Comptes Rendus
Mathematical Problems in Mechanics
Another approach to linearized elasticity and Korn's inequality
Comptes Rendus. Mathématique, Volume 339 (2004) no. 4, pp. 307-312.

We describe and analyze an approach to the pure traction problem of three-dimensional linearized elasticity, whose novelty consists in considering the linearized strain tensor as the ‘primary’ unknown, instead of the displacement itself as is customary. This approach leads to a well-posed minimization problem, constrained by a weak form of the St Venant compatibility conditions. It also provides a new proof of Korn's inequality.

On décrit et analyse une approche du problème de traction pure en élasticité linéarisée tridimensionnelle, dont la nouveauté consiste à considérer le tenseur linéarisé des déformations comme l'inconnue principale, au lieu du déplacement lui-même selon l'habitude. Cette approche conduit à un problème bien posé de minimisation sous contraintes, celles-ci consistant en une forme affaiblie des conditions de compatibilité de St Venant. Cette approche conduit aussi à une nouvelle démonstration de l'inégalité de Korn.

Received:
Published online:
DOI: 10.1016/j.crma.2004.06.021
Philippe G. Ciarlet 1; Patrick Ciarlet 2

1 Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
2 École Nationale Supérieure de Techniques Avancées, 32, boulevard Victor, 75015 Paris, France
@article{CRMATH_2004__339_4_307_0,
     author = {Philippe G. Ciarlet and Patrick Ciarlet},
     title = {Another approach to linearized elasticity and {Korn's} inequality},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {307--312},
     publisher = {Elsevier},
     volume = {339},
     number = {4},
     year = {2004},
     doi = {10.1016/j.crma.2004.06.021},
     language = {en},
}
TY  - JOUR
AU  - Philippe G. Ciarlet
AU  - Patrick Ciarlet
TI  - Another approach to linearized elasticity and Korn's inequality
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 307
EP  - 312
VL  - 339
IS  - 4
PB  - Elsevier
DO  - 10.1016/j.crma.2004.06.021
LA  - en
ID  - CRMATH_2004__339_4_307_0
ER  - 
%0 Journal Article
%A Philippe G. Ciarlet
%A Patrick Ciarlet
%T Another approach to linearized elasticity and Korn's inequality
%J Comptes Rendus. Mathématique
%D 2004
%P 307-312
%V 339
%N 4
%I Elsevier
%R 10.1016/j.crma.2004.06.021
%G en
%F CRMATH_2004__339_4_307_0
Philippe G. Ciarlet; Patrick Ciarlet. Another approach to linearized elasticity and Korn's inequality. Comptes Rendus. Mathématique, Volume 339 (2004) no. 4, pp. 307-312. doi : 10.1016/j.crma.2004.06.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.06.021/

[1] R.A. Adams Sobolev Spaces, Academic Press, 1975

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

[3] S.S. Antman Ordinary differential equations of nonlinear elasticity I: Foundations of the theories of non-linearly elastic rods and shells, Arch. Rational Mech. Anal., Volume 61 (1976), pp. 307-351

[4] P.G. Ciarlet, P. Ciarlet Jr., Linearized elasticity and Korn's inequality revisited, in preparation

[5] P.G. Ciarlet; F. Laurent Continuity of a deformation as a function of its Cauchy–Green tensor, Arch. Rational Mech. Anal., Volume 167 (2003), pp. 255-269

[6] P.G. Ciarlet; C. Mardare On rigid and infinitesimal rigid displacements in three-dimensional elasticity, Math. Models Methods Appl. Sci., Volume 13 (2003), pp. 1589-1598

[7] P.G. Ciarlet; C. Mardare An estimate of the H1-norm of deformations in terms of the L1-norm of their Cauchy–Green tensors, C. R. Acad. Sci. Paris, Ser. I, Volume 338 (2004), pp. 505-510

[8] P.G. Ciarlet, C. Mardare, Recovery of a manifold with boundary and its continuity as a function of its metric tensor, J. Math. Pures Appl., in press

[9] P. Ciarlet Jr., Potentials of vector fields in Lipschitz domains, in preparation

[10] G. Duvaut; J.L. Lions; G. Duvaut; J.L. Lions Les Inéquations en Mécanique et en Physique, Inequalities in Mechanics and Physics, Dunod, 1972 (English translation, 1976, Springer-Verlag)

[11] V. Girault The gradient, divergence, curl and Stokes operators in weighted Sobolev spaces of R3, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 39 (1992), pp. 279-307

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

[13] J. Nečas Les Méthodes Directes en Théorie des Equations Elliptiques, Masson, 1967

[14] L. Schwartz Cours d'Analyse, Deuxième Partie, École Polytechnique, 1959

[15] Y.G. Reshetnyak Mappings of domains in Rn and their metric tensors, Siberian Math. J., Volume 44 (2003), pp. 332-345

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

Cited by Sources:

Comments - Policy


Articles of potential interest

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

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

C. R. Math (2006)


Another approach to linear shell theory and a new proof of Korn's inequality on a surface

Philippe G. Ciarlet; Liliana Gratie

C. R. Math (2005)


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

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

C. R. Math (2006)