Comptes Rendus
Mathematical Problems in Mechanics
Recovery of a displacement field from its linearized strain tensor field in curvilinear coordinates
[Reconstruction d'un champ de déplacements à partir de son tenseur des déformations linéarisées en coordonnées curvilignes]
Comptes Rendus. Mathématique, Volume 344 (2007) no. 8, pp. 535-540.

Nous montrons que, si un champ de matrices symétriques défini sur un ouvert simplement connexe vérifie les équations de Saint Venant en coordonnées curvilignes, alors c'est le tenseur des déformations linéarisées associé à un champ de déplacements. Notre démonstration fournit un algorithme explicite de reconstruction d'un tel champ de déplacements, qui peut être considéré comme la version linéarisée de la reconstruction d'une immersion à partir d'une métrique riemannienne plate.

We establish that, if a symmetric matrix field defined over a simply-connected open set satisfies the Saint Venant equations in curvilinear coordinates, then its coefficients are the linearized strains associated with a displacement field. Our proof provides an explicit algorithm for recovering such a displacement field, which may be viewed as the linear counterpart of the reconstruction of an immersion from a given flat Riemannian metric.

Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.03.012
Philippe G. Ciarlet 1 ; Cristinel Mardare 2 ; Ming Shen 1

1 Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
2 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, Place Jussieu, 75005 Paris, France
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     title = {Recovery of a displacement field from its linearized strain tensor field in curvilinear coordinates},
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Philippe G. Ciarlet; Cristinel Mardare; Ming Shen. Recovery of a displacement field from its linearized strain tensor field in curvilinear coordinates. Comptes Rendus. Mathématique, Volume 344 (2007) no. 8, pp. 535-540. doi : 10.1016/j.crma.2007.03.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.03.012/

[1] P.G. Ciarlet; P. Ciarlet Another approach to linearized elasticity and a new proof of Korn's inequality, Math. Models Methods Appl. Sci., Volume 15 (2005), pp. 259-271

[2] P.G. Ciarlet, L. Gratie, C. Mardare, M. Shen, Saint Venant compatibility equations on a surface – application to intrinsic shell theory, preprint, 2006, in press

[3] P.G. Ciarlet; C. Mardare On rigid and infinitesimal rigid displacements in three-dimensional elasticity, Math. Models Methods Appl. Sci., Volume 13 (2003), pp. 1589-1598

[4] P.G. Ciarlet, C. Mardare, M. Shen, Saint Venant compatibility equations in curvilinear coordinates, Anal. Appl., in press

[5] S. Mardare On isometric immersions of a Riemannian space with little regularity, Anal. Appl., Volume 2 (2004), pp. 193-226

[6] S. Mardare, Sur quelques problèmes de géomètrie différentielle liés à la théorie de l'élasticité, Doctoral Disertation, Université Paris 6, 2003

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