Legendre–Fenchel duality in elasticity

Philippe G. Ciarlet; Giuseppe Geymonat; Françoise Krasucki

doi:10.1016/j.crma.2011.03.007

Mathematical Problems in Mechanics

Legendre–Fenchel duality in elasticity

Presented by: Philippe G. Ciarlet

Philippe G. Ciarlet ¹; Giuseppe Geymonat ²; Françoise Krasucki ³

¹ Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
² LMGC, UMR – CNRS 5508, université de Montpellier II, place Eugène-Bataillon, 34095 Montpellier cedex 5, France
³ I3M, UMR – CNRS 5149, université de Montpellier II, place Eugène-Bataillon, 34095 Montpellier cedex 5, France

Comptes Rendus. Mathématique, Volume 349 (2011) no. 9-10, pp. 597-602.

Abstract (Original language)
Résumé

We show that the displacement and strain formulations of the displacement–traction problem of three-dimensional linearized elasticity can be viewed as Legendre–Fenchel dual problems to the stress formulation of the same problem. We also show that each corresponding Lagrangian has a saddle-point, thus fully justifying this new approach to elasticity by means of Legendre–Fenchel duality.

On montre que les formulations en déplacements et en déformations du problème de lʼélasticité linéarisée tri-dimensionnelle avec des conditions aux limites mixtes peuvent être vues comme des problèmes duaux de Legendre–Fenchel de la formulation en contraintes de ce même problème. On montre également que chacun des Lagrangiens correspondants a un point-selle, justifiant ainsi complètement cette nouvelle approche de lʼélasticité au moyen de la dualité de Legendre–Fenchel.

Received: 2011-03-04
Accepted: 2011-03-09
Published online: 2011-04-02

DOI: 10.1016/j.crma.2011.03.007

Author's affiliations:

Philippe G. Ciarlet ¹; Giuseppe Geymonat ²; Françoise Krasucki ³

¹ Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
² LMGC, UMR – CNRS 5508, université de Montpellier II, place Eugène-Bataillon, 34095 Montpellier cedex 5, France
³ I3M, UMR – CNRS 5149, université de Montpellier II, place Eugène-Bataillon, 34095 Montpellier cedex 5, France

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     author = {Philippe G. Ciarlet and Giuseppe Geymonat and Fran\c{c}oise Krasucki},
     title = {Legendre{\textendash}Fenchel duality in elasticity},
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Philippe G. Ciarlet; Giuseppe Geymonat; Françoise Krasucki. Legendre–Fenchel duality in elasticity. Comptes Rendus. Mathématique, Volume 349 (2011) no. 9-10, pp. 597-602. doi : 10.1016/j.crma.2011.03.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.03.007/

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