Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions

Abstract

This paper is concerned with two-dimensional, linearly elastic, composite materials made by mixing two isotropic components. For given volume fractions and average strain, we establish explicit optimal upper and lower bounds on the effective energy quadratic form. There are two different approaches to this problem, one based on the “Hashin-Shtrikman variational principle” and the other on the “translation method". We implement both. The Hashin-Shtrikman principle applies only when the component materials are “well-ordered", i.e., when the smaller shear and bulk moduli belong to the same material. The translation method, however, requires no such hypothesis. As a consequence, our optimal bounds are valid even when the component materials are not well-ordered. Analogous results have previously been obtained by Gibianski and Cherkaev in the context of the plate equation.