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

Abstract

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.

A geometrically and materially non‐linear piezoelectric three‐dimensional‐beam finite element formulation including warping effects

“…Accordingly, Equations (20) can be expressed alternatively as With respect to small strains, the Green-Lagrangian strains are taken into account and are derived using the tangent vectors given in Equation (19), see also [37],…”

Hybrid finite element formulation for electrostrictive materials: static and buckling analysis of beam

“…Finite element method is a potential tool for solving such complex problems for more general types of loading. Linear finite element methods for piezoelectric materials are relatively mature [6,7,8,9] whereas finite element formulations for electrostrictive materials are relatively new and there are only very few research papers are available on finite element formulation of electrostrictive materials [10,11,13,15,16,17]. Winzer et all [10] approximated the electrostrictive behavior in multilayered actuators with finite element analysis.…”

“…Jocelyne Coutte et all [16] have extended the Jean -Claude Debus et all static finite element formulation to a dynamic finite element formulation. Pablo et all [17] have presented a plate variational formulation for active vibration control using current driven and voltage driven electrostrictive actuators. In their current driven actuator formulation, electric displacements in the transverse direction need to be known prior in each time step which is not practical because usually voltages are the inputs and are known prior.…”

“…A simple technique for obtaining the initial guess of the solution for Newton -Raphson technique is also proposed which gives faster convergence of the solution. Since the polarization is an explicit independent variable in the present formulation, the assumption, made in most of the finite element formulations [15,16,17,18], that polarization is approximately equal to electric displacement has been relaxed. The developed static finite element formulation has been extended to solve buckling problems of electrostrictive beams.…”

Modeling of Plate Structures Equipped with Current Driven Electrostrictive Actuators for Active Vibration Control

“…The use of plate finite elements to simulate the behavior of such structures thus seems to be more suitable. As no plate electrostrictive finite element, with direct a priori plate assumptions, has been up to now presented (to our knowledge), we have presented in a previous paper (Pablo et al, 2001a) the elaboration of such a finite element for electrostrictive patches used as actuators.…”