Mathematical Problems in Mechanics
Recovery of a displacement field on a surface from its linearized change of metric and change of curvature tensors
Comptes Rendus. Mathématique, Volume 344 (2007) no. 9, pp. 597-602.

We establish that the components of the linearized change of metric and change of curvature tensors associated with a displacement field of a surface in R3 must satisfy compatibility conditions, which are the analogues ‘on a surface’ of the Saint Venant equations in three-dimensional elasticity.

We next show that, conversely, if two symmetric matrix fields of order two satisfy these compatibility conditions over a simply-connected surface SR3, then they are the linearized change of metric and change of curvature tensors associated with a displacement field of the surface S.

On montre que les composantes des tenseurs linéarisés de changement de métrique et de changement de courbure associés à un champ de déplacements d'une surface de R3 doivent satisfaire certaines relations de compatibilité, qui sont les analogues « sur une surface » des relations de Saint Venant en élasticité tri-dimensionnelle.

On montre ensuite que, inversement, si deux champs de matrices symétriques d'ordre deux satisfont ces mêmes relations de compatibilité sur une surface SR3 simplement connexe, alors ce sont les tenseurs linéarisés de changement de métrique et de changement de courbure d'un champ de déplacements de la surface S.

Accepted:
Published online:
DOI: 10.1016/j.crma.2007.03.019
Ciarlet, Philippe G. 1; Gratie, Liliana 2; Mardare, Cristinel 3; Shen, Ming 1

1 Department of Mathematics, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong
2 Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, 83, Tat Chee Avenue, Kowloon, Hong Kong
3 Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, 4, place Jussieu, 75252 Paris cedex 05, France
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Ciarlet, Philippe G.; Gratie, Liliana; Mardare, Cristinel; Shen, Ming. Recovery of a displacement field on a surface from its linearized change of metric and change of curvature tensors. Comptes Rendus. Mathématique, Volume 344 (2007) no. 9, pp. 597-602. doi : 10.1016/j.crma.2007.03.019. http://www.numdam.org/articles/10.1016/j.crma.2007.03.019/

[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] Ciarlet, P.G.; Gratie, L. A new approach to linear shell theory, Math. Models Methods Appl. Sci., Volume 15 (2005), pp. 1181-1202

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

[4] Ciarlet, P.G.; Mardare, C. On rigid and infinitesimal rigid displacements in shell theory, J. Math. Pures Appl., Volume 83 (2004), pp. 1-15

[5] Mardare, S. On Pfaff systems with Lp coefficients and their applications in differential geometry, J. Math. Pures Appl., Volume 84 (2005), pp. 1659-1692

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