Three-point bounds and other estimates for strongly nonlinear composites

A variational procedure due to Ponte Casta\~neda et al. [Phys. Rev. B 46, 4387 (1992)] is used to determine three-point bounds and other types of estimates for the effective response of strongly nonlinear composites with random microstructures. The variational procedure makes use of estimates for the effective properties of ``linear comparison composites'' to generate corresponding estimates for nonlinear composites. Several equivalent forms of the variational procedure are derived. In particular, it is shown that the mean-field theory of Wan et al. [Phys. Rev. B 54, 3946 (1996)], which also makes use of a linear comparison composite, together with a certain ``decoupling approximation,'' leads to results that are precisely identical to those that can be obtained from the earlier variational procedure. Finally, three-point bounds and other estimates are computed for power-law composites with cell-type microstructures, and the results are compared with random resistor network simulations available from the literature.