A theorem on the effective conductivity of a two-dimensional heterogeneous medium

Consider a two-dimensional heterogeneous system with a conductivity of σ (x,y) =σ0η (x,y), where σ0 is a scalar constant and η (x,y) is a dimensionless tensor function. Then the effective conductivity has the form σ* (σ) =σ0η* (η), where the tensor η* is also dimensionless. In this paper we prove that η* satisfies the relation ηR* (ηR−1) η* (η) =1̂, where 1̂ is the unit tensor and the subscript R indicates the tensor obtained by rotating the coordinate system through 90°. If the system is locally, but not necessarily statistically, isotropic and x and y are the principal axes of the effective conductivity, then the above relation becomes ηx* (η) ηy* (η−1) =1.