Compensated compactness for homogenization and reduction of dimension: The case of elastic laminates

Abstract

The aim of this paper is to extend, to the linear elasticity system, the asymptotic analysis by compensated compactness previously developed by the authors for the linear diffusion equation. For simplicity, we restrict ourselves to stratified media. In the case of sole homogenization we recover the classical result of W.H. Mc Connel, deriving explicitly the effective elasticity tensor for stratified media. Here we give a new proof of his result, based on compensated compactness and on a technique of decomposing matrices. As for the case of simultaneous homogenization and reduction of dimension, we perform the asymptotic analysis, as the thickness tends to zero, of a three-dimensional laminated thin plate having an anisotropic, rapidly oscillating elasticity tensor. The limit problem is presented in three different ways, the final formulation being a fourth-order problem on the two-dimensional plate, with explicitly given elasticity tensors and effective source terms.