Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-16T18:04:20.905Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact estimates of the conductivity of a binary mixture of isotropic materials*

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 Visiting the Department of Solid Mechanics, the Technical University of Denmark, Lyngby, Denmark in March 1985
Get access Rights & Permissions [Opens in a new window]

Synopsis

This paper describes the set GmU of effective conductivity tensors of mixtures generated by two isotropic materials taken in prescribed proportions m1 and m2 We describe microstructures which realise any point of GmU for n-dimensional space.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 104 , Issue 1-2 , 1986 , pp. 21 - 38
Copyright
Copyright © Royal Society of Edinburgh 1986

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. and Cherkaev, A. V.. Exact estimates of conductivity of a binary mixture of isotropic compounds. Preprint 894, A. F. Ioffe Physical-Technical Institute (Leningrad: Acad. of Sc. USSR, 1984).Google Scholar
2Lurie, K. A. and Cherkaev, A. V.. Accurate estimates of the conductivity of mixtures formed of two materials in a given proportion (two-dimensional problem). (In Russian.) Dokl. Akad. Nauk SSSR 264 (1982), 1128–1130.Google Scholar
3Lurie, K. A. and Cherkaev, A. V.. Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion. Proc. Roy. Soc. Edinburgh Sect. A 99 (1984), 71–87.CrossRefGoogle Scholar
4Morrey, C. B.. Quasiconvexity and the lower semicontinuity of multiple integrals. Pacific J. Math. 2 (1952), 25–53.CrossRefGoogle Scholar
5Ball, J. M., Currie, J. C. and Olver, P. J.. Null Lagrangians, weak continuity and variational problems of arbitrary order. /. Fund. Anal. 41 (1981), 135–174.CrossRefGoogle Scholar
6Kohn, R. V. and Strang, G.. Optimal design and relaxation of variational problems. Comm. Pure Appl. Math. 39 (1986), 113–137; 139–182; and to appear.CrossRefGoogle Scholar
7Tartar, L.. Compensated compactness and applications to partial differential equations. In Non-Linear Analysis and Mechanics (Heriot-Watt Symp. IV), ed Knops, R. J. (London: Pitman, 1979).Google Scholar
8Lurie, K. A., Cherkaev, A. V. and Fedorov, A. V.. Regularisation of optimal design problems for bars and plates. I, II. J. Optim. Theory Appl. 37 (1982), 499–543.CrossRefGoogle Scholar
9Murat, F.. Control in coefficients. Encyclopedia of Systems and Control (Oxford: Pergamon, 1983).Google Scholar
10Tartar, L.. Estimations fines de coefficients homogénéisées. To be published in Ennio DeGiorgi Colloquium, ed. Krée, P.. Research Notes in Mathematics 125, pp. 168–187 (London: Pitman, 1985).Google Scholar
11Zhikov, 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
12Lurie, 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; corrig. to appear in the same journal.CrossRefGoogle Scholar
13Ball, J. M.. Convexity conditions and existence theorems in non-linear elasticity. Arch. Rational Mech. Anal. 63 (1977), 337–407.CrossRefGoogle Scholar
14Hashin, Z. and Shtrikman, S.. A variational approach to the theory of elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11 (1963), 127–140.CrossRefGoogle Scholar
15Dacorogna, B.. Weak continuity and weak lower semi-continuity for non-linear functionals. Lecture Notes in Mathematics 922 (Berlin: Springer, 1982).Google Scholar
16Lurie, K. A. and Cherkaev, A. V.. G-closure for some particular sets of admissible material characteristics for the problem of the bending of thin elastic plates. J. Optim. Theory Appl. 42 (1984), 305–316; corrig. to appear in the same journal.CrossRefGoogle Scholar