INTERNATIONALJOURNALFORNUMERICALMETHODSINENGINEERINGInt.J.Numer.Meth.E - PDF document

INTERNATIONALJOURNALFORNUMERICALMETHODSINENGINEERINGInt.J.Numer.Meth.EngngEmbeddedcrackmodel.PartII:CombinationwithsmearedcracksMilanJirandThomasZimmermannLaboratoryofStructuralandContinuumMechanicsDepartmentofCivilEngineeringSwissFederalInstituteofTechnology,CH-1015LausanneThepaperinvestigatesthebehaviourofniteelementswithembeddeddisplacementdiscontinuitiesthatrepresentcracks.Examplesoffracturesimulationsshowthatanincorrectseparationofnodesduetoalocallymispredictedcrackdirectionleadstoaseverestresslocking,whichproducesspurioussecondarycracking.Asapossibleremedythepaperadvocatesanewconceptofamodelwithtransitionfromasmearedtoanembedded(discrete)crack.Anadditionalimprovementisachievedbyreformulatingthesmearedpartasnon-local.Variouscriteriaforplacingthediscontinuityarecompared,andtheoptimaltechniqueisidentied.Remarkableinsensitivityoftheresultingmodeltomesh-induceddirectionalbiasisdemonstrated.Itisshownthatthetransitiontoanexplicitdescriptionofawidelyopeningcrackasadisplacementdiscontinuityimprovesthebehaviourofthecombinedmodelandremediescertainpathologiesexhibitedbyregularizedcontinuummodels.Copyright2001JohnWiley&Sons,Ltd.KEYWORDS:fracture;damage;cracking;localization;embeddeddiscontinuities;non-localcontinuum;stresslocking1.MULTIPLEEMBEDDEDCRACKSInPartIofthispaper[1]wehavedevelopedthetheoreticalformulationandnumericalalgo-rithmsfortriangularniteelementswithembeddedcracks(strongdiscontinuities)describedbyadamage-typetraction-separationlaw.Theseelementsshallnowbetestedonseveraltypicalfrac-tureproblems,andtheresultsshallbecomparedtothoseobtainedwithsmearedcrackmodelsimplementedintostandardconstant-straintriangularelements(CST).Astherstexample,considerthethree-point-bendspecimeninFigure1(a).Theload{displacementdiagraminFigure1(b)andthecrackpatternsinFigures1(c)and1(d)show Correspondenceto:MilanJirasek,LaboratoryofStructuralandContinuumMechanics(LSC),SwissFederalInstituteofTechnology(EPFL),CH-1015,Lausanne,SwitzerlandE-mail:Milan.Jirasek@EPFL.chgrantsponsor:SwissCommissionforTechnologyandInnovation,contractgrantnumber:CTI-3201.1InFigures1(c)and1(d),thethicknessoflinesrepresentingcracksisproportionaltotheinelasticstrain,inordertoshowwhichcracksdominate.Received9November19982001JohnWiley&Sons,Ltd.Revised5February1999 M.JIRASEKANDT.ZIMMERMANN Figure1.Three-point-bendspecimen:(a)geometryandloading;(b)load{displacementdiagrams;(c)crackpatternforstandardelementsandsmearedrotatingcrack(RC)model;(d)crackpatternforstandardelementsandsmearedrotatingcrackmodelwithtransitiontoscalardamage(RC-SD).theresultsobtainedwithordinaryCSTelementsusing,respectively,thestandardrotatingcrack(RC)model[2]andtherotatingcrackmodelwithtransitiontoscalardamage(RC-SD),recentlyproposedbyJirasekandZimmermann[3].Thedimensionsandmaterialparameterscorrespondtotypicalconcretefracturespecimens;theirexactvaluesarenotimportant,becausethepurposeofthisexampleistodemonstratecertainpathologicalfeaturesthatareofageneralnature.AsexplainedinReference[4],unlessthemeshisalignedsuchthatthemacroscopiccrackcanprop-agateinabandrunningparalleltothemeshlines(sidesofniteelements),theRCmodelleadstoseverestresslocking.Atypicalconsequenceisanunrealisticload{displacementdiagramwithanon-negligibleresistingforceevenatverylatestagesofthedegradationprocess,whenthecrackshouldbealmostcompletelystress-free.Spuriousstressestransferredbythecrackingbandleadtoadditionalcrackingonbothsidesofthebandandthustoanoverestimatedenergydissipation.Thepresenttypeoflockingcanbealleviatedbytransitiontoascalardamagemodelinthoseelementswherethecrackopeninghasreachedacertaincriticalvalue.ThisisdocumentedbytheimprovedperformanceofthecombinedRC-SDmodel,forwhichtheload{displacementdiagramhasareasonableshapeandcrackingoccursonlyinonezig-zaglayerofelements(seeFigure1(d)).Theproblemhasbeenre-analysedusingtheelementswithembeddedcracksdescribedinPartIofthispaper.AsillustratedinFigure2(a),theorientationofthediscontinuitylinepredicted2001JohnWiley&Sons,Ltd.Int.J.Numer.Meth.Engng EMBEDDEDCRACKMODEL.PARTII Figure2.Evolutionofcrackpatternforathree-point-bendspecimen:(a),(b)elementswithembeddedcracks(EC)anddiscontinuouscrackpath;(c)elementswithembeddedcracks(EC)andcontinuouscrackpath;(d)elementswithdelayedembeddedcracks(DEC;tobedescribedinSection2).fromthelocalcriterionofmaximumprincipalstressisnotalwayswellalignedwiththeactualmacroscopiccrack.Sometimestheembeddedcracksfailtoproperlyseparatenodesthatshouldbeontheoppositesidesofthebandofcrackingelements.ThisisthecaseforexampleforthetopmostcrackinFigure2(a),whichshould(butdoesnot)separatethenodesmarkedbycircles.Inamodelwithonlyoneembeddeddiscontinuityperelementthisleadstoaseverelockingbecausethemacroscopiccrackcannotopenproperly.Itisthereforeessentialtoallowtheformationofanotherdiscontinuitylinewithinthesameelementifthemaximumprincipalstressexceedsthetensilestrengthandtherotationofthecorrespondingprincipalaxiswithrespecttotheprimarydiscontinuityexceedsacertainthresholdangle.Iftheprimaryembeddedcracklocks,thesecondaryonecanreleasespuriousstressesbycorrectlyseparatingthenodesandallowingthepropagationofthecrackingbandinthecorrectoveralldirection.Typically,thesecondarydiscontinuitydominates InFigure2,alllinesrepresentingembeddedcrackshavethesamethickness,independentofthecrackopening.2001JohnWiley&Sons,Ltd.Int.J.Numer.Meth.Engng M.JIRASEKANDT.ZIMMERMANNthefailureprocesswhiletheoriginaloneformedinawrongdirectioneitherclosesoratleastdoesnotopenveryfast.Figure2(b)showsthattheproblemwithincorrectseparationofnodesmaylaterreappearinanotherelement.Whenmultiplecrackingisallowed,themodelcanreproducetheoverallfracturepatterninasatisfactoryway.Noexcessivespuriousstresstransferoccurs,andtheload{displacementdiagramisveryclosetothatobtainedwiththeRC-SDmodel;seethecurvewithlabelembeddedcrack(EC)inFigure3.However,theintroductionofasecondarycrackis(inthepresentcontext)onlyanarticialremedy|itrelaxesthestressinasituationthatwouldotherwiseleadtolocking.Thesecondarycrackhasnothingtodowiththephysicalnatureoftheprocessbecauseitappearsonlyifthedirectionoftheprimarycrackismispredicted.Multiplecrackingcomplicatesthenumericalalgorithmandhasanadverseeectonitsrobustness.Notethatenforcementofacontinuouscrackpathwouldnotremedytheproblem.Infact,inthepresentexamplethecrackseparatingnodesincorrectlywouldappearevenearlier(seeFigure2(c)).2.DELAYEDEMBEDDEDCRACKS2.1.TransitionfromsmearedtoembeddedcrackThedirectionofanembeddeddiscontinuity,onceintroduced,isnormallykeptxed.Recently,someresearchershaveproposedmodelswithrotatingdiscontinuities[5{7].Itisquestionablewhethersuchformulationscanleadtoecientandrobustnumericalalgorithms.Accordingtothepresentauthors'experience,thepassageofanodefromonesideofthediscontinuitytotheothergeneratesnumericalinstabilities,andaspecialtreatmentisrequiredtohandlesuchsitua-tions.Also,somedoubtshavebeenraisedregardingthethermodynamicadmissibilityofarotatingdiscontinuity.Nevertheless,thisideacertainlydeservesafurtherinvestigation.Inthepresentstudyweshallexploreanalternativewayofallowinganadjustmentofthecrackpath.Observingtheevolutionofthecrackpatternsimulatedwithasmearedcrackmodel,e.g.withtheRC-SDmodel,wenotethattheinitialmispredictionofthecrackdirectionisoftencorrectedasthecrackgrows,becausetheoverallevolutionofthecrackingbandforceseachindividualcracktorotateintoapositioninwhichitseparatesnodescorrectly(cf.Figure1(d)).Soitcanbeexpectedthattheresolutionofthemacroscopiccrackshallbeimprovedifwedonotintroducethedisplacementdiscontinuityimmediatelyattheonsetofcrackingbutwestartwithasmearedcrackdescriptionandactivateanembeddedcrackonlyafteracertaincriticalcrackopening,say,hasbeenreached.Foreasyreference,letuscallsuchanapproachthedelayedembeddedcrackmodel(DEC),asopposedtothe`pure'embeddedcrackmodel(EC)withdisplacementdiscontinuitiesintroducedrightattheonsetofcracking.TheDECmodelcanbedescribedasasmearedcrackmodelwithtransitiontoanembeddedcrack.Ateverymaterialpoint,theinitialstageofcrackingismodeledinasmearedmanner,i.e.intermsofcrackingstrain.Whenthecrackopening(denedasthecrackingstrainmultipliedbytheeectiveelementsize[82])reachesacriticalvalue,adisplacementdiscontinuityisintroducedintotherespectiveniteelement.Fromthatmomenton,damageinthebulkofthematerialisfrozen,andfurtherdegradationisdescribedbydamageattheembeddedinterface.Thismeansthat,aftertransitiontotheembeddedcrack,thebulkmaterialisconsideredaslinearelastic,however,withamaterialstinessmatrixthatcorrespondstothesecant(unloading)stinessofthesmearedmodelatthetimeoftransition.Ofcourse,itisnecessarytotuneupthematerialparametersof2001JohnWiley&Sons,Ltd.Int.J.Numer.Meth.Engng EMBEDDEDCRACKMODEL.PARTII Figure3.Load{displacementdiagramsforthethree-point-bendspecimen.Thecurveshavebeenobtainedwiththede-layedembeddedcrackmodel(DEC),smearedrotatingcrackmodelwithtransitiontoscalardamage(RC-SD),andembeddedcrackmodel(EC).Figure4.Matchingofmateriallawsforthesmearedanddiscreteparts.bothcomponents|thecontinuumlawforthesmearedpartandtheinterfacelawforthediscretepart|suchthatthetransitionissmoothandthetotalenergydissipationiscorrect.ThematchingoftheconstitutiverelationsisillustratedinFigure4.ThediagraminFigure4(a)correspondstothestress{separationlawexploitedbythesmearedmodel.Toconstructthestress{strainlaw,thecrackopening,,istransformedintothecrackingstrain,,whereistheeectiveelementsize.Thisistheusualtechniqueappliedinsimulationsoflocalizationduetostrainsofteninginordertoavoidpathologicalsensitivitytotheelementsize[89].AstandardsmearedcrackmodelwouldfollowthecompletesofteningcurveA{B{C;theareaunderthiscurvecorrespondstothefractureenergyofthematerial.ThesmearedpartofaDECmodelfollowstheinitialpartA{Bofthesofteningcurve,butatpointBthediusedamageisfrozen,i.e.thesmearedcracksarenotallowedtogrowanymore.Thebulkofthematerialisnowtreatedaslinearelasticwithareducedstiness,andthedependencebetweenthesmearedcrackingstrainandthestressnormaltothecracksisdescribedbythestraightlineO{B{E.Atthesametime,adiscretecrackisembeddedintotheelement,governedbythetraction{separationlawplottedinFigure4(b).ThesofteningcurveBisconstructedasthedierencebetweencurveB{CandstraightlineB{Ointhetopdiagram.Consequently,curvedtrianglesOBCandOexactlythesamearea,andtheenergydissipatedbytheembeddedcrackisequaltothedierencebetweenthefractureenergy(areaofOABC)andtheenergydissipatedbythesmearedcrack(areaofOAB).Asthetractiontransmittedbytheembeddedcrackdecreases,thestressnormaltothesmearedcrackmustalsodecrease,andthesmearedpartofthemodelfollowsthesegmentB{OratherthanB{E.Strictlyspeaking,alltheseconsiderationsareexactonlyinthesimpleuniax-ialcase,butnumericalexperienceshowsthattheresultsarereasonableeveninmorecomplex2001JohnWiley&Sons,Ltd.Int.J.Numer.Meth.Engng M.JIRASEKANDT.ZIMMERMANN Figure5.Wedge-splittingtest:(a)specimengeometryandloading;(b)load{displacementdia-grams;(c)crackpatternforelementswithembeddedcracks(EC);(d)crackpatternforelementswithdelayedembeddedcracks(DEC).2.2.ResultsofsimulationsThecrackpatterninFigure2(d),obtainedwiththeDECmodel,indicatesthatthedelayindeedleadstoanimprovementintheorientationofindividualdiscontinuities.Thecompletefailurepatternhasformedwithoutanysecondarycracking,withasinglecrackperelement.Theload{displacementdiagraminFigure3isverysimilartothoseobtainedwiththeRC-SDmodelandwiththe(multiple)ECmodel.Thesimulationshavebeenrepeatedforawedge-splittingspecimentestedbySlowik[10].Thematerialparametersforthesmearedmodels(RCandRC-SD)havebeentakenfrom[3]:Young's=50GPa,Poisson'sratio2,tensilestrength9MPa,andfractureenergy=70Nm.Theembeddedcrackmodelhasusedthesamebasicpropertiesandthesametypeofsofteninglaw(bilinear)asthesmearedmodel.Theratioofreferencestinesseshasbeentakenas1,andfortheDECmodeltransitiontookplaceatcrackopening=16m.Figure5showstheload{displacementdiagramsforvariousmodelsandthecrackpatternsforECandDECelements.Again,weobservethattheembeddedcracksalleviatestresslocking,eventhoughtheECmodelstillgivesasomewhattoostiresponseinthelatestagesofsoftening.TheDECmodelleadstoamoreregularcrackpatternwithalesssignicantamountofsecondarycracking.2001JohnWiley&Sons,Ltd.Int.J.Numer.Meth.Engng EMBEDDEDCRACKMODEL.PARTII Figure6.Four-pointsheartest:(a)specimengeometryandloading;(b)load{displacementdia-grams;(c)crackpatternforelementswithembeddedcracks(EC);(d)crackpatternforelementswithdelayedembeddedcracks(DEC).Figure6showstheresultsofasimulationofthefour-pointshearspecimentestedbyArreaandIngraea[11].Thematerialparametersforthesmearedmodels(RCandRC-SD)havebeentakenfromReference[3]:=30GPa,5MPa,andfractureenergy=140NTheembeddedcrackmodelhasusedthesamebasicproperties.Theratioofreferencestinesseshasbeentakenas1,andfortheDECmodeltransitiontookplaceatcrackopening=26m.Inordertoavoidtroubleswithmesh-induceddirectionalbiaswehaveusedameshthatoersthecrackanopportunitytopropagatethroughalayerofelementsarrangedalongasmoothcurvewhichapproximatelycorrespondstotheexperimentallyobservedtrajectory(dashedcurveinFigure6(a)).Theload{displacementdiagraminFigure6(b)conrmsthattheembeddedcrackmodelsdonotexhibitlockingevenifthemacroscopiccracktrajectoryiscurved.ThecrackpatternsareshowninFigures6(c)and6(d).TheECandDECelementsgivealmostidenticalresultsbutconvergenceproblemsappearfortheECapproachduetomultiplecrackingatlatestagesoftheloadingprocess.TheDECapproachismorerobustandallowstocontinuethesimulationuptotheformationofacompletecrack.2001JohnWiley&Sons,Ltd.Int.J.Numer.Meth.Engng M.JIRASEKANDT.ZIMMERMANN Figure7.Four-pointsheartestsimulatedonalessfavourablemesh:(a)crackpatternforstandardelementsandsmearedrotatingcrackmodel(RC);(b)crackpatternforelementswithdelayedembeddedcracks(DEC).3.NON-LOCALFORMULATION3.1.MotivationThemodelcombiningtheembeddedcracktechniquewiththesmearedcrackapproachreducestheamountofsecondarycrackingbutstillsuersbymesh-induceddirectionalbias.ThisisillustratedinFigure7,whichshowsthenalcrackpatternsforthefour-pointsheartestsimulatedonalessfavourablemesh.Inthiscase,themeshwasgeneratedwithouttakingintoaccounttheexpectedcracktrajectory,andthecrackbandcannotpropagatealongasmoothcurve.Initially,thebandofcrackingelementsstartsfromthenotchandfollowsthepreferredmeshlinesthatareclosetothecorrectdirection.However,intheregionwherethemeshisnotalignedwiththeactualcracktrajectory,thecomputationalcrackisattractedbymeshlinesrunningapproximatelyintheverticaldirection.Forthestandardrotatingcrackmodel(RC),thisisaccompaniedbyseverestresslocking,becausetheinitialbranchhastoopeninamixedmode,whichisnotcorrectlyreproducedbythestandardniteelementkinematics.Duetolocking,secondarycrackingisproducedandthecrackingbanddevelopsadditionalbranchesthatareclearlynon-physical(Figure7(a)).TheDECmodel,usingthesmearedrotatingcrackatearlystagesandexplicitlymodelleddisplacementdiscontinuityembeddedintoniteelementsatlatestagesofthefractureprocess,doesnotsuerbylockingandprovidesacleancrackingband(Figure7(b)),butthedirectionofthebandisobviouslybiasedbytheniteelementmesh.Dependenceofthecracktrajectoryontheorientationofmeshlinescanbealleviatedifthesmearedpartofthemodelisreformulatedasnon-local.Inallexamplesbelow,thesmearedpartisrepresentedbyanon-localisotropicdamagemodel.3.2.Descriptionofnon-localmodelThemateriallawexploitedhereisamodiedversionofthenon-localdamagemodelproposedbyPijaudier-CabotandBazant[12].Thestress{strainlawiswritteninthecompacttensorialnotationas=(1=(12001JohnWiley&Sons,Ltd.Int.J.Numer.Meth.Engng 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 EMBEDDEDCRACKMODEL.PARTIIisascalardamageparameter,andistheeectivestress.InReference[12],theevolutionofdamagewascontrolledbythedamageenergyreleaserate,2.Thisbasicversionoftheisotropicdamagemodelisnotappropriateforconcrete,whichhasverydierentpropertiesintensionandincompression.Werestrictourattentiontopredominantlytensilefailureandassumeunlimitedcompressivestrength.Theevolutionofdamageisdrivenbytheequivalent Ep wheretheMcAuleybracketsdenotethe`positivepart'operator.Forascalar,thepositivepartisdenedas=max(0).Forasymmetricsecond-ordertensor,thepositivepartisawiththesamepricipalaxesasandwithprincipalvaluesobtainedbyextractingthe(scalar)positivepartofthecorrespondingprincipalvaluesofInitially,thedamageparameterisequaltozero,andtheresponseofthematerialislinearelastic.Whentheequivalentstrainreachesacertainthresholdvalue,,thedamageparameterstartsgrowing,whichre ectsthegraduallossofintegrityofthematerial.Duetothespecialchoiceoftheexpressionforequivalentstrain(2),theelasticdomainintheprincipalstressspaceissimilartotheonedescribedbytheRankinecriterionofmaximumprincipalstress,withasmoothround-ooftheedgesandvertexcorrespondingtobiaxialandtriaxialtension.Forgeneralapplications,coveringalsocompressivefailure,itwouldbenecessarytorenethemodel,e.g.usingtheideasfromReference[13].Duringunloading,characterizedbyadecreasingvalueof~,thedamageparameterremainsconstant.Thiscanbetakenintoaccountbymakingdependentonasofteningparameter,~whichisdenedasthemaximumvalueof~reachedintheprevioushistoryofthematerialuptothecurrentstate.Thedamageevolutionisthendescribedbytheexplicitrelation)(3)andfunctioncanbeidentiedfromtheuniaxialtensilestress{straincurve.Herewetakeasimplelaw[14]0if~ ~�~0 if~whichcorrespondstolinearelasticbehaviouruptothepeakstress,,followedbyexpo-nentialsoftening.Theparameteraectsductilityoftheresponseandisrelatedtothefractureenergyofthematerial.Inthenon-localversionofthemodel,the`local'equivalentstrain,~,isreplacedbyitsweightedaverageoveracertainneighbourhoodofthegivenpoint,Thenon-localweightfunctionisusuallydenedas 2001JohnWiley&Sons,Ltd.Int.J.Numer.Meth.Engng M.JIRASEKANDT.ZIMMERMANN if00ifisabell-shapedfunctiondecayingwiththedistancebetweenpoints.For,thesupportof)(i.e.closureofthesetofpointsatwhich)isnon-zero)isacircleorsphereofradiuscenteredat.In(6),thescalingfactorensuresthatthenormalizingcondition=1(9)holdseveninthevicinityoftheboundary,wherethesupportof)protrudesoutsidethe.Parametercanbecalledtheinteractionradiusbecausethenon-localstrainatagivenpointisaectedonlybypointscloserthan.Theinteractionradiusisdirectlyrelatedtothenon-localcharacteristiclengthofthematerial,whichinturndependsonthemicrostructure,e.g.onthesizeandspacingofmajorinhomogeneities.3.3.SimplesimulationsNumericalsimulationsperformedwiththenon-localdelayedembeddedcrackmodelhaverevealedtheimportanceofthecriterionforplacingthediscontinuityline.Thedirectionofthediscontinuitylineisassumedtobeperpendiculartothedirectionofmaximumprincipalstressorstrain.However,theexactpositionofthediscontinuityisnotfullydeterminedevenwhenthedirectionisknown.Itisnecessarytosupplyanadditionalpieceofinformation,e.g.selectonepointofthediscontinuityline.Inprinciple,thiscouldbeanypointofthecorrespondingniteelement.Considerrstthecasewhenthepositionofeachembeddedcrackisdeterminedbytheelementcentreandwhenthecrackisperpendiculartothedirectionofmaximumprincipalstress(whichcoincideswiththedirectionofmaximumprincipallocalstrain)atthetimeoftransitionfromsmearedtoembeddedcrackformulation.Inthiscase,thecracksegmentsinneighbouringelementsingeneraldonotformacontinuouspath.Plate1(a){(c)showstheevolutionofthefractureprocesszoneinathree-point-bendspecimenfromFigure1(a).ThemeshissomewhatnerthanintheoriginalexamplefromFigure1,tomakesurethatthenon-localinteractionisproperlycaptured(thedistancebetweenneighbouringintegrationpointsmustbesmallerthantheinteractionradiusofthenon-localmodel).Onlytheopeningembeddedcracksareexplicitlyvisualizedasdiscontinuities(whitelines)whilethediusedamageinthecontinuouspartofthemodelisrepresentedbythecolorscale.Severestresslockingisobservedatlaterstagesoftheloadingprocess.Thisis NotethatPlate1displaystherawresultsdirectlyobtainedfromtheniteelementinterpolation.Astheelementsareconstantstraintriangles,thestressstateandthedamageparameterineachelementareconstant,withjumpsbetween2001JohnWiley&Sons,Ltd.Int.J.Numer.Meth.Engng EMBEDDEDCRACKMODEL.PARTII Figure8.(a)Incorrectand(b)correctseparationofnodes.documentedinPlate1(d),whichshowsthedistributionofthemaximumprincipalstressinthestatecorrespondingtoPlate1(c).Thesourceoflockingisrevealedintheclose-upinFigure8(a)|theembeddedcracksdonotseparatethenodescorrectly.Thetwonodesmarkedbylledcirclesareobviouslyonthesamesideofthebandofcrackingelementswhilethenodemarkedbyanemptycircleisontheoppositeside.However,thediscontinuitylineinthecorrespondingelementfailstoseparateoneofthelledcirclesfromtheemptycircle,andthemacroscopiccrackcannotopenproperly.Paradoxically,thealleviationofmeshbiashasratheranadverseeect.Thesmearedcracksdonotrotateintoapositioninwhichtheywouldseparatethenodescorrectlybecausetheyremainalignedwiththemacroscopiccracktrajectory.Theremedyistoenforcecontinuityoftheembeddedcrackpath.Insteadofplacingeachdiscontinuitylineintothecentreofthecorrespondingelement,itischeckedwhetheroneoftheneighbouringelementsalreadyhasanembeddeddiscontinuitythatintersectsthecommonside.Ifthisisthecase,thenewdiscontinuitylineisplacedsuchthatitintersectsthecommonsideatthesamepoint.Figure8(b)demonstratesthatthistechniqueproperlyseparatesthenodes,sothatthecrackcanopenwithoutanyspuriousstresstransfer.Whencontinuityofthecrackpathisenforced,lockingdisappearsandtheoverallcrackdirectioniscorrectbutlocallythecrackistortuous;seeFigure9(a).Meshrenementdoesnotalwayshelp,asisclearfromthebottompartofthegure.Aperfectlystraightcracktrajectoryinexactlythecorrectdirectionisobtainedifthecrackpathcontinuityisenforcedandif,inaddition,thedirectionofthecracksegmentineachelementistakenasorthogonaltothedirectionofmaximumstrain;seeFigure9(b).Theresultsareconrmedbysimulationonanemeshwithaclearlymarkedpreferreddirectiondeviatingonlyslightlyfromthecorrectcracktrajectory.ThebottompartofFigure9(b)demonstratesaremarkableinsensitivityofthesimulatedtrajectorytotheorientationofthemesh.Theevolutionofthefractureprocesszoneandthegradualtransitionfromacontinuousrep-resentationofcrackingtoadiscontinuousoneisillustratedinPlate2.TheresultsshowninPlate2(a),(c)havebeenpostprocessedbyasmoothingprocedure.Thecolorscorrespondtovar-iouslevelsoftotaldamageinthebulkofthematerial(Plate2(a))andofthemaximumprincipalstress(Plate2(c)).Plate2(b)providesinformationontheincrementsofdamage(notpostpro-cessed).Theredcolorindicatesgrowingdamage(non-zeroincrements)whiletheregionwheredamageremainsconstantisplottedinblue.Itisobviousthatthebulkmaterialinthewakeoftheembeddedcrackisunloading.Diusedamagekeepsgrowinginalimitedregion,whichissituatedmainlyaheadoftheembeddedcracktip.Intheremainingpartofthefractureprocesszone,thedamageparameterdoesnotevolveanymore.Thepresentmodelisthereforecapableof2001JohnWiley&Sons,Ltd.Int.J.Numer.Meth.Engng M.JIRASEKANDT.ZIMMERMANN Figure9.Embeddedcracktrajectoryforamodelwithenforcedcontinuityofthecrackpath,withdirectiondetermined:(a)fromthelocalstrain;(b)fromthenon-localstrain.describingthetransitionfromdiusetohighlylocalizeddamage.Letusemphasizethat,atanystageoftheloadingprocess,themodelcontainssimultaneouslyadiscretecrack(embeddedin-niteelements)andregionswithgrowingandclosingsmearedcracks.Thisconceptismore exibleandbetterre ectstheactualphysicalprocessesthantheapproachpreviouslyproposedbyMazarsandPijaudier-Cabot[1516],whichjumpsfromthecontinuumrepresentationtothediscreteone(orviceversa)atallmaterialpointssimultaneously.3.4.ApplicationexampleTheprecedingexamplesclearlydemonstratethatthenon-localformulationofthesmearedpartofthemodelsubstantiallyimprovestheinsensitivityofthenalembeddedcracktotheorientationoftheniteelementmesh.However,itisnotclearyetwhethertheincorporationofembeddeddiscontinuitiesintothecombinedmodelhasalsosomebenecialeectonitsoverallperformance.Originally,theembeddeddiscontinuitiesweremotivatedbybetterresolutionofthekinematicsofawidelyopeningcrack.Sincestresslockingdoesnotseemtobeaseriousproblemfornon-localcontinuummodels(unlessitisinducedbyaninappropriatenon-localformulation,seeReference[17]),onemaybetemptedtothinkthatthetransitiontoembeddedcracksisonlyanelegantbutdispensableadditiontotheconventionalapproach.However,thereareanumberofsituationsinwhichtheimprovedrepresentationofhighlylocalizedfractureisindeedessential.Oneofthem2001JohnWiley&Sons,Ltd.Int.J.Numer.Meth.Engng EMBEDDEDCRACKMODEL.PARTIIisdescribedinthenalexample,whileanotherhasbeendiscussedatarecentconference[18].Thegrowthandcoalescenceofmicrocracksinthefractureprocesszoneeventuallyleadtotheformationofastress-freemacroscopiccrack.Thisistosomeextentre ectedbytheevolutionofthesimulatedprocesszoneiftheunderlyingmodelisbasedonnon-localdamagemechanics.InReference[19]itwasshownthatthezoneofincreasinglocalstraingetsthinnerastheloadingprocesscontinues,whilethezoneofincreasingnon-localstrainkeepsanapproximatelyconstantwidth.Thesecantmaterialstinessandtheresidualstrengthtendtozeroinanitezonewhosethicknessisequaltothesupportdiameterofthenon-localweightfunction(twicetheinteraction).Thiszonemustalwayscontainseveralniteelements,otherwisethenon-localinteractionwouldnotbeproperlycapturedbythenumericalmodel.Thecompletedegradationoftheresidualstrengthinabandcontainingseveralelementsacrossitsthicknessiscertainlynotrealisticanditcausesseriousnumericalproblemsifthestructureissubjectedtodistributedbodyforces.Thisisthecase,e.g.intheanalysisofgravitydams,wheretheeectofgravityforcesisessentialandcannotbeneglected.Thepresenceofgravityforcesinsidethefractureprocesszoneleadstospuriousshiftingofthecentreofthezoneandnallytothedivergenceoftheequilibriumiterationprocess,documentedinPlate3.Agravitydamwithdimensionscorrespondingtothewell-knownKoynaDam(struckbyanearthquakein1967)isloadedbyitsownweight,fullreservoirpressure,andanincreasinghydro-staticpressureduetoreservoirover ow(Plate3(a)).Thecomputationalmesh,consistingof4295lineartriangularelements,isshowninPlate3(b).Inasimulationusingthenon-localrotatingcrackmodel[3]withexponentialsofteningandparameters=25GPa=1MPa,and=1m,theinitiallydiusecrackingzoneattheupstreamface(Plate3(c))eventuallylocalizesandpropagatesintotheinteriorofthedam(Plate3(d)).Theclose-upregiondepictedinPlate3(c){(f)ismarkedinPlate3(a)byasmalldottedrectangle.ThelargerdashedrectanglecorrespondstotheregionthatshallbeshowninPlate4.Blueandredrectanglesrepre-sentopeningandclosingsmearedcracks,respectively.Whenthemouthofthemacroscopiccrackopenssucientlywide,thematerialbecomestooweaktoresistthe(constant)bodyforces.Theequivalentnodalforcesstartpullingdownthenodessurroundedbytheweakestmaterialaroundthecentreoftheprocesszone,whichclosesthecracksinthebottompartofthezoneandaccel-eratesopeningofthecracksinthetoppart.Bynon-localinteraction,theopeningcrackslocatedclosetotheupperboundaryoftheprocesszoneinducedamageintheoriginallysanemateriallayerlocatedjustabovetheprocesszone.Byadominoeect,theactivepartofthezonetravelsupwards(Plate3(e),(f))untilconvergenceislost.Needlesstosay,thismechanismthatshiftsthenumericalcrackperpendiculartoitstrajectoryiscompletelynon-physicalandhasitsoriginintheassumptionofaconstantnon-localinteractiondistance.Anelegantremedyisagradualtransitionfromasmearedtoadiscretedescriptionofmaterialfailure.Itcanbearguedthat,asfractureprogresses,long-rangeinteractionbetweenmaterialpointsbecomesmoredicultandnallyimpossible.Thiswouldbebestre ectedbyanon-localmodelwithanevolving(decreasing)characteristiclength.However,suchamodelwouldbecomputa-tionallyveryexpensive,sincetheinteractionweightsforallinteractingpairsofintegrationpointswouldhavetobecontinuouslyrecomputed.Theapproachproposedinthepresentpapercanbeconsideredasareasonableapproximation,withaconstantinteractionradiusatlowlevelsofdam-ageandadiscretecrackdescription(correspondingtoazerointeractionradius)athighlevelsofThesimulationofagravitydamhasbeenrepeatedusingthemodelwithtransitionfromnon-localrotatingcrackstoembeddeddisplacementdiscontinuities(discretecracks).Thetransition2001JohnWiley&Sons,Ltd.Int.J.Numer.Meth.Engng M.JIRASEKANDT.ZIMMERMANNtakesplacewhenthecrackingstrainreachesacertaincriticalvalue,inthepresentexampletakenas500,andthesofteninglawsdescribingthesmearedanddiscretepartsofthemodelarematchedsuchthatasmoothtransitionandcorrectenergydissipationareensured.Theinitialcrackingpatternisthesameasbefore,butsoonafterlocalizationtherstdiscretecrackappearsandpropagatesalongthecenterofthefractureprocesszone(seethewhitecurveinPlate4).Thepartiallydamagedmaterialinthewakeofthediscretecrackunloads,anditsresidualstrengthdoesnotdiminishanymore.Therefore,thecorrectcracktrajectorycanbecapturedwithoutanynumericalinstabilities.Notethatthematerialinthewakeoftheembeddedcrackisingeneralunloading,becausethetractioncarriedbythediscontinuityisrelievedasthedisplacementjumpincreases.However,aclusterofactive(opening)smearedcracksfarbehindthetipofthediscretecrackcanbedetectedinPlate4(d).Thisisnotaspuriouseectproducedbystresslocking,butanaturalindicationofincipientbranchingoftheprimarymacroscopiccrack.Thisphenomenonhasalreadybeenobservedinsimulationsusingapurenon-localrotatingcrackmodel[20]andhasalsobeendiscussedandillustratedinReference[21,p.375andfrontcover].4.CONCLUDINGREMARKSNumericaltestinghasrevealedthattheconventionalembeddedcrackapproach,whichintroducesadisplacementdiscontinuityrightattheonsetofcracking,oftenleadstoamispredictionofthedis-continuitydirection.Asthedirectionhastoremainxed,thereisnochanceforitsadjustment.Thisinevitablyleadstostresslockingthatmustberelaxedbyasecondarycrackinthesameelement.Multiplecrackingcomplicatesthenumericalalgorithmandcanleadtoconvergenceproblems.Ithasthereforebeenproposedtouseacombinedmodelthatrepresentstheearlystageofcrackinginasmearedmannerandintroducesadiscontinuityonlywhenthecrackopenssucientlywide.Ifthesmearedpartismodeledbytherotatingcrackapproach,thecrackhasachancetoreadjustitsdirection,andthereisnoneedforsecondarycracking.Thecombinationofthesmearedandembeddeddescriptionsofcrackingisappealingfromthephysicalpointofview.Itisintuitivelyclearthatdiusedamageatearlystagesofmaterialdegradationisadequatelydescribedbyamodeldealingwithinelasticstrain,whilehighlylocalizedfractureisbetterrepresentedbyadisplace-mentdiscontinuity.Examplesofsimulationsoffracturespecimensdemonstratethepotentialofthedevelopedtechnique.Asanadditionalimprovement,thesmearedpartofthecombinedmodelhasbeenreformulatedasnon-local.Itturnsoutthatforthealleviationoflockingitisessentialtoenforcecontinuityoftheembeddedcracktrajectory.Optimalperformanceintermsofinsensitivitytomesh-induceddirectionalbiasisobtainediftheorientationoftheembeddedcrackisineachelementdeterminedfromtheprincipaldirectionsofnon-local(ratherthanlocal)strain.Transitiontotheexplicitdescriptionofacrackasadisplacementdiscontinuityturnsouttobeanecientremedytopathologicalshiftingoftheprocesszone,exhibitedbythenon-localdamagemodelwithaconstantcharacteristiclength.Tosummarize,theembeddedcrackapproachseemstobeaveryappealingtechniquethatcertainlyhasanumberofimportantadvantagescomparedtomoretraditionalapproaches.Ofcourse,alargeamountofworkstillremainstobedone.Forexample,achallenginggoalwouldbeanextensionofthemodeltothreedimensions.2001JohnWiley&Sons,Ltd.Int.J.Numer.Meth.Engng EMBEDDEDCRACKMODEL.PARTIIFinancialsupportoftheSwissCommissionforTechnologyandInnovationunderprojectCTI.3201.1isgratefullyacknowledged.1.JirasekM,ZimmermannTh.Embeddedcrackmodel:I.Basicformulation.InternationalJournalforNumericalMethodsinEngineering,thisissue.2.RotsJG.Computationalmodelingofconcretefracture.Ph.D.Thesis,DelftUniversityofTechnology,Delft,TheNetherlands,1988.3.JirasekM,ZimmermannTh.Rotatingcrackmodelwithtransitiontoscalardamage.JournalofEngineeringMechanicsASCE1998;:277{284.4.JirasekM,ZimmermannTh.Analysisofrotatingcrackmodel.JournalofEngineeringMechanicsASCE1998;842{851.5.TanoR.Localizationmodellingwithinnersofteningbandniteelements.LicentiateThesis,LuleaUniversityofTechnology,Sweden,August,1997;26.6.TanoR,KlisinskiM,OlofssonTh.Stresslockingintheinnersofteningbandmethod:astudyoftheoriginandhowtoreducetheeects.InComputationalModellingofConcreteStructures,deBorstR,BicanicN,MangH,MeschkeG(eds).Balkema:Rotterdam,1998;329{335.7.SluysLJ,BerendsAH.2D3Dmodellingofcrackpropagationwithembeddeddiscontinuityelements.In:ModellingofConcreteStructures,deBorstR,BicanicN,MangH,MeschkeG(eds).Balkema:Rotterdam,1998;399{408.8.PietruszczakS,MrozZ.Finiteelementanalysisofdeformationofstrain-softeningmaterials.InternationalJournalforNumericalMethodsinEngineering9.BazantZP,OhBH.Crackbandtheoryforfractureofconcrete.eriauxetConstructions10.SlowikV.BeitragezurexperimentellenBestimmungbruchmechanischerMaterialparametervonBeton.MaterialsReportsNo.3,ETHZurich,1995.11.ArreaM,IngraeaAR.Mixed-modecrackpropagationinmortarandconcrete.DepartmentofStructuralEngineering81-13,CornellUniversity,Ithaca,NY,1981.12.Pijaudier-CabotG,BazantZP.Nonlocaldamagetheory.JournalofEngineeringMechanicsASCE1987;13.SaouridisC,MazarsJ.Predictionofthefailureandsizeeectinconcreteviaabi-scaledamageapproach.:329{344.14.OliverJ,CerveraM,OllerS,LublinerJ.Isotropicdamagemodelsandsmearedcrackanalysisofconcrete.InAidedAnalysisandDesignofConcreteStructures,BicanicN,MangH(eds).PineridgePress:Swansea,UK,1990;945{957.15.MazarsJ,Pijaudier-CabotG.Fromdamagetofracturemechanicsandconversely:acombinedapproach.JournalofSolidsandStructures16.BodeL,TailhanJL,Pijaudier-CabotG,LaBorderieCh,ClementJL.FailureanalysisofinitiallycrackedconcreteJournalofEngineeringMechanicsASCE1997;:1153{1160.17.JirasekM.Nonlocalmodelsfordamageandfracture:comparisonofapproaches.InternationalJournalofSolidsand:4133{4145.18.Jirasek,M.Nonlocaldamagemodels:practicalaspectsandopenissues.Proceedingsofthe13thASCEEngineeringMechanicsDivisionSpecialtyConference,JohnsHopkinsUniversity,Baltimore,13{16June1999.(CD-ROM)19.Jirasek,M.Comparisonofnonlocalmodelsfordamageandfracture.LSCInternalReport02,SwissFederalInstituteofTechnologyatLausanne,Switzerland,1998.20.JirasekM,ZimmermannT.Rotatingcrackmodelwithtransitiontoscalardamage:I.Localformulation,II.Nonlocalformulationandadaptivity.LSCInternalReport01,SwissFederalInstituteofTechnologyatLausanne,Switzerland,21.BazantZP,PlanasJ.FractureandSizeEectinConcreteandOtherQuasibrittleMaterials.CRCPress:BocaRaton,2001JohnWiley&Sons,Ltd.Int.J.Numer.Meth.Engng