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Bounds on the effective conductivity of two-dimensional composites made of n ≧ 3 isotropic phases in prescribed volume fraction: the weighted translation method

Published online by Cambridge University Press:  14 November 2011

V. Nesi
V. Nesi
Affiliation:
Dipartimento di Matematica Pura ed Applicata, Universita' degli Studi de L'Aquila, 67010 L'Aquila, Italy
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Abstract

We establish lower and upper bounds which are valid for the overall conductivity of twodimensional composites. They are based on a method which modifies the so-called translation method in a way which makes it effectively much more flexible. When specialised to composites of n > 2 isotropic phases, the new bounds are often strictly better than all the previously known ones. From the mathematical point of view, the improvement is due mainly to a new regularity result in p.d.e.s [2]. From the physical point of view the latter can be interpreted as a result bounding in a suitable sense the fluctuations of the ‘electric field’.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 6 , 1995 , pp. 1219 - 1239
Copyright
Copyright © Royal Society of Edinburgh 1995

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References

1Avellaneda, M., Cherkaev, A. V., Lurie, K. A. and Milton, G. W.. On the effective conductivity of polycrystals and a three dimensional phase-interchange inequality. J. Appl. Phys. 63(10), (1988), 4989–5003.CrossRefGoogle Scholar
2Bauman, P., Marini, A. and Nesi, V.. Univalent solutions of an elliptic system of partial differential equations arising in homogenization (Preprint).Google Scholar
3Francfort, G. and Murat, F.. Optimal bounds for conduction in two-dimensional, two phase, anisotropic media. In Non-classical Continuum Mechanics, eds. Knops, R. J. and Lacey, A. A., London Mathematical Society Lecture Note Series 122, 197–212 (Edinburgh: Cambridge University Press, 1987).Google Scholar
4Hashin, Z. and Shtrikman, S.. A variational approach to the theory of effective magnetic permeability of multiphase materials. J. Appl. Phys. 33 (1962), 3125.CrossRefGoogle Scholar
5Kohn, 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. L., Kinderlehrer, D., Kohn, R. and Lions, J. L., 97–125 (New York: Springer, 1986).Google Scholar
6Lurie, K. A. and Cherkaev, A. V.. Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportions. Proc. Roy. Soc. Edinburgh Sect. A 99(1984), 71–87.Google 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.Google Scholar
8Meyers, N.. An L p-estimate for the gradient of solutions of second order elliptic divergence form equations. Ann. Scuola Norm. Sup. Pisa cl. Sci. (3) 17 (1963), 189–206.Google Scholar
9Murat, F. and Tartar, L.. Calcul des variations et homogénéisation. In Les methodes d'homogénéisation: théorie et applications en physique, Coll. de la Dir. des Etudes et Recherches d'Electricité de France, 319 (Paris: Eyrolles, 1985).Google Scholar
10Nesi, V.. Using quasiconvex functionals to bound the effective conductivity of composite materials. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), 633–79.CrossRefGoogle Scholar
11Tartar, L.. Compensated compactness and applications to p.d.e. in nonlinear analysis and mechanics. Heriot-Watt Symposium, Vol. IV, ed. Knops, R. J., Research Notes in Mathematics 39, 136–212 (Boston: Pitman, 1979).Google Scholar
12Tartar, L.. Estimation fines des coefficients homogeneises. In Ennio De Giorgi's Colloquium, ed. Kree, P., Research Notes in Mathematics 125, 168 (London: Pitman, 1985).Google Scholar
13Zhikov, V. V.. Estimates for the averaged matrix and the averaged tensor. Russian Math. Surveys 46(3) (1991), 65–136.Google Scholar