Homogenisation and spectral convergence of a periodic elastic composite with weakly compressible inclusions

TL;DR

This paper investigates the homogenisation and spectral convergence of a periodic elastic composite with weakly compressible inclusions, revealing how microscopic properties influence macroscopic behavior and spectrum.

Contribution

It derives the two-scale limit problem for the composite and characterizes the spectral convergence, linking microscopic and macroscopic spectral properties.

Findings

Microscopic part solves a Stokes type problem.

No microscopic oscillations under irrotational body forces.

Spectrum converges to the two-scale limit spectrum.

Abstract

A two phase elastic composite with weakly compressible elastic inclusions is considered. The homogenised two-scale limit problem is found, via a version of the method of two-scale convergence, and analysed. The microscopic part of the two-scale limit is found to solve a Stokes type problem and shown to have no microscopic oscillations when the composite is subjected to body forces that are microscopically irrotational. The composites spectrum is analysed and shown to converge, in an appropriate sense, to the spectrum of the two-scale limit problem. A characterisation of the two-scale limit spectrum is given in terms of the limit macroscopic and microscopic behaviours.