Combined Effects of Homogenization and Singular Perturbations : A Bloch Wave Approach

TL;DR

This paper investigates how homogenization of periodic media with singular perturbations can be understood through Bloch wave analysis, revealing different regimes and deriving the homogenized tensor from Bloch eigenvalues.

Contribution

It introduces a Bloch wave approach to homogenization in media with singular perturbations, identifying regimes and explicitly computing the homogenized tensor.

Findings

Different homogenization regimes depending on perturbation strength and heterogeneity scale

Homogenized tensor expressed via first Bloch eigenvalue

Higher Bloch modes do not influence the homogenization limit

Abstract

We study Bloch wave homogenization of periodically heterogeneous media with fourth order singular perturbations. We recover different homogenization regimes depending on the relative strength of the singular perturbation and length scale of the periodic heterogeneity. The homogenized tensor is obtained in terms of the first Bloch eigenvalue. The higher Bloch modes do not contribute to the homogenization limit.