Finite Element Modeling of Piezoelectric Actuators and Sensors: Local Analysis of the Ferroelectric and Ferroelastic Eff
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Finite element modeling of piezoelectric actuators and sensors: Local analysis of the ferroelectric and ferroelastic effects M. Elhadrouz, T. Ben Zineb and E. Patoor LPMM, UMR CNRS 7554, ENSAM Metz 4, rue Augustin Fresnel 57078 Metz cedex 03 France [email protected] ABSTRACT A new electromechanical finite element is implemented in the Abaqus F.E. code. It takes into account the ferroelectric and ferroelastic behavior. It is represented by a developed phenomenological constitutive law using two internal variables (remnant ferroelastic strain and remnant polarization) and two loading surfaces (electrical and mechanical ones). Saturation phenomena and depolarization under compression mechanical loading are taken into account. In order to lead to a more accurate description of the domain switching mechanisms, a micromechanical single-crystal ferroelectric and ferroelastic constitutive law was also developed. The self consistent scale transition technique will be adopted to derive the polycristalline behavior. The results that were obtained by the micromechanical and phenomenological approaches will be compared in order to identify material parameters that will be introduced in the F.E. modeling. Keywords: Piezoelectricity, Ferroelectric, Ferroelastic, Finite Element, Electromechanical actuators and sensors INTRODUCTION With the development of finite element codes, specific elements integrating an electric degree of freedom are programmed [1, 2]. They take into account the electromechanical coupling only in the linear behavior area. Since the 1980s, research tasks have been carried out in order to describe and to understand the nonlinear part of the electromechanical behavior better [3-6]. These developments can be exploited in order to optimize piezoelectric application performances. When the material is requested to an overall loading below the coercive stress and coercive electric field, there are localised areas (electrode points, areas with high geometrical variation) where concentrations of electric field or stress induce a domain switching. The behavior becomes ferroelectric. The piezoelectric finite elements, available in the widely known F.E. codes, do not take into account such a behavior. In order to contribute to the development of numerical design tools integrating the ferroelectric effect, an adapted finite element is proposed. It is implemented in the F.E. Abaqus code via UEL subroutine. It is based on a developed phenomenological ferroelectric constitutive law derived from a thermodynamic formulation and presents two internal variables. The first corresponds to the ferroelastic strain and the second to the remnant polarization. Two loading surfaces, electrical and mechanical, make it possible to manage the 90° and 180° domain switching. This constitutive law, even if it is well adapted to the implementation in F.E. codes, allows only an averaged and simplified description of switching domains. A micromechanical model is proposed in order to lead to a better description of this mechanis
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