1|PageCopyrighte-Xstreamengineering,2009Multi-ScaleModelingofCompositeMaterialsandStructureswithDIGIMATtoANSYSDocumentVersion1.0,February2009Copyright,e-Xstreamengineering,2009infoe-Xwww.e-XMaterials:EngineeringPlastics,ReinforcedPlastics.e-XstreamTechnology:DIGIMAT,Digimat-MF,Digimat-FE,DigimattoANSYS,MAP.ComplementaryCAETechnology:Moldflow,Moldex3D,SigmaSoft,ANSYS.Industry:MaterialSuppliers,Automotive,Aerospace,Consumer&IndustrialProducts.TABLEOFCONTENTEXECUTIVESUMMARY.2MaterialMulti-ScaleModeling:anintroduction.2FEHomogenization:anapplicationtonanocomposites.5ModelingFillerClustering,atypicalnanoeffect.5ResultComparison.7FE/MFHCoupledComputation:anapplicationtoanindustrialpart.9ProblemDescription.9MaterialModeling.10SimulationResults.11Bibliography.12LegalNotice.eX,eXdigimatande-Xstreamengineeringareregisteredtrademarksofe-XstreamengineeringSA.Theotherproductandcompanynamesandlogosaretrademarksorregisteredtrademarksoftheirrespectiveowners.2|PageCopyrighte-Xstreamengineering,2009EXECUTIVESUMMARYInthispaper,webrieflyintroducetwomulti-scalemodelingapproaches,namelytheMean-Field(MFH)andFiniteElementHomogenization(FEH)methods.Thesepowerfultechniquesrelatethemicroscopicandmacroscopicstressandstrainfieldswhenmodelingmaterialbehaviorsandhencecancapturetheinfluenceofthematerialmicrostructure(i.e.fiberorientation,fibercontent,fiberlength,etc.)onitsmacroscopicresponse.Toillustratethesetechniques,wealsopresent(i)anapplicationoffiniteelementhomogenizationtoananostructureand(ii)thestudyofaninjectedglassfiberreinforcedplasticneonlightclaspusingfiniteelementcomputationsatthemacroscalecoupledwithMFhomogenizationatthemicroscale.MaterialMulti-ScaleModeling:anintroductionAsamotivatingexample,letusconsideraplasticpartmadeupofathermoplasticpolymerreinforcedwithshortglassfibers.Astypicaloftheinjectionmoldingmanufacturingprocess,thefiberdistributioninsidethefinalproductwillvarywidelyintermsoforientationandlength,seeFigure1.Thecompositematerialwillbebothanisotropicandheterogeneous,whichmakesitextremelydifficulttoperformareliablesimulationoftheproductusingaclassicalapproachbasedonmacroscopicconstitutivemodels.However,apredictivesimulationispossibleviaamulti-scaleapproach,whichcanbedescribedinarathergeneralsettingasfollows.Figure1:Fiberorientationdistributioninaninjectedglassfiber-reinforcedplasticclutchpedal.CourtesyofRhodia&Trelleborg.Letusstudyaheterogeneoussolidbodywhosemicrostructureconsistsofamatrixmaterialandmultiplephasesofso-called“inclusions”,whichcanbeshortfibers,platelets,particles,micro-cavitiesormicro-cracks.Ourobjectiveistopredicttheresponseofthebodyundergivenloadsandboundaryconditions(BCs),basedonitsmicrostructure.Wecandistinguishtwoscales,themicroscopicandmacroscopiclevels,respectively.Theformercorrespondstothescaleoftheheterogeneities,whileatthemacroscale,thesolidcanbeseenaslocallyhomogeneous；seeFigure2.Inpractice,itwouldbecomputationallyimpossibletosolvethemechanicalproblematthefinemicroscale.Therefore,weconsiderthemacroscaleandassumethateachmaterialpointisthecenterofarepresentativevolumeelement(RVE),whichcontainstheunderlyingheterogeneousmicrostructure.Classicalsolidmechanicsanalysisiscarriedoutatthemacroscale,exceptthatateachcomputationpoint,strainorstressvaluesaretransmittedasBCstotheunderlyingRVE.Inotherwords,anumericalzoomisrealizedateachmacropoint.TheRVEproblemsaresolvedandeachofthemreturnsstressandstiffnessvalues,whichareusedatthemacroscale.3|PageCopyrighte-Xstreamengineering,2009Figure2:Illustrationofthemulti-scalematerialmodelingapproach,afterNemat-NasserandHori(1).Nowtheonlydifficultyinthistwo-scales(andmoregenerallymulti-scale)approachistosolvetheRVEproblems.ItcanbeshownthatforaRVEunderclassicalBCs,themacrostrainsandstressesareequaltothevolumeaveragesovertheRVEoftheunknownmicrostrainandstressfieldsinsidetheRVE.Inlinearelasticity,relatingthosetwomeanvaluesgivestheeffectiveoroverallstiffnessofthecompositeatthemacroscale.InordertosolvetheRVEproblem,onecanusethewell-knownfiniteelement(FE)method,seeFigures7to10.Thismethodofferstheadvantagesofbeingverygeneralandextremelyaccurate.However,ithastwomajordrawbackswhichare:seriousmeshingdifficultiesforrealisticmicrostructuresandalargeCPUtimefornonlinearproblems,suchasforinelasticmaterialbehaviour.Anothercompletelydifferentmethodismean-fieldhomogenization(MFH),whichisbasedonassumedrelationsbetweenvolumeaveragesofstressorstrainfieldsineachphaseofaRVE；seeFigure3.ComparedtothedirectFEmethod,andactuallytoallotherexistingscaletransitionmethods,MFHisboththeeasiesttouseandthefastestintermsofCPUtime.However,twoshortcomingsofMFHarethatitisunabletogivedetailedstrainandstressfieldsineachphaseanditisrestrictedtoellipsoidalinclusionshapes.Figure3:Mean-fieldhomogenizationprocess:(i)localstrainsarecomputedbasedonthemacrostrains,(ii)localstressesarecomputedbasedonthelocalstrainsandaccordingtoeachphaseconstitutivemodel,and(iii)macrostressesarecomputedbyaveragingthelocalstresses.4|PageCopyrighte-Xstreamengineering,2009AtypicalexampleofMFHistheMori-Tanakamodel(2)whichissuccessfullyapplicabletotwo-phasecompositeswithidenticalandalignedellipsoidalinclusions.ThemodelassumesthateachinclusionoftheRVEbehavesasifitwerealoneinaninfinitebodymadeoftherealmatrixmaterial.TheBCsinthesingleinclusionproblemcorrespondtothevolumeaverageofthestrainfieldinthematrixphaseoftherealRVE.ThesingleinclusionproblemwassolvedanalyticallybyJ.D.Eshelby(3)inalandmarkpaper,whichisthecornerstoneofMFHmodels.Figure4:SchematicoftheMori-Tanakahomogenizationprocedure.Mori-TanakaandotherMFHmodelsweregeneralizedtoothercases,suchasthermoelasticcoupling,two-phasecompositeswithmisalignedfibers(usingamulti-stepapproach)ormulti-phasecomposites(usingamulti-levelmethod).ThepredictionshavebeenextensivelyverifiedagainstdirectFEsimulationofRVEsorvalidatedagainstexperimentalresults.Asageneralconclusion,itwasfoundthatinlinear(thermo)elasticity,MFHcangiveextremelyaccuratepredictionsofeffectiveproperties,althoughfordistributedorientations,progressinclosureapproximationwillbewelcomed.NotealsothatMFHcanbeusedforUD,andforeachyarninwovencomposites.AnimportantandstillongoingeffortbothintheoreticalmodelingandincomputationalmethodsisthegeneralizationofMFHtothematerialorgeometricnonlinearrealms.Suchextensioninvolvessomemajordifficulties.Thefirstoneislinearization,whereconstitutiveequationsatmicroscaleneedtobelinearizedontolinearelastic-orthermoelastic-likeformat.Thesecondissueisthedefinitionofso-calledcomparisonmaterialswhicharefictitiousmaterialsdesignedtopossessuniforminstantaneousstiffnessoperatorsi