Effective conductivity of two−phase material with cylindrical phase boundaries

Abstract

A two−phase material in which the phase boundaries are cylindrical surfaces is considered. The generators of the phase boundaries are parallel to the z axis and the x and y axes are the principal axes of the effective conductivity tensor. If the two phases have conductivities σ₁ and σ₂, then the effective conductivity σ_x* (σ₁,σ₂) (in the x direction) of the material and the effective conductivity σ_y* (σ₂,σ₁) (in the y direction) of the material obtained by interchanging the two phases are related by σ_x* (σ₁,σ₂) σ_y* (σ₂,σ₁) =σ₁,σ₂. For a statistically isotropic material σ_x* (σ₁,σ₂) =σ_y* (σ₁,σ₂) =σ* (σ₁,σ₂). If, in addition, the material is statistically symmetric so that interchange of the two phases yields again the same material, then σ* (σ₁,σ₂) =σ* (σ₂,σ₁) = (σ₁σ₂)^1/2.