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

Abstract

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, K 2 and μ 2 , respectively, and volume fraction φ 2 , dispersed randomly throughout a matrix phase, with bulk and shear moduli, K 1 and μ 1 , respectively, and volume fraction φ 1 . We tabulate the two fundamental microstructural parameters I 1 and L 1 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 φ 2 . The third-order bounds are very restrictive at low φ 2 values (e.g., φ 2 = 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.