[Sur les conditions de compatibilité de Saint Venant et le lemme de Poincaré]
Le théorème de Saint Venant constitue une caractérisation classique de champs de matrices réguliers comme des champs de tenseurs de déformation linéarisés. Ce théorème a été étendu aux champs de matrices avec des composantes dans par le second auteur et P. Ciarlet, Jr. en 2005. Un objectif de cette Note est d'étendre cette caractérisation aux champs de matrices dont les composantes sont seulement dans . Un autre objectif est de démontrer que le théorème de Saint Venant n'est autre que l'analogue matriciel du lemme de Poincaré.
Saint Venant's theorem constitutes a classical characterization of smooth matrix fields as linearized strain tensor fields. This theorem has been extended to matrix fields with components in by the second author and P. Ciarlet, Jr. in 2005. One objective of this Note is to further extend this characterization to matrix fields whose components are only in . Another objective is to demonstrate that Saint Venant's theorem is in fact nothing but the matrix analog of Poincaré's lemma.
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@article{CRMATH_2006__342_11_887_0,
author = {Amrouche, Cherif and Ciarlet, Philippe G. and Gratie, Liliana and Kesavan, Srinivasan},
title = {On {Saint} {Venant's} compatibility conditions and {Poincar\'e's} lemma},
journal = {Comptes Rendus. Math\'ematique},
pages = {887--891},
publisher = {Elsevier},
volume = {342},
number = {11},
year = {2006},
doi = {10.1016/j.crma.2006.03.026},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2006.03.026/}
}
TY - JOUR AU - Amrouche, Cherif AU - Ciarlet, Philippe G. AU - Gratie, Liliana AU - Kesavan, Srinivasan TI - On Saint Venant's compatibility conditions and Poincaré's lemma JO - Comptes Rendus. Mathématique PY - 2006 SP - 887 EP - 891 VL - 342 IS - 11 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2006.03.026/ DO - 10.1016/j.crma.2006.03.026 LA - en ID - CRMATH_2006__342_11_887_0 ER -
%0 Journal Article %A Amrouche, Cherif %A Ciarlet, Philippe G. %A Gratie, Liliana %A Kesavan, Srinivasan %T On Saint Venant's compatibility conditions and Poincaré's lemma %J Comptes Rendus. Mathématique %D 2006 %P 887-891 %V 342 %N 11 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2006.03.026/ %R 10.1016/j.crma.2006.03.026 %G en %F CRMATH_2006__342_11_887_0
Amrouche, Cherif; Ciarlet, Philippe G.; Gratie, Liliana; Kesavan, Srinivasan. On Saint Venant's compatibility conditions and Poincaré's lemma. Comptes Rendus. Mathématique, Tome 342 (2006) no. 11, pp. 887-891. doi : 10.1016/j.crma.2006.03.026. http://www.numdam.org/articles/10.1016/j.crma.2006.03.026/
[1] C. Amrouche, P.G. Ciarlet, L. Gratie, S. Kesavan, New formulations of linearized elasticity problems, based on extensions of Donati's theorem, 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] Another approach to linearized elasticity and a new proof of Korn's inequality, Math. Models Methods Appl. Sci., Volume 15 (2005), pp. 259-271
[4] Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2: Functional and Variational Methods, Springer, 1988
[5] G. Geymonat, F. Krasucki, Some remarks on the compatibility conditions in elasticity, Accad. Naz. Sci. XL Mem. Math. Appl., in press
[6] Beltrami's solutions of general equilibrium equations in continuum mechanics, C. R. Acad. Sci. Paris, Ser. I, Volume 342 (2006), pp. 359-363
[7] Finite Element Methods for Navier–Stokes Equations, Springer-Verlag, Heidelberg, 1986
[8] On Poincaré's and J.-L. Lions' lemmas, C. R. Acad. Sci. Paris, Ser. I, Volume 340 (2005), pp. 27-30
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