Embedded Corrector Problems for Homogenization in Linear Elasticity

Multiscale Modeling and Simulation, vol. 23, pp. 1236–1273

Abstract

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 nontrivial 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. One of the approximations we propose is based on a self-consistent formulation, for which proving, in the general case, the existence of a solution is challenging. We show that, by restricting ourselves to a class of heterogeneous isotropic materials for which the homogenized elasticity tensor is known to be isotropic, it is possible to introduce a weaker self-consistent formulation for which both existence of a solution and convergence of that approximation to the homogenized tensor can be shown.

Authors 4

1. Virginie Ehrlacher

   Institut Polytechnique de Paris · Institut national de recherche en sciences et technologies du numérique · CERMICS · MATHERIALS: MATHematics for MatERIALS

   Affiliation as printed

   CERMICS, ENPC, Institut Polytechnique de Paris, Marne-la-Vallée, France and MATHERIALS project-team, Inria, Paris, France

2. Frédéric Legoll

   Centre National de la Recherche Scientifique · Institut Polytechnique de Paris · Institut national de recherche en sciences et technologies du numérique · Université Gustave Eiffel · École nationale des ponts et chaussées · Laboratoire Navier · MATHERIALS: MATHematics for MatERIALS

   Affiliation as printed

   Navier, ENPC, Institut Polytechnique de Paris, Univ Gustave Eiffel, CNRS, Marne-la-Vallée, France and MATHERIALS project-team, Inria, Paris, France

3. Benjamin Stamm

   University of Stuttgart

   Affiliation as printed

   Chair of Numerical Mathematics for High-Performance Computing (NMH), University of Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany

4. Shuyang Xiang Aachen Department of Mathematics Institut Polytechnique de Paris

   RWTH Aachen University · Institut Polytechnique de Paris · CERMICS

   Affiliation as printed

   CERMICS, ENPC, Institut Polytechnique de Paris, Marne-la-Vallée, France and MATHCCES, Department of Mathematics, RWTH Aachen University, Schinkelstrasse 2, D-52062 Aachen, Germany