Correlation functions for predicting properties of heterogeneous materials. III. Effective elastic moduli of two-phase solids

Abstract

The Beran-Molyneux bounds on the effective bulk modulus and the McCoy bounds on the effective shear modulus of a statistically homogeneous and isotropic heterogeneous material require a knowledge of several three-point correlation functions. These correlation functions are contained within integrals which must be evaluated. Both sets of bounds are here evaluated for two-phase materials, using the correlation functions developed in Paper II of this series. The method of evaluation includes a simplification of the integrals containing the correlation functions and a numerical evaluation of the simplified integrals. The resultant bounds are compared with experimental data on the effective shear modulus of three two-phase systems: aluminum-lead, iron-lead, and tungsten-lead. The bounds are also compared with the less restrictive Hashin-Shtrikman bounds which do not require any knowledge of correlation functions. It is shown that both sets of bounds are a significant improvement over the Hashin-Shtrikman bounds, and that they are in agreement with existing experimental data.