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首页 Kinematic hardening model suitable for ratchetti…

Kinematic hardening model suitable for ratchetting with steady-state.PDF

Kinematic hardening model suita…

天涯囚徒
2012-07-05 0人阅读 0 0 0 暂无简介 举报

简介:本文档为《Kinematic hardening model suitable for ratchetting with steady-statepdf》,可适用于人文社科领域

KinematichardeningmodelsuitableforratchettingwithsteadystateMAbdelKarim,NOhno*DepartmentofMechanicalEngineering,NagoyaUniversity,Chikusaku,Nagoya,JapanReceivedinfinalrevisedformAprilAbstractAnewkinematichardeningmodelusefulforsimulatingthesteadystateinratchettingisdevelopedwithintheframeworkofthestrainhardeninganddynamicrecoveryformatThemodelisformulatedtohavetwokindsofdynamicrecoveryterms,whichoperateatalltimesandonlyinacriticalstate,respectivelyThemodelisexaminedonthebasisofnonproportionalexperimentsofModifiedCr–Mosteelat�CandINLCat�CTheexperimentsincludemultiaxial,aswellasuniaxial,ratchetting,multiaxialcyclicstressrelaxation,andnonproportionalcyclicstrainingalongabutterflytypestrainpathItisshownthatthemodelissuccessfulinsimulatingtheexperiments,andthatthemodelisfeaturedbythecapabilityofrepresentingappropriatelythesteadystateinratchettingundermultiaxialanduniaxialcyclicloading#ElsevierScienceLtdAllrightsreservedKeywords:BConstitutivebehaviorBCyclicloadingBElastic–viscoplasticmaterialIntroductionSinceratchettingisconcernedwiththeaccumulationofsecondarydeformationproceedingcyclebycycle,itisnoteasytodescribeitaccuratelyItledtopoorcapabilitiesofcyclicplasticitymodelsinsimulatingratchettinguptoabout,aswasreviewedbyOhno(,)EspeciallythewellknownmodelofArmstrongandFrederick(),whichisbasedonthemechanismofstrainhardeninganddynamicrecoveryofbackstress,turnedouttooverpredictratchettingVigorousstudiesinthelastdecadehoweverhavecontributedtoovercomingthedicultyinsimulatingratchettingMostofsuchstudieshavedealtwithmodificationsInternationalJournalofPlasticity()–wwwelseviercomlocateijplas$seefrontmatter#ElsevierScienceLtdAllrightsreservedPII:S()*CorrespondingauthorTel:fax:Emailaddress:ohnomechnagoyauacjp(NOhno)Administrator文本框OhnoKarim模型ofthedynamicrecoverytermintheArmstrongandFrederickmodel(Ohno,)Chaboche()showedthatathresholdfordynamicrecoveryofbackstressise€ectiveforcontrollingratchettinginsimulationOhnoandWang()introducedacriticalstateofdynamicrecovery,andtheyshowedthatthecriticalstateenablesustoexpressnoorlittleratchettingunderuniaxialcyclicloadingwithintheframeworkofthestrainhardeninganddynamicrecoveryformatNonlinearformsofthedynamicrecoverytermwerethendiscussedforsimulatingratchettingappropriately(OhnoandWang,,Chaboche,McDowell,JiangandKurath,JiangandSehitoglu,)DecayingratchettingwasthusdiscussedindetailwithsuccessfromtheviewpointofconstitutivemodelingbyJiangandSehitoglu()andbyJiangandKurath()ThesteadystateinratchettinghoweverhasnotbeentargetedexplicitlyinconstitutivemodelingofcyclicplasticitysofarThesteadystatewasobservedinratchettingexperimentsofModifiedCr–Mosteelat�C(Tanakaetal,),INLCat�C(Ziebsetal,),carbonsteelsatroomtemperatureinthestabilizedstateofcyclichardening(HassanandKyriakides,Hassanetal,),andsoforthMoreoversteadystateratchettingcanberegardedasanidealizedorapproximatedmodeforsomematerialsifratchettingratedoesnotchangegreatlyForexample,uniaxialratchettingofstainlesssteelandFRsteelatroomtemperaturedidnotdecaysignificantlyunderarelativelylargenumberofstresscyclingexceptfortheearlycyclesinwhichviscoplasticitywasamaindrivingforceforratchetting(Yoshida,Mizunoetal,),sothatthesteadystateinratchettingmayberegardedasprevailingapproximatelyinsuchmaterialsItisthereforeworthwhiletoconsidersteadystateratchettinginthecontextofconstitutivemodelingofcyclicplasticityInthiswork,forthepurposeofsimulatingthesteadystateinratchetting,theArmstrongandFrederickmodelisfurtherfurnishedwiththecriticalstateofdynamicrecoveryintroducedbyOhnoandWang()Theresultingmodel,whichhastwokindsofdynamicrecoveryterms,iscombinedwithaviscoplasticequationandappliedtosimulatingnonproportionalexperimentsofModifiedCr–Mosteelat�C(Tanakaetal,)andINLCat�C(Ziebsetal,)Itisthusshownthatthekinematichardeningmodeldevelopedhasthecapabilityofrepresentingproperlythesteadystateinmultiaxial,aswellasuniaxial,ratchettingConstitutivemodelItisassumedthatstrain"isdividedadditivelyintoelasticpart"eandinelasticpart"p,"ˆ"e‡"p…†andthattheelasticpartobeysHooke’slaw"eˆ‡�E�ÿ�E…tr�†I…†MAbdelKarim,NOhnoInternationalJournalofPlasticity()–whereEand�indicateelasticconstants,�andIstresstensorandtheunittensorofranktwo,respectively,andtrthetraceFortheinelasticpart,weemployaviscoplasticequationwithbackstress,":pˆf…�eff†sÿa�eff…†wheresandaindicatethedeviatoricpartsofstress�andbackstress�respectively,andf…�eff†representsamaterialfunctionofe€ectivestress�effdefinedas�effˆ…sÿa†:…sÿa†��=:…†Hereandfromnowon()denotesthedi€erentiationwithrespecttotimet,and(:)theinnerproductbetweensecondranktensorsItisassumedfurtherthatbackstressconsistsofMpartsasfollows(ChabocheandRousselier,):aˆXMiˆai:…†Backstressistheninterpretedusingmultisurfaces(OhnoandWang,)Inordertoexpressthechangeofai,letusconsideracombinationoftheArmstrongandFrederickmodelandthefirstversionoftheOhnoandWangmodelThetwomodelshaveadi€erencewithrespecttothedynamicrecoveryofaiIntheArmstrongandFrederickmodel,thedynamicrecoveryofaioperatesatalltimesinproportiontoaiandaccumulatinginelasticstrainratep:ˆ":p:":p��=:…†InthefirstversionoftheOhnoandWangmodel,ontheotherhand,thedynamicrecoveryofaiisassumedtotakeplaceonlyinacriticalstate,whichistakentobeahypersphereofradiusriinthespaceofaisuchasfiˆai:aiÿriˆ:…†LetusassumethetwokindsofdynamicrecoverytermsmentionedaboveThen,withHeaviside’sstepfunctionHandMacaulay’sbrackethi,theevolutionequationofaicanbeexpressedasa:iˆ�iri":pÿ�iaip:ÿH…fi†hl:iiai��…†MAbdelKarim,NOhnoInternationalJournalofPlasticity()–where�iand�iarematerialparameters,andl:iisdeterminedtohavethefollowingformusingtheconsistencyconditionf:iˆ:l:iˆ":p:airiÿ�ip::…†ThesecondandthirdtermsintherighthandsideinEq()expressthedynamicrecoveryofaibasedontheArmstrongandFrederickmodelandtheOhnoandWangmodel,respectively,whereasthefirsttermisresponsibleforstrainhardeningFeaturesofkinematichardeningmodelAsaforementioned,Eq()isasuperpositionofthetwomodels,theArmstrongandFrederickmodela:iˆ�iri":pÿaip:��…†andthefirstversionoftheOhnoandWangmodela:iˆ�iri":pÿH…fi†":p:airi��ai��:…†Thematerialparametertocombinethetwomodelsis�iinEq()Letusdiscussthee€ectof�ibyconsideringthetwocasesof�iˆand�iˆIf�iˆ,Eq()isreducedtoEq(),inwhichthedynamicrecoveryofaioccursonlyinthecriticalstatefiˆOhnoandWang()showedthatEq()givesmuchlessratchettingunderbothmultiaxialanduniaxialcyclicloadingconditionsthantheArmstrongandFrederickmodelespeciallyunderuniaxialcyclicloading,Eq()describesnoratchettingexceptforthee€ectofviscoplasticityOntheotherhand,if�iˆ,Eq()becomesidenticaltoEq(),ietheArmstrongandFrederickmodelThisisbecausethefirstandsecondtermsintherighthandsideinEq()allowaitoapproachthecriticalsurfacefiˆasymptotically,sothatthedynamicrecoverytermbasedontheOhnoandWangmodelneverbecomesactiveinEq()(OhnoandAbdelKarim,)NowwerememberthattheArmstrongandFrederickmodelgivesingeneralverysignificantratchetting,aswasstatedintheIntroductionItisthensuggestedthatifalargervalueisassignedto�i,Eq()predictsmoresignificantratchettingIncidentally,Eq()with�iyieldsnouniaxialratchettingexceptforthee€ectofviscoplasticity,ashasbeenstatedaboveItis,however,noticedthatmultiaxialratchettingdoesoccurevenwhen�iˆinEq()Thisisbecausemultiaxialanduniaxialratchettingsarecausedbydi€erentmechanismsiemultiaxialratchettingisaresultofinelasticflowtakingplaceinthedirectionofsÿawhereasuniaxialratchettingisinducedbyhysteresisloopopeningexceptforthee€ectofviscoplasticityMAbdelKarim,NOhnoInternationalJournalofPlasticity()–DeterminationofmaterialparametersModifiedCr–Mosteelat�CandINLCat�Cexhibitednegligibleisotropichardeningsofteninguntilthenumberofcyclesbecamerelativelylarge(Tanakaetal,Ziebsetal,)HenceletusignoreisotropichardeningsofteningforsimplicityThenthematerialparametersintheconstitutivemodeldescribedinSectioncanbedeterminedonthebasisofmonotonictensileexperimentsSinceEq()hasauniaxialexpression":pˆf…�ÿ�†,viscoplasticflowstressinuniaxialtensiledeformationatconstantstrainratecanbeexpressedapproximatelyinaform��fÿ…":†‡�…"p†…†wherefÿindicatestheinversefunctionoff,and�…"p†denotestheevolutionofbackstress�intensiledeformationUsingEqs()and(),�…"p†isrepresentedas(OhnoandAbdelKarim,)�ˆXMiˆriÿhÿÿexp…ÿ�i�i"p†�ii��:…†As�i!,theaboverelationbecomes�ˆXMiˆriÿhÿ�i"pi‰Š…†whichismultilinearandhascorners,asillustratedinFigThus,inthecaseof�iˆ�iandriarerelatedwiththecoordinatesofthecorners,�…i†and"p…i†,asfollows(JiangandKurath,JiangandSehitoglu,):�iˆ"p…i†…†riˆ�…i†ÿ�…iÿ†"p…i†ÿ"p…iÿ†ÿ�…i‡†ÿ�…i†"pP…i‡†ÿ"p…i†"#"p…i†…†where"p…†and�…†ˆThen,byapproximatingmultilinearlyatensilestress–straincurveataconstantstrainrate,andbynoticingthatbackstressisresponsiblefortheincreaseoftensilestressbeyondproportionallimit,thevaluesof�iandriaredeterminedtentativelybyuseofEqs()and()moreover,usingEqs()and(),thefunctionalformoff…�ÿ�†canbesoughtThevaluesof�iandriandtheformoff…�ÿ�†arethenadjustedtofitbestmonotonictensileexperimentsatconstantstrainratesThisprocedureisvalidif�iissmallandinfluenceslittleonsimulatingmonotonictensileexperimentsMAbdelKarim,NOhnoInternationalJournalofPlasticity()–ApplicationoftheprocedureabovetothetensileexperimentsofModifiedCr–Mosteelat�CandINLCat�CshowninFigsandresultedinthematerialparametersgiveninTablesandUsingtheseparametersalongwith�iˆthetensileexperimentsaresimulatedwell,asdepictedbythesolidlinesinthefiguresFigshowsthee€ectof�ionthemonotonictensilerelationscomputedusingFigChangeofbackstress�anditspartsunderuniaxialtensileloadinginthecaseof�iˆ(Mˆ)FigTensilestress–straincurvesofmodifiedCr–Mosteelat�CatconstantstrainratesMAbdelKarim,NOhnoInternationalJournalofPlasticity()–allotherparameterslistedinTablesandItisseenfromthefigurethatthee€ectisfairlysmallevenif�iˆInthefollowingsection,�iwillbetakentobeaparametertoinvestigatethee€ectofthetwokindsofdynamicrecoverytermsinEq()onratchettingandcyclicstressrelaxationItwillbethusshownthatsmallvaluesof�iareappropriateforthetwomaterialsFigTensilestress–straincurvesofINLCat�CatconstantstrainratesTableViscoplasticfunctionandmaterialparametersforModifiedCr–Mosteelat�CaElasticE=�,v=Viscoplasticf…�eff†=�ÿ(�eff)�=�,r=�=�,r=Kinematichardening�=�,r=�=�,r=(Mˆ)�=�,r=�=�,r=�=�,r=�=�,r=aStress(MPa),strain(mmmm),time(s)TableViscoplasticfunctionandmaterialparametersforINLCat�CaElasticE=�,v=Viscoplasticf…�eff†=�ÿsinh(�eff=)�=�,r=�=�,r=Kinematichardening�=�,r=�=�,r=(Mˆ)�=�,r=�=�,r=�=�,r=�=�,r=aStress(MPa),strain(mmmm),time(s)MAbdelKarim,NOhnoInternationalJournalofPlasticity()–ResultsofsimulationofexperimentsThissectiondescribestheresultsofsimulationofexperimentsofModifiedCr–Mosteelat�C(Tanakaetal,)aswellasINLCat�C(Ziebsetal,)TheexperimentswereperformedbyemployingthinwalledtubularspecimensFromnowon,�and"indicatetensilestressandstrain,and�andshearstressandstrain,respectivelymoreover,Ndenotesthenumberofcycles,and�tensilebackstressequalto()azz,wherezindicatestheaxialcoordinateoftubularspecimensModifiedCr–Mosteelat�CFigs(a)–(d)dealwithmultiaxialratchettingundercombinedconstanttensilestressandcyclicshearstrainingofModifiedCr–Mosteelat�CTheexperimentsinthefiguresweredonebyprescribingfoursetsofconstanttensilestress�andcyclicshearstrainrange�ataconstantshearstrainrateof:jj=pˆ�ÿ=sAsseenfromthefigures,alltheexperimentsaresimulatedverywellifthematerialparameter�iistakentobeItisemphasizedthatthisvalueof�ienablesustosimulateproperlythesteadystateinratchettinginalltheexperimentsTheresultsofsimulationwithrespectto�iˆand�iˆarealsogiveninthefiguresIf�iˆ,theexperimentsareunderestimated,andthesteadystateinratchettingisnotexpressedespeciallywhen�isrelativelysmallFig(a)and(b)Thisisbecausethedynamicrecoveryofaionlyinthecriticalstatefiˆmayallowtensilebackstress�toapproach�withtheincreaseoftensile,ratchettingstrainseeFig(b)belowIf�iˆ,thesteadystateispredictedbutoverpredictedbecausethedynamicrecoveryaiistoomuchaccordingtotheArmstrongandFrederickmodelFigE€ectof�ionuniaxialtensilecurvesMAbdelKarim,NOhnoInternationalJournalofPlasticity()–FigMultiaxialratchettingofModifiedCr–Mosteelat�Cundercombinedconstanttensilestressandcyclicshearstrainingatshearstrainrateof:jj=pˆ�ÿ=sMAbdelKarim,NOhnoInternationalJournalofPlasticity()–Wethereforecansayasfollows:InEq()itisthedynamicrecoverytermbasedontheArmstrongandFrederickmodel,�i�iaip:,thatisresponsibleforexpressingthesteadystateinratchetting,butsuchasmallvalueof�iasisappropriateforrepresentingproperlythesteadystateobservedexperimentallyInotherwords,asmallamountofdynamicrecoveryofeachaiinsidethecriticalsurfacefiˆleadstorealisticsimulationofmultiaxialratchettingofModifiedCr–Mosteelat�CFigUniaxialratchettingofModifiedCr–Mosteelat�CMAbdelKarim,NOhnoInternationalJournalofPlasticity()–Figs(a)and(b)compareuniaxialratchettingexperimentswiththeresultsofsimulationpredictedbyassumingthethreevaluesof�idiscussedintheaboveIntheexperimentsstresswascycledbetweenmaximumandminimumvalues,�maxand�min,ataconstantstressrateof�:jj=MPasItisseenfromthefigurethatthevalueofFigMultiaxialratchettingofINLCat�Cundercombinedconstanttensilestressandcyclicshearstraining:(a)increaseoftensilestrain,(b)evolutionoftensilebackstressMAbdelKarim,NOhnoInternationalJournalofPlasticity()–�iˆ:,whichhasbeenfounde€ectiveforthemultiaxialratchettingexperiments,givesfairlygoodagreementstotheuniaxialratchettingexperiments,tooThisimpliesthatthekinematichardeningmodelexpressedasEq()mayhavegoodperformanceforbothuniaxialandmultiaxialratchettingwithoutchangingthevalueof�iFigMultiaxialstressrelaxationofINLCat�Cundercyclicshearstrainingfollowingtensileprestrain:(a)relaxationoftensilestress,(b)changeoftensilebackstressMAbdelKarim,NOhnoInternationalJournalofPlasticity()–INLCat�CFigsanddealwithmultiaxialratchettingandmultiaxialcyclicstressrelaxationtestsofINLCat�C,respectivelyInthemultiaxialratchettingtest,aconstanttensilestressof�=MPawascombinedwithcyclicshearstraining,leadingtothedevelopmentoftensilestrain"withtheincreaseofNInthemultiaxialcyclicstressrelaxationtest,ontheotherhand,atensileprestrainof"ˆ:wasfollowedbycyclicshearstraining,resultinginthecyclicrelaxationofthetensilestressinducedbythetensileprestrainThecyclicshearstrainingwasprescribedtobeidenticalinthetwotestsie�=p=and:jj=p=sTherefore,sincethetwotestsarethesameaseachotherexceptforthetensileloadingconditions,weexpectthatthetwotestscanbesimulatedwellbyassumingonesetofmaterialparametersLetusdiscussfirstmultiaxialratchettingTheincreaseoftensilestrainandtheevolutionoftensilebackstress�areshowninFigs(a)and(b),respectively,wheretheresultsofsimulationaregivenwithrespecttothreevaluesof�iIf�iˆ,themultiaxialratchettingceaseseventuallyfromdeveloping,becausetensilebackstress�growsuntilitbecomesequaltotheprescribedtensilestressof�=MPa,asshowninFig(b)ItisobviousfromEq()thatwhen�ÿ�ˆnofurtherincreaseoftensileratchettingstrainoccursIf�iˆ,tensilestrainincreasestoorapidlyincomparisonwiththeexperimentalresultsincetheevolutionof�saturatesatalevelmuchlowerthan�=MPaasaconsequenceoftoolargedynamicrecoveryofaiexpressedbytheArmstrongandFrederickmodelIf�i=,theincreaseoftensilestrainataconstantrateintheexperimentalresultisreproducedwell,and�approachesalevelwhichisalittlelowerthan�=MPaFigCyclicstrainingalongbutterflytypestrainpathMAbdelKarim,NOhnoInternationalJournalofPlasticity()–NowwediscusscyclicstressrelaxationFigs(a)and(b)If�iˆ,thedecreaseoftensilestress�stopsataboutMPabecause�hardlyrelaxesfromaboutMPainduced

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