Exact Results for Conductivity of 2D Isotropic Heterophase Systems

TL;DR

This paper derives exact formulas for the effective conductivity of 2D isotropic heterophase systems with any number of phases, revealing fixed points and hyperplanes where conductivity remains constant, validated through various approximations.

Contribution

It provides new exact results for effective conductivity in multi-phase systems, generalizing known cases and identifying hyperplanes of constant conductivity.

Findings

Exact values of sigma_e at fixed points of duality transformations.

Existence of hyperplanes with constant sigma_e for N > 3.

Validation of results through various approximation methods.

Abstract

The duality relation for the effective conductivity sigma_{e} of 2D isotropic heterophase systems is used for obtaining the exact results for sigma_{e} at arbitrary number of phases N. The exact values of sigma_{e} correspond to the fixed points of the duality transformations. The new exact results for sigma_{e}, generalizing the well-known exact values of sigma_{e} for N = 2,3 at equal phase concentrations, are found. It is shown that for N > 3 there exist the whole hyperplanes in the space of phase concentrations, on which sigma_{e} takes constant values. These results are checked in the framework of various approximations for different random heterophase systems.