Hashin_Failure_Criteria_for_Unidirectional_Fiber_C

Z.Hashin2ProfessorandChairman,DepartmentofSolidMechanics,Materials,andStructures,TelAvivUniversity,TelAviv,IsraelFellowASMEFailureCriteriaforUnidirectionalFiberComposites1Three-dimensionalfailurecriteriaofunidirectionalfibercompositesareestablishedintermsofquadraticstresspolynomialswhichareexpressedintermsofthetransverselyisotropicinvariantsoftheappliedaveragestressstate.Fourdistinctfailuremodes—ten­sileandcompressivefiberandmatrixmodes—aremodeledseparately,resultinginapiecewisesmoothfailuresurface.1IntroductionOneoftheoutstandingproblemsintheanalysisofmechanicalpropertiesoffibercompositesisthepredictionoffailureduetoaspecifiedstateofstress.Themicrostructuralaspectsoffailureareofsuchcomplexitythatthereislittlehopeofresolutionofthisproblemonthebasisofmicromechanicsmethods.Suchmethodswouldrequireanalyticaldetectionofsuccessivemicrofailuresintermsofmicrostressanalysisandmicrofailurecriteriaandpredictionofthecoalescenceofsomeofthemtoformmacrofailureswhichisanintractabletask.Amorehopefulapproachistoassumetheexistenceofathree-dimensionalfailurecriterionintermsofmacrovariables,suchasav­eragestressesorstrains.Aconvenientmathematicalrepresentationofsuchcriteriaisintermsofpolynomials..Itisthennecessarytode­terminethecoefficientsofthepolynomialintermsoftestresultswhichcanbeconvenientlyobtainedinthelaboratory,suchasuniaxialtensionorcompression,pureshear,etc.Theprobleminthissenseissuperficiallyreminiscentofthatoftheconstructionofaninitialyieldcriterionforanelasto-plasticmaterial.Indeed,inoneofthefirstcontributionstothesubjectTsai[1]assumedthatthefailurecriterionofaunidirectionalfibercompositehasthesamemathematicalformastheyieldcriterionofanorthotropicideallyplasticmaterialasgivenbyHill[2]intheformA(rjn-T22)2+B(o-22-033)2+C(T33-Oil)2+2£o-22+2Eal3+2Fa\l••1(1)Theadvantageoftheform(1)isthatitpermitseasydeterminationofthecoefficientsintermsofone-dimensionaluniaxialandshearfailurestresses.Butitsdisadvantageisthatitisbasedontheas­sumptionthatisotropicstresshasnoeffect.Itisforthatreasonthatonlysquaresofnormalstressdifferencesenterintothecriterion.1SupportedinpartbytheU.S.NavalResearchLaboratoryunderContractN00173-78-C-0170withMaterialsSciencesCorporation,BlueBell,Pa.2Also,ScientificAdvisor,MaterialsSciencesCorporation.ContributedbytheAppliedMechanicsDivisionforpublicationintheJOURNALOFAPPLIEDMECHANICS.DiscussiononthispapershouldbeaddressedtotheEditorialDepartment,ASME,UnitedEngineeringCenter,345East47thStreet,NewYork,N.Y.10017,andwillbeaccepteduntilSeptember1,1980.ReaderswhoneedmoretimetoprepareadiscussionshouldrequestanextensionfromtheEditorialDepartment.ManuscriptreceivedbyASMEAppliedMechanicsDivision,April,1979;finalrevision,July,1979.Whilesuchanassumptionmaybeagoodapproximationforinitialyieldingofametal,itiscertainlynotvalidforisotropictensionofafibercomposite.Hoffman[3]hasmodified(1)byaddinglinearstresstermstoitsoastoaccountforunequalfailurestressesintensionandcompressionwithasinglequadraticexpression.TsaiandWii[4]representedthefailurecriterionasageneralquadraticinthestresses,therebyeliminatingthespecialdependenceonthenormalstressesof(1).IntheirnotationwhereFijTijj+FiOi=11,2,3,4,5,6(2)1=11,2=22,3=33,4=31,5=23,6=12.Foranorthotropicmaterialwithunequalstrengthsintensionandcompression1ivfjvr111NtNJFRR——(3a)(36)(3c)whereNJ,N^aretensileandcompressivestrengths,respectively,in1-directionandSQisshearstrengthinthe12plane.ThecoefficientsF22,^33,F44,F55,F2,F3,aregivenbyanalogousexpressions.Sincethefailureofthematerialisinsensitivetoachangeofsignofshearstress(incontrasttoachangeofsignofnormalstress)alltermscontainingashearstresstofirstpowermustvanish.Therefore,theonlysurvivingunknowncoefficientsareF12,F23,andF13.TsaiandWuproposetodeterminethesecoefficientsbybiaxialfailuretests.Unfortunately,suchtestsarecomplicatedandexpensive.Thecriterion(2)isadefiniteimprovementoverpreviouscriteriabecauseofitsgeneralityandversatilityandhasalsoprovidedgoodfitwithtestdata.Nevertheless,itsutilizationandinterpretationraisesomeintrinsicproblems.Underlyingtheseisthefactthatafibercompositeconsistsofmechanicallyverydissimilarphases:stiffelasticbrittlefibersandacompliantyieldingmatrix.Consequently,theJournalofAppliedMechanicsJUNE1980,VOL.47/329Copyright©1980byASMEDownloaded18Mar2010to192.38.67.112.RedistributionsubjecttoASMElicenseorcopyright;see(2).Anhexagonalcrystalhasmacroscopicallythesamesymmetryasatransverselyisotropicfibercomposite,yetthefailuremodesofthesematerialsareentirelydifferent.Itwouldhardlybepossibletobringforththesedifferencesbymathematicallysimilarsmoothfailurecriteriawhichonlydifferinthenumericalvaluesofthecoeffi­cients.Topointoutsomedifficultiesspecifictofibercompositeslet(2)bespecializedtoaunidirectionallyreinforcedlaminainplanestress.Thenthefailurecriterionis-Fllff?!+.F22