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首页 general theory of uniqueness and stability in e…

general theory of uniqueness and stability in elastic-plastic solids.pdf

general theory of uniqueness an…

silly 2010-05-24 评分 0 浏览量 0 0 0 0 暂无简介 简介 举报

简介:本文档为《general theory of uniqueness and stability in elastic-plastic solidspdf》,可适用于工程科技领域,主题内容包含AGENERALTHEORYOFUNIQUENESSiNl)STABILITYINELASTICPLASTICSOLIDSSCOlCJlr‘I’rl符等。

AGENERALTHEORYOFUNIQUENESSiNl)STABILITYINELASTICPLASTICSOLIDSSCOlCJlr‘I’rl:‘JJ:theoryofpartlypIastitsolitlsisc~spc~iallyc~tlc*crnc~lwiththehIlowingboulldnr~XlWproblenAtayencric~stageinaproccsofcluasistnticdistortionthemrrcntshapeofthebodyalldtheinternalclistributionofstrmsarc*suIqxMxtoIinebcchdetcrmineclalready,togetherwiththestingstateofhardeningaudmechanicApropertiesgcncrally‘I’hcillrremcntalchangesinallthcscvariableshavenowtoecalculatedforafurtherinlinitcsimalvariationofthesurfaceloadsandgeometricalconstraintsVhcnthetotalstrainremainssmall,:lJldthepositionalchangesandrotatiotwofmaterialclcmentsareneglected,thebol~~~t~ar~~al~~eprobfcmaiwayshasauniquesolutionwhenthe~vorkl~~~r~leIlingis~r~oIlot(~~~icandtheyieldfunctionandplasticpotenti~~lareidcnt,icat(XELASI!>asomewhatmorerigorousandgeneralproofcanbefoundiuHILLI!o,ppG)However,thereallytypicalplastic*problemsinvolvechangesingeometrythatcannotbedisregarded,andherethequestionofuniquenesshasremainedopenuntilveryrecently,when:nianswerwasgivenforrigidplasticsolids(HILLSa)Broadlyspeaking,itisfoundthatauniquesolutioniscertainonlywhentherateofhardeningexceedsadeterminablecritical‘Llue,dependingontheparticularsituationTnthepresentpaperasimilarresultisprovedforanextensiveclassofelasticplasticsolids,includin,ametals,withmechanicalpropertieswiderinimportantrespectsthanthoseusuallycontemplatedxover,thesolution,whenitisunique,ischaracterizedbyancxtremumprinciple,analogoustothatforrigiclplasticsolids(HILLSb)AgeneraltheoryofuniquenessandstabilityinelasticplasticsolidsInelasticsolidstheproblemofinstabilityiscloselyrelatedtothatofuniqueness(HILL~)inplasticsolidstheconnexionislooser,butstillsignificantForthisreasonwealsoinvestigatethequestionofstabilityhere,andtreatitfromthemostgeneralstandpoint,namelyviatheenergycriterionPreviousworkisunsatisfactoryinthateitherthemerhanicalproperties,thefieldequationsortheboundaryconditions,areoftennot,properlyformulatedforasituationwhcrcrotationsofmaterialelementsarelargecomparedwiththestrainsFurthermore,theoccurrenceofinstabilityhasinvariablybeentakenassynonymouswiththeexistenceofinfinitesimallynearpositionsofequilibriumthismaybequiteunjustifiedwhenthesystemisnoninearornonconservativeForthesereasonsaloneitishardlysurprisingthatthtoreticalpredictionsoftheonsetofinstabilityshouldoftenhavedisagreedmarkedlywitheachotherandwithexperimentItisnot,however,thepresentintentiontoreviewaconfusedliteraturenortoattemptanycorrelationwithexperimentbuttomakeafreshstartandestablishabroadbasictheoryfreeatleastfromtheobjectionsmcntioncdRsrsFORAmr~YsrsOFSTRESSThetypicaltractionboundaryconditionconsideredhereisthatthechangeintheIoadrectoronaninfinitesimalsurfaceelementisassigned,irrespectiveofchangesinitsareaandorientationThus,atanystagetduringaquasistaticprocessofdeformationthereisgivennottherateofchangeofthe‘true’tractionbuttherateofchangeofthe‘nominal’tractionbasedontheconfigurationatthatinstantToformulatethisboundaryconditionweintroducetheunsymmetrical‘nominalstress’tensorszj(t)associatedwithcertainfixedrectangularaxesAtanystagetStsubsequenttot,(sijSSij)AS’isthejthcomponentoftheforceontheplaneelementwhichattwasperpendiculartotheithaxisandofinfinitesimalareaAS’Correspondingly,theforceattStonaplaneareawhosevectorialareawaslidSatthascomponents(FiFj)dSwhereFj=isij,SFi:=liSs,Fiisthe’nominaltraction’basedontheconfigurationattTheboundaryconditionisthenthatpj=i,()isprescribedInequilibriumandintheabsenceofbodyforceswherex,marksthepositionofelementsatstagetTheangularequationsofequilibriumaresatisfiedthroughtheassociationofsiiwiththesymmetric‘truestress’tensoraijoreferredtothesameaxesIfbuOdenotesitsrateofchangefollowingthesameelement,whereviisthevelocityThisisanimmediatespecializationofMurnaghan’swellknownformulaconnectingthetrueandnominalstresstensorsRHmNowthematerialpropesfoundbycsl~~rirrtentarcgiten,inthtfirstinstanwintermsofthet,rucstresstensoroii~swt~dwithrectangularaxesrotatingwiththeelement,andsohavingspinfurlvIomthetransformationrukfortensorsitiseasilyshownthatwherenInanclementwhoseresponset,oanyfurtheril~finitrsimalc~hangeofstressispllfrlyclastir(ieanclementthathascithcrncvryirldcdorisnowfXHIylo~tiafterbeingplastir)thrisothermalrelatioikslwrenstressrateandstzailtratearctakentobel~c~ln~~ell~(~usandlinear,antarcwrittrtlasAgeneraltheoryofuniquenessandstabilityinelasticplasticsolidsrecoverablestrainisfinite,andindeedthereisnoneedsotorestrictKinthrsubsequentanalysis*Foraplast,icelementthesimplesthypothesisisirJ=K(EEP)()wherecpistheplast,icpartofthestrainrate,andKisasymmetricmatrixdependingonlyonstrainhistorySupposethatEP(“In(Bo)nwhenn(i’)>GOwherehisapositivescalarmeasureofthecurrentrateofworkhardeninganddependsoncraswellast,hestrainhistoryThe‘direction’oftheplasticstrainrateisspecifiedbytheunitvectorn,whichalsodependsoncandstrainhistorynisstipulatedtobedeviatoricsothatthereisnopermanentvolumechangeSinceBisdirectlyexpressibleintermsofstressratethrought,heelasticmoduli,theequationn(i’~)=cffertivelydefinesthosestressratevectorsthataretangentialtotheyieldsurfaceatthecurrentstresspoint,andtherebythesurfaceclementThisdoesnothavenasitsnormalandsoyieldingisaffectedbythehydrostaticpartofthetruestressForpresentpurposesnorestrictionsneedbeplacbedontheshapeofthesurface,andinparticularwedonotmakethecustomaryassumptionofconvexityNotethat,inviewofthedefinitionofir,equation()conformstotheminimumrequirementthatitissatisfiedidenticallybyarigidbodyspinoftheelementtogetherwiththestressstateToinverttherelations(),with(s),wvcfirstremarkthatwhennh'c=n(Bo)((hlnKn)n(irOff)><oprovidedimlnKn>Thisinequalitywesatisfyn,fortioriforanyh>byrequiringnKn>inaplasticelement()Thatis,theKquadraticformispositivefordevintoricvect,orsthishastheinterpretationthatastrainrateindirectionnconstitutesoadngThen,inaplasticelement,inKc~(nh'ndh)linwhenI,~u=~EnKE>O()<Iftherateofhardeningiszero()isreplacedby(ynwhereY>,whenn(Bu)=Ep=I(’)<*Itisofmorethanacadrmicinterestthattheeventualuniquenesstheoremandextremumprinciplethenapplyalsotosome’unreal’materialsForitcanhappenthataboundaryvalueproblrmissolvableinthefirstinstancemoreeasilyforsomespecialkindofunrealmaterialthissolutioncanthenbeofuseinfindingthatfortherealmaterialRHILLwhile(IO)st,illholds,withh=NowinfacttheonlyelasticplasticsolidsthathavebrcnstudiedexperimentallyarctherommonnlctnlsThrsehareelastic*modulisogreatcomparedwithanystresssupportablebythematerialthatthcrrisperhapsnopointinretainingthetermnorinpostulatinganyparticularstressdependenceofKitselfintheelasticrange,thoughwemaystillenvisagemeasurablechangesinKthetoplasticstraining,espcriallythroughchangesinthrstateofanisotropyEquations()()and()thenreducrtothegenerallyacceptrelations”withnastheunitnormaltotheyieldsurfac>c(nowindepcndcnt~ofliylrostatkstress)Weshalllaterneedtorefertothequantity()whichisafunctionofvelocitygradientequaltoiiiki~by(:)Xorcovcrinviewof()and(),r‘It)=(jEXi,,iii=““(J),ihSi)Thattheleft,handdifferentialispcrfccatcaodalsoherccognisedinadvancefromthesymmetry,withrcsprctointerchangeofijand~ofthematrixofcocfficxicntsintherelationsbetweenkijand‘s,NowctE*,EdenotedistirirtstrainratesandCr*trthecorrespondingstressratesinanelementinagivclrirenditionandstateofstressWcwriteAE~~~E*EandsimilarlyforthediffercnccsofotherstnrrcdandunstarredquantitirsNotethatwhentheclementisplasticAandAErrgartlctlassiqlcactorsdonotnecrssarilycorrespondinthrscnscofkingrclntc~tlhy()I,cwmuIInanelasticclcnlentthVrcisttlciiumr~tliatcitlentit,yA(b‘~Bu)AEACKAC(")Inaplasticselement,forh,Thislemmaisusedinderivingthruniqucncsscrit,crion(Secon)IJroqfTheequalityin()holtlswhenbothstrainratrscallforadditionalloading,from()andthesymmetryofKTh(~inqualit,yholdswhennKE*IoandnKE<l’orthentheleftsitkof()isandthisexreedstherightsideFinally,ifbothstrainrat,csproduceunloadingtheirdifferencesatisfies()andhcncr()nfortiori(withequalityonlywhennKAC=)AgeneraltheoryofuniquenessandstabilityinelasticplasticsolidsLemmalInanelasticelementthereistheeasilyverifiedidentity(~So)AhAEKAe=A(crBo)lInaplasticelement,forh>,()‘L(BBu)AE(nK“”A<KAenKnh<A(cJ)c()Thislemmaisusedinobtainingtheextremumprinciple(Section)weakerinequalitysufticedintheexistingproofofthisprinciplewhengeometrychangesareneglected(Kbeingpositivedefinite):inthateventthesquarebracket,whichispositivebyCauchy’sinequality,isomittedfromtheleftsideof()ProofWhennKE>theleftsideof()isequaltoAEKE!“KErnKnhIby()andthesymmetryofKIfnKE*>thisisjusttherightsideof()whichisthereforeanequalityIf,however,nKE*<otherightsideexceedstheleftby(nKE*):!,‘(nKnh)WhennKE<theleftsideof()isequaltoA(EKE):“)IfnKE*>therightsideof()isA(EKE)whwhichexceedstheleftsince(nKAE)~>(nKE*)~IfnKE<therightsideof()isjustA(cKE)andisnotlessthantheleft(beingequalwhennKAE=O)UNIQUENESSCRITERIONWeregardthecurrentdistributionofstressinabodyasgiven,togetherwiththematerialpropertiesateverypointForsimplicitybodyforcesareomittedsincetheirmodeofinclusionissufticientlyobviousThenominaltractionratePisspecifiedonapartS,ofthecurrentsurfaceandthevelocityvontheremainderSFTheseconditionsandthefieldequations(),(),()and()setaboundaryvalueproblemfortheinternalvelocityfieldSupposethattherecouldbetwodistinctsolutions(notdifferingmerelybyarigidbodymotionwhenSD=)anddenotetheirdifferencebyAvThen,from()and(‘L),withSandVasthepresentsurfaceandvolume,O=A$‘AvdS==lls,A(~)ricJ‘=A(tiBo)AedL’Z(Av)swhere(v)=dvI(Il:IsuflicientconditionforuriiqucncssistlirrdorrthatA(i’’~)ACII’S(Av)’Ageneraltheoryofuni(luenessandStStyinelasticplasticsolidsI’,,theuniquenessconditioncanberearrangedtogiveexplicitlyh>fwhere=Max(gj)inclassw()theubramaximumbeingimpliedbkTHElllUhl)HINC!IPLEWhenthecriterion()issatisfiedtheuniquesolutionischaracterizedbythefoilowingextremumpropertyIntheclassofcontinuousvelocityfieldstakingthegivenvaluesonSothefunctionalE(v)dL’his,J()hasanabsoluteminimurnwhenvJ‘hASAfrom()andtheusualtransformationWithAV:==v*v,wherev*isanydistinctfieldoftheclass,AJ‘tiv=‘PAVtSIbeinggiveninthelefthandintegralFrom()thisbecomesNowby()and()i()AE~VH(A~)<AThereisalsotheidentityCombiningthelastthreeequationsandusing(),whichholdsforw=Avbyhypothesis,AIJ‘($)EdV~(V)fivC>OI‘ThisprovestheextremumprincipleWhengeometrychangesarenegiigiblr(ZandBtermsdiscarded,trueandnominalstressratesnotdistinguished)itreducestoaknownresult(NILL,ppB)Theminimumisalsoanalytic,aswellasabsolute,inthesensethat,foranyinfinitesimalvariationV,thefirstvariationof()vanisheswhenvistheactualsolutionFortheinequalityrestsonthesecondordertermH(v)z’(V)and(byinspectionoftheproofoflemmaII)ontheintegralof(nrif(nKnh)overthatpartoftheplasticzonewhereneitheroftheactualandvaricbtlfieldsproduceloadingForthesamereasons,evenwhenthesolutionisnotuniqueand()isnotKIIILLsatisfied,thevariationprincipleexistsinrespectofeachsolution,thoughtheextremumprinciplemaynotItisessentialtokeepinmindthedistinctionbetweenthefunctionalsin()and()Intlieelasticzoneandinthepartoftheplastic,zonewherenh’E,,(u)EisequaltoEKEOnlyintheloadingpartoftheplasticzonewherenKE>,isitequaltoBycontrast,intheIniquenessfunctiondtheterIncontaininghtiastobeincluded~z~yzereintheplasticzone,irrespectiveofthesignofnKtjFurtherinsightintothec~onnexionbctwc~c~nthecxtremumprincipleanduniqucnesscriterion(abegainedbystudyingthematterfIastillmoregeneralclassofsolidsWCtles~ribethebasicideahercbonlyinoutlineIAthe(nonlinear)relationsbetweennominalstressrateandvelocitygradientbewhereEisafunctionofthevelocitgradientdependingalsoonstrainhistory,withcontinuousfirstderivativesuitlsectiondlycontinuoussecondderivativesSupposingtherecouldbetwodistinc~tsolutionstothel,outld:trvvalIIeprohlc~mAsufkientconditionforuniquenessisttierefurethatforallpairsofcontinuousvelocityfieldstakingtheprcscrihedvaluesonSoUsspecializing()forpairsdifferingonlyinlinit~esitnally,weseethat,it,inlpliescorrespondingtoa,zyvelocitylicttlNow,bytheIllCallvaluethCYJWllltinuousfunctionofvelocitygmdicntwhereAvisnowt,hediffereuceofcrrytwofields,andthebardenotesiivdue’someveocitygradientonthe’join’oftlloscfurtlirtwofieldsFront()atld(l),withtij:A(kjXi),Ageneraltheoryofunicluenessandstabilityinelasticplasticsolids(WhenCijk,isdiscontinuousanextensionoftheconventionalmeanvaluetheoremhastobeinvoked,andacorrespondinginterpretationplacedon())If,now,vistheactualuniquesolutionatransf(jrmationofthelastequationgivestheextremumprinciple()whichhastherebybeenderivedasaconsequenceof()ThisisanextensionofthesystematicprocedureofHILL(a)for’convex’functionsE’,inthat()canbesatisfiedwithouttheintegranditselfbeingalwayspositiveThisextensionwasgiveninessencebyHILL(c,pp)inconnexionwithrubberlikesolidsforwhichEisaquadraticformwhosesecondderivativesarctheelasticmoduliForelasticplasticsolidsEisgivenby()inconjunctionwith()and()itssecondderivativesarediscontinuousforvelocitygradientssuchthatnKE=STABILITYCRITERIONSupposethatapartofthesurfaceofthebodyisrigidlyconstrainedandthatconstantnominaltractions(deadloading)aremaintainedontheremaindert~ou~~lout,anymovementfromthepositionofequilil~riunlatstagetLetubeanarbitraryvirtualdisplacement,,duringthecourseofwhichthenominalstresschangesfromsijtosijS,(stillreferredtostaget)InafurtherinfinitesimaldisplacementdutheincreaseininternalenergyexceedstheworkoftheexternalforcesbytheamountwhereS,Vandxiarethesurface,volumeandpositionatstagetThetotalexcessinreachingthepositionuistheintegralofthisoverthewhole‘path'Asufficientconditionforstabilityatstagetisthereforethat()wheretheinnerintegralistakenalonganypathleadingtoeachgeometricallypossibleconfigurationinfinitesimallyneartheinitialone,Theseconfigurationsneednotofcoursebepositionsofequilibriumsincetheyareenvisagedasbeingattained(ifatall)byfreemotionfolIowingatransitorydynamicaldisturbanceHowever,itmayhappenthatthereisacertainpaththatcanbetraversedquasistaticallyunderthedeadloadsinthiseventtheintegralvanishes,asmaybeshownbydirecttransformationorequivalentlyfromthevirtualworkprincipleOfcoursetheintegralcanvanishinothercircumstancesalsoInaplasticsolidthechangeinstressisnotasinglevaluedfunctionofthetotal~lisplacementgradientandSO(aifonlyforthisreason)theintegralin()isstronglypathdependent=mongthepathsleadingtoanyonefinalpositionthemostcriticalforstabilityisthatwhichcallsforthesmallestincreaseininternalenergyinthebodyasawholeInattemptingtodeterminethisitseemsdesirable,attheoutset,toapproachtheproblembycalculatingtheleastincreaseinenergywIfILlinbringingeachelementofthebody~~~~ara~~il~toitsfinalstateOfdistortionsincbctlliswouldgenerallyviolatethec~ontinuityoftilt:msterial,theafuaIItxstvaluemaybeuntlcrestimetedantitheresultingstabilitycriterionbcOversutiixtIknvever,weshalllatergiveareitsnwhythisisnotthe(~:hereInanelementafterafinitedistortionthereihauniquetriadOftriutunllylx’rpendicul~trdirectionswhichwerealsoperpet~flidnrinitidly,thoughnotncf*cssaril>soduringt,hestrainingThesearethe’principaltlircf~tiotis’inthusc~kntentLetA,(r=,,)hcthennturallogarithmsoftheratiosoft,ltfLlinnlatiflinitiallengthsinthesecliref~tionsTheLinaldistortionc~~ulfltxrcaclifdnlotqapltliat:everystageofwhidithetrue,sttxitir:ttc(basedonthef~onfigurationatthatstage)hasprincipalaxescoinfdingwiththepritif~ipaidireknsantiprincipal~Ol~~I~onentsproportiotialtothe,I,‘Itseemsveryprobablethatthisisactualtheoptimumpath(atalleventsto~e~~~~l~~urj~~states)butarigorOusproofhastttbeenfunflWeadoptithereasahypothf~sistsecondorder,wherepisthedensityItitermsOff*ompOncntsreferredtotftcriaxesthisisCQjt(Sq,‘JijSpjphijwhere,tosecondorder,Xij=l’ijeikejkandfOrtheworkOfflistortionperunitinitidvolurtteonthe

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