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Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion. (English) Zbl 0564.73079
This paper is a sequel e.g. to the authors’ article in (*) J. Optimization Theory Appl. 42, 283-304 (1984; Zbl 0504.73060). 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 \(\nabla \cdot D\cdot \nabla.\) of the elements \(D\in U\) [V. V. Zhikov, S. M. Kozlov, O. A. Olejnik and Kha T’en Ngoan, Russ. Math. Surv. 34, 65-133 (1979; Zbl 0445.35096)].
The G-closure problem for an arbitrary initial set U in the two- dimensional case has already been solved (*). 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 \(G_ mU\) 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 \(0<u_- <u_+<\infty\) and \(E(=ii+jj)\) is a unit tensor.

MSC:
74P99 Optimization problems in solid mechanics
74E30 Composite and mixture properties
35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
74A40 Random materials and composite materials
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