Saint Venant compatibility equations on a surface - Application to linear intrinsic shell theory
Résumé
We establish that the linearized change of metric and linearized change of curvature tensors associated with a displacement field of a surface S immersed in R3 must satisfy compatibility conditions that may be viewed as the linear version of the Gauss and Codazzi-Mainardi equations. These compatibility conditions, which are the analogous in two-dimensional shell theory of the Saint Venant equations in three-dimensional elasticity, constitute the Saint Venant equations on the surface S. We next show that these compatibility conditions are also sufficient, i.e., they in fact characterize the linearized change of metric and the linearized change of curvature tensors in the following sense: If two symmetric matrix fields of order two defined over a simply-connected surface S ⊂ R3 satisfy the above compatibility conditions, then they are the linearized change of metric and linearized change of curvature tensors associated with a displacement field of the surface S, a field whose existence is thus established. The proof provides an explicit algorithm for recovering such a displacement field from the linearized change of metric and linearized change of curvature tensors. This algorithm may be viewed as the linear counterpart of the reconstruction of a surface from its first two fundamental forms.
Domaines
Analyse numérique [math.NA]
Origine : Fichiers produits par l'(les) auteur(s)
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