Transient thermal stress analysis of an edge crack

InternationalJournalofFracture107:73–98,2001.©2001KluwerAcademicPublishers.PrintedintheNetherlands.TransientthermalstressanalysisofanedgecrackinafunctionallygradedmaterialZ.-H.JINandGLAUCIOH.PAULINODepartmentofCivilandEnvironmentalEngineering,UniversityofIllinoisatUrbana-Champaign,Urbana,IL61801,USA(Authorforcorrespondence:fax:[phone];e-mail:[e-mail])Received24January2000;acceptedinrevisedform20June2000Abstract.Anedgecrackinastripofafunctionallygradedmaterial(FGM)isstudiedundertransientthermalloadingconditions.TheFGMisassumedhavingconstantYoung’smodulusandPoisson’sratio,butthethermalpropertiesofthematerialvaryalongthethicknessdirectionofthestrip.Thusthematerialiselasticallyhomoge-neousbutthermallynonhomogeneous.ThiskindofFGMsincludesomeceramic/ceramicFGMssuchasTiC/SiC,MoSi2/Al2O3andMoSi2/SiC,andalsosomeceramic/metalFGMssuchaszirconia/nickelandzirconia/steel.Amulti-layeredmaterialmodelisusedtosolvethetemperaturefield.ByusingtheLaplacetransformandanasymptoticanalysis,ananalyticalfirstordertemperaturesolutionforshorttimesisobtained.Thermalstressintensityfactors(TSIFs)arecalculatedforaTiC/SiCFGMwithvariousvolumefractionprofilesoftheconstituentmaterials.ItisfoundthattheTSIFcouldbereducedifthethermallyshockedcrackededgeoftheFGMstripispureTiC,whereastheTSIFisincreasedifthethermallyshockededgeispureSiC.Keywords:Fracturemechanics,stressintensityfactor,functionallygradedmaterial,thermalstress,heatconduc-tion.1.IntroductionCeramicmaterialsrepresentoneofthemostpromisingmaterialsforfuturetechnologiesofaerospace,nuclearandotherengineeringapplicationsbecauseoftheirexcellentpropertiesathightemperaturesandtheirsuperiorcorrosionandwearresistance.Onemajorlimitationofceramicsistheirinherentbrittlenessthatcanresultincatastrophicfailureunderseverethermalshockloads.Toovercomethisdisadvantage,considerableeffortshavebeenmadetotoughenceramicswithsomesuccess.Ontheotherhand,forceramicsinhightempera-tureapplications,onemayspecificallydesignthematerialtoreducethermalstresseswhensubjectedtoathermalshock.Thisisoneoftheobjectivestobefulfilledbytheconceptoffunctionallygradedmaterials(FGMs)(Koizumi,1993;Hirai,1996;SureshandMortensen,1998).AnFGMisamulti-phasematerialwiththevolumefractionsoftheconstituentsvaryinggraduallyinapre-determinedprofile,thusgivinganon-uniformmicro-structureinthema-terialwithcontinuouslygradedmacro-thermomechanicalproperties.Byintroducingthermalconductivitygradient,HasselmanandYoungblood(1978)showedthatsignificantreductionsinthemagnitudeofthetensilethermalstressinceramiccylinderscouldbeachieved.Thermalresidualstressesmayberelaxedinametal-ceramiclayeredmaterialbyinsertingafunctionallygradedinterfacelayerbetweenthemetalandtheceramic(KawasakiandWatanabe,1987;Drakeetal.,1993;Giannakopoulosetal.,1995).Whensubjectedtothermalshocks,FGM74Z.-H.JinandGlaucioH.Paulinocoatingsmaysuffersignificantlylessdamagethanconventionalceramiccoatings(Kurodaetal.,1993;Takahashietal.,1993).TheknowledgeoffracturebehaviorofFGMsisimportantinordertoevaluatetheirin-tegrity.TheexistinganalyticalstudiesinthisaspecthavebeenmainlyrelatedtocrackgrowthbehaviorinFGMswithspecificmaterialproperties.Byassuminganexponentialspatialvari-ationofelasticmodulus,DelaleandErdogan(1983),Erdogan(1995)andErdoganandWu(1997)solvedcrackproblemsundermechanicalloadingconditions.GuandAsaro(1997a)calculatedthestressintensityfactor(SIF)forasemi-infinitecrackinbothisotropicandorthotropicmaterials.TheyalsostudiedcrackdeflectionprobleminFGMs(GuandAsaro,1997b).HoneinandHerrmann(1997)studiedconservationlawsforinhomogeneouselasticmaterialsandobtainedtheSIFforasemi-infinitecrackbyusingthepath-independentintegralthattheyproposed.Sincethematerialpropertiesinthosestudieswerespecificallyassumed,theSIFconceptcouldbewelldefined.Forgeneralinhomogeneousmaterials,SchovanecandWalton(1988)andJinandNoda(1994a)showedthatthecracktipfieldsareidenticaltothoseinhomogeneousmaterialsprovidedthatthematerialpropertiesarecontinuousandpiece-wisedifferentiable.Hence,theSIFcanstillbeusedtostudyfracturebehaviorofFGMsaslongasthecracktipnonlineardeformationsandprocesszonesarecompletelyincludedwithintheregiondominatedbytheSIF.TheSIFdominant(K-dominant)zone,however,maybereducedsignificantlyifthemodulusgradientisverylarge.JinandBatra(1996a)gavearoughestimateontheeffectofmodulusgradientsintheK-dominantzone.Theyalsostudiedcrackgrowthresistancecurve(R-curve)inFGMsusingbothruleofmixturesandcrack-bridgingconcepts.TheeffectofloadingconditionsandspecimensizeontheR-curveandresidualstrengthbehaviorwerealsoinvestigated(JinandBatra,1996a;1998).CaiandBao(1998)performedacrack-bridginganalysistopredictcrackpropagationinFGMcoatings.Usingafiniteelementmethod,BaoandWang(1995)studiedmulti-crackinginanFGMcoating.Carpenteretal.(1999)presentedatestingprotocalandanalysisforasub-scaleFGMspecimensuitableforexperimentalmeasurementofcriticalSIFandJ–Rcurve.Paulinoetal.(1999)analyzedanantiplaneshearcrackinanFGMinthecontextofgradientelasticity.Forthermalloadingproblems,byassumingexponentialvariationsinbothelasticandther-malproperties,NodaandJin