A relation on the effective conductivity of composites

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 $\sigma=[p(\sigma_1~-~\sigma_2)+~\sigma_2+\sigma_1\sigma_2] [1+\sigma_1-p(\sigma_1-\sigma_2)]^{-1}$.