Setting Of Some Criteria for Piezoelectric Materials Stabilizability

Romziath B, Abdou S and Aboubacar M

Published on: 2023-02-14

Abstract

In the present paper we study the stabilization of piezoelectric bodies via an electromagnetic field. To this end, by using Lyapunov’s stability criteria, we have identified 11 types of piezoelectric bodies which can be stabilized by only utilizing an electromagnetic field. Indeed, the feedback controls we design deal with the current density on the boundary of the body, and the current density and the electric charge density in the body. To construct such control laws, we use a priori estimation techniques

Keywords

Piezoelectric materials; Dielectrics; Electromagnetic; Biomechanics

Introduction

Piezoelectricity is an electromechanical interaction. That is, piezoelectric materials are dielectrics that deform under the effect of an electromagnetic field and produce polarisation under the effect of deformations. This last phenomenon is called "direct effect" for a purely historical reason, given its reversible aspect. These piezoelectric phenomena were explained experimentally in 1880 by Jacques and Pierre Curie [6]. Then, various mathematical models have been derived, as in [1], [2], [4], and [9].

These properties are used in industry to exert active control over certain elastic structures. In this direction, by using a distributed system of sensors and actuators spread inside or on the surface of the material, one can attenuate or even stop the vibrations of a structure. They are also used in the control of the shape of propellers, aircraft wings, telescope mirrors, as well as in the control of the fatigue of materials, artificial organs in biomechanics and many other contexts. For more applications, one can consult [1], [8], [12], [14], [15], and [17].

However, most of the utilizations of piezoelectric materials as actuators exclude the magnetic field, assuming that it can be neglected. On the one hand, in general, the displacement vector is handled by using the Kirchhoff, Euler Bernoulli, or Mindlin-Timoshenko linearization adapted to small displacements; see [1], [7], [11], and [16]. On the other hand, the electromagnetic field variables are handled through the four Maxwell equations; the latter equations including the mechanical variables in there constitutive laws. In order to see particular cases where the magnetic field should not be neglected, refer to [10] where it is highlighted that in a context of high frequency of electromagnetic vibrations, the effect of that field might be important for the mechanical behaviour of a piezo-electric body. One can also find in [13] detailed criteria for quasi-static approximation which leads to neglecting the magnetic potential in piezoelectric models. In the present paper we consider the deformations of piezoelectric bodies taking into account the whole electromagnetic field. In 2003, Raoul et al [9] published the first paper that presented a deformation model of a 3D piezoelectric body taking into account the magnetic field, and then derived a 2D corresponding model. Here, we exploit that 3D model to control the de-formations of a piezoelectric body using the magnetic field. In addition to this introduction, the paper consists of two sections. Section 2 is devoted to the notations and the presentation of the stabilization problem. Section 3 gives the intermediate and final results of the Stabilizability of the displacements of 11 types of piezoelectric bodies through electromagnetic actions.

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