A relation on the effective conductivity of composites

TL;DR

This paper establishes a relation between the effective conductivity tensors of certain random and deterministic 2D composite materials with inclusions of two different conductivities.

Contribution

It proves that the effective conductivity tensor of a three-phase random composite equals that of a two-phase deterministic composite with a specific effective conductivity value.

Findings

Effective conductivity tensor equivalence proven

Relation simplifies analysis of composite materials

Applicable to 2D composites with random inclusions

Abstract

Consider a 2D composites with non-overlapping equal inclusions imbedded in a host material of the normalized unit conductivity. The conductivity of inclusions takes two values $\sigma_1$ and $\sigma_2$ with the probabilities $p$ and $1-p$, respectively. We prove that the effective conductivity tensor of the considered three-phase random composite is equal to the effective conductivity tensor of the two-phase deterministic composite with the same inclusions of the conductivity $\sigma=[p(\sigma_1~-~\sigma_2)+~\sigma_2+\sigma_1\sigma_2] [1+\sigma_1-p(\sigma_1-\sigma_2)]^{-1}$.