crack tip stress fields for interface cracks in st

\PERGAMONJournaloftheMechanicsandPhysicsofSolids360888#0852
19099911!4985:88:,!seefrontmatter0888ElsevierScienceLtd[AllrightsreservedPII]S9911!498588#99993!5Afamilyofplanestraincracktipstress_eldsforinterfacecracksinstrength!mismatchedelastic
perfectlyplasticsolidsT[!L[Shama\\J[Lib\J[W[HancockbaDepartmentofMechanicalEn`ineerin`\AeronauticalEn`ineerin`andMechanics\RensselaerPolytechnicInstitute\097thStreet\Troy\NY01079!2489\U[S[A[bDepartmentofMechanicalEn`ineerin`\UniversityofGlas`ow\ScotlandG017QQ\U[K[Received12March0887AbstractAsymptoticcracktipstress_eldsaredevelopedforaplanestraincracklocatedontheinterfacebetweentwoincompressible\elastic
perfectlyplasticsolids\withmatchingelasticproperties\butmismatchedyieldstrengths[Thecompleterangeofnormalversussheartractionsontheinterfaceatthecracktipisanalyzed[Inthelimitastheyieldstrengthmismatchvanishestheasymptoticstress_eldsfortheinterfacecrackapproachestablishedcracktip_eldsforhomogeneousbodies[Thecracktipstress_eldsareassembledfromcombinationsofcenteredfans\constantstresssectorsandelasticsectors[Allthestresseswithineachmaterialregionarecontinuouswhiletractioncontinuityismaintainedacrosstheinterface[Theseasymptoticstress_eldsareveri_edbycomparisonwithsmall!scaleyielding_niteelementsolutionsobtainedforvariousremoteloadmixityusingboundarylayerformulationsRice\0856#[0888ElsevierScienceLtd[Allrightsreserved[Keywords]A[Cracktipplasticity^Fracture^B[Crackmechanics^C[Asymptoticanalysis^Finiteelements0[IntroductionDefectassessmentproceduresforweldedjointsarecriticaltotheintegrityofengineeringstructures[Insteelstructurestheelasticpropertiesoftheweldmetalandthebaseplateareusuallycloselysimilar\butsigni_cantmismatchesintheyieldstrengthmayarisebetweentheparentplateandtheheata}ectedzoneortheweldmetal[TheproblemofacracklyingontheinterfacebetweentwostrengthmismatchedCorrespondingauthor[E!mail]shamtrpi[eduT[!L[Shametal[:JournaloftheMechanicsandPhysicsofSolids360888#0852
19090853materialswithsimilarelasticpropertiesisthusoffundamentalandpracticalsigni_!cance[Fracturemechanicshaslargelybeendevelopedbyattemptingtocharacterizethenatureofelastic
plasticcracktip_eldsinhomogeneoussolids[Insightintothenatureofthese_eldshasbeenobtainedbyusingasimpli_edmaterialresponseinwhichthematerialisenvisagedtobeelastic
perfectlyplastic[Underplanestrainconditionsthisallowsthecracktip_eldstobeinterpretedintheformofplanestrainslip!line_eldsforrigid
perfectlyplasticsolidsHill\0849#[Importantexamplesofsuch_eldsariseastheelastic
perfectlyplasticlimitingcasesoftheHRR!_eldsidenti_edbyHutchinson0857a#andRiceandRosengren0857#[ThisincludestheimportantcaseofthePrandtl_eldRice\0857a#\andtheModeII_eldidenti_edbyHutchinson0857b#whichareillustratedinFigs0and1[Intheseanalyses\andthegeneralmixedmodeloadingproblemsaddressedbyShih0863#\plasticityisassumedtosurroundthecracktipatallangles[However\amoregeneralclassofsolutionsidenti_edbyNemat!NasserandObata0873#allowstheasymptoticcracktip_eldstocomprisebothelasticandplasticsectors[AfamilyofModeIcontainedyielding_eldsofthistype\hasbeendiscussedbyDuandHancock0880#[The_eldsexhibitarangeofcracktipconstraintswhichareassociatedwiththenon!singularT!stressWilliams\0846#intheouterelastic_eld[UnderModeIFig[0[ThePrandtl_eldRice\0857a#forahomogeneousbody[Fig[1[TheModeII_eldHutchinson\0857b#forahomogeneousbody[T[!L[Shametal[:JournaloftheMechanicsandPhysicsofSolids360888#0852
19090854loading\thePrandtl_eldisassociatedwithtensileT!stresses[WhentheT!stressiscompressive\theModeI_eldscompriseincompletePrandtl_elds\withelasticwedgesonthecrack~anks[Similarlyinplanestrainmixedmodeloading\_eldswhichcomprisedistortionsofthemodi_edModeI_eldshavebeenanalyzedbyLiandHancock0888#[Interfacecrackswithelasticmismatchhavereceivedconsiderableattentionundersmall!scaleyieldingconditionsZywiczandParks\0889\0881#[Asigni_cantfeatureoftheseproblemsisthatclosetothecracktip\theinterfaceisinherentlysubjecttoamixedmodeloading\whichdoesnotcorrespondinasimplemannertothemixityoftheremoteloading[Thisisalsoanimportantfeatureofinterfacecracksinyieldstrengthmismatchedbi!materialswhereanysimplesymmetryoranti!symmetryoftheremotelyapplied_eldsislostinthelocal_eldatthecracktip[ProgressinunderstandingthisclassofproblemhasbeenmadebyGantiandParks0886#andZhangandco!workers0886#underremoteModeIloading[ForremoteModeIloading\GantiandParks0886#proposedafamilyof_eldswhicharemodi_cationsoftheincompletePrandtl_eldsdiscussedbyDuandHancock0880#foryieldstrengthmismatchesuptoaratioof0[310[Atgreatermismatchesthe_eldcomprisesacenteredfanandacrack~ankelasticsectorinthesoftphaseandadistortedincompletePrandtl_eldintheharderphase[TheseresultsallowedGantiandParks0886#toinvestigatethee}ectsofcracktipconstraintonthestrength
mismatchedinterfacecrackunderremoteModeIloadingconditions[Inthepresentwork\theasymptoticcracktipstress_eldswhicharerelatedtothemoregeneralcaseofremotemixedmodeloadingaredeveloped[Astheyieldstrengthmismatchesvanish\thesecracktipstress_eldsarereducedtoknownplanestrainsolutionsforhomogeneousbodies]thePrandtl_eldRice\0857a#\theM