Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion

Abstract

SynopsisThis paper is a sequel to [1-4]. We consider the problem of G-closure, i.e. the description of the set GU of effective tensors of conductivity for all possible mixtures assembled from a number of initially given components belonging to some fixed set U. Effective tensors are determined here in a sense of G-convergence relative to the operator ∇· D · ∇, of the elements DeU ∈ [5, 6].The G-closure problem for an arbitrary initial set U in the two-dimensional case has already been solved [3, 4]. It remained, however, unclear how to construct, in the most economic way, a composite with some prescribed effective conductivity, or, equivalently, how to describe the set GmU of composites which may be assembled from given components taken in some prescribed proportion. This problem is solved in what follows for a set U consisting of two isotropic materials possessing conductivities D+ = u+E and D− = u−E where 0<u−<u+<∞ and E ( = ii+jj) is a unit tensor.