JOURNALOFTHEORETICALANDAPPLIEDMECHANICS55,1,pp.55-68,Warsaw2017DOI:10.

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Document on Subject : "JOURNALOFTHEORETICALANDAPPLIEDMECHANICS55,1,pp.55-68,Warsaw2017DOI:10."— Transcript:

1 JOURNALOFTHEORETICALANDAPPLIEDMECHANICS5
JOURNALOFTHEORETICALANDAPPLIEDMECHANICS55,1,pp.55-68,Warsaw2017DOI:10.15632/jtam-pl.55.1.55CRACKANALYSISINBIMATERIALINTERFACESUSINGT-SPLINEBASEDXIGASadamHoucineHabib,IdirBelaidiUniversityofM'hamedBougara,DepartmentofMechanicalEngineering,Boumerdes,Algeriae-mail:hb houcine@yahoo.fr;idir.belaidi@gmail.comAnalysis-suitableT-splinesareusedforthemodelingandanalyzingofcracksinbimaterialinterfaceswithintheframeworkofanextendedisogeometricanalysis(XIGA).Thecracktipenrichmentfunctionsofbimaterialinterfacecracksareimplementedtoreproducesingular elds,andthesigneddistancefunctionsareusedtotreatthecrackfaceandtheinterfaceinthemodels.Acompatiblelocalre nementalgorithmisappliedtore nelocationofthecrackandtheinterface,whichhelpsonetoavoidproduceexcessivepropagationofcontrolpoints.Themixedmodestressintensityfactors(SIFs)whichareevaluatedbytheinteractionintegral(M-integral)areusedasanalysisparameters.Numericalsimulationsareperformedtoanalyzetheproblemandtoexaminetheeciencyoftheproposedmethod.Theobtainedresultsarecomparedwithotheravailableresults.Keywords:extendedisogeometricanalysis,T-splines,bimaterialinterfacecracks,enrichmentfunctions,localre nement1.IntroductionAsitsnameindicates,acompositematerialismadeupoftwoormoredi erentconstituents;ithaspropertiesthatcannotbeobtainedtogetherbyoneoftheindividualconstituents,suchashighspeci cstrengthandsti ness,gooddurabilityandgoodcorrosionresistance.Duetotheirproperties,compositeshavebeendevelopedandusedinvariousindustrialandengineeringap-plications,likethoseinaerospace,aircraft,automotiveindustries,etc.However,thesematerialsarenotimmunetomanufacturedefectsespeciallyfromthosewhicharecreatedasinterfacialcracks.Thisproblemgreatlyin\ruencesthebehaviorofstructuresandcancausebrutalfracture.Themechanicalbehaviorofcompositematerialsneedsmoreunderstanding,especiallyinthepresenceofstrongandweakdiscontinuities.ManyanalyticalstudieswereperformedbasedupontheworkofWilliams(1959)forunderstandingtheproblemofbimaterialinterfacecracks,suchas(Erdogan1963;RiceandSih,1965;S

2 unandJih,1987;Hutchinsonetal.,1987;Ri-ce
unandJih,1987;Hutchinsonetal.,1987;Ri-ce,1988;Evansetal.,1990).However,thecomplexityofanalyticalsolutionsevenforsimplecasesrequiresthemodellingofmechanicalbehaviorofthisproblemusinge ectivenumericalmethods.Severalinvestigationshavebeendevelopedinthisdomain,viatheboundaryelementmethod(BEM)(LeeandChoi,1988;YuukiandXu,1994;Miyazakietal.,1993), niteelementmethod(FEM)(Ikedaetal.,2006),elementfreeGalerkinmethod(EFGM)(Pantetal.,2011),extended niteelementmethod(XFEM)(Nagashimaetal.,2003;Liuetal.,2004;BelytschkoandGracie,2007)andothermethods(Zhouetal.,2013,2014;Anetal.,2013).Recently,alarge eldwasopenedbyHughesetal.(2005)o eringthepossibilityofintroducingcomputeraideddesign(CAD)toolsintheanalysismethodsusingtheisoparametricconcept.Theba-sicideaofthisnovelalternativemethod,calledisogeometricanalysis(IGA),istoexploitthetechnologiesofcomputationalgeometryasshapebasestodescribethegeometryexactly,alsofortheapproximationofunknown elds.Followingthisdiscovery,severalresearchesinvarious 56S.H.Habib,I.Belaidi eldshavebeenconductedbythismethod,including:\ruid-structureinteraction(Bazilevsetal.,2006),compositematerials(Pekovietal.,2015),elastic-plasticanalysis(Kalalietal.,2016),electromagneticproblems(Bu aetal.,2010),turbulent\row(BazilevsandAkkerman,2010),contactproblems(Temizeretal.,2011),aero-dynamics(Hsuetal.,2011),heattransfer(Andersetal.,2012)and\ruidmechanics(EvanandHudhes,2013).FormoredetailsaboutIGA,arecentreviewhasbeenpublished,seeNguyenetal.(2015).Infracturemechanicsproblems,IGAhasbeenalsoappliedindi erentstudies(Verhooseletal.,2011;Bordenetal.,2012;NguyenandNguen-Xuan,2013;Nguyenetal.,2014;Pengetal.,2014),howeverBensonetal.(2010)andDeLuyckeretal.(2011)proposedextendedisogeometricanalysis(XIGA)formodellingcracks.InthismethodthegeneralprincipleoftheXFEMisusedinIGAbyincludingtheasymptoticandsigneddistanceenrichmentfunctions.Therefore,thismethodhastheadvantagesofbothXFEMandIGA,whicharesummarizedbytheabilitytorepresentcomplexgeometriesinde-pendentlyofanydiscontinuitiesandwithoutexplicitmeshi

3 ngtoobtainsolutionswithhigherorders.Some
ngtoobtainsolutionswithhigherorders.SomeapplicationsinfracturemechanicshavebeencheckedbytheXIGA,suchasinthecasesofhomogeneousmaterials(Ghorashietal.,2012;BhardwajandSingh,2015),functionallygradedmaterial(Bhardwajetal.,2015a,c)andbimaterialinterfaces(Bhardwajetal.,2015b;Jiaetal.,2015),wherethenon-uniformrationalB-splines(NURBS)areused.Also,orthotropicmediahavebeenstudiedusingT-splinebasedXIGA(Ghorashietal.,2015).TherearemanyCADbasisfunctionsthatcanbeusedinIGA,wheretheNonUniformRationalB-splines(NURBS)arewidelyusedduetotheirproperties,likecontinuity,smoothness,variationdiminishing,convexhullandpossibilityofusingknotinsertionanddegreeelevationre nements.Theyhavetheabilitytodescribeexactlyallconicsectionsbuttheyhavedicultiesincertaincomplexgeometrieswhichcannotbeavoidedevenbyusingmultiplepatches,whereNURBSgenerateacomplicatedmeshwhichleadstoproducesuper\ruouscontrolpoints.Inordertohandlethesedisadvantages,Sederbergetal.(2003)proposedaT-splineasageneralizedtoolofNURBS,inwhichtheindexspace(T-mesh)locallyre nedusingT-junctions(Sederbergetal.,2004).Therefore,themajoradvantagesofthistechniquearethelocalre nementandtheabilitytorepresentcomplexgeometrieswithaminimalnumberofcontrolpointscomparedwiththoseusedinNURBS.Accordingtotheirabilityinengineeringdesign,T-splineshavebeenusedbyanalysistoserveasbasisfunctionsforIGAinmanyadvancedsearches.HoweverT-splinebasesarenotalwaysvalidtobeusedinanalysisfordi erentgeometriccon gurations,becausethelinearindependenceandpartitionunitypropertiesarenotalwaysensured.Lietal.(2012)introducedanalysis-suitableT-splines,whereforanychoiceofknotvectorstheblendingfunctionsarelinearlyindependent.LikeNURBSbases,analysis-suitableT-splinebaseshavethepropertiesoftheanalysisbasisfunctions.Moreover,theyprovideanecientalgorithmwhichallowsmakinghighlylocalizedre nement(Scottetal.,2012).Inthispaper,theinterfacecrackinthecaseof2DcompositesisanalyzedusingT-splinebasedXIGA;theaccuracyofthisapproachis rsttestedinisotropicmaterials.Theanalysis-suitableT-splineanditsre nementalgo

4 rithmarehighlighted.2.Analysis-suitableT
rithmarehighlighted.2.Analysis-suitableT-splinesAnanalysis-suitableT-splineisfoundedwhentheT-mesh(T-meshisameshofrectangularelementsthatisde nedbythelinescorrespondingtoknotvaluesoftheparametricvectors)providesarestrictedtopologythathasnointersectingT-junctionextensions.TheT-junctionextensionisde nedineachT-junctionvertexbyanintervalwhichincludestwodistances.The rstdistanceisbetweentheT-junctionandthetwonextadjoiningedgesorverticesinthedirectionofmissingedge,whiletheseconddistanceisbetweentheT-junctionandoneedge CrackanalysisinbimaterialinterfacesusingT-splinebasedXIGA57 orvertexintheotherdirection,asshowninFig.1b.TheT-meshthatshowsallT-junctionextensionscanbecalledextendedT-mesh.AnemptyextendedT-meshmeanstherearenointersectionsbetweenT-junctionextensions(seeFig.1c),whichmeanstheT-meshisanalysis--suitable. Fig.1.Anexampledepicts:(a)T-mesh,(b)extendedT-meshand(c)emptyextendedT-meshInordertomakelocalre nementofanalysis-suitableT-splinespaces,Scottetal.(2012)introducedanalgorithmconsistingofthefollowingsteps:createthere nedT-meshT2fromtheoriginalanalysis-suitableT-meshTs1,formtheextendedT-meshofT2.iftheextendedT-meshofT2hasintersectingT-junctionextensions,oneedgemustbeinsertedintoT2insuchawaythatreducesthenumberoftheintersections,repeatstep3untiltheextendedT-meshhasnointersectingT-junctionextensions,computethere nementmatrixM.Formoredetails,see(Scottetal.,2012).3.Extendedisogeometricanalysis(XIGA)XIGA(Bensonetal.,2010;DeLuyckeretal.,2011)usesthesamemethodologyoftheextended niteelementmethod(XFEM)forthemodellingofdiscontinuitiesbutwithbasisfunctionsderivedfromgeometrylikeinisogeometricanalysis(Hughesetal.,2005).Forcrackproblems,XIGAprovidesthepossibilityofmodellingthecrackindependentlyofthemeshandwithinexactlypresentedgeometry.Uncommonly,inthisstudy,T-splinesareadoptedinXIGAusinganalysis-suitableT-splinestoapproximatethedisplacementinanypoint=(;)asfollowsu()=nsXi=1Ri()ui+ncfXj=1Rj()H()aj+nctXk=1Rk() 4X`=1F`()b`k!+niXt=1Rt()()ct(3.1)whereRistheT-splinebasisfunctionext

5 ractedfromanemptyextendedT-mesh,HistheHe
ractedfromanemptyextendedT-mesh,HistheHeavisidefunctionusedforthemodellingofthecrackface,ittakesvalue1abovethecrackand1belowthecrack,Farethecrack-tipenrichmentfunctions,ui,aj,bkandctarethedisplacementvectorscorrespondingtons,ncf,nctandntcontrolpoints,respectively.Thefourthtermisusedwhenthereisnocoincidencebetweentheinterfaceandthe niteelementmeshforthemodellingofweakdiscontinuity.TheenrichmentfunctionofMoesetal.(2003)canbeused()=XRI()jIj XRI()I (3.2)whereisthesigneddistancevalueoftheinterfacecontrolpoints. 58S.H.Habib,I.Belaidi TheenrichmentfunctionsofbimaterialinterfacecrackswerederivedbySukumaretal.(2004)asfF`(r;)g12`=1=np rcos("logr)e"sin 2;p rcos("logr)e"cos 2;p rcos("logr)e"sin 2;p rcos("logr)e"cos 2;p rcos("logr)e"sin 2sin;p rcos("logr)e"cos 2sin;p rsin("logr)e"sin 2;p rsin("logr)e"cos 2;p rsin("logr)e"sin 2;p rsin("logr)e"cos 2;p rsin("logr)e"sin 2sin;p rsin("logr)e"cos 2sino(3.3)4.NumericalsimulationsHere,theanalysis-suitableT-splineisusedinXIGAtosimulatethecrackinhomogeneousisotropicandbimaterialinterfaces.Twonumericalexamplesareconsideredforeachmaterialtypeinplanestaticproblems,wheremodeIandmodeIISIFsareevaluatedandcomparedwithothernumericalandanalyticalresults.First,theisotropicmaterialisconsideredinarectangularplatewithanedgecrackinordertostudytheconvergenceandthedomainindependenceinthecomputationsofSIF,alsoanisotropicsquareplatewithacentercrackisanalyzedfordi erentcrackanglestoverifytheaccuracyoftheproposedapproach.Then,numericalapplicationsintheformofparametricstudiesareconsideredforedgeandcenterinterfacecracksin niterectangularplates.Inallgeometricmodels(NURBSandT-splines)thecubicorderisusedinbothparametricdirections,wheretheweightsaretakenasunity.Intheedgecrackproblems,thegeometryisre nedlocallyonce,whileforthecentercrackproblemsthegeometryisre nedlocallytwice.Fourtypesof niteelementsaredistinguishedintheseexamplesaccordingtotheirpositionswithrespecttothecrack,thestandardelementcontains33Gausspoints.Theelementhaving

6 tipenrichedcontrolpointscontains77Gauss
tipenrichedcontrolpointscontains77Gausspointsandthesub-triangletechnique(Ghorashietal.,2011)isusedforthetip-elementby13Gausspointsineachtriangle,howeverthesplitelementcontains66Gausspointsforthehorizontalcrackproblemsandthesub-triangletechniqueisusedby13Gausspointsineachtrianglefortheinclinedcrackproblems.TheSIFsareevaluatedusinginteractionintegral(YauandWang,1984),whereinthecracktipelementisnotconsideredinthecalculation.4.1.HomogeneousisotropicmaterialInthiscase,wesimulatea niterectangularplatecontaininganedgecrack(Fig.2a)andasquareplatecontaininganinclinedcentralcrack(Fig.2b),subjectedtounituniaxialtensioninplanestressstate.Theconvergenceoftheproposedapproachisstudiedfortheedgecrackpro-blemwithnormalizedM-integralradiusequalto1using vedi erentcontrolnetcon gurations(200,296,362,754and1800controlpoints),allshowninFig.3.TheerrorsofthenormalizedSIFvaluesobtainedfromtheproposedapproachwhichareshowninTable.1arecomputedusingthefollowingequation KI=KI p a=TIa L(4.1)whereTI(a=L)istheanalyticalformulawhichcorrespondstomodeI,itcanbecomputedas(Tadaetal.,2000)TI=1:1220:231a L+10:55a L221:71a L3+30:382a L4 CrackanalysisinbimaterialinterfacesusingT-splinebasedXIGA59 Fig.2.Geometriesandloadingofthehomogeneousisotropicexamples(a)rectangularplatewithanedgecrackand(b)squareplatewithacenterinclinedcrack Fig.3.Di erentmeshcon gurationsusedintheconvergencestudy:(a)200points,(b)296points,(c)362points,(d)788pointsand(e)1800pointsTable1.ConvergenceoftheSIFforvariouscontrolnets Controlpoints KI Error[%] 200 2.1275 1.0257 296 2.1189 0.6173 362 2.1121 0.2944 754 2.1098 0.1852 1800 2.1131 0.3419 The rstandthelastmeshesinFig.3representaspecialcaseofT-splineswhichisNURBS.AccordingtoTable1,analysissuitableT-splinesgiveuspreciseresultsforadi erentnumberofcontrolpoints(meshes2,3and4),evenfortheminimalnumberofcontrolpointscomparedtoNURBS(mesh4comparedtomesh5)andthatattributedtothelocalre nementproperty.Table2comparestheresultsofthenormalizedSIFfordi erentradiustostudythedomainindependenceinT-splin

7 emeshes.WeobservethattheSIFvaluesarealmo
emeshes.WeobservethattheSIFvaluesarealmostnotsensitivetotheradiusoftheM-integral.ThecontourplotsofthenormalstresscomponentyyandtheverticaldisplacementuyareillustratedinFig.4.Forthesquareplate,weusedameshconsistingof788controlpointsand689elements(Figs.5aand5b)toevaluatethenormalizedmixedmodeSIFfora=0:5indi erentinclinedangles.TheexactSIFsofthisproblemcanbeobtainedbythefollowingequationsKI=0p acos2 KII=0p asin cos (4.2) 60S.H.Habib,I.Belaidi Table2.Domainindependencestudy Radius Mesh2 Mesh3 Mesh4 KI Error[%] KI Error[%] KI Error[%] 0.6 2.1131 0.3419 2.1280 1.0494 2.1245 0.8832 0.7 2.1256 0.9355 2.1202 0.6790 2.1142 0.3941 0.8 2.1256 0.9355 2.1209 0.7123 2.1122 0.2992 0.9 2.1218 0.7550 2.1177 0.5603 2.1122 0.2992 1.0 2.1189 0.6173 2.1121 0.2944 2.1098 0.1852 1.1 2.1146 0.4131 2.1118 0.2802 2.1109 0.2374 Fig.4.Graphicalvisualization:(a)normalstressand(b)verticaldisplacement Fig.5.Themeshesusedfortheisotropicsquareplate:(a)T-splinecontrolnet(788points),(b)elementscorrespondingtotheT-splinecontrolnet(689elements)and(c)NURBScontrolnet(4625points) Fig.6.Thecracktip(redsquares)andthecrackface(bluecircles)enrichedpointsof:(a)T-splinecontrolnetand(b)NURBScontrolnet,inthecase =0Figure6adepictstheenrichedcontrolpointsthatcorrespondtothecrackfaceandcracktipelements.Figure7showsacomparisonbetweenthenormalizedSIFscalculatedbythepropo-sedapproachandthosederivedfromtheexactsolutionandNURBS-basedXIGA.AuniformNURBSmeshisusedFig.6c,itsenrichedcontrolpointsarepresentedinFig.6b.Asseeninbothmodes,thereisaverycloseagreementbetweentheT-splineresultsandtheotherresults. CrackanalysisinbimaterialinterfacesusingT-splinebasedXIGA61 Fig.7.VariationsofnormalizedmodeIandIISIFswithrespecttodi erentcrackanglesusingtheanalysis-suitableT-splines,NURBSandexactsolutionforthesquareplateproblem Fig.8.Geometriesandloadingofthebimaterialinterfaceexamples:(a)interfacecentercrackand(b)interfaceedgecrack4.2.BimaterialinterfacecrackWeconsidertwo niterectangularplatessubjectedtouniaxialtensionsinplanestresscon-ditions,each

8 oneconstitutedoftwodissimilarmaterialsan
oneconstitutedoftwodissimilarmaterialsandcrackedintheinterfaceasshowninFig.8.Di erentratiosofYoung'smodulus(E1=E2=2,3,4,10and100)with xedPois-sonratios(1=2=0:3)aretakeninthesimulation.SimilarproblemsweresolvedbeforebyMiyazakietal.(1993)utilizingtheboundaryelementmethod(BEM),Nagashimaetal.(2003)utilizinganextended niteelementmethod(XFEM),Matsumtoetal.(2000)makinguseoftheinteractionenergyreleaseratesandBEMsensitivityandLiuetal.2004)usingXFEMfordirectevaluationofthemixedmodeSIF.Forthecentercrackproblem(Fig.8a),weuseameshconsistingof3132controlpointsand2925elementsasshowninFigs.9aand9b.Fortheedgecrackproblem(Fig.8b),weuseameshconsistingof1446controlpointsand1235elementsasshowninFigs.9cand9d.Theenrichedcontrolpointsarede nedinFig.10.Inordertoverifytheaccuracyoftheobtainedresults,thenormalizedSIFsarecomparedwiththoseobtainedby 62S.H.Habib,I.Belaidi othermethodsinFig.11forthecentercrackproblem(2a=L=0:4)andinFig.12fortheedgecrackproblem(a=L=0:3).Figures13and14illustratevariationsofthenormalizedSIFsintermsofcracklengthsforthecenterandedgecrackproblems,respectively.Formoredetails,checkTable3andTable4. Fig.9.T-splinemeshesusedforthebimaterialinterfaceexamples:(a)controlnetfortheinterfacecentercrack(793points),(b)meshfortheinterfacecentercrack(688elements),(c)controlnetfortheinterfaceedgecrack(566points)and(d)meshfortheinterfaceedgecrack(431elements) Fig.10.Cracktip,crackfaceandinterfaceenrichedcontrolpoints:(a)interfacecentercrackand(b)interfaceedgecrack Fig.11.VariationsofnormalizedmodeIandIISIFswithrespecttodi erentYoung'smodulusratiosusingT-splinebasedXIGA,BEMandXFEMforthecenterinterfacecrack(2a=L=0:4) CrackanalysisinbimaterialinterfacesusingT-splinebasedXIGA63 Fig.12.VariationsofnormalizedmodeIandIISIFswithrespecttodi erentYoung'smodulusratiosusingT-splinebasedXIGA,BEMandXFEMfortheedgeinterfacecrack(a=L=0:3) Fig.13.Thee ectofYoung'smodulusratioonthenormalizedSIFsforthecenterinterfacecrack Fig.14.Thee ectofYoung'smodulusratioonthenormalizedSIFfortheedgeinterfacecrack 64S.H.Habib,I.Bela

9 idi Table3.Resultsofnormalizedstressinte
idi Table3.Resultsofnormalizedstressintensityfactorsforthecenterinterfacecrack E1=E2 2a=L Presentstudy Matsumtoetal.(2000) Miyazakietal.(1993) KI KII KI KII KI KII 0.1 1.006 0:0731 0.995 0:072 1.001 0:072 0.2 1.0245 0:0713 1.019 0:07 1.02 0:071 2 0.3 1.0572 0:071 1.053 0:072 1.053 0:071 0.4 1.1056 0:0725 1.104 0:073 1.104 0:073 0.5 1.1814 0:0764 1.18 0:077 1.181 0:077 0.1 0.9993 0:1097 0.987 0:106 0.993 0:107 0.2 1.0179 0:1072 1.013 0:105 1.012 0:106 3 0.3 1.0504 0:1068 1.044 0:105 1.045 0:106 0.4 1.0981 0:1089 1.095 0:108 1.096 0:109 0.5 1.1726 0:1145 1.172 0:115 1.171 0:115 0.1 0.9934 0:1314 0.981 0:128 0.987 0:129 0.2 1.0121 0:1284 1.006 0:126 1.006 0:127 4 0.3 1.0443 0:1279 1.037 0:126 1.031 0:127 0.4 1.0916 0:1303 1.088 0:131 1.089 0:13 0.5 1.1649 0:1368 1.163 0:136 1.163 0:137 0.1 0.972 0:1764 0.962 0:172 0.968 0:174 0.2 0.9906 0:1729 0.987 0:168 0.986 0:171 10 0.3 1.0224 0:1708 1.017 0:171 1.018 0:17 0.4 1.0712 0:1745 1.065 0:172 1.066 0:173 0.5 1.1418 0:1838 1.135 0:181 1.136 0:182 0.1 0.9488 0:2086 0.943 0:207 0.946 0:206 0.2 0.967 0:2043 0.964 0:201 0.964 0:201 100 0.3 0.9979 0:201 0.994 0:198 0.994 0:2 0.4 1.0435 0:204 1.039 0:2 1.039 0:203 0.5 1.1088 0:2129 1.104 0:208 1.104 0:21 TheresultsoftheproposedmethodareclosertotheBEMresultsthantheXFEMresults,asshowninFigs.11and12.Young'smodulusratiohasaslighte ectontheSIFs,asshowninTables3and4.Asitisobviousintheprecedentexamples,thelocalre nementpropertyofanalysis-suitableT-splinesallowsincreasingtheaccuracyoftheresultsandusinglessDOFs.Finally,wenotethattheevaluationofshapefunctionsinXIGAisslowerthansomemethodssuchasXFEM.5.ConclusionInthisstudy,theanalysis-suitableT-splinehasbeenusedinXIGAtoapproximatethesolutionincrackedbimaterialinterfacesinordertoconstructgeometryandtomakelocalre nementaroundthediscontinuities.Furthermore,ithelpsavoidingtheemergenceofsuper\ruouscontrolpointsduringthelocalre nementprocess.Theasymptoticcrack-tipenrichmentfunctionsandt

10 heinteractionintegralmethodcorresponding
heinteractionintegralmethodcorrespondingtobimaterialinterfacecrackshavebeenusedtoevaluatethestressintensityfactors.Theresultsobtainedbytheproposedmethodhavebeencomparedwiththeresultsformliterature,whereagoodagreementhasbeenregardeddemonstratingtheaccuracyoftheapproach. CrackanalysisinbimaterialinterfacesusingT-splinebasedXIGA65 Table4.Resultsofnormalizedstressintensityfactorsfortheedgeinterfacecrack E1 E2 2a L Presentstudy Matsumtoetal.(2000) Miyazakietal.(1993) Liuetal.(2004) KI KKII KI KII KI KII KI KII 0.1 1.1899 0:1299 1.19 0:127 1.195 0:129 { { 0.2 1.3682 0:1352 1.367 0:137 1.368 0:137 1.374 0:137 2 0.3 1.6619 0:1576 1.657 0:156 1.659 0:158 1.669 0:159 0.4 2.1198 0:1975 2.109 0:195 2.11 0:198 2.125 0:198 0.5 2.8423 0:2678 2.819 0:268 2.882 0:267 2.844 0:267 0.1 1.1974 0:1988 1.198 0:195 1.203 0:197 { { 0.2 1.369 0:2049 1.368 0:208 1.368 0:207 1.375 0:208 3 0.3 1.6603 0:2379 1.655 0:235 1.656 0:239 1.668 0:240 0.4 2.116 0:2977 2.102 0:298 2.105 0:298 2.121 0:299 0.5 2.8351 0:403 2.812 0:402 2.814 0:402 2.839 0:402 0.1 1.2216 0:343 1.222 0:336 1.229 0:34 { { 0.2 1.3719 0:3461 1.366 0:348 1.369 0:349 1.379 0:354 10 0.3 1.6547 0:3976 1.648 0:394 1.648 0:399 1.661 0:403 0.4 2.1023 0:4945 2.09 0:491 2.09 0:494 2.109 0:5 0.5 2.8103 0:6649 2.789 0:661 2.789 0:663 2.819 0:668 0.1 1.2422 0:4286 1.251 0:424 1.251 0:424 { { 0.2 1.3744 0:4252 1.376 0:429 1.370 0:428 1.381 0:434 100 0.3 1.6491 0:4842 1.647 0:47 1.642 0:485 1.657 0:494 0.4 2.0895 0:5975 2.083 0:569 2.078 0:597 2.101 0:608 0.5 2.7888 0:7972 2.772 0:793 2.77 0:797 2.804 0:813 References1.AnX.,ZhaoZ.,ZhangH.,HeL.,2013,Modelingbimaterialinterfacecracksusingthenumericalmanifoldmethod,EngineeringAnalysiswithBoundaryElements,37,464-4742.AndersD.,WeinbergK.,ReichardtR.,2012,Isogeometricanalysisofthermaldi usioninbinaryblends,ComputationalMaterialsScience,52,182-1883.BazilevsY.,AkkermanI.,2010,LargeeddysimulationofturbulentTaylor-Couette\rowusingisogeometri

11 canalysisandtheresidual-basedvariational
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