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1、Chapter12HomogenizationMethodsandMultiscaleModelingTarekI.ZohdiUniversityofCalifornia,Berkeley,CA,USAmanymodernengineeringdesigns,materialswithhighly1Introduction407complexparticulatemicrostructuresarenowinuse.The2FundamentalMicroMacroConcepts409macroscopicresponseofsuchmicroscopicallymodi?edbas
2、ematerialsistheaggregatebehavioroftheassemblage3TestingProcedures409ofdifferentpurecomponents,forexample,several4TheHillReussVoigtBounds411particlesor?berssuspendedinabindingmatrixmaterial5MoreRe?nedMicroMacroApproximations412(Figure1).Intheconstructionofsuchmaterials,thebasic6ComputationalHomog
3、enization413philosophyistoselectmaterialcombinationstoproduce7MicrogeometricalManufacturingaggregateresponsespossessingdesirablepropertiesfromIdealizations414eachcomponent.Forexample,instructuralengineering8NumericalDiscretization416applications,theclassicalchoiceisaparticulatephasethat9OverallT
4、estingProcess:Numericalservesasastiffeningagentforabasematrixmaterial,Examples417althoughinmanymoderndaydesignstheconcernsarewide10IncreasingSampleSize420ranging,fromincreasingfracturetoughnesstotailoringthermal,acoustical,andpiezoelectricalproperties.Itis11Hierarchical/MultiscaleInterpretations
5、421nowcommonlyacceptedthatnumericalsimulationisan12ClosingCommentsandFutureDirections425importantcomponentintheanalysisanddesignoftheNotes427micromacrostructuralresponserelationshipsofsuchReferences427materials[2].FurtherReading430Ifoneweretoattempttoperformadirectnumericalsim-ulation,forexample
6、,ofthemechanicalresponseofanentiremacroscopicengineeringstructurecomposedofamicro-1INTRODUCTIONheterogeneousmaterial,incorporatingallofthemicroscaledetails,anextremely?nespatialdiscretizationmesh,forexample,thatofa?niteelementmesh,wouldbeneededtoAkeytothesuccessofmanymodernstructuralcapturetheef
7、fectsofthemicroscaleheterogeneities.Thecomponentsisthetailoredbehaviorofthematerial.Arelativelyinexpensivewaytoobtaindesiredstructuralresultingsystemofequationswouldcontainliterallybil-scale(macroscopic)materialresponsesisto
