Generalized Clausius-Mossotti formula for random composite with circular fibers

An important area of materials science is the study of effective dielectric, thermal and electrical properties of two phase composite materials with very different properties of the constituents. The case of small concentration is well studied and analytical formulas such as Clausius-Mossotti (Maxwell-Garnett) are successfully used by physicists and engineers. We investigate analytically the case of an arbitrary number of unidirectional circular fibers in the periodicity cell when the concentration of the fibers is not small, i.e., we account for interactions of all orders (pair, triplet, etc.). We next consider transversely-random unidirectional composite of the parallel fibers and obtain a closed form representation for the effective conductivity (as a power series in the concentration v). We express the coefficients in this expansion in terms of integrals of the elliptic Eisenstein functions. These integrals are evaluated and the explicit dependence of the parameter d, which characterizes random position of the fibers centers, is obtained. Thus we have extended the Clausius Mossotti formula for the non dilute mixtures by adding the higher order terms in concentration and qualitatively evaluated the effect of randomness in the fibers locations. In particular, we have proven that the periodic array provides extremum for the effective conductivity in our class of random arrays ("shaking" geometries). Our approach is based on complex analysis techniques and functional equations, which are solved by the successive approximations method.