Optimal bounds and microgeometries for elastic two-phase composites

Consider two isotropic, linearly elastic materials with elasticity tensors C//1, C//2 and assume C//1 less than C//2. We prove that the class of effective tensors of composites built from these materials with given volume fractions that are invariant under a given symmetry group is such that each of its elements is bounded above and below by tensors in the same class corresponding to finite-rank laminates. This implies that for any imposed uniform strain or stress field, optimal bounds on the effective strain or stress energy per unit volume are attained by finite-rank laminates. Explicit bounds on the strain energy, with no symmetry assumptions, are given in dimensions 2 and 3. These bounds are more stringent than the classical Voigt-Reuss bounds. Finally, explicit bounds and microgeometries are given for the effective moduli of composites with cubic symmetry.