Effective properties of two-phase disordered composite media: II. Evaluation of bounds on the conductivity and bulk modulus of dispersions of impenetrable spheres

Abstract

We evaluate third-order bounds on the effective conductivity ${\ensuremath{\sigma}}_{e}$ and effective bulk modulus ${K}_{e}$ of a random dispersion of equal-sized impenetrable spheres in a matrix up to sphere volume fractions near the random close-packing value. The third-order bounds, which incorporate an integral ${\ensuremath{\zeta}}_{2}$ that depends upon the three-point probability function of the two-phase medium, are shown to significantly improve upon second-order Hashin-Shtrikman bounds, which do not utilize this information, for a wide range of phase property values and volume fractions. The physical significance of the microstructural parameter ${\ensuremath{\zeta}}_{2}$ for general microstructures is briefly discussed. The third-order bounds on ${\ensuremath{\sigma}}_{e}$ and ${K}_{e}$ are found to be sharp enough to yield good estimates of the bulk properties for a wide range of sphere volume fractions, even when the phase property values differ by as much as two orders of magnitude. Moreover, when the spheres are highly conducting or highly rigid relative to the matrix, the third-order lower bound on the respective effective property provides a useful estimate of it for a wide range of sphere volume fractions.