Optimal lower bounds on the elastic energy of a composite made from two non-well-ordered isotropic materials
This paper is a continuation of our previous work [AK] concerning optimal bounds on the effective behavior of a mixture of two linearly elastic materials. While in [AK] we restricted our attention to the case of two well-ordered components, here we focus on the case of two non-well-ordered and isotropic ones, i.e., the case when the smaller shear and bulk moduli do not belong to the same material. For given volume fractions and average strain, we establish an optimal lower bound on the effective energy quadratic form. We give two proofs of this result: one based on the Hashin-Shtrikman-Walpole variational principle, the other on the translation method. 0. Introduction. This paper is a continuation of our previous work [AK], which was concerned with optimal bounds on the effective behavior of a mixture of two well-ordered elastic materials. Here, we establish an optimal lower bound on the elastic energy of a mixture of two non-well-ordered, isotropic elastic materials. The macroscopic properties of a linearly elastic composite material are described by its tensor of effective moduli (Hooke's law) a*. This fourth-order tensor depends on the microgeometry of the mixture as well as on the elastic properties of the components. Suppose that a* arises by mixing two materials ct, and a2 with prescribed volume fractions 0X and 02 respectively, but with an unknown microstructure. For any given strain £ (a symmetric, second-order tensor), an optimal lower (resp. upper) bound is a function /_ — f_{ax, o2, d{, d2, £) (resp. f ) such that