Homogenisation of thin periodic frameworks with high-contrast inclusions

Kirill Cherednichenko, James A. Evans

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Abstract

We analyse a problem of two-dimensional linearised elasticity for a two-component periodic composite, where one of the components consists of disjoint soft inclusions embedded in a rigid framework. We consider the case when the contrast between the elastic properties of the framework and the inclusions, as well as the ratio between the period of the composite and the framework thickness increase as the period of the composite becomes smaller. We show that in this regime the elastic displacement converges to the solution of a special two-scale homogenised problem, where the microscopic displacement of the framework is coupled both to the slowly-varying “macroscopic” part of the solution and to the displacement of the inclusions. We prove the convergence of the spectra of the corresponding elasticity operators to the spectrum of the homogenised operator, which has a band-gap structure.
Original languageEnglish
Pages (from-to)658-679
Number of pages22
JournalJournal of Mathematical Analysis and Applications
Volume473
Issue number2
Early online date11 Dec 2018
DOIs
Publication statusPublished - 15 May 2019

Keywords

  • Band-gap spectrum
  • High-contrast composites
  • Loss of uniform ellipticity
  • Periodic homogenisation
  • Thin structures
  • Two-scale convergence

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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