Bulk properties of composite media. II. Evaluation of bounds on the shear modulus of suspensions of impenetrable spheres

We evaluate third-order bounds due to Milton and Phan-Thien on the effective shear modulus G_e of a random dispersion of identical impenetrable spheres in a matrix up to sphere-volume fractions near the random close-packing value. The third-order bounds, which incorporate two parameters, ζ₂ and η₂, that depend upon the three-point probability function of the composite medium, are shown to significantly improve upon the second-order Hashin-Shtrikman (or, more general, Walpole) bounds which do not utilize this information, for a wide range of volume fraction and phase property values. The physical significance of the microstructural parameter η₂ for general microstructures is briefly discussed. The third-order bounds on G_e are found to be sharp enough to yield good estimates of the effective shear modulus for a wide range of sphere-volume fractions, even when the individual shear moduli differ by as much as two orders of magnitude. Moreover, when the spheres are highly rigid relative to the matrix, the third-order lower bound on the effective property provides a useful estimate of it. The third-order bounds are compared with experimental data for the shear modulus of composites composed of glass spheres in an epoxy matrix and the shear viscosity of suspensions of bituminous particles in water. In general, the third-order lower bound (rather than the upper bound) on G _e tends to provide a good estimate of the data.