strainrate sensitivity of unstable localized phase transformation phenomenon in shape memory alloys

edinohsityocalinesofovlcoterirmathechanicty,shaloys(Smtheo-electlylargcivildergoanunstablephasetransformationwhichcausesamacro-scalesofteninginthemechanicalbehaviorandhencelocalizationofdeformationorphasetransformation(IadicolaandShaw,2004).ThiscanbeobservedintheglobalresponseofSMAsbyaloadpeakandavalley(nucleationofA¢Mtransformation)duringforwardandreversetransformationsfollowedbyaplateauofconstantloadhileChangetal.dynamicalesh-depeToinvestigeffectsofreversephasetransformation,Azadietal.(2007)‘‘totaltransformationstrain’’constitutivemodel,andHea(2009,2010)usedagradient-basednonlocalmodelwithinthecontinuumGinzburg–Landauelasticframeworkandanalyzedmodelsensitivitytomateriallength-scalesandgeometricalchar-acteristics.DetailedexperimentswereconductedbyZhangetal.(2010)onnucleationandpropagationoftransformationfrontsinSMAsatdifferentstrainrates.Duvaletal.(2011)proposedanon-localgradientmodelinordertodescribelocalizationofphase⇑Correspondingauthor.Tel.:+982161112258;fax:+982166403808.E-mailaddress:smoham@ut.ac.ir(S.Mohammadi).InternationalJournalofSolidsandStructures63(2015)167–183Contentslistsavailabof.elsengineeringapplications,betterunderstandingoftheirbehaviorhasbeenachallengingtargetofexperimentalandtheoreticalresearch.Ithasbeenacceptedwithinthescientificcommunitythatunderuniaxialloading,SMAsespeciallyNi-Tibasedalloysundergomartensitemicrostructureinelasticmaterials,w(2006)developeda1-Dstrain-gradientthermo-forSMAwirestoavoidthepotentialproblemofmresultedfromlocalizationphenomenoninSMAs.http://dx.doi.org/10.1016/j.ijsolstr.2015.02.0490020-7683/�2015ElsevierLtd.Allrightsreserved.modelndencyatetheusedandSunsubsequentlyrecoverfromitthroughsolid–solidphasetransformation.Generally,SMAshavetwophaseswithdifferentpropertiesandhencedifferentbehaviors.Oneisaustenite(A)whichisstableathighertemperaturesandtheotherismartensite(M)whichisstableatlowertemperatures.SincetheintroductionofSMAsaspotentialmaterialsforintoaccounttheeffectsofthermo-mechanicalcouplingduringinstabilityoftransformationandsimulatedproblemsatdifferentloadingrates.ThedependencyoftheproposednumericalschemetodifferentboundaryconditionswasinvestigatedbyIadicolaandShaw(2004).Idesmanetal.(2005)proposedaphase-fieldmodelbasedonstrainsofteningtocapturetheevolutionofmulti-variant1.IntroductionBecauseoftheuniquethermo-meacteristicsincludingpseudo-elasticibio-compatibility,shapememoryalavarietyofapplicationsrangingfroinbiomedicalengineeringandmicr(MEMS)technologytotherelativedampingdevicesinaerospaceandfeatureofSMAsisthattheycanunalandqualitativechar-pememoryeffectandMAs)arebeingusedinveryfine-sizeddevicesro-mechanicalsystemser-scaleactuatorsandengineering.Themainlargedeformationand(propagationofA¢Mtransformation).Bothexperimentalobser-vationsandcomputersimulationshavebeenusedtoinvestigatethisSMAphenomenoninthepastdecade.ShawandKyriakides(1997a)establishedanexperimentalproceduretomonitortheunstablephasetransformationinpseudo-elasticresponseofSMAs.Then,ShawandKyriakides(1997b)usedatri-linearup-down-upplasticity-basedmodeltoinvestigatetheeffectsoflocalizeddeformationinSMAsnumericallyandpointedoutonsomesimilaritiesbetweenthedevelopmentofmartensitebandsinSMAstoLudersbandsinfine-grainedsteels.Shaw(2000)tookStrain-ratesensitivityofunstablelocalizphenomenoninshapememoryalloysusHosseinAhmadian,SaeedHatefiArdakani,SoheilMHighPerformanceComputingLaboratory(HPCLab),SchoolofCivilEngineering,UniverarticleinfoArticlehistory:Received14March2014Receivedinrevisedform5October2014Availableonline5March2015Keywords:ShapememoryalloysPhasetransformationSofteningThermo-mechanicalcouplingabstractInthiswork,theoriginallosofteningfunctiontoexamphenomenoninthinsheeteningmodelisadoptedtostrongthermo-mechanicathecouplingeffectsonmabyunstablephasetransfohavebeenverifiedagainstInternationalJournaljournalhomepage:wwwphasetransformationganon-localmodelammadi⇑fTehran,Tehran,IranmodelofBoydandLagoudas(1996)hasbeenextendedbyproposinganewthelocalizedmacrobehaviorresultedfromunstablephasetransformationshapememoryalloys(SMAs).Inaddition,anintegraltypenon-localsoft-ercomethemeshdependencyproblemoflocalsofteningmodels.DuetouplinginSMAs,astaggeredschemehasbeenusedtotakeintoaccountalbehavior.Inclusionofsucheffectshasrevealedthatinstabilityinducedtioncouldbesurmountedbyincreasingthestrainrate.Thesimulationsavailableexperimentalandnumericaldata,showingagoodagreement.�2015ElsevierLtd.Allrightsreserved.leatScienceDirectSolidsandStructuresevier.com/locate/ijsolstrtransformationinSMAwiresandthinfilms.HallaiandKyriakides(2013)experimentallyandnumericallyinvestigatedtheinitiationandpropagationoflüdersbandsinsteelandNiTiSMA.Recently,BechleandKyriakides(2014)andReedlunnetal.(2014)experimentallymonitoredandcomparedthedifferentlocalizationmodesofSMAsduringtension,compressionandbending.MeshdependencyisonetheconcernsofresearchesstudyingtheunstablephasetransformationinSMAs,asobservedintheworksofShawandKyriakides(1997a),Shaw(2000)andAzadietal.(2007).Whilethemeshdependencyphenomenoncouldbemildinsomecases(ShawandKyriakides,1997b;HallaiandKyriakides,2011,2013),someothersstronglyrecommendeditsresolution(HeandSun,2009,2010;Changetal.,2006;Duvaletal.,2011).Inthisregard,theexistingnonlocal(HeandSun,2009,2010)andstraingradient(Changetal.,2006)modelsareeithercomputationallyintensiveforcomplexstructuralcalculationoronlylimitedto1-Dcases.WiththeincreasingapplicationofSMAsinindustry,itiscriticaltodevelopamulti-dimensional168H.Ahmadianetal./InternationalJournalofSmodelthattakesintoaccountunstablephasetransformationsandissimpleenoughtobeappliedinavarietyofapplications.Forinstance,Duvaletal.(2011)proposedanonlocalgradientmodelinordertodescribelocalizationofphasetransformationinSMAwiresandthinfilms.Strain(loading)rateisalsoanimportantsubjectofinvestiga-tion.AccordingtoShaw(2000)andIadicolaandShaw(2004),theheat-releaseofSMAsduringforwardtransformation,resultedfromthermo-mechanicalcoupling,hasaratherstabilizingeffectontheinstabilityofmaterialresponse.Fig.1schematicallyillustratesthismaterialstabilityphenomenon,wherethelocalisothermalexponentialsofteningcurve(whichwillbederivedinSection3)turnsintoadifferentresponseatnearadiabaticcondi-tionsorhigherstrainrates.Therefore,beforeanydesigntotakeplaceonanSMA-baseddevice,analysisisrequiredonthematerialresponseatdifferentimposedloadingrates.Mostofthemodelsavailableinliteraturelackacertaincharac-teristicsnecessaryforsimulationofunstablephasetransformationofSMAs.Whileanumberofmodelsaremesh-dependent,mostofthemdonottakeintoaccountthethermo-mechanicalcouplingeffectsforafullcycleoftransformationandmanyofthemarelim-itedto1-Dcases.Inthispaper,theoriginalmodelofBoydandLagoudas(1996)isextendedtoaccountforthermo-mechanicalcouplingandtransformationsoftening.Fortheproblemssimulatedinthispaper,theeffectofmartensitereorientation,whichisthesubjectofanindependentwork,isnotconsidered.Aclassicaldam-agemechanicsformulationwithexponentialsofteningisproposed.Fig.1.Typicalstress–straincurvesinisothermalandadiabaticconditions.Thethermo-mechanicalcouplingequationsaresolvedusingastaggeredalgorithm,whichdecouplestheoriginalproblemintotwoproblems.Ineachstep,firstthemechanicalproblemissolvedisothermallyandthenthetemperaturevariationsduetophasetransformationaretakenintoaccountbysolvingtheheatequation.Thisiterativeprocedurecontinuesuntiltheproblemcon-verges.Toremedythepotentialmeshdependencyissues,aclassicnonlocalintegralformulation(JirásekandBazant,2002)isimplemented.Theperformedsimulationshaveproperlycapturednucleationandpropagationoflocalizedphasetransformation.2.ConstitutivemodelInthissection,theconstitutivemodelofBoydandLagoudas(1996)isdiscussedbriefly.Theconstitutivestress–strainrelationisgivenby:e¼C:rþaðT�T0Þþetrð1ÞwhereCandaaretheeffectivecomplianceandtheeffectivether-malexpansiontensors,respectively.Themodelstatevariablesr,e,T,T0andetraredefinedasthestress,thetotalstrain,thetempera-ture,thereferencetemperatureandthetransformationstrain,respectively.TheflowruleoftheincrementalconstitutivemodelofthemartensiticphasetransformationinSMAsduringforwardandreversetransformationscanbedefinedas:_etr¼K_nð2Þwherenisdefinedasthemartensiticvolumefractionwhichisasca-larparameterdeterminingphasemixtures,rangingfrom0to1,where0standsforthefullyausteniticstateand1representsthefullymartensiticstate.Kisthetransformationtensor,K¼32emaxr0ffiffiffiffiffiffiffiffiffiffi32kr0k2p;_n>0emaxetr�Rffiffiffiffiffiffiffiffiffiffiffiffiffiffi23ketr�Rk2p;_n<08><>:ð3Þwherethematerialparameteremaxisthemaximumuniaxialtrans-formationstrain.r0isthedeviatoricstresstensorandetr�Risthetransformationstrainatthereversalpoint.Thisisverysimilartotheclassicvon-Misesflowruleinplasticity,whichwelljustifiesuti-lizingthereturnmappingalgorithmfornumericalimplementationpurposes.Here,theclosestpointprojectionimplicitreturnmappingalgorithm,proposedbyQidwaiandLagoudas(2000),hasbeenadopted.DetailscanbefoundinQidwaiandLagoudas(2000).First,thetransformationsurfaceU¼0whichdeterminesthewayphasetransformationtakesplaceeitherinforwardorreversestatesisdefinedas,U¼p�Y;_n>0�p�Y;_n<0(ð4ÞwhereYisamaterialparameterandpisthethermodynamicsforce,p¼r:Kþ12r:DC:rþr:aðT�T0Þ�qDcðT�T0Þ�TlnTT0����þqDs0T�qDu0�@fðnÞ@nð5Þwhereq,c,s0andu0arethedensity,theeffectivespecificheat,theeffectivespecificentropyatthereferencestateandtheeffectivespecificinternalenergyatthereferencestate,respectively,andDisthedifferenceofaspecificpropertybetweenthetwophases.olidsandStructures63(2015)167–183ThefunctionfðnÞintheoriginalmodelofBoydandLagoudas(1996)standsforthetransformationhardening,butitisredefinedheretoconsiderthesofteningbehaviorandwillbediscussedlater.byLagoudasetal.(1996)forunifyinghardeningmodels.ItrequiresthecriA1¼qDs0ðMs�MfÞSfðe�Sf�1ÞA2¼qDs0ðAs�AfÞSfðe�Sf�1ÞB¼0:5ðA1ðe�Sf�1ÞþA2ð1�e�SfÞÞqDu0¼0:5ðqDs0ðMsþAsÞþA1SfþA2SfÞY¼qDs0Ms�qDu0þA1SfþBð15ÞSubstitutingtheabovematerialparametersintothemodel,astress–straincurve,typicallypresentedinFig.2,canbeobtained.4.Thermo-mechanicalcouplingToinvestigatethestrongthermo-mechanicalcouplinginSMAs,theweakformofthefullycoupledheatequationofSMAs,asgivenbyLagoudas(2008),isderived,::T����_ofSolidsandStructures63(2015)167–183169pðr;T;nÞ¼�Yatr¼0;T¼Af;n¼0ð11ÞleadingtoqDs0Ms�qDu0þA1SfþB¼YqDs0Mf�qDu0þA1Sfe�SfþB¼YqDs0As�qDu0þA2Sf�B¼�YqDs0Af�qDu0þA2Sfe�Sf�B¼�Y8>>><>>>:ð12ÞItisobservedthatapplyingthethermodynamicsconstraintsaddsanothertwounknownsofYandqDu0whichrequireaddi-tionalequations.Withrespecttothereversibilityofphasetrans-formationinSMAs,allvariables,includingstress,strainandmartensiticvolumefraction,shouldreturntotheirinitialvaluesafterthematerialundergoesafulltransformationcycle.Byreturn-ingthematerialtoitsinitialstate,theGibbsfreeenergy(13)needstoremainunchanged(Lagoudasetal.,2012).Gðr;T;n;etrÞ¼�12qr:C:r�1qr:aðT�T0Þþetr��þcðT�T0Þ�TlnTT0�����s0Tþu0þ1qfðnÞð13Þwhichrequiresthat:Z10@fðnÞ@njA!MdnþZ01@fðnÞ@njM!Adn¼0ð14Þpðr;T;nÞ¼�Yatr¼0;T¼As;n¼1ð10Þ4-Endofreversetransformationatzerostress:pðr;T;nÞ¼Yatr¼0;T¼Mf;n¼1ð9Þ3-Beginningofreversetransformationatzerostress:pðr;T;nÞ¼Yatr¼0;T¼Ms;n¼0ð8Þ2-Endofforwardtransformationatzerostress:beuseofindependentequationsbyapplyingthetransformationterionatfourdifferentthermo-mechanicalstates:1-Beginningofforwardtransformationatzerostress:TheeffectivematerialpropertiesincludingC,a,c,s0andu0aredeterminedusingtheruleofmixtures:CðnÞ¼CAþnDCaðnÞ¼aAþnDacðnÞ¼cAþnDcs0ðnÞ¼sA0þnDs0u0ðnÞ¼uA0þnDu0ð6ÞwheresuperscriptsAandMdenotetheausteniticandmartensiticphases,respectively.3.ModificationofthemodeltoaccountforsofteningToincludesofteningintheoriginalhardeningthermo-dynami-calmodelofBoydandLagoudas(1996),thefollowingsofteningfunctionisproposed:@fðnÞ@n¼�A1Sfe�Sfn�B;ðA!MÞ�A2Sfe�Sfð1�nÞþB;ðM!AÞ(ð7ÞwhereSfisthesofteningrateandA1,A2andBaretransformationparameters,whichcanbederivedsimilartothemethodproposedH.Ahmadianetal./InternationalJournalUsingthesystemofEqs.(12)and(14),A1,A2,B,qDu0andYcandetermined:Ta:rþqcTþ�pþTDa:r�qDclnT0þqDs0Tn¼�r:qð16Þwherethetermontheright-sideofEq.(16)isrelatedtotheheattransferprocessbytheheatfluxq.Consideringtheconvectiveheattransferbetweenthedeviceandenvironment,theusualboundaryconditionsandinitialvaluesassociatedwiththeheatequationaredefinedas:T¼T0onCTdofsq:n¼hðT�T0Þon@CTdofsTðt¼0Þ¼T0ð17ÞwhereCTdofsand@CTdofsaretheboundariesinwhichtemperature(Drichlet)andconvective(Neumann)boundaryconditionsareapplied,respectively.tisthetimeandndenotesthenormalvectortotheboundarywithconvectiveheattransfer.UsingtheFourier’slawofheatconduction,q¼�krT,theconvectiveboundarycondi-tioncanberewrittenas:�ðkrTÞ:n¼hðT�TextÞð18ÞMoreover,byconsideringthattheeffectivespecificheatandthethermalexpansioncoefficientsofthetwophasesareidentical,theheatequationyieldsas:Ta:_rþqc_Tþð�pþqDs0TÞ_n¼r:ðkrTÞð19ÞPerformingtheconventionalfiniteelementdiscretization,theheatEq.(19)canbecompactedtoanordinarydifferentialequation,Fig.2.Typicalstress–straincurvewithexponentialsoftening.ðKþKhþKrþKnÞ:TþM:_T¼Fð20Þorinanincrementalformintime,KþKhþKnþ1rþKnþ1n :Tnþ1þM:Tnþ1�TnDt¼Fnþ1ð21Þwherethesuperscriptnistheindexoftheincrement,Dtisthetimeofeachincrement,Kistheheatconductionmatrix,Khistheheatconvectionmatrix,Kristhematrixoflatentheatduetostresschanges,Knisthematrixoflatentheatduetomartensitevolumefractionchanges,MistheheatcapacitymatrixandFisanequiva-lentloadvector,ðKÞij¼ZXkrNirNjdXðKhÞij¼ZShNiNjdSðKrÞij¼ZXa:_rNiNjdXZð22Þthisproblem,nonlocalmodelshavebeendevelopedbasedonapriorivalueforthewidthoftheprocesszone,preventingthemeshdependencyoflocalizationofphasetransformationinaverynarrowzone.Inthisresearchanintegral-typeformulationisadoptedwhichisdiscussedbrieflyinthissection.Unlikethelocalmodelsinwhichtheeffectsoflocalstatevariablenateachgausspointsareconsideredindependently,non-localmodelsutilizeatypeofweightfunctiontoincludetheeffectsofotherintegrationpoints.Therefore,thefinalgoalistoreplacethelocalvariablewithitsnonlocalvalue.AccordingtoJirasek(2007),thenonlocalfieldfðxÞofthelocalfieldfðxÞovertheVdomaincanbedefinedas:�fðxÞ¼ZVaðx;sÞfðsÞdsð24Þwhere,aðx;sÞistheweightfunction,aðx;sÞ¼a0ðkx�skÞRVa0ðkx�fkÞdfð25Þmartensitevolumefractionnisofinterest(Fig.3(b)),andcanbe170H.Ahmadianetal./InternationalJournalofSolidsandStructures63(2015)167–183ðKnÞij¼XqDs0_nNiNjdXðMÞij¼ZXqcNiNjdXðFÞi¼ZShTextNidSþZXp_nNidXwhereNi,Njarethefiniteelementshapefunctions.Aftersomemanipulations,thetemperatureatincrementnþ1isgivenas:Tnþ1¼Knþ1eq �1Fnþ1eq Knþ1eq¼KþKhþKnþ1rþKnþ1nþMDt��Fnþ1eq¼Fnþ1þMDtTn��ð23ÞwhereKeqandFeqaretheequivalentstiffnessmatrixandforcevec-tor,respectively.5.NonlocalmodelSolvingtheconstitutiveequationsbasedonthelocalsofteningmodelleadstoinhomogenousdeformation.Consequently,inhomogenousphaseevolutionoccursinthestructurewhichmayleadtopotentialnumericalmeshdependency.ToovercomeFig.3.(a)QuarticweightfunctionwiththeinteractionradiusofR,(b)IntegrationprocefromtheoriginalideabyBazantandJirásek(2002)).definedas,�nðxÞ¼ZVa0ðx;sÞnðsÞdsð27Þ6.Numericalsimulations6.1.Tensiontestondog-bonesamples6.1.1.MaterialpropertiesandmodelcalibrationThematerial,geometryanddimensionsofthesamplearethesameastherecentlyreportedexperimentaldatabyZhangetal.(2010).Thetestswerecarriedoutonathinspecimenwith60mmtotallength,30mmgaugelength,2.6mmwidthandsandxarethesourceandtargetpoints,respectively,anda0ðkx�skÞisadecreasingnonnegativefunctionofthedistancer¼kx�sk.Here,aquarticformoftheweightfunction(Fig.3(a))isutilized:a0ðrÞ¼1�r2R2��2ð26ÞwhereRiscalledtheinteractionradiusandneedstobedeterminedasapredefinedvalueofthenonlocalmodel.Thenonlocalvalueofduretoconsidertheeffectsofgausspointswithintheinteractionradius(adopted0.5mmthickness.Theambientandinitialtemperatureswereequalto23�C.Theexperimentswereconductedinstagnantair,soacommonvalueof4W=ðm2KÞforaverageconvectioncoefficientwasassumed.Also,theconductioncoefficientandheatcapacityofthematerialwere18.3W/(mK)and3:2�106J=ðm3KÞ,respectively.TheconstitutivemodelparameterSfwaschosentobe1.SincethevaluesofcriticaltemperaturesincludingMs,Mf,AsandAfarenotreportedinZhangetal.(2010),thelineartransformationsurfacesinLagoudas(2008)areusedtocalculatethesevalues,rMs¼CMðT�MsÞrMf¼CMðT�MfÞrAs¼CAðT�AsÞrAf¼CAðT�AfÞð28ÞwhererMs,rMf,rAsandrAfarethestartandfinishstressesofforwardandreversetransformations,respectively.TisthematerialtemperatureandCMandCAaretheso-calledstressinfluencecoeffi-cients.Sobysimplyhavingthestartandfinishstresses,valuesofMs,Mf,AsandAfcouldbecalculated.Itisobservedthatinthecaseofunstablephasetransformation,whichisdealtwithinthisresearch,MsandAfarelowerthanMfandAswhichcontradicttheexperimentalDSCtests(atesttocalculatethevaluesofMs,Mf,AsandAfinzero-stressconditions).Itshouldbenoted,however,thatthesearejusttheoriginalterminologiesassociatedwiththepioneeringworkofBoydandLagoudas(1996),andtheyareusedwithdifferentinterpretationsinthepresentresearchbecauseofextendingtheoriginallyhardeningmodeltoaccountforthesofteningresponse.Therefore,toavoidinconsistenciestheyarecalledmartensitenucleationtemperature(MN),martensitecomple-tiontemperature(MC),austenitenucleationtemperature(AN)andaustenitecompletiontemperature(AC).ThefinalmaterialpropertiesusedinthesimulationsarelistedinTable1.6.1.2.MeshdependencyanalysisToevaluatetheproposednonlocalapproachseveralproblemswithdifferentmeshdensitieshavebeensimulated.AsitwasstatedintheworksofShaw(2000)andIadicolaandShaw(2004)andalsoisinvestigatedlaterinthisresearch,thethermo-mechanicalcou-plinghasaratherstabilizingeffectontheoverallmaterialbehaviorathigherstrainrates.Asimpleexplanationisthatbyincreasingtheloadingrate,thetemperatureofthematerialraisesandhencethetransformationsurfacebecomeslarger,whichdecreasesthenegativeslopeofthetransformationregionoflocalstress–strainresponseandsubstantiallyreducesinstabilityandmeshdepen-dency.Onthecontrary,thesituationbecomesworseatlowstrainrates.Moreover,ithasbeennumericallyobservedthatinadditiontothemeshsize,themeshregularitycouldaffectnotonlyontheglobalresponsebutalsothetiming,locationandorientationofnucleationbandsandthedeformationpatternswhichcouldjeopardizetheconventionaldesignproceduresfordevicesmadeTable1Definitionofmaterialproperties.EM40�109PamA¼mM0.3EA40�109PaqcA¼qcM3:2�105J=ðm3KÞMN�24.7�CCA¼CM8:5�106Pa=KMC�18.9�Cemax0.048AN10.6�Ck18:3W=ðmKÞAC6.5�ChAir4W=ðm2KÞH.Ahmadianetal./InternationalJournalofSolidsandStructures63(2015)167–183171Fig.4.GeometryofthesampleofSMA