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.
Published online:
Philippe G. Ciarlet 1; Patrick Ciarlet 2
@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},
}
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] Sobolev Spaces, Academic Press, 1975
[2] Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czech. Math. J., Volume 44 (1994), pp. 109-140
[3] 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] Continuity of a deformation as a function of its Cauchy–Green tensor, Arch. Rational Mech. Anal., Volume 167 (2003), pp. 255-269
[6] On rigid and infinitesimal rigid displacements in three-dimensional elasticity, Math. Models Methods Appl. Sci., Volume 13 (2003), pp. 1589-1598
[7] An estimate of the -norm of deformations in terms of the -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] Les Inéquations en Mécanique et en Physique, Inequalities in Mechanics and Physics, Dunod, 1972 (English translation, 1976, Springer-Verlag)
[11] The gradient, divergence, curl and Stokes operators in weighted Sobolev spaces of , J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 39 (1992), pp. 279-307
[12] Finite Element Methods for Navier–Stokes Equations, Springer-Verlag, 1986
[13] Les Méthodes Directes en Théorie des Equations Elliptiques, Masson, 1967
[14] Cours d'Analyse, Deuxième Partie, École Polytechnique, 1959
[15] Mappings of domains in and their metric tensors, Siberian Math. J., Volume 44 (2003), pp. 332-345
[16] St. Venant's compatibility conditions, Tensor (N.S.), Volume 28 (1974), pp. 5-12
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