能量释放率理论

ChapterThreeEnergyreleaseratetheoryEnergyreleaserateGisanotherimportantconceptinfracturemechanics.ThereisarelationshipbetweenGandK.Oneisknown.Anothercanalsobeknown.Sometime,itisdifficulttoobtainKfactor.However,itisperhapseasytocalculateG.3-1Conceptofenergyreleaserate(1) BasicconceptsStrainenergyWorkdonebytheexternalforceischangedtothestrainenergystoredintheelasticdeformation.Thestrainenergycanbereleasedandtheelasticdeformationthendisappears.Strainenergydensityw:strainenergyperunitvolume.w二Jeij(5*ds*ijij1-gs2ijij,(forlinearelasticbody)TotalstrainenergyU(Internalforcepotential):totalenergystoredinthevolumeVU-JwdV(2) VExternalforcepotentialUP:thenegativevalueofvirtualworkdonebytheexternalforce.AssumethatthebodyforceisBandthesurfaceforceisTonthestressiiUPisTudS)iiboundary.TheexternalforcepotentialU-(JBudV+JTudS)PViiSiigBudV+JViiSgTotalpotential口：1. 口-U+U-JwdV-(JPVEnergyreleaserateGThebodyforceB,thesurfaceforceTonSandthedisplacementuonSiigiuaregiven.Assumethatthecracksizeischangedfromatoa+Aa.Accordingly,thedisplacement,strain,stress,stainenergydensity,internalforcepotential,externalforcepotentialandtotalpotentialarealsochanged.Thetotalpotential口ischangedton+An.AnistheincrementcausedbythecrackgrowthAa.Assumethattheplatethicknessist.AS=t-Aadenotesthesinglesurfaceareaincrement.TheenergyreleaserateGisdefinedas-limAnAa0ASdndSIftheplatethicknesstisaconstant,theenergyreleaserateGisG二lim-=-1dnAaot.AatdaConstantforceandconstantdisplacementconditions(1)ConstantforceconditionAplatewithacrackisappliedbyaconstantforceFasshowninFig.3.1(b).TheexternalforcevirtualworkWandexternalforcepotentialUP,respectively,areW二F,U=W=FPThetotalstrainenergyU(internalforcepotential)isU=1F2Then,thetotalpotentialnisNow,W=2U,n=U.TheenergyreleaserateGundertheconstantforceconditioncanbewrittenasG=一=(),forconstantF.dsasfInthiscase,dU0,thestrainenergyinthebodyinfactincreasesratherthanreleaseswiththecrackgrowth.Gcannotbecalledasthestrainenergyreleaserate.(2)ConstantdisplacementconditionAfteradisplacement5occurs,theplateisclamped.ThisistheconstantdisplacementconditionasshowninFig.3.1(c).Inthiscase,thereisW=0,口二U-W二UG=-r-()5,forconstantdisplacement5.Itisseenthatonlyfortheconstantdisplacementcondition,Gcanbecalledasthestrainenergyreleaserate.SinceW=0,theenergyneededbythecrackgrowthcomesfromthereleaseofstrainenergystoredinthebody.Thatisthestrainenergystoredinthebodydecreaseswiththecrackgrowth.Foetheconstantforcecondition,theincrementofexternalworkisdW=Fd5inwhichapartisusedtoincreasestrainenergydUwhiletheotherpartisusedforcrackgrowth.However,thevaluesofGfortwocasesareequal.Constantforcecase:.Cisthecomplianceoftheplate.asConstantdisplacementcase:口i52小辱、aa2i52-1acp2acU=F5=,G=()=()=22cas5as2c2c2as2asItisseenthatthevaluesofGfortwocasesareequal.3-2RelationbetweenGandKTherelationshipbetweenGandKisoneofthemostimportantrelationsinfracturemechanics.betweenKandGcanbeestablished.Assumethatasegmentofcracklengthisclosure.Tothisend,adistributedstressCyyisapplied.ThisdistributedstressCyyisthestressfieldinthevicinityinthecracktipfor0=0,r=x,i.e.Inaddition,fortheplanestraincase,thedisplacementproducedinthecrackclosureprocesscanbeknownfromtheasymptoticsolutionofdisplacementfieldinthevicinityofthecracktip.Thepointoistakenastheorigi0anpoint=aFox,thedisplacementalongy-axisisv(x)=空K王2(1-v)=K.尸E小兀E小2兀Assumethattheplatethicknessis1.TheappliedforceFinthearea1-dxisF二c-1-dxyAsshowninFig.3.2,thetotalcrackclosuredisplacementis6=2v.Sincetheactualworkdonebytheexternalforcecyinthelengthdxisequaltothestrainenergy,thereis11U=F5=c-1-dx-2v=cv-dx22yyyyThecrackclosurelengthisasothatthechangeofsystemstrainenergyisJacvdx0yThechangeofsystemstrainenergybyclosingtheunitcrackareais巴=丄人vdxdS1-a0yI,canalsobewrittenasG=-辽=1Jacvdx1daa0yTheexpressionofcyandvcanbeinsertedintotheaboveeq.togive=丄JaGa0vdx=-Ja厶4(1-VKJ二dxya0：2心Ei2兀1-V22Jaax,1-V2K2Jadx=K2E1a兀0xE1i.e.1-V2G=K2,forplanestraincaseIEI2. Fortheplanestresscase,thereis1G=K2,forplanestresscase.IEIModeIIcrackForthemodeIIcrack,therelativeslidedisplacementofcracksurfacesalongtheis2u.TheenergyreleaserateGIIisx-axisbytheshearstresstxyG=1J先udxIIa0xyTheshearstresstxyanddisplacementuarerespectivelyknownasKt=,。二0,r=xxy2兀xu=4(1V2)K口,0二0,r=a-xE叫2兀Wehave1V22a.ax1V2.GE=K2Jadx=K2,forplanestraincase3. 11a兀0xEii1G=一K2,forplanestresscase.IIEIIModeIIIcrackTzyForthemodeIIIcrack,therelativedisplacementbytheanti-planeshearstressalongthez-axisis2w.TheenergyreleaserateGiiicanalsobewrittenas1aG=IaTwdxiiia0zyTheanti-planeshearstressTandanti-planedisplacementwareknownaszyTzy4(1+V)w=Ea-X,6二0,r=a-xWehaveGIIIHla0：2兀E4(K三dx=1+VK2iii2兀EiiiForthecomplexcrack,K产0,K产0,K萨0,thetotalenergyreleaserateis1-V21+vG=(K2+K2)+K2iEiiiEiii3-3Bi-cantileverbeamproblemItisamodeIcrackproblem.1.Longcrackcase:lhWhenlh,thecracksegmentisequivalenttoacantileverbeam.25=2匕3EI1I=12th3(rectangularsection)d=8Fl3Eth3UP=-Fd,口=U+U=-Fd=-P2Eth3G=-1巴tdlEt2h3Fortheplanestressproblem,thereis1G=K2EIThestressintensityfactorcanbeknownasK=2込旦1th3/22.Shortcrackcase:Inthiscase,thecracksegmentisequivalenttoadumpybeam(短粗梁).Thesheardeformationofthecracksegmentandthedeformationintheuncrackedsegmentmustbeconsidered.Thedisplacementdcanbedividedintothreeparts.d=d+d+d123Fig.3-5d1isproducedbybendandd2byshear.Itisknownthatdx13,dx1212d3isthebodydisplacementinducedbythedeformationoftheuncrackedsegment.dx13d=a13+a12+a1123Anormalizedcompliancecoefficient九isintroduced.九二d(tE)二afh3)3+&h2(-)2+tE/h(-)FF1hF2hF3h二a(-)3+a1h22+a3Thecoefficientsa,aandacanbedeterminedbytheoryorexperiment.Ifthe123normalizedcompliance九isknown,thedisplacementdcanbedetermined.F111d二a()3+a()2+a()tE1h2h3hTotalpotentialofthesystemis口7=-2Fd=-2等%)3+a2(h)2+a1Theenergyreleaserateisg一1dEtdl1 F2Et2l3a13+2a22+a3Fortheplanestressproblem,thestressintensityfactorcanbedeterminedfrom1G=K2EIthatG)2LF3a(L)3+2a(-)2+aandl2Therearethreespecimens.Thecracklengthsarel,Fversusdcurvescanbeobtainedbytest.Fanddarethegeneralizedforceandv2lt3respectively.Threeh2h3displacement.Inlightof九=(tE)=a(存+a(存+awecanhavethreeequationsEt=a(1)3+a(J)2+aF1h2h31Et=a(J)3+a(L)2+a(L)F1h2h3hEt匕=a(L)3+a(L)2+aF1h2h3Thecoefficientsa,aandanowcanbeknown.Then,wecalculatetheenergy123releaserateGandthestressintensityfactorK.3-4DeterminationofSIFbyexperimentThemethodinthelastsectioncanbeappliedtothegeneralcase.a,a,a,wecanobtainagroupofFversusdstraightlinesbyexperiment.12nForaconstantappliedforceF,wecanhavethedisplacementvaluesd,d,d.12nThetotalpotentialcanbecalculatedfordifferentcracksizes.1口二一一Fd,fora,i=1,2,ni2iiFinally,agroupofdata(na)isobtained.Theniversusaicanbeplotted.-n（a）Fig.証FortheModeIandplanestressproblem,inviewofG二1巴二丄K2tdaEIthestressintensityfactorKis1dn)dnwhere一-datdaistheslopeatthepointa.Anothertreatment:Assumethatthecompliance九(a)ofthesystemdependsonthecracksizea.ThegeneralizeddisplacementqandforceFarerelatedby九asd=九(a)F11n二Fd二F2九(a)22WhenF=constant,thederivativeisdn1d九(a)=一一F2-da2dak=F,；zai1tda2tdaForaconstantappliedforceF,wecanhavethedisplacementvaluesd,d,d12n九二d/F,agroupofdata(九,a)canbeobtained.The九iiiiversusacurvecanbedepicted.Byusingtheslopeatthepointa,wecancalculatethestressintensityfactorforthecracksizeaandtheappliedloadF.3-5AninfiniteplatewithacentralcrackunderuniaxialtensionThisproblemhasbeensolvedinchaptertwo.Nowweresolveitbytheenergyapproach.Case(a)canbedividedintocase(b)pluscase(c).K(a)=K(b)+K(c)IIICase(b)isthesameasnocrackcase,K)二0,suchthatIK(a)=K(c)IICase(a)isequaltocase(c).1.Solutionforcase(c)VoFig.3-12Forthecase(c),assumethatthedisplacementdistributionontheuppercracksurfaceisanellipse.(上)2+(X)2=1va0Therelationx=a一rissubstitutedtogive/Vx、(ar)2a22ar+r2()2=1(_)2=1=1vaa2a20Whenthedistancerisverysmall,rTheexpressionofenergyreleaserateGInwhatfollows,theexpressionofenergyreleaserateGisderivedfromitsdefinition.Then,thedisplacementVcanbedeterminedandthesolutionfortheproblemcanbeocanbeignored.Onthecracksurfacenearthecracktip,thedisplacementdistributionisaparabola.,2rV=J：万Ithasbeenknowninchaptertwothatthedisplacementasymptoticsolutionisv=isin(k+12cos2)2p2兀22Ontheuppercracksurface,0=k,wehave2ro*aForplanestress,E2(1+v),thestressintensityfactorcanbeobtainedthatandG=丄K2=EIkEv24aoThedisplacementVremainsdetermined.oknown.Thesystemtotalpotentialisn二一U二4-favQtdx20n=-2Qtvfa、a2一x2dx=-Qtav0a020Notethatvisrelatedto0a.n=-Qtav(a)20dn兀/Qt(v+a2_daG=-dn=dSdn1dn兀/dv、=Q(v+ao)d(2ta)2tda40daTheearlyresultisAdifferentialequationcanbeobtainedthatdvvE0+0=v2,BernoulliequationdaaQa20ThegeneralformoftheBernoulliequationisy+P(x)y二q(x)ynThesolutioncanbefoundinthemathematichandbook.Theresultis2cav二0E+2Qa2CCisanintegralconstant.Whenafg,thedisplacementvfg.Thisrequiresthat0C=0.2Qav=0=殛2oaFinally,thesolutionisK=g兀a,v=v1-(兰)2,v二2oaI0a0Itisidenticaltotheearlyresultinchaptertwo.3-6ModeIcrackinthegeneralcasesolvedbyenergyapproach1.SymmetriccaseForthiscase,itisdifficultforthedirectanalyticalmethod.Thecase(b)isthesameasnocrackcase.K(a)=K(c)IICase(c)canalsobedividedintocase(1)andcase(2).Case(1)istheuniformappliedloadingcase.K(a)二K(c)二K+KIII1I2Khasbeenknown.Theproblemistosolvethecase(2).I1TheenergyreleaserateGiswrittenas卜K.+KJ21I1I2=丄(K2+2KK+K2)=G+G+?KKEI1I1I2I212EI1I211G=K2,1EI11Inaddition,theenergyreleaserateGcanbederivedfromthedefinition.n=-U=一4(丄fapvtdx)=-2fa(p+p)(v+v)tdx2001212=-2fapvtdx-2fapvtdx-(2apvtdx+2fapvtdx)=n+口-4fapvtdx022、o12o21,120217equal,Bettitheoremdn1d/.fa7G+G一(-4Jpvtdx)2tda122tda021=G+G+2d(fapvdx)12da021G=-巴dS11=-1ComparisonwithG=G+G+KK2EI1I21resultsI1I2=2(Japvdx)da021KI2E1KdaI1d(fapvdx)=pv(a)+fapdV1dx021K2102daI1v(a)=0,thedisplacementatthecracktipvanishes.KI2EfadvI p1dx,K02daIIConclusion:thestressintensityfactorKcanbedeterminedfromthestressintensityI2factorKforanothercasethathasbeenknown.I1Example:Thesolutionforcase(1)hasbeenknown.KI12piv=1：a2一x21E1KI3I1巴dx,p(x)=jp,x=bda30,x主bKI3Edv|I1E=px=bpv兀a斗D=JP20=daE耳兀a、：a2一b2ThestressintensityfactorKI2canbederivedfromcase(3)byintegraloperation.p(x)dx2adK=-I2aP(b)dbcK=皂J12i兀a0、：a2一b2p(x)dxcIfthedistributionp(x)2isknown,theSIFKcanbedetermined.Therefore,weI2canknowthestressintensityfactorK(a)forcase(a).IK(a)=K(c)=K+KIII1I22.Generalcase:asymmetrictensionK(a)=K(c)=K+KIII1I2Inthesameway,wecanderivethatki2=J:叮x)I1l2dv(x,l,l)dxdl1121KnowthatKI1二pvKa,1(a,x)=2a2一x2,a=(l-1),lE2221KI3E1KI1jl1p(x)2vl23al11(x,l,l)dx,1P3(X)p,x二a+1+b20,x主a+1+b2KI3Eipvkai1I11(x,Jx=a+l2bnaa-bdKI2p(b)db:a+b二一兀aabByintegration,wehave5哙!：p凹搭db第三章完U=*1F5=仝C,G=(aU)22asF1.ModeIcrackByconsiderationofstressandenergyfieldsaheadofthecracktip,therelation