## 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

### Citations:

Zbl 0504.73060; Zbl 0445.35096
Full Text:

### References:

 [1] Murat, Ann. Scuola Norm. Sup. Pisa 5 pp 489– (1978) [2] Murat, In Proc. of International Meeting on Recent Methods in Nonlinear Analysis pp 245– (1979) [3] Raitum, Soviet Math. Dokl. 19 pp 1342– (1978) [4] DOI: 10.1070/RM1979v034n05ABEH003898 · Zbl 0445.35096 [5] Marino, Ann. Scuola Norm. Sup. Pisa 23 pp 657– (1969) [6] Lurie, Dokl. Akad. Nauk SSSR 259 pp 328– (1981) [7] DOI: 10.1007/BF00934300 · Zbl 0504.73060 [8] DOI: 10.1007/BF00934299 · Zbl 0505.73060 [9] DOI: 10.1007/BF00934954 · Zbl 0464.73110 [10] Schulgasser, J. Phys. C10 pp 407– (1977) [11] DOI: 10.1063/1.329385 [12] DOI: 10.1007/3-540-11202-2_9 [13] DOI: 10.1063/1.1728579 · Zbl 0111.41401 [14] Rayleigh, Phil. Mag. 34 pp 481– (1892) [15] Gantmakher, Theory of matrices (1959) · Zbl 0050.24804 [16] Tartar, Lect. Notes Econom. and Math. Systems 107 pp 420– (1975) [17] Bensoussan, Asymptotic analysis for period structures (1978) [18] Lurie, Dokl. Akad. Nauk SSSR 264 pp 1128– (1982)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.