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.
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.
Published online:
Philippe G. Ciarlet 1; Cristinel Mardare 2; Ming Shen 1
@article{CRMATH_2007__344_8_535_0,
author = {Philippe G. Ciarlet and Cristinel Mardare and Ming Shen},
title = {Recovery of a displacement field from its linearized strain tensor field in curvilinear coordinates},
journal = {Comptes Rendus. Math\'ematique},
pages = {535--540},
publisher = {Elsevier},
volume = {344},
number = {8},
year = {2007},
doi = {10.1016/j.crma.2007.03.012},
language = {en},
}
TY - JOUR AU - Philippe G. Ciarlet AU - Cristinel Mardare AU - Ming Shen TI - Recovery of a displacement field from its linearized strain tensor field in curvilinear coordinates JO - Comptes Rendus. Mathématique PY - 2007 SP - 535 EP - 540 VL - 344 IS - 8 PB - Elsevier DO - 10.1016/j.crma.2007.03.012 LA - en ID - CRMATH_2007__344_8_535_0 ER -
%0 Journal Article %A Philippe G. Ciarlet %A Cristinel Mardare %A Ming Shen %T Recovery of a displacement field from its linearized strain tensor field in curvilinear coordinates %J Comptes Rendus. Mathématique %D 2007 %P 535-540 %V 344 %N 8 %I Elsevier %R 10.1016/j.crma.2007.03.012 %G en %F CRMATH_2007__344_8_535_0
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] 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] 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] 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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