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 $\sigma =[p({\sigma }_1-{\sigma }_2)+{\sigma }_2+{\sigma }_1{\sigma }_2][1+{\sigma }_1-p({\sigma }_1-{\sigma }_2){]}^{-1}$.

The paper contains an error which should be corrected. The error is in the estimation of the expected value of the effective conductivity