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1、Chapter10Damage,MaterialInstabilities,andFailureRenedeBorst´DelftUniversityofTechnology,Delft,TheNetherlandspolymers,andthelocalizedaccumulationofmicrocracksin1Introduction335concreteandrocksunderlowconfiningpressures.Basically,therearetwowaystomodelfailure,eith
2、erina2DamageMechanics336discreteorinacontinuumformat.Thelatterapproachhas3MaterialInstabilitiesandMeshSensitivity341theadvantagethattheearlystagesofthefailureprocess,4Cohesive-ZoneModels349forexampletheonsetofmicrocrackingorvoidnucleation,5EnhancedContinuumDesc
3、riptions355canbemodeledadequately.Variousclassesofconstitutive6DiscreteFailureModels362modelsareavailable,withdamageandplasticitymodels7ConcludingRemarks370beingthemostprominent.Attemptstocaptureshearbands,References370cracks,andotherlocalizationphenomenawithnu
4、mericalmethodsforcontinuumdamageandcontinuumplasticitymodelshavestartedinthemid-1980s.Theseattemptsfailedinthesensethatthesolutionappearedtobedetermined1INTRODUCTIONfullybythefinenessandthedirectionofthediscretization.Theunderlyingreasonappearedtobealocalchangeo
5、fcharacterofthegoverningdifferentialequationsthePreventionoffailureofstructuresandstructuralcompo-equilibriumequations,kinematicequationsandconstitutivenentshasalwaysbeenamajorissueinengineering.Whileequationsatacertainlevelofdamageaccumulation.Intherearemanypo
6、ssiblecausesforstructuralfailure,muchthestaticcase,theellipticcharacterofthesetofpartialattentionhasbeengiventofractureofmaterialsoverthedifferentialequationsislost,while,ontheotherhand,inthepastdecades.Uponacloserobservationofafracture,itdynamiccase,wetypicall
7、yobservealossofhyperbolicity.appearsthatitisoftenprecededbytheformationofaIneithercase,therateboundaryvalueproblemceasestofractureprocesszone,inwhichdamageandotherinelasticbewellposedandnumericalsolutionssufferfromasevereeffectsaccumulateand,accordingly,inwhich
8、high-straindiscretizationsensitivity.gradientsprevail.ThisphenomenoniscommonlydenotedTosolvethisproblem,onemusteitherintroduceaddi-asstrainlocalization,orlocalizationofdefor
