A Lagrangian approach to intrinsic linearized elasticity

Philippe G. Ciarlet; Patrick Ciarlet; Oana Iosifescu; Stefan Sauter; Jun Zou

doi:10.1016/j.crma.2010.04.011

Mathematical Problems in Mechanics

A Lagrangian approach to intrinsic linearized elasticity
[Une approche lagrangienne de l'élasticité linéarisée intrinsèque]

Présenté par : Philippe G. Ciarlet

Philippe G. Ciarlet ¹ ; Patrick Ciarlet ² ; Oana Iosifescu ³ ; Stefan Sauter ⁴ ; Jun Zou ⁵

¹ Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
² Laboratoire POEMS, École nationale supérieure de techniques avancées, 32, boulevard Victor, 75739 Paris cedex 15, France
³ Département de mathématiques, université de Montpellier II, place Eugène-Bataillon, 34095 Montpellier cedex 5, France
⁴ Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
⁵ Department of Mathematics, Lady Shaw Building, The Chinese University of Hong Kong, Shatin, Hong Kong

Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 587-592.

Résumé
Abstract

On considère le problème en déplacement pur et le problème en traction pure de l'élasticité linéarisée tri-dimensionnelle. On montre que, dans chaque cas, l'approche intrinsèque conduit à un problème de minimisation quadratique avec des contraintes semblables à celles de Donati. Utilisant la condition inf–sup de Babuška–Brezzi, on montre ensuite que, dans chaque cas, le minimiseur du problème de minimisation avec contraintes trouvé dans une approche intrinsèque est le premier argument du point-selle d'un lagrangien approprié, ce qui fait que le second argument de ce point-selle est le multiplicateur de Lagrange associé aux contraintes correspondantes.

We consider the pure traction problem and the pure displacement problem of three-dimensional linearized elasticity. We show that, in each case, the intrinsic approach leads to a quadratic minimization problem constrained by Donati-like relations. Using the Babuška–Brezzi inf–sup condition, we then show that, in each case, the minimizer of the constrained minimization problem found in an intrinsic approach is the first argument of the saddle-point of an ad hoc Lagrangian, so that the second argument of this saddle-point is the Lagrange multiplier associated with the corresponding constraints.

Accepté le : 2010-04-01
Publié le : 2010-04-24

DOI : 10.1016/j.crma.2010.04.011

Affiliations des auteurs :

Philippe G. Ciarlet ¹ ; Patrick Ciarlet ² ; Oana Iosifescu ³ ; Stefan Sauter ⁴ ; Jun Zou ⁵

¹ Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
² Laboratoire POEMS, École nationale supérieure de techniques avancées, 32, boulevard Victor, 75739 Paris cedex 15, France
³ Département de mathématiques, université de Montpellier II, place Eugène-Bataillon, 34095 Montpellier cedex 5, France
⁴ Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
⁵ Department of Mathematics, Lady Shaw Building, The Chinese University of Hong Kong, Shatin, Hong Kong

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     author = {Philippe G. Ciarlet and Patrick Ciarlet and Oana Iosifescu and Stefan Sauter and Jun Zou},
     title = {A {Lagrangian} approach to intrinsic linearized elasticity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {587--592},
     publisher = {Elsevier},
     volume = {348},
     number = {9-10},
     year = {2010},
     doi = {10.1016/j.crma.2010.04.011},
     language = {en},
}

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Philippe G. Ciarlet; Patrick Ciarlet; Oana Iosifescu; Stefan Sauter; Jun Zou. A Lagrangian approach to intrinsic linearized elasticity. Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 587-592. doi : 10.1016/j.crma.2010.04.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.04.011/

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