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

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 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 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 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 3 . We establish here that the mapping defined in this fashion is continuous, for ad hoc metrizable topologies.

Reçu le :
Publié le :
DOI : 10.1016/S1631-073X(02)02504-9
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
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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://archive.numdam.org/articles/10.1016/S1631-073X(02)02504-9/

[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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