Non-existence of minimizers for a nonlinear membrane plate under compression

Karim Trabelsi

doi:10.1016/j.crma.2003.09.008

Mathematical Problems in Mechanics

Non-existence of minimizers for a nonlinear membrane plate under compression
[Non existence de minimiseurs pour une plaque membranaire non linéaire comprimée]

Présenté par : Philippe G. Ciarlet

Karim Trabelsi ¹

¹ Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, BP 187, 75252 Paris cedex 05, France

Comptes Rendus. Mathématique, Volume 337 (2003) no. 8, pp. 553-558.

Résumé
Abstract

Les équations classiques de plaques membranaires non linéairement élastique, constituées d'un matériau de Saint Venant–Kirchhoff ont été justifiées par Fox et al. (Arch. Rational Mech. Anal. 124 (2) (1993) 157–199). On montre que le problème de minimisation associé à une telle plaque comprimée n'admet pas de solution. La preuve fait appel à un résultat de non existence de minimiseurs de fonctionnelles non convexes dû à Dacorogna et Marcellini (Arch. Rational Mech. Anal. 131 (4) (1995) 359–399). On généralise l'application de leur résultat de l'élasticité plane aux plaques membranaires.

The classical equations of a nonlinearly elastic membrane plate, made of Saint Venant–Kirchhoff material, have been justified by Fox et al. (Arch. Rational Mech. Anal. 124 (2) (1993) 157–199). We show that, under compression, the associated minimization problem admits no solution. The proof is based on a result of non-existence of minimizers of non-convex functionals due to Dacorogna and Marcellini (Arch. Rational Mech. Anal. 131 (4) (1995) 359–399). We generalize the application of their result from plane elasticity to membrane plates.

Reçu le : 2003-05-28
Accepté le : 2003-09-03
Publié le : 2003-10-15

DOI : 10.1016/j.crma.2003.09.008

Affiliations des auteurs :

Karim Trabelsi ¹

¹ Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, BP 187, 75252 Paris cedex 05, France

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Karim Trabelsi. Non-existence of minimizers for a nonlinear membrane plate under compression. Comptes Rendus. Mathématique, Volume 337 (2003) no. 8, pp. 553-558. doi : 10.1016/j.crma.2003.09.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.09.008/

Bibliographie
Cité par

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[11] K. Trabelsi, Non-existence of minimizers for the nonlinear plate membrane under compression, Preprint

Karim Trabelsi On nonlinearly elastic membranes under compression., Chinese Annals of Mathematics. Series B, Volume 27 (2006) no. 4, pp. 379-392 | DOI:10.1007/s11401-005-0182-0 | Zbl:1117.74033
Karim Trabelsi Nonlinearly elastic thin plate models for a class of Ogden materials. I: The membrane model, Analysis and Applications (Singapore), Volume 3 (2005) no. 2, pp. 195-221 | DOI:10.1142/s0219530505000534 | Zbl:1105.74017
Karim Trabelsi Incompressible nonlinearly elastic thin membranes, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 340 (2005) no. 1, pp. 75-80 | DOI:10.1016/j.crma.2004.11.005 | Zbl:1094.74038
Karim Trabelsi Non-existence of minimizers for a nonlinear membrane plate under compression., Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 337 (2003) no. 8, pp. 553-558 | DOI:10.1016/j.crma.2003.09.008 | Zbl:1113.74375

Cité par 4 documents. Sources : zbMATH

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