An intrinsic approach and a notion of polyconvexity for nonlinearly elastic plates

Philippe G. Ciarlet; Sorin Mardare

doi:10.1016/j.crma.2011.11.001

Mathematical Problems in Mechanics

An intrinsic approach and a notion of polyconvexity for nonlinearly elastic plates

Presented by: Philippe G. Ciarlet

Philippe G. Ciarlet ¹; Sorin Mardare ²

¹ Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
² Laboratoire de mathématiques Raphaël-Salem, université de Rouen, avenue de lʼuniversité, 76801 Saint-Etienne-du-Rouvray, France

Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, pp. 111-116.

Abstract
Résumé

Let ω be a domain in $R^{2}$ . The classical approach to the Neumann problem for a nonlinearly elastic plate consists in seeking a displacement field $η = (η_{i}) \in V (ω) = H^{1} (ω) \times H^{1} (ω) \times H^{2} (ω)$ that minimizes a non-quadratic functional over $V (ω)$ . We show that this problem can be recast as a minimization problem in terms of the new unknowns $E_{α β} = \frac{1}{2} (\partial_{α} η_{β} + \partial_{β} η_{α} + \partial_{α} η_{3} \partial_{β} η_{3}) \in L^{2} (ω)$ and $F_{α β} = \partial_{α β} η_{3} \in L^{2} (ω)$ and that this problem has a solution in a manifold of symmetric matrices $E = (E_{α β})$ and $F = (F_{α β})$ whose components $E_{α β} \in L^{2} (ω)$ and $F_{α β} \in L^{2} (ω)$ satisfy nonlinear compatibility conditions of Saint-Venant type. We also show that such an “intrinsic approach” naturally leads to a new definition of polyconvexity.

Soit ω un domaine de $R^{2}$ . Lʼapproche classique du problème de Neumann pour une plaque non linéairement élastique consiste à chercher un champ de déplacements $η = (η_{i}) \in V (ω) = H^{1} (ω) \times H^{1} (ω) \times H^{2} (ω)$ qui minimise une fonctionnelle non quadratique sur $V (ω)$ . Nous montrons que ce problème peut être ré-écrit comme un problème de minimisation en termes des nouvelles inconnues $E_{α β} = \frac{1}{2} (\partial_{α} η_{β} + \partial_{β} η_{α} + \partial_{α} η_{3} \partial_{β} η_{3}) \in L^{2} (ω)$ et $F_{α β} = \partial_{α β} η_{3} \in L^{2} (ω)$ et que ce problème a une solution dans une variété de matrices symétriques $E = (E_{α β})$ et $F = (F_{α β})$ dont les composantes $E_{α β} \in L^{2} (ω)$ et $F_{α β} \in L^{2} (ω)$ satisfont des conditions non linéaires de compatibilité du type de Saint-Venant. Nous montrons également quʼune telle « approche intrinsèque » conduit naturellement à une nouvelle définition de polyconvexité.

Accepted: 2011-10-31
Published online: 2011-11-16

DOI: 10.1016/j.crma.2011.11.001

Author's affiliations:

Philippe G. Ciarlet ¹; Sorin Mardare ²

¹ Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
² Laboratoire de mathématiques Raphaël-Salem, université de Rouen, avenue de lʼuniversité, 76801 Saint-Etienne-du-Rouvray, France

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     title = {An intrinsic approach and a notion of polyconvexity for nonlinearly elastic plates},
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Philippe G. Ciarlet; Sorin Mardare. An intrinsic approach and a notion of polyconvexity for nonlinearly elastic plates. Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, pp. 111-116. doi : 10.1016/j.crma.2011.11.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.11.001/

References
Cited by

[1] J. Ball Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., Volume 63 (1977), pp. 337-403

[2] P.G. Ciarlet; P. Destuynder A justification of a nonlinear model in plate theory, Comput. Methods Appl. Mech. Engrg., Volume 17/18 (1979), pp. 227-258

[3] P.G. Ciarlet; S. Mardare Nonlinear Saint-Venant compatibility conditions for nonlinearly elastic plates, C. R. Acad. Sci. Paris, Ser. I, Volume 349 (2011) no. 23–24, pp. 1297-1302

[4] P.G. Ciarlet, S. Mardare, Saint-Venant compatibility conditions and a notion of polyconvexity in nonlinear plate theory, in preparation.

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