Modelization and numerical approximation of piezoelectric thin shells: Part I: The continuous problems Original Research Article

This paper comprises three parts mainly directed to the obtention of a two-dimensional formulation (Part I), to the analysis of an approximation by finite element methods and to some numerical experiments (Part II) and to the use of piezoelectric components in order to realize active structures (Part III). In this first part, the general three-dimensional equations of piezoelectricity are recalled; they use a representation of the three-dimensional body by a system of three curvilinear coordinates. An existence and uniqueness result is proved. Next, under appropriate assumptions on the mechanical and on the electrical behaviour of the shell during the deformation, the integration of the three-dimensional equations through the thickness leads to a set of two-dimensional equations which are themselves simplified by using an energy criterion. Finally, it is proved that these reduced two-dimensional equations have one and only one solution