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The G-closure of two well-ordered, anisotropic conductors

Published online by Cambridge University Press:  14 November 2011

Yury Grabovsky
Affiliation:
Courant Institute, 251 Mercer Street, New York, NY 10012, U.S.A

Synopsis

We give a complete solution to the G-closure problem for mixtures of two well-ordered possibly anisotropic conductors. Both the G-closure with fixed volume fractions and the full G-closure are computed. The conductivity tensors are considered in a fixed frame and no rotations are allowed.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1993

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References

1Bensoussan, A., Lions, J. L. and Papanicolaou, G.. Asymptotic analysis of periodic structures (Amsterdam: North-Holland, 1978).Google Scholar
2Hashin, Z. and Shtrikman, S.. A variational approach to the theory of the effective magnetic permeability of multiphase materials, J. Appl. Phys. 33 (1962), 31253131.CrossRefGoogle Scholar
3Kohn, R. V. and Milton, G. W.. On bounding the effective conductivity of anisotropic composites. In Homogenization and Effective Moduli of Materials and Media, eds Ericksen, J. et al., pp. 97125 (Berlin: Springer, 1986).CrossRefGoogle Scholar
4Luríe, 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) 7187.CrossRefGoogle Scholar
5Lurie, K. A. and Cherkaev, A. V.. Exact estimates of a binary mixture of isotropic components. Proc. Roy. Soc. Edinburgh Sect. A 104 (1986), 2138.CrossRefGoogle Scholar
6Milton, G. W.. On characterizing the set of possible effective tensors of composites: the variational method and the translation method. Comm. Pure Appl. Math. 43 (1990), 63125.CrossRefGoogle Scholar
7Milton, G. W. and Kohn, R. V.. Variational bounds on the effective moduli of anisotropic composites. J. Mech. Phys. Solids 36 (1988), 597–629.CrossRefGoogle Scholar
8Nesi, V.. Using quasi-convex functionals to bound the effective conductivity of composite materials. Proc. Roy. Soc. Edinburgh Sect. A (1993, in print).CrossRefGoogle Scholar
9Tartar, L.. Estimation fines des coefficients homogeneises. In Ennio de Giorgi's Colloquium, ed. Kree, P., pp. 168187 (London: Pitman, 1985).Google Scholar
10Wiener, W.. Abhandlungen der Mathematisch Physichen Klasse der Koniglischen Sachsischen Gesellschaft der Wissenschaften 32 (1912), 509.Google Scholar