Comptes Rendus
no. 10
Mathematical Problems in Mechanics
New formulations of linearized elasticity problems, based on extensions of Donati's theorem
[Nouvelles formulations de problèmes d'élasticité linéarisée, basées sur des généralisations du théorème de Donati]

Présenté par : Robert Dautray

Cherif Amrouche1 ; Philippe G. Ciarlet2 ; Liliana Gratie3 ; Srinivasan Kesavan4
1 Laboratoire de mathématiques appliquées, université de Pau et des pays de l'Adour, avenue de l'université, 64000 Pau, France
2 Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
3 Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
4 The Institute of Mathematical Sciences, CIT Campus Taramani, Chennai – 600113, India
Comptes Rendus. Mathématique, Volume 342 (2006) no. 10, pp. 785-789.

Le théorème classique de Donati sert à caractériser les champs de matrices réguliers qui sont des champs de déformation linéarisés. Dans cette Note, on donne plusieurs généralisations de ce théorème, en particulier à des champs de matrices dont les composantes sont seulement dans H−1. On montre ensuite que de telles généralisations conduisent à de nouvelles formulations des problèmes d'élasticité linéarisée tridimensionnelle, comme des problèmes de minimisation quadratique où les déformations sont les inconnues principales.

The classical Donati theorem is used for characterizing smooth matrix fields as linearized strain tensor fields. In this Note, we give several generalizations of this theorem, notably to matrix fields whose components are only in H−1. We then show that our extensions of Donati's theorem allow to reformulate in a novel fashion linearized three-dimensional elasticity problems as quadratic minimization problems with the strains as the primary unknowns.

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Accepté le :
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DOI : 10.1016/j.crma.2006.03.027
Cherif Amrouche 1 ; Philippe G. Ciarlet 2 ; Liliana Gratie 3 ; Srinivasan Kesavan 4

1 Laboratoire de mathématiques appliquées, université de Pau et des pays de l'Adour, avenue de l'université, 64000 Pau, France
2 Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
3 Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
4 The Institute of Mathematical Sciences, CIT Campus Taramani, Chennai – 600113, India 
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     author = {Cherif Amrouche and Philippe G. Ciarlet and Liliana Gratie and Srinivasan Kesavan},
     title = {New formulations of linearized elasticity problems, based on extensions of {Donati's} theorem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {785--789},
     publisher = {Elsevier},
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     year = {2006},
     doi = {10.1016/j.crma.2006.03.027},
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Cherif Amrouche; Philippe G. Ciarlet; Liliana Gratie; Srinivasan Kesavan. New formulations of linearized elasticity problems, based on extensions of Donati's theorem. Comptes Rendus. Mathématique, Volume 342 (2006) no. 10, pp. 785-789. doi : 10.1016/j.crma.2006.03.027. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.03.027/

[1] C. Amrouche, P.G. Ciarlet, L. Gratie, S. Kesavan, On Saint Venant's compatibility conditions and Poincaré's lemma, C. R. Acad. Sci. Paris, Ser. I, in press

[2] C. Amrouche, P.G. Ciarlet, L. Gratie, S. Kesavan, On the characterizations of matrix fields as linearized strain tensor fields, J. Math. Pures Appl., in press

[3] C. Amrouche; V. Girault Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J., Volume 44 (1994), pp. 109-140

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

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

[6] G. Geymonat, F. Krasucki, Some remarks on the compatibility conditions in elasticity, Accad. Naz. Sci. XL Mem. Math. Appl., in press

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

[8] J.J. Moreau Duality characterization of strain tensor distributions in an arbitrary open set, J. Math. Anal. Appl., Volume 72 (1979), pp. 760-770

[9] J. Peetre Espaces d'interpolation et théorème de Soboleff, Ann. Inst. Fourier, Volume 16 (1966), pp. 279-317

[10] L. Tartar, Topics in Nonlinear Analysis, Publications Mathématiques d'Orsay No. 78.13, Université Paris-Sud, 1978

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

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