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 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 , 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 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 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.
Published online:
Ciarlet, Philippe G. 1 ; Gratie, Liliana 2 ; Mardare, Cristinel 3 ; Shen, Ming 1
@article{CRMATH_2007__344_9_597_0,
author = {Ciarlet, Philippe G. and Gratie, Liliana and Mardare, Cristinel and Shen, Ming},
title = {Recovery of a displacement field on a surface from its linearized change of metric and change of curvature tensors},
journal = {Comptes Rendus. Math\'ematique},
pages = {597--602},
year = {2007},
publisher = {Elsevier},
volume = {344},
number = {9},
doi = {10.1016/j.crma.2007.03.019},
language = {en},
url = {https://numdam.org/articles/10.1016/j.crma.2007.03.019/}
}
TY - JOUR AU - Ciarlet, Philippe G. AU - Gratie, Liliana AU - Mardare, Cristinel AU - Shen, Ming TI - Recovery of a displacement field on a surface from its linearized change of metric and change of curvature tensors JO - Comptes Rendus. Mathématique PY - 2007 SP - 597 EP - 602 VL - 344 IS - 9 PB - Elsevier UR - https://numdam.org/articles/10.1016/j.crma.2007.03.019/ DO - 10.1016/j.crma.2007.03.019 LA - en ID - CRMATH_2007__344_9_597_0 ER -
%0 Journal Article %A Ciarlet, Philippe G. %A Gratie, Liliana %A Mardare, Cristinel %A Shen, Ming %T Recovery of a displacement field on a surface from its linearized change of metric and change of curvature tensors %J Comptes Rendus. Mathématique %D 2007 %P 597-602 %V 344 %N 9 %I Elsevier %U https://numdam.org/articles/10.1016/j.crma.2007.03.019/ %R 10.1016/j.crma.2007.03.019 %G en %F CRMATH_2007__344_9_597_0
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
[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] 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] On rigid and infinitesimal rigid displacements in shell theory, J. Math. Pures Appl., Volume 83 (2004), pp. 1-15
[5] On Pfaff systems with coefficients and their applications in differential geometry, J. Math. Pures Appl., Volume 84 (2005), pp. 1659-1692
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