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Almost-everywhere injectivity in nonlinear elasticity

Published online by Cambridge University Press:  14 November 2011

Tang Qi
Tang Qi
Affiliation:
Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, U.K.
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Synopsis

This paper gives a sufficient condition for almost-everywhere injectivity for nonlinear three dimensional elasticity similar to that of Claret-Necas [8], namely.

We prove that this relation is maintained under the weak convergence of minimising sequences for nonlinear elasticity problems. The existence and partial regularity of an “inverse” function are proved.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 109 , Issue 1-2 , 1988 , pp. 79 - 95
Copyright
Copyright © Royal Society of Edinburgh 1988

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References

1Antman, S. S.. Ordinary differential equations of nonlinear elasticity. II. Existence and regularity theory for conservative boundary value problems. Arch. Rational Mech. Anal. 61 (1976), 353–393.CrossRefGoogle Scholar
2Ball, J. M.. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977), 337–403.CrossRefGoogle Scholar
3Ball, J. M.. Global invertibility of Sobolev functions and the interpenetration of matter. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 315–328.CrossRefGoogle Scholar
4Ball, J. M.. Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos. Trans. Roy. Soc. London Ser. A 306 (1982), 557–612.Google Scholar
5Ball, J. M., Currie, J. C. and Olver, P. J.. Null Lagrangians, weak continuity and variational problems of arbitrary order. J. Fund. Anal. 41 (1981), 135–174.CrossRefGoogle Scholar
6Ciarlet, P. G.. Elasticite Tridimensionnelle (Paris: Masson, 1985).Google Scholar
7Ciarlet, P. G. and Necas, J.. Unilateral problems in nonlinear three-dimensional elasticity. Arch. Rational Mech. Anal. 87 (1985), 319–338.CrossRefGoogle Scholar
8Ciarlet, P. G. and Necas, J.. Injectivité presque partout, auto-contact et non-interpénétrabilité en elasticité nonlinèaire tridimensionnelle. C. R. Acad. Sci. Paris, Ser. A 301 (1985), 621–624.Google Scholar
9Giorgi, E. De, Ambrosio, L. and Buttazo, G.. Integral representation and relaxation for functionals denned on measures (to appear).Google Scholar
10Demengel, F. and Temam, R.. Convex functions of measures and applications (Prepublication d'Orsay, 83T10).Google Scholar
11Dieudonne, J.. Treatise on Analysis (New York: Academic Press, 1972).Google Scholar
12Evans, L. C.. Quasiconvexity and partial regularity in the calculus of variations (to appear).Google Scholar
13Federer, H.. Geometric Measure Theory (Berlin: Springer, 1969).Google Scholar
14Knops, R. J. and Stuart, C. A.. Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity. Arch. Rational Mech. Anal. 86 (1984), 233–249.CrossRefGoogle Scholar
15Marcus, M. and Mizel, V. J.. Transformations by functions in Sobolev spaces and lower semicontinuity for parametric variational problems. Bull. Amer. Math. Soc. 79 (1973), 790–795.CrossRefGoogle Scholar
16Morrey, Ch. B.. Multiple Integrals in the Calculus of Variations (Berlin: Springer, 1966).CrossRefGoogle Scholar
17Necas, J.. Les Methodes Directes en Théories des Equations Elliptiques (Praha: Academia, 1967).Google Scholar
18Schwartz, L.. Cours d'Analyse (Paris: Hermann, 1967).Google Scholar
19Sverak, V.. Regularity properties of deformations with finite energy. Arch. RationalMech. Anal. (to appear).Google Scholar
20Temam, R.. Problèmes Mathematiques en Plasticité (Paris: Gauthier-Villas, 1983).Google Scholar
21Neumann, J. Von. Functional Operators, Vol. 1 (Princeton: Princeton University Press, 1950).Google Scholar