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1、Comput.MethodsAppl.Mech.Engrg.190$2001)6641±6662www.elsevier.com/locate/cmaSPHelasticdynamicsaa,*bJ.P.Gray,J.J.Monaghan,R.P.SwiftaEpsilonLaboratory,DepartmentofMathematicsandStatistics,MonashUniversity,Clayton3800,Vic.3168,AustraliabLosAlamosNationalLaboratory,LosAlamos,NM8
2、7545,USAReceived5July2000AbstractThestandardsmoothedparticlehydrodynamics$SPH)formulationof¯uiddynamicscanexhibitaninstabilitycalledthetensileinstability.Thisinstabilitymayoccurwithbothpositiveandnegativepressure.Usuallytheeectsaresmall,butinthecaseofelasticorbrittlesolidst
3、heeectsmaybesevere.Undertension,abrittlesolidcanfracture,butitisdiculttodisentanglethephysicalfractureandfragmentationfromthenonphysicalclumpingofSPHparticlesduetothetensileinstability.Recently,oneofus$JJM)hasshownhowthisinstabilitycanberemovedbyanarti®cialstresswhichintrod
4、ucesnegligibleerrorsinlong-wavelengthmodes.Inthispaperweshowhowthealgorithmcanbeimprovedbybasingthearti®cialstressonthesignsoftheprincipalstresses.Wedeterminethepa-rametersofthearti®cialstressfromthedispersionrelationforelasticwavesinauniformmaterial.Weapplythealgorithmtoos
5、-cillatingbeams,collidingringsandbrittlesolids.Theresultsareinverygoodagreementwiththeory,andwithotherhigh-accuracymethods.Ó2001ElsevierScienceB.V.Allrightsreserved.1.IntroductionSmoothedparticlehydrodynamics$SPH)isagridlessLagrangianparticlemethodwhichreplacesthecontinuume
6、quationsof¯uiddynamicsbyparticleequations.TheinteractionsbetweentheparticlesaredeterminedbyinterpolationfrominformationattheSPHparticles$forreferencessee[11]).WhenSPHisappliedtosolidstheSPHparticlesmimicthebehaviouroftheatoms.Ifthesolidiscompressedtheatomsrepeleachother.Ifi
7、tisstretchedtheatomsattracteachotherandopposethestretch.TheSPHparticlesbehavethesamewayexceptthat,whenthematerialisstretched,theattractioncanresultintheSPHparticlesformingclumps.Thisinstabilitywas®rststudiedindetailbySwegleetal.[19],whorelatedittothesignofthepressureandthes
8、ignofthesecondderivativeoftheinterpolatingkernel.Itiscommonlycalledthetensileinsta
