A high order model for piezoelectric rods: An asymptotic approach

This paper is devoted to the study of an electromechanical model for a linear transversely inhomogeneous anisotropic rod made of piezoelectric symmetry class 2 materials. The materials in this class include, among many others, Barium Titanate (BT), Lead Zirconium Titanate (PZT) and Polyvinylidene Fluoride (PVDF). For that purpose, we use asymptotic analysis in order to derive from the 3D piezoelectricity problem a reduced model, without making any a priori assumptions of geometrical or mechanical nature. This process involves considering the diameter of the cross section h as small parameter, the assumption of existence of asymptotic expansions of the unknowns (displacements field and electric potential) and the characterization of their respective leading terms as the unique solutions of limit reduced coupled equations. Finally, the validity of the limit reduced model is supported by providing rigorous convergence results.