Comptes Rendus
Existence of a solution for an unsteady elasticity problem in large displacement and small perturbation
Comptes Rendus. Mathématique, Volume 334 (2002) no. 6, pp. 521-526.

In this Note we present a model for an unsteady pure traction problem in large displacement and small perturbation for an elastic body in dimension 2, and we show the existence of a solution to the associated problem. The weak formulation of this nonlinear problem involves test-functions depending on the solution, which is not standard. We then study the dynamic of the translation, of the rotation, and of the perturbation associated to the deformation of the body. We prove the existence of a weak solution using a Galerkin method.

Nous présentons dans cette Note la modélisation et l'analyse d'un problème d'élasticité instationnaire en grands déplacements et petites perturbations pour un corps non-encastré en dimension 2. La formulation faible de ce problème non-linéaire utilise des fonctions-tests dépendant de la solution. Nous étudions alors la dynamique de la translation, de la rotation et de la perturbation associées à la déformation du corps élastique. Nous montrons l'existence d'une solution faible au problème par une méthode de Galerkin.

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DOI: 10.1016/S1631-073X(02)02300-2
Céline Grandmont 1; Yvon Maday 2; Paul Métier 2

1 CEREMADE, Université Paris Dauphine, place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France
2 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, BC 187, 175, rue du Chevaleret, 75252 Paris cedex 05, France
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Céline Grandmont; Yvon Maday; Paul Métier. Existence of a solution for an unsteady elasticity problem in large displacement and small perturbation. Comptes Rendus. Mathématique, Volume 334 (2002) no. 6, pp. 521-526. doi : 10.1016/S1631-073X(02)02300-2. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02300-2/

[1] V.I. Arnold Mathematical Methods of Classical Mechanics, Graduate Texts in Math., 60, Springer-Verlag, New York, 1989 (Translated from the Russian by K. Vogtmann and A. Weinstein)

[2] P.G. Ciarlet Mathematical Elasticity, Volume 1: Three-Dimensional Elasticity, North-Holland, Amsterdam, 1988

[3] J.L. Lions Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauthier-Villars, Paris, 1969

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