Homogenisation of thin periodic frameworks with high-contrast inclusions

TL;DR

This paper studies the homogenisation of a two-dimensional elastic composite with high-contrast inclusions, revealing a coupled two-scale limit and spectral band-gap structure as the composite's microstructure shrinks.

Contribution

It introduces a new homogenisation framework for high-contrast periodic frameworks with inclusions, capturing the coupled microscopic and macroscopic displacements.

Findings

Convergence of elastic displacement to a coupled two-scale homogenised problem

Spectral convergence to an operator with band-gap structure

Identification of the limit behavior as the composite's period diminishes

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 with a band-gap structure.