Since the effective properties of a composite depend in a complicated way on the microstructure, it is useful to have realistic model composites for which the effective properties can be computed exactly. One such model, called the coated sphere assemblage, was introduced by Hashin (1962) as a model of a composite comprised of spherical grains of one phase embedded in a matrix of a second phase. This model and its generalizations are the subject of this chapter.

The coated sphere assemblage

Hashin (1962) found that the effective bulk modulus of the assemblage could be computed exactly for all volume fractions of the phases. Using a similar analysis, Hashin and Shtrikman (1962) found an exact expression for the effective conductivity of the assemblage. To see how their argument works, consider a coated sphere consisting of a spherical core of phase 1 of isotropic conductivity σ1I fitting snugly inside a concentric spherical shell of phase 2 of isotropic conductivity σ2I, with a core radius c and exterior radius e. This coated sphere is inserted as an inclusion in an infinite matrix of conductivity σ0I within which a uniform current field flows from infinity. Suppose for the moment that σ1 > σ2. From a physical standpoint it is clear that when σ0 = σ1 the inclusion has lower conductivity than its surroundings and current will tend to flow around this obstacle.