Embedded corrector problems for homogenization in linear elasticity

HAL (Le Centre pour la Communication Scientifique Directe)

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.

Authors 4

1. Virginie Ehrlacher

   Institut national de recherche en sciences et technologies du numérique · École nationale des ponts et chaussées · CERMICS · MATHERIALS: MATHematics for MatERIALS

   Affiliation as printed

   CERMICS , École des Ponts , Marne-La-Vallée , France

   MATHERIALS project-team , Inria , Paris , France

   CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (6 et 8 avenue Blaise Pascal Cité Descartes - Champs sur Marne 77455 Marne la Vallée Cedex 2 - France)

   MATHERIALS - MATHematics for MatERIALS (France)

2. Frédéric Legoll

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

   Affiliation as printed

   MATHERIALS project-team , Inria , Paris , France

   Navier , École des Ponts , Univ Gustave Eiffel , CNRS , Marne-La-Vallée , France

   MATHERIALS - MATHematics for MatERIALS (France)

   NAVIER UMR 8205 - Laboratoire Navier (École des Ponts ParisTech, 6-8 avenue Blaise Pascal, 77455 Champs-sur-Marne - France)

3. Benjamin Stamm Aachen Department of Mathematics

   RWTH Aachen University · FH Aachen

   Affiliation as printed

   MATHCCES , Department of Mathematics , RWTH Aachen University , Schinkelstrasse 2 , D-52062 Aachen , Germany

   CCSE - Center for Computational Engineering Science [Aachen] (Germany)

4. Shuyang Xiang Aachen Department of Mathematics

   École nationale des ponts et chaussées · CERMICS · RWTH Aachen University · FH Aachen

   Affiliation as printed

   CERMICS , École des Ponts , Marne-La-Vallée , France

   MATHCCES , Department of Mathematics , RWTH Aachen University , Schinkelstrasse 2 , D-52062 Aachen , Germany

   CCSE - Center for Computational Engineering Science [Aachen] (Germany)

   CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (6 et 8 avenue Blaise Pascal Cité Descartes - Champs sur Marne 77455 Marne la Vallée Cedex 2 - France)