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

Si le tenseur de Riemann–Christoffel associé à un champ de classe 𝒞2 de matrices symétriques définies positives d'ordre trois s'annule sur un ouvert connexe et simplement connexe Ω3, alors ce champ est celui du tenseur métrique associé à une déformation de classe 𝒞3 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 𝒞2 of positive definite symmetric matrices of order three vanishes in a connected and simply connected open subset Ω3, then this field is the metric tensor field associated with a deformation of class 𝒞3 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

Philippe G. Ciarlet 1, 2 ; Florian Laurent 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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Philippe G. Ciarlet; Florian Laurent. Up to isometries, a deformation is a continuous function of its metric tensor. Comptes Rendus. Mathématique, Volume 335 (2002) no. 5, pp. 489-493. doi : 10.1016/S1631-073X(02)02504-9. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02504-9/

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

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

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

[4] P.G. Ciarlet; F. Larsonneur 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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