The pure displacement problem in nonlinear three-dimensional elasticity: intrinsic formulation and existence theorems

Philippe G. Ciarlet; Cristinel Mardare

doi:10.1016/j.crma.2009.03.020

Mathematical Problems in Mechanics

The pure displacement problem in nonlinear three-dimensional elasticity: intrinsic formulation and existence theorems

Presented by: Philippe G. Ciarlet

Philippe G. Ciarlet ¹; Cristinel Mardare ²

¹ Department of Mathematics, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong
² Université Pierre et Marie Curie, Laboratoire Jacques-Louis Lions, 4, place Jussieu, 75005 Paris, France

Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 677-683.

Abstract
Résumé

In this Note, the equations of nonlinear three-dimensional elasticity corresponding to the pure displacement problem are recast either as a boundary value problem, or as a minimization problem, where the unknown is in both cases the Cauchy–Green strain tensor, instead of the deformation as is customary. We then show that either problem possesses a solution if the applied forces are sufficiently small and the stored energy function satisfies specific hypotheses. The second problem provides an example of a minimization problem for a non-coercive functional over a Banach manifold.

Dans cette Note, les équations de l'élasticité non linéaire tri-dimensionnelle correspondant au problème en déplacement pur sont ré-écrites, soit comme un problème aux limites, soit comme un problème de minimisation, l'inconnue étant dans les deux cas le tenseur des déformations de Cauchy–Green, au lieu de la déformation comme il est usuel. On montre ensuite que l'un et l'autre problème ont au moins une solution si les forces sont suffisamment petites et si la densité d'énergie satisfait certaines hypothèses naturelles. Le second problème constitue un exemple de problème de minimisation d'une fonctionnelle non coercive sur une variété de Banach.

Accepted: 2009-03-19
Published online: 2009-04-17

DOI: 10.1016/j.crma.2009.03.020

Author's affiliations:

Philippe G. Ciarlet ¹; Cristinel Mardare ²

¹ Department of Mathematics, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong
² Université Pierre et Marie Curie, Laboratoire Jacques-Louis Lions, 4, place Jussieu, 75005 Paris, France

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     title = {The pure displacement problem in nonlinear three-dimensional elasticity: intrinsic formulation and existence theorems},
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Philippe G. Ciarlet; Cristinel Mardare. The pure displacement problem in nonlinear three-dimensional elasticity: intrinsic formulation and existence theorems. Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 677-683. doi : 10.1016/j.crma.2009.03.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.03.020/

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