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首页 Kinematic hardening model suitable for ratchett…

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

Kinematic hardening model suita…

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 model suitable for ratchetting with steady-statepdf》,可适用于人文社科领域,主题内容包含KinematichardeningmodelsuitableforratchettingwithsteadystateMAbdelKarim,NO符等。

KinematichardeningmodelsuitableforratchettingwithsteadystateMAbdelKarim,NOhno*DepartmentofMechanicalEngineering,NagoyaUniversity,Chikusaku,Nagoya,JapanReceivedinfinalrevisedformAprilAbstractAnewkinematichardeningmodelusefulforsimulatingthesteadystateinratchettingisdevelopedwithintheframeworkofthestrainhardeninganddynamicrecoveryformatThemodelisformulatedtohavetwokindsofdynamicrecoveryterms,whichoperateatalltimesandonlyinacriticalstate,respectivelyThemodelisexaminedonthebasisofnonproportionalexperimentsofModifiedCr–MosteelatCandINLCatCTheexperimentsincludemultiaxial,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–MosteelatC(Tanakaetal,),INLCatC(Ziebsetal,),carbonsteelsatroomtemperatureinthestabilizedstateofcyclichardening(HassanandKyriakides,Hassanetal,),andsoforthMoreoversteadystateratchettingcanberegardedasanidealizedorapproximatedmodeforsomematerialsifratchettingratedoesnotchangegreatlyForexample,uniaxialratchettingofstainlesssteelandFRsteelatroomtemperaturedidnotdecaysignificantlyunderarelativelylargenumberofstresscyclingexceptfortheearlycyclesinwhichviscoplasticitywasamaindrivingforceforratchetting(Yoshida,Mizunoetal,),sothatthesteadystateinratchettingmayberegardedasprevailingapproximatelyinsuchmaterialsItisthereforeworthwhiletoconsidersteadystateratchettinginthecontextofconstitutivemodelingofcyclicplasticityInthiswork,forthepurposeofsimulatingthesteadystateinratchetting,theArmstrongandFrederickmodelisfurtherfurnishedwiththecriticalstateofdynamicrecoveryintroducedbyOhnoandWang()Theresultingmodel,whichhastwokindsofdynamicrecoveryterms,iscombinedwithaviscoplasticequationandappliedtosimulatingnonproportionalexperimentsofModifiedCr–MosteelatC(Tanakaetal,)andINLCatC(Ziebsetal,)Itisthusshownthatthekinematichardeningmodeldevelopedhasthecapabilityofrepresentingproperlythesteadystateinmultiaxial,aswellasuniaxial,ratchettingConstitutivemodelItisassumedthatstrain"isdividedadditivelyintoelasticpart"eandinelasticpart"p,"ˆ"e‡"p…†andthattheelasticpartobeysHooke’slaw"eˆ‡EÿE…tr†I…†MAbdelKarim,NOhnoInternationalJournalofPlasticity()–whereEandindicateelasticconstants,andIstresstensorandtheunittensorofranktwo,respectively,andtrthetraceFortheinelasticpart,weemployaviscoplasticequationwithbackstress,":pˆf…eff†sÿaeff…†wheresandaindicatethedeviatoricpartsofstressandbackstressrespectively,andf…eff†representsamaterialfunctionofe€ectivestresseffdefinedaseffˆ…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:iˆiri":pÿiaip:ÿH…fi†hl:iiai…†MAbdelKarim,NOhnoInternationalJournalofPlasticity()–whereiandiarematerialparameters,andl:iisdeterminedtohavethefollowingformusingtheconsistencyconditionf:iˆ:l:iˆ":p:airiÿip::…†ThesecondandthirdtermsintherighthandsideinEq()expressthedynamicrecoveryofaibasedontheArmstrongandFrederickmodelandtheOhnoandWangmodel,respectively,whereasthefirsttermisresponsibleforstrainhardeningFeaturesofkinematichardeningmodelAsaforementioned,Eq()isasuperpositionofthetwomodels,theArmstrongandFrederickmodela:iˆiri":pÿaip:…†andthefirstversionoftheOhnoandWangmodela:iˆiri":pÿH…fi†":p:airiai:…†ThematerialparametertocombinethetwomodelsisiinEq()Letusdiscussthee€ectofibyconsideringthetwocasesofiˆandiˆIfiˆ,Eq()isreducedtoEq(),inwhichthedynamicrecoveryofaioccursonlyinthecriticalstatefiˆOhnoandWang()showedthatEq()givesmuchlessratchettingunderbothmultiaxialanduniaxialcyclicloadingconditionsthantheArmstrongandFrederickmodelespeciallyunderuniaxialcyclicloading,Eq()describesnoratchettingexceptforthee€ectofviscoplasticityOntheotherhand,ifiˆ,Eq()becomesidenticaltoEq(),ietheArmstrongandFrederickmodelThisisbecausethefirstandsecondtermsintherighthandsideinEq()allowaitoapproachthecriticalsurfacefiˆasymptotically,sothatthedynamicrecoverytermbasedontheOhnoandWangmodelneverbecomesactiveinEq()(OhnoandAbdelKarim,)NowwerememberthattheArmstrongandFrederickmodelgivesingeneralverysignificantratchetting,aswasstatedintheIntroductionItisthensuggestedthatifalargervalueisassignedtoi,Eq()predictsmoresignificantratchettingIncidentally,Eq()withiyieldsnouniaxialratchettingexceptforthee€ectofviscoplasticity,ashasbeenstatedaboveItis,however,noticedthatmultiaxialratchettingdoesoccurevenwheniˆinEq()Thisisbecausemultiaxialanduniaxialratchettingsarecausedbydi€erentmechanismsiemultiaxialratchettingisaresultofinelasticflowtakingplaceinthedirectionofsÿawhereasuniaxialratchettingisinducedbyhysteresisloopopeningexceptforthee€ectofviscoplasticityMAbdelKarim,NOhnoInternationalJournalofPlasticity()–DeterminationofmaterialparametersModifiedCr–MosteelatCandINLCatCexhibitednegligibleisotropichardeningsofteninguntilthenumberofcyclesbecamerelativelylarge(Tanakaetal,Ziebsetal,)HenceletusignoreisotropichardeningsofteningforsimplicityThenthematerialparametersintheconstitutivemodeldescribedinSectioncanbedeterminedonthebasisofmonotonictensileexperimentsSinceEq()hasauniaxialexpression":pˆf…ÿ†,viscoplasticflowstressinuniaxialtensiledeformationatconstantstrainratecanbeexpressedapproximatelyinaformfÿ…":†‡…"p†…†wherefÿindicatestheinversefunctionoff,and…"p†denotestheevolutionofbackstressintensiledeformationUsingEqs()and(),…"p†isrepresentedas(OhnoandAbdelKarim,)ˆXMiˆriÿhÿÿexp…ÿii"p†ii:…†Asi!,theaboverelationbecomesˆXMiˆriÿhÿi"pi‰Š…†whichismultilinearandhascorners,asillustratedinFigThus,inthecaseofiˆiandriarerelatedwiththecoordinatesofthecorners,…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,thevaluesofiandriaredeterminedtentativelybyuseofEqs()and()moreover,usingEqs()and(),thefunctionalformoff…ÿ†canbesoughtThevaluesofiandriandtheformoff…ÿ†arethenadjustedtofitbestmonotonictensileexperimentsatconstantstrainratesThisprocedureisvalidifiissmallandinfluenceslittleonsimulatingmonotonictensileexperimentsMAbdelKarim,NOhnoInternationalJournalofPlasticity()–ApplicationoftheprocedureabovetothetensileexperimentsofModifiedCr–MosteelatCandINLCatCshowninFigsandresultedinthematerialparametersgiveninTablesandUsingtheseparametersalongwithiˆthetensileexperimentsaresimulatedwell,asdepictedbythesolidlinesinthefiguresFigshowsthee€ectofionthemonotonictensilerelationscomputedusingFigChangeofbackstressanditspartsunderuniaxialtensileloadinginthecaseofiˆ(Mˆ)FigTensilestress–straincurvesofmodifiedCr–MosteelatCatconstantstrainratesMAbdelKarim,NOhnoInternationalJournalofPlasticity()–allotherparameterslistedinTablesandItisseenfromthefigurethatthee€ectisfairlysmallevenifiˆInthefollowingsection,iwillbetakentobeaparametertoinvestigatethee€ectofthetwokindsofdynamicrecoverytermsinEq()onratchettingandcyclicstressrelaxationItwillbethusshownthatsmallvaluesofiareappropriateforthetwomaterialsFigTensilestress–straincurvesofINLCatCatconstantstrainratesTableViscoplasticfunctionandmaterialparametersforModifiedCr–MosteelatCaElasticE=,v=Viscoplasticf…eff†=ÿ(eff)=,r==,r=Kinematichardening=,r==,r=(Mˆ)=,r==,r==,r==,r=aStress(MPa),strain(mmmm),time(s)TableViscoplasticfunctionandmaterialparametersforINLCatCaElasticE=,v=Viscoplasticf…eff†=ÿsinh(eff=)=,r==,r=Kinematichardening=,r==,r=(Mˆ)=,r==,r==,r==,r=aStress(MPa),strain(mmmm),time(s)MAbdelKarim,NOhnoInternationalJournalofPlasticity()–ResultsofsimulationofexperimentsThissectiondescribestheresultsofsimulationofexperimentsofModifiedCr–MosteelatC(Tanakaetal,)aswellasINLCatC(Ziebsetal,)TheexperimentswereperformedbyemployingthinwalledtubularspecimensFromnowon,and"indicatetensilestressandstrain,andandshearstressandstrain,respectivelymoreover,Ndenotesthenumberofcycles,andtensilebackstressequalto()azz,wherezindicatestheaxialcoordinateoftubularspecimensModifiedCr–MosteelatCFigs(a)–(d)dealwithmultiaxialratchettingundercombinedconstanttensilestressandcyclicshearstrainingofModifiedCr–MosteelatCTheexperimentsinthefiguresweredonebyprescribingfoursetsofconstanttensilestressandcyclicshearstrainrangeataconstantshearstrainrateof:jj=pˆÿ=sAsseenfromthefigures,alltheexperimentsaresimulatedverywellifthematerialparameteriistakentobeItisemphasizedthatthisvalueofienablesustosimulateproperlythesteadystateinratchettinginalltheexperimentsTheresultsofsimulationwithrespecttoiˆandiˆarealsogiveninthefiguresIfiˆ,theexperimentsareunderestimated,andthesteadystateinratchettingisnotexpressedespeciallywhenisrelativelysmallFig(a)and(b)Thisisbecausethedynamicrecoveryofaionlyinthecriticalstatefiˆmayallowtensilebackstresstoapproachwiththeincreaseoftensile,ratchettingstrainseeFig(b)belowIfiˆ,thesteadystateispredictedbutoverpredictedbecausethedynamicrecoveryaiistoomuchaccordingtotheArmstrongandFrederickmodelFigE€ectofionuniaxialtensilecurvesMAbdelKarim,NOhnoInternationalJournalofPlasticity()–FigMultiaxialratchettingofModifiedCr–MosteelatCundercombinedconstanttensilestressandcyclicshearstrainingatshearstrainrateof:jj=pˆÿ=sMAbdelKarim,NOhnoInternationalJournalofPlasticity()–Wethereforecansayasfollows:InEq()itisthedynamicrecoverytermbasedontheArmstrongandFrederickmodel,iiaip:,thatisresponsibleforexpressingthesteadystateinratchetting,butsuchasmallvalueofiasisappropriateforrepresentingproperlythesteadystateobservedexperimentallyInotherwords,asmallamountofdynamicrecoveryofeachaiinsidethecriticalsurfacefiˆleadstorealisticsimulationofmultiaxialratchettingofModifiedCr–MosteelatCFigUniaxialratchettingofModifiedCr–MosteelatCMAbdelKarim,NOhnoInternationalJournalofPlasticity()–Figs(a)and(b)compareuniaxialratchettingexperimentswiththeresultsofsimulationpredictedbyassumingthethreevaluesofidiscussedintheaboveIntheexperimentsstresswascycledbetweenmaximumandminimumvalues,maxandmin,ataconstantstressrateof:jj=MPasItisseenfromthefigurethatthevalueofFigMultiaxialratchettingofINLCatCundercombinedconstanttensilestressandcyclicshearstraining:(a)increaseoftensilestrain,(b)evolutionoftensilebackstressMAbdelKarim,NOhnoInternationalJournalofPlasticity()–iˆ:,whichhasbeenfounde€ectiveforthemultiaxialratchettingexperiments,givesfairlygoodagreementstotheuniaxialratchettingexperiments,tooThisimpliesthatthekinematichardeningmodelexpressedasEq()mayhavegoodperformanceforbothuniaxialandmultiaxialratchettingwithoutchangingthevalueofiFigMultiaxialstressrelaxationofINLCatCundercyclicshearstrainingfollowingtensileprestrain:(a)relaxationoftensilestress,(b)changeoftensilebackstressMAbdelKarim,NOhnoInternationalJournalofPlasticity()–INLCatCFigsanddealwithmultiaxialratchettingandmultiaxialcyclicstressrelaxationtestsofINLCatC,respectivelyInthemultiaxialratchettingtest,aconstanttensilestressof=MPawascombinedwithcyclicshearstraining,leadingtothedevelopmentoftensilestrain"withtheincreaseofNInthemultiaxialcyclicstressrelaxationtest,ontheotherhand,atensileprestrainof"ˆ:wasfollowedbycyclicshearstraining,resultinginthecyclicrelaxationofthetensilestressinducedbythetensileprestrainThecyclicshearstrainingwasprescribedtobeidenticalinthetwotestsie=p=and:jj=p=sTherefore,sincethetwotestsarethesameaseachotherexceptforthetensileloadingconditions,weexpectthatthetwotestscanbesimulatedwellbyassumingonesetofmaterialparametersLetusdiscussfirstmultiaxialratchettingTheincreaseoftensilestrainandtheevolutionoftensilebackstressareshowninFigs(a)and(b),respectively,wheretheresultsofsimulationaregivenwithrespecttothreevaluesofiIfiˆ,themultiaxialratchettingceaseseventuallyfromdeveloping,becausetensilebackstressgrowsuntilitbecomesequaltotheprescribedtensilestressof=MPa,asshowninFig(b)ItisobviousfromEq()thatwhenÿˆnofurtherincreaseoftensileratchettingstrainoccursIfiˆ,tensilestrainincreasestoorapidlyincomparisonwiththeexperimentalresultsincetheevolutionofsaturatesatalevelmuchlowerthan=MPaasaconsequenceoftoolargedynamicrecoveryofaiexpressedbytheArmstrongandFrederickmodelIfi=,theincreaseoftensilestrainataconstantrateintheexperimentalresultisreproducedwell,andapproachesalevelwhichisalittlelowerthan=MPaFigCyclicstrainingalongbutterflytypestrainpathMAbdelKarim,NOhnoInternationalJournalofPlasticity()–NowwediscusscyclicstressrelaxationFigs(a)and(b)Ifiˆ,thedecreaseoftensilestressstopsataboutMPabecausehardlyrelaxesfromaboutMPainduced

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