Exact Relations for the Effective Properties of Nonlinearly Elastic Inhomogeneous Materials

This study is concerned with the effective behavior of nonlinearly elastic materials, which are locally inhomogeneous in one, two, or three directions and whose prototypes are layered, fiber reinforced, matrix-inclusion composites or polycrystals. A systematic method based on the implicit function theorem is proposed to find conditions for the existence of locally uniform strain fields and to exactly determine the overall stress response of such a material to a macroscopic strain associated with a locally uniform strain field. General exact connections are established between the effective elastic tangent moduli evaluated at each macroscopic strain inducing a locally uniform strain field. These results are applied to a cubic polycrystal whose elastic constitutive relation is the most general one, and to power-law fiber-reinforced composites. In particular, it is proven that the overall nonlinear elastic stress response of a cubic polycrystal to an isotropic strain is identical to that of a cubic monocrystal. This conclusion constitutes a nonlinear extension of a well-known result of Hill (1952).