Optimal bounds on the effective behavior of a mixture of two well-ordered elastic materials

We consider linearly elastic composite materials made by mixing two possibly anisotropic components. Our main hypothesis is that the Hooke's laws of the two components be well-ordered. For given volume fractions and average strain, we present optimal upper and lower bounds on the elastic energy quadratic form. We also discuss bounds on sums of energies and bounds involving complementary energy rather than elastic energy. Our arguments are based primarily on the HashinShtrikman variational principle; however, we also discuss how the same results arise from the "translation method", making use of the analysis of Milton. Our bounds are equivalent to those established by Avelleneda and closely related to the "trace bounds" established by Milton and Kohn. The optimal energy bounds, however, are presented here as the extreme values of certain convex optimization problems. The optimal microgeometries are determined by the associated first-order optimality conditions. A similar treatment for mixtures of two incompressible, isotropic elastic materials has previously been given by Kohn and Lipton.