A mathematical framework for reducing the domain in the mechanical analysis of periodic structures
作者： S. T. Pinho , P. Robinson , N. V. De Carvalho
DOI:
关键词: Combinatorics 、 Domain (mathematical analysis) 、 Mathematics 、 Topology 、 Unit (ring theory) 、 Reduction (mathematics)
摘要: AbstractAtheoreticalframeworkisdeveloppedleadingtoasoundderivationofPeriodicBoundaryConditions(PBCs) for the analysis of domains smaller then Unit Cells (UCs), named reduced Cells(rUCs),byexploitingnon-orthogonaltranslationsandsymmetries. AparticulartypeofUCs,Offset-reduced (OrUCs) are highlighted. These enable reduction domain ofthe traditionally defined UCs without any loading restriction. The relevance framework anditsapplicationtoanyperiodicstructureisillustratedthroughtwopracticalexamples: 3Dwovenandhoneycomb.1. IntroductionNumericalanalysisofperiodicmaterialsandstructureshasproventobeanextremelypowerfultool.Itprovidesdetailedinformation,suchasfailureinitiationsitesandstress-strainatsmallerscales(meso/micro)Ithasbeensuccessfullyusedtodeterminehomogenisedproperties,studythedetailedstress-strainfieldsatnano-andmicroscopicscalestoobtainstructuraldamageinitiationconditionsandsites,aswellastosimulatedamagedevelopmentandassociateddeteriorationofthehomogenisedmechanicalproperties[1]. Severalworkscanbefounddiscussingtheapplicationofperiodicboundaryconditionstorepresentativeregions,e.g. [2–5]. Forperiodicstructures,theUnitCell(UC)isusedastherepresentativeregion,andtheanalysisisperformedbyapplyingperiodicdisplacementboundaryconditions. ThetopologicalcomplexityofmanyUCsfoundinpractice,suchasintypicalwovencomposites,oftenleadstounpracticalmodellingandanalysistimes. Forthisreason,internalsymmetriesoftheUCsmustwheneverpossiblebeexploitedtoreducetheanalysis
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