A 2ND-ORDER THEORY FOR NONLINEAR COMPOSITE-MATERIALS

Motivated by a small-contrast perturbation expansion for the effective potential of a nonlinear composite material, due to Suquet and Ponte Castaneda (1993), this Note proposes a new, approximate method for est imating the effective behaviour of nonlinear composite materials with arbitrary phase contrast. The key idea is to write down a second-order Taylor series expansion for the phase potentials, about appropriately defined phase-average strains. The resulting estimates, which are exa ct to second order in relation to the contrast, involve the ''tangent moduli'' of the nonlinear potential, and reduce the problem for the no nlinear composite to a linear problem for an anisotropic thermoelastic composite. Making use of Levin's result for two-phase thermoelastic c omposites, together with estimates of the Hashin-Shtrikman type for li near elastic composites, explicit results are given for two-phase comp osites with power-law constitutive behaviour and statistically isotrop ic particulate microstructures. Like the small-contrast estimates, the new estimates are found to depend on the determinant of the strain, b ut unlike the small-contrast estimates, which diverge for shear loadin g in the non-hardening limit, the new estimates remain bounded and red uce to the classical lower bound in this limiting case.