Statistical theory of effective electrical, thermal, and magnetic properties of random heterogeneous materials. V. One− and two−dimensional systems

Abstract

A perturbation theory is developed for the effective permittivity of one− and two−dimensional random heterogeneous materials that are statistically homogeneous. Closely following the formulations for three−dimensional systems presented in Papers I−IV in this series of work, we derive the formal perturbation solutions for two−dimensional systems, evaluate the second−order and third−order terms for cell materials, and determine upper and lower bounds of the effective permittivity. Here statistical isotropy is not necessarily required. Several approximate perturbation solutions for completely random systems (which are statistically isotropic) are obtained by summing some selected partial series of the perturbation expansion up to an infinite order and numerical results are illustrated. We analyze validity of approximations by means of diagram representation of the perturbation series. The effective−medium theory serves as a good approximation for two−dimensional systems and gives the critical percolation concentration correctly. For one−dimensional materials, the effective−medium approximation turns out to be an exact solution.