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

Philippe G. Ciarlet; Florian Laurent

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]

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

Comptes Rendus. Mathématique, Volume 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 :

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/

Bibliographie
Cité par

[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

Philippe G. Ciarlet The continuity of a surface as a function of its two fundamental forms., Journal de Mathématiques Pures et Appliquées. Neuvième Série, Volume 82 (2003) no. 3, pp. 253-274 | DOI:10.1016/s0021-7824(03)00017-5 | Zbl:1042.53003
Philippe G. Ciarlet A surface is a continuous function of its two fundamental forms, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 335 (2002) no. 7, pp. 609-614 | DOI:10.1016/s1631-073x(02)02538-4 | Zbl:1029.53008

Cité par 2 documents. Sources : zbMATH

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