Homogenization of perforated laminates

Abstract

This paper deals with the homogenization of perforated laminates which are used for attaining optimal bounds in conduction or elasticity. A perforated laminate is composed of the periodic overlapping of homogeneous layers in different directions and at different ordered scales, one of which is void. We study the homogenization of a conduction problem with Neumann boundary condition on the boundary of the perforated domain occupied by the laminate. Assuming the perforated laminate domain is connected, we prove the H-convergence of the laminate matrix (i.e. the conductivity matrix of the laminate) to a positive definite constant matrix as well as a corrector result in which the corrector has a laminate structure. The proof is based on both the Tartar energy method and the multiscale convergence method of Nguetseng-Allaire, since the construction of an extension operator seems technically not feasible

Keywords:

• homogenization
• laminate
• Neumann's problem
• perforated domain