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Kinematic hardening rules with critical state of dynamic recovery, part I.pdf

Kinematic hardening rules with …

上传者: 天涯囚徒 2012-07-05 评分 0 0 0 0 0 0 暂无简介 简介 举报

简介:本文档为《Kinematic hardening rules with critical state of dynamic recovery, part Ipdf》,可适用于人文社科领域,主题内容包含InternationalJournalofPlasticity,Vol,pp,$PrintedintheUSACopyrightPergamonP符等。

InternationalJournalofPlasticity,Vol,pp,$PrintedintheUSACopyrightPergamonPressLtdKINEMATICHARDENINGRULESWITHCRITICALSTATEOFDYNAMICRECOVERY,PARTI:FORMULATIONANDBASICFEATURESFORRATCHETTINGBEHAVIORNOHNOandJDWANGNagoyaUniversity(CommunicatedbyJLChaboche,ONERAORCE)AbstractKinematichardeningrulesformulatedinahardeningdynamicrecoveryformatareexaminedforsimulatingratchettingbehaviorTheserules,characterizedbydecompositionofthekinematichardeningvariableintocomponents,arebasedontheassumptionthateachcomponenthasacriticalstateforitsdynamicrecoverytobeactivatedfullyDiscussingtheirbasicfeatures,theauthorsshowthattheycanpredictmuchlessaccumulationofuniaxialandmultiaxialratchettingstrainsthantheArmstrongandFrederickruleComparisonswithmultilayerandmultisurfacemodelsaremadealso,resultinginafindingthatthesimpleoneinthepresentrulesissimilartothemultilayermodelwithtotalstrainratereplacedbyinelastic(orplastic)strainratePartIIofthisworkdealswithapplicationstoexperimentsINTRODUCTIONConstitutivemodelingofcyclicplasticityandviscoplasticityhasadvancedsogreatlyinthelasttwodecadesthatvariousaspectsinsuchdeformationnowcanbedescribedwithconsiderableaccuracyforengineeringpurposesHowever,westillhaveproblemsintheconstitutivemodeling,asseeninarecentreviewbyOHNOSimulationofratchettingbehaviorundermultiaxialanduniaxialcyclicloadingremainsoneofthemostdifficultproblemsSinceratchettingbehaviorisconcernedwithsecondarydeformationaccumulatingcyclebycycleinthedirectionofnonzeromeanstress,itisnoteasytodescribeitquantitativelyInfact,mostoftheexistingconstitutivemodelshavepoorcapabilityofpredictingtheaccumulationofratchettingstrainunderuniaxialandmultiaxialloadingconditions(eg,INOUEetal,)Itis,however,importanttoconsiderratchettingbehaviorindesigningstructuralcomponentslikereactorcomponentsConstitutivemodelingofratchettingbehavior,therefore,hasbeenstudiedeveninrecentworks(eg,BURLETCAILLETAUDBLANCHARDLEMOINECHABOCHENOUAILHASCHABOCHEb,INOUEIMATANIFREEDWALKERHASSANKYRIAKIDESHASSANetal)ModelingofkinematichardeningisresponsibleforsimulationofratchettingbehaviorThenonlinearkinematichardeningruleofARMSTRONGandFREDERICK(),usedextensivelybyChabocheandhiscoworkers(eg,CHABOCHEa),iswellknownHowever,italsogivespoorpredictionswithrespecttoratchettingbehavioritusuallyoverpredictstheaccumulationofratchettingstrainunderbothuniaxialandmultiaxialloadingconditions(INOUEetal,CHABOCHENOUAILHASFREEDWALKERCHABOCHE)Nevertheless,theconceptemployedinthisrule,ie,Administrator文本框OhnoWang模型NOHNOandJDWANGhardeninganddynamicrecoveryofkinematichardening,issimpleandphysicallysoundModificationsoftheArmstrongandFrederickrulehavebeenthereforediscussedrecentlyinseveralworks(eg,BURLETCAILLETAUDCHABOCHENOUAILHASFREEDWALKERCHABOCHEHASSANetal)CHABOCHE,especially,introducedathresholdofdynamicrecoverytosuppresseffectsofdynamicrecoveryInthepresentwork,kinematichardeningrulesformulatedinahardeningdynamicrecoveryformatareexaminedforapplicabilitytoratchettingbehaviorTheserules,characterizedbydecompositionofthekinematichardeningvariableintocomponents,arebasedontheassumptionthateachcomponenthasacriticalstateforitsdynamicrecoverytobeactivatedfullyThispaperdescribesuniaxialandmultiaxialfeaturesoftherules,theirthermodynamicadmissibility,andcomparisonsofthemtoclassicalmodels,suchastheARMSTRONGandFREDERICKrule,themultilayermodelbyBESSELING,andthemultisurfacemodelbyMRozPartIIofthisworkdealswithapplicationsofthepresentrulestoexperimentsofModifiedCrlMosteeldonebyTANAKAetalandanonproportionalexperimentbyLAMBAandSIDEBOTTOMIIKINEMATICHARDENINGRULESInthepresentwork,sinceweareconcernedwithsmallstrain,straineisadditivelydecomposedintoelasticandinelastic(orplastic)parts:~=I~Ip()FortheelasticpartF,e,weassumeHooke'slawlv~efftr(a)l,()EwherevandEareelasticconstants,aandIdenotethestresstensorandtheunittensorofranktwo,respectively,andtrindicatesthetraceFromnowon,(')indicatesthederivativewithrespecttotime(:)theinnerproductbetweensecondranktensors,ie,a:b=tr(ab)()theMacauleybracket,ie,(x)=xifx>and(x)=ifx<HtheHeavisidestepfunction,ie,H(x)=ifx>andH(x)=ifx<IIArmstrongandFrederickmodel(AFmodel)Whenmetallicmaterialsaredeformedinelastically,variousinternalstressesactingondislocationsaregeneratedHenceweassumethatthekinematichardeningvariableaiscomposedofMcomponents(CHAaOCHEetalCHABOCHEaROUSSELIER):M=~~i()i=TheArmstrongandFrederickmodelassumes(strain)hardeninganddynamicrecoveryfortheevolutionofkinematichardeningTheevolutionequationofotiisthenwrittenasOti=hi~P~iolip,()Kinematichardeningrulesforratchettingbehavior,Partwherehiand~(i=,M)arematerialparameters,andpdenotestheaccumulatedinelastic(orplastic)strain:p=(~p:~p)l()Wemayintroducethesocalledtargetpointofa,a*,definedas:Por*=~ri,()Pwhereri=hi~iEquation()thentakesanalternativeformi=~i(ot*oti)p()Itisnoticedthatriisthematerialparameter,whichthemagnitudeofai,Oli=(~i:ai),()approachesundermonotonicloadingGeneralizationsofeqn()werediscussedbyMOOSBRUGGERandMcDoWELL,Modelwithcriticalstateofdynamicrecovery(ModelDCrossslipofdislocationspileduptoobstaclescanbeamicromechanismofthedynamicrecoveryofkinematichardeningInfcccrystals,especially,adislocationthatisextendedonitsslipplaneduetothestabilityrecombineintoaperfectonewhenitcrossslips,butthisrequiresenergy(eg,DIETER)Wenowassumethatthedynamicrecoveryofaiisactivatedfullyonlywhenitsmagnitude(Xiattainsacriticalvalue,resultingfromtheenergyrequiredforcrossslipLetusrepresentthecriticalstateofdynamicrecoverybyasurfaceJr,=r=O()WiththeHeavisidestepfunctionH,then,anevolutionequationofaicanbewritteninthefollowingforminwhichthedynamicrecoverytermoperatesonlywhenOli=ri'Oti="~hi~pH(fi)Aiai()riTheunknownparameterinthisequation,Ai,isdeterminedusingtheconditionf~=,representingthatt~,remainsinthecriticalstatefi=ifcurrentlyf~=and~P:a,>(Fig):)~i=hi(~p:ki),()wherekidenotesthedirectionofOi,ie,ki=:oli()NOHNOandJDWANG•<~P:ki>ispaceofIFigEvolutionof~ionthesurfacef,=Equation()thustakestheform~=h~~~PH(fi)(~P:ki)ti"r~Substitutinghi=~ir~intheequationabove,wehaveOti=~iri~pH(fi)(~P:ki)otiEvolutionofuibyeqn()isdiscussedindetailinSectionIII(a)(b)ExtensionofModelI(ModelII)Wemaygeneralizeeqn()byallowingthedynamicrecoverytermtooperateinsidethesurfacef~Letusassumethatthedynamicrecoveryof~becomessignificantnonlinearlyas~iapproachesthesurfacef~=Thenwemayextendeqn()as~ti:~iI~ri~P(~ii)'n'(~p:ki)oti,(b)wheremg(i=,M)areconstantsEquation()isreducedtoeqn(),whenmiooInModelsIandII,(~P:ki),ratherthanp,contributestothedynamicrecoveryofaiThisisincontrasttopreviousformulationsofnonlinearkinematichardeninginwhichpisassumedtoexpressthedynamicrecoveryofoti(ARMSTRONGFREDERICKCHABOCHENOUAILHASMOOSBRUGGERMcDOWELL,FREEDWALKERCHAaOCHE)Since(gP:ki)<PundernonproportionalKinematichardeningrulesforratchettingbehavior,Partloading,<~P:ki>makesthedynamicrecoveryofetilessactiveandenablesustopredictlessaccumulationofmultiaxialratchettingstrainthanp(OHNOaWANC)HIFEATURESOFMODELSiANDIInthissection,featuresofModelsIandIIie,eqns()and()formulatedinthelastsectionarediscussedincomparisonwiththeArmstrongandFrederickmodellie,eqn()IIIUniaxialloadingWediscussfirsttheevolutionofacomponent~i,expressedbyModelIunderuniaxialtensileloading(Fig)Wheni<r~ineqn(),ot~developsonlyduetothehardeningtermbecauseofnoactivationofthedynamicrecoverytermwhen~i=ri,thedynamicrecoverytermgetsactivatedandbalanceswiththehardeningterm,resultingini=ModelIthusgivesabilinearchangeofaiunderuniaxialtensileloadingSincetherisingpartinthisbilinearchangeisexpressedasi=~'iriepusingeqn(b),thecornerpointhasEP=~,()i=ri()InModelII,ontheotherhand,sincethedynamicrecoverytermbecomesactivenonlinearlyas~iapproachesri,thechangeofaiisnonlinear,asshownbythedashedlineinFigIfmi>>,ModelIIdifferslittlefromModelI,sothatthechangeofaiisnearlybilinearModelIandModelIIwithmi>>thusgivethechangesoft~i,whicharequitedifferentfromthatoftheAFmodelIntheAFmodel,thedynamicrecoverytermisproportionaltooti,sothatthismodelgivesagradualapproachofitoriundermonotonicloadingasfollows:i=riexp(~i~P)Figureshowsthec~vsePrelationsbyModelsIandIIunderuniaxialtensileloadingHere~isdefinedas=~ot:t~wSincet~isobtainedbysuperposinga,tot~M,theversusePrelationsbyModelsIandIIarepiecewiselinearandnonlinear,asshownbythesolidanddashedlinesinFig,respectivelyLetussupposethat~tl,riaiI""ModelIModelII'AFmodelI~ipFigChangeof~iunderuniaxialtensileloadingNOHNOandJD~VANGCt(k)(k}f()ModelPcPe(k)FigChangeof~underuniaxialtensileloadingatatMarenumberedtosatisfytherelation~'<~"<<~M(Fig)Then,substitutingi=~iri~p(~P<~,)ori=ri(eP~'~)into=Zigivenbyeqn(),andusingeqn(),weobtainthekthcornerpointonthepiecewiselinearrelationinFigasfollows:e~x)=~x,()kM(x)=~,,ri~a(~ik)ri()i=i=klFigure(ac)illustrateshowatchangesundercyclicstraininginsertedintotensilestrainingModelIprovidesaclosedhysteresisloopforthechangeofatunderanycyclicstraining,becausethechangeofatbyeqn()ispiecewiselinearduetonoactivationofthedynamicrecoveryterminsidethesurfacefi=ModelII,ontheotherhand,doesnotprovideacompletelyclosedhysteresisloopasshowninFigb,becausethedynamicrecoverytermineqn()operatesinsidethesurfacef=atanyrateItisnoticed,however,thatifmiissufficientlylarge,thehysteresisloopbyModelIIineffectclosesTheAFmodel,inwhichthedynamicrecoverytermisproportionaltoati,doesnotallowthehysteresislooptocloseatall(Figc)Thediscussionabovemeansthefollowing:ModelIcanpredictclosedstressstrainhysteresisloops(ie,noratchettingstrain)underuniaxialcyclicloadingwithanymeanstress,althoughviscosityofmaterialsmayinduceratchettingstrain(seeCHABOCHEQtictl~C~kEPctkFigNumberingof~iKinematichardeningrulesforratchettingbehavior,PartI(a)ModelICt{l(b)ModelII~o~o~a(c)AFmodelloE~FigChangeof~tiunderloading,unloading,reverseloading,andreloading(a)ModelI,(b)ModelII,(c)AFmodelNOUAILHAS)ModelII,ontheotherhand,expressesratchettingstrain,whichispredictedtobelargerbyasmallervalueofmiIncidentally,thebilinearchangeof,xiunderuniaxialloadingexpressedbyModelI,showninFig,maybedescribedbyalimitingcaseofthemodelCHABOCHEbasedonathresholdofdynamicrecoveryItsuggeststhattheuniaxialformofModelImayberegardedasalimitingcaseofhismodelHowever,themultiaxialformofModelI,eqn(),cannotbederivedfromhismodelThisisbecauseModelIhas(~P:ki>inthedynamicrecoveryterm,whereasisassumedinhismodelMultiaxialloadingFiguredealswithnonproportionalchangesofacomponent(itipredictedbyModelI,ModelII,andtheAFmodelAccordingtoModelI,iiscollinearwith~Pwhileaimovesinsidethesurface~=whenoiisonthesurfacef~=,itmovestangentiallytothissurfacebecauseeqn()satisfies,:ai=duetotheactivationofthedynamicrecoverytermInthecaseofModelII,ichangesitsdirectioncontinuously,asshownbythedashedlineinFigThisisbecausethedynamicrecoverytermineqn()beNOntoandJDWAN(~<FigNonproportionalchangeof~icomesactivenonlinearlyasogiapproachesthesurfacef,=TheAFmodel,ontheotherhand,directstothetargetpointor*,asseenfromeqn(),whichisanalternativeformofeqn()Hence,forazigzagpathofePextendinginthetensiledirection(Fig),thethreemodelsgivetheresponsesofoishowninFig(ac),respectivelyThezigzagpathhastwodirectionsof~PThepointsPandQinFigarethetargetpointsdefinedbyeqn()forthesetwodirectionsof~PTheAFmodelpredictsthat~idevelopsuptothelevelPQinthetensiledirection(Figc),whereasModelsIandIIallowotgtodevelopbeyondthelevelPQ(Figaandb)Inotherwords,ModelsIandIIexpressstrongerresistanceinmultiaxialratchettingdeformationthantheAFmodelIVTHERMODYNAMICDISCUSSIONHere,weshowthatModelsIandIIarethermodynamicallyadmissibleLetusconsiderafreeenergy~r:~(~e,T,~i),()FigZigzagpathinplasticstrainspaceKinematichardeningrulesforratchettingbehavior,PartI~fi=O(a)ModelI(ai)~~(~i)(b)ModelKL~(~i)~~fi=(c)AFmodelFigChangeofaiunderzigzagstrainingshowninFig(a)ModelI,(b)ModelII,(c)AFmodelwhereTdenotestemperatureandBi(i=,,,M)arestrainlikekinematichardeningvariablesThesecondprincipleofthermodynamicsthenrequiresthattheintrinsicdissipationbepositive(eg,CHABOCHE):Nowwetake~itobe:L()~iOi()hiandassumethefollowingequationsforq"and~P,respectively:=~'e(~'e,T)hi•i:#i,"=()NOHNOandJDWANGSOr~:P=~p,()Oeffwhereq,eisthethermoelasticpartofq',sindicatesthedeviatorofstressa,andoeff=(sot):(sa)'Then,notingthat~P,ot~and~iaredeviatorictensors,andusingeqns()togetherwitheqns(),()and(),weobtainthefollowingequationsforeqns(a)and(a),respectively:MD=O'eff,bZH(f)(~P:oti),()i=lM(~,~,,b=Oeff~"Z(~P:Oi>"i=,r~!()ModelsIandIIthereforesatisfythethermodynamicrequirement>Incidentally,ifhiineqns(a)and(a)istemperaturedependentundernonisothermalconditions,eqns()and()arederivedifthetemperaturerateterm(aihi)(ahiaT)Tisaddedtotherighthandsidesofeqns(a)and(a)ThistermwasconsideredbyWALKERandderivedfromthethermodynamicspointofviewbyCHABOCHEEffectsofsuchatermwerediscussedbyWANGandOHNOVCOMPARISONWITHCLASSICALMODELSIfModelIisbasedonrateindependentplasticity,itprovidespiecewiselinearstressstrainhysteresisloopsthatcloseirrespectiveofmeanstressunderuniaxialcyclicloading,asdiscussedinSectionIIIThispredictionisthesameasthemultilayermodelofBESSELINGandthemultisurfacemodelbasedonthefieldofworkhardeningmoduli(MRoz)Here,wecompareModelIwiththesetwoclassicalmodelsVMultilayermodelThemultilayermodelconsistsofparallelelementsthatarestrainedequally(BESSELINC)Ifweassumetheincompressibilityofinelastic(orplastic)deformationineachelement,wehavethefollowingrelationswithrespecttothedeviatoricpartsofstressandstrainrate,sand~:Ms=~~isi,()i=l=~e~f,()where()idenotestheithelement,and~bi(i=,M)arethevolumefractionsoftheelementsLetusassumeHooke'slawandperfectplasticityforefand~ip,respectively:si=Gee,()#ip=H(gi)izisi,()Kinematichardeningrulesforratchettingbehavior,PartIwhereG=E(v),iisanundeterminedscalar,andgiindicatesthefollowingyieldfunctionoftheithelementpossessingtheconstantyieldstressri:sK~()gi=I:SiThen,fromeqns(),wehavesi=G~H(gi)fzisi()Substitutingeqns()and()intotheconsistencyconditiongi=,andnoticingthatLi~~duetothepositivedissipationineachelement(ie,si:~p>),weobtain~i=(~:si)~Therefore,eqn()becomesas=Ge~H(g~):()Ifweintroducesi=~isiandii=~iKi,eqns()and()taketheformsMs=)si,()i=Si:CiKin(~,i):Si,()whereci=GK~and~i=~~i:ai~()Equation()hasthefollowingfeatures:TotalstrainrateecontrolstheevolutionofgiWhensigrowstosatisfytheconditiongi=,thedynamicrecoveryterminthisequationisactivatedOnlytheprojectionofthecontrollingstrainratetothedirectionofgicontributestothedynamicrecoverytermofsi,iftheprojectionispositiveThelasttwofeaturesintheabovewereseenineqn()Inotherwords,ModelIandthemultilayermodelofeqn()havethesamestructureexceptthatdifferentstrainratescontroltheevolutionofinternalstressesIncidentally,VAZANISusedtotalstraintoformulatetheendochronictheoryofviscoplasticity,andKREMPLetalassumedtotalstrainrateforthehardeningterminthegrowthruleoftheequilibriumstressintheirmodelVMultisurfacemodelInordertoderivemultisurfaceformsofnonlinearkinematichardeningrulesinwhichthekinematichardeningvariableatisdecomposedintocomponentsat(i=,,,M)aseqn(),OIaNO

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