Finite Element Analysis of the non Linear Behavior of a Multilayer Piezoelectric Actuator
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Finite element analysis of the non linear behavior of a multilayer piezoelectric actuator M. Elhadrouz, T. Ben Zineb, and E. Patoor Laboratoire de Physique et Mécanique des Matériaux UMR CNRS 7554 Ecole Nationale Supérieure d’Arts et Métiers – CER de Metz 4 rue Augustin Fresnel Metz Technopôle 2000 57078 Metz Cedex - France ABSTRACT A constitutive law for ferroelectric and ferroelastic piezoceramics is implemented in ABAQUS Standard using the subroutine user element. A linear solid element is defined: it is an eight-node hexahedron having the mechanical displacement components and the electric potential as degrees of freedom for each node. The element is formulated for static analysis and it needs the definition of the contribution of this element to the Jacobian (stiffness) and the definition of an array containing the contributions of this element to the right-hand-side vectors of the overall system of equations The subroutine is called for each element that is of a user-defined element type each time element calculations are required. As an example, the element is used for the simulation of a multilayer actuator made of piezoceramics. In this case, the piezoelectric equations are not valid since the electric loading induces non linear phenomena, which are captured through the constitutive law implemented in the user element.
INTRODUCTION As elementary components of sensors and actuators, ferroelectric ceramics have received more attention in recent years. The applications of piezoelectric materials have expanded widely in electromechanical and microelectromechanical sensors and actuators [1, 2]. Recently, the finite element analysis becomes very attractive for many researchers and is used for modeling piezoelectric sensors and actuators [3, 4]. However, the shortcoming of such developments is the fact that the behavior taken into account is the linear piezoelectric effect. But Ferroelectrics belong to a subgroup of piezoelectrics, when in the form of a single domain, processes a spontaneous polarization that can be reversed by applying a high electric field or a large stress [5]. It means that the response is no longer linear when they are subjected to excessive electric field or stress. In addition, the structural reliability of applications using piezoelectric ceramics is an important concern due to their brittleness and susceptibility to fracture. Stress and electric field concentrations appear to play a key role [6]. Then, the finite element method can be adopted as a powerful tool for analyzing a complicated system.
FINITE ELEMENT ANALYSIS Constitutive model for ferroelectric and ferroelastic piezoceramics [7] Consider a deformable dielectric material not subjected to body force or electric charge. By using, in the Cartesian coordinates xi (i = 1,2,3), the Einstein summation convention over repeated indices and the comma as a symbol of the partial differentiation with respect to the coordinate xi , the mechanical and electric equilibriums require that
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