Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T03:31:07.135Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  14 November 2011

K. A. Lurie  and
A. V. Cherkaev
K. A. Lurie
Affiliation:
Academy of Sciences of the U.S.S.R., A. F. Ioffe Physical Technical Institute, Leningrad, U.S.S.R.
A. V. Cherkaev
Affiliation:
Academy of Sciences of the U.S.S.R., A. F. Ioffe Physical Technical Institute, Leningrad, U.S.S.R.
Get access Rights & Permissions [Opens in a new window]

Extract

This 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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 99 , Issue 1-2 , 1984 , pp. 71 - 87
Copyright
Copyright © Royal Society of Edinburgh 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Lurie, K. A., Federov, A. V. and Cherkaev, A. V.. Regularization of optimal design problems for bars and plates, Parts 1 and 2. J. Optim. Theory Appl. 37 (1982), 499–521, 523–543.CrossRefGoogle Scholar
2Lurie, K. A., Fedorov, A. V. and Cherkaev, A. V.. On the existence of solutions to some problems of optimal design for bars and plates. J. Optim. Theory Appl. 42 (1984), 247–282.CrossRefGoogle Scholar
3Lurie, K. A. and Cherkaev, A. V.. G-closure of a set of anisotropically conducting media in the two-dimensional case. J. Optim. Theory Appl. 42 (1984), 283–304.CrossRefGoogle Scholar
4Lurie, K. A. and Cherkaev, A. V.. G-closure of a set of anisotropically conducting media in case of two dimensions. Dokl. Akad. Nauk SSSR 259 (1981), 328–331.Google Scholar
5Marino, A. and Spagnolo, S.. Un tipo di approssimazione dell'operatore σiσj∂/∂xiaij(x)∂/∂xj con operatori σ∂/∂xia(x)∂/∂xi. Ann. Scuola Norm. Sup. Pisa 23 (1969), 657–673.Google Scholar
6Zhikov, V. V., Kozlov, S. M., Oleinik, O. A. and Ngoan, Kha Thieng. Averaging and G-convergence of differential operators. Russian Math. Surveys 34 (1979), 69–147.CrossRefGoogle Scholar
7Raitum, U. E.. The extension of extremal problems connected with a linear elliptic equation. Soviet Math. Dokl. 19 (1978), 1342–1345.Google Scholar
8Lurie, K. A. and Cherkaev, A. V.. Accurate estimates of the conductivity of mixtures formed of two materials in a given proportion (two-dimensional problem). Dokl. Akad. Nauk SSSR 264 (1982), 1128–1130.Google Scholar
9Murat, F.. Compacite par compensation. Ann. Scuola Norm. Sup. Pisa 5 (1978), 489–507.Google Scholar
10Murat, F.. Compacite par compensation II. In Proc. of International Meeting on Recent Methods in Nonlinear Analysis, pp. 245–256 (ed. Giorgi, E. De, Magenes, E. and Mosco, U.) (Bologna: Pitagora Editrice, 1979).Google Scholar
11Bensoussan, A., Lions, J. L. and Papanicolaou, G.. Asymptotic analysis for period structures (Amsterdam: North Holland 1978).Google Scholar
12Tartar, L.. Problemes de controle des coefficients dans des equations aux derivees partielles. Lect. Notes Econom. and Math. Systems 107, 420–426 (Berlin: Springer, 1975).Google Scholar
13Gantmakher, F. R.. Theory of matrices (New York: Chelsea, 1959).Google Scholar
14Rayleigh, Lord. Phil. Mag. (5) 34 (1892), 481.CrossRefGoogle Scholar
15Hashin, Z. and Shtrikman, S.. A variational approach to the theory of the effective magnetic permeability of multiphase materials. J. Appl. Phys. 33 (1962), 3125–3131.CrossRefGoogle Scholar
16Kohler, W. and Papanicolaou, G.. Bounds for the effective conductivity of random media. Macroscopic Properties of Disordered Media, ed. Burridge, R., Childress, S. and Papanicolaou, G.. Lecture Notes in Physics 154, 111–130 (Berlin: Springer, 1982).CrossRefGoogle Scholar
17Milton, G. W.. Bounds on the complex pennittivity of a two-component composite material. J. Appl. Phys. 52 (1981), 5286–5293.CrossRefGoogle Scholar
18Schulgasser, K.. Bounds on the conductivity of statistically isotropic polycrystals. J. Phys. C10, (1977), 407–417.Google Scholar