Some remarks on the application of the QCA to the determination of the overall elastic response of a matrix/inclusion composite

Abstract

A ‘‘multiple scattering’’ formulation for the elastic response of a matrix containing a random distribution of inclusions is considered. The ‘‘ordinary’’ quasicrystalline approximation (QCA) is reviewed, together with a recently proposed ‘‘self‐consistent’’ QCA. It is shown that the basic ‘‘self‐consistent’’ postulate can be implemented without making an extra approximation in relation to scattering from the ‘‘background’’ material and that, when this extra approximation is dispensed with, the alternative self‐consistent prescription that results is capable in principle of generating the overall properties exactly. In practice, there is still the need to close a hierarchy by making some assumption about interactions between inclusions. When the QCA is adopted, the alternative self‐consistent prescription yields precisely the same estimates for overall behavior as the ‘‘ordinary’’ QCA. The formulation is given explicitly for elastostatics but the conclusions depend only upon the algebraic structure of the problem and apply equally well to problems of wave propagation.