SAINT VENANT COMPATIBILITY EQUATIONS ON A SURFACE APPLICATION TO INTRINSIC SHELL THEORY

Abstract

We first establish that the linearized change of metric and change of curvature tensors, with components in L2and H-1respectively, associated with a displacement field, with components in H1, of a surface S immersed in ℝ3must satisfy in the distributional sense compatibility conditions that may be viewed as the linear version of the Gauss and Codazzi-Mainardi equations. These compatibility conditions, which are analogous to the familiar 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. that 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 ⊂ ℝ3satisfy 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 and second fundamental forms.Finally, we show how these results can be applied to the "intrinsic theory" of linearly elastic shells, where the linearized change of metric and change of curvature tensors are the new unknowns. These new unknowns solve a quadratic minimization problem over a space of tensor fields whose components, which are only in L2, satisfy the Saint Venant compatibility conditions on a surface in the sense of distributions.