Exact second-order estimates for the effective mechanical properties of nonlinear composite materials

Motivated by previous small-contrast perturbation estimates, this paper proposes a new method for estimating the effective behavior of nonlinear composite materials with arbitrary phase contrast. The key idea is to write down a second-order Taylor expansion for the phase potentials, about appropriately defined phase average strains. The resulting estimates, which are exact to second order in the contrast, involve the ''tangent'' modulus tensors of the nonlinear phase potentials, and reduce the problem for the nonlinear composite to a linear problem for an anisotropic thermoelastic composite. Making use of a well-known result by Levin for two-phase thermoelastic composites, together with estimates of the Hashin-Shtrikman type for linear elastic composites, explicit results are generated for two-phase nonlinear composites with statistically isotropic particulate microstructures. Like the earlier small-contrast asymptotic results, the new estimates are found to depend on the determinant of the strain, but unlike the small-contrast results that diverge for shear loading conditions in the nonhardening limit, the new estimates remain bounded and reduce to the classical lower bound in this limiting case. The general method is applied to composites with power-law constitutive behavior and the results are compared with available bounds and numerical estimates, as well as with other nonlinear homogenization procedures. For the cases considered, the new estimates are found to satisfy the restrictions imposed by the bounds, to improve on the predictions of prior homogenization procedures and to be in excellent agreement with the results of the numerical simulations. Copyright (C) 1996 Elsevier Science Ltd