AssessmentoftheapplicabilityofXFEMinAbaqusformodelingcrackgrowthinrubberLuigiGigliottiSupervisor: Dr. MartinKroonMasterThesisStockholm,Sweden2012KTHSchoolofEngineeringSciencesDepartmentofSolidMechanicsRoyalInstituteofTechnologySE-10044Stockholm-SwedenAbstractTheeXtendedFiniteElementMethodisapartitionofunitybasedmethod,particularlysuitableformodellingcrack propagation phenomena, without knowing a priori the crack path. Its numerical implementation is mostlyachievedwithstand-alonecodes.TheimplementationoftheeXtendedFiniteElementMethodincommercialFEAsoftwaresisstilllimited,andthemostfamousoneincludingsuchcapabilitiesisAbaqusTM. However,duetoitsrelativelyrecentintro-duction, XFEM technique in Abaqus has been proved to provide trustable results only in few simple benchmarkproblemsinvolvinglinearelasticmaterialmodels.Inthis work, wepresent anassessment of theapplicabilityof theeXtendendFiniteElement MethodinAbaqus, todeal withfracture mechanics problems of rubber-like materials. Results are providedfor bothNeo-HookeanandArruda-Boycematerialmodels,underplanestrainconditions.In the rst part of this work, a static analysis for the pure Mode-I and for a 45omixed-Mode load condition,whose objective has been to evaluate the ability of the XFEM technique in Abaqus, to correctly model the stressanddisplacementeldsaroundacracktip, hasbeenperformed. OutcomesfromXFEManalysiswithcoarsemeshes have been compared with the analogous ones obtained with highly rened standard FEM discretizations.Noteworthy, despitetheremarkablelevel of accuracyinanalyzingthedisplacementeldatthecracktip,concerningthestresseld,theadoptionoftheXFEMprovidesnobenets,ifcomparedtothestandardFEMformulation. Theonlyremarkableadvantageisthepossibilitytodiscretizethemodel withoutthemeshcon-formingthecrackgeometry.Furthermore, thedynamicprocessofcrackpropagationhasbeenanalyzedbymeansoftheXFEM. A45omixed-Modeanda30omixed-Modeloadconditionareanalyzed. Inparticular, threefundamental aspectsofthecrackpropagationphenomenonhavebeeninvestigated,i.e. theinstantatwhichapre-existingcrackstartstopropagatewithinthebodyundertheappliedboundaryconditions,thecrackpropagationdirectionandthepredictedcrackpropagationspeeds.Accordingtotheobtainedresults,themostinuentparametersarethoughttobetheelementssizeatthecrack tip h and the applied displacement rate v. Severe diculties have been faced to attain convergence. Somereasonablemotivationsoftheunsatisfactoryconvergencebehaviourareproposed.Keywords: FractureMechanics;eXtendedFiniteElementMethod;Rubber-likematerials;AbaqusAcknowledgementsNowthatmymasterthesisworkiscompleted,therearemanypeoplewhomIwishtoacknowledge.First of all, my deepest gratitude goes to my supervisor Dr. Martin Kroon for his guidance, interesting cuesandforgivingmethepossibilitytoworkwithhim.Secondly,IamgratefultoDr. ArtemKulachenkoforhissuggestionsandstimulatingdiscussions.My big brother PhD student Jacopo Biasetti deserves a special thanks for being my scientic mentor and,most of all my friend, during my period at the Solid Mechanics Department of the Royal Institute of Technology(KTH).Lastbutnotleast,Iwillneverbesucientlythankfultomyfamilyfortheirconstantsupportandencour-agementduringmystudies.Stockholm,June2012LuigiGigliottiContentsAbstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiMotivationandoutline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv1 Fundamentals: literaturereviewandbasicconcepts 11.1 Rubberelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Kinematicsoflargedisplacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Hyperelasticmaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.3 IsotropicHyperelasticmaterialmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 FractureMechanicsofRubber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.1 Fracturemechanicsapproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.2 Stressaroundthecracktip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.3 Tearingenergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2.4 Qualitativeobservationofthetearingprocess . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.5 Tearingenergyfordierentgeometries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3 eXtendedFiniteElementMethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.2 Partitionofunity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3.3 eXtendedFiniteElementMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.3.4 XFEMimplementationinABAQUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.3.5 LimitationsoftheuseofXFEMwithinAbaqus . . . . . . . . . . . . . . . . . . . . . . . . 272 Problemformulation 292.1 Geometricalmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 Solutionprocedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 StaticAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4 DynamicAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5 Damageparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Numericalresults-StaticAnalysis 353.1 Stresseldaroundthecracktip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.1 PureMode-Iloadingcondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.2 Mixed-Modeloadingcondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2 Displacementeldaroundthecracktip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.1 PureMode-Iloadingcondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.2 Mixed-Modeloadingcondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 Numericalresults-DynamicAnalysis 454.1 Convergencecriteriafornonlinearproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1.1 Linesearchalgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 Crackpropagationinstant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3 Crackpropagationangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.4 Crackpropagationspeed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.5 Remarksonconvergencebehaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 Summary 65Bibliography 67MotivationandoutlineTheaimofthestudyinthepresentmasterthesishasbeentoassesstheapplicabilityoftheXFEMimplementationinthecommercial FEAsoftwareAbaqus, tohandlefracturemechanicsproblemsinrubber-likematerials. Inparticular,wefocusonthecapabilitiesofXFEManalyseswithcoarsemeshes,tocorrectlymodelthestressanddisplacementeldsaroundthecracktipunderpureMode-Iandundera45omixed-Modeloadconditioninstaticproblems. Suchabilitieshave been investigated in crack growth processes as well, and the eects of the most relevant parameters are emphasized.TheintroductionoftheeXtendedFiniteElementMethod(XFEM)representsundoubtedly,themajorbreakthroughinthecomputationalfracturemechanicseld,madeinthelastyears. Itisasuitablemethodtomodelthepropagationof strongandweakdiscontinuities. Theconceptbehindsuchtechniqueistoenrichthespaceof standardpolynomialbasis functions withdiscontinuous basis functions, inorder torepresent thepresenceof thediscontinuity, andwithsingularbasisfunctionsinordertocapturethesingularityinthestresseld. Byutilizingsuchmethod, theremeshingprocedureof conventional niteelement methods, is nomoreneeded. Therefore, therelatedcomputational burdenandresultsprojectionerrorsareavoided. FormovingdiscontinuitiestreatedwiththeXFEM, thecrackwill followasolution-dependentpath. Despitetheseadvantages,theXFEMarenumericallyimplementedmostlybymeansofstand-alonecodes. However,duringthelastyearsanincreasingnumberofcommercialFEAsoftwareareadoptingtheXFEMtechnique;amongthese,themostfamousandwidelyemployedisAbaqus,producedbytheDassaultSyst`emesS.A..TheimplementationofXFEMinAbaqusisaworkinprogressanditsapplicabilityhasbeenstillevaluatedonlyforparticularlysimplefracturemechanicsproblemsinplanestressconditionsandforisotropiclinearelasticmaterials. Noattempts have been made in considering more complex load conditions and/or material models. This work aims, at leasttosomeextent,atlimitingthelackofknowledgeinthiseld. Forthesereasons,theapplicationofXFEMinAbaqustomodelcrackgrowthphenomenainrubber-likematerials,hasbeeninvestigated.This report is organized as follows. In Chapter 1, a background of the arguments closely related to the present work,isprovided. Ashortreviewofthecontinuummechanicsapproachforrubberelasticity, alongwithconceptsoffracturemechanicsforrubberisgiven. Boththeoretical andpractical foundationsoftheeXtendedFiniteElementMethodarepresented. Inconclusionofthischapter,theapplicationofsuchmethodtolargestrainsproblemsandthemainfeaturesofitsimplementationinAbaqusarediscussed.InChapter2,theproblemformulationathandispresented. Furthermore,detailsabouttheperformednumericalsimulationsareprovided.Numerical results of the static analysis are summarized in Chapter3. The outcomes from coarse XFEM discretiza-tionsarecomparedtothoseobtainedwiththestandardFEMapproach, inordertoevaluatetheirabilitytodeal withfracturemechanicsproblemsinrubber-likematerials.In conclusion, numerical results of the crack growth phenomenon in rubber, analyzed by means of the XFEM techniqueinAbaqusarepresentedanddiscussedinChapter4. Threefundamentalaspectsofthecrackpropagationprocessareinvestigated,i.e. theinstantinwhichthepre-existingcrackstartstopropagateundertheprescribedloadingconditions,thedirectionof crackpropagationanditsspeed. Attheendof thischapter, conclusiveremarksontheconvergencebehaviourinnumerical simulationsofcrackgrowthphenomenabyutilizingtheeXtendedFiniteElementMethodare,indicated.ivChapter1Fundamentals: literaturereviewandbasicconcepts1.1 RubberelasticityRubber-likematerials, suchasrubberitself, softtissuesetc, canbeappropriatelydescribedbyvirtueof awell-knowtheoryinthecontinuummechanics framework, namedHyperelasticityTheory. For this purpose, inthis sectionthefundamental aspects of this theory- albeit limitedtothecaseof isotropicandincompressiblematerial - as well asnite displacements and deformations theory will be expounded [1]. Lastly,a description of the hyperelastic constitutivemodelsadoptedinthisworkisproposed.1.1.1 KinematicsoflargedisplacementsThemaingoalofthekinematicstheory, istostudyanddescribethemotionofadeformablebody, i.etodetermineitssuccessivecongurationsunderageneraldenedloadcondition,asfunctionofthepseudo-timet.A deformable body, within the framework of 3D Euclidean space, R3, can be regarded as a set of interacting particlesembedded in the domain R3(see Fig. 1.1). The boundary of this latter, often referred to as = is split up in twodierent parts, ualong which the displacement values are prescribed and where the stress component values have tobe imposed. A problem is said to be well-posedif these two dierent boundary conditions are not applied simultaneouslyonthesameportionoffrontier. Moreover,theboundaryshouldbecharacterizedbyasucientsmoothness(atleastpiece-wise),in order to dene uniquely the outward unit normal vector n;last but not least,it must be highlighted that,auniquesolutionoftheboundaryvalueproblemisachievableonlyifu ,= , suchthatall therigidbodymotionsareeliminated.Figure1.1: Initialanddeformedcongurationsofasoliddeformablebodyin3DEuclideanspace.Amongallpossiblecongurationsassumedbyadeformablebodyduringitsmotion,ofparticularimportanceisthereference(or undeformed) conguration, denedat axedreferencetimeanddepictedinFig. 1.1withthedashedline. Withexcessofmeticulousness,itisworthnotingthataso-calledinitialcongurationatinitialtimet = 0,canbedenedandthat,whilstforstaticproblemssuchcongurationcoincideswiththereferenceone,indynamicstheinitialcongurationisoftennotchosenasthereferenceconguration. Inthereferenceconguration,everyparticlesof12L.Gigliotti CHAPTER1. FUNDAMENTALS:LITERATUREREVIEWANDBASICCONCEPTSthedeformablebodyaresolelyidentiedbythepositionvector(orreferential position)xdenedasfollowsx = xiei x =__x1x2x3__(1.1)where eirepresent the basis vectors of the employedcoordinate system. The motionof abodyfromits referencecongurationiseasilyinterpretableastheevolutionofitsparticlesanditsresultingpositionattheinstantoftimetisgivenbytherelationx= t(x) (1.2)Atthispoint, themajordierencewithrespecttothecaseofsmall displacementsstartstoplayasignicantrole.Indeed, in case of deformable body, it is required to take account of two sets of coordinates for two dierent congurations,i.e. theinitial congurationandtheso-calledcurrent(ordeformed)conguration(=()). Undertheseassumptions, itisnowstraightforwardtowritethepositionvectorsforanyparticlesconstitutingthedeformablebodyinbothitsreferenceandcurrentcongurations:x = xiei; x= xi ei(1.3)beingeiandeitheunit basevectors inreferenceandcurrent conguration, respectively. Withintheframeworkofgeometricallynonlineartheory, inwhichlargedisplacementsandlargedisplacementgradientsareinvolved, thesimpli-cationusedinthelineartheory, namelythatnotonlythetwobasevectorsarecoincident, butalsothetwosetsofcoordinatesinbothreferenceandcurrentcongurations,cannolongerbeexploited. Inthisregard,itisthenrequiredtouniquelydenethecongurationwithrespecttowhich,theboundaryvalueproblemwillbeformulated.Forthisaim,thechoicehastobemadebetweentheLagrangianformulationinwhichalltheunknownsarereferredtothecoordinatesxi1inthereferencecongurationandtheEulerianformulation, inwhichspecularly, theunknownsaresupposedtodependuponthecoordinatesxi2inthedeformedconguration. Inuidmechanicsproblemsthemostappropriate formulation is the Eulerian one, simply because the only conguration of interest is the deformed one and, atleast for Newtonian uids, the constitutive behaviour is not dependent upon the deformation trajectory; on the contrary,theLagrangianformulationappears tobebetter suitablefor solidmechanics problems sinceit refers tothecurrentconguration. Furthermore, asitisnecessarytoconsiderthecompletedeformationtrajectory, itishencepossibletodene the corresponding evolution of internal variables and resulting values of stress within solid materials. Of particularinterestisamixedformulationcalledarbitraryEulerian-Lagrangianformulation,especiallywellsuitableforinteractionproblems,suchasuid-structureinteraction;inthiscase,theuidmotionwillbedescribedbytheEulerianformulationwhilethesolidevolutionbytheLagrangianone.1.1.1.1 DeformationgradientThe motion of all particles of a deformable body might be described by means of the point transformation x= (x); x .Let x=()with representing the parametrizationshowninFig. 1.2, a material (or undeformed)curveindependentwithrespecttotime. Thislatterisdeformedintoaspatialordeformedcurvex=(, t)=((), t) atanytimet. Atthispoint,itisusefultodeneamaterialtangentvectordxtothematerialcurveFigure1.2: Deformationofamaterialcurve intoaspatialcurve .andaspatial tangentvectordxtothespatialcurve.dx=(, t)d ; dx = ()d (1.4)1Oftencoordinatesxiarelabelledasthematerial(orreferential)coordinates.2Asabove,usuallywithxiarereferredtoasspatial(orcurrent)coordinates.KTHRoyal InstituteofTechnology-SolidMechanicsDepartment1.1. RUBBERELASTICITY L.Gigliotti 3Thetangent vectorsdxanddxareusuallylabelledasmaterial (orundeformed) lineelementandspatial (ordeformed) line element, respectively. Furthermore, for anymotiontakingplace inthe Euclideanspace, alargedisplacementvectorcanbeintroducedasfollowsd(x) = xdi(x) = jijxi(1.5)AccordingtoFig.1.3,andbymeansofbasicnotionsofalgebraoftensors,itispossibletoinferthat:Figure1.3: Totaldisplacementeld.x+ dx= x + dx +d(x + dx) =dx= dx +d(x + dx) d(x) (1.6)Thetermd(x + dx)canbecomputedbyexploitingTaylorseriesformula,andtruncatingthemafterthelinearterm:d(x + dx) = d(x) +d(x)dx + o(|dx|) (1.7)where, disthelargedisplacementgradient,orinindexnotationdi(x + dx) = di(x) +dixi(x)dxi + o(|dx|) (1.8)Atthispoint,itissucienttocompareEq. 1.6andEq. 1.7,toexpressthespatialtangentvectordxasfunctionofitscorrespondingmaterialtangentvectorandofthetwo-pointtensorFnamedasdeformationgradienttensor:dx= (I +d)..Fdx dx= Fdx ; [F]ij:=xixj+dixj=ixj(1.9)oralternatively,F =ixjei ej; F = (1.10)withthegradientoperator =xjei.It is apainless tasktodemonstrate howthe deformationgradient, not onlyprovides the mappingof ageneric(innitesimal) material tangent vector into the relative spatial tangent vector,but also controls the transformation of aninnitesimalsurfaceelementoraninnitesimalvolumeelement(seeFig.1.4).Figure1.4: Transformation of innitesimal surface and volume elements between initial and deformedconguration.KTHRoyal InstituteofTechnology-SolidMechanicsDepartment4L.Gigliotti CHAPTER1. FUNDAMENTALS:LITERATUREREVIEWANDBASICCONCEPTSLet dA be an innitesimal surface element, constructed as the vector product of two innitesimal reciprocally orthog-onalvectors,dxanddy;theoutwardnormalvectoris,asusual,denedasn = (dx dy)/ |dx dy|. BymeansofthedeformationgradientF,thenewextensionandorientationinthespaceofthesurfaceelementcanbeeasilydetermineddAn:= dxdy= (Fdx) (Fdy) = (det [F] FT)(dx dy). .dAn= dA(cof [F])n (1.11)whichisoftenreferredtoasNansonsformula.Once the Nansons formula has been derived, it is straightforward to obtain the analogous relation for the change of aninnitesimalvolumeelementoccurringbetweentheinitialandthedeformedconguration. Describingtheinnitesimalvolumeelementinthematerial congurationasthescalarproductbetweenaninnitesimal surfaceelementdAandtheinnitesimalvectordzandexploitingresultsinEq. 1.11,thefollowingrelationholdsdV:= dz dAn= FdzJFTdAn = JdzdAn = JdA (1.12)inwhichJ= det [F]iswell-knownastheJacobiandeterminant(orvolumeratio).As stated in Eq. 1.9, the deformation gradient F, is a linear transformation of an innitesimal material vector dx intoitsrelativespatial dx; suchtransformationaectsall parameterscharacterizingavector, itsmodulus, directionandorientation. However, thedeformationisrelatedonlytothechangeinlengthofaninnitesimal vector, andtherefore,itresultstobehandytoconsidertheso-calledpolardecomposition,byvirtueofwhichthedeformationgradientcanbe written as a multiplicative split between an orthogonal tensor R,an isometric transformation which only changes thedirectionandorientationofavector, andasymmetric, positive-denitestretchtensorUthatprovidesthemeasureoflargedeformation.F = RU ; RT= R1; UT= U ; |dx| = |Ux| (1.13)Inotherwords, thesymmetrictensorU, yetreferredinliteraturetoastheright(ormaterial)stretchtensor,producesadeformedvectorthatremainsintheinitial conguration(nolargerotations). Inindexnotation, thelargerotationstensorR,andtherightstretchtensorU,canberespectivelyexpressedasfollowsR = Rijei ej; U = Uijeiej(1.14)An alternative form of the polar decomposition can be provided, by simply inverting the order of the above mentionedtransformationsandintroducingtheso-calledleft(orspatial)stretchtensorVF = VR ; V = Vijei ej(1.15)Insuchcase,alargerotation,representedbyR,isfollowedbyalargedeformation(tensorV).1.1.1.2 StrainmeasuresTheoretically, apart form the right and the left stretch tensors Uand V, an innite number of other deformation measurescanbedened; indeed, unlikedisplacements, whicharemeasurablequantities, strainsarebasedonaconcept thatisintroducedasasimplicationforthelargedeformationanalysis.Fromacomputationalpointofview,thechoiceorUorVtocalculatethestressvaluesisnotthemostappropriateone, as it requires, rst, toperformthepolar decompositionof thedeformationgradient. Hence, it is necessarytointroducedeformationmeasures that candirectly, without anyfurther computations, provideinformationabout thedeformationstate.Wemayconsidertwoneighbouringpointsdenedbytheirpositionvectorsxandyinthematerialdescription;withreferencetoFig.1.5onthenextpage,itispossibletodescribetherelationbetweenthesetwo,sucientlyclosepoints,i.e.y = y + (x x) = x +y xy xy x= x + dx (1.16)dx = da and d =y x, a =y xy x(1.17)Intheaboveequations, it is clear that thelengthof thematerial lineelement dxis denotedbythescalar valuedandthattheunitvectora, witha=1, representsthedirectionof theaforesaidvectoratthegivenpositioninthereferenceconguration. AsstatedinEq. 1.9,thedeformationgradientFallowstolinearlyapproximateavectordxinthematerial description, withitscorrespondingvectordxinthespatial description. Thesmallerthevectordx, thebettertheapproximation.Atthispoint,itisthenpossibletodenethestretchvectora,inthedirectionoftheunitvectoraandatthepointx asa(x, t) = F(x, t)a (1.18)withitsmodulusknownasstretchratioorjuststretch. Thislatterisameasureofhowmuchtheunitvectorahasbeen stretched. In relation to its value, < 1, = 1 or > 1, the line element is said to be compressed, unstretchedKTHRoyal InstituteofTechnology-SolidMechanicsDepartment1.1. RUBBERELASTICITY L.Gigliotti 5Figure1.5: Deformationofamaterial lineelementwithlengthdintoaspatial elementwithlengthd.orextended, respectively. Computingthesquareof thestretchratio, thedenitionof theright Cauchy-GreentensorCisintroduced2= a a= FaFa = aFTFa = aCa ,C = FTF or CIJ= FiIFiJ(1.19)OftenthetensorCisalsoreferredtoastheGreendeformationtensoranditshouldbehighlightedthat, sincethetensor C operate solely on material vectors, it is denoted as a material deformation tensor. Moreover, C is symmetricandpositivedenite x :C = FTF = (FTF)T= CTand uCu > 0 u ,= 0 (1.20)TheinverseoftherightCauchy-Greentensoristhewell-knownPioladeformationtensorB,i.e. B = C1.Toconcludetheroundupof material deformationtensors, thedenitionof thecommonlyusedGreen-LagrangestraintensorE,ishereprovided:12_(d)2d2_ =12_(da)FTF(da) d2_ = dxEdx ,E =12_FTF I_ =12 (CI) or EIJ=12 (FiIFiJ IJ)(1.21)whosesymmetricalnatureisobvious,giventhesymmetryof CandI.Inananalogousmanneroftheoneshownabove,itispossibletodescribedeformationmeasuresinspatialconguration,too;thestretchvectorainthedirectionofa,foreachx mightthusbedeneas:1a(x, t) = F1(x, t)a(1.22)where, thenormof theinversestretchvector 1a is calledinversestretchratio1or simplyinversestretch.Moreover, the unit vector amay be interpreted as a spatial vector, characterizing the direction of a spatial line elementdx. ByvirtueofEq. 1.22,computingthesquareoftheinversestretchratio,i.e.2= 1a1a= F1aF1a = aFTF1a = ab1a (1.23)wherebistheleftCauchy-Greentensor,sometimesreferredtoastheFingerdeformationtensorb = FFTor bij= FiIFjI(1.24)Likeitscorrespondingtensorinthematerialconguration,theGreendeformationtensor,theleftCauchy-Greentensorbissymmetricandpositivedenite x :b = FFT= (FTF)T= bTand ubu > 0 u ,= 0 (1.25)Lastbutnotleast,thewell-knownsymmetricEuler-Almansi straintensoreishereintroduced:12[d 2(1d )2] =12[d 2(d a)FTF1(d a)] = dx edx,e =12(I FTF1) or eij=12(ij F1KiF1Kj)(1.26)wherethescalarvalued isthe(spatial)lengthofaspatiallineelementdx= xy.KTHRoyal InstituteofTechnology-SolidMechanicsDepartment6L.Gigliotti CHAPTER1. FUNDAMENTALS:LITERATUREREVIEWANDBASICCONCEPTS1.1.1.3 StressmeasuresDuringaparticulartransformation, themotionanddeformationwhichtakeplace, makeaportionofmaterial interactwiththerestof theinteriorpartof thebody. Theseinteractionsgiverisetostresses, physicallyforcesperunitarea,whichareresponsibleofthedeformationofmaterial.GivenadeformablebodyoccupyinganarbitraryregionintheEuclideanspace, whoseboundaryisthesurfaceatthespecictimet,letusassumethattwotypesofarbitraryforces,somehowdistributed,actrespectivelyontheboundarysurface(external forces)andonanimaginaryinternalsurface(internal forces).Letthebodybecompletelycutbyaplanesurface;therebytheinteractionbetweenthetwodierentportionsofthebodyisrepresentedbyforcestransmittedacrossthe(internal)planesurface. Undertheactionofthissystemofforces,Figure1.6: Tractionvectorsactingoninnitesimalsurfaceelementswithoutwardunitnormals.thebodyresultstobeinequilibriumconditionsbut,oncethebodyiscutintwoparts,bothofthemarenomoreunderthese equilibrium conditions and thus,an equivalent force distribution along the faces created by the cutting process hastobeconsidered, inordertorepresenttheinteractionbetweenthetwopartsofthebody. Theinnitesimal resultant(actual)forceactingonasurfaceelementdf,isdenedasdf = tdA= TdA (1.27)Here, t=t(x, t, n)isknownintheliteratureastheCauchy(ortrue)tractionvector(forcemeasuredperunitsurfaceareainthecurrent conguration),whilethe(pseudo)tractionvectorT = T(x, t, n)representstherstPiola-Kirchho(ornominal)tractionvector(forcemeasuredperunitsurfaceareainthereferenceconguration).In literature Eq. 1.27 is referred to as the Cauchyspostulate. Moreover, the vectors t and T acting across surfaceelements dA and dAwith the corresponding normals n and n, can be also dened as surfacetractions or, accordingtoothertexts,ascontactforcesorjustloads.Theso-calledCauchysstresstheoremclaimstheexistenceoftensoreldsandPsothatt(x, t, n) = (x, t)nor ti= ijnjT(x, t, n) = P(x, t)n or Ti= PiInI_(1.28)wherethetensordenotesthesymmetric3Cauchy(ortrue)stresstensor(orsimplytheCauchystress)andPisreferredtoastherstPiola-Kirchho(ornominal)stresstensor(orsimplythePiolastress.)Therelationlinkingtheabovedenedstresstensorsistheso-calledPiolatransformation,obtainedbymergingEq.1.27andEq. 1.28andexploitingtheNansonsformula:P = JFT; PiI= JijF1Ij(1.29)orinitsdualexpression= J1PFT; ij= J1PiIFjI= ji(1.30)Alongwiththestresstensorsgivenabove,manyothershavebeenpresentedinliterature;inparticular,themajorityof themhavebeenproposedinorder toeasenumerical analyses for practical nonlinear problems. Oneof themostconvenientistheKirchhostresstensor,whichisacontravariantspatialtensordenedby:= J ; ij= Jij(1.31)3Symmetryof the Cauchystress is satisedonlyunder the assumption(typical of the classical formulationof continuummechanics)thatresultantcouplescanbeneglected.KTHRoyal InstituteofTechnology-SolidMechanicsDepartment1.1. RUBBERELASTICITY L.Gigliotti 7In addition,the so-called secondPiola-KirchhostresstensorS has been proposed,especially for its noticeableusefulness in the computational mechanics eld, as well as for the formulation of constitutive equations; this contravariantmaterialtensordoesnothaveanyphysicalinterpretationintermsofsurfacetractionsanditcanbeeasilycomputedbyapplyingthepull-backoperationonthecontravariantspatialtensor:S = F1FTor SIJ= F1IiF1Jjij(1.32)The secondPiola-KirchhostresstensorS can be, moreover, related to the Cauchy stress tensor by exploiting Eqs.1.29,1.32and1.31:S = JF1FT= F1P = STor SIJ= JF1IiF1Jjij= F1IiPiJ= SJI(1.33)as consequence, the fundamental relationship between the rst Piola-Kirchho stress tensor P and the symmetric secondPiola-KirchhostresstensorSisfound,i.e.P = FS or PiI= FiJSJI(1.34)Aplethoraofotherstresstensorscanbefoundinliterature; amongthemtheBiotstresstensorTB, thesymmetryccorotatedCauchystresstensoruandtheMandel stresstensordeservetobementioned[2].1.1.2 HyperelasticmaterialsThecorrectformulationofconstitutivetheoriesfordierentkindsofmaterial,isaveryimportantmatterincontinuummechanics,inparticularwithregardstothedescriptionofnonlinearmaterials,suchasrubber-likeones.The branch of continuum mechanics, which provides the formulation of constitutive equations for that category of ma-terials which can sustain to large deformations, is called nite (hyper)elasticity theory or just nite (hyper)elasticity.Inthistheory,theexistenceoftheso-calledHelmoltzfree-energyfunction,denedperunitreferencevolumeoralternatelyperunitmass, ispostulated. Inthemostgeneral case, theHelmoltzfree-energyfunctionisascalar-valuedfunctionof thetensorFandof thepositionof theparticularpointwithinthebody. Restrictingtheanalysistothecaseofhomogeneousmaterial,theenergysolelydependsonthedeformationgradientF,andassuch,itisoftenreferredtoasstrain-energyfunctionorstored-energyfunction=(F). Byvirtueofwhathasbeenshownintheprevioussection,thestrainenergyfunctioncanbeexpressedasfunctionofseveralotherdeformationtensors,e.g.therightCauchy-GreentensorC,theleftCauchy-Greentensorb.Ahyperelasticmaterial,orGreen-elasticmaterial,isasubclassofelasticmaterialsforwhichtherelationexpressedinEq. 1.35holdsP = G(F) =(F)For PiI=FiI(1.35)Manyotherreducedformsofconstitutiveequations,equivalenttothelatter,forhyperelasticmaterialsatnitestrainscanbederived;whilenotwishingtoreporthereallthedierentformsavailableinliterature,considerforthispurpose,thederivativewithrespecttotimeofthestrainenergyfunction(F): = tr__(F)F_TF_ = tr__(C)C_C_ == tr_(C)C_FTF +FTF__ = 2tr_(C)CFTF_(1.36)Given the symmetry of the tensor C, and the resulting symmetry of the tensor valued scalar function (C), it followsimmediatelythat:_(F)F_T= 2(C)CFT(1.37)1.1.2.1 IsotropichyperelasticmaterialsWithinthecontextof hyperelasticity, atypologyofmaterialsof unquestionableimportance, ofwhichrubberisoneofthemostrepresentativeexamples, consistsof theso-calledisotropicmaterials. Fromaphysical pointof view, thepropertyof isotropyisnothingmorethantheindependenceintheresponseof thematerial, intermsof stress-strainrelations,withrespecttotheparticulardirectionconsidered.Let us consider apoint withinanelastic, deformablebodyoccupyingtheregionandidentiedbyits positionvectorx. Furthermore, letthebody, inthereferenceconguration, undergoatranslational motionrepresentedbythevectorcandrotatedthroughtheorthogonaltensorQ(seeFig.1.7onthefollowingpage):x= c +Qx (1.38)The deformation gradient F that links the material conguration , to the spatial conguration might be computedKTHRoyal InstituteofTechnology-SolidMechanicsDepartment8L.Gigliotti CHAPTER1. FUNDAMENTALS:LITERATUREREVIEWANDBASICCONCEPTSFigure1.7: Rigid-bodymotionsuperimposedonthereferenceconguration.bymakinguseofthechainruleandEq. 1.38,leadingtoF =xx=xxQ = FQ or FiI=xixI=xixJQJI= FiJQJI(1.39)Amaterialissaidtobeisotropicif,andonlyif,thestrainenergiesdenedwithrespecttothedeformationgradientsFandFarethesameforallorthogonalvectorsQ;thus,itmightbewrittenthat:(F) = (F) = (FQT) (1.40)whichistheunavoidableconditiontorefertoamaterialasisotropic.(C) = (FTF) = (QFTFQT) = (C) (1.41)Hence,ifthis latterrelationisvalidforallsymmetrictensorsCandallorthogonal tensorsQ,the strainenergyfunction(C), is a scalar-valuedisotropictensorfunction solely of the tensor C. Under these assumption, the strain energymightbeexpressedintermsofitsinvariants,i.e. = [I1 (C) , I2 (C) I3 (C)]or,equivalently,ofitsprincipalstretches = (C) = [1, 2, 3].1.1.2.2 IncompressiblehyperelasticmaterialsAcategoryofrubber-likematerialswidelyusedinpracticalapplicationsandthereforeparticularlyattractive,especiallywith regard to the corresponding computational analysis by means of numerical codes, are the so-called incompressiblematerials,whichcansustainnitestrainswithoutshowanyconsiderablevolumechanges. InreferencetoEq. 1.12,itmighttobestatedthat,theincompressibilityconstraintcanbeexpressedas:J= 1 (1.42)Theincompressibilityconstraint is widelyknowninliteratureas aninternal constraint andamaterial subjectedtosuchconstraintiscalledconstrainedmaterial. Inordertoderiveconstitutiveequationsforageneralincompressiblematerial,itisnecessarytopostulatetheexistenceofaparticularstrainenergyfunction: = (F) p(J 1) (1.43)dened exclusively for J= det(F) = 1. In such expression, the scalar parameter p, is referred to as Lagrangemultiplier,whose value can be determined by solving the equations of equilibrium. As proven in the previous sections, it is sucient,assumingthatthisispossible, todierentiatethestrainenergyfunctioninEq. 1.43withrespecttothedeformationgradientF, toobtainthethreefundamental constitutiveequations intermsoftherstandthesecondPiola-Kirchhostresses,i.e. PandS,andoftheCauchystresstensor. Fortheparticularcaseofincompressiblematerials,theymaybewrittenasP = pPT+(F)FS = pF1FT+F1(F)F= pC1+ 2(C)C= pI +(F)FFT= pI +F_(F)F_T(1.44)KTHRoyal InstituteofTechnology-SolidMechanicsDepartment1.1. RUBBERELASTICITY L.Gigliotti 9Additionally, ithasbeendemonstratedformerlythat, inthecaseof isotropicmaterial, thestrainenergyfunctioncanbeexpressedas functionof theright CauchyGreentensor C, theleft Cauchy-Greentensor bandtheir invariants.However, if thematerial isatthesametimeincompressibleandisotropic, itisalsotruethatI3=det C=det b=1andconsequently,thethirdinvariantisnolongeranindependentdeformationvariablelikeI1andI2. Consequently,therelationstatedin1.43canbereformulatedasfollows = [I1(C), I2(C)] 12p(I31) = [I1(b), I2(b)] 12p(I31) (1.45)Thus,theassociatedconstitutiveequationsarewrittenasS = 2(I1, (I1)C[p(I31)]C= pC1+ 2_I1+I1I2_I 2I2C= pI + 2_I1+I1I2_b 2I2b2= pI + 2I1b 2I2b1(1.46)Lastly,ifthestrainenergyfunctionisexpressedasafunctionofthethreeprincipalstretchesi,itholdsthatSi= 12ip +1ii, i = 1, 2, 3 (1.47)Pi= 1ip +i, i = 1, 2, 3 (1.48)i= p +ii, i = 1, 2, 3 (1.49)forwhomtheconstraintofincompressibility,i.e. J = 1takesthefollowingform:133= 1 (1.50)1.1.3 IsotropicHyperelasticmaterialmodelsDuetothegreater dicultyinthemathematical treatment of hyperelasticmaterials, thereareseveral examples inliteratureaboutpossibleformsofstrainenergyfunctionsforcompressible,aswellasforincompressiblematerials.Inthe followingsections, the twomodels adoptedinthe present work, i.e. the Arruda-Boyceandthe Neo-Hookeanmodel, will bedescribed. Itmustbestressedhoweverthat, exclusivelyisotropicincompressiblematerialmodelsunderisothermal regimehavebeentreated. Manyothermodelshavebeenproposedinliterature, e.g. Ogdenmodel[4,5],Mooney-Rivlinmodel,[11],Yeohmodel[16],Kilian-VanderWaalsmodel[20]amongthemostfamous.1.1.3.1 Neo-HookeanmodelTheNeo-Hookeanmodel[6]canbereferredtoasaparticularcaseoftheOgdenmodel. Itsmathematicalexpressionisthefollowingone = c1_21 +22 +233_ = c1 (I13) (1.51)Byvirtueoftheconsistencycondition[7],itfollowsthat =2_21 +22 +233_(1.52)whereindicatestheshearmodulusinthereferenceconguration.Theneo-Hookeanmodel,rstlyproposedbyRonaldRivlinin1948,issimilartotheHookeslawadoptedforlinearmaterials; indeed, thestress-strainrelationshipisinitiallylinearwhileatacertainpointthecurvewill level out. Theprincipaldrawbackofsuchmodelisitsinabilitytopredictaccuratelythebehaviourofrubber-likematerialsforstrainslargerthen20%andforbiaxalstressstates.Itcannowbeproventhat, eveniffromamathematicalpointofview, theNeo-Hookeanmodelmaybeseenasthesimplest case of the Ogden model, it might be also justied within the context of the Gaussianstatisticaltheory[8, 9]ofelasticity,whichisbasedontheassumptionthatonlysmallstrainswillbeinvolvedinthecourseofthedeformation4.Briey, rubber-like materials are made up of long-chain molecules, producing one giant molecule, referred to as molecular4Thisfactisafurthervalidationoftheadequacyofneo-Hookeanmodelforstrainsupto20%;themorerenednon-Gaussianstatistical theory,ofwhichanexampleisbasedontheLangevindistributionfunctionisneeded,inordertoobtainamoreaccuratemodelforlargestrains.KTHRoyal InstituteofTechnology-SolidMechanicsDepartment10L.Gigliotti CHAPTER1. FUNDAMENTALS:LITERATUREREVIEWANDBASICCONCEPTSnetwork [10]; starting from the Boltzmann principle, and under the assumptions of incompressible material and anemotion, the entropy change of this network, generated by the motion, can be readily computed as function of the numberNofchainsinaunitvolumeofthenetworkitselfandoftheprincipalstretchesi, i = 1, 2, 3= 12Nr20inr2out_21 +22 +233_(1.53)where = 1.381023Nm/Kisthewell-knownBoltzmannsconstant;atthesametime,theparameterr2outandr20inarethemeansquarevalueof theend-to-enddistanceof detachedchainsandof theend-to-enddistanceof cross-linkedchainsinthenetwork,respectively. Forisothermalprocesses_ = 0_,theLegendretransformationleadstothefollowingexpressionfortheHelmholtzfree-energyfunction =12Nr20inr2out_21 +22 +233_(1.54)Inconclusion,iftheshearmodulusisexpressedasproportionaltotheconcentrationofchainsN,itholdsthat: = Nr20inr2out(1.55)Byvirtueofthislatterresult,theequivalencebetweenEqs. 1.51and1.54isdemonstrated.1.1.3.2 Arruda-BoycemodelThesecondmaterialmodeladoptedinthisworkformodelingtheresponseofrubber-likematerialsistheArruda-Boycemodel [14], proposedin1993; inthismodel, alsoknownastheeight-chainmodel, theassumptionthatthemolecularnetworkstructurecanberegardedasarepresentingcubicunitvolumeinwhich, eightchainsaredistributedalongthediagonaldirectionstowardsitseightcorners, ismade. TheArruda-Boycemodelisparticularlysuitabletocharacterizeproperties of carbon-black lled rubber vulcanizates;such a notable category of elastomers are reinforced with llers likecarbonblackorsilicaobtainingthus,asignicantimprovementintermsoftensileandtearstrength,aswellasabrasionresistance. Byvirtueofthesereasons,thestress-strainrelationistremendouslynonlinear(stieningeect)atthelargestrains.Unliketheneo-Hookeanmodel, theArruda-Boycemodel isbasedonthenon-Gaussianstatistical theory[15]andconsequentlyisadequatetoapproximatetheniteextensibilityofrubber-likematerialsaswellastheupturneectathigherstrainlevels. Thestrainenergyfunctionforthemodelconsideredherein,maybepresentedas = Nn_chainnln_sinh __(1.56)Thecoecientsintheabovewrittenequation,areeasilydenedasfollowschain=_I1/3 and = L1_chainn_(1.57)where,LisknownasLangevinfunction;obviously,forcomputationalreasonsthelatterfunctionisapproximatedwithaTaylorseriesexpansion. BymakinguseoftherstvetermsoftheTaylorexpansionoftheLangevinfunction,adierentanalyticalexpressionisgivenby = c1_12 (I13)1202m_I21 9_+1110504m_I31 27_+1970006m_I41 81_+5196737508m_I51 243__(1.58)wheremisreferredtoaslockingstretch,representingthestretchvalueatwhichtheslopeofthestress-straincurvewill risesignicantlyandthus, wherethepolymerchainnetworkbecomeslocked. Theconsistencyconditionallowstodenetheconstantc1asc1=_1 +352m+991754m+5138756m+42039673758m_ (1.59)Lastly, itoughttobestressedthatthestrainenergyfunctionintheArruda-Boycemodel dependsonlyupontherstinvariantI1; fromaphysical pointofview, thismeansthattheeightchainsstretchuniformlyalongall directionswhensubjectedtoageneraldeformationstate.AcomparisonofthequalityofapproximationfordierentmaterialmodelsisdepictedinFig.1.8onthenextpage;accordingtothis plot, it is inferablethat not all material models showthesamelevel of accuracyinpredictingthestress-strainbehaviorofrubber-likematerials. Inparticular,somemodels,i.e. Neo-HookeanmodelandMooney-Rivlinmodel,exhibittheincapacitytomodelthestieningeectatthehighstrains.KTHRoyal InstituteofTechnology-SolidMechanicsDepartment1.2. FRACTUREMECHANICSOFRUBBER L.Gigliotti 11Figure1.8: Stress-straincurvesforuniaxialextensionconditions-Comparisonamongvarioushypere-lasticmaterialmodels.1.2 FractureMechanicsofRubberTheextensionof fracturemechanicsconceptstoelastomershasalwaysrepresentedaproblemof majorinterest, sincetherstworkinthiseldhasbeenpresentedbyRivlinandThomasin1952[21]. Inthiscornerstoneworktheauthorshaveshownhowlargedeformations of rubber render thesolutionof theboundaryvalueproblemof acrackedbodymadeof rubber, aquitecompoundedtask. Byvirtueof theaforementionednonlinearnatureof constitutivemodelsandduetothecapacityofrubber-likematerialstoundergonitedeformations,LEFMresultscannotbe,withoutpriormodications, extendedtothiscategoryofmaterialsandthus, aslightlydierentapproachhastobeadopted. Inthissection, some of the most relevant results achieved in the fracture mechanics of elastomers eld,along with experimentalresults,arebrieydescribedanddiscussed.1.2.1 FracturemechanicsapproachTheintroductionoffracturemechanicsconceptsgoesbacktoGrithsexperimentalworkonthestrengthofglass[22].Grithnoticedthat thecharacteristictensilestrengthof thematerial was highlyaectedbythedimensions of thecomponent; byvirtueoftheseobservations, hepointedoutthatthevariabilityoftensilestrengthshouldberelatedtosomethingdierentthanasimpleinherentmaterial property. Previously, Inglishaddemonstratedthatthecommondesignprocedurebasedonthetheoreticalstrengthofsolid,wasnolongeradaptandthatthismaterialpropertyshouldhavebeenreduced,inordertotakeintoaccountthepresenceofawswithinthecomponent5.Grith [22] hypothesized that, in an analogous manner of liquids, solid surfaces are characterized by surface tension.Havingthisborneinmind, forthepropagationof acrack, orinordertoincreaseitssurfacearea, itisnecessarythatthesurfacetension,relatedtothenewpropagatedsurface,islessthantheenergyfurnishedfromtheexternalloads,orinternallyreleased. Alternatively,theGrith-Irwin-Orowantheory[24][25][26]claimsthatacrackwillrunthroughasoliddeformablebody, assoonastheinputenergy-ratesurmountsthedissipatedplastic-energy; denotingwithWtheworkdonebytheexternalforces,withUesandUpstheelasticandtheplasticpartofthetotalstrainenergy,respectively,andwithUthesurfacetensionenergy,wemaywritethusWa=Uesa+Upsa+Ua(1.60)Thisexpressionmightthenberewrittenintermsofthepotentialenergy = Ues W,i.e.a=Upsa+Ua(1.61)5Inotherwords,thecomparisonoughttobemadebetweenthetheoreticaltensilestrengthandtheconcentratedstressandnotwiththeaveragestresscomputedbyusingtheusual solidmechanicstheory, basedontheassumptionof theabsenceof internaldefects.KTHRoyal InstituteofTechnology-SolidMechanicsDepartment12L.Gigliotti CHAPTER1. FUNDAMENTALS:LITERATUREREVIEWANDBASICCONCEPTSwhichrepresentsastabilitycriterionstatingthat,thedecreasingrateofpotentialenergyduringcrackgrowthmustequal therateof dissipatedenergyinplasticdeformationandcrackpropagation. Furthermore, Irwindemonstratedthattheinputenergyrateforaninnitesimalcrackpropagation,isindependentoftheloadapplicationmodalities,e.g.xed-grip condition or xed-force condition, and it is referred to as strain-energy release rate G, for a unit length increaseinthecrackextension.For the particular case of brittle materials, the plastic termUpsvanishes andthe followingexpressionmight bededuced:G = a= 2s(1.62)wheresisthesurfaceenergyandtheterm2iseasilyjustiedgiventhepresenceoftwocracksurfaces.Inoneof hissuccessiveworks, Grithcomputed, inthecaseof aninniteplatewithacentral crackof length2asubjected to uniaxial tensile load (see Fig. 1.9), the strain energy needed to propagate the crack,showing that it is equaltotheenergyneededtoclosethecrackundertheactionoftheactingstressFigure1.9: Inniteplatewithcentralcrackoflength2a,subjectedtoanuniaxialstressstate. = 4_a0uy (x) dx =2a22E

G = a=a2E

(1.63)wherethecoecientE

isdenedbelowE

=___E Plane stressE12Plane strain(1.64)beingEtheYoungsmodulus.CombiningEqs. 1.62and1.63itisstraightforwardtoobtainthecriticalstressforcrackingascr=_2E

sa(1.65)andthecriticalstressintensityfactorKC6isgivenbyKC= cra (1.66)Thecrackgrowthstabilitymaybeassessedbysimplyconsideringthesecondderivativeof ( +U); namely, thecrackpropagationwillbeunstableorstable,whentheenergyatequilibriumassumesitsmaximumorminimumvalue,respectively[27]2( +U)a2=___< 0 unstable fracture= 0 stable fracture> 0 neutral equilibrium(1.67)Withcertainmodications, inordertoconsidertheirdierentbehaviour, e.g. theplasticdeformationareainthevicinityof thecracktip, Griththeoryhasbeenextendedtofractureprocessesof metallicmaterials. Hence, LEFMbecameapowerfultoolforpost-mortemanalysistopredictmetalsfracture,tocharacterizefatiguecrackextensionrate,along with the identication of the threshold or lower bound below which fatigue and stable crack growth will not occur.6AccordingtosomeauthorsKCisreferredtoasfracturetoughness.KTHRoyal InstituteofTechnology-SolidMechanicsDepartment1.2. FRACTUREMECHANICSOFRUBBER L.Gigliotti 131.2.2 StressaroundthecracktipAtthispoint, itisworthwhiletoprovideaconcisedescriptionofthecrackbehaviour. Albeitinpracticalapplications,extremelycomplicatedloadconditionsmayoccurinpresenceofacrack,allofthesecanbeconsideredasacombinationofthree,muchlesscomplicatedcasesofloadingconditionsorcrackopeningsmodes(seeFig.1.10): ModeI,whichdescribesasymmetriccrackopeningwithrespecttothex zplane; ModeII,whichdenotesanantisymmetricseparationofcracksurfacesduetorelativedisplacementinx-direction,i.e. normaltothecrackfront; ModeIII,whichischaracterizedbyaseparationduetorelativedisplacementinz-direction,i.e. tangentialtothecrackfront.Figure1.10: Crackopeningmodes.Theregionclosetothecrackfront, inwhichmicroscopicallycomplexprocessesof bondbreakingoccur, isnamedprocesszoneandingeneral,cannotbecompletelydescribedbymeansoftheclassicalcontinuummechanicsapproach.Basedonthis,ifitisneededtousethislatterforthedescriptionoftheremainingcrackedbody,itoughttobeassumedthattheprocesszoneextensionisnegligiblysmall, ifcomparedtoall othermacroscopicdimensionsofthebody. Thishighlocalizationfeaturemightbeobservedinmostofmetallicmaterials,forthemajorityofbrittlematerials,aswellasforrubber-likematerials.Inall fracturemechanicsproblems, it isbeof particularinteresttodetermine, whenpossible, theanalytical for-mulationofcracktipelds, namelystressandstrainsdistributionswithinasmall regionofradiusRaroundthecracktip.Figure1.11: Cracktipregion-Coordinatessystemcentredatthecracktip.For planeproblems, i.e. planestress andplanestrain, byexploitingthecomplex variable method, thefollowingexpressionmightbederived + ir=

(z) +

(z) +z

(z) +

(z) z/z= Ar1ei(1)+Ar1ei(1)++A( 1) r1ei(1)+Br1ei(+1)(1.68)wherevaluesofA,BandAcanbeobtainedbyimposingtheboundaryconditionequation + ir= 0alongthecrackfaces = 7. Moreover,randdenethepolarcoordinatesystemcentredatthecracktipandwelldepictedinFig.1.11.Thestressesijanddisplacementsui, wherei, j =x, y, canbeexpressedasthesumof theeigenfunctionscorre-spondingtotheeigenvaluesoftheeigenproblemposedabove,i.e.ij= r1/2 (1))ij() + (2))ij() +r1/2 (1))ij() +. . .uiui0= r1/2 u(1))i() +r u(2))i() +r3/2 u(1))ij() +. . .(1.69)7Theanglestemsfromthehypothesisofastraightcrack;ifaV-shapedcrack,forminganangleequalto2( )ispresent,aboveformulatedboundaryconditionshastobeappliedfor = .KTHRoyal InstituteofTechnology-SolidMechanicsDepartment14L.Gigliotti CHAPTER1. FUNDAMENTALS:LITERATUREREVIEWANDBASICCONCEPTSHere, ui0representsaneventual rigidbodymotionwhile, forr 0, thedominatingtermistherstoneandthusasingularityinthestresseldisobtainedatthecracktip. Awidelyadoptedprocedureistosplitthesymmetricsin-gulareld, correspondingtoMode-Icrackopening, fromtheantisymmetricone, relatedtotheMode-IIcrackopening.Accordingtothislatterconsideration,stressanddisplacementeldsatthecracktipforbothMode-IandMode-IIcanbewrittenasfollowsMode-I :___xyxy___ =KI2rcos (/2)___1 sin(/2) sin(3/2)1 + sin(/2) sin(3/2)sin(/2) cos(3/2)____uv_ =KI2G_r2( cos ())_cos(/2)sin(/2)_(1.70)Mode-II :___xyxy___ =KII2r___sin(/2)[2 + cos(/2) cos(3/2)]sin(/2) cos(/2) cos(3/2)cos(/2)[1 sin(/2) sin(3/2)]____uv_ =KII2G_r2_sin(/2)[ + 2 + cos()]cos(/2)[ 2 + cos()]_(1.71)whereplane stress : = 3 4, z= 0plane strain : = (3 )/(1 +), z= (x +y)(1.72)AccordingtoEqs. 1.70and1.71, theamplitudeof thecracktipeldsiscontrolledbythestress-intensityfactorsKIandKII;theirvaluesdependonthegeometryofthebody,includingthecrackgeometry,andonitsloadconditions.Indeed, providedthestressesanddeformationsareknown, itispossibletodeterminetheK-values: forexample, fromEqs. 1.70and1.71onemightinferthatKI= limr02ry ( = 0) , and KII= limr02rxy ( = 0) (1.73)In conclusion, it ought to be stressed that for larger distances from the crack tip, the higher terms in Eq. 1.69 cannotbeneglectedandtheeectofremainingeigenvalueshastobetakenintoaccount. Moreover,ithasbeenobservedthat,inmost of thecrackproblems thecharacteristicstress singularityis of theorder r1/2; however,dierent singularityorders for the stress eldmight also come tolight. As general remark,the stress singularities are of the type ij r1,havingdenotedwiththesmallesteigenvalueintheeigenproblemformulatedinEq. 1.68.1.2.3 TearingenergyTheoretically, Griths approach is suitable to predict the fracture mechanics behaviour of elastomers, since no limitationstosmall strainsorlinearelasticmaterial responsehavebeenmadeinitsderivation. Manyattemptshavebeencarriedout throughout theyears, tondacriterionfor thecrackpropagationinrubber-likematerials; however this taskischaracterizedbyoverwhelmingmathematical dicultiesindeterminingthestresseldinacrackedbodymadeof anelasticmaterial, duetolargedeformationsatthecracktippriorfailure. Inaddition, sincehighstressesdevelopedarebounded within a limited region surrounding the crack tip, their experimental measurements cannot be promptly carriedout.Basedonthermodynamicconsiderations,Griththeorydescribesthequasi-staticcrackpropagationasareversibleprocess; ontheotherhand, forrubber-likematerialsthedecreaseof elasticstrainenergyisbalancednotonlybytheincreaseof thesurfacefreeenergyof thecrackedbody, as hypothesizedfor brittlematerials, but it is alsopartiallyconverted into other forms of energy, i.e. irreversible deformations of the material. Such other forms of dissipated energyappeartoberelevantonlyinproximityofthecracktip,i.e. inportionsofmaterial,relativelysmallifcomparedtotheoveralldimensionsofthecomponent. Ithasbeenobservedthat,forathinsheetofarubber-likematerial,inwhichtheinitialcracklengthislargeifcomparedtoitsthickness,suchenergylossesareproportionaltotheriseofcracklength.Inaddition,theyarereadilycomputablejustasfunctionofthedeformationstateintheneighbourhoodofthecracktipatthetearinginstant,whilebasicallyindependentofthespecimentypeandgeometry,andoftheparticularmannerinwhichthedeformingforcesareappliedtothecrackedbody. Evenifaslightdependencewiththeshapeofthecracktipisobserved,suchenergyisacharacteristicpropertyofthetearingprocessofrubber-likematerials.Letusdeform,underxed-gripconditions,athinsheetofrubber-likematerialcutbyacrackoflengthaandwhosethicknessist. Inordertoobservethecracklengthincreasesof da, aworkTcr t dahastobedone, whereTcristhecritical energyfortearingandisacharacteristicpropertyofthematerial:KTHRoyal InstituteofTechnology-SolidMechanicsDepartment1.2. FRACTUREMECHANICSOFRUBBER L.Gigliotti 15Tcr= 1t_Usa_l(1.74)Intheaboveexpression, thesuxl indicatesthatthedierentiationisperformedwithconstantdisplacementoftheportions of theboundarywhicharenot force-free. Physically, thecritical energyfor tearingTcrrepresents thewholedissipatedenergyasresultof fracturepropagation(of which, incertaincases, surfacetensionmaybeaminorcomponent). Therefore,this critical energy has to be compared with the tearing energy calculated from the deformationstateatthecracktipandwhosevalue,asfunctionofthenotchtipdiameterdiswrittenasT= d_ 20U0scos () d (1.75)where,thetermU0sisthestrainenergydensityatthenotchtipfor= 0.Lastly,iftheaveragestrainenergydensityUbsisintroduced,Eq. 1.75issimpliedasfollowsT = d Ubs(1.76)wherethelinearcorrelationofTwiththenotchdiameterdisproven.ConcerningthephysicalmeaningofUbs,thiscanbeinterpretedastheenergyrequiredtofractureaunitvolumeundersimpletensionconditionsandtherefore,itisanintrinsicmaterialproperty.1.2.4 QualitativeobservationofthetearingprocessIn[21], aformidablenumberof experimentshavebeencarriedout, inordertoassesstheeectivenessof thetearingcriterion expressed in Eq. 1.74; further information regarding vulcanizate materials adopted and experimental modalitiesaregiveninthecitedwork. Irreversiblebehaviourisobservedexclusivelywithintheneighbourhoodregionofthecracktip, wherethematerial undergoeslargedeformations; inaddition, ifexperimental testsareperformedatasucientlyslowrateofdeformation,thesearenotaectedbythetestspeed.To present a qualitative description of the tearing process, we may now consider a thin sheet of vulcanizate in which apre-existent crack is present. Experimental observations show how, even relatively small forces lead to considerable valuesofthetearingenergyand,inaddition,thetearingprocessceasesassoonasthedeformationprocessisinterrupted. Thecrack propagation process can be readily described since its earlier stages: as the deformation continues,the crack growsuptoafewhundredthsofmillimetres. Oncethisconditionisreached,catastrophicfailureoccursandthecracklengthabruptlygrowsbyafewmillimetres. Suchpropagationmechanismisrepeatedasthedeformationfurther increases,leadingtoacatastrophicruptureofthecrackedbody.As always, in fracture mechanics analysis, noticeable information might be deduced from the observation of the cracktipgeometry. Intheprocessof crackgrowthinelastomers, duringthestagesprecedingthecatastrophicrupture, thecracktipisinitiallyblunted,whilst,asthetragicruptureoccurs,thecracktipassumesanincreasinglyirregularshape.Last but not least, it has to be stressed that the instant at which the catastrophic rupture commences, is by denition,takenasthetearingpoint.1.2.5 Tearingenergyfordierentgeometries1.2.5.1 Thetrouserstest-pieceThe trousers specimen (see Fig. 1.12) has been widely used for the determination of out-of-plane mode-III critical tearingenergyforelastomers. Historically,isoneoftherstspecimensintroducedforthedeterminationoffracturepropertiesofelastomers.Figure1.12: Trouserstest-piece.TheenergybalanceinthespecimenmightbewrittenasWa=Ta+Usa(1.77)KTHRoyal InstituteofTechnology-SolidMechanicsDepartment16L.Gigliotti CHAPTER1. FUNDAMENTALS:LITERATUREREVIEWANDBASICCONCEPTSwhereWistheworkdonebytheappliedforces, TistheenergyrequiredfortearingandUsisthetotal internalstrainenergy.Next,assumingthatthestretchratiointhespecimen,whosethicknessisindicatedwithtandthewidthwithw,isequalto = 1 + (u/L) 1undertheappliedforceF,Eq. 1.77canbereformulatedasfollows2F = Tt + wt (1.78)In addition, since = 1 in the reference conguration, according to the normalizationcondition, the strain energyvanishes;therefore,thefollowingexpressionofthetearingenergyTmaybeinferredT=2Ft(1.79)showingthelineardependenceofthetearingenergyontheappliedforceF.1.2.5.2 Theconstrainedtension(shear)specimenThis specimen, also called pure shear test-piece [28], is constituted of a longstrip of (rubber-like) materials which containsa symmetrically located cut (see Fig. 1.13). Let the strip be clamped along its parallel sides,and make them move apartof a distance v0in the y-direction, in correspondence of which the material starts to crack and then held at this position.Figure1.13: Constrainedtension(shear)specimen.Ifboththestripandthecrackaresucientlylong,threedierentregionsaredistinguished,namely: Region1,whichremainsunstressedandwhoserelatedstrainenergyU1svanishes; Region 2, the region containing the crack-tip and in which the strain energy U2sis an unknown complicated functionofxandy; Region3, characterized by an uniform stress distribution and within which, the strain energy U3s= U0sis constant.The constant strain energy U0smight be computed as function of the relative clamp displacement v0 and of constitutivematerial properties. Atthispoint, itisworthwhileremarkingonthat, asthecrackpropagatesbyacertainlengthda,Region2simplymoves withthe cracktipwhile the strainenergyvalue U2sremains constant. Inother words, theextensionof theunstressedRegion1growswhereascontemporary, Region2becomeslargerasthecrackpropagates.ThenetvariationoftheoverallstrainenergyUsisgivenbydUs= U0sht da (1.80)By virtue of Eq. 1.74, together with Eq. 1.80, the expression of the tearing energy expression in the case of constrainedtension(shear)specimenisobtainedT= 1t_Usa_l= U0sh (1.81)1.2.5.3 ThetensilestripspecimenAnother well-known specimen adopted for the characterization of fracture mechanics properties of rubber-like materials,isthetensilestripspecimen. Thisspecimen, consistsinathinsheetof rubber, containingacrack, whoselengthaissmallifcomparedtothelengthLofthetestpiece. Thestraindistributioninasmallregionsurroundingthecracktipisinhomogeneous,whileinthecenterofthesheet,farfromthecracktip,thespecimenmightbereasonablyassumedtobeinsimpleextensionconditions. Inaddition, theregionindicatedinFig. 1.14onthenextpagewithA, namelytheareaattheintersectionofthecutandthefreeedgeofthespecimen,resultstobeunstretched.Giventhecomplexityofthestraindistributionaroundthecracktip, in[21] dimensional considerationsallowustostatethat,ifatestpieceiscutbyanideallysharpcrackinitsundeformedconguration,thevariationintheelasticallystoredenergyduetoitspresence,willbeproportionaltoa2. Suchevidenceisstrictlyvalidonlyforideallysharpcrackandsemi-innitesheet,butitcanbeeasilyextendedforotherpracticalcases,providedthattheradiusofcurvatureatthenotchissmall if comparedwiththecracklengtha. Thevariationof suchelasticallystoredenergy, causedbytheintroductionofthecrackinthespecimenisexpressedbythefollowingrelationKTHRoyal InstituteofTechnology-SolidMechanicsDepartment1.2. FRACTUREMECHANICSOFRUBBER L.Gigliotti 17Figure1.14: Tensilestripspecimenwithacrackoflengtha.U

sUs= k

a2t (1.82)wheretheelasticallystoredenergyintheabsenceofthecrackisdenotedbyU

sandtheconstantofproportionalityk

isfunctionoftheextensionrate. TheproportionalitybetweenU

sUsandthethicknesstholdsonlyifplanestressconditionsareemployed,i.e. t a.Atthispoint,itisstraightforwardtoobservethat,giventheproportionalityofthespecimenelongationwith 1,the energy variation U

sUswill be proportional to (1)2or, in other words, to the strain energy density . By virtueoftheaboveconsiderations,Eq. 1.82canbereformulatedasU

sUs= ka2t (1.83)wherekisafunctionof.BydierentiatingEq. 1.83withrespecttothecracklengtha, nallytheexpressionof thetearingenergyforthetensilestripspecimengettheform_Usa_l= 2kat T= 1t_Usa_l= 2ka (1.84)Concerningthedependanceoftheconstantkwiththeextensionrate, several experimentsandFEAsimulationshavebeenperformed[30, 31] showingtheproportionalityof kwiththeinverseof thesquareroot of , throughtheconstant,i.e.k=3(1.85)Althoughthethreetestspecimenspresentedsofar, arewidelyusedinexperimental procedures, compressionandshear areencounteredmuchmoreofteninengineeringapplications, becauseunder theseloadconditions rubber-likematerialscanbefullyusedwithoutrisksofcrackgrowth.1.2.5.4 Thesimplesheartest-pieceAmathematicallysimpleexpressionforthetearingenergyTinsimpleshearspecimens(seeFig.1.15)hastheformT= kh (1.86)Figure1.15: Simplesheartest-piecewithanedgecrack.KTHRoyal InstituteofTechnology-SolidMechanicsDepartment18L.Gigliotti CHAPTER1. FUNDAMENTALS:LITERATUREREVIEWANDBASICCONCEPTSInsuchcase, theconstant of proportionalityk, commonlyassumes thevalueof 0.4, but its rangeof variationisbetween0.2and1.0,dependingonthecongurationandsizeofthecrack. Inpracticalsituations,itisreallydemandingto carry out simple shear experiments since, as the crack grows, it tends to change direction to Mode-I crack opening. Inaddition,relationEq. 1.80holdsonlyifthecrackisshort,andthisimpliesdicultiesinthedeterminationofthestressconcentration.1.2.5.5 TheuniaxialcompressiontestpieceUniaxial compression specimen slightly diers from those discussed in the previous sections; indeed, the strain distributionis highly inhomogeneous even without cracks within the component. For the strain energy of such test-piece, providedstrainsareoftensmallenough,thefollowinglinearapproximationholds =12Ece2c(1.87)wheree2cisthecompressivestrain,whilethecompressionmodulusEcisdenedasEc= 2G_1 + 2S2_(1.88)whereGisthesmall strainshearmodulus.Thefactor SinEq. 1.88referredtoasashapefactor, istheratiobetweentheloadedareaandtheforce-freesurface,i.e.S=D2/4Dh=D4h(1.89)According to [29], when a bonded rubber unit is cyclically loaded in compression, an approximately parabolic surfaceisgeneratedandthecrackinitiatesattheintersectionofsuchsurfacewiththecoreofthespecimen(seeFig.1.16)Figure1.16: Typical stages of crackgrowthincompression: a) unstrainedb) compressed- crackinitiationatbondedgesc)compressed- bulgeseparatefromcored)unstrained- showingparaboliccracklocus.Undertheseassumptions, thetearingenergyforuniaxial compressiontest-pieceisgivenbytheapproximatedex-pression,validforS> 0.5andstrainsbelow50%T=12h =14Ece2ch (1.90)1.3 eXtendedFiniteElementMethod1.3.1 IntroductionResults presentedinthe previous section, concerningthe analytical treatments of fracture mechanics problems areaectedbycertainlimitations,amongwhichthemostconstrainingareundoubtedlythefollowingones: Thematerialdomainisalwaysconsideredinnite,inordertoneglectedgeeectsinthemathematicalderivationofstressanddisplacementdistributions; Inthemajorityofcases,thematerialisassumedtobehomogeneousandisotropic; Onlysimpleboundaryconditionsareconsidered.However, it is easy to guess that in practical problems of complex structures, containing defects of nite sizes, subjectto complicated boundary conditions and whose material properties are much more complicated than those related to theideal linear, homogeneousandisotropicmaterial model, asatisfactoryfracturemechanicsanalysiscanbecarriedoutexclusively by means of numerical methods. Among these, the most widely adopted in practical engineering applicationsis theniteelement method[33]; for this reason, several softwarepackages basedontheFEMtechniquehavebeendevelopedthroughouttheyears[34]. Althoughtheniteelementmethodhasshowntobeparticularlywell-suitedforfracture mechanics problems [35, 36, 37], the non-smooth crack tip elds in terms of stresses and strains can be capturedKTHRoyal InstituteofTechnology-SolidMechanicsDepartment1.3. EXTENDEDFINITEELEMENTMETHOD L.Gigliotti 19onlybyalocallyrenedmesh. Thisleadstoanabruptincreaseofthenumberofdegreesoffreedomandsuchdefectisworsenedin3D-problems. Concerningthecrackpropagationanalysis, itstill remainsachallengeforseveral industrialmodellingproblems. Indeed, sinceitisrequiredtotheFEMdiscretizationtoconformthediscontinuity, formodellingevolvingdiscontinuities, themeshhastoberegeneratedateachtimestep. Thismeansthatthesolutionhastobere-projectedforeachtimestepontheupdatedmesh,causingadramaticriseintermsofcomputationalcostsandtoalossofthequalityofresults[38]. Becauseoftheselimitations, several numerical approachestoanalyzefracturemechanicsproblemshavebeenproposedduringlastyears. Themethodbasedonthequarter-pointniteelementmethod[39],theenrichedniteelementmethod[40, 41], theintegral equationmethod[42], theboundarycollocationmethod[43], thedislocation method [44, 45], the boundary nite element method [46], the body force method [47] and mesh-free methods[48, 49], e.g. free-element Galerkin method [50, 51], represent the most valuable examples. In order to overcome the needofremeshing,dierenttechniqueshavebeenintroducedoverthelastdecades,e.g. theincorporationofadiscontinuousmodeonanelementlevel [52], amovingmeshtechnique[53] andanenrichmenttechnique, basedonthepartitionofunity,laterreferredtoastheeXtendedFiniteElementMethod(XFEM)[54,56,55].1.3.2 PartitionofunityGiven a Cmanifold M, with an open cover Ui, a partition of unity subject this latter, is a collection of n nonnegative,smoothfunctionsfisuchthat,theirsupportisincludedinUiandthefollowingrelationholdsn

i=1fi(x) = 1 (1.91)Oftenitisrequiredthat, thecoverUihavecompactclosure, whichcanbeinterpretedasnite, orbounded, opensets. If thisconditionislocallyveried, anypointxinMhasonlynitelymanyi withfi(x) ,=0. Itcanbeeasilydemonstratedthat,thesuminEq. 1.91doesnothavetobeidenticallyunitytowork;indeed,foranyarbitraryfunction(x)itisveriedthatn

i=1fi(x)(x) = (x) (1.92)Furthermore, itmightbeinferredthatthepartitionof unitypropertyisalsosatisedbythesetof isoparametricniteelementshapefunctionsNj. i.e.m

j=1Nj(x) = 1 (1.93)1.3.2.1 PartitionofunityniteelementmethodToincreasetheorder of completeness of aniteelement approximation, theso-calledenrichment proceduremaybeexploited. Inother words, the accuracyof solutioncanbe ameliorated, bysimplyincluding inthe nite elementdiscretization, the apriori analytical solution of the problem. For instance, in fracture mechanics problems, an improve-mentinpredictingcracktipeldsisachieved, if theanalytical cracktipsolutionisincludedintheframeworkof theisoparametricniteelementdiscretization. Computationally,thisinvolvesanincreaseinnumberofthenodaldegreesoffreedom.Thepartitionof unityniteelementmethod(PUFEM)[57] [58], usingtheconceptof enrichmentfunctionsalongwith the partition of unity property in Eq. 1.93, allows to obtain the following approximation of the displacement withinaniteelementuh(x) =m

j=1Nj(x)_uj+n

i=1pi(x)aji_(1.94)where, pi(x)aretheenrichmentfunctionsandajiaretheadditional unknownsordegreesoffreedomassociatedtotheenrichedsolution. Withmandnthetotal numberof nodesof eachniteelementandthenumberof enrichmentfunctionspi,areindicated.ByvirtueofEqs.1.92and1.93,foranenrichednodexk,Eq. 1.94mightbewrittenasuh(xk) =_uk +n

i=1pi(xk)aji_(1.95)whichis clearlynot aplausible solution. Toovercome this defect andsatisfyinterpolationat nodal point, i.e.uh(xi) = ui,aslightlymodiedexpressionfortheenricheddisplacementeldisproposedbelowuh(x) =m

j=1Nj(x)_uj+n

i=1(pi(x) pi(xj)) aji_(1.96)KTHRoyal InstituteofTechnology-SolidMechanicsDepartment20L.Gigliotti CHAPTER1. FUNDAMENTALS:LITERATUREREVIEWANDBASICCONCEPTS1.3.2.2 GeneralizedniteelementmethodAbreakthroughinincreasingtheorderof completenessof aniteelementdiscretizationisprovidedbytheso-calledgeneralized nite element method (GFEM) [59, 60], in which two separate shape functions are employed for the ordinaryandfortheenrichedpartoftheniteelementapproximation,i.e.uh(x) =m

j=1Nj(x)uj+m

j=1Nj(x)_n

i=1pi(x)aji_(1.97)whereNj(x)aretheshapefunctionsassociatedwiththeenrichmentbasisfunctionspi(x).Forthereasonexplainedintheprevioussection,Eq. 1.97shouldbemodiedasfollowsuh(x) =m

j=1Nj(x)uj+m

j=1Nj(x)_n

i=1(pi(x) pi(xj)) aji_(1.98)1.3.3 eXtendedFiniteElementMethodTheeXtendedFiniteElementMethodisapartitionof unitybasedmethodinwhich, asforPUFEMandGFEM, theclassicalniteelementapproximationisenhancedbymeansofenrichmentfunctions. However,inPUFEMandGFEM,theenrichmentprocedureinvolvestheentiredomain, whilstitisemployedonalocal level fortheXFEM. Thus, onlynodesclosetothecracktip, aswell astheonesrequiredforthecorrectlocalizationof thecrack, areenriched. Thisevidentlyentailsatremendouscomputationaladvantage.TheXFEMmethodwasrstlyintroducedbyBelytsckhoandBlackin1999[61]. Theirwork,inwhichamethodforenriching nite element approximation in such a way that crack growth problems can be solved with minimal remeshing,represents a milestone in the XFEM history. Later on, much more elegant formulations, including the asymptotic near-tipeldandtheHeavisidefunctionH(x)intheenrichmentscheme,havebeenproposed[62,63,64]8. TheeXtendedFiniteElementmethod, furthermore, hasbeendemonstratedtobewell suitedforthreedimensional crackmodelling[65]. Inthis latter work, geometric issues associated with the representation of the crack and the enrichment of the nite elementapproximationhavebeenaddressed. Amajorstepforwardhasbeenthenachievedwhenageneralizedmethodologyforrepresenting discontinuities,located within the domain indipendetely from the mesh grid,has been proposed [66, 62]. Insuch manner, the eXtended Finite Element Method allows to alleviate much of the burden related to the mesh generation,astheniteelementmeshisnotsupposedtoconformthecrackgeometryanymore. Thisrepresentscertainly,oneofthemajoradvantagesprovidedbytheXFEMusage. TheXFEMcapabilitiescanbeextendedif employedinconjunctionwiththeLevel SetMethod(LSM)[67, 68, 69]. Suchmethodpermitstorepresentthecrackposition, aswell asthelocationofcracktips. Withinthiscontext,theXFEMhasalsobeenemployedinconcertwithaparticulartypeoflevelset method named Fast Marching Method [70, 71]. Accuracy,stability and convergence of XFEM, along with dicultiesinusingthestandardGaussianquadraturehavebeeninvestigated[72,73]; asolutiontothislatterdrawbackhasbeenproposedin[74].1.3.3.1 EnrichmentfunctionsIntwo-dimensionalproblems,crackmodellingisobtainedbymeansoftwodierenttypesofenrichmentfunctions: TheHeavisidefunctionTheHeavisidefunctionH(x), isemployedtoenrichelementscompletelycutbythecrack. Thesplittingof thedomainbythecrack,causesajumpinthedisplacementeldandtheHeavisidefunctionprovidesatremendouslysimplemathematicaltooltomodelsuchbehaviour.Given a continuous curve , representing a crack within the deformable body , let us consider a point x(x, y) .Thewholeaimistodeterminethepositionofsuchpointwithrespecttothecracklocation. Inthiscontext,iftheclosest point belonging to is denoted with x(x, y) and the outward normal vector to in x with n (see Fig. 1.17onthefacingpage),theHeavisidefunctionmightbedenedasfollowsH (x, y) =_1 for (x x)n > 01 for (x x)n < 0(1.99)Ifnouniquenormalisdened,thentheH(x)functionwillassumeapositivevalueif(x x)belongtotheconofnormalsinx(seeFig.1.17onthenextpage) Asymptoticnear-tipeldfunctionsIn case of not completely cracked element, the Heaviside function cannot be used to approximate the displacementeldovertheentireelementdomain, sincetheelementcontainsthecracktip. In[51] ithasbeenproventhatthedisplacementeldfromLEFMtheoryinEqs. 1.70and1.71,isincludedwithinthespanofthefollowingfourfunctions,expressedintermsofthelocalcracktipcoordinatesystem(r, )|Fi(r, )4i=1=_r cos_2_,r sin_2_,r sin_2_sin ()r cos_2_sin ()_(1.100)8In[64],thenameeXtendedFiniteElementMethodhasbeenusedforthersttime.KTHRoyal InstituteofTechnology-SolidMechanicsDepartment1.3. EXTENDEDFINITEELEMENTMETHOD L.Gigliotti 21(a)Smoothcrack. (b)Kinkcrack.Figure1.17: EvaluationoftheHeavisidefunction.Byusingtheenrichment functions inEq. 1.100, four dierent additional degrees of freedomineachdirectionforeachnodeareaddedtothoserelatedtothestandardniteelementdiscretization. Itshouldbestressedthatamongtheaforementionedenrichingfunctions(seeFig. 1.18), onlythesecondterm r sin_2_isdiscontinuousalongcracksurfaces andhence, is responsibleof thediscontinuityintheapproximationalongthecrack. Theremainingthreefunctionsareusedtoenhancethesolutionapproximationintheneighborhoodofthecracktip.(a) r cos

2

. (b) r sin

2

.(c) r sin

2

sin (). (d) r cos

2

sin ()Figure1.18: Near-tipenrichmentfunctions.Lastly, itshouldbehighlightedthat, referringtoEq. 1.100, therequiredsingularityinthestresseld, of order1ris,therefore,readilyintroduced.Byvirtueoftheabovediscussedenrichmentfunctions,thefollowingexpressionfortheXFEMapproximationmightbeformulateduh(x) = uFEM (x) +uENR (x) ==

iINi (x) ui+

jJNj [H (x)] aj+

kK1Nk (x)_4

l=1bl1kF1l(x)_++

kK2Nk (x)_4

l=1bl2kF2l(x)_(1.101)or,toeliminatethelackofinterpolationpropertyKTHRoyal InstituteofTechnology-SolidMechanicsDepartment22L.Gigliotti CHAPTER1. FUNDAMENTALS:LITERATUREREVIEWANDBASICCONCEPTSuh(x) =

iINi (x) ui+

jJNj [H (x) H (xj)] aj++

kK1Nk (x)_4

l=1bl1k_F1l(x) F1l(xk)__+

kK2Nk (x)_4

l=1bl2k_F2l(x) F2l(xk)__(1.102)where, Jindicatesthesetof nodeswhosesupportdomainiscompletelycutbythecrackandthusenrichedwiththeHeavisidefunctionH (x),K1andK1arethesetsofnodesassociatedwiththecracktips1and2intheirinuencedomain, respectively, andwhoserespectivecracktipenrichment functions areF1l(x) andF2l(x). Moreover, uiarethestandarddegreesof freedom, whileaj, bl1kandbl2kindicatethevectorsof additional nodal degreesof freedomformodellingcrackfacesandthetwocracktips,respectively.1.3.3.2 LevelsetmethodformodellingdiscontinuitiesIn several cases, numerical simulations involve time-varying objects, like curves and surfaces on a xed cartesian grid, e.g.interfaces, discontinuities, etc. Theirmodellingandtrackingisparticularlycumbersomeandisbasedonthecomplexmathematicalproceduredenominatedparametrization.TheLevelSetMethod[68](oftenabbreviatedasLSM)isanelegantnumericaltechniquethatallowstogetoverthesediculties. Thekey-point of suchmethodis torepresent discontinuities as azerolevel set function. For thispurpose,tofullycharacterizeacrack,twodierentlevelsetfunctionsaredened1. Anormallevelsetfunction,(x)2. Atangentiallevelfunction, (x).At this point, for the evaluation of the signed distance functions, let cbe the crack surface (see Fig. 1.19) and x thepoint in which it is sought to evaluate the (x) function. In an analogous manner of what done in the foregoing section,thenormallevelsetfunctionmightbedenedas = (x x)n (1.103)wherexandnassumethepreviouslystatedmeanings(seeFig.1.17ontheprecedingpage)9.Figure1.19: Constructionoflevelsetfunctions.InFig.1.20onthenextpage,theplotofthenormalsignedfunction(x)foraninteriorcrackisprovided.Thetangential level setfunction (x)iscomputedbyndingtheminimumsigneddistancetothenormal atthecracktip;incase of aninterior crack,twodierentfunctions canbe individuated. However,auniquetangential levelsetfunctioncanbedenedas (x) = max |1 (x) , 2 (x) (1.104)Inconclusion,referringtoFig.1.19,itmaybewrittenwhatfollows_for x cr(x = 0) and (x 0)for x tip(x = 0) and (x = 0)(1.105)9AccordingtothedenitiongiveninEq.1.103,incaseofinteriorcrack,thenormallevelsetfunctioniscomputableonlywithintheregiondelimitedbythenormalstothecracktips; inordertodene(x)overthewholedomain, bothcracktipsshouldbevirtuallyextended.KTHRoyal InstituteofTechnology-SolidMechanicsDepartment1.3. EXTENDEDFINITEELEMENTMETHOD L.Gigliotti 23(a)2Dcontourof(x). (b)3Dcontourof(x).Figure1.20: Normallevelsetfunction(x)foraninteriorcrack.(a)2Dcontourof (x). (b)3Dcontourof (x).Figure1.21: Tangentiallevelsetfunction (x)foraninteriorcrack.wheretipindicatesthecracktipslocation.1.3.3.3 BlendingelementsDiscussingdierent methods exploitingthepartitionof unityproperty, e.g. PUFEMandGFEM, theadvantageintermsofcomputationalcostrelatedtotheXFEMhasbeenintroduced. Indeed,unlikePUFEMandGFEM,inXFEMtheenrichingfunctionsareintroducedonlyinalocal partofthedomain, inordertocapturethenon-smoothsolutioncharacteristics. Elements, whose all nodes have beenenrichedare namedreproducing elements since theyallowtoreproducetheenrichmentfunctionsexactly. Besidesthese,therearetheso-calledblendingelements,whoseroleistoblendtheenrichedsub-domainwiththerestofthedomain, where, beingthesolutionsmooth, standard(notenriched)niteelementsareemployed. Onlysomeofnodesinblendingelementsareenriched. Enrichedniteelements,blendingelements and standard nite elements partition the whole domain in three dierent parts, an enriched domain, a blendingdomainandastandarddomain,respectively,asshowninFig.1.22onthenextpage.Twoimportantdrawbacksaectblendingelements,i.e. Enrichmentfunctionscannotbereproducedexactlyinblendingelements,sincethepartitionofunitypropertyisnotsatisedwithinthem; Theseelementsproduceunwantedtermsintheapproximation,whichcannotbecompensatedbytheFEpart;forinstance, if theenrichmentintroducesnon-linearterms, alinearfunctioncannolongerbeapproximatedwithinblendingelements.Unliketherst one, whichdoesnot represent adramaticproblemintheXFEM, theseconddrawbackimpliesasignicantreductionoftheconvergencerateforgeneralenrichmentfunctions[72];thus,suboptimalrateofconvergenceinXFEMmaybecausedbyproblemsinblendingelements[73]. ArelativelystraightforwardmethodtocircumventthisdefectistoexploitablendingrampfunctionRoverthetransitionregionconnectingdomainswithandwithoutenrichment,i.e.uh(x) = uFEM(x) +RuENR(x) (1.106)KTHRoyal InstituteofTechnology-SolidMechanicsDepartment24L.Gigliotti CHAPTER1. FUNDAMENTALS:LITERATUREREVIEWANDBASICCONCEPTSFigure1.22: Standard,enrichedandblendingelements.whereRissetequalto1ontheenrichmentboundary, andequalto0onthestandardniteelementdiscretizationboundary. This linear blending function R ensures the continuity in the displacement eld but not in the strain eld. Toachievethislatter, higherorderblendingfunctionsshouldbeused. Otherapproachesprescribetouseenhancedstraintechniquesorp-renementinblendingelements[75],ortoadjusttheorderoftheFEshapefunctionsdependingontheenrichment[76]. Recently, aninterestingsolutionhasbeenproposedin[77]; inparticular, inthisworktheenrichmentfunctionshavebeenmodiedsuchthattheyvanishinstandardelements, unchangedinelementswithall theirnodesbeingenrichedandcontinuouslyvaryingwithinblendingelements. Insuchapproachallnodesofblendingelementsareenriched.1.3.3.4 XFEMdiscretizationConsideraregularregionboundedbyasmoothcurveinthereferenceconguration, andletthislatterbesplitintwoportions, namelytandu. Essential boundaryconditions areimposedonuwhiletractionboundaryconditionsareappliedalongt. Lettheregionbecutbyacrack,whosesurfacescraretractionfree. ThestrongformoftheinitialboundaryvalueproblemcanbewrittenasDivP+Bf= 0 u in u = u on uPN = o on crPN = T on t(1.107)whereBfand0 udenotethereferencebodyforceandtheinertiaforceperunitereferencevolume,respectively,Nthe outward unit normal to ,u the prescribed displacement vector and T the prescribed rst Piola-Kirchho tractionvector.Thedisplacementeldumustsatisfyalltheessentialboundaryconditionsandthesmoothnessproperty,sothatuiscontinuous_C0_in,i.e.u |, |=_u[u C0except on cr, u = u on u_(1.108)Atthesametime,testfunctionsvaredenedbyv |0, |0=_v[v C0except on cr, v = 0 on u_(1.109)Thus,usingtheaboveconcepts,theweakformoftheequilibriumequationandtractionboundaryconditionsmightbeformulatedas_S : EdV=_(Bf 0 u)udV+_tT udA (1.110)Inengineeringproblems, thenonlinear boundary-valueprobleminEq. 1.110is numericallysolvedbymeans oftheiterativeprocedureknownasNewton-Raphsonmethod. Themainideabehindsuchmethodistolinearizeallquantitiesassociatedwiththeproblemathand, replacingitwithaseriesoflinearproblems, whoseresolutionismuchmore undemanding. The linearization procedure is the foundation of nite element methods [33, 78]. Here we present theXFEMdiscretizationofprobleminEq. 1.110fortheparticularcaseofhybridelementsandincompressiblehyperelasticmaterialmodel. Hybridelementsusethewell-knownincomputationalmechanicseldHu-Washizumixedform[79,80].Theresultingelementsexhibitthecapabilitiestoovercomethelockingeects. Insuchelementsthestrainand/orstresseldisinterpolatedindependentlyof thedisplacementeld. Theutilizationof mixedformsisparticularlyprotablefor constrainedproblems. Inaddition, the incompressibilityconstrainedis modeledbymakinguse of the Lagrangemultipliers(Cfr. Sec. 1.1.2.2).Theresultinglinearizedequations,maybeexpressedasfollowsKTHRoyal InstituteofTechnology-SolidMechanicsDepartment1.3. EXTENDEDFINITEELEMENTMETHOD L.Gigliotti 25_J GTG 0_ _uhp_ =_rvGTpvGTuhv_(1.111)Intheabovematrixrelation,theJacobianmatrixJ,andthematrixGcanberespectivelywrittenasJ =sDt2M+Kint+Kext, G =gu(1.112)WerstconsidertheJmatrix. InthersttermontheRSH, whichispresentonlyindynamicproblems, istheparameterof theNewmark-method, Misthemassmatrixandtisthetimestepincrement. Thesecondandthethirdtermarerespectivelythetangent stinessmatrices forinternal andexternal nodal forces; thislattercanbeeasilyexpressedasKext=fextu.Inthesuppositionthatonlyonecracktipispresentwithinthebody, thevectorofnodal parametersuhisdenedasfollowsuh= |u a b1b2b3b4T(1.113)Concerningtheglobal tangentstinessmatrixforinternal forcesKintandtheglobal vectorofexternal forcesfext,theseareobtainedbyassemblingthetangentstinessmatrixforinternalforcesKint eijandthevectorofexternalforcesfext eiforeachelementeoftheXFEMdiscretization,i.e.Kint eij=__KuuijKuaijKubijKauijKaaijKabijKbuijKbaijKbbij__(1.114)fext ei=_fuifaifb1ifb2ifb3ifb4i_T(1.115)withKrsij=_e[(Bri)T: CSE: Bsj+I(Bri)T: (S pC1) : Bsj]d , r, s = u, a, b (1.116)fui=_etNiftdA +_etNifbdVfai=_etNiHftdA +_etNiHfbdVfbi=_etNiFftdA +_etNiFfbdV , = 1, 2, 3 and 4(1.117)InEq. 1.116, CSEisaforthordertensoroftenreferredtoastangent modulustensor denedthroughtherelationS = CSE:E,whileBistheshapefunctionderivativesmatrix,i.e.Bui=__Ni,x00 Ni,yNi,yNi,x__(1.118)Bai=__[Ni(H() H(i))],x00 [Ni(H() H(i))],y[Ni(H() H(i))],y[Ni(H() H(i))],x__(1.119)Bbi=_Bb1iBb2iBb3iBb4i_Bbi=__[Ni (FFi)],x00 [Ni (FFi)],y[Ni (FFi)],y[Ni (FFi)],x__, = 1, 2, 3 and 4(1.120)where is the local curvilinear (mapping) coordinate system. Inaddition, Brepresents the columnmatrixofderivativesofshapefunctionsIgivenby[B]iI=NIxiinthereferenceconguration.Furthermore, in Eq. 1.111, p is the Lagrangian multipliers vector, uhis the incremental displacement in the iterativeNewtonprocedure, uhthestepincrement andpthevirtual pressureeld. ConcerningtheLagrangemultipliersmethod, Gisthematrixof material derivativesof constraints, i.e. G=gu. Inconclusion, ristheresidual columnmatrix and the subscript vrepresent the iteration number. In this formulation the Lagrange multiplier p can be regardedasthephysicalhydrostaticpressure.The above discussed formulation leads to a two-elds mixed nite element implementation in which the displacementeldandthe hydrostatic pressure are treatedas independent eldvariables. Lastly, it is fundamental tohighlightthatsuchformulationrequirestoconsideranadditional numbersof unknown, i.e. equal tothenumberof Lagrangemultiplierssought. AlongwiththeadditionalDOFsintroducedbytheXFEMtechnique,thisrendersthetreatmentofincompressiblematerialswithsuchmethod,extremelycostlyfromacomputationalperspective.KTHRoyal InstituteofTechnology-SolidMechanicsDepartment26L.Gigliotti CHAPTER1. FUNDAMENTALS:LITERATUREREVIEWANDBASICCONCEPTS1.3.3.5 NumericalintegrationandconvergenceThoughtheGaussquadraturehasbeenproventobeexactforpolynomial integrands, fornon-polynomial onesitmaycause a reductioninthe accuracyof results. Introducing anarbitraryorienteddiscontinuityinthe nite elementdiscretizationtransformsdisplacementsandstressesintohighlynon-lineareldswhichcannotbecorrectlyintegrated.Tocircumventsuchproblems, asubtriangulationprocedure(seeFig. 1.23), inwhichelementsedgesconformthecrackfaceshasbeenproposed[62];withintheseelementsthestandardGaussintegrationproceduremightbeexploited.Figure1.23: Subtriangulationofelementscutbyacrack.Ithastobehighlightedthat, forelementscontainingthecrack, thereforeincludingthesingularstresseldatthecracktip,thisproceduremightresulttobeinaccurateifGausspointsofsub-trianglesaretoocloseatthecracktip.Crackmodelingwiththestandardniteelementmethod, isperformedbyre-meshingthedomainsothatelementsboundariesmatchthecrackgeometry; furthermore, newcreatedelementshavetobewell conditionedandnotbadlyshaped. Accordingly, re-meshingprocedureisacumbersomeandcomputationallycostlyoperation. Onthecontrary,since the sub-triangulation is performed only for integration purposes,no additional degrees of freedom are added to thesystemand,sub-trianglesarenotforcedtobewellshaped.Itisworthmentioningthatanalternativemethodbasedontheeliminationof quadraturesub-elementshasbeenrecentlyproposed[74]. Insuchapproach,ratherthenpartitionelementscutbyacrack,discontinuousnon-dierentiablefunctionsarereplacedwithequivalentpolynomialfunctionsandconsequentlytheGaussquadraturecanbecarriedoutoverthewholeelement. Althoughthismethodwouldallowasensiblereductionintermsof computational costs, itisstill aectedbysomedrawbacks, e.g. itislimitedtoelementcompletelycutbyastraightcrack, noadditional rulesareprescribedforelementscontainingthecracktipandevenif thesolutionisaccuratefortriangularandtetrahedralelements,dicultiesarestillencounteredindealingwithquadrilateralelements.Inconclusion, as previouslystated, the XFEMmethodensures more accurate results thanclassical FEMone.However, therateof convergencedoesnotimproveasthemeshparameterhgoestozeroduetothepresenceof thesingularity[73]anditislowerthantheoneexpectedbyusingtheclassicalFEMmethodinsmoothproblems10. Severalmethodshavebeenproposedduringlastdecadestoachieveanoptimal rateof convergence, e.g. XFEMwithaxedenrichment area, high-order XFEM [72, 81] as well as a modied construction of blending elements. In [82] the robustnessofthemethodhasbeenenhancedbymeansofanewintegrationquadratureforasymptoticfunctionsandimplementingapreconditioningschemeadaptedtoenrichmentfunctions.1.3.3.6 XFEMfornitestrainfracturemechanicsNumerical methodsbasedof theFiniteElementMethod, devotedtothecomputational analysisof crackpropagationphenomenainnon-linearfracturemechanicsisstillanopenproblem[83,84].EvenmorecumbersomeistheapplicationoftheeXtendedFiniteElementMethodtolargestrainproblems;indeed,veryfewworks have beenintendedtoextendthe XFEMcapabilities tomaterials exhibitinganonlinear behaviourandabletoundergolargedeformations, especiallyif associatedwiththevolumetricincompressibilityconstraint. Thesatisfactionof suchconstraintwithintheenrichedapproximationisthoughttobe, inthiscontext, themostdicultissueandrequirestobefurtherinvestigated. Forthispurpose, theenrichmentofastandardbiquadratic-displacementbilinear pressure for node quadrilateral element has been proposed [85]. Nevertheless, even if very good results have beenachievedintermsofstressanddisplacementeldsprediction, theextenttowhichtheincompressibilitylimithasbeensatisedoughttobebetterexamined. Aproceduretomodelthecrackgrowthwithacohesivelawonthediscontinuitysurfaceshasbeendeveloped,forcompressiblehyperelasticmaterials,in[86]. Theincompressibilitylimit,alongwiththehighstresslocalizationatthecracktip, hasbeenproventoleadtosevereproblemsofmeshlocking. In[88] amethodtocircumventthisdrawbackhasbeenproposed. Moreindetail, insuchwork, shearlockinginMindlin-Reissnerplatefracture has been reduced, but not completely eliminated, by enriching the four-node T1 element of Hughes and Tezduyar[87]. Anenhancedassumedstrainmethodtocircumventlockingissueoccurringinplanestrainproblemsandeliminatethespuriousstressoscillationsalongthefailuresurfaceinnearlyincompressiblematerialshasbeendevelopedin[90].Theanalysisof theasymptoticdisplacementeldnearthecracktipinplanestressconditionsforaNeo-Hookeanmaterial model has been performed in [89], where the importance of the right choice for the singularity enriching functions10Ithasbeenproventhattherateofconvergenceisoforderh.KTHRoyal InstituteofTechnology-SolidMechanicsDepartment1.3. EXTENDEDFINITEELEMENTMETHOD L.Gigliotti 27hasbeeninvestigated. Inparticular, withreferencetothiswork, itmustbestressedthat, asexpected, thebestchoicefortheenrichmentfunctionsistheonebasedontheanalytical results. Moreover, theauthorspointedoutthatduetothe blunting process,it is better to avoid the enrichment rather then use an inappropriately chosen enrichment function.Thehigh-speedcrackgrowthinrubber-likematerials, includingtheinertiaeects, viscoelasticityandnitestrainsinplanestressconditionshasbeenanalyzedbyusingtheextendedniteelementmethodin[91].1.3.4 XFEMimplementationinABAQUSTherstformulationoftheeXtendedFiniteElementMethodgoesbacktothe1999and,duetothis,thereisashortageof commercial codesthathaveimplementedsuchmethod. However, seeingtheenormouscapabilitiesprovidedbytheXFEMimplementation,amongwhichthemostimportantoneisundoubtedlythatthemeshhasnottoconformexactlythe crack surfaces, several attempts have been done throughout the years to include them in both, stand-alone codes andmulti-purposecommercial FEMsoftwares. Amongtheselatter, themostfamousonesareundoubtedlyLS-DYNAandAbaqus[92], thoughotherminorcodeslikeASTERandMorfeoincludeanalogousfacilities. ThecoreimplementationofXFEMisavailableinANSYS,neverthelessitisstillnotavailablefortheuser. Thiscanbejustiedbythestillpoorqualityofsuchimplementation.XFEMfunctionalitiesappearforthersttimewithintheAbaqus/CAEframework, in2009withtheAbaqus6.9release[93]. TheXFEMimplementationinAbaqus/Standardisbasedonthephantomnodesmethod[94, 95]; insuchmethod, these phantomnodes are superposedtothe standardones, toreproduce the presence of the discontinuity.Roughlyspeaking,phantomnodesaretiedtotheircorrespondingrealnodeswhentheenrichedelementisintact. Thissituationholdsuntil theelementisnotcutbyacrack; assoonastheelementiscracked, itisdividedintwoseparateparts,eachofthemincludingbothrealandphantomnodes(seeFig.1.24).Figure1.24: Phantomnodesmethod.Inparticular, theseparationprocedureoccurs whentheequivalent strainenergyreleaserateexceeds thecriticalstrainenergyreleaserateatthecracktipinanenrichedelement. Oncethisconditionhasbeensatised,everyphantomnodeisnomorerestrainedtoitscorrespondingrealoneandthus,theycanfreelymoveapart.1.3.5 LimitationsoftheuseofXFEMwithinAbaqusAsaforementioned, duetoitsrelativelyrecentintroduction, theXFEMimplementationinAbaqusisstill aectedbycertainrelevantlimitations[96]. Amongthemostimportant,wemaylistthefollowingones: OnlyGeneralStaticandImplicitDynamicanalysescanbeperformed; Onlylinearcontinuumelementscanbeused,withorwithoutreducedintegration; Parallelprocessingofelementsisnotallowed; Fatiguecrackgrowthphenomenoncannotbemodeled; Onlysingleornon-interactingcrackscanbecontainedinthedomain; Nocrackbranching; Acrackcannotturnmorethan90degreeswithinanelement; StillnotavailableinAbaqus/Explicit.In addition to this, it is of remarkable concern to analyze the enrichment procedure of the nodes executed in Abaqus.Briey,Abaqus/Standarddistinguishesbetweentwokindsofcrackwhichcouldbepresentwithinacertaindomain,i.e.stationarycracksandpropagatingcracks. Itappearsobviousthatintherstcasethecrackisnotallowedtopropagatewithinthebodyandonlyastaticanalysismightbecarriedout.Forstationarycracks, thecracktipcanbelocatedwhereverwithinanelementwhilst, forapropagatingcrackitisrequired that the crack itself, completely cut an element and, as consequence, the crack tip cannot be located everywhereinthemodelbutonlyalonganelementedge. Undertheseconsiderations,itmaybeinferredthat,onceacrackstartstopropagate,itwillkeepcuttingcompletelyeachoftheelementsthatitwillgothrough. Inotherwords,inapropagatingcracks,thecracktipmotioncannotarrestwithinanelement.KTHRoyal InstituteofTechnology-SolidMechanicsDepartment28L.Gigliotti CHAPTER1. FUNDAMENTALS:LITERATUREREVIEWANDBASICCONCEPTS(a)Propagatingcracks. (b)Stationarycracks.Figure1.25: EnrichmentprocedureinAbaqus/Standard.Themaindierencebetweentheaforementionedcasesof propagatingandstationarycracks, isintheenrichmentprocedure;moreindetail,itisdierentthenumberofenrichednodesandtheenrichmentfunctionsadopted.For the propagating cracks typology, the asymptotic near-tip singularity functions are not included in the enrichmentschemeandonlytheHeavisidefunctionisused;inthisway,asseenbefore,thecrackcanbelocatedeverywhereinthemodel while the crack tip is forced to lie on an element edge11. On the other hand,for a stationary crack both HeavisideandcracktipsingularityfunctionsareincludedintheXFEMdiscretization. AccordingtowhatdepictedinFig. 1.25,thenodesof theelementscompletelycutbythecrackareenrichedonlywiththeHeavisidefunction, whilethesingleelementcontainingthecracktiphasitsnodesenrichedwithboththeHeavisidefunctionandtheasymptoticnear-tipsingularityfunctions. Inordertoimprove theaccuracyofresults,itispossibletoenrich,logicallyonlywiththenear-tipsingularityfunctions, alsoacertainnumberof elementswithinthedistancenamedasenrichment r