Pp. Castaneda, A 2ND-ORDER THEORY FOR NONLINEAR COMPOSITE-MATERIALS, Comptes rendus de l'Academie des sciences. Serie II. Mecanique, physique, chimie, astronomie, 322(1), 1996, pp. 3-10
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.