Homogenization of two-dimensional elasticity problems with very stiff coefficients

Abstract

In this paper we study the asymptotic behaviour of a sequence of two-dimensional linear elasticity problems with equicoercive elasticity tensors. Assuming the sequence of tensors is bounded in L 1 , we obtain a compactness result extending to the elasticity the div–curl approach of [M. Briane, J. Casado-Díaz, Two-dimensional div–curl results. Application to the lack of nonlocal effects in homogenization, Comm. Partial Differential Equations 32 (2007) 935–969] for the conduction. In the periodic case this compactness result is refined replacing the L 1 -boundedness by a less restrictive condition involving the oscillations period. We also build a sequence of isotropic elasticity problems with L 1 -unbounded Lamé's coefficients, which converges to a second gradient limit problem. This loss of compactness shows a gap in the limit behaviour between the very stiff problems of elasticity and those of conduction. Indeed, in the conduction case a compactness result was proved in [M. Briane, J. Casado-Díaz, Asymptotic behaviour of equicoercive diffusion energies in dimension two, Calc. Var. Partial Differential Equations 29 (4) (2007) 455–479] without assuming any bound from above for the conductivities.