Saint Venant compatibility conditions in curvilinear coordinates and applications to elasticity

Abstract

In this thesis, we study Saint Venant compatibility equations in curvilinear coordinates on a three-dimensional elastic body and on a surface. In the case of the three-dimensional elastic body, we establish that the linearized strains in curvilinear coordinates associated with a given displacement field necessarily satisfy compatibility conditions that constitute “Saint Venant equations in curvilinear coordinates”. Furthermore, we show that these equations are also sufficient, in the following sense: 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. In addition, our proof provides an explicit algorithm for recovering such a displacement field from its linear strains in curvilinear coordinates. We also show how these Saint Venant compatibility in curvilinear coordinates may be used for defining a new approach to existence theory in three-dimensional linearized elasticity in curvilinear coordinates.

In the case of the surface, 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 analog 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 in the following sense: If two symmetric matrix fields of order two defined over a simplyconnected 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. We also show how these Saint Venant compatibility on a surface may be used for defining a new approach to existence theory in linear shell theory.