1142 IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 5, MAY 2011 Finite Element Harmonic Modeling of Magnetoelectric Effect Thu Trang Nguyen, Xavier Mininger, Frédéric Bouillault, and Laurent Daniel Laboratoire de Génie Electrique de Paris, CNRS UMR8507, SUPELEC, UPMC Univ Paris 6, Univ Paris-Sud, Gif-sur-Yvette, France The magnetoelectric (ME) effect in composite materials results from the combination of the magnetostrictive effect and the piezoelec- tric effect via elastic interaction. This work focuses on the modeling of multilayer structures under dynamic excitation. The calculated ME coefficient versus frequency shows the enhancement of the ME effect at mechanical resonance in accordance with experimental measurements. The impact of electric conductivity is investigated. Applications on the ME sensor and tunable inductor are proposed. Index Terms—Finite element formulation, frequency effect, magnetoelectric effect, magnetostriction, piezoelectricity . I. INTRODUCTION R ESEARCH on magnetoelectric materials has increased rapidly in recent years due to many applications of such materials as magnetic sensors, memory devices, variable induc- tances [1]. The magnetoelectric phenomenon consists in the ex- istence of a magnetization induced by an electric polarization, or conversely an electric polarization induced by a magnetiza- tion. Such a coupled property is characterized by ME coeffi- cients. ME coefficients are larger in composite materials than in homogeneous materials [2]. Moreover, the enhancement of the ME coefficients at mechanical resonance frequencies [3] is useful for many smart devices. Therefore, the design and opti- mization for ME devices require accurate and compact numer- ical modeling. Up to now, there are several numerical models: the frequency effect is integrated in the model of Liu et al. [4] but only from the mechanical point of view. The model of Galopin et al. [5] takes into account the nonlinear behavior of the magnetostric- tive phase but only under quasi-static loadings. The purpose of this work is to build a model based on the finite element method and accounting for the nonlinearity of magnetostriction and for the frequency effect (leading to eddy currents). In a first part, the formulation based on a thermody- namical approach is introduced. The model is then applied to a magnetoelectric sensor and a comparison to experimental data on a tunable inductor is proposed. II. EQUILIBRIUM EQUATIONS A. Mechanical Equilibrium The mechanical equilibrium is given by (1) where is the stress tensor, the driving force, the displace- ment and the mass density. Manuscript received May 31, 2010; accepted September 13, 2010. Date of current version April 22, 2011. Corresponding author: T. T. Nguyen (e-mail: thutrang.nguyen@supelec.fr). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2010.2081356 B. Electromagnetic Equations The electromagnetic equations are given by (2) (3) where is the magnetic field, the current density, the elec- tric induction, the charge density . III. CONSTITUTIVE LAWS ME composite materials often consist in multilayers of piezo- electric (pz) and magnetostrictive (ms) materials. A. Electroelastic Behavior Considering that piezoelectric materials are usually prepolar- ized, the constitutive law is assumed to be linear around the po- larization point (4) where is the strain tensor, the electric field, the stiffness tensor at constant electric field, the electric permittivity at constant strain, and the piezoelectric coefficients. We note the small variation of around a polarization point (5) B. Magnetostrictive Behavior 1) General Form: Unlike the piezoelectric material, the mag- netostrictive material is not prepolarized. Its constitutive law is strongly nonlinear and has to be accurately analyzed. The total strain is divided into the elastic strain and the magnetostriction strain , [6]. According to Hooke’s law, the total stress is expressed by (6) where is the usual stiffness tensor of the magnetostrictive material under static loading. In the case of an isotropic material, (6) can be written using Lamé coefficients and (7) is the Kronecker symbol ( if , 0 if ). We assume that the magnetostriction phenomenon is iso- choric and isotropic, and that the magnetostriction strain can 0018-9464/$26.00 © 2011 IEEE