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Homogenisation of a class of fourth order equations with application to incompressible elasticity

Published online by Cambridge University Press:  14 November 2011

G. A. Francfort
G. A. Francfort
Affiliation:
Laboratoire Central des Ponts et Chaussees, 58 Boulevard Lefebvre, 75732 Cedex 15, France
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Synopsis

Upon formalising an analogy between two-dimensional Stokes flow and two-dimensional isotropic conductivity, we exhibit a class of fourth order equations which behave “isomorphically” like isotropic conductivity from the standpoint of homogenisation and from that of corresponding bounding methods on possible effective behaviours. In particular, Lipton's result on the G-closure problem for mixtures of two incompressible elastic materials is recovered in the two-dimensional case.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 120 , Issue 1-2 , 1992 , pp. 25 - 46
Copyright
Copyright © Royal Society of Edinburgh 1992

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References

1Francfort, G. and Murat, F.. Optimal Bounds for conduction in Two-Dimensional Two-Phase, Anisotropic Media. In Proceedings of the Durham Symposium on Non-Classical Continuum Mechanics, July 1985, ed. Knops, R. J., pp. 197–212 (Cambridge: Cambridge University Press, 1987).Google Scholar
2Francfort, G. and Murat, F.. Homogenization and optimal bounds in linear elasticity. Arch. Rational Mech. Anal 94 (1986), 307–334.CrossRefGoogle Scholar
3Hashin, Z. and Shtrikman, S.. A variational approach to the theory of the elastic behaviour of multiphase materials. Mech. Phys. Solids 11 (1963), 127–140.CrossRefGoogle Scholar
4Knops, R. J. and Payne, L. E.. Uniqueness Theorems in Linear Elasticity, Tracts in Natural Philosophy 19 (Berlin: Springer, 1971).Google Scholar
5Kohn, R. V.. Recent progress in the mathematical modeling of composite materials. In Composite Material Response: Constitutive Relations and Damage Mechanisms, eds. Shi, G.. et al., 155–177 (New York: Elsevier, 1988).Google Scholar
6Kohn, 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. et al., IMA Volumes in Mathematics and its Applications 1, pp. 97–125 (New York: Springer, 1986).Google Scholar
7Kohn, R. V. and Milton, G. W.. Variational bounds on the effective moduli of anisotropic composites. J. Mech. Phys. Solids 36 (1988), 597–629.Google Scholar
8Le, H.Dret. An example of H1-unboundedness of solutions to strongly elliptic systems of partial differential equations in a laminated geometry. Proc. Roy. Soc. Edinburgh Sect. A 105 (1987), 77–82.Google Scholar
9Lipton, R.. On the effective elasticity of a two-dimensional homogenised incompressible elastic composite. Proc. Roy. Soc. Edinburgh Sect. A 110 (1988), 45–61.CrossRefGoogle Scholar
10Lurie, K. A. and Cherkaev, A. V.. G-closure of some particular sets of admissible material characteristics for the problem of bending of thin elastic plates. J. Optim. Theory Appl. 42 (1984), 305–316.CrossRefGoogle Scholar
11Milton, 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), 63–125.CrossRefGoogle Scholar
12Murat, F.. H-convergence. Séminaire d'Analyse Fonctionnelle et Numérique, 1977/1978 (Université d'Alger, multigraphed).Google Scholar
13Murat, F.. Control in coefficients. In Systems and Control Encyclopedia: Theory, Technology, Applications (New York: Pergamon, 1985).Google Scholar
14Murat, F.. Compacité par compensation. Ann. Scuola Norm. Sup. Pisa 5 (1978), 489–507.Google Scholar
15Murat, F.. A survey on compensated compactness. In Contributions to Modern Calculus of Variations, ed. Cesari, L., Pitman Research Notes in Mathematics 148, pp. 145–183. (Harlow: Longman, 1987).Google Scholar
16Spagnolo, S.. Una caratterizzazione degli operatori differenziali autoggiunti del 2° Ordine a coefficienti misurabili e limitati. Rend. Sem. Mat. Padova, 39 (1967), 56–64.Google Scholar
17Spivak, M.. A comprehensive introduction to differential geometry 1, 2nd edn (Wilmington: Publish or Perish, 1979).Google Scholar
18Tartar, L.. Problème de contrôle des coefficients dans des équations aux dérivées partielles. In Control Theory, Numerical Methods and Computer Systems Modelling, International Symposium, Rocquencourt, June 17–21, INRIA-LABORIA, eds. Bensoussan, A. and Lions, J. L., Lecture Notes in Economics and Math. Sys. 107, pp. 420–426 (Berlin: Springer, 1975).Google Scholar
19Tartar, L.. Cours Peccot, Collège de France, 1977, unpublished.Google Scholar
20Tartar, L.. Estimations de coefficients homogénéisés. In Computer Methods in Applied Sciences and Engineering, Proc. 1977INRlA PARIS, ed. Glowinski, R. and Lions, J. L., Lecture Notes in Mathematics 704, pp. 364–373 (Berlin: Springer, 1979).Google Scholar
21Tartar, L.. Compensated compactness and applications to partial differential equations. In Non Linear Mechanics and Analysis, Heriot-Watt Symposium, ed. Knops, R. J., Pitman Research Notes in Mathematics 39, pp. 136–212 (London: Pitman, 1976).Google Scholar
22Tartar, L.. Estimations fines de coefficients homogénéisés. In Ennio De Giorgi's colloquium, ed. Kree, P., Pitman Research Notes in Mathematics 125, pp. 168–187 (London: Pitman, 1985).Google Scholar