamics (CDD)primarilyaimstoaddressamajorlimitationoftheDDDtechnique: directproportionalityofthecomputationalcostwiththenumberofdislocation segments[39,40].Moreover,itfacilitatesstraightforwardintegrationofthedislocationplasticitywiththeelasticresponseofthematerialtoachieveaunifiedcontinuumframework.However,compressingtherichandcomplex informationofintricateanddiscretedislocationnetworksintoaveragedensitylikevariablesinacontinuumsettingfacesnumerouschallenges.Totacklethe averagingproblemindiscrete-to-continuumtransitions,variousapproacheshave beendevelopedtosuccinctlyrepresentdislocationnetworks.Thefoundational worksofNye[41]andKröner[42,43],whichwaslaterextendedtoarate-dependentformbyMura[44],characterizeddislocationnetworksintermsofadislocationdensitytensor,derivedfromthegeometricalbasisofplastic distortion.However,theirdescriptionaccountedonlyforgeometricallynecessary dislocations,makingitsensitivetothesizeofthechosenvolumeelement.To overcomethislimitation,Hochraineretal.[45]extendedthedensitydescription intohigher-dimensionalconfigurationspaces,wherethedislocationdensities alsocapturedinformationaboutthelineorientation.Inadditiontoaimingforamorecompleterepresentationofdislocations,thesemodelsgenerallyincorporate dislocationtransportbetweenintegrationpointsandarethereforeclassifiedas non-localmodels.Suchmodelsareparticularlyimportantwhenintragranular variationsinphysicalquantities,suchasdislocationpile-upsatgrainboundaries,areofprimaryinterest.Duetothehighretentionofinformationduringdiscreteto-continuumconversion,thecomputationalcostassociatedwiththisapproach remainssignificantlyhigh.Toaddressthis,lower-orderdescriptionshavebeen proposed,andstillremainsanactiveareaofresearch[46,47,48].Thistechniqueiscommonlyclassifiedwithinthebroaderclassofcrystalplasticityformulations andrequiresextensiveparameterizationtocapturedislocationmotionand complexinteractions. 8INTRODUCTIONCrystalPlasticity ThefieldofCDDaimstoretainthemaximumamountofinformationfrom discretedislocationnetworksobservedatthemicroscale,andthereforeserves asabridgebetweenmicroscaleandmacroscaledescriptionsofthematerial. Crystalplasticity,ontheotherhand,seekstoabstracttheexplicitdescription ofdislocationsegmentsinto,forinstance,scalar,density-likevariables[49,24]. Sinceintragranularvariationofquantitiesisnotoftentheprimaryfocus,the constitutivedescriptionsaretypicallylocalinnature.Adistinctfeatureofcrystal plasticityisthenotionofslipsystems,whereinplasticdeformationofagrain isrestrictedtocontributionsexclusivelyfromaspecificsetofcrystallographic planesanddirections.Throughthesesimplifications,crystalplasticitystrikes aneffectivebalancebetweenthelevelofphysicaldetailandtheassociated computationalcost.MacroscopicPlasticity Althoughtheclassicalcontinuummacroscopicplasticityapproachisbeingpresentedlastinthepresentlistoftechniques,effortstomodeltheinelastic responseofmacroscalemetalcomponentsdatebackmorethantwocenturies [50,51,52].Notably,theconceptofayieldsurfacepredateseventhetheoryof dislocations.SincetheworksofTresca[53]andvonMises[54],numerousyield criteriahavebeenproposedoverthepastcenturytoempiricallyfitthedistinct behaviorsofindividualmaterialsobservedinexperiments[55,56].Anextensive databaseofmacroscopicmaterialproperties,includingparametersforvarious yieldcriteria,hasbeendevelopedoverseveraldecadesandisnowroutinely usedinmacroscalesimulationsofengineeringcomponents.Amajoradvantage ofthesimulationsatthisscaleisitsabilitytorealisticallyreproducethe boundaryconditions,whichcanoftenbecapturedusingsensordata.However, thisapproachrequiresextensivecalibrationofmaterialparameterswhichare mostlyempirical,employshighlysimplifieddescriptionofthematerial,andlackspredictivecapabilitiesatsmallcomponentscalesorinthepresenceof microstructuralheterogeneities.1.3KnowledgeGapatMesoscale Intheprevioussection,materialbehaviormodelingstrategiesdeveloped specificallytoaccountforfeaturesobservedatdifferentlengthandtimescales werediscussed.Itisevidentfromthediscussionthatnosinglemodeling KNOWLEDGEGAPATMESOSCALE9 techniquecanfullydescribematerialbehavioracrosstheentirerangeoflength andtimescales.Whileatomisticsimulationsexplicitlycharacterizeindividual defectsandareabletocapturetheirpotentialinteractionconfigurations(forexample,dislocation–grainboundaryinteraction),thistechniqueisnotpracticallyscalabletomodelcomponentbehaviorinvolvingpotentiallyoverbillionsofinteractions.Althoughthefirst-principlesmodelsareenvisagedtobeextremelyaccurate,theapplicationofthesemodelstoengineeringscales is—atthispointintime—notonlyunlikelybutarguablyunnecessary[57].At theoppositeendofthespectrum,theclassicalmacroscopicplasticitymodels havebeenroutinelyemployedtomodelmacroscopicbehaviorofengineeringcomponents.Thesemodelsarecomputationallyefficient;however,despite theextensiveamountofmaterialtestinginvolvedinempiricallytailoringthe behavior,theylackpredictiveabilitiesbecauseoftheomissionofintrinsicmicrostructuralfeaturesthatgoverntheirbehavior.Thisiscritical,asthe behaviorofengineeringcomponentsatmacroscaleissignificantlyimpactedby microscaleheterogeneitiessuchastexture,grainboundariesanddefects[58]. Thisclearlystressestheneedtobridgethetwoextremes:aknowledgegapthat mesoscalemodelingstrategiesaimtoaddress[58]. Sincemesoscaletechniquesaimtobridgethegapbetweenmicroscaleand macroscalestrategies,thisrequiresanexhaustivedescriptionofthematerial, resultinginahighnumberofinputparameters,asdepictedinFig.1.2.While larger-scaletechniquesareabletoreproducecalibratedmaterialbehaviorwith arelativelylownumberof—mostlyempirical—inputparameters,first-principles calculationsrequirenoempiricalinputparametersbesidesthefundamental materialproperties.Thehighnumberofinputparametersisalsoapotential reasonwhypredictionsofmesoscalemodelsexhibitahighdegreeofvariance,andarethereforeconsideredtobetheareaofhighestuncertainty[18].Formodeling mechanicalbehavior,crystalplasticityisaprominenttechniqueusedatthe mesoscale.Thetechniquestrikesaneffectivebalancebetweenthecomplexityof materialdescriptionandcomputationalcost,andenablesseamlessintegration withcomponent-scalesimulations. Toaddresstheextensiveinputrequirementsofmesoscalemodels,recentadvances ontheexperimentalfrontcanbeintegratedintomodelingworkflows.For instance,high-energyX-raycharacterizationtechniquesnowallowextraction ofmicrostructuralinformationfromthebulksamplesduringinsitutesting [18].Whilecrystalplasticitymodelsarereasonablyeffectiveatreproducingthe macroscopicresponse,theyoftenfallshortincapturingspatialvariationsofstressorstrain,especiallywhensuchlocalvariationsareofparticularinterest[59,60].Thisleavessubstantialscopeforimprovementincurrentmodelingstrategiestoeventuallyachievehigh-fidelityandcomputationally efficientdigitaltwinsofpolycrystals.Furthermore,obtainingsuchhigh-fidelity 16INTRODUCTION theprimaryobjectiveisnotmerelytoreplicateexperimentalobservations,the goalistoachieveefficientmesoscalemodels,thatarereliableevenbeyondthe domainofcalibration.Achievingthisenablesrigorousmicrostructure–property linkagestudies,providingdeeperinsightsintophysicalmechanismsthatareotherwisedifficulttoaccessthroughexperimentsalone.Althoughreal-timemonitoringofphysicalcomponentsbelowthemacroscaleremainsadistantgoal,intermittenton-fieldmeasurementsatthemacroscalecouldbeusedtoperiodicallyupdatethedigitaltwins,therebysupportingmorereliablerisk assessmentandlifetimepredictionofthephysicalsystem.1.6OutlineoftheThesis Thepresentchapterprovidedanoverviewofthebroadercontextandhighlighted areasofapplicationwherethefindingsofthisthesisadvancetheexistingstate ofunderstanding.Theremainderofthethesis,whichaddressesthestatedresearchquestionsinpursuitoftheoverarchingobjective,isstructuredasfollows.Chapter2presentsthenecessarybackgroundknowledgerequiredto clearlyunderstandthecorecontentofthethesis.Chapter3utilizesalarge-scale experimentaldatasetonanickel-basedsuperalloytocalibrate,simulate,andvalidatetwofundamentallydifferentconstitutivemodels.Throughthis,thestudiesaimtoaddressresearchquestionsRQ1andRQ2.Inthefollowingchapter(Chapter4),theoperationallyefficientphenomenologicalmodelis enhancedtoimproveitspredictivecapabilitiesbeyondthecalibrationdomain, therebyaddressingresearchquestionRQ3.Chapter5againemploysthe operationallyefficientphenomenologicalmodelwithinsightsfromChapter4,this timeforamaterialthatexhibitsstrongelasticandplasticanisotropy.Another multiscaleexperimentaldataset,acquiredinsituduringmechanicaltestingofatitaniumalloy(Ti-6Al-4V),servesasthebasisformultilevelcalibration,whichisthefocusofresearchquestionRQ4.ThecalibratedmaterialmodelforTi-6Al-4VfromChapter5isthenemployedinChapter6toperformaqualitativemicrostructure–mechanicalpropertylinkagestudyinthecontext oftheabrasivewheelcuttingstepinvolvedinspecimenpreparation.Here,the effectsofelastoplasticdeformationmechanismsthatmayhavecontributedtoa particularlyinterestingresidualstressstateinthespecimenareinvestigated throughcrystalplasticitysimulations,thusaddressingresearchquestionRQ5. Finally,Chapter7summarizesandconcludestheresearchcoveredinthisthesisandplacesthecontributionswithinthebroadercontextofachievinghigh-fidelity digitaltwins.Additionally,potentialnewavenuesforresearchinthedirection ofimprovingstrategiesarealsodiscussed. Chapter2Background Theaimofthischapteristoprovidethereaderwiththenecessarybackground knowledgerequiredtoclearlyunderstandthecorecontentofthisthesis.Itbegins withanintroductiontocrystalplasticity,coveringtopicsfromthedefinitionof fundamentalquantitiestotheformulationofconstitutivemodels.Thechapter alsoexplainshowdifferentdeformationmechanismsinelastoplasticityare formulatedwithintheframeworkofcontinuummechanics.Additionally,abrief overviewisprovidedoftheexperimentaltechniquesthroughwhichthedata wereacquiredandconsideredasthegroundrealityformodelingandvalidation. MoredetaileddiscussionsofthetopicspresentedinSection2.1canbefound instandardcontinuummechanicstextbooks,suchasthoseauthoredbyAsaro andLubarda(2006)[118],BonetandWood(2008)[119],Spencer(2012)[120], andLubliner(2013)[121].2.1ContinuumMechanics Atveryfinelengthscales,matterisinherentlydiscontinuous,primarilydueto thediscretenatureofatomicarrangement,andthepresenceofdefectsandvoids.Thesemicrostructuralfeaturesplayafundamentalroleinenablingmaterialsto exhibitdiverseproperties,whichmakesthemasubjectofsignificantscientific interest.However,atthelengthscalesrelevanttoengineeringapplications, matteristypicallyassumedtobeacontinuum.Abodyissaidtobemodeledasacontinuumif,inanyconfiguration,itoccupiesaregion ℝ3 inEuclideanspace suchthateverymaterialpointintheregionisoccupiedbyaparticleofthe 17 18BACKGROUND body[121].Thissimplificationallowsfortheapplicationofthemathematical frameworkofcontinuummechanics.2.1.1FiniteStrainKinematics Afoundationalprincipleincontinuummechanicsistheidealizationofabodyasaninfinitecollectionofmaterialpoints,eachrepresentinganinfinitesimallysmall volumeofmaterial.Anychangeinrelativepositionofthepointscorresponds tothedeformationinthematerialbody.Mathematically,thedeformationcan beformulatedasamappingbetweenthereference(initial)configuration ℬ0 andthedeformed(current)configuration ℬ ,suchthatanygivenmaterialpoint, representedbyitspositionvectorinEuclideanspace, 𝐱∈ℬ0 hasaunique correspondentpoint 𝐱∈ℬ ,andthemappingconservestheinfinitecollectionof points.Thedisplacementofamaterialpointisthendefinedas𝐮≔𝐱−𝐱. Afunctionthatmaps ℬ0→ℬ canbeformulatedbyconsideringaninfinitesimalfiberelementd 𝐱 in ℬ0 thathasdeformedtod 𝐱 in ℬ .Itisknownthat 𝐱=𝐱(𝐱)andthisimpliesd𝐱=𝜕𝐱(𝐱)𝜕𝐱d𝐱=𝐅d𝐱,(2.1) where 𝐅 isthesecond-ordereddeformationgradienttensorthatcaptureshow materialpointsmoveinrelationtotheneighboringpoints,andtherebycaptures deformation.Thedeformationgradienttensorfrequentlyappearsinmaterial mechanicsandisrelatedtothedisplacementgradient 𝜕𝐮𝜕𝐱 throughtherelation 𝐅=𝜕𝐮𝜕𝐱+𝐈. Since 𝐅 isasecondordertensorandisinvertible,apolardecompositioncanbe appliedintwoways,eachofwhichresultsintwophysicallymeaningfultensors: 1. Adecomposition 𝐅=𝐑𝐔 resultsinanorderedapplicationofmaterial (pure)deformationthroughthesymmetricstretchtensor 𝐔 andrigidbody rotationthroughtheorthogonalrotationtensor 𝐑 .Sincedeformationcomesbeforetherotationintheorderofoperations,andconsequentlyappearsontherightside, 𝐔 isgenerallyreferredastherightstretch tensor.2. Alternatively,adecomposition 𝐅=𝐕𝐑 switchestheorderofoperations, withleftstretchtensor𝐕operatingonarotatedconfiguration. Although 𝐅 isveryimportantintheanalysisofdeformation,itisnotby itselfasuitablemeasureofdeformation.Thereasoncanbeunderstoodfrom thepreviousparagraphwhereitwasshownthat 𝐅 capturesallaspectsof CONTINUUMMECHANICS19 deformation:thisnotjustincludestheactualdeformationbutalsorigidbody rotations.Forinstance,let 𝐐 beanyproperorthogonaltensorthatinducesarigidbodyrotationthattransforms 𝐱 to 𝐱=𝐐𝐱 .Inthiscase,computing 𝐅 similartoEq.(2.1)leadsto 𝐅=𝐐 .Thisshowsthatthecomponentsof 𝐅 changeinvalueforadisplacementofmaterialpointsthatresultedfromapure rigidbodyrotation. Forthisreason,alternatemeasuresofdeformationthatareafunctionof 𝐅 are adopted.Amongthem,rightandleftCauchy–Greendeformationtensorsare popularandaredefinedas 𝐂≔𝐅T𝐅 and 𝐁≔𝐅𝐅T ,respectively.Incontrast to 𝐅 ,thevaluesof 𝐂 and 𝐁 donotchangeduringarigidbodyrotationas 𝐐T𝐐=𝐐𝐐T=𝐈 .Duringadeformation,itisoftendesirabletoknowhow muchmaterialpointshavemovedfromtheirinitialpositionsratherthanjust theirabsolutedisplacement.Tocapturethisrelativedisplacement,themeasures ofdeformationmentionedabovearetypicallyconvertedtostrainstensors.MeasuresofStrain Evenifthescopeofdiscussionisrestrictedtoasimpleuniaxialcase,there areinfinitelymanywaystodescribestraininamaterial—allofwhich,attheminimum,shouldquantitativelydescribehowmuchthematerial’slengthhas changed[118].Consider,forinstance,acaseofstretchingafiberwhichwasinitiallyoflength 𝑙0 to 𝑙 .Theuniaxialstraincanbedescribedwiththehelp ofstretch 𝜆 thatdefinedas 𝜆≔𝑙/𝑙0 ,inthefollowingwaysalthoughthelistis non-exhaustive:1.nominalstrain:𝑙−𝑙0𝑙0=𝜆−1.2.naturalstrain:𝑙−𝑙0𝑙=1−1𝜆.3.Lagrangianstrain:12𝑙2−𝑙02𝑙02=12(𝜆2−1).4.Eulerianstrain:12𝑙2−𝑙02𝑙2=12(1−1𝜆2).5.Logarithmicortruestrain:ln𝑙𝑙0=ln𝜆. Analogously,thethree-dimensionalstraintensorsarecomputedfromthe deformationdatastoredinstretchtensors 𝐔 or 𝐕 ,ormoreprecisely 𝐂 or 𝐁 whicharesimply 𝐔2 or 𝐕2 ,respectively.Thechoiceleadsuptotwodifferent approachesfordescriptionofthedeformation: 20BACKGROUND• Lagrangianapproachdescribesmaterialbehaviorintermsofquantities describedinthereferenceconfiguration.Thisistypicallythepreferred approachinsolidmechanicsandcrystalplasticitymodelingalthoughthere existsafewexceptions[122,123].• Eulerianapproachdescribesmaterialbehaviorintermsofquantities associatedwiththecurrentconfiguration. Therefore 𝐂 and 𝐁 ,andthestrainmeasuresderivedfromthem,canbeclassified intothecategoriesofLagrangianandEuleriantensorfields,respectively. ThemostcommonlyusedLagrangianstraintensorsarethespecialcasesof generalform 1𝑚(𝐔𝑚−𝐈) andaretheextensionoftheaforementioneduniaxial strains[124,121].Threeofthemarelistedbelow:1.Theconventionalstraintensor:𝐔−𝐈.2. Green–Saint-VenantstraintensororGreen–Lagrangestraintensoror simplytheLagrangianstraintensor:12(𝐔2−𝐈).3.Logarithmicstraintensor:ln𝐔. Amongtheonesmentionedabove,theGreen–Lagrangestraintensorisconsideredtobeanalyticallythemostconvenientchoice.Thesamecanbe writtenintermsofdisplacementgradients(𝜕𝐮𝜕𝐱or𝑢𝑖,𝑗)inindexnotationas𝐸𝑖𝑗=12(𝑢𝑖,𝑗+𝑢𝑗,𝑖+𝑢𝑘,𝑖𝑢𝑘,𝑗).(2.2) Thesquarematrix 𝑢𝑖,𝑗 canbedecomposedintosymmetric (𝑢𝑖,𝑗)sym=12(𝑢𝑖,𝑗+𝑢𝑗,𝑖) andantisymmetric (𝑢𝑖,𝑗)Asym=12(𝑢𝑖,𝑗−𝑢𝑗,𝑖) matrices.Consequently,the Eq.(2.2)canberewrittenas𝐸𝑖𝑗=(𝑢𝑖,𝑗)sym+12((𝑢𝑖,𝑘)sym(𝑢𝑗,𝑘)sym−(𝑢𝑖,𝑘)sym(𝑢𝑗,𝑘)Asym−(𝑢𝑖,𝑘)Asym(𝑢𝑗,𝑘)sym+(𝑢𝑖,𝑘)Asym(𝑢𝑗,𝑘)Asym). Itisevidentfromtheaboveequationthatif ∣(𝑢𝑖,𝑗)sym∣≪1 and ∣(𝑢𝑖,𝑗)sym∣≪1 ,thequadratictermscanbeconsiderednegligible.Giventhat, (𝑢𝑖,𝑗)sym becomesanapproximationoftheGreen–Lagrangestrainandisknownastheinfinitesimal CONTINUUMMECHANICS21 straintensor,whichisthetypicalstrainmeasureinsmalldeformationstudies. Theantisymmetriccomponentisthenknownastheinfinitesimalrotationtensor.However,inthepresentcontextoffinitedeformationstudies,the Green–Lagrangestraintensoristhepreferredchoice.VelocityGradient Ifmaterialpointswithinabodydisplacewithrespecttotime( 𝑡 ),anon-zero velocityfield 𝐯 associatedwiththisdisplacementfieldoriginatesandcanbe writtenas𝐯=d𝐮d𝑡=𝐮also𝐯=d(𝐱−𝐱)d𝑡=𝐱. Thenthevelocitygradient 𝐋 isthespatialgradientof 𝐯 withrespecttothe currentconfiguration(cf.Eulerianapproach)𝐋=𝜕𝐯𝜕𝐱, andquantifiesrelativevelocitybetweentwomaterialpointsinthecurrent configuration.Thiscanbefurthersimplifiedas𝐋=𝜕𝐯𝜕𝐱=𝜕𝐯𝜕𝐱⋅𝜕𝐱𝜕𝐱=𝜕𝐱𝜕𝐱⋅𝐅−1=dd𝑡𝜕𝐱𝜕𝐱⋅𝐅−1=𝐅𝐅−1.(2.3)2.1.2MeasuresofStress Continuummechanicsnotjustdealswithchangeofrelativepositionsofmaterial pointswithinacontinuumbutalsotheforcesthatinflictedthechange.In general,theforcesactingonabodycanbeclassifiedintotwocategories[121]:1.bodyforce:long-rangeforcesthatactthroughoutthebody,and2. surfacetraction:short-rangeforcesthatactascontactforcesonthe boundarysurfaceofthebody. Whiletheformerisusuallydescribedasadensityofforce(withunitNm −3 ), thelatterisdescribedasavectorforceacting(withunitNm −2 )onasurface elementthatischaracterizedbyitssurfacenormal. Consider,forinstance,anorthonormalbasis 𝐞𝑖 inthecurrentconfiguration(Eulerianapproach)madefromvectors 𝐞1 , 𝐞2 and 𝐞3 .Let 𝐭(𝐧) beatraction vectoractingonanarbitrarysurfaceelementwithnormal 𝐧 ,andtheareaof 22BACKGROUND thesurfaceelementisinfinitesimallysmall.Thetractionvector 𝐭(𝐧) withan arbitrarydirectioncanberesolvedintocomponents 𝐭1 , 𝐭2 and 𝐭3 alongthebase vectorsof𝐞𝑖.ApplyingCauchy’slawofmotion[121],itcanbederivedthat𝐭(𝐧)=𝐭1𝑛1+𝐭2𝑛2+𝐭3𝑛3 orsimply 𝐭(𝐧)=𝐭𝑖𝑛𝑖 intheindexnotation.A3×3matrix 𝜎𝑖𝑗 cannowbe definedwhosecomponentssatisfy,forinstance,𝐭1=𝜎11𝐞1+𝜎12𝐞2+𝜎13𝐞3 andalsofortheothertwotractionvectors.Ageneralfromofthisiswritten easilyintheindexnotationas 𝐭𝑖=𝜎𝑖𝑗𝐞𝑗 .Onecanalsowritethesameequation inanotherformas 𝜎𝑖𝑗=𝐭𝑖⋅𝐞𝑗 using 𝐞𝑖⋅𝐞𝑗=𝛿𝑖𝑗 (Kronecker’sdelta).Finally, thisleadstotherelation𝑡(𝐧)𝑗=𝜎𝑖𝑗𝑛𝑖or𝐭(𝐧)=𝐧⋅𝝈(2.4) where 𝝈 iscalledtheCauchystresstensorwhichisofsecondorderandis symmetric. Asalreadymentionedabove,theorthonormalbasisusedforthederivationoftherelationshipisinthecurrentconfiguration,whichsuggeststhattheCauchystress isameasureofforceperunitareainthedeformedconfiguration.Thismeans thatthestressstatechangescontinuouslyasthematerialdeforms.However, sometimes,ameasurethatisindependentofthechangingmaterialstructure anddependentonlyonthereferenceconfiguration(Lagrangianapproach)isdesirable.Tothisend,alternativemeasuresofstress,listedbelow,thatare thework-conjugatepairofcertaindeformationmeasuresareemployed,thereby facilitatingconvenientcomputationofthevirtualwork[119].1.FirstPiola–Kirchhoffstresstensordefinedas𝐏=(det𝐅)𝝈𝐅-T(2.5) where 𝐅 isthedeformationgradientthattransformsfromthereference tothecurrentconfiguration.Therefore, 𝑃𝑖𝑗 isatwo-pointtensorthat canbeinterpretedasthe 𝑖th componentoftheforcevectoractingonan areaelementthathaditsnormal 𝑗 directionandhadaunitareainthe referencestate[118].Itshouldbenotedthat 𝐏 isnotsymmetricas 𝝈andalsonotexclusivelydependentonthereferenceconfiguration.2.SecondPiola–Kirchhoffstresstensordefinedas𝐒=(det𝐅)𝐅−1𝝈𝐅-T(2.6) completelytransfersthespatialcomponentofforcebypullingitbackto thereferenceconfiguration.Moreover,𝐒issymmetric. SIMULATIONFRAMEWORK233. Incrystalplasticity,oneoftenencountersMandelstresstensor[125] thatisdefinedas𝐌=𝐂𝐒(2.7) thatisconvenientlydefinedinanintermediateconfiguration(thiswill bediscussedinthenextsection)with 𝐂 beingtherightCauchy–Green deformationtensor. 𝐌 isasymmetricandadoptedprimarilyforthe convenienceassociatedwithhandlingitswork-conjugatepair—thevelocity gradienttensor.2.2SimulationFramework Thissectionwillcoverhowdifferentdeformationmechanismsinvolvedinan elastoplasticdeformationareformulatedincontinuummechanismsframework, andhowtheircouplingistreatedinaself-consistentmanner.Thetopics presentedinthisandthefollowingsectionsareadaptedfromtheconventions followedinthematerialbehaviorsimulationtoolkitDAMASK,asitwillbeemployedforallstudiespresentedinthelaterchapters.Consequently, comprehensivedescriptionsofthetopicscoveredinthesetwosectionscanalso befoundintheworksofRotersetal.[101,24]andDiehl[126].2.2.1Decompositionof𝐅 Atthebeginningofthissection,thekinematictreatmentofdeformationusing thedeformationgradient 𝐅 ,whichmapstheundeformedconfigurationtothedeformedconfiguration,waspresented.However,thiswasdonewithoutconsideringhowcrystallinematterphysicallyrealizesuchdeformations.Itis knownthattheunderlyinglatticeofacrystallinemattercandeformeitherby smalldisplacementsofatomsfromtheirequilibriumpositionsorviapermanent rearrangementofatomsfromoneequilibriumpositiontoanother.Theformer resultsinelasticdeformationcharacterizedbystretchandrotationofthecrystal lattice,whilethelatter—whenrestrictedtoperfectdislocationplasticity—only producesshapechange.Thetranslation-invariantkinematicsofplasticslip ensuresthatthelatticecoordinatesystemremainsunchangedfromthereference. Furthermore,theplasticslippersistsevenaftertheexternalloadisremoved unliketheelasticdeformation,andthelatticeisatastress-freestateafterthe plasticdeformation. Theseinsightshaveleadtothechoiceofamultiplicativedecompositionofthe overalldeformationwhichisalsoknownastheKröner–Leemultiplicative decomposition[43,127,128],andformsthefundamentalbaseformany 24BACKGROUND Figure2.1:Schematicrepresentationofthree-stepmultiplicativedecomposition oftheoveralldeformationgradient 𝐅 withintermediateconfigurations.Figure reproducedfromRotersetal.[24]. theoriesinfiniteplasticity.Anextensiontotheinitiallyproposeddecomposition withtwocomponentsi.e.,plastic 𝐅p andelastic 𝐅e ,hasbeensubsequently adopted[24]byincludinganadditionalcomponent 𝐅i .Thedecompositionnow reads𝐅=𝐅e𝐅i𝐅p(2.8) withanintroductionoftwointermediateconfigurationsbetweentheundeformed (orreference)andthedeformedstates.Thisisschematicallyrepresentedin Fig.2.1.Equation(2.8)canbeelaboratedinthesequenceofoperationswith1.𝐅p thelattice-preservinginelasticdeformationgradienttensorthatmaps thecurrentconfigurationtoanintermediateplasticconfiguration.2.𝐅i thelattice-distortinginelasticdeformationgradienttensorthatmaps fromtheplasticconfigurationtoanotherintermediateconfigurationthat isnamedeigenstrainconfigurationinthefigure.3.𝐅e thelattice-distortingelasticdeformationtensorthatfinallymapsfrom theeigenstrainconfigurationtothedeformedconfiguration. Hereon, 𝐅p , 𝐅i and 𝐅e willbereferredasplastic,eigenorintermediate,and elasticdeformationgradients,respectively. TheelasticandplasticpartsinEq.(2.8)canbedecomposedusingpolar decompositionas𝐅=𝐕e𝐑e𝐅i𝐕p𝐑p(2.9) SIMULATIONFRAMEWORK25 where 𝐕 and 𝐑 arepurestretchandrotation,respectively.Previouslyinthis section,itwasexplainedhowplasticdeformationthroughslipcanbeperceived astranslationinvariantanddoesnotalterthelatticecoordinatesystem.This propertyisconvenientlyusedtointentionallyrotateallgrainsinapolycrystal withtheircorrespondingmisorientationthrough 𝐑p suchthattheirlattice coordinatesystemcoincideswiththefixedglobalcoordinatesystem(likecube orientation).Theadvantageofperformingtheaboveoperationisapparentin Fig.2.1,whereintheundeformedconfiguration,thevectorsassociatedwithaslipsystem—theslipplanenormal 𝐧 andtheslipdirection 𝐬 —areatanorientationfromtheglobalbasis 𝐞𝑖 .Aftertherotationthrough 𝐑p ,thetwocoordinateframescoincideandthereforeavoidsthecumbersomeoperations oftransformationandrotationoftheseslipvectorsofeachgrainfromglobal basistograin-specificlatticecoordinateframe.Theadditionalintermediate deformationgradient 𝐅i isusedtoaccommodatephenomenasuchasthermal expansion,crackopeningetc.,anditwillbepresentedlaterinChapter3how thiscanbeusedtoincorporateresidualstrains.Finally,elasticstretchingand rotationsarecapturedby 𝐕e𝐑e wheretheintentionalrotationsperformedin thefirststeparealsocompensatedforinthisstep.2.2.2TimeIntegration Itiswellestablishedthatthemechanicalbehaviorofcrystallinematteris sensitive—thoughtovaryingdegrees—totherateofdeformation.Thismeans thattheconstitutivemodels,especiallytheinelasticones,aretypicallyderivedinrateform.Thisrequirestimeintegrationofthekinematicquantities(describedintheprevioussection)andmaterialstates(whichwillbeexplainedinthenextsection)fromtime 𝑡0 to 𝑡=𝑡0+Δ𝑡 .Tothisend,Eq.(2.3)isappliedtoplastic andinelasticdeformationgradientsthatresultsin𝐅p=𝐋p𝐅p(2.10)and𝐅i=𝐋i𝐅i.(2.11) Equation(2.10),forinstance,canbenumericallysolvedbyapproximatingata fixedmaterialstateas𝐅p(𝑡)−𝐅p(𝑡0)Δ𝑡=𝐋p(𝑡)𝐅p(𝑡). Thiscanbefurtherwritteninasimplifiedformwherethecurrentplastic deformationiscomputedas𝐅p(𝑡)=(𝐈−Δ𝑡𝐋p(𝑡))−1𝐅p(𝑡0).(2.12) 32BACKGROUND However,ifoppositedislocationscomeveryclosetoeachother,withinthe criticaldistanceforannihilation 𝑑dipole ,thedislocationsannihilateandtherefore mobiledensityisreduced(minussignforthethirdterm)becauseofthisevent. ThisdistanceisconsideredtobearealnumbermultipleofBurgerslength 𝑑dipole=𝐷a𝑏. Analogously,evolutionofdipoledislocationdensityisdependentonconversion ofmobiletodipoledislocations,spontaneousannihilationofdipoles(withother mobiledislocations),andannihilationofdipolesduetodislocationclimb.The relationnowreads𝜌d=2(𝑑dipole−𝑑anni)𝑏𝜌|𝛾|−2𝑑anni𝑏𝜌d|𝛾|−4𝑣climb𝑑dipole−𝑑anni𝜌d,(2.24) wherethefirsttermwithmobiledislocationdensityappearsagain(fromEq.(2.23))butwithpositivesignaccountsforincreaseddipoledensity.The secondandthethirdterminEq.(2.24)accountforannihilationofdislocation dipoleswithothermobiledislocationsandthroughdislocationclimbmechanism, respectively[24]. 𝑣climb intheaboveequationrepresentsdislocationclimb velocitycomputedaccordingtotheworkofArgonandMoffatt[139].2.3.2PhenomenologicalModel Thephenomenologicalmodellumpsalltheinformationpertainingtodislocation stateofaslipsysteminamaterialtoasinglevariable:thecriticalvalueofresolvedshearstress 𝜉 .Thisreferstotheresolvedshearstressvaluebelow whichnoslipwilloccur.Thepresentmodel[24]isinspiredbythewidelyused modelofJ.W.Hutchinson[140]. Unlikethephysics-basedOrowan’slawinthedislocationdensity-basedmodel, theshearrateonaslipsystem 𝑖 inthisphenomenologicalmodelisrelatedto thestateofthematerial 𝜉 andtheappliedstimulus 𝜏 throughapowerlaw relationthatreads𝛾𝑖=𝛾0∣𝜏𝑖𝜉𝑖∣𝑛sgn(𝜏𝑖),(2.25) where 𝛾0 isthereferenceshearrate,and 𝑛 istheinverseoftheratesensitivity exponent. Theresolvedshearstressonaslipsystem 𝑖 ,i.e.,thestateofthematerial,evolvesfromaninitialvalue 𝜉0 toasaturationvalue 𝜉∞ ,andisgovernedby theshearratesonallslipsystemsandreads𝜉𝑖=ℏ0𝑖∣1−𝜉𝑖𝜉𝑖∞∣𝑎sgn(1−𝜉𝑖𝜉𝑖∞)𝑁𝑠∑𝑗=1ℎ𝑖𝑗∣𝛾𝑗∣,(2.26) CONSTITUTIVEMODELING33 where ℎ𝑖𝑗 arethecomponentsofthehardeningmatrixandaccountforforest hardeningduetoslip(moreinformationonthiswillbeprovidedinChapter4), ℏ0 and 𝑎 aresystem-dependentfittingparameters,and 𝑁𝑠 isthetotalnumber ofslipsystems.2.3.3IsotropicPlasticitywithDilatation Thismodelisbasedontheconstitutivedescriptionproposedby[141]tosimulate emptyspacesinthematerialvolume.Asthenamesuggests,themodeldoes notaimtoaccountfordirectionallydependentmaterialproperties(isotropic) [24].Tothisend, 𝐋p forthismodelwouldlookdifferentfromthetypicalcrystal plasticitymodels(cf.Eq.(2.16))andreads𝐋p=𝛾p𝐌devp‖𝐌devp‖F, where 𝐌devp isthedeviatoriccomponentoftheMandelstressand ‖⋅‖F istheFrobeniusnorm.Theplasticshearrateatamaterialpoint 3 isrelatedtothecriticalvalueofstress 𝑀𝜉 following 𝐽2 (secondinvariantof 𝐌devp )plasticity theoryas𝛾p=𝛾0(√3𝐽2𝑀𝜉)𝑛=𝛾0(√32‖𝐌devp‖F𝑀𝜉)𝑛,(2.27) where 𝑀 istheTaylorfactor.Thisallowsforisochoricpermanentdeformation duetodeviatoriccomponentoftheappliedstress.Thesameiscomputedin thelatticeconfigurationandhencethesubscriptp.Additionally,thematerial isalsoallowedtoundergonon-isochoricpermanantdeformationthrough 𝐅i in theintermediateconfiguration.Thecorrespondingvelocitygradient 𝐋i ,with thesubscriptirepresentingtheintermediateconfiguration,isformulatedas𝐋i=𝛾i𝐈=𝛾0(|𝑝|𝑀𝜉)𝑛𝐈sign(𝑝),(2.28) where 𝑝=tr(𝐌sphi)/3 isthehydrostaticpressurecomputedfromthespherical componentoftheMandelstress.Theabilitytodeforminelasticallyevenfora hydrostaticloadcase(dilatation)isthefundamentaldifferencebetweenthis andtheclassicalisotropicmodelswithvonMisesyieldcriterion.InDAMASK, thechoiceofwhethertoincludeinelasticdilatationmodecanbespecifiedinthematerialdescriptionfilethatgoesasaninputtothesolver.Thescalarvariable 𝜉 intheaboveequationsparameterizestheinternalstateofthematerialand 3scalarvalueasthenotionofslipsystemisnotadopted. 34BACKGROUND thereforeevolvesfromaninitialvaluetoasaturationvaluesimilarinthesimilar fashionasinthephenomenologicalmodel[24]. Theprimaryapplicationofthismodelwithdilatationisnottodescribeelastoplasticbehaviorofapolycrystalbuttoaddresscertainshortcomingsoftheemployedspectralsolver.InChapter3,thisconstitutivedescriptionwillbeusedtomodelathermosetmaterialandbreaktheperiodicityofthe volumeofinterest.Amongotherapplications,suchaconstitutivedescription canalsobeusedtomodelvoidsinamaterialandalsoallowstousenon-compact geometriesinaspectralsolver[141].2.4ExperimentalCharacterizationTechniques Figure2.3:Featuresofacrystallinematerialasseenatdifferentlengthand timescales.FigureadaptedfromKochmannandAmelang[142]andreproduced withpermissionfromSpringerNature. Polycrystallinematerialscanbemechanicallycharacterizedatawiderangeof lengthandtimescales.Thefeaturesobservedatdifferentscales,alongwithsomeoftheextensivelyusedcharacterizationandmodelingtechniques,are depictedinFig.2.3.Sincethecurrentfocusistomodelmaterialbehavioratthe scaleofindividualgrainsandbeyond,theappropriatelengthscalerangesfrom sub-microndimensionsandupward.Amongthecharacterizationtechniques EXPERIMENTALCHARACTERIZATIONTECHNIQUES35 establishedatthisscale,electron-basedtechniquessuchastransmission electronmicroscopy(TEM),scanningelectronmicroscopy(SEM),andelectron backscatterdiffraction(EBSD)areparticularlypopular[143].Despitethe veryhighspatialresolutionofTEM(typicallyontheorderofatenthof nm ), thetechniquesuffersfromlimitedfieldofviewandstrictsamplethickness requirements[143].Acommonlimitationacrosselectronmicroscopytechniquesarisesfromthelimitedpenetrationdepthofelectronbeams,thatpreventsthem fromextractinginformationfromthebulkofthespecimen[143]. Numerousstudieshavedemonstratedthatrelyingsolelyonsurfacefeaturesare insufficienttomodelbulkmaterialbehavior[144,145,146,147].Generating3Dmicrostructurefromexperimentallycharacterizedsurface(orientationsfromEBSD)mapsoftenrequiresassumptionsbasedonstatisticaltheories, orsimplificationofthegrainstructurealongthethicknessdirections.However, suchapproachesreducefidelityofthemodelscalibratedfromtheseassumptions.Oneworkaroundistocharacterizematerialswithtailoredmicrostructures,suchastheoneshavingonlycolumnargrainsWhilethisimprovesthecorrespondence betweenexperimentsandsimulations,itislimitedtoidealizedcases;inreality, microstructuresarecertainlymorecomplex. OneapproachtoaddressthislimitationinvolvessuccessiveEBSDscanningofthematerialsurfacealongthethicknessdirection,achievedthroughserialsectioning [148,149].Thismaterialreconstructiontechniqueisknownas3D-EBSD.Thismethod,however,waspreviouslyrestrictedtoverysmallsamplesizesduetotheneedforthinlayersduringserialsectioning,whichwasprimarily manualandthereforelabor-intensive[150].Recently,thislimitationhasbeen overcomethroughautomationusingroboticarmsandadvancementsinserial sectioningtechniques,includingtheadoptionofplasmafocusedionbeam(FIB) milling[150].Byadoptingtheaboveprocedure,Tsaietal.[150]wereableto characterizeavolumeaslargeas 5003µm3 ,typicallycontaining 100 to 100000grains,therebyenablingstatisticalrelevance. Despiteitsabilitytoreconstructreasonablylarge3Dmicrostructure,thetechniqueisinherentlydestructiveduetotheserialsectioningprocess.Thishaslimitedconsequencesforstudiesfocusedexclusivelyongrainstructuresandmicroplasticdeformation.However,todevelophigh-fidelitymechanical models,thecharacterizedsamplemustsubsequentlybepreparedformechanical testingtoobtaindataformodelcalibration.Additionally,anon-destructive characterizationwouldnotonlyenablematerialcalibrationbutfacilitatestudies onmicrostructure–propertylinkages. Amongthealternativesfornon-destructivecharacterization,techniquesbased onX-rayarepopular.X-raysalsoprovidegreaterpenetrationdepththanthe electronbeams:high-energyX-raybeams(energy ≥50keV )atsynchrotron 36BACKGROUND facilitiescanpenetratespecimenswithmillimeter-to-centimeterthicknesses[19]. Additionally,duetofasterdataacquisitionandprocessingtimes,theycanalso beemployedinsituduringexperiments. Thetechniquesdiscussedinthefollowingsubsectionsarerestrictedtosynchrotron-basedhigh-energyX-rays,asconventionallab-scaleexperimentssufferfromattenuationduetolimitedbeamenergies[151].However,there havebeensignificanteffortsontheexperimentalfronttobringcharacterizationtechniques,previouslyexclusivetosynchrotronfacilities,intolaboratorysettings [20].Someofthesynchrotron-basedX-raycharacterizationtechniquesinclude differentialapertureX-raymicroscopy(DAXM),high-energyX-raydiffraction microscopy(HEDM),high-energyX-raydiffraction(HEXRD),dark-fieldX-ray microscopy(DFXM).2.4.1FundamentalsofX-RayDiffraction Figure2.4:IllustrationofBragg’slaw.FiguretakenfromWasedaetal.[152] withpermissionfromSpringerNature. X-raysareelectromagneticradiationofexactlythesamenatureaslightbutofshorterwavelength[153].ThefundamentalprincipleofX-raydiffractionisbasedonBragg’sLawofdiffraction[154]whichstatesthatconstructiveinterferenceofadiffractedbeamoccurswhendistancebetweentheadjacent planesareintegermultiple 𝑛 ofthewavelength 𝜆 ofthebeamasdepictedin Fig.2.4.Thiscanbewrittenmathematicallyas𝑛𝜆=2𝑑′(hk.l)sin(𝜃(hk.l)),(2.29) EXPERIMENTALCHARACTERIZATIONTECHNIQUES37 where (hk.l) istheMiller–Bravaisrepresentationofagenericplane, 𝑑′(hk.l) is theinterplanardistancebetweenadjacent (hk.l) planes,and 𝜃(hk.l) istheangle ofincidence.2.4.2High-EnergyX-rayDiffraction Figure2.5:SchematicrepresentationofHEXRDtechniquewiththesamplein transmissiongeometry.Ontheright,thegraphshowsevolutionoflatticestrains ingrainswherespecificlatticeplanessatisfyBragg’sconditionofdiffraction. FiguretakenandadaptedfromY.-D.Wangetal.[155]. Inhigh-energyX-raydiffraction(HEXRD)experiments,monochromatichighenergyX-raysaretransmittedthroughthesampleofthicknesstypicallyinthe orderofmillimeters[156,157,158].SincetheX-rayshavetotravelthroughthe samplethickness,theyshouldbeofhighenergy,andthereforetheexperiment istypicallycarriedoutatsynchrotronfacilities.Thediffractionitselfwould lookdifferentfromFig.2.4wheretheX-raysseemtoreflectfromthesurface (reflectiongeometry),whereasinthiscase,asdepictedinFig.2.5,theX-rays penetratethroughthespecimenthicknessandgetdiffracteddifferentlybygrainsofdifferentorientations.ThediffractedX-raysarecapturedona2D (area)detectorwhichisplacedperpendiculartothedirectionofthebeam[158]. AsdepictedinFig.2.5,thediffractedbeamformstheso-calledDebyerings(orsimplydiffractionrings)onthedetector.TheseringscanbeconsideredanalogoustoBragg’speaksoftenreportedfromlab-scaleX-raydiffractionexperiments.Eachdiffractionringcanbeassociatedwithanensembleof grainswhenapolycrystalisilluminated,andcorrelateswithMillerindicesof thegrains.Thankstothefastsignalacquisitiontimeinthistechnique,thesamplecanbecharacterizedinsituduringanexternalloadingsuchas,forinstance,uniaxialtension.Duringtheloading,asthematerialdeforms,theelasticdeformationincreasesinterplanardistanceswhichiscorrelatedwith 38BACKGROUND diffractionringmovement[157].Asuitablepost-processingtechnique[159,160] isappliedonthediffractiondatatoextractevolutionofaveragelatticestrain inallgrainsofspecificorientationsoverappliedtensileload.Toachievehigh spatialresolutionsuchthattheconcentricringsareadequatelyspacedfromeach otherinordertoavoidoverlaps,thesample-to-detectordistanceinthissetup istypicallyaround 1 m.Theexperimentaldatausedinthestudydescribedin Chapter5isbasedonHEXRDandfurtherspecificdetailsregardingthesame areprovidedinthatchapter.2.4.3High-EnergyX-rayDiffractionMicroscopy Figure2.6:SchematicrepresentationofHEDMsetupwithdetectorsatdifferent sample-to-detectordistancescapturingdiffractiondataatbothnear-fieldand far-fieldmodalities.FigurereproducedfromH.F.Poulsenetal.[19]with permissionoftheInternationalUnionofCrystallography. Thehigh-energyX-raydiffractionmicroscopy(HEDM,oralsoknownas3DXRD) [19,18]techniquealsouseshigh-energymonochromaticX-rayintransmission geometrysimilartoHEXRD.However,HEDMisfundamentallydifferentfrom HEXRDinthefollowingtwoaspects:one,incorporationofrotationalscanning strategyofthesampleaboutitsaxis;andtwo,deploymentofmultipledetectors tocapturedifferentaspectsofthediffractiondata.Aschematicrepresentation ofHEDMtechniqueispresentedinFig.2.6.AplanarX-raybeamhitsthe polycrystalsampleandgetsdiffracted,andthisdataiscapturedbydetectors. Thesampleisrotatedincrementallyandperpendiculartotheplaneofthebeam(about 𝑧 directioninFig.2.6).Oncesufficientdataisacquiredfromtherotation EXPERIMENTALCHARACTERIZATIONTECHNIQUES39 ( 𝜔 inFig.2.6),thescanningstrategyisrepeatedbyshiftingtheplanarbeam along𝑧direction. Duringeachrotationperscanningstep,thediffractiondatainamultimodal strategycanbecapturedthroughdetectorsattwomodes:nearfield(nf)and farfield(ff).Thenear-fielddetectorsaretypicallyplacedclosetothesampleatadistanceontheorderofafew mm ,whilethefar-fielddetectorsarelocatedat distances ≥1000mm [161].Thecloseproximityofthenear-fielddetectorsto thesamplefacilitatesrecoveryofgrainorientationandshapethroughforward modeling.Tothisend,thediffractiondataarecapturedinnf-modeattwo ormoresample-to-detectordistancesperscaninordertoresolvetheoriginof thediffractedbeam.Thediffractiondata,inthiscase,arespotsseenonthe detectorwiththeirsizeandshapecorrelatingwiththeprobedgrains.Onthe otherhandintheff-mode,thesample-to-detectordistanceislargesuchthatthe diffractiondataarenolongerseentobespotsbutformDebye–Scherrerrings albeitdiscontinuous.Thedeviationsofdiffractionringsfromtheirtheoretical locationsareusedincomputationofcenterofmassandelasticstrainofthe grains[162].Appropriatepostprocessingstepsappliedonthediffractiondata capturedthroughdifferentmodalitiesallowsfor3Dreconstructionofgrain volumeandelasticstrainfieldduringaninsitutest.However,incomparison toHEXRD,thedataacquisitiontimesarelonger,whichnecessitatestheinsitutesttobepausedatregularintervals.Theexperimentaldatausedinthestudy describedinChapter3isbasedonmulti-modalHEDMtechniqueandfurther detailsonthecharacterizationwillbeprovidedthere. Chapter3ComparisonofFull-FieldCrystalPlasticitySimulationstoHigh-EnergyDiffractionMicroscopyExperimentsThischapterwaspreviouslypublishedas: N.PrabhuandM.Diehl.“ComparisonofFull-FieldCrystalPlasticitySimulationstoSynchrotronExperiments:DetailedInvestigationofMispredictions”. In:IntegratingMaterialsandManufacturingInnovation(2024).doi: 10.1007/s40192-024-00359-1andreproducedherewithpermissionfromSpringerNature.41 48COMPARISONOFFULL-FIELDCRYSTALPLASTICITYSIMULATIONSTOHIGH-ENERGYDIFFRACTIONMICROSCOPYEXPERIMENTSStrain(%)Stress(MPa)Loadstate0.00S00.053100S10.102200S20.1513000.128250S30.353600.334310S40.53740.493324S51.04030.982353S6 Table3.3:Keydatapointsinthetensilestress-straincurveobtainedfromthe experiment.Thecyanfontindicatesvaluesthatwerenotexplicitlymentioned inFig.2ofMenascheetal.[161]butinferredfromtherawtensiletestdata [185]. behaviorisdisallowedtoaccountfortheisochoriccharacterofdislocationglide (cf.Section2.3.3).Bydesign,thismodelcanbeusedwithparametersofthephenomenologicalcrystalplasticitymodel—heretakenfromD.Maetal.[193]—afterdefinitionoftheTaylorfactor 𝑀 .Theemployedparametersare giveninTables3.1and3.2b.3.2.3BoundaryConditions Basedontheexperimentaldata(cf.Section3.2.1),thespecimenwasloadedin uniaxialtensionalongthey-axis(Fig.3.1)atanominalrateof 1×10−4 s −1 during theinsituexperiment.Whilestresswasrecordedforthewholespecimen,the strainvaluesareobtainedviadigitalimagecorrelationonlyfortheconsidered regionofinterest.ThemicrostructurewascharacterizedbyHEDMintheunloadedconfiguration(S0)andduring 6 loadstatestermedS1toS6.For characterizationatstatesS3toS6thesamplewasunloadedfromahigherload toavoidcreep.ThetargetstrainsinloadingsectionsareshowninTable3.3. Tosimulateauniaxialtensiletestalongy-axis,they-componentsofthe deformationgradienttensors 𝐹y ,whichwerecalculatedfromthetargetstrains 𝜺y throughtherelation 𝐹y=√2𝜺y+1 ,wereprescribed.Theembeddinginthe void-likematerialallowstoexplicitlyprescribeallthecomponentsofaverage SIMULATIONSETUP49deformationtensorforagivenstrainstepintheform𝐅=⎡⎢⎣1√𝐹y000𝐹y0001√𝐹y⎤⎥⎦,(3.1) i.e.effectivelyensuringstressfreesurfacesonthemicrostructurewithout specifyingstressboundaryconditions[141].3.2.4InitialElasticStrain Theundeformedmicrostructureisnotfreeofelasticstrains,andtheincorporationofthese“eigenstrains” 5 isaprerequisitefortheaccuraterepresentationoftheinitialcondition[187].Theconsequencesofneglectingresidualstressesonoverallmaterialperformancehavebeendemonstratedby numerousexperimental[196]andcomputationalstudies[181].Toaddressthis issueandtobetterrepresenttheunderlyingphysics,studiesincorporatingtype-1residualstresses 6 havebeenconductedbyP.TurnerandC.Tomé [197],MusinskiandD.L.McDowell[198],andMcNelisetal.[199].Although type-2andtype-3(inter-andintragranular,respectively)residualstressescandiffersignificantlyfromthetype-1stresses,incorporatingonlytype-1 stressesisinsufficientwhenexplicitmicrostructuralinformationisavailable.In thisdirection,PokharelandR.A.Lebensohn[200]proposedamethodology involvingthecomputationofgrain-averagedeigenstrainscorrespondingtothe observedelasticstrains,basedonEshelby’sapproximation[85]foranellipsoidal inclusioninahomogenizedmatrix.Theper-graineigenstrainthuscalculated wasprescribed,followedbyanequilibrationloadsteptoensurebothstrain compatibilityandstressequilibrium.Theircalculationswerebasedonasmallstrainformulationandincludedanadditional“corrector”matrixintheoriginalEshelby’sapproximationtoiterativelyimprovethestatisticalagreementbetween experimentalandsimulatedresidualstressstatesafterequilibration.Following this,KapoorandSangid[201]proposed,withinafinitestrainformulation, theincorporationofeigenstrainsviaanintermediatedeformationgradientinthemultiplicativesplitoftheoveralldeformation(Eq.(2.8)inSection2.2.1). Whilestressequilibriumandstraincompatibilitywereenforcedthoughan equilibrationstep,theelasticstrainstateafterequilibrationwasnotreported intheirwork.Thisomissioniscritical,asneighboringgrainsmayimposeconstraintsduringequilibrationthatcannoticeablyaltertheaverageelastic 5Hereon,initial(elastic)strainsandeigenstrainsareusedinterchangeably.6 Actingatlengthscalescomparabletothecomponent’sphysicaldimensions;seeChapter6 forfurtherdiscussion. 50COMPARISONOFFULL-FIELDCRYSTALPLASTICITYSIMULATIONSTOHIGH-ENERGYDIFFRACTIONMICROSCOPYEXPERIMENTS strainstateofagivengrain.Additionally,althoughinitialelasticstrainscanbe directlymeasuredfromthematerialresponse,residualstressesarecomputed throughthetheoryoflinearelasticity,whichdemandsanaccurateknowledge ofsingle-crystalproperties.Asaresult,theenforcementofstressequilibrium, evenwhenusinghigh-confidence,experimentallyobservedelasticstrains,is subjecttouncertainty. Inthepresentdataset,averageinitialelasticstrainsareonlyknowninthechallengegrains,andconsequently,theresultingstressfield—ifinitialized directlywiththeknownelasticstrains—willnotbeinmechanicalequilibrium. Eveniftheaverageelasticstrainswereknownforallthegrainsinthe microstructure,becausetheyaregrain-wiseaveragedquantities[202,181,161], themicrostructurevolumewouldstillnotbeinequilibriumwheninitiated withtheexactvalues[201].Thisreflectsalimitationoftheexperiment:itcan onlycapturetype-2residualstresses,andtheinformationontheintragranular variationisnotretained.Consequently,anequilibrationstepbecomesnecessary toaccuratelyreproducetheactualstateofthematerial,asthisinherentlyinvolvesintragranularvariationsintherelevantfields.Nevertheless,recent advancesinsynchrotroncharacterizationtechniqueshavenowmadeitpossibletocaptureintragranularvariations[203].Toaddresstheissueofstressequilibrium, H.Zhouetal.[204],inarecentstudy,proposedanoptimizationmethodinvolvingHodgedecompositiontoachieveanequilibratedstressfieldwhenavoxelized,non-equilibratedstressfieldfromsynchrotronmeasurementsis available.Whilethismethodrequiresinitialelasticstraindataforallthegrains, thepresentdatasetcontainsthisinformationonlyforthechallengegrains. Theapproachadoptedinthisworkutilizestheintermediatedeformationgradienttoincorporatetheexperimentallyobservedelasticstrainsinthegrains.Aswas discussedinSection2.2.1,DAMASKutilizesfinitestrainkinematicswitha multiplicativedecompositionofthedeformationgradientas:𝐅=𝐅e𝐅i𝐅p, with 𝐅e , 𝐅i ,and 𝐅p accountingforelastic,eigenandplasticcontributions,respectively.Attheundeformedstate,i.e.at 𝑡=0 ,theoveralldeformationgradientequalstoidentity: 𝐅=𝐈 .Toincludeinitialelasticstrains,aleft polardecompositionoftheelasticdeformationgradient 𝐅e intostretch 𝐕e and rotation𝐑eisperformed:𝐅e=𝐕e𝐑e. Thisallowstoincludeinitial(elastic)strainsvia 𝐕e inadditiontotheinitializationofthecrystallographicorientation 𝐎 via 𝐅p=𝐎 and 𝐑e=𝐎T asoutlinedinRotersetal.[24].Notethatthisrequires 𝐅i=𝐎𝐕e−1𝐎T to retain𝐅=𝐈. RESULTSANDDISCUSSION51 Whiletheinitialelasticstraincanbeprescribedintermsof 𝐕e ,thesubsequent equilibrationloadstepmaycausetheaveragestraintodeviatefromthe prescribedvalue.Toaddressthisissue,anapproachisemployedtodeterminethevalueoftheinitialelasticstrain 𝜺0e inthechallengegrainsforwhichtheaverage oftheelasticstrain 𝜺e intherespectivegrainreachesthetargetvalue 𝜺targete afterequilibration.Incontrasttomethodsthataimatmatchingthestressfield inthewholemicrostructure[204],thisapproachisspecificallydesignedforthe givensituationwheretheinitialelasticstrainisonlyknowninafewgrains. TheproposedapproachisbasedonNewton’smethod.Ititerativelyadjusts 𝜺0einvectorforeachgrainaccordingto:𝑛+1𝜺0e=𝑛𝜺0e+𝑛𝐉−1(𝜺targete−𝑛𝜺e),(3.2) where 𝑛 isthecurrentiterationnumberand 𝐉∶=𝜕𝜺e𝜕𝜺0e theJacobianwith dimension6×6calculatedbymeansoffinitedifferences. Intheemployedlargestrainformulations,theelasticdeformationisdescribed by 𝐕e .Undertheassumptionofsmallelasticstrains,i.e. 𝜺e=12(𝐕2e−𝐈) ,the elasticstretchiscalculatedas 𝐕e=√(2𝜺e+𝐈) ,Eq.(3.2)isthereforeexpressed intermsof𝐕e:𝑛+1𝐕0e=𝑛𝐕0e+𝑛𝐉−1(𝜺targete−𝑛𝜺e),(3.3)with𝐉∶=𝜕𝜺e𝜕𝐕0e. Undertheassumptionthattheinteractionsbetweenthestrainfieldsofthe grainsarenegligible,thisrequiressevenequilibrationstepsperiteration—one forthecurrentsolutionandsixforthefinitedifferenceperturbation.3.3ResultsandDiscussion Inthefollowing,thesimulationresultsarepresentedanddiscussed.First,theperformanceofthenovelschemeforinclusionofinitialelasticstrainsinindividualchallengegrainsatS0isinvestigated.Then,theabilitiesofthe crystalplasticitymodelstoreproducetheexperimentalresultsatloadstates S1–S6areexamined.Tothisend,theoverallpredictionqualityusingthemetric oftheAFRLAMmodelingchallengeisconsideredfollowedbyanin-detaildiscussionofthreeaspects—monotonyoftheelasticstrain,stressevolution,andvolumetricstrain—forwhichadisagreementbetweensimulationsand experimentisobserved. 52COMPARISONOFFULL-FIELDCRYSTALPLASTICITYSIMULATIONSTOHIGH-ENERGYDIFFRACTIONMICROSCOPYEXPERIMENTS3.3.1IncorporationofInitialElasticStrain Here,theaccuracyandnumericalperformanceofthepresentedapproachforinclusionofeigenstrainsinselectedgrainsarediscussed.Animportant observationisthathighvaluesof 𝐕0e inEq.(3.3)canleadtostressesthatinitiateplasticdeformationwhichinturncausesconvergenceissues.Toovercometheseissues,plasticityisnotconsideredintheiterativeschemeoutlinedinSection3.2.4. Thefullelasticstrainfieldforwhichthemicrostructureisinstaticequilibrium, andtheaverageinallchallengegrainsagreeswiththeexperimentallymeasured valuesisthenusedtoinitializethesimulations.Theresolvedshearstressesfor thisinitialconditionarewellbelowtheupdatedcriticalthresholdtoinitiateplasticdeformation,whichindicatesanappropriateparametrizationofthe employedconstitutivemodelsasthemeasuredinitialstrainsare,bydefinition, purelyelastic. 369121518iteration10−1110−1010−910−810−710−610−510−4 max(|𝜺targete−𝑛𝜺e|) iteration6 1 Figure3.2:Grain-wiseevolutionofmaximumofabsolutedifferencebetween experimentalandsimulatedinitialelasticstraincomponentsoverthesuccessive iterationsintheNewton’smethod.Ontherightarethegrainpairs(withIDs) thatexhibitslowerconvergence.Thedoughnutchartwithintheplotrepresentsproportionsoftheplottedquantityatiteration6.Thecolorcodinginthecharts haveone-to-onecorrespondencewiththatofthegrainsontheright.Thegrain pairsarenotdrawntoscale. Figure3.2showstheconvergencebehaviorofEq.(3.3)forallchallengegrains exemplarily.Theresidual(ontheverticalaxisofFig.3.2)isquantifiedas RESULTSANDDISCUSSION53 themaximumnormofthedifferencebetweentargetandachieved(simulated) initialelasticstrain,andvalueof 1×10−8 wassetastheconvergencecriterion. Alllinesintheplotshowadownwardtrendwiththenumberofiterations. However,itcanbenotedthatfourgrainstakehighernumberofiterationstoachievetargetstrains.Acloserlookrevealsthattheyaremerelytwodistinct pairsofmutuallyinteractinggrainscomprisingofchallengegrains9and14,and12and28,respectively 7 ,whicharedisplayedontherightinFig.3.2.A noticeableaspectoftheirinteractionsisthatthesecondpairofgrainsismore intricatelyinterconnectedthanthefirstpair,whichseemstocausealowerconvergencerate.AlsoincludedwithinFig.3.2isadoughnutchartshowing thecontributionsofthefourabove-mentionedgrainstothetotalvalueofthe residualatiteration6,i.e.whenthedeviationsofallothergrainsarebelowthe threshold.Thegrains12and28contributesto 91.3% and 6.9% ofthetotal residualvalue,respectively,leavingaportionofonly 0.36% totheremaining 24grains. AsseeninFig.3.2,attheendoftheeighteenthiteration,theresidualerrorsinallgrainsarebelowthespecifiedtolerance.Thismeansthattheaverage initialelasticstrainsinthechallengegrainsattheunloadedstatenowprecisely matchtheaveragevaluesreportedintheexperiment.Furthermore,bothstress equilibriumandstraincompatibilityconditionsinthemicrostructurevolume aresatisfied.Thesimulatedmodelalsocapturesintragranularvariationsthat areabsentintheexperimentaldata.Throughtheincorporationofinitialelastic strains,anaccuraterepresentationofthesample’sinitialstateisachieved, providingareliablebasisforfurtherstudies.3.3.2OverallPredictionQuality Onceanaccuraterepresentationoftheinitialconditionisachievedbyinitializing challengegrainswiththemeasuredeigenstrains,themicrostructureisloadedaccordingtotheloadstatesS1toS6asmentionedinTable3.3.Theaverage stress–strainresponseobtainedfromthesimulationsusingbothcrystalplasticity modelsisjuxtaposedwithexperimentaldatainFig.3.3.Itisapparentthattheparameteridentificationaccuratelyreproducestheglobal(macroscopic) mechanicalresponse,asthecalibrationwasperformedtofitthemacroscopic data.Toassessthepredictivequalityofthesimulationapproach,thegrain-averaged elasticstrainofthechallengegrains—amesoscalequantity—istreatedasabenchmark.Tothisend,the ℓ2 -normofthedifferencebetweensimulated 7 Grainsarenumberedfrom1to28intheordergivenin[185],i.e.#1isequivalentwith global#145and#28isequivalentwithglobal#27757. 54COMPARISONOFFULL-FIELDCRYSTALPLASTICITYSIMULATIONSTOHIGH-ENERGYDIFFRACTIONMICROSCOPYEXPERIMENTS 0.0000.0020.0040.0060.0080.010𝜀01234 𝜎(Pa) ×108S0S1S2S3S4S5S6FromtensiletestLoadstatephenomenologicaldislocationdensity-based 1 Figure3.3:Comparisonoftheexperimentaluniaxialtensiletestdataagainst itsdigitalcounterpart. andpredictedelasticstraintensorsin6-componentVoigtnotation, Δ𝜺e∶=𝜺expe−𝜺sime ,iscalculated.Figure3.4showsthisscoreseparatelyforallchallenge grainsatallloadstates.Sincetherewasnonoticeabledifferencebetween bothcrystalplasticitymodels,onlytheresultsforthephenomenologicalmodel areshown.Thecloseagreementofthepredictionsfrombothmodelsisalsoobviousfromthebarplotontopoftheheatmap,whichshowsthesummed residualofallgrainsperloadstateseparatelyforbothmodelsinblueandyellow. Thegraybarsinthesameplotrepresentacasewherethephenomenologicalcrystalplasticitymodelwasusedwithoutincorporationoftheinitialelasticstrains.Incomparisontotheothertwocases,theresidualsintheelasticity-dominatedloadingregimeinthiscaseissignificantlyhigher,emphasizingtheimportanceofaddressingtheinitialelasticstrainsincrystalplasticity simulations.SimilarobservationsweremadebyT.J.Turneretal.[181],who performedanexperiment–simulationjuxtapositiontostudytheconsequences RESULTSANDDISCUSSION55 0.010.02 ℓ2-norm dislocationdensity-basedphenomenologicalnoeigenstrainsS0S1S2S3S4S5S6Macroscopicloadstates12723512177112422621254156131020191622188281439 GrainIDIncreasingvolumeofthegrain 0.00.71.42.22.9×1e-34.85%2.61%6.55%4.98%6.87%2.39%2.58%2.35% 1 Figure3.4:Aheatmapofindividual ℓ2 -normresidualsofthephenomenological model.ThegrainIDsontheverticalaxisareorderedbasedonthevolumeof thegrain.Grainsthataremajor(red)andminor(green)contributorstothe residualarehighlightedontheleftaxisandtheircorrespondingcontributions aspercentageoftheoverallresidualarementionedontherightaxis.Thebar graphdisplayedonthetopcomparesstate-wisecumulative ℓ2 -normresiduals betweentheemployedmodels.Thecolorcodingofthemacroscopicstrain statesdistinguishesbetweenelasticity-dominatedregion(olive)andplasticitydominatedregion(magenta). 56COMPARISONOFFULL-FIELDCRYSTALPLASTICITYSIMULATIONSTOHIGH-ENERGYDIFFRACTIONMICROSCOPYEXPERIMENTS ofneglectingresidualstressincorporationinamodelsimulatingelasticity-dominatedpartofthestress–strainresponse.Theirstudyshowedthatthe influenceofinitialresidualstresseswasnotableintheearlyelasticityregime, butdiminishedwithincreasingdeformationasotherdeformationmodesbegan todominate.Inthepresentcase,intheplasticity-dominatedloadingregime, thedisparityamongthebarsinFig.3.4decreasesincomparisontothatinthe elasticity-dominatedloadingregime,revealingasimilarlydiminishinginfluence oftheinitialstrainstate.Thesumoftheresultsofallloadstates—excludingthe initialstateS0—isthequantityusedinchallenge4oftheAFRLAMmodeling challengeseries:𝑟𝜀∶=6∑𝑠=128∑𝑔=1||Δ𝜀e||2.(3.4) Thevalueof 𝑟𝜀=0.7 (separatedvaluesforeachloadstatearegiveninTable3.4 forbothconstitutivemodels)isclosetothevaluesreportedinotherstudies [187,188].StatePhenomenologicalDislocationdensity-basedS10.00570.0057S20.00640.0064S30.00710.0071S40.01220.0123S50.01790.0179S60.02040.0202sum0.06970.0695 Table3.4:Theresidual ∑28𝑔=1||Δ𝜀e||2 ofthetwoconstitutivemodelsforthe loadstatesmentionedinTable3.3. Figure3.4showshowthedifferencebetweensimulatedandexperimentalelasticstraindependsontheload.Initializationofchallengegrainswiththemeasured eigenstrainsensuresnearzeroresidualatS0.Whileanalmostconstantvalue isseenfortheelasticity-dominatedregime(S1toS3),acontinuousincreaseis seenduringplasticdeformation.Threefeaturesarenoteworthy:• Thegrainsdonotcontributeequallytotheresidual.Grains1,4,10,and 21represent 14% ofall28challengegrainsbutcontributetotheresidual with23%.• Thedeviationtotheexperimentofindividualgrainsisnotmonotonically increasing,i.e.grain1and21showalargerdeviationforloadstate5 thanforloadstate6. RESULTSANDDISCUSSION57• Nocorrelationbetweenthecontributionofagraintotheoverallresidual anditsvolumecanbeseen.ThisisalsoconfirmedbyFig.A.1where contributionsofallchallengegrainsareplottedagainstarelativemeasure oftheirvolume. Tofurtherinvestigatethedifferencesbetweengrainswithalowandhigh agreementwithexperimentalresults,allcomponentsoftheelasticstrainsare plottedasafunctionoftheloadstate.InFig.3.5thegrainscontributingmosttotheresidual(withIDs1,4,10,and21)arecontrastedtothegrainswiththebestmatch(IDs16and9).Theplotsforallchallengegrainsaregivenintheappendix(Figs.A.2andA.3).Fromtheseplotsitcanbeseen thatthesimulations—independentlyoftheselectedmodel—predictalwaysa monotonousincreaseoftheelasticstrainwhilenon-monotonousbehavioris observedintheexperiment.Despitethiscommonfeature,specificaspectscan beseenforthedifferentgrains:Grain10,whichhashighestcontributiontotheresidual,showsaparticularlylargedeviationatS6whichiscausedby elasticshearstrainsandnormalstrainsinthetransversedirections( 𝑥 and 𝑧 ). ThedeviationatS6contributeswith 60% totheoverallresidualofthisgrain. Grain21showsafundamentallydifferentbehavior:Here,allstraincomponents deviateatS5becausethesimulationshowsamonotonousincreaseincontrast tothe“jump”observedforallstraincomponentsintheexperiment.Thisstate alonecontributestoapproximately 50% tothedeviationmeasuredingrain21. Thebehaviorofgrain4isagaindifferent:Forgrain4,amismatchbetween simulationsandexperimentofthetwotransverseelasticstrains( 𝑥 and 𝑧 )and the 𝑥𝑦 shearstrainisobservedforallloadstates,leadingtoamarginalbutsteadyincreaseofthedeviationasseeninFig.3.4.Finally,thebehaviorofgrain1canbedescribedasacombinationofasystematicdeviationofthetransversecomponentsandashearcomponent,incombinationwithajump.Nevertheless,thisgraindisplaysanalmostperfectagreementfortheelastic straininloadingdirection.Thedeviationingrainswithagoodagreementis mainlycausedbyindividualcomponents,i.e. 𝑧 ingrain16and 𝑦 ingrain9. Investigatingseveralfeaturesofthegrainsintheneighborhoodofthechallenge grainsdidnotrevealanycorrelationbetweentheinvestigatedfeaturesandthe contributiontotheresidual.FiguresA.6toA.8inthesupplementarymaterial summarizetheseinvestigations.3.3.3MonotonyoftheElasticStrainEvolution Thenon-monotonousbehaviorobservedin,e.g.,grains21and1,hasbeen attributedtointermittentplasticity,i.e.avalanche-likemotionofdislocations [188].Dislocationavalanchesarecausedbythecollectiveandcooperativemotion 64COMPARISONOFFULL-FIELDCRYSTALPLASTICITYSIMULATIONSTOHIGH-ENERGYDIFFRACTIONMICROSCOPYEXPERIMENTS3.3.5VolumetricStrain Anotherdifferencebetweenexperimentandsimulationthatisnotreadily visiblefromFig.3.5isthevolumetricelasticstrain.Figure3.9shows tr(𝜺)=𝜀x+𝜀y+𝜀z exemplarilyforgrains10,21,16,and19.Themoststrikingfeature— whichcorrespondstothehigher(negative)strainofthetransversecomponents seenfortheexperimentinFig.3.5—isthesystematicoverestimationofthevolumetricstrainsbythesimulation.Theonlycaseamongall28challenge grainsinwhichthevolumetricstrainintheexperimentexceedsthatpredicted bythesimulationisatS5forgrain21.Asdiscussedbefore,thisisoneexampleforanon-monotonousstrainevolutionintheexperimentandS5differssignificantlyfromS4andS6.Asecondaspectforwhichasystematic deviationisseen,isthedecreaseof tr(𝜺) fromS0toS1whichisobservedinthe experimentaldatawhilethesimulationspredictforalmostallgrainsasteady increase.Figure3.10presentsthisdatainadifferentviewinspiredbythe Poissonratio 𝜈 .Here,thenegativeaverageofthetransversestrainsdividedby thestraininloadingdirectionisshownforexperimentandsimulation.Theexpectedvalueforpolycrystalwithrandomcrystallographictextureandthestiffnessconstantsusedinthesimulationisalsogiven.Itcanbeseenthat thecontractionpredictedbythesimulationscattersaroundthevalueexpected foranisotropicallyequivalentmaterialloadedinuniaxialtension,whilethe experimentalresultsaresystematicallyhigher. Onepotentialreasonforthesystematicdifferenceof tr(𝜺) istheuseofwrong stiffnessvaluesinthesimulationsetup.However,toreachtheexperimentally observedvaluesforthecontraction −0.5(𝜀z+𝜀x)/𝜀y ,elasticconstantsthat deviatesignificantlyfromvaluesreportedintheliteraturearerequired.Another explanationisthecrystallographicorientationofthechallengegrains.Ifthey haveapreferredorientation,assumingaPoissonratioof 𝜈=0.31 isnot justified.However,ifthatwouldbethecase,itisdifficulttoexplainwhythe simulationpredictsanaveragevaluecloseto 𝜈=0.31 asthethecrystallographic orientationisincludedinthesimulation.Withoutadetailedanalysisoftherawexperimentaldata,adefiniteanswertothereasonfortheobservationoflargevolumetricstraincannotbegiven.Butsinceitisknownthatthe sample-to-detectordistancesignificantlyinfluencesthevalueobtainedforthe normalcomponentsoftheelasticstraintensors[210],itisreasonabletoassume thatthecurrentdatasetisaffectedbythisissue.Inthiscontext,itshouldbenotedthatCockeetal.[187]subtractedtheaverageinitialelasticstrain amongallgrainsfromthegrain-wisevaluesusedforinitialization.Whilethis modificationdecreasedthemismatchbetweensimulationandexperiment,the givenreason—thepresenceofapre-stress[213]—isnotmentionedinthedataset description[161]. CONCLUSIONSANDOUTLOOK65 3.77 4.08 S1 S2 S3 S4 S5 S6 Macroscopic load states -0.31 0.00 0.31 0.63 1.88 1 2−ϵz −ϵx ϵy experiment simulation Figure3.10:Aboxplotdifferentiatingthecontractionbehaviorofgrainsas seeninexperimentandsimulation.Everyboxextendsfromthefirstquartile (Q1,bottom)tothethirdquartile(Q3,top)ofitscorrespondingdatasetand thewhitehorizontallineinsiderepresentsthemedian.Thewhiskersextenduntilthefarthestdatapointlyingwithin1.5timesoftheinterquartilerange ( IQR=Q3−Q1 )oneithersideandthedatapointsoutsidetherangeofwhiskers arerepresentedascircularmarkers.Theredlineindicatesthevalueexpected forapolycrystalwithrandomtexture.3.4ConclusionsandOutlook ThetensiletestofanadditivelymanufacturedIN625specimencharacterizedin situbyHEDMwassimulated.Tothisend,twocrystalplasticityconstitutive modelswereemployed:aphenomenologicalmodelandadislocationdensity-basedone.Modelparameterswerecalibratedusingglobalstress–straindataobtainedduringthetensiletestwhichcapturedthemacroscopicresponseof thespecimen. Anoveliterativeapproachtoincorporateeigenstrains,andtherebyresidualstresses,inthefull-fieldsimulationwasdevised.Basedonthisstrategy, 66COMPARISONOFFULL-FIELDCRYSTALPLASTICITYSIMULATIONSTOHIGH-ENERGYDIFFRACTIONMICROSCOPYEXPERIMENTS theproposedmethodspecificallyaddressesscenarioswhereexperimentallycharacterizedinitialelasticstrainsareunavailableforallgrainsinthe microstructure.Themethodnotonlyinitializesthecharacterizedgrainsusing grain-averagedstrainvalues,butalsoenforcesstressequilibriumandstrain compatibilityconditions.Asaresult,theimposedelasticstrainisredistributed amongneighboringgrains,allowingintragranularvariationsinresidualstress tobecapturedevenwhenonlygrain-averagedvaluesareprovidedbytheexperiment.Thisapproachfortheinclusionoftheeigenstrainsinselected grainsleadstoaverycloseagreementtotheexperimentallymeasuredvalues atloadstateS0.ThisdirectlyaddressesresearchquestionRQ1posedinthe introductorychapter(Section1.5)andfacilitatesanaccuraterepresentationof thematerial’sinitialstate,therebyofferingareliablefoundationforsubsequent analyses. Accuraterepresentationofresidualstressesthroughafaithfulreplicationoftheinitialmaterialstateisparticularlyimportant.Theirinfluenceisclearlynoticeableintheelasticity-dominatedloadingregimebuttendstodiminish onceplasticitymechanismsbegintodominatetheresponse.Thisconsideration becomesespeciallycriticalinhigh-cyclefatiguepredictionsimulations,where analysesbasedonthefirstfewloadingcycles—typicallywithintheelastic regime—areoftenextrapolatedtoestimatethematerial’sfatiguelifetime[214]. Next,addressingresearchquestionRQ2,theabilitiesofthecrystalplasticity modelstoreproducetheexperimentallyobservedmaterialbehavioratmesoscaleareexaminedusingtheinsituexperimentaldata.TheoverallpredictionqualityusingthemetricoftheAFRLAMmodelingchallengeisconsideredfollowedbyanin-detaildiscussionofthreeaspects,i.e.monotonyoftheelasticstrain,stress evolution,andvolumetricstrain,forwhichadisagreementbetweensimulations andexperimentisobserved.Thefollowingconclusionaredrawn:• Overall,closeagreementwithexperimentallymeasuredelasticstrain valuesatsubsequentloadstatesafterinitializationisachieved,confirming observationsfromearlierstudies[187,188,189].Evenatthemesoscopic level,thetwodistinctlyformulatedcrystalplasticitymodelsperformednearlyindistinguishablybasedonthecurrentevaluationmetrics(cf. Table3.4).Itshouldbenotedthattheseevaluationswereconductedwithin thedatadomainusedformaterialcalibration,wherephenomenological modelsaregenerallyexpectedtoperformwell.• Thenon-monotonousevolutionoftheelasticstrainsseeninthe experimentsinsomegrainsforcertainloadstatescannotbereproduced byconventionalcrystalplasticitymodels. CONCLUSIONSANDOUTLOOK67• Enforcingtheexperimentallyobservedjumpoftheelasticstraininasinglegrainandadjustingtheplasticstateslightlyincreasestheagreementwith thenextloadstate.• Whenconsideringindividualtensorcomponents,adeviationbetweenthe evolutionofelasticstrainandstressisseen.• Asystematicdeviationbetweenexperimentsandsimulationsisobserved forthevolumetricstrain. Basedonthesefindings,twosuggestionsaremadeforfuturecrystalplasticity benchmarkproblemsobtainedfromsynchrotrondata:First,theuncertaintyof themeasuredelasticstrainsshouldbequantifiedtosetrealisticexpectations. Ideally,theroutinesforprocessingtheexperimentaldatacanbeimprovedto mitigatemeasurementartifactsandreducetheuncertainty.Second,further experimentsareneededtobetterunderstandthestrainjumps.Sincedislocationavalanches,whichareassumedtobethereasonforthenon-monotonousevolution oftheelasticstrain,arestochasticphenomenathiswouldrequireadatasetin whichmoregrainsaretracked.Suchadataset,inwhichideallyallgrainsare tracked,wouldalsoallowtosystematicallyinvestigatewhetherfeaturesofthe neighborhoodsuchasgrainsize,grainshape,andcrystallographicorientation influencethestrainmonotony. Whilethisstudyassessesthecapabilitiesoftwostate-of-the-artcrystalplasticity models,therebyaddressingresearchquestionRQ2posedinSection1.5,the mainfeaturethatismissingintheemployedconstitutivelawsistheprediction ofdiscreteslipevents.Modelsforlocalizedslip[215]couldpredictsuchbehavior inastatisticalsense,butitisimportanttorealizethattheuseofsuchmodels wouldnotimprovethepredictionofthebehavioratindividualgrains.Thisfurtherraisesanimportantquestionthatneedstobeaddressed:asdatarichexperimentsbecomeincreasinglycommonandenablethedevelopmentof higher-fidelitysimulationsatthegrainscale,howcanoneachieveareliable assessmentofmodelpredictionsagainstexperimentalobservations,especially whenstochasticmechanismsbegintodominatetheextractedbehavior? Otheraspectssuchassize-effectsorkinematichardening(duringunloading) thatarenotincludedintheusedcrystalplasticitylawsseemtoplay—atleast fortheconsideredsample—noimportantrole. Chapter4 IncorporationofPhysics-based StrengtheningCoefficients intoPhenomenologicalCrystal PlasticityModels Thischapterisbasedonthefollowingpublicationwhichiscurrentlyunder reviewprocessinajournal: N.PrabhuandM.Diehl.Incorporationofphysics-basedstrengtheningcoefficientsintophenomenologicalcrystalplasticitymodels.2025.doi: 10.48550/ARXIV.2502.1043069 70INCORPORATIONOFPHYSICS-BASEDSTRENGTHENINGCOEFFICIENTSINTOPHENOMENOLOGICALCRYSTALPLASTICITYMODELS4.1Introduction Inthepreviouschapter,itwasshownhowtwodistinctcrystalplasticity models—onephysics-basedandanotherphenomenological—canbecalibrated againstareferenceexperimentaldatausinganexperimentally-characterized3D microstructure.Thecalibrationprocessnotjustinvolvedadjustingmaterial constitutiveparametersagainstthemacroscopictensilebehaviorofthesample butalsoincludedinitializingasubsetofgrainswiththeirobservedelastic strainstatestoaccuratelyrepresenttheinitialstateofthematerial.Afterthe calibration,themodels’performanceswereevaluatedbycomparingamesoscale quantity—thegrain-averagedelasticstraintensor—atseveralstatesduringthe mechanicalloading. Thestudyrevealedthat,despitethedifferingcomplexityofthemodels,theirperformanceswereonparwitheachother.Importantly,itshouldbenotedthattheevaluationswerecarriedoutwithinthecalibratedregimeofthematerialbehavior.Itisquitestraightforwardtoinferthatincreasing modelcomplexityraisescomputationalcosts,asmorefloating-pointoperations areneededtocomputeandhandleadditionalstatevariables.Inadditiontolowercomputationalcosts,theoperationallyefficientcalibrationofthe phenomenologicalmodelstypicallyinvolvesfittingonlyahandfulofparameters ofadhocfunctionstoamacroscopicmaterialbehavior.Theaboveadvantages positionphenomenologicalmodelsascompellingalternativestophysics-based models. Althoughphenomenologicalmodelsareoperationallyefficient,thephysics-based modelsholdtwodistinctadvantagesoverthem[126]:1. theyhavepredictivecapabilitiesoutsideofthedatarangeusedto determinetheirparameters,and2. theycanbeformulatedandparameterizedwiththehelpofsimulations methodsforsmallerscalessuchasdensityfunctionaltheory(DFT)and discretedislocationdynamics(DDD). AsdescribedpreviouslyinSection2.3,thephysics-basedmodelsincrystal plasticitytypicallydescribematerialstateintermsofdislocationdensitieswhich iswhytheyarealsogenerallyreferredasdislocationdensity-basedmodels.A plethoraofparametrizationsofdislocationdensitieshavebeenproposedwith thesimplestonestreatingthedensityasascalarvariable[217,218,138]while othersophisticatedonescharacterizeitasatensorialquantity[41,42,46,219]. Themovementofdislocationsandinteractionsamongthemduringaplasticdeformationresultinginstrainhardeningarealsocaptureddifferentlyinthe INTRODUCTION71 models.Whilethesimplemodelsrelyontheevolutionofagenericdislocation densityperslipsystem[49],thesophisticatedapproachesarebasedonmultiple dislocationdensitiesandconsidertheircharacteristicsformechanism-based evolutionrules[220,174]. Strainhardeninginmostdislocationdensity-basedmodelsisderivedfromthe so-calledTaylorrelation[67],whichimpliesthatthecriticalresolvedshear stress 𝜏crit (CRSS)isproportionaltothesquarerootofthedislocationdensity ( 𝜌 ).Therelationship 1𝜏crit=𝛼𝜇𝑏√𝜌 ,withshearmodulus 𝜇 andmagnitudeoftheBurgersvector 𝑏 ,heldtrueformostmetalswithlowlatticefrictionwhere dislocation–dislocationinteractionssolelydeterminedthestrainhardening[221].Thevalueofthescalarcoefficient 𝛼 thatdefinedaveragestrengthof distributionofdislocationswasconsideredtorangefrom 0.2 to 0.5 [222].The valuewastypicallycalculatedanalytically[223,224]andadditionallydetermined againstexperimentalshearflowstressvs.totaldislocationdensityplots[224, 225] 2 .However,thevaluesdeterminedthroughexperimentsoftensuffered fromuncertaintieswithmeasuringdislocationdensitiesandhencewerelimited tosimplecases.Additionally,thepredictivecapabilitiesofthebasicTaylor-likemodelswerelimitedasitlumpedalltheinformationaboutdislocation arrangementsanddensitiesintoasinglevariable𝜌. TheshortcomingswereaddressedthroughextensionsoftheTaylorrelationthat decomposethetotaldensitytosumofdensitiesperslipsystem.Tothisend, twodifferentapproacheswereproposed:1. chronologically,firstbyF.LavrentevandY.Pokhil(1975)[227,226,224] whichread𝜏𝑖crit=𝑁∑𝑗𝜇𝑗𝑏𝑗𝛼𝑖𝑗√𝜌𝑗,(4.1)2.andfollowedbyFranciosietal.(1980)[136,228]whichread𝜏𝑖crit=𝜇𝑖𝑏𝑖√√√⎷(𝑁∑𝑗(𝛼𝑖𝑗)2𝜌𝑗).(4.2) Intheaboveequations,thescalarcoefficient 𝛼 fromtheTaylor’srelationisnowrewritteninamatrixformwithcomponents 𝛼𝑖𝑗 whicharereferredasstrengtheningcoefficients.InEq.(4.2),thesquaredcomponentsofthe strengtheningmatrix (𝛼𝑖𝑗)2 areoftenrepresenteddirectlyas 𝑎𝑖𝑗 whichisthe 1 ThisrelationwasderivedfromTaylor’soriginalwork[67]althoughnotexplicitlywritten init.2 Thedislocationdensitywasmeasuredinitiallythroughetch-pitcountandlateronby TEMobservationsondeformedsinglecrystalsandpolycrystals. 72INCORPORATIONOFPHYSICS-BASEDSTRENGTHENINGCOEFFICIENTSINTOPHENOMENOLOGICALCRYSTALPLASTICITYMODELS so-calledinteractionmatrixwithrelation (𝛼𝑖𝑗)2=𝑎𝑖𝑗 .Whencomparedtothe originalformulations,theadditionalcontributionoflatticeresistanceto 𝜏crit is omittedintheaboveequations,whichis,ingeneral,justifiedforFCCmetals. Animportantdifferencebetweenthetwoproposedapproachesisthestageatwhichthestrengtheningcoefficientsaremultipliedwiththedecomposeddislocationdensities:whileinEq.(4.1)thematrixismultipliedafterthesquareroot,inEq.(4.2)theinteractioncoefficientsaremultipliedandthesummationoverslipsystemsiscomputedwithinthesquareroot.Sinceitwasfirstproposed,theextensionbyFranciosietal.(cf.Eq.(4.2))hasbeenakeyingredientindislocationdensity-basedcrystalplasticitymodelsthataccountfordensitiesonindividualslipsystemsandstrengtheningduetodislocation–dislocationinteractionsbetweensystems.Forthesakeofbrevity inthesubsequentdiscussions,theEq.(4.2)willberewrittenasfollowsinthe formthatexplicitlycontainstheinteractionmatrix𝜏𝑖crit=𝜇𝑖𝑏𝑖√√√⎷(𝑁∑𝑗𝑎𝑖𝑗𝜌𝑗).(4.3) Theentries 𝑎𝑖𝑗 oftheinteractionmatrix 𝐚 areameasureoftheaverageinteractionstrengththatdislocationsonsystem 𝑗 exertondislocationsfrom system 𝑖 .Earlyinvestigationsattemptedtoindirectlyidentifythesecoefficients bothanalyticallyandexperimentallytoobtainaroughestimate.Theexperimentalapproach,i.e.thelatenthardeningtests,involvedconducting tensiletestsattwolevelsandwerepredominantlyonsinglecrystals[136,229]: first,apreloadingstepwhereideallyasinglesystemisactivatedensuringthatlatentsystemsgetadequatelyhardened.Asecondloadingstepensues alonganotherdirectionsuchthatapreviouslylatentsystemisnowloadedina singleslipcondition.Thelatenthardeningratioswerethencomputedfromthe shearstress–straindatafromexperimentswithmultiplesampleorientations. F.LavrentevandY.Pokhil[227,226,224]employedanintegratedexperimental– analyticalapproachwherecalculationofthestrengtheningcoefficientswerebasedonenergygainduetopairwiseinteractionofdislocations.However, theabovemethodssufferedeitherfromdifficultieswithactivatingasingleslip systemorfromtheuncertaintiesassociatedwithmeasuringdislocationdensities. Astheaveragestrengthsofvariousdislocationinteractionsaredifficulttocomputethroughtheoreticalcalculationsortomeasuredirectlyfromexperiments,aneffectivealternativeforsuchevaluationsisthediscretedislocationdynamics(DDD)simulationapproach.Ingeneral,thetechnique modelsthedislocationmovementbydiscretecrystallographicstepsofconnected segmentsmovingthroughanelasticcontinuum.Theroutineforcalculatinga setofinteractioncoefficientsforagivenmaterialtypicallyinvolvesperforming INTRODUCTION73 multipleDDDsimulations,inwhichdislocationsonaprimaryslipsystemare driven—byanexternallyappliedforce—tocutthroughaforestdislocationsystem,resultingindistinctinteractions[230].TheinteractioncoefficientsarethendeducedfromEq.(4.3)withsimultaneousmeasurementsofapplied resolvedshearstressandforestdislocationdensities.Theforestdensitiesare usuallykeptconstantduringtheprocesstoavoidmutualreaction[231].Initially, isotropicshearmoduluswasassumedforcomputationaltractability[230,231]; however,recentstudiessuggestimprovementsthoughadoptionofhigher-order approximationsoftheanisotropicshearmodulus[232].Overall,thevaluesofmostoftheseinteractioncoefficientscanbeobtaineddirectlyfromDDD simulations[230]independenceofmaterial[232]anddislocationdensity[231, 233].Thesevaluescandirectlybepluggedintodislocationdensity-basedmodels (cf.Eq.(4.3))toaccuratelycaptureactiveandlatenthardeningphenomenaat thelevelofslipsystems. Despitethelimitationsofphenomenologicalmodelsagainstthedislocationdensity-basedonesthatwereoutlinedbefore,theformerfindswidespreadapplicationsbecauseofitssimplicity:asinglevariable 𝜏crit (perslipsystem)parametrizesthestateofthematerial.TheCRSSofslipsystem 𝑖 , 𝜏𝑖crit is formulatedasanadhocfunctionwherethedetailsofthephysicalphenomena intermsofdislocationdensitiesareomitted.Suchmodelscommonlydescribe evolutionoftheCRSSthroughVocelaw[234,235]extendedtomultipleslip systems[236,237]orpowerlaw[96].Inthecaseofpowerlaw(cf.Section2.3.2), theevolutionfollowstheform3𝜏𝑖crit=ℏ0∣1−𝜏𝑖crit𝜏crit,∞∣𝑎𝑁∑𝑗ℎ𝑖𝑗∣𝛾𝑗∣,(4.4) where ℏ0 and 𝑎 arefittingparameters, 𝜏crit,∞ isthesaturationvalueof 𝜏crit , 𝛾istheshearrate,and𝐡isthehardeningmodulimatrix[140,238]. Despiteaclearanalogybetween 𝐚 inEq.(4.3)and 𝐡 inEq.(4.4),typically ℎ𝑖𝑗 arechosentobeeither 1.0 or 1.4 [239,240,241].Thischoicedatesback—tothebestknowledgeoftheauthor—toalimitedsetofexperimentalobservationsreportedbyU.F.Kocks[164]whichmotivatedPeirceetal.[96]tousevaluesof 1.0 forallcoplanarinteractionsand 1.4 otherwise.Asimplificationofthisparametrization isfrequentlyusedwhere 1.0 isusedonlyforselfinteractionsandnotforother coplanarinteractionswhichalsohaveavalueof 1.4 [242].Althoughtheinitial motivationforusingsuchvaluesdatesbackoverfourdecades—rootedinthe experimentalinsightsavailableatthetime—areassessmentoftheircontinued usageislongoverdue.Withsignificantadvancementsinbothexperimentalcharacterizationandcomputationaltechniques,moreaccurateinsightsinto 3 Incomparisontotheresistancetosliprepresentedas 𝜉 inEq.(2.26),thesameis representedas𝜏critinEq.(4.4)forconvenience. 80INCORPORATIONOFPHYSICS-BASEDSTRENGTHENINGCOEFFICIENTSINTOPHENOMENOLOGICALCRYSTALPLASTICITYMODELS (1938)[75].However,intheveryparagraphin[164],healsodiscussedpossibilitiesofcertainlatentsystemshardeninglesserthantheactive systems.Inthecontextofphenomenologicalcrystalplasticity,theinitial adoptionofU.F.Kocks’proposalwasseenintheworkofPeirceetal. (1982)[96];theauthorsfurthernarroweddowntherangeoftheratioto 1.3–1.4.However,theparametrizationsuffersfromasimplificationthat failstocapturethedifferenceininteractionstrengthsamongnon-coplanar interactions.• Simplified:Allmodulihaveavalueof 1.4 exceptfortheselfinteraction whichhasavalueof1.0[242]. Thisisafurthersimplificationfromthetraditionalparametrizationwhere alllatenthardeningsystems,irrespectiveofwhethercoplanarornot,areassumedtohardenatahigherratethantheactivesystem.The complexityofthisparametrization,intermsofthelevelofdetailaccounted forbetweeninteractions,liesinbetweenTaylor’sisotropichardening[75] andU.F.Kocks’proposal[164].• Strengtheningcoefficients( 𝛼 ):Strengtheningcoefficientsaredirectly obtainedfromDDDsimulations[233,232]andarescaledsuchthat ℎ0=1.0.Thestrengtheningcoefficientsdirectlyactasweightsforthesquareroot ofdislocationdensitythatisdecomposedontoslipsystemsasseeninEq.(4.1)followingtheapproachofF.F.LavrentevandY.A.Pokhil [227].• Interactioncoefficients( 𝑎 ):Thesevaluesareusedindislocationdensitybasedconstitutivelaws(andconstitutetheinteractionmatrix 𝑎𝑖𝑗 that appearsinEq.(4.3))andarethesquareofthestrengtheningcoefficients [232].TheresultingindependentvaluesforthefourcasesareshowninTable4.4.4.4MaterialParametersCalibration CalibratedmaterialdescriptionofIN625basedonthephenomenologicalmodel presentedinTable3.2aofthepreviouschapterisreusedhere.Asthehardening modulimatrixwillnowemploydifferentparametrizations,thisnecessitates slightadjustmentsofotherparametersintheplasticconstitutivelawinorder forthemodelstoexhibitidenticalmacroscopicresponses.Tothisend,thethree plasticconstitutiveparameters ℏ , 𝜏crit,0 ,and 𝑎 fromEq.(4.4),areadjusted MODELPERFORMANCEASSESSMENT81ℎ0ℎcoplaℎcoliℎ1ℎ2ℎ∗2ℎ3traditional1.01.01.41.41.41.41.4simplified1.01.41.41.41.41.41.4strengtheningcoeff.1.001.002.350.600.910.861.17interactioncoeff.1.001.005.510.360.840.741.38 Table4.4:Independentvaluesofthehardeningmodulimatrix 𝐡 ofthephenomenologicalmodelusedintheparameterstudy.ThestrengtheningandinteractioncoefficientsaretakenfromMadecandL.P.Kubin[232]and scaledsuchthatℎ0=1.0. forallfourcasesindividuallysuchthattheaveragestress–strainbehaviorofthereferencematerialisreproduced.Tothat,asimplifiedpolycrystalmodelconsistingof 10×10×10 materialpointswith 1000 grainsisused,i.e.each “grain”discretizedbyasinglevoxel.Twomonotonicloadcases,whichwillbe discussedinthefollowingsection,areusedforcalibration.Theparametersare optimizedusingtheNelder–MeadalgorithmavailableinSciPy[255,256,257]. Thelossiscomputedasthedifferenceinstressinloadingdirectionfromall fourcombinationsoftexture(randomandrolling,cf.Section4.5.1)andload case(uniaxialtensionandsimpleshear,cf.Section4.5.2)whichareconsidered withequalweights.Thecalibratedvaluesoftheparametersarereportedin Table4.5.𝜏crit,0ℏ𝑎UnitMPaMPa-traditional158.59411.5simplified158.29061.5strengtheningcoeff.158.011001.5interactioncoeff.156.611221.5 Table4.5:Calibratedvaluesofplasticityparametersfordifferentparametrizationsobtainedfromoptimizingagainstthereferencematerial behaviors.4.5ModelPerformanceAssessment Theassessmentofthemodelsiscarriedoutacrosstwofacetsoftheirperformance.Thissectionconcernsevaluatingeffectivenessofmodelstoreproducemesoscopic 82INCORPORATIONOFPHYSICS-BASEDSTRENGTHENINGCOEFFICIENTSINTOPHENOMENOLOGICALCRYSTALPLASTICITYMODELS quantitiesintermsofhardeninganddeformationbehaviorsgiventhattheyallexhibitidenticalmacroscopicbehaviors.Consequently,theevaluationsareperformedwithinthecalibratedregimeofthephenomenologicalmodels,wheretheyareanticipatedtoperformtotheirstrengths.Theotherfacetofperformanceevaluationcanalsobeenvisagedasevaluationofpredictive capabilitiesofthemodel.MoreinformationonthisisprovidedinSection4.7. Throughoutthisstudy,thematerialbehaviorsproducedbyadislocationdensity-basedmodelareconsideredasgroundtruthbenchmarks.Thecalibratedmodel descriptionthatisbasedondislocationdensityfromthepreviouschapteris reusedhere.InthismodeltheinteractionsaredescribedbyEq.(4.3),i.e.the valuesoftheinteractionmatrixareindependentofthedislocationdensity,and directlyemploysthevalueslistedinTable4.3.4.5.1MicrostructuresandTextures Twodifferentmicrostructuresaresyntheticallygenerated,onewithglobular grainsandarandomtextureandonewithflattenedgrains(aspectratio1:1:3) andanidealizedFCCrollingtexture.Bothmicrostructuresarediscretizedby 48×48×48 voxels,contain 1000 grains,andareperiodicallyrepeatedattheboundaries.TherollingtexturewascreatedusingtexturecomponentswithGaussianscatterfollowingtheideasofHelming[258].Itconsistsof 30% Sorientation( {123}⟨634⟩ ), 20% Cuorientation( {112}⟨111⟩ ), 15% Brassorientation( {011}⟨211⟩ ), 10%α -fiber( ⟨110⟩∥ ND),andarandom background[259,260].Ascatterwithafullwidthathalfmaximumof 50° and 20° wasusedforthetexturecomponentsandforthefiber,respectively.The microstructureswithgrainorientationsmappedontoanIPFcolortriangleis displayedinFig.4.1(A&B).Alongside(inFig.4.1C),a {111} polefigureoftherollingtextureispresentedandqualitativelyresemblestheonesreportedin [259,260].4.5.2CalibrationLoadCases Thetwomonotonicloadcasesthatwereconsideredforthematerialparameter calibrationareuniaxialtensionandsimpleshear,bothtoastrainof approximately 415% atastrainrateof 1×10−3 s −1 ;thetwo(uniaxialtension) orthree(simpleshear)undefinednormalcomponentsofstressaresetto 0.0MPa toenablevolumechangesduetocontractionintheelasticregime.Thevaluesof 4 TheloadisdefinedintermsofthedeformationgradientFandafinalvalueof 1.15 and 0.15 wasprescribedforrespectivedirectionofthetensile( 𝐹11 )andshear( 𝐹21 )load, respectively. MEASUREMENTOFCORRELATION83 thecomponentsarevolumeaveragesandperiodicboundaryconditions,which areinherenttothespectralmethod,areapplied.4.6MeasurementofCorrelation Theagreementbetweenthereferencesolutionandthedifferentparametrizations ofthephenomenologicalmodelatmesoscalecanbeconvenientlyvisualizedin correlationplots.Foraperfectagreement,allpointsarelocatedonalinewith aslopeof 1.0 ( 45° inclination)thatgoesthroughtheoriginwhichiscalledthe 1:1line.Foranimperfectagreement,twokindofdeviationsfromthisideal situationcanbeobserved:1. thebest-fitlinemightdeviatefromthisline.Thisiscalledalackof accuracy[261].2. theindividualpointsscatteraroundthebest-fitline.Thisiscalledalack ofprecision[261]. Theconcordancecorrelationcoefficient 𝜌c proposedbyL.I. - K.Lin[261]isused inthisstudytomeasuretowhichdegreedatapairsfallonthe1:1line.It accountsconvenientlyforboth,precisionandaccuracy,andisdefinedas:𝜌c∶=1−𝔼[𝜖2]𝔼[𝜖2]|𝜌=0.(4.5) InEq.(4.5),thenumerator 𝔼[𝜖2] istheexpectedsquareofthedistancefromthe1:1line.Thedenominatoristhesamemeasurebutforthecasethatassumesthetwovariablestobecompletelyuncorrelated,i.e.Pearson’scorrelationcoefficient 𝜌 iszero.Aftersimplification, 𝜌c canbecomputedusingstandarddeviation (𝜎𝑧and𝜎𝑧),mean(𝜇𝑧and𝜇𝑧)andcovariance(𝜎𝑧𝑧)asfollows:𝜌c=2𝜎𝑧𝑧𝜎2𝑧+𝜎2𝑧+(𝜇𝑧−𝜇𝑧)2=𝜎𝑧𝑧𝜎𝑧𝜎𝑧21𝜎∗+𝜎∗+𝑢2=∶𝜌𝐶b,(4.6) where 𝜎∗=𝜎𝑧𝜎𝑧 , 𝑢=𝜇𝑧−𝜇𝑧√𝜎𝑧𝜎𝑧 ,and 𝑧 and 𝑧 denotematerialpointdatafromphysics-basedandphenomenologicalmodels,respectively.L.I. - K.Lin[261]multiplicativelydecomposed 𝜌c intotwocontributionsasseeninEq.(4.6): Pearson’scorrelationcoefficient 𝜌 whichmeasuresthedegreeoflinearcorrelation, andbiascorrectionfactor 𝐶b whichmeasureshowfarthebest-fitlinedeviates fromthe1:1line.Interpretationofvalueof 𝜌c isanalogoustothatofPearson’s correlationcoefficient𝜌: 84INCORPORATIONOFPHYSICS-BASEDSTRENGTHENINGCOEFFICIENTSINTOPHENOMENOLOGICALCRYSTALPLASTICITYMODELS•1indicatesperfectagreement(𝑧=𝑧),•−1indicatesperfectdisagreement(𝑧=−𝑧),• andadeparturefromtheseterminalvaluestowardszeroindicatelowering degreeofagreement. Although 𝜌c iscommonlyusedasasingleindexwhencomparingmultiplemodelsagainstasolereferencedataset[262],thedecompositionproducts 𝜌 and 𝐶b arealsoshown.Inaddition,themodelingefficiencycoefficient[263] MEC∶=1−MSE𝜎2𝑧whereMSE=∑𝑁𝑖=1(𝑧−𝑧)2,iscalculated.4.7EvaluationofPredictiveCapabilities AstouchedonbrieflyinSection4.5,thesecondfacetoftheperformanceevaluationisconductedinascenariowherethebehaviorreproducedbythe phenomenologicalmodelsisoftentakenwithadegreeofskepticism,particularly inloadstatesthatareexternaltothedatarangeoftheircalibration.Tothis end,twonon-monotonic,non-calibratedloadcasesareapplied,andcomparisons aremadeamongthefourparametrizations.Thetwoloadcasesthataimto inflictloadpathchangescascadethreemonotonicloadingsandareasfollows:• Loadcase1:Subsequentapplicationsofuniaxialtensionalong 𝑥 and 𝑦directions,andsimpleshearonthe𝑦𝑧plane.⎡⎢⎣𝜀𝑥𝑥000∗000∗⎤⎥⎦→⎡⎢⎣∗000𝜀𝑦𝑦000∗⎤⎥⎦→⎡⎢⎣∗000∗𝜀𝑦𝑧0𝜀𝑦𝑧∗⎤⎥⎦• Loadcase2:Subsequentapplicationsofsimpleshearon 𝑥𝑦 plane, uniaxialtensionalong𝑧direction,andsimpleshearon𝑦𝑧plane.⎡⎢⎣∗𝜀𝑥𝑦0𝜀𝑥𝑦∗000∗⎤⎥⎦→⎡⎢⎣∗000∗000𝜀𝑧𝑧⎤⎥⎦→⎡⎢⎣∗000∗𝜀𝑦𝑧0𝜀𝑦𝑧∗⎤⎥⎦ Inadditiontotheirdescription,theloadcasesarerepresentedbyapproximate volume-averagedstraintensors.Theasterisksymbolindicatesthattheparticular componentisallowedstress-freedeformation. Thepredictivecapabilitiesaretestedonthemacroscopicfrontbycomparing stress–strainresponsealongtheloadingdirections.Inparallel,thetestonthe mesoscopiclevelconsistsacomparisonofpercentageofthetotalslipsystems thatareactiveinthemicrostructureattheendofeachloading. RESULTSANDDISCUSSION854.8ResultsandDiscussion Theresultsfromthesimulationsarepresentedandcomplementedbybrief discussionsinthissection.Asaprerequisiteforanobjectivecomparisonofthe differentparametrizations,ithastobeestablishedthatthephenomenological modelreproducesthemacroscopicbehaviorofthedislocationdensity-based modelindependentlyofthechosenparametrization.Tothisend,theequivalent stress–straincurvesforbothmicrostructures(equiaxed(A)androlled(B))loadedintensionandinsimpleshearareshowninFig.4.1.Thereference 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 ε 0 200 400 600 800 1000 1200 σ(MPa) globular/random: UT flattened/rolling: UT globular/random: SS flattened/rolling: SS (A) (B) TD RD (C) 1 3 5 8 10 13 15 18 20 23 Figure4.1:Stress–straincurveinloadingdirectionforthefourcombinations ofmicrostructure/textureandload.Thesolid(uniaxialtension,UT)anddashed(simpleshear,SS)linesindicatethereferenceresultsobtainedwith thedislocationdensity-basedmodelandtheshadedbackgroundrepresentsthe rangeobtainedfromthefourdifferentparametrizationsofthephenomenologicalmodel.Theblackdotsmarkstrainlevelsof ≈8% andtheblackcrossesindicate thefinalloadof≈15%. 86INCORPORATIONOFPHYSICS-BASEDSTRENGTHENINGCOEFFICIENTSINTOPHENOMENOLOGICALCRYSTALPLASTICITYMODELS resultsareshownasasolidlineandtheminimumandmaximumamongallfourparametrizationsisshownasashadedarea.Itcanbeseenthatallparametrizationsofthephenomenologicalconstitutivelawreproducethe macroscopicbehaviorofthephysics-basedformulation.Inthisplot,thestrain levels( 𝜀≈0.08 and 𝜀≈0.15 )atwhichthecorrelationofmesoscopicquantities isinvestigatedarealsoshown. Asmentionedpreviously,thecorrelationbetweenthereferencesolutionandthedifferentparametrizationsatmesoscaleisdeterminedindependentlyfor thehardeningbehaviorandforthedeformationbehavior.Forthehardening behavior,theCRSSincreaseperslipsystemofallmaterialpoints, Δ𝜏crit∶=𝜏crit−𝜏crit,0 ,isexamined.Forthedeformationbehavior,theaccumulatedshear 𝛾 perslipsystemisexamined.Sinceitisknownonlyafewslipsystemsneedtobeactivatedtoachievetheapplieddeformation[75],onlyactiveslipsystems,i.e.systemswith 𝛾>1×10−3 areconsideredforthedeformationbehavioranalysis. Table4.6givesthefractionofactiveslipsystemsforwhich 𝛾>1×10−3 and showsthatabout 35% ,i.e. 4 outofthe 12 slipsystems,areactiveforthe consideredcases.𝜀≈0.08𝜀≈0.15traditional0.330.340.320.370.380.36simplified0.330.340.320.370.380.36strengtheningcoeff.0.340.350.330.380.390.37interactioncoeff.0.340.340.330.370.390.36dislocationdensity-based0.320.330.310.360.370.35 Table4.6:Fractionofslipsystemsthatareactive,i.e.haveaccumulatedslip 𝛾>1×10−3 forthedifferentparametrizations.Thevaluesgiveninnormal fontaretheaverageofallfourcombinationsofmicrostructure/textureandload whicharecomplementedbytheirminimuminsubscriptandtheirmaximumin superscript.4.8.1HardeningBehavior CorrelationplotsoftheCRSSincrease Δ𝜏crit perslipsystemofallmaterial pointsareshownatappliedstrainsofapproximately 8% and 15% (cf.Fig.4.1) inFigs.4.2and4.3,respectively.Thedataisplottedindividuallyforthe fourconsideredparametrizationsasdensitycontourplotswiththecolorcode representingpointdensityintermsofnumberofdatapointsonalogarithmic RESULTSANDDISCUSSION87 scale 5 .Thatis,thedarkertheshadeofaregion,thehighertheclusteringof datapoints.Thisservesasaneffectivealternativetoascatterplot,fromwhichitisdifficulttoinfertheintensityofpointclusters,especiallywhenthenumber ofdatapointislarge. 0 400 800 0 400 800 dislocation density-based (a) traditional 0 400 800 (b) simplified 0 400 800 (c) strengthening coeff. 0 400 800 (d) interaction coeff. 100 101 102 103 104 105 phenomenological Figure4.2:DensityplotshowingthecorrelationbetweentheincreaseinCRSS( Δ𝜏crit in MPa )obtainedwiththedifferentparametrizationsofthe phenomenologicalmodelandthedislocationdensity-basedmodelatanapplied strainofapproximately8%inloadingdirection. 0 500 1000 0 500 1000 dislocation density-based (a) traditional 0 500 1000 (b) simplified 0 500 1000 (c) strengthening coeff. 0 500 1000 (d) interaction coeff. 100 101 102 103 104 105 phenomenological Figure4.3:DensityplotshowingthecorrelationbetweentheincreaseinCRSS( Δ𝜏crit in MPa )obtainedwiththedifferentparametrizationsofthe phenomenologicalmodelandthedislocationdensity-basedmodelatanapplied strainofapproximately15%inloadingdirection. Thefiguresrevealthatthetraditionalandsimplifiedparametrizationsperform worsethanthephysics-informedparametrizationandtheuseofinteraction parametersleadstotheclosestagreement.Detailedinspectionoftherawdata [264]revealsthat Δ𝜏crit oftheprimary,i.e.themostactive,slipsystemisina 5 asinglelevelofcolorintensityshiftcancorrespondtoapointdensityshiftofapowerof ten. 88INCORPORATIONOFPHYSICS-BASEDSTRENGTHENINGCOEFFICIENTSINTOPHENOMENOLOGICALCRYSTALPLASTICITYMODELS significantlybetteragreementthantheremainingslipsystemsforthetraditionalandsimplifiedparametersets.Hence,besidesacleardeviationofcontoursfrom 1:1line,ahintofbimodaldistributioncanbeseeninFig.4.2(a)and(b)and Fig.4.3(a)and(b).Suchbehaviorisnotseenwhenusingthephysics-informedparametrizations.Thereisalsoacleardifferencebetweentheresultsfromusing strengtheningcoefficientstothatfromusinginteractioncoefficients:Whilethe formerhaveadistributionthatislocatedabovethe1:1line,indicatingless hardeningofthephenomenologicalmodelthanthereferenceresult,usingthe interactioncoefficientsleadstoanalmostsymmetricdistributionofhigh-density contourregionsaboutthelineofequality.𝜀≈0.08𝜌𝐶𝑏𝜌cMECtraditional0.450.490.420.680.710.660.310.320.280.090.130.06simplified0.460.510.440.670.700.650.310.330.290.110.140.09strengtheningcoeff.0.870.880.860.760.800.720.660.690.630.470.520.39interactioncoeff.0.870.890.860.960.960.960.840.850.820.560.610.50𝜀≈0.15𝜌𝐶b𝜌cMECtraditional0.410.440.380.710.740.690.300.310.270.120.160.08simplified0.420.460.400.700.720.680.300.310.270.140.160.11strengtheningcoeff.0.830.840.820.810.840.760.670.690.640.510.550.44interactioncoeff.0.840.860.830.970.980.970.820.840.810.540.590.50 Table4.7:Concordancecorrelationcoefficient 𝜌c anditsdecomposedproducts: Pearson’scorrelationcoefficient 𝜌 andbiascorrectionfactor 𝐶b ;andmodeling efficiencycoefficientMECmeasuringthedegreeofagreementbetweenthereferenceresultsandthedifferentparametrizationsfortheincreaseinCRSS Δ𝜏 .Thevaluesgiveninnormalfontaretheaverageofallfourcombinationsof microstructure/textureandloadwhicharecomplementedbytheirminimumin subscriptandtheirmaximuminsuperscript. Tofurtherquantifythevisualimpressions,themeasuresofcorrelationintroducedinSection4.6aregiveninTable4.7forthetwoconsideredstrainlevels.Inthis table,theaverageamongthefourcombinationsofmicrostructure/textureand RESULTSANDDISCUSSION89 loadiscomplementedbytheminimumandmaximumamongthem.Fromthis table,thefollowingobservationscanbemade:• ThevisualimpressionofFigs.4.2and4.3isreaffirmedandthehighestdegreeofagreementisseenforinteractioncoefficients,followedby strengtheningcoefficients.• Lowvaluesofthecorrelationcoefficient 𝜌 forsimplifiedandtraditional casesindicatehigherscatter(imprecisioncf.Section4.6)aboutthebest-fit line.•𝐶b≈1.0 inthecaseofinteractioncoefficientindicatethatthebest-fit lineagreeswiththe1:1line.• Independentlyofthechosenparameterization,theagreementseemsto largelyremainconsistentwithincreasingstrainlevel:whileMECsuggestsamarginalimprovementat 𝜀≈0.15 over 𝜀≈0.08 (mainlyfortraditional, simplifiedandstrengtheningcoeff.cases), 𝜌c hintsatslightdeterioration illustratingthatthetwosimilaritymeasuresadoptedcapturedistinct featuresofthedataset.Furthermore,MECisalwayslowerthan𝜌c.• Similarresultsareobtainedbyallfourcombinationsofmicrostructure/- textureandload.•𝜌c indicatesequalqualityforthesimplifiedandthetraditional parametrizationswhileMECislowerforthetraditionalcase.4.8.2DeformationBehavior Forstudyingtheimpactofdifferentparametrizationsonthedeformationbehavior,correlationplotscomparingplasticshearonactiveslipsystemsataglobalstrainlevelof 15% arepresentedinFig.4.4 6 .Overall,the correlationplotsdepictreasonablesymmetryaboutthe1:1lineandthewidth ofcontourregionsfromthislineishigherforthesimplifiedandtraditional parametrizationsincomparisontotheparametrizationsusingstrengtheningor interactioncoefficients. Toquantifythedegreeofagreement,thevaluesforthecorrelationmetricsarereportedintheTable4.8.Anubiquitous 𝐶b valueof 1.0 indicatesthat thelineofbestfitexactlyoverlapswith1:1line.Inthisscenario, 𝜌c issolely determinedby 𝜌 .Asthespreadofdatapointsaboutthe1:1lineincreases,thecorrelationcoefficientdecreases;maximumcorrelationisfoundforthe 6Theplotfor≈8%appliedstrainisnotpresentedhereasitlooksidenticaltoFig.4.4 96INCORPORATIONOFPHYSICS-BASEDSTRENGTHENINGCOEFFICIENTSINTOPHENOMENOLOGICALCRYSTALPLASTICITYMODELS lengthscalesimulationsor,wherepossible,latenthardeningexperiments,to enhancemodelaccuracy.Furthermore,theuseofphysics-informedhardening parametersisparticularlyimportantforsimulationsinvolvingloadpathchanges,e.g.inthecaseofcyclicloading,thatleadtotheactivationofpreviouslyinactive (buthardened)slipsystems. SinceDDDparametrizationsexistsformostFCC[230,231,233,232,270]and body-centeredcubic[222,232]metalsandforsomehexagonalclosepacked[271, 243]materials,thisfindingsallowforeasyimprovementsofphenomenological crystalplasticitymodelparametrizations.Itisalsohypothesizedthatthe findingspresentedherearealsoapplicabletootherphenomenologicalhardening models,e.g.thosesuitableforloadpathchanges[267]. Tofurtherimprovethecapabilitiesofthephenomenologicalformulation,itmightberequiredtodecreasethemagnitudeoftheinteractionparameters withincreasingdeformation.However,implementingtheapproachpresentedbyDevincreetal.[231]isnotstraightforwardasitrequirestohavethedislocation densityasaninternalvariable.Furthermore,itmustbequestionedwhetherfurthercomplicationstothephenomenologicalmodel,whichispreferentially usedbecauseofitssimplicity,aredesired. Chapter5DeterminationofCrystalPlasticityParametersfromSynchrotronDiffractionExperiments Thischaptercoverscombinedexperimental–computationalstudythatwasresultedfromajointcollaborationbetweenthedepartmentofmaterialsengineeringandthedepartmentofcomputerscienceatKULeuven.Thecollaborationledtoapublicationthatiscurrentlyunderreviewprocessina journal. P.P.Dhekne,N.Prabhu,M.Bönisch,M.Seefeldt,M.Diehl,andK.Vanmeensel.DeformationmechanismsofL-PBF-processedTi-6Al-4V investigatedusingacombinedexperimentalandsimulationapproach.2025.doi: 10.48550/ARXIV.2508.16367 TheexperimentaldatausedinthisstudyiscontributedbyDr.ir.PushkarPrakashDhekne,duringhisPhDatKULeuvenunder thesupervisionofProf.dr.ir.KimVanmeensel.97 98DETERMINATIONOFCRYSTALPLASTICITYPARAMETERSFROMSYNCHROTRONDIFFRACTION EXPERIMENTS5.1Introduction Intheprevioustwochapters,calibration,validation,andenhancementstudies performedonIN625materialsystemwerepresented.IN625isasolid-solution-strengthened,nickel-basedsuperalloypredominantlymadeupofaphaseexhibitingface-centeredcubic(FCC)symmetry.Sincethefocushereisonlyondislocationslip,FCCmetalscanbeconsideredtobeonthelowerendoftheplasticanisotropyspectrumbecauseoftheirtwelvehighlysymmetric {111}⟨110⟩ octahedralslipsystems(cf.Table4.1).However,not allcommercially-relevantmaterialsexhibitsuchhighorderofsymmetryin plasticslip.Forinstance,inbody-centeredcubic(BCC)metals—whicharealso cubicbutnotascloselypackedasFCC—thelackofwell-definedglideplanes resultsintheslipoccurringalong ⟨111⟩ directionson {110} , {112} and {123} latticeplanes[273].Therefore,ascrewdislocationmayglidealong ⟨111⟩ inahaphazardwayonacombinationofthelatticeplanesthataredetermined dynamicallybythestateofappliedstressontheplanes[21].Thisrendersthe taskofuniquelyidentifyingcrystalplasticityparameterstobetediousbecause theparameterspertainingtodifferentslipfamiliesarestronglycoupledina non-linearway[274]. Furtheralongthespectrumofasymmetrycomethehexagonalclose-packed (HCP)metals,whichdisplayconsiderablemechanicalanisotropyduetounequallatticespacingsalongthe 𝐚 and 𝐜 directions.TheslipinHCPmetalscanoccurindifferentslipfamilies:basal ⟨𝑎⟩ (representedas 𝑏 ),prismatic ⟨𝑎⟩ (representedas 𝑝 ),first-orderpyramidal ⟨𝑐+𝑎⟩ (representedas 𝜋 ),andsecond-orderpyramidal ⟨𝑐+𝑎⟩ ,eachdemonstratingdistinctbehavior 1 [275].Duetotheirreducedsymmetry,thedifferenceincriticalresolvedshearstress(CRSS)amongtheplausibleslipmodes—especiallybetweenfamilieswith 𝐜+𝐚 and 𝐚 Burgersvectors—issubstantialinHCPmetals[239,278]thanincomparisontothe BCCmetals[274]. NumerousstudieshaveexploitedthesizeabledifferenceinresistancesofferedtoslipbydifferentslipmodestoexperimentallyidentifythevalueofCRSSinHCPmetals.Earliestexperiments[279,280,281]includedorientingpureHCPsinglecrystalsamplessuchthattheslipmodeofinteresthadthehighestSchmidfactorunderasimpletensileorcompressivemacroscopicloading.Amajorimpediment toperformsuchexperimentalcharacterizationisgrowinglargesinglecrystals ofhighlyalloyedsystem.Thislimitationensuedalternativeapproachesthatleveragebothexperimentalandcomputationaltechniques.Pioneeringworks 1 Thequestionontheindependenceofpyramidal ⟨𝑎⟩ slipmoderemainspersistent,asthe strainachievedthroughtheiractivationcanalsobeachievedthroughacombinedactivation ofbasalandprismatic ⟨𝑎⟩ systems[275].However,theirexistencehasalsofoundsignificant supportintheliterature[276,277]. INTRODUCTION99 ofP.TurnerandC.Tomé(1994)[197]onazirconiumpolycrystalcomparedlatticestrainsofseveralplanesobtainedfromneutrondiffractiontoaself-consistentelastoplasticmodel.Theabovework,alongwithsimilarstudies[88],aimedtodeterminetheCRSSvaluesofdifferentslipmodesinvolvedin deformationusingamean-fieldsimulationapproach.Asexplainedpreviously inSection1.4.1,thisapproach,however,doesnotexplicitlyconsidergrain shape,itsinteractionswiththesurrounding,orintergranularstressvariations, whichareespeciallyimportantinlowsymmetrymaterialssuchasHCPmetals. Thelimitationsposedbyself-consistentmodelinthemean-fieldsimulation approachwereamelioratedafteradoptingfull-fieldcrystalplasticitysimulation techniques[282].EarlieststudiesemployedFEM[96,283,284]but,duringthe lasttwodecades,spectralmethodbasedonFFT[106],thatcanbedirectly appliedonexperimentallycharacterizedimagedata,hasgainedpopularity.The spectralmethodalsoeliminatestheneedformeshingthemicrostructurewhich ismandatoryinFEM. Anotherclassofcombiningexperimentalandcomputationaltechniquesinvolves performingandsimulatingnanoindentationonasmallregionwithinindividual grainsofapolycrystal.Zambaldietal.[239]employedasimplexsearchoptimizationalgorithmtocalibrateinitialandsaturationvaluesofCRSSof basal,prismaticandpyramidalslipmodesbyminimizingalossfunctionthatwas basedontopographicaldata.Atotalof 1424 indentsweremadethatfollowed agridpatternonthesurfaceoftwopolycrystalsamplesofcommerciallypure titanium,andthereforecoveredarangeofcrystalorientations.Althoughanaxisymmetricsphericalindenterwasused,theindentationprocessimposesacomplexstressandstrainfieldslocallyintheregionwhichmakeitharder toisolatetheactivityofindividualslipmodestoquantitativelyextracttheir CRSS.Strategiestoovercomethislimitationincludeusingfocusedionbeam tomachinecompressionpillarsorcantileverbeamswithinindividualgrainsof aspecificorientationinapolycrystal.However,thecriticalstressvaluesthus obtainedsufferfromsizeeffectsandstochasticnatureofplasticflowthatis observedasstrainbursts(ordislocationavalanches).ThelattermechanismwasalsohypothesizedtohavematerializedintheHEDMexperimentaldatautilized inChapter3,wherecertaingrainsexperiencedabruptchangesintheiraverage elasticstrainsduringamonotonicmacroscopicloading(seeSection3.3.3). Thetechniquesmentionedabovearenotexhaustive,andrecentinvestigations alsoincludeincorporatingDDDsimulationstocorrelatewithexperiments [285],conductinghigh-temperatureexperimentalstudiesetc[278].Regardless ofthetechniqueused,theseinvestigationsonHCPmetalshelpunravelthecomplexinterplayofslipmodes,therebycontributingtotheimprovementofmicromechanicalplasticitymodelsandadvancingthedevelopmentofhigh-fidelitydigitaltwins.Whileeachtechniquepresentsitsownchallenges,a 100DETERMINATIONOFCRYSTALPLASTICITYPARAMETERSFROMSYNCHROTRONDIFFRACTIONEXPERIMENTS clearunderstandingoftheunderlyingmechanismsisincreasinglyvitalasHCP metalsfindbroaderapplicationsinhigh-performanceengineering,whereextreme precisionandoptimalityareparamount. Inthepresentchapter,amultilevelmaterialparametercalibrationacrossscalesofanelastoplasticconstitutivemodeliscarriedoutusingalarge-scalesynchrotronexperimentaldata.Themultilevelcalibrationinvolvesnotjust fittingparameterstoasinglemacroscopicstress–strainresponse,butalsofittingtotheaveragemesoscopicbehaviorofmultiplegrainensemblesthatarespatially distributedwithinthematerialvolume.Thecalibrationperformedherediffers fromtheprevioustwochaptersinthatmesoscaledata,whichwereusedonly forvalidationandassessmentinearlierchapters,aredirectlyincorporatedtocomplementthecalibrationeffortsinthisstudy.Thematerialsystem underinvestigationcomprisesadditivelymanufacturedTi-6Al-4Vspecimens, characterizedattwostagesofthemanufacturingprocess:theas-builtcondition andtheheat-treatedcondition.Thespecimensarecharacterizedinsituduring uniaxialtensiletestsusinghigh-energyX-raydiffractiontechniques.Aninversemodelingstrategy,detailedinSection5.4,isemployedintwostagesthroughanoptimizationroutine.Ineachiterationofasingleoptimizationloop,mechanicalloadingissimulatedonarepresentativemicrostructure,andtheresultingmaterialresponsedataisextractedforcomparisonwithexperimentalobservations.Theformerisacceleratedthroughtheadoption ofthespectralmethod-basedFFTsolverinDAMASK.Additionally,insights fromChapter4regardingtheadoptionofphysics-backedinteractioncoefficients intothehardeningmodulimatrixisincorporated.Backgroundonthematerial system,microstructure,andexperimentaldatasetisprovidedinSections5.2 and5.3.Inthefollowingsection(Section5.4),detailsregardingthecalibration throughinversemodelingapproacharepresented.Theresultsobtainedfrom thecalibratedmodelarepresentedanddiscussedinSection5.5.Tothisend, thecalibratedmodelsarequalitativelyvalidatedbycomparingtheevolutionofdiffractionringintensitiesobservedintheinsituexperimentwiththose extractedfromthe“virtualexperiment”.Furthermore,insightsaregainedinto thesequenceofslipsystemactivation,contributionsofvariousslipfamiliesto theoveralldeformation,andtheinfluenceofslipactivityongrainreorientation.Thismechanisticunderstandingofdeformationprocesses—otherwiseinaccessible throughexperimentsalone—ismadepossiblebytheintegratedexperimental– computationalapproach. BACKGROUND1015.2Background AmongtheHCPmetals,titanium(Ti)isthesecondmostabundantstructural metalontheearth’scrustaftermagnesium[286].Sinceitsdiscoveryin1790s andextractioninthepurifiedforminearly1900s,titaniumfounditswidespread applicationsonlyinthesecondpartofthetwentiethcentury.Itexhibits remarkableversatilityinitsmechanicalproperties.Forinstance,puretitanium isonlyabout 60% denseasthatofsteelorsuperalloys,andthroughalloyingand deformationprocessing,itcanbemadeequallystrong[287].Itshighcorrosion resistance,stiffness,andrelativestrength,andbiocompatibilityallowforits use—eitherincommerciallypureoralloyedforms—inapplicationsrangingfrom gasturbinebladestobiomedicalimplants.Inthecaseofthelatter,titaniumis generallyregardedasoneofthemostbiocompatiblemetalsavailableforclinical applications[288]. Beingamemberofthefirsttransitionseries,titaniumisideallysituatedintheperiodictablewithafavorabledensitythatismidwaybetweenironand aluminum.Itsnear-medianatomicdiametergreatlyfacilitatesalloyformation withadiverserangeofcommerciallyimportantmaterials[289].Additionally, titaniumisallotropic:itsexhibits 𝛼 (HCP)phaseatroomtemperatureand 𝛽 (BCC)atelevatedtemperatures.Thecombinationoftwocrystalstructures andamyriadoffavorablealloyingelementsallowsforabroaddesignspaceto developcomplexalloys.5.2.1Ti-6Al-4V Amongthebroadspectrumofdifferentgradesoftitaniumalloys,Ti-6Al-4V—with ≈6% aluminum(Al)and ≈4% vanadium(V)asmajoralloying elements—hasbeentheflagshipalloysincethecommercialusageoftitanium. Duringthesecondhalfofthetwentiethcentury,Ti-6Al-4Vcommandedanunmatched 50% shareoftitaniumapplicationsinthemarket[287],andstillcontinuestodoso.Thealuminumandvanadiuminthealloyactas 𝛼 phase and 𝛽 phasestabilizersandthereforeTi-6Al-4Vcanbecategorizedasan 𝛼+𝛽alloy. Despitethesuperiorpropertiesexhibitedbytitaniumalloys,itswidespreaduseislimitedbyitshighercostincomparisontocompetingmaterials.A majorcontributortohighercostsisthedifferentstepsinvolvedinconventional manufacturingtechniques,whereasthecostofmetalextractionfromtheoreisbarelyafractionofthetotalfabricationcost[290,291,292].Forinstance,intitaniumcomponentsfabricatedforaerospaceapplications,itisgenerally acceptedthatamachiningstepwouldsimplydoublethecostofthecomponent. 102DETERMINATIONOFCRYSTALPLASTICITYPARAMETERSFROMSYNCHROTRONDIFFRACTIONEXPERIMENTS Inaworst-casescenario,thecostofmaterialwastageintermsofbuy-to-fly 2 ratiocanreachashighas40:1[291,293].Thehighfabricationcostsassociated withtheconventionalmanufacturingmethodshascompelledthetitanium communitytoembracepowdermetallurgytechniquesthatcandirectlyproduce near-net-shapecomponents.5.2.2AdditiveManufacturingofTi-6Al-4V Additivemanufacturingisarelativelynovelpowdermetallurgytechniquethat employsalaseroranelectronbeamtomeltthemetalfeedstock(powderorwire) thatisdepositedonabaseplatelayerbylayertoproduceanear-net-shaped component.Thebuy-to-flyratioforanadditivelymanufacturedcomponentis significantlylowerthanconventionalmethods[294]eventhoughthetheoretically idealscenarioof1:1withrepeatedrecyclingofbuildmaterialisnottypically achieved.Inthecaseoftitaniumalloys,additivemanufacturingstartedtogainprominenceinthelastquarterofthetwentiethcenturywithwellover 90% ofthedevelopmentaleffortsexpendedexclusivelyonthe“workhorse” Ti-6Al-4Valloy[287,295].Sincethen,thetitaniumcommunityhasembraced theadditivemanufacturingtechniquenotjustforitscost-effectivenessbutalso foritsflexibilityindesignmodificationsanditsabilitytofabricategeometrically complexparts.Althoughas-builtcomponentspresentsub-optimalproperties, subsequentpost-processingtreatmentscanenhancetheirmechanicalproperties tobeonparwithorevensuperiortothoseofconventionalcastorwrought alloys[292].5.2.3PhasesinAMTi-6Al-4V Thecyclic,layer-by-layermaterialdepositionintheAMprocessexposesthe materialbeneaththebuildsurfacetorepeatedthermalcyclesofrapidheating andcooling.Theimpactofthisphenomenonisespeciallyprevalentinparts fabricatedfromlaser-basedAMtechniquesthanelectron-basedones[296].The remeltingofthelayersunderneathbecauseofthedirectedlocalheatsourcepromotessolidtoliquidtransformationthatpresentsepitaxialgrowthof 𝛽 grainsalongthebuilddirection[297,298].Thisisfollowedbyarapidcooling stagewherethe 𝛽 grainstransformintomartensitic 𝛼′ (HCP)structuresthat exhibitacolumnarmorphology[299].Athighcoolingratesthethermally-assisteddiffusionofatomstorearrangeintoanewconfigurationisinhibited, 2 Usedpredominantlyinaerospaceindustry,itistheratiooftheweightoftheinitialraw materialtotheweightofthefinishedcomponentthatgoesontheaircraft. BACKGROUND103 andthereforethephasetransformationoccursthoughspontaneousshearof latticeplanesresultinginthemetastable𝛼′phase[300,301]. Asthemartensitic 𝛼′ isanon-equilibriumphase,apost-builtheattreatment istypicallyfollowedtostabilizethemicrostructureandenhancemechanical properties,mainlyductility.Thesetreatmentspromotethedecompositionof 𝛼′ intoamorestablemixtureof 𝛼 and 𝛽 phases[302].Adetailedoverviewofstandardheattreatmentpracticesandtheirimpactonthemechanical performanceofAMTi-6Al-4VispresentedinAppendixB. Dependingonthechosenheat-treatmenttechnique,avarietyofmicrostructural configurationscanbeachieved,includinglamellar 𝛼+𝛽 coloniesembeddedwithincolumnarorequiaxedprior 𝛽 grains[302].Beyondfacilitatingthe decompositionofthebrittle 𝛼′ phase,theheattreatmentalsohelpsrelievethe residualstressbuild-upinthecomponents[303].Inlinewiththis,theresidual stressesinTi-6Al-4Vcomponentsaretheprimaryfocusofthenextchapter (Chapter6). Traditionally,thepresenceofthemartensitic 𝛼′ phaseinthemicrostructureis considereddetrimentaltothemechanicalperformanceofacomponent[299].However,recentstudiessuggestthatthismaynotalwaysbethecase(see AppendixB).Incontrasttothegeneralperception,thepresenceofthe 𝛼′ phase inthemicrostructure—withsuitableprocessingconditions—hasbeenobserved tocontributepositivelytoboththestrengthandductilityofthecomponent [304,305,306]. Insummary,notonlythe 𝛼 phasebutalsothe 𝛼′ phasecanleadtosignificant improvementsinacomponent’smechanicalperformance,therebybroadeningtheapplicabilityofTi-6Al-4Valloystonewdomains.Moreover,thespatialvariationinphasesalongthebuilddirection 3 hasmadeitnecessarytocharacterizethe 𝛼′ phaseinordertoincorporatethesedetailswhensimulatingthematerialbehaviorofanAMcomponent[307,308].Consequently,aphase-specificunderstanding— enabledbymechanicalcharacterizationandmaterialcalibrationofindividual microstructuralphases—cansupportthedevelopmentofaccuratedigitaltwins, ultimatelycontributingtomoreefficientproductdevelopmentcycles.5.2.4MechanicalCharacterizationofthePhases Despiteexhibitingthesamecrystalstructure, 𝛼 and 𝛼′ phasespresentconsiderablydifferentmechanicalproperties.Thetechniques,mentioned previouslyintheintroductorysection(Section5.1),tomechanicallycharacterize deformationbehaviorofHCPmetalshavealsobeenemployedextensively,in 3Asaconsequenceofvaryingthermalhistory 104DETERMINATIONOFCRYSTALPLASTICITYPARAMETERSFROMSYNCHROTRONDIFFRACTIONEXPERIMENTS particular,for 𝛼 phaseofTi-6Al-4Valloy.I.JonesandW.Hutchinson[309]carriedoutmechanicalcharacterizationofheavilytexturedalloysamplesat differentorientationstoextractCRSSondifferentslipfamiliesthroughanatomic hardspheremodel;theyobservedthat ⟨𝑎⟩ dislocationsglideatalmostequal stresslevelsirrespectiveoftheslipplane.Duringthenextdecade,MedinaPerilla andGilSevillano[310]conductedmacroscopicloadingexperimentsonaTi6Al-4Vsheetwithastrong“edge-type”textureandtrackedtextureevolution duringthedeformationprocess;theyattemptedtoreplicatetheobservedtextureevolutionwithTaylor-typemodelsandconcludedthatCRSSofpyramidalfamilywasonlywithintwotimesthemagnitudeofprismaticfamily.Subsequently,with advancementsinbothcharacterizationandcomputationalmethods,numerous integratedandsophisticatedtechniquesemerged.Thelimitationsofmean-field modelswereaddressedbyfull-fieldapproaches(FEM[283,311,312,284,313, 314]orspectralmethod).Ontheexperimentalfront,theuseofx-raydiffraction technologiesleadtoinsituacquisitionofsub-surfacematerialdataduringexternalloading[312,313].Moreover,inrecentyears,improvedaccessibility tobeamlinetimeatsynchrotronfacilitiesworldwidehasfurthercatalyzedthe surgeinstudiesthatextractorientation-basedgrainfamily-leveldatafrombulk specimens[314,315]. Integrationofsuchhigh-fidelitydatasetswithfull-fieldsimulationtechniques hasfacilitatedin-depthinvestigationsintothecompetingdeformationmodes inTi-6Al-4Valloys.Although,theaforementionedstudies,inisolation,seem quiteconvincingindecipheringdeformationmodesinthe 𝛼 phase,thevalues ofslipactivityparameters—mainlyCRSS—reportedbythemexhibitsizeable dispersion.ThedispersioninvaluesreportedintheliteratureisplottedandanalyzedinFig.5.8.Ontheotherhand,theinvestigationsondeformationmodesof 𝛼′ arelimitedasitisconsideredasatemporaryphase;however, recentadvancements[305,306]havedemonstrateditsapplicationsandthereby underscoretheimportancetobemechanicallycharacterizedtodevelopaccurate models.5.3ExperimentalData TheexperimentaldatapresentedinthischapterwerecontributedbyDhekne etal.;acomprehensivedescriptionofthesamecanbefoundelsewhere[316,317].Inwhatfollows,appropriatedetailsthatarenecessaryforaholistic understandingofthecomputationalstudyisbrieflyrecapitulated. AnHEXRDcharacterizationwasperformedinsituduringtensiletestingof additivelymanufacturedTi-6Al-4Vspecimens,withabeamenergyof 100keV , EXPERIMENTALDATA105 atDESY(DeutschesElektronen-Synchrotron).Theworkingprincipleofsuch acharacterizationtechniquehasalreadybeendescribedinSection2.4.2.The diffractedbeamfromtheilluminatedvolumeofthespecimenintransmission modewascapturedbyaPerkinElmerareadetector,producingdiffractionrings.Thisprocedurewascarriedoutinsituduringthemechanicalexperiment,thereby enablingobservationoftheevolutionofdiffractionpatternsoverthecourse ofthetest.Aschematicrepresentationofthediffractionpatternshowingthe DebyeringscanbeseeninFig.2.5.Eachringrepresentstheaveragebehavior ofanensembleofgrainsthatarerelatedtoeachotherbytheirorientations. Throughthedisplacementoftheringsfromtheirreferencepositions,average directionallatticestrainofeachgrainensemblewascomputedviaRietveld refinementperformedintheMAUDsoftwarepackage[318].Thelatticestrain ofagivenplane,denotedbyitsMiller-Bravaisindices,iscomputedfollowing thedefinitionofengineeringstrainas𝜀{ℎ𝑘.𝑙}=𝑑({ℎ𝑘.𝑙})−𝑑0({ℎ𝑘.𝑙})𝑑0({ℎ𝑘.𝑙}), where 𝑑({ℎ𝑘.𝑙}) representsthecurrentinterplanardistanceofthe {ℎ𝑘.𝑙} plane and 𝑑0({ℎ𝑘.𝑙}) representsthesamequantitybeforethematerialisdeformed. Giventhatthenormaltothediffractingplane {ℎ𝑘.𝑙} makesanangle 𝜓 with theloadingdirection,thelatticespacinginthatplaneisformulatedas𝑑({ℎ𝑘.𝑙})=𝑑hyd({ℎ𝑘.𝑙})[1+(1−3cos2𝜓)𝑄({ℎ𝑘.𝑙})], where 𝑑hyd({ℎ𝑘.𝑙}) isthelatticeplanespacingifthestressstatewerepurely hydrostatic,andtheremainingpartoftheequationtoitsrightaccountsfor angularvariationoflatticestrainduetonon-hydrostaticstressstate[319,320]. Anisotropicbehaviorinelasticityisaccountedforbytheparameter 𝑄({ℎ𝑘.𝑙}) . Thevariable 𝑄({ℎ𝑘.𝑙}) alongwith 𝑑hyd({ℎ𝑘.𝑙}) intheaboveequationare treatedasfittingparameterstoobtaintheinterplanarspacingdistance. Theinsitucharacterizationwasperformedontwosamplesobtainedattwo differentstagesintheadditivemanufacturingprocess:1.As-built:composedpurelyofmartensitic𝛼′phase;and2.Heat-treated:consistingof94%𝛼and6%𝛽phases. Consequently,theinsituexperimentsresultedinextractionofmaterialbehavior ofeachsampleattwolevels:1. Global(macroscopic)data:averagestress-strainresponseofthespecimen alongtheloadingdirection,and 112DETERMINATIONOFCRYSTALPLASTICITYPARAMETERSFROMSYNCHROTRONDIFFRACTIONEXPERIMENTS5.4.4FormulationofLossFunctions Ateachiteration,theoptimizercomputesthelossbyjuxtaposingthereference experimentaldata(exp)withthesimulatedmaterialresponse(sim).Thefive independentelasticconstantsassociatedwithhexagonalsymmetryinthelinear elasticmodelarecalibratedagainsttheexperimentalmacroscopicstressversus latticestraindataset.Thediscrepancyfromthereferenceisquantifiedasa scalarlossvalueandisformulatedasfollows:𝐿e={ℎ𝑘.𝑙}∑∣∣𝜀LS,exp−𝜀LS,sim∣∣2𝑁,(5.1) where 𝑁 isthetotalnumberofdatapointsandisusedfornormalization,and LSstandsforlatticestrain. 𝜀 isthelatticestrainofaplanefamily {ℎ𝑘.𝑙} , computedatmatchingvaluesofthemacroscopicstress(betweenexpandsim), and‖⋅‖2istheℓ2-norm. Inthefirst“global”stepoftheplasticparameterscalibrationroutine(cf.Section5.4.3),thetotalloss 𝐿p,G —wherethesubscripts p and G standforplasticandglobal,respectively—notonlycomprisescontributionsfromthe actualvalues(representedas0)butalsofromtheirfirstderivatives(represented as1)andreads𝐿p,G=𝑤p,G0𝐿p,G0+𝑓G𝑤p,G1𝐿p,G1.(5.2) SincethesecondterminEq.(5.2)accountsforthelossinthefirstderivatives,itsmagnitudeisexpectedtobesignificantlyhigherthanthatofthefirstterm. Toconsistentlybridgethisdifferenceinscalebetween 𝐿p,G0 and 𝐿p,G1 ,ascalingfactor 𝑓G isautomaticallycalculatedduringtheveryfirstiteration.Thisensures thatthecomputedlossisnotdisproportionatelyinfluencedbyasingleterm. Additionally,thecontributionsoftheactualvaluesandtheirfirstderivatives arerelativelyweightedusingthevariables𝑤p,G0and𝑤p,G1. Inthesecondstepoftheplasticparameterscalibrationroutine,inaddition tothemacroscopic(global)responseofthesample,latticestrainsofthefour planefamiliestogetherconstitutethereferencedataset.Thetotalplasticloss functionnowconsiststwoadditionalterms—onthetopofEq.(5.2)— 𝐿p,LS0 and 𝐿p,LS1 ,andtwomorescalingfactorstobridgethedifferencesinmagnitudes. Thetotallossofplasticfitting𝐿pthenreads𝐿p=𝑤p,G𝐿p,G+𝑓LS𝑤p,LS𝐿p,LS.(5.3)𝐿p,LS iscomputedinasimilarmannerasfortheelasticparametersinEq.(5.1); however,thekeydifferenceisthatthelosscomputationnowalsoincludesthe differenceinthefirstderivativeofthelatticestrainversusmacroscopicstress curves. INVERSEMODELING1135.4.5InitialGuessValuesandConstraints Astheinitialsimplexplaysavitalroleindeterminingthesearchdirectionofthe direct-searchNelder-Meadmethod,theinitialguessvaluesoftheparameters shouldbechosencarefully.Forthefiveindependentelasticparameters,theinitialvaluesareadoptedfromtheworkofWielewskietal.(2017)[324]and arelistedinTable5.1.Aminordistinctioninthepresentapproachisthat 𝐶33 isincludedasanindependentparameterintheoptimizationroutine.Intheir optimizationstudy,Wielewskietal.[324]introducedaconstraintontheelastic constants, 𝐶11+𝐶12=𝐶13+𝐶33 ,whichassumesanisotropicbulkresponse andtherebyreducesthenumberofindependentparameterstofour.Incontrast, thepresentapproachdoesnotimposesuchconstraintsapriori.Instead,theoptimizerisallowedgreaterflexibilitytonavigatethefullfive-dimensional parameterspaceinordertoaccuratelyreproducetheexperimentallyobserved elasticbehavior. Inthefirststepofplasticparameteroptimization,wherethesolitarydataset ofmacroscopicstress–strainbehaviorisusedasthesolereference,onlythree fittingparametersoutofthetotaltwelve 4 aretreatedasindependent.Theseare 𝑎p , ℏ0,p and 𝜉0,p representingthefittingexponent,initialhardeningrate,and initialslipresistance,respectively,fortheprismaticslipfamily.Theprismatic slipparametersareconsideredasthebase,andtheparametersassociatedwith thebasalandpyramidalslipfamiliesaretreatedasdependentonthoseofthe prismaticsystem.Inparticular,fixedratios 𝜉0,b/𝜉0,p and 𝜉0,𝜋/𝜉0,p ,areassumed fortheinitialslipresistancesofthebasalandpyramidalfamilies,respectively. Theseratiosareadoptedfromexistingliterature;however,considerablescatter existsamongthereportedvalues. NumerousratiosofCRSSamongtheslipfamiliesofTi-6Al-4Vhavebeen previouslyreportedintheliterature;thistopicwill,infact,beakeypointof discussioninthelatersections.Inthepresentscenario,theseratioshaveto bechosencarefully,astheywillhaveasignificantinfluenceonlocalbehavior ofthemateriali.e.,onthelevelofslipsystems.Toarriveatthisdecision, sevenuniaxialloadingsimulationswereperformedonarandomlyorientedRVE, wheretheonlydifferencebetweenthesimulationswastheallowableplasticslip activityoftheindividualslipfamilies.Toachievesuchacontrolledvariation, theCRSSofslipfamilyexcludedfromagivensimulation(asindicatedby“0”inFig.5.2)wasassignedanextremelyhighvalue,effectivelyeliminating itscontributiontotheplasticresponse.Figure5.2illustratestheinfluenceof plasticactivityinthethreeslipfamiliesonthelatticestrainsofthefourplane familieschoseninthisstudy.Thedottedlinesrepresentthecorresponding 4𝑎 , ℏ0 , 𝜉0 ,and 𝜉∞ forthethreeslipfamilies( 𝑏 , 𝑝 ,and 𝜋 )constitutethetwelveindependent plasticityparametersconsideredinthisstudy. 114DETERMINATIONOFCRYSTALPLASTICITYPARAMETERSFROMSYNCHROTRONDIFFRACTIONEXPERIMENTS behaviorwhenthematerialismodeledpurelyaselastic.Itisapparentthat thelatticestrainresponseofthe {00.2} planefamilydeviatesdownwardfrom linearityonlywhenslipviafirst-orderpyramidalslipfamilyispermitted.A contrastingbehavior,i.e.,the {00.2} curveexhibitinganupperinflectionwhen thefirst-orderpyramidalslipsystemisinactive,isalsoconsistentlyobserved. Thelattertrend,namelythesignificant“load-bearing”roleofthe {00.2} grainensemble,isobservedintheexperimentalresultsfortheas-builtsample, whichwillbepresentedlaterwhenside-by-sidecomparisonsaremadebetween simulationsandexperiments.However,thistypeofloadbearinginthe {00.2} grainaggregateisnotseenfortheheat-treatedsamplesuggestingadifference infirst-orderpyramidalslipactivity.Accordingly,differentCRSSratiosforthe pyramidal-to-prismaticsystems 𝜉0,𝜋/𝜉0,p areadoptedforthetwosamples:for theheat-treatedone,avalueof 1.68 ischosenfollowingtheworkofDickand Cailletaud[325];fortheas-builtsample,ahighervalueof 3 ischosen,based ontheworkofDunstandMecking[326],andMayeurandD.McDowell[327]. Finally,thebasal-to-prismaticratios 𝜉0,b/𝜉0,p forbothsamplesaretakentobe 1.05[325,328]. 0.002 0.004 0.006 0.008 0.010 Nsl = [3,0,0]Nsl = [3,0,0]Nsl = [3,0,0]Nsl = [3,0,0] Nsl = [0,3,0]Nsl = [0,3,0]Nsl = [0,3,0]Nsl = [0,3,0] Nsl = [0,0,12]Nsl = [0,0,12]Nsl = [0,0,12]Nsl = [0,0,12] Nsl = [3,3,0]Nsl = [3,3,0]Nsl = [3,3,0]Nsl = [3,3,0] 600 900 0.002 0.004 0.006 0.008 0.010 Nsl = [0,3,12]Nsl = [0,3,12]Nsl = [0,3,12]Nsl = [0,3,12] 600 900 Nsl = [3,0,12]Nsl = [3,0,12]Nsl = [3,0,12]Nsl = [3,0,12] 600 900 Nsl = [3,3,12]Nsl = [3,3,12]Nsl = [3,3,12]Nsl = [3,3,12] Nsl = [b, p,π] {10.0}{00.2}{10.1}{10.2} Latticestrain Stress(MPa) 1 Figure5.2:Evolutionoflatticestrainsoffourplanefamiliesduringuniaxialloadingsimulation.Ontheleftcornerofeachplot,thearray 𝑁𝑠𝑙 representsthenumberofoperableslipsystemsofbasal,prismaticandfirst-orderorder pyramidal ⟨𝑐+𝑎⟩ families(inthegivenorder).Dottedlinesrepresentpurely elasticresponseoftheplanes(absolutelynoplasticity). Nowthattheratiosamonginitialslipresistancesofdifferentfamilyare RESULTSANDDISCUSSION115 established,thenextlogicalstepwouldbetodeterminetheinitialguessvalue of 𝜉0,p foreachsample.Thisiscalculatedfromtheglobalstress–strainresponseofthesamples.Tothisend,theyieldstress( 𝜎y )andtheultimatetensilestress ( 𝜎UTS )areidentifiedfromthemacroscopicbehaviorofthesample.Theinitial guessvalueof 𝜉0,p wasthensetto 𝜎y2.65 ,wherethefactor 2.65 istheaverage Taylorfactorfromiso-stressandiso-strainconditions.Furthermore,theratio 𝜎UTS𝜎y wasusedasamultiplierto 𝜉0 todetermine 𝜉∞ .Theinitialguessvalues providedforthisfirstoptimizationsteparelisteddowninTable5.2. Inthesecondandfinaloptimizationstep,wherelatticestraindataare additionallyinvolved,thereferencedatasetnowcapturesthedifferencesinplastic activityacrossthevariousslipfamilies.Thefixed-ratioconstraintsimposedin theprevioussteparelifted,resultingintwelveindependentparameterstobe calibrated: 𝑎 , ℏ0 , 𝜉0 and 𝜉∞ foreachslipfamily.Theoptimalvaluesobtained fromthefirstoptimizationstepofplasticparametersserveastheinitialguesses forthisstageoftheoptimization.5.5ResultsandDiscussion Theresultsfromthecalibrationoftheelastoplasticmaterialparametersare presentedanddiscussedinthissection.Thisisfollowedbyblindpredictions 5 usingthecalibratedmodels,whicharecomparedside-by-sidewithadditional experimentaldatatogaindeeperinsightintothecompetingslipmechanisms activeinthematerial.Finally,thereorientationofthegrainsasafunctionof thedominantplasticdeformationmechanismisinvestigated. Asafirststeptowardinterpretingtheoutputsofthevirtualexperiments andcomparingthemwiththeexperimentalobservations,thegrainensembles contributingtotheextractedlatticestraindataarevisualized.Tothisend, thepositionsandorientationsofgrainswithintheRVEthatsatisfytheBragg diffractionconditionsforthefourselectedlatticeplanefamiliesaredisplayed inFig.5.3.Thelatticestraindatapresentedinthesubsequentsectionsare nothingbuttheaverageelasticresponseofthesegrainsalongthemacroscopic loadingdirection.Acloserexaminationofthegrainensemblesrevealsthatthe individualgrainswithineachensemblearespatiallydispersedandisolatedfrom eachother.Furthermore,thenumberofgrainscontributingtoeachensemble varieswiththelatticeplanefamilyandwillbefurtherexploredlaterinthe discussionofdiffractionintensitiesinSection5.5.3.5 itistermedas“predictions”solelybecausethisdatawasnotpartofthematerial parameterscalibrationprocess. 116DETERMINATIONOFCRYSTALPLASTICITYPARAMETERSFROMSYNCHROTRONDIFFRACTIONEXPERIMENTS A{0 0 .2} B{1 0 .2} C {1 0 .1} D{1 0 .0} EF 1 Figure5.3:A:ArandomlytexturedRVEofthesamplespresentedasanillustrativeexample.B–E:Grainensemblesofdifferentplanefamilies.The grainsarecoloredbasedonX-axis(loadingdirection)IPFcoloringscheme(F). 5.5.1ElasticConstants Theinitialguessvaluesofthefiveindependentelasticconstantsarementioned inTable5.1.Thispoint,infivedimensions,formsoneofthesixverticesof thefirstNelder-Meadsimplex(cf.Section5.4.3).Theoptimalvalueofelastic constantsobtainedaftercalibrationagainstlatticestraindatasetarealsolisted inthesametable. AcomparisonamongthethreesetsofelasticconstantsispresentedinFig.5.4, wheretheevolutionoflatticestraininthediffractinggrainsoffourplanefamilies isplottedagainstmacroscopicstress.Theexperimentaldatafromthisstudy, representedbycircularmarkers,revealthatthedirectionalstiffnessofbothphasesdecreaseswithincreasingmisalignmentofthe 𝑐 -axisfromtheloading direction.Theorientationsofthediffractinggrainscorrespondingtoeachplane familyarelaterplottedontoanIPFtriangleinSection5.5.5toreinstatethis RESULTSANDDISCUSSION117Initialguess[324]Heat-treatedAs-built𝐶11169.0147.7150.8𝐶1289.061.283.2𝐶1362.066.959.2𝐶4443.059.755.7𝐶33196.0203.5187.0 Table5.1:Initialguess(left,takenasisfromWielewskietal.[324])andoptimal (centerandrightforheat-treatedandas-builtcases,respectively)valuesof independentelasticconstants(inGPa)ofHCPphaseofTi-6Al-4Valloy. trend.Thisobservationisconsistentwithgeneralbehaviorseeninhexagonal phasesoftitaniumalloys[286].Furthermore,thebroaderrangeoflatticestrain valuesobservedinthe 𝛼′ phasecomparedtothe 𝛼 phasesubtlyhintsthatthe 𝛼′ phaseexhibitsahigherdegreeofelasticanisotropy.Additionally,thelarger magnitudeoflatticestrainsatagivenmacroscopicstressfurthersuggeststhat the𝛼′phaseiselasticallylessstiffthanthe𝛼phase. 0 100 200 300 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 InitialguessInitialguessInitialguessInitialguess0 100 200 300 As-builtAs-builtAs-builtAs-built0 100 200 300 Heat-treatedHeat-treatedHeat-treatedHeat-treatedsimulationexperiment{10.0}{00.2}{10.1}{10.2}Stress(MPa) Latticestrain 1 Figure5.4:Latticestrainevolutioninthediffractinggrainsofthefourplane familiesaftercalibratingtheelasticconstantsagainsttheexperimentaldataof thetwosamples(centerandright).Theboldlinesrepresentsimulatedbehavior fittedagainsttheexperimentsthatarerepresentedascircularmarkers.In additiontothefittedresults,latticestrainevolutionwhenthevaluesofelastic constantstakenasisfrom[324]isplottedontheleft. TheelasticconstantstakenfromWielewskietal.[324]wereoriginallycalibratedusinglatticestrainpolefiguresofthe 𝛼 phaseofTi-6Al-4Vundergoinguniaxial tension.Therefore,theexperimentaldatashownintheplottotheleft 118DETERMINATIONOFCRYSTALPLASTICITYPARAMETERSFROMSYNCHROTRONDIFFRACTIONEXPERIMENTS correspondtothe 𝛼 phase.ThesamesetofvaluesreportedbyWielewski etal.[324]werefedastheinitialguesstotheoptimizerinbothcases,i.e.,for 𝛼 and 𝛼′ phases.WhenthereferencesetofelasticconstantsfromWielewski etal.[324]isemployedinthepresentsimulation,asignificantdeviationfrom experiment,notinmagnitude,butinorderingofthelatticestrainevolutionin thediffractinggrainsofthefourplanefamiliesisobserved(seeFig.5.4left).Inparticular,theexpectedtrendofdecreasingstiffnesswiththeincreasing 𝑐 -axismisalignmentfromtheloadingdirectionisnotfollowedwhentheseelastic constantsareused. Awiderangeofvalueshavebeenreportedforelasticconstantsof 𝛼 and 𝛼′ phasesofTi-6Al-4V[329,330,331].Dumontetetal.[330]performedatomistic simulations(densityfunctionaltheory)todeterminetheindependentelastic constantsof𝛼and𝛼′phases.Theirstudyrevealedthatthe𝛼′phaseexhibits lowerstiffnessthanthe 𝛼 phaseandiselasticallymoreanisotropic.Theformer isalsoevidentlaterinFigure5.6,whilethelattercanbequantifiedusingtheuniversalelasticanisotropyindex 𝐴U formulatedbyRanganathanandOstoja-Starzewski[332].TheanisotropyindexiscomputedusingVoigtand Reuss(superscripts V and R ,respectively)estimatesofshear( 𝜇 )andbulk( 𝐾 ) moduliandreads𝐴U=5𝜇V𝜇R+𝐾V𝐾R.(5.4) TheVoigtandReussestimatesforarandom-texturedcrystalaggregatewere computedfollowingtheequationslaidoutin[333].Computing 𝐴U forboth 𝛼′ and 𝛼 phasesyield 0.36 and 0.17 ,respectively,wherethehighervaluefor 𝛼′ phaseindicateshigheranisotropy,whichisconsistentwiththefindingsfrom otherstudies[330].5.5.2PlasticConstitutiveParameters Asmentionedpreviously,thecalibrationoftheplasticconstitutiveparameters inatwelve-dimensionalparameterspaceiscarriedoutthroughinversemodeling intwostagesofoptimization.OptimizationStep1:FixedRatiosandMacroscopicResponse Theresultsfromthefirststepofplasticparameterscalibrationarereported inthissection.Inthisstep,fixedratiosbetweenslipfamilieswereenforcedto calibratethethreeparameters 𝑎p , ℏ0,p ,and 𝜉0,p ,usingthesolitarymacroscopicstress–strainresponseofthetwosamples.Theinitialguessesandoptimalvalues ofthethreeparametersforeachsamplearelistedinTable5.2. RESULTSANDDISCUSSION119ParameterHeat-treated(𝛼)As-built(𝛼′)UnitInitialguessOptimalvalueInitialguessOptimalvalue𝑎p2.0000.9132.0001.500-ℏ0,p2.0002.6452.0002.636GPa𝜉0,p328.2325.7324.6319.5MPa Table5.2:Initialguessesandoptimizedvaluesforbothsamplesfromthefirststagecalibrationusingonlymacroscopicstress–straindata,withfixedratios enforcedbetweenslipfamilies.SensitivityAnalysisofthePlasticParameters SinceNelder-Meadmethodisanon-gradient-basedlocaloptimizer,theresulting optimalvaluesareindeedsensitivetotheinitialguesses.Tounderstandthe influenceoftheinitialguesses,asensitivityanalysisisconductedwhereeach parameter’sinitialguessvalueisperturbedby+ 10% andtheresultingchange (inpercentage)inoptimalvaluesisquantified.Theresultsfromthissensitivity studyisrepresentedintheformofheatmapinFig.5.5forbothsamples. apℏ0,pξ0,papℏ0,pξ0,p Changeintheresultingoptimalvalue(%) -3-1-1-7-2-4000As-builtapℏ0,pξ0,p-188-41922000Heat-treated-20%-10%0%10%20%Perturbationofinitialguessvalueby10 % 1 Figure5.5:Sensitivitystudyontheinitialguessvaluesofplasticparameters inthefirst-stage“global”(macroscopic)optimization.Theparametersonthe X-axisareperturbedby+ 10% ,andtheresultingchangeinoptimalvalues(in %)relativetothereferenceareshownontheY-axis. 120DETERMINATIONOFCRYSTALPLASTICITYPARAMETERSFROMSYNCHROTRONDIFFRACTIONEXPERIMENTS Despitethemultipleperturbationsperformedforbothsamples,themaximum relativedeviationintheoptimizedlossvaluesacrossallcasesisfoundtobelessthan 0.3% .Thisindicatesthat,regardlessoftheinitialguessesandresultingoptimizedparameters,theoptimizerconsistentlyachievesanearidenticalmacroscopicresponse. Itisimmediatelyclearfrombothheatmapsthata 10% perturbationininitial guessvalueofanyofthethreeparametershasnoinfluenceontheoptimizedvalueof 𝜉0,p .Thisindicatesthatthemacroscopicresponseofthematerialis profoundlysensitiveto 𝜉0,p andtheoptimizerisabletoarriveatsimilaroptimal valuesirrespectiveoftheperturbationintheinitialguesses.However,thisisnotthecasefortheinitialslopeofhardening ℏ0,p andtheexponent 𝑎p thatjointlydetermineshapeofthehardeningcurve.Forthem,aperturbationof anyofthethreeparametersresultsinasizeabledeviationfromthereference optimalvalue.Nocleartrendinthedeviations,i.e.,eitherpositiveornegative, isapparentfromthisstudy. Betweenthetwosamples,themagnitudeofrelativedeviationsisobservedto behigherintheheat-treatedcasecomparedtotheas-builtcase.Thiscanbeattributedtodifferencesintheextenttowhichaslipsystemisallowedtoharden.ReferringtotheexperimentalmacroscopicbehaviorpresentedinFig.5.6,theheat-treatedsampleyieldsatastresslevelof 870MPa and ultimatelyhardensupto 990MPa .Similarly,theas-builtsampleyieldsaround 860MPa andultimatelyhardensupto 1096MPa .Thisbehaviorestablishesatrend:thegreaterthestressrangeoverwhichaslipsystemhardens,thelowerthepercentagedeviationfromthereferenceoptimalvalues,asseenin Figure5.5. Thedeviationsobservedinthepresentsensitivitystudysuggestthatthesystemisrelativelylesssensitiveto 𝑎p and ℏ0,p ,andthatmultiplecombinationsofthese parametervaluescanproduceamacroscopicresponsesimilartothereference. AstudycarriedoutbySedighianietal.[323]investigatedthesensitivityof adjustableparametersinthephenomenologicalcrystalplasticitymodelemployed here.Theirfindingsareconsistentwiththepresentobservations:criticalvalue oftheslipresistancescanbeuniquelyidentified,whiletheoptimalvaluesforthe exponent(𝑎p)andtheinitialhardeningslope(ℏ0,p)exhibitincreasingscatter, inthatorder.Anovelobservationmadeinthepresentstudyisthedependence ofmagnitudeofdeviationfromoptimalvaluesontheultimatestresslevelto whichthematerialisallowedtoharden. Itisimportanttonoteherethatalthoughdifferentcombinationsofparameter valuesmayyieldnearlyidenticalmacroscopicresponses,theunderlyinglocal behaviorattheslipsystemlevelcanvarysignificantly.Thistendencywasclearly demonstratedinChapter4,wheremultipleparametrizationsofthehardening RESULTSANDDISCUSSION121 modulimatrixresultedinindistinguishablemacroscopicstress–strainbehaviors,yetproducedsubstantialdiscrepanciesingrain-scalebehaviorandmaterialstate representation.Theissuewasfurtheramplifiedwhenloadcasesbeyondthe datadomainofcalibrationwereapplied.Suchobservationsemphasizeamajor shortcomingofmaterialparametercalibrationtechniquesthatrelysolelyona singlemacroscopicdataset:parametersthatappearlessinfluentialinshaping thesolitaryglobalbehaviormaynotbeuniquelyidentified.Furthermore,when solelyasingledatasetisusedforcalibration,eventheinfluentialparameters mayassumemultiplecombinationsthatyieldindistinguishablemacroscopic behavior.Thisemphasizesaneedforimprovedmaterialcalibrationstrategies whichutilizeexperimentaldatathatcapturedetailsacrossmultiplelength scales.Here,anattemptinthisdirectionismadebyintroducinglatticestrain evolutiondatafrommultipleplanefamiliesintothecalibrationprocess,thereby introducingmesoscaleconstraintstotheoptimizersuchthatthecompeting deformationmechanismsoftheinvolvedslipfamiliesarecaptured.OptimizationStep2:RelaxedratiosandCombinedMacro–MesoResponse Thereferenceexperimentaldatasets,plottedagainstthecalibratedmaterial behaviorsatbothscales,arepresentedinFig.5.6andFig.5.7.Withtheinclusionofadditionalgrain-levelreferencedata,thefixedratiosamongslip familiesarerelaxedinthisstepoftheoptimization.Theremovalofthisrigid constraintiscompensatedbytheinclusionofadditionalreferencedatainthe objectivefunction,whichallowstheoptimizertochoosetheratiosbasedonthereferencedataset.Additionally,theslipsystem-specificparameter 𝜉∞ , whichwaspreviouslycoupledto 𝜉0 throughafixedratio,isnowtreatedasan independentparameter.Theresultingoptimizedvaluesofalltwelveindependent parameters,obtainedfromthecombinedmacro–mesocalibrationapproach,are presentedinTable5.3. 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CurriculumVitaePersonaldetailsNameNikhilPrabhuBorn21/07/1996inKota,Karnataka,IndiaORCiD0000-0003-0843-3903Emailnikhilprabh[email protected]Publicprofilewww.linkedin.com/in/nikhil-prabhu-np2107Employment 01-2025to10-2025Researchassociateatdepartmentofcomputer science,KULeuven,Belgium 01-2021to12-2024PhDresearcheratdepartmentsofcomputerscience andmaterialsengineering,KULeuven,Belgium 02-2020to09-2020MasterthesisstudentatIKEAComponentsAB, Älmhult,Sweden 09-2019to01-2020TeachingandlabassistantforthecourseIntroduction toComputationalMechanicsatLinköpingUniversity,SwedenEducation01-2021to10-2025 Doctorofengineeringscience:materialsengineeringatKULeuven, Belgium233