Explicit Hashin-Shtrikman bounds in 3D linearized elasticity

Abstract

In this paper, we are dealing with Hashin-Shtrikman bounds in the context of linearized elasticity. We give an explicit computation of the upper bound on the primal energy in three space dimensions and the corresponding microstructure that saturates this bound. Due to equivalency between the upper Hashin-Shtrikman bound on the primal energy and the lower Hashin-Shtrikman bound on the complementary energy, the explicit calculation of the latter arises. These calculations have straightforward applications in evaluation of experimental results and numerical schemes regarding composite elastic materials, as well as structural optimization problems.