Macroscopic conductivity of random inhomogeneous media. Calculation methods

The macroscopic conductivity of nonhomogeneous media such as polycrystals, composites, etc., is discussed. One of the major global parameters of a randomly inhomogeneous medium (RIM), the effective (macroscopic) conductivity tensor (ECT), is considered. Based on functional analysis ideas, particularly the projector and orthogonal reduced field concepts, a method for obtaining bounds on ECT components is developed. A criterion is given by which a position of each iteration solution relative to the preceding one is determined. It is shown that previous model ECT results fall within, and are in fact special cases, of the scheme proposed by the author. The importance of the auxiliary, zero-fluctuation parameter σ^c in constructing convergent series for quantities of interest is established. Extended versions of the Hashin—Shtrikman variational principles and of Keller's theorem are obtained. The structural parameters introduced for describing an RIM are expressed, in the proposed method, in terms of the n-point probabilities (n-point interactions for the random local conductivity field). The piecewise uniform 'polarised' field approximation is combined with classical energy theorems to obtain the ECT bounds best achievable within the n-point RIM description.