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

@article{CRMATH_2010__348_9-10_587_0,
     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},
}

TY  - JOUR
AU  - Philippe G. Ciarlet
AU  - Patrick Ciarlet
AU  - Oana Iosifescu
AU  - Stefan Sauter
AU  - Jun Zou
TI  - A Lagrangian approach to intrinsic linearized elasticity
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 587
EP  - 592
VL  - 348
IS  - 9-10
PB  - Elsevier
DO  - 10.1016/j.crma.2010.04.011
LA  - en
ID  - CRMATH_2010__348_9-10_587_0
ER  -

%0 Journal Article
%A Philippe G. Ciarlet
%A Patrick Ciarlet
%A Oana Iosifescu
%A Stefan Sauter
%A Jun Zou
%T A Lagrangian approach to intrinsic linearized elasticity
%J Comptes Rendus. Mathématique
%D 2010
%P 587-592
%V 348
%N 9-10
%I Elsevier
%R 10.1016/j.crma.2010.04.011
%G en
%F CRMATH_2010__348_9-10_587_0

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/

Bibliographie
Cité par

[1] C. Amrouche; P.G. Ciarlet; L. Gratie; S. Kesavan On the characterizations of matrix fields as linearized strain tensor fields, J. Math. Pures Appl., Volume 86 (2006), pp. 116-132

[2] I. Babuška The finite element method with Lagrange multipliers, Numer. Math., Volume 20 (1973), pp. 179-192

[3] F. Brezzi On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér., Volume R-2 (1974), pp. 129-151

[4] F. Brezzi; M. Fortin Mixed and Hybrid Finite Element Methods, Springer, 1991

[5] P.G. Ciarlet; P. Ciarlet Another approach to linearized elasticity and a new proof of Korn's inequality, Math. Mod. Meth. Appl. Sci., Volume 15 (2005), pp. 259-271

[6] P.G. Ciarlet, P. Ciarlet Jr., O. Iosifescu, S. Sauter, Jun Zou, Lagrange multipliers in intrinsic elasticity, in preparation

[7] G. Duvaut; J.L. Lions Les Inéquations en Mécanique et en Physique, Dunod, 1972

[8] G. Geymonat; F. Krasucki Some remarks on the compatibility conditions in elasticity, Acad. Naz. Sci., Volume XL 123 (2005), pp. 175-182

[9] G. Geymonat; F. Krasucki Beltrami's solutions of general equilibrium equations in continuum mechanics, C. R. Acad. Sci. Paris, Ser. I, Volume 342 (2006), pp. 359-363

[10] V. Girault; P.A. Raviart Finite Element Methods for Navier–Stokes Equations, Springer, 1986

[11] T.W. Ting St. Venant's compatibility conditions, Tensor N.S., Volume 28 (1974), pp. 5-12

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Legendre–Fenchel duality in elasticity

Philippe G. Ciarlet; Giuseppe Geymonat; Françoise Krasucki

C. R. Math (2011)

On Saint Venant's compatibility conditions and Poincaré's lemma

Cherif Amrouche; Philippe G. Ciarlet; Liliana Gratie; ...

C. R. Math (2006)

New formulations of linearized elasticity problems, based on extensions of Donati's theorem

Cherif Amrouche; Philippe G. Ciarlet; Liliana Gratie; ...

C. R. Math (2006)