A Plate Electrostrictive Finite Element - Part I: Modeling and Variational Formulations

Piezoelectric plates have been widely used for the vibration reduction and noise control of structures. Due to power forces considerations, electrostrictive patches present a growing interest. It is thus the purpose of the present research to contribute to modeling aspects of thin structures integrating such actuators. Many three-dimensional finite element models have then been elaborated to simulate these structures behavior but no two-dimensional models have been presented up to now. The aim of this article is thus to set out the elaboration of a thin plate electrostrictive finite element for PMN-PT type ceramics used as actuators. This element is developed for dynamic purposes and thus takes into account phenomena induced by applying to the patch a cycling electric field. The finite element formulation is based on electromechanical constitutive equations derived in a previous paper, mechanical and electrical considerations and direct a priori plate assumptions. The electrostrictive finite element is here derived using techniques inspired from a piezoelectric finite element. This method has the particular property of reducing the initial electromechanical problem to a purely mechanical problem based on a modified elastic constitutive law. The electrical unknowns are then explicitly derived from the mechanical displacements. This method considerably simplifies the resolution of the problem since classical finite elements for laminated plates can be used to model the electrostrictive plate with a modified constitutive law. This paper is focused on the plate modeling and corresponding variational formulations.