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The exact coupled bounds for effective tensors of electrical and magnetic properties of two-component two-dimensional composites1

Published online by Cambridge University Press:  14 November 2011

A. V. Cherkaev  and
L. V. Gibiansky
A. V. Cherkaev
Affiliation:
Courant Institute, 251 Mercer Street, New York, NY 10012, U.S.A
L. V. Gibiansky
Affiliation:
Courant Institute, 251 Mercer Street, New York, NY 10012, U.S.A
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Synopsis

This paper describes the effective properties of a two-dimensional two-component dielectric composite. Each component is supposed to be isotropic and is characterised by a pair of constants of electrical and magnetic permeabilities. Effective properties of an arbitrary anisotropic composite depend on its microstructure and are characterised by a pair of symmetric tensors of electrical and magnetic permeabilities. Here we construct the so-called Gm-closure which is the set of all possible values of these pairs, corresponding to all possible microstructures with prescribed volume fractions of the given components. The exact bounds of the Gm-closure are obtained and the microstructure corresponding to each point of the Gm-closure set is determined.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 122 , Issue 1-2 , 1992 , pp. 93 - 125
Copyright
Copyright © Royal Society of Edinburgh 1992

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References

1Avellaneda, M.. Optimal bounds and microgeometries for elastic two-phase composites. SIAM J. Appl. Math. 47 (1987), 1216–1228.CrossRefGoogle Scholar
2Avellaneda, 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
3Bergman, D. J.. The dielectric constant of composite material – a problem in classical Physics. Phys. Rep. C 43 (1988), 377.CrossRefGoogle Scholar
4Berryman, J. G.. Effective medium theory for elastic composites. In Elastic wane scattering and propagation, eds Varadan, V. K. and Varadan, V. V., 111 (Ann Arbor, Mich: Ann Arbor Science, 1982).Google Scholar
5Christensen, R. M.. Mechanics of Composite Materials, (New York: Wiley-Interscience, 1979).Google Scholar
6Dacorogna, B.. Weak continuity and weak lower semicontinuity of non-linear functionals (Berlin: Lecture Notes in Mathematics 922 (Berlin: Springer, 1982).CrossRefGoogle Scholar
7Dykhne, A. M.. Conductivity of the two-dimensional two-phase system. Zh. Èksper. Teoret. Fiz., 7(59) (1970), 110–116 (in Russian); Soviet Phys. JETP 32 (1971), 53–65.Google Scholar
8Francfort, G. and Murat, F.. Homogenization and optimal bounds in linear elasticity. Arch. Rational Mech. Anal. 94 (1986), 307–334.CrossRefGoogle Scholar
9Francfort, G. and Murat, F.. Optimal bounds for conduction in two-dimensional, two-phase, anisotropic media. In Non-Classical Continuum Mechanics, eds Knops, R. and Lacey, A. A., 197–212 (Cambridge: Cambridge University Press, 1987).CrossRefGoogle Scholar
10Gantmacher, F. R.. Theory of Matrices (Moscow: Nauka, 1966, in Russian).Google Scholar
11Gibiansky, L. V. and Cherkaev, A. V.. Design of composite plates of extremal rigidity (Preprint 914, Ioffe Physicotechnical Institute, Leningrad, 1986).Google Scholar
12Gibiansky, L. V. and Cherkaev, A. V.. Microstructures of composites of extremal rigidity and exact estimates of provided energy density (Preprint 1115, Ioffe Physicotechnical Institute, Leningrad, 1987).Google Scholar
13Hashin, Z. and Shtrikman, S.. A variational approach to the theory of the effective magnetic permeability of multiphase materials. J. Appl. Phys. 35 (1962) 3125–3131.CrossRefGoogle Scholar
14Hashin, Z. and Shtrikman, S.. A variational approach to the theory of the elastic behavior of multiphase materials. Mech. Phys. Solids 11 (1963), 127–140.CrossRefGoogle Scholar
15Kohn, R. V. and Milton, G. W., Variational bounds on the effective moduli of anisotropic composites. Mech. Phys. Solids 36(6) (1988), 597–629.Google Scholar
16Kohn, R. V. and Kipton, R.. Optimal bounds for the effective energy of a mixture of isotropic, incompressible, elastic materials. Arch. Rational Mech. Anal. 102 (1988), 331–350.CrossRefGoogle Scholar
17Kohn, 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. 97–125 (Berlin: Springer, 1986).CrossRefGoogle Scholar
18Kohn, R. V. and Strang, G.. Optimal design and relaxation of variational problems. Comm. Pure Appl. Math. 39 (1986), 113–137; 139–182; 353–377.CrossRefGoogle Scholar
19Keller, J. B.. A theorem on the conductivity of a composite medium. Math. Phys. 5 (1964), 548–549.CrossRefGoogle Scholar
20Lipton, R.. On the effective elasticity of a two-dimensional homogenized incompressible elastic composite. Proc. Roy. Soc. Edinburgh Sect. A (to appear).Google Scholar
21Lurie, K. A. and Cherkaev, A. V.. G-closure of a set of anisotropically conducting media in the two-dimensional case. Optim. Theory Appl. 42 (1984), 283–304 (Errata 53 (1987), 317).CrossRefGoogle Scholar
22Lurie, K. A. and Cherkaev, A. V.. G-closure of some particular sets of admissible material characteristics for the problem of bending of thin plates. Optim. Theory Appl. 42 (1984), 305–316.CrossRefGoogle Scholar
23Lurie, 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 (first version, Preprint 783, Ioffe Physicotechnical Institute, Leningrad, 1982).CrossRefGoogle Scholar
24Lurie, K. A. and Cherkaev, A. V.. Exact estimates of the conductivity of a binary mixture of isotropic materials. Proc. Roy. Soc. Edinburgh. Sect. A 104 (1986), 21–38 (first version, Preprint 894, Ioffe Physicotechnical Institute, Leningrad, 1984).CrossRefGoogle Scholar
25Lurie, K. A. and Cherkaev, A. V.. The effective characteristics of composite materials and optimal design of constructions. Adv. in Mech 9(2) (1986), 3–81.Google Scholar
26Matveyev, A. N.. Electrodynamics (Moscow: Visshaya Shkola, 1980, in Russian).Google Scholar
27Mikhlin, S. G.. Variational Methods in Mathematical Physics (Moscow: Nauka, 1970, in Russian).Google Scholar
28Milgrom, M. and Shtrikman, S.. Linear response of polycrystals to coupled fields: exact relations among the coefficients (Preprint WIS-89/4/Jan.-Ph).Google Scholar
29Milton, G. W.. Bounds on the transport and optical properties of a two-component composite material. J. Appl. Phys. 52(8) (1981), 5294–5304.CrossRefGoogle Scholar
30Milton, G. W.. Bounds on the complex permittivity of a two-component composite material. Appl. Phys. 52(8) (1981), 5286–5293.CrossRefGoogle Scholar
31Milton, G. W.. Bounds on the electromagnetic, elastic, and other properties of two-component composites. Phys. Rev. Lett. 46(8) (1981), 542–545.CrossRefGoogle Scholar
32Milton, G. W. and Golden, K.. Thermal conduction in composities. In Thermal Conductivity 18, eds. Ashworth, T. and Smith, D. R., 571–582 (New York: Plenum Press, 1985).CrossRefGoogle Scholar
33Milton, G. W.. Modeling the properties of composites by laminates. In Homogenization and Effective Moduli of Materials and Media, eds Erickesn, J. L., Kinderlehrer, D., Kohn, R., and Lions, J.-L., 150–174 (New York: Springer, 1986).CrossRefGoogle Scholar
34Milton, G. W.. A proof that laminates generates all possible effective conductivity functions of two dimensional two-phase materials. In Advances in Multiphase Flow and Related Problems, ed. Papanicolaou, G., 136–146 (Philadelphia: SIAM, 1986).Google Scholar
35Milton, G. W.. Multicomponent composites, impedance networks and new types of continued fraction I, II. Comm. Math. Phys. 111 (1987), 281–327, 329–372.CrossRefGoogle Scholar
36Milton, G. W.. On characterizing the set of possible effective tensors of composites: The variational method and the translation method. Comm. Pure Appl. Math. XLIII (1990), 63–125.CrossRefGoogle Scholar
37Milton, G. W.. The field equation recursion method. In Composite Media and Homogenization Theory, eds Maso, G. Dal and Antonio, G. Dal, vol. 5, Progress in Non-linear Differential Equations and their Applications, series ed. Brezis, H., 223–245 (Boston: Birkhauser, 1991).CrossRefGoogle Scholar
38Murat, F. and Tartar, L.. Calcul des variations et homogeneisation. In Les Methodes de I'Homogeneisation: Theorie et Applications en Physique, 319–370 (Paris: Coll. de la Dir. des Etudes et Recherches de Electr. de France, Eyrolles, 1985).Google Scholar
39Mendelson, K. S.. A theorem on the effective conductivity of a two-dimensional heterogeneous medium. J. Appl. Phys. 46(11) (1975) 4740–4741.CrossRefGoogle Scholar
40Prager, S.. Improved variational bounds on some bulk properties of two-phase random medium. J. Chem. Phys. 50 (1969), 4305.CrossRefGoogle Scholar
41Reuss, A.. Berechnung der Fliebgrenze von Mischkristallen auf Grund d.Plastizitatsbedingung fur Rinkristalle. Z. Angew. Math. Mech. 9 (1929), 49–58.CrossRefGoogle Scholar
42Sanchez-Palencia, E.. Nonhomogeneous Media and Vibration Theory, Lecture Notes in Physics 127 (Berlin: Springer, 1980).Google Scholar
43Schulgasser, K.. Concerning the effective transverse conductivity of a two-dimensional two-phase material. Internat. J. Heat Mass Transfer 20 (1977), 1273–1289.CrossRefGoogle Scholar
44Tartar, L.. Estimations fines des coefficients homogeneises. In Ennio de Giorgi Colloquium,, ed. Kree, P., Pitman Research Notes in Mathematics 125, 168–187 (London: Pitman, 1985).Google Scholar
45Tartar, L.. Compensated compactness and applications to partial differential equations. In Nonlinear Analysis and Mechanics, Heriot-Watt Symposium VI, ed. Knops, R. J. (London: Pitman, 1979).Google Scholar
46Voigt, W.. Lehrbuch der Kristallphysik (Berlin: Teubner, 1928).Google Scholar