Periodic Structures and Periodic Structures with Random Imperfections

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1、11PeriodicStructuresandPeriodicStructureswithRandomImperfections11.1GeneralAnalysisofPeriodicStructuresandPeriodicStructureswithRandomImperfectionsInmanyproblemsthematerialmaypossessaperiodicmicrostructureformedbythespatialrepetitionofsmallmicrostructures,orunitcells.The

2、periodicstructureofcompositesisveryattractivebecauseitprovidesanestimationofinteractioneffectsforaninfinitenumberofinclusions.Thisinteractiongreatlyinfluencescompositeelasticproperties,especiallyforstronglyheterogeneousma-terialswithahighinclusionconcentration.Theperiodicit

3、yofstructuresoffersthepossibilityoffindingananalytical(ornumerical)solutionofthecorrespond-ingperiodicboundary–valueproblemwithcontrolledaccuracy.Forhomogeneousboundaryconditionsperiodicityofacompositestructureleadstoaperiodicityofthesolutionforthestressandstraindistributi

4、onswiththesameperiodasthatofthemicrostructure.Periodicityofthesolutionallowsonetoexploreatypeofmodelingcompletelydifferentfromtheaverageschemesofrandommedia(consideredinChapters8–10)suchasasymptoticexpansionaswellastheFourierseriesexpansiontofieldinasolidwithperiodicmicros

5、truc-ture[806].Thisasymptotichomogenizationmodelingcoupledthefiniteelementanalysis(FEA)hasbeensupposedtobemosteffectivetoolespeciallyfortheinvestigationofthemicrostructuraleffectontheoverallproperties(see,e.g.,[25],[48],[507],[532],[960]).Suchaperfectlyregulardistribution,o

6、fcourse,doesnotexistinactualcases,althoughtheperiodicmodelingcanbequiteuseful,sinceitprovidesrigorousestimationswithaprioriprescribedaccuracyforvariousmaterialproperties.Forperiodicstructurecompositestherearealsoknownsemianalyticalmeth-odsforsolvingthecellproblem.Themeth

7、odofEshelbytransformationstrain,takingaccountofthevariabilityofthefieldoftransformationstrainwithintheinclusion,hasbeenproposedin[807].However,becausetheyusedveryslowlyconvergingseries,theirapproachishardlyapplicable.Usingtheperiodicfunda-mentalsolutionforanisotropicmediu

8、m[442](seealso[1147])numericalvaluesoftheeffectivecharacteristicsofdispersionallyreinforcedcompositeswit