Effective energy of nonlinear elastic and conducting composites: Approximations and cross-property bounds

By using similar techniques that were employed in an earlier article on nonlinear conductivity [L. V. Gibiansky and S. Torquato, J. Appl. Phys. 84, 301 1998], we find approximations for the effective energy of nonlinear, isotropic, elastic dispersions in arbitrary space dimension d. We apply our results for incompressible dispersions with rigid or liquid inclusions and, more generally, with a power-law-type shear energy. It is shown that the new approximations lie within the best available rigorous upper and lower bounds on the effective energy. We also develop bounds on the effective energy of nonlinear conducting media with voids or cracks, purely in terms of the effective and phase elastic moduli of the media.