Up to isometries, a deformation is a continuous function of its metric tensor

Ciarlet, Philippe G.; Laurent, Florian

doi:10.1016/S1631-073X(02)02504-9

Up to isometries, a deformation is a continuous function of its metric tensor
[Aux isométries près, une déformation est une fonction continue de son tenseur métrique]

Ciarlet, Philippe G.^1,² ; Laurent, Florian ³

¹ Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France
² Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
³ Radial Soft, 12 rue de la Faisanderie, 67381 Ingelsheim, France

Comptes Rendus. Mathématique, Tome 335 (2002) no. 5, pp. 489-493.

Résumé
Abstract

Si le tenseur de Riemann–Christoffel associé à un champ de classe $𝒞^{2}$ de matrices symétriques définies positives d'ordre trois s'annule sur un ouvert connexe et simplement connexe $Ω \subset ℝ^{3}$ , alors ce champ est celui du tenseur métrique associé à une déformation de classe $𝒞^{3}$ de l'ensemble $Ω$ , déterminée de façon unique à une isométrie de $ℝ^{3}$ près. On établit ici la continuité de l'application ainsi définie, pour des topologies métrisables convenables.

If the Riemann–Christoffel tensor associated with a field of class $𝒞^{2}$ of positive definite symmetric matrices of order three vanishes in a connected and simply connected open subset $Ω \subset ℝ^{3}$ , then this field is the metric tensor field associated with a deformation of class $𝒞^{3}$ of the set $Ω$ , uniquely determined up to isometries of $ℝ^{3}$ . We establish here that the mapping defined in this fashion is continuous, for ad hoc metrizable topologies.

Reçu le : 2002-06-20
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02504-9

Affiliations des auteurs :

Ciarlet, Philippe G. ^{1, 2} ; Laurent, Florian ³

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Ciarlet, Philippe G.; Laurent, Florian. Up to isometries, a deformation is a continuous function of its metric tensor. Comptes Rendus. Mathématique, Tome 335 (2002) no. 5, pp. 489-493. doi : 10.1016/S1631-073X(02)02504-9. http://www.numdam.org/articles/10.1016/S1631-073X(02)02504-9/

Bibliographie
Cité par

[1] Antman, S.S. Nonlinear Problems of Elasticity, Springer-Verlag, Berlin, 1995

[2] Blume, J.A. Compatibility conditions for a left Cauchy–Green strain field, J. Elasticity, Volume 21 (1989), pp. 271-308

[3] Ciarlet, P.G. Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity, North-Holland, Amsterdam, 1988

[4] Ciarlet, P.G.; Larsonneur, F. On the recovery of a surface with prescribed first and second fundamental forms, J. Math. Pure Appl., Volume 81 (2002), pp. 167-185

[5] P.G. Ciarlet, F. Laurent, On the continuity of a deformation as a function of its Cauchy–Green tensor, 2002, to appear

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