Embedded corrector problems for homogenization in linear elasticity

TL;DR

This paper extends embedded corrector problems to linear elasticity, providing convergent approximations for homogenized elastic tensors in heterogeneous materials, with special focus on isotropic cases.

Contribution

It introduces a novel approach for approximating homogenized elastic properties using embedded corrector problems specific to elasticity, with proven convergence.

Findings

Approximations converge to the true homogenized tensor as domain size increases.

Special considerations are made for isotropic elastic materials.

The method extends previous scalar diffusive equation results to elasticity.

Abstract

In this article, we extend the study of embedded corrector problems, that we have previously introduced in the context of the homogenization of scalar diffusive equations, to the context of homogenized elastic properties of materials. This extension is not trivial and requires mathematical arguments specific to the elasticity case. Starting from a linear elasticity model with highly-oscillatory coefficients, we introduce several effective approximations of the homogenized tensor. These approximations are based on the solution to an embedded corrector problem, where a finite-size domain made of the linear elastic heterogeneous material is embedded in a linear elastic homogeneous infinite medium, the constant elasticity tensor of which has to be appropriately determined. The approximations we provide are proven to converge to the homogenized elasticity tensor when the size of the embedded domain tends to infinity. Some particular attention is devoted to the case of isotropic materials.