Mathematical Problems in Mechanics
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.

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.

Accepted:
Published online:
DOI: 10.1016/j.crma.2007.03.012
Ciarlet, Philippe G. 1; Mardare, Cristinel 2; Shen, Ming 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
@article{CRMATH_2007__344_8_535_0,
     author = {Ciarlet, Philippe G. and Mardare, Cristinel and Shen, Ming},
     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},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2007.03.012/}
}
TY  - JOUR
AU  - Ciarlet, Philippe G.
AU  - Mardare, Cristinel
AU  - Shen, Ming
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
UR  - http://www.numdam.org/articles/10.1016/j.crma.2007.03.012/
DO  - 10.1016/j.crma.2007.03.012
LA  - en
ID  - CRMATH_2007__344_8_535_0
ER  - 
%0 Journal Article
%A Ciarlet, Philippe G.
%A Mardare, Cristinel
%A Shen, Ming
%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
%U http://www.numdam.org/articles/10.1016/j.crma.2007.03.012/
%R 10.1016/j.crma.2007.03.012
%G en
%F CRMATH_2007__344_8_535_0
Ciarlet, Philippe G.; Mardare, Cristinel; Shen, Ming. 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. http://www.numdam.org/articles/10.1016/j.crma.2007.03.012/

[1] Ciarlet, P.G.; Ciarlet, P. Jr. 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] Ciarlet, P.G.; Mardare, C. 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] Mardare, S. 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

Cited by Sources: