Bounds on the properties of fiber-reinforced composites

Abstract

The microstructure of a fiber-reinforced composite material can be characterized in terms of a set of n-point matrix probability functions Sn which give the probability of finding n points all in the matrix phase. From a knowledge of these functions one can place rigorous upper and lower bounds on effective thermal, electrical, and mechanical properties of the composite. Third- and fourth-order bounds derived by Milton involve threefold integrals over the three-point matrix function S3. Exact analytic determination of S3 is impossible except in idealized cases, and the highly oscillatory nature of the integrands makes numerical integration costly and time consuming. In this paper we discuss an approximation to S3 in terms of the two-point function S2, which is often quite easy to determine. We show that in this approximation the bounds can be expressed in terms of twofold integrals for which the integrands are mostly of one sign, and which can be evaluated very cheaply to a high degree of precision. For a model microstructure in which the inclusions consist of fully penetrable cylinders, we compare our approximate results with the exact results. Agreement is excellent.