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Comptes Rendus Mathématique
Volume 344, n° 8
pages 535-540 (avril 2007)
Doi : 10.1016/j.crma.2007.03.012
accepted : 2 Mars 2007
Recovery of a displacement field from its linearized strain tensor field in curvilinear coordinates
Reconstruction dʼun champ de déplacements à partir de son tenseur des déformations linéarisées en coordonnées curvilignes
 

Philippe G. Ciarlet a , Cristinel Mardare b , Ming Shen a
a Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong 
b Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, Place Jussieu, 75005 Paris, France 

Abstract

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. To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).

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Résumé

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 cʼest le tenseur des déformations linéarisées associé à un champ de déplacements. Notre démonstration fournit un algorithme explicite de reconstruction dʼun tel champ de déplacements, qui peut être considéré comme la version linéarisée de la reconstruction dʼune immersion à partir dʼune métrique riemannienne plate. Pour citer cet article : P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007).

The full text of this article is available in PDF format.


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