Third-order bounds on the effective bulk and shear modulus of a dispersion of fully penetrable spheres

We evaluate the third-order Beran-Molyneux bounds on the effective bulk modulus K_e and the third-order McCoy bounds on the effective shear modulus μ_e of a model of a two-phase composite in which one of the phases consists of spherical inclusions (or voids), with bulk and shear moduli, K₂ and μ₂, respectively, and volume fraction φ₂, dispersed randomly throughout a matrix phase, with bulk and shear moduli, K₁ and μ₁, respectively, and volume fraction φ₁. We tabulate the two fundamental microstructural parameters I₁ and L₁ required to evaluate the bounds, which depend upon the three-point matrix probability function of the model, for the aforementioned fully-penetrablesphere model. We compare the third-order bounds on K_e and μ_e to the second-order bounds due to Hashin and Shtrikman and to Walpole. We find that the third-order bounds for our model are always more restrictive than the corresponding second-order bounds. When the moduli of the phases differ by an order of magnitude, the third-order bounds are sharp enough to provide quantitatively useful estimates of K_e and μ_e for all φ₂. The third-order bounds are very restrictive at low φ₂ values (e.g., φ₂ = 0.1) where they remain useful for cases in which the moduli of the phases differ by two orders of magnitude. Experimental values of μ_e measured by Corson for a tungstenlead composite are found to lie within the McCoy bounds for our model, with the lower bound giving a good estimate of the data.