A note on the $C^{2}$-term of the effective conductivity for random dispersions

Abstract

The paper is devoted to the study of the effective conductivity $\varkappa^*$ of a random dilute dispersion of spheres. A special attention is paid to the $c^2$-coefficient $a_2$ in the expansion of $\varkappa^*$ in powers of the volume fraction $c$ of the soheres. The functional dependence of $a_2$ upon the radical distribution function is discussed and it is shown, using simple arguments, that $a_2$ is a sum of a constant and a linear functional of the said function. Tha analytical form and certain estimates for the kernal of this functional are obtained in the two-dimensional case (fiber-reinforces material)