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

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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,acompositematerialismadeupoftwoormoredierentconstituents;ithaspropertiesthatcannotbeobtainedtogetherbyoneoftheindividualconstituents,suchashighspecicstrengthandstiness,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,thecomplexityofanalyticalsolutionsevenforsimplecasesrequiresthemodellingofmechanicalbehaviorofthisproblemusingeectivenumericalmethods.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)oeringthepossibilityofintroducingcomputeraideddesign(CAD)toolsintheanalysismethodsusingtheisoparametricconcept.Theba-sicideaofthisnovelalternativemethod,calledisogeometricanalysis(IGA),istoexploitthetechnologiesofcomputationalgeometryasshapebasestodescribethegeometryexactly,alsofortheapproximationofunknownelds.Followingthisdiscovery,severalresearchesinvarious 56S.H.Habib,I.Belaidi eldshavebeenconductedbythismethod,including:\ruid-structureinteraction(Bazilevsetal.,2006),compositematerials(Pekovi�etal.,2015),elastic-plasticanalysis(Kalalietal.,2016),electromagneticproblems(Buaetal.,2010),turbulent\row(BazilevsandAkkerman,2010),contactproblems(Temizeretal.,2011),aero-dynamics(Hsuetal.,2011),heattransfer(Andersetal.,2012)and\ruidmechanics(EvanandHudhes,2013).FormoredetailsaboutIGA,arecentreviewhasbeenpublished,seeNguyenetal.(2015).Infracturemechanicsproblems,IGAhasbeenalsoappliedindierentstudies(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-splinebasesarenotalwaysvalidtobeusedinanalysisfordierentgeometriccon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,alsoanisotropicsquareplatewithacentercrackisanalyzedfordierentcrackanglestoverifytheaccuracyoftheproposedapproach.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vedierentcontrolnetcon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.Dierentmeshcon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-splinesgiveuspreciseresultsforadierentnumberofcontrolpoints(meshes2,3and4),evenfortheminimalnumberofcontrolpointscomparedtoNURBS(mesh4comparedtomesh5)andthatattributedtothelocalrenementproperty.Table2comparestheresultsofthenormalizedSIFfordierentradiustostudythedomainindependenceinT-splin

7 emeshes.WeobservethattheSIFvaluesarealmo
emeshes.WeobservethattheSIFvaluesarealmostnotsensitivetotheradiusoftheM-integral.ThecontourplotsofthenormalstresscomponentyyandtheverticaldisplacementuyareillustratedinFig.4.Forthesquareplate,weusedameshconsistingof788controlpointsand689elements(Figs.5aand5b)toevaluatethenormalizedmixedmodeSIFfora=0:5indierentinclinedangles.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.VariationsofnormalizedmodeIandIISIFswithrespecttodierentcrackanglesusingtheanalysis-suitableT-splines,NURBSandexactsolutionforthesquareplateproblem Fig.8.Geometriesandloadingofthebimaterialinterfaceexamples:(a)interfacecentercrackand(b)interfaceedgecrack4.2.BimaterialinterfacecrackWeconsidertwoniterectangularplatessubjectedtouniaxialtensionsinplanestresscon-ditions,each

8 oneconstitutedoftwodissimilarmaterialsan
oneconstitutedoftwodissimilarmaterialsandcrackedintheinterfaceasshowninFig.8.DierentratiosofYoung'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.VariationsofnormalizedmodeIandIISIFswithrespecttodierentYoung'smodulusratiosusingT-splinebasedXIGA,BEMandXFEMforthecenterinterfacecrack(2a=L=0:4) CrackanalysisinbimaterialinterfacesusingT-splinebasedXIGA63 Fig.12.VariationsofnormalizedmodeIandIISIFswithrespecttodierentYoung'smodulusratiosusingT-splinebasedXIGA,BEMandXFEMfortheedgeinterfacecrack(a=L=0:3) Fig.13.TheeectofYoung'smodulusratioonthenormalizedSIFsforthecenterinterfacecrack Fig.14.TheeectofYoung'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'smodulusratiohasaslighteectontheSIFs,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,Isogeometricanalysisofthermaldiusioninbinaryblends,ComputationalMaterialsScience,52,182-1883.BazilevsY.,AkkermanI.,2010,LargeeddysimulationofturbulentTaylor-Couette\rowusingisogeometri

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12 ence-conformingB-splinesforthesteadyNavi
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