Keywords: finite element, magnetoelectric composites, laminated beam.

Magneto-Electric composites have been proposed as effective for building smart structures having valuable performance with respect to standard actuation and sensing materials. They show a wide range of potential applications because of the capability of passively converting energy in the elastic, electric and magnetic forms. Laminated magneto-electric structures have emerged as the composite configuration that shows the strongest magneto-electric coupling [1]. As a result of the inherent multi-fields nature, efficient analysis and design tools are required to predict the interaction of the electric, elastic and magnetic fields characterizing the smart behaviour of such composites.

In this paper a new finite element is developed subject to the hypothesis of linear magnetoelectroelasticity for a straight generally layered beam. The governing equations are derived under the assumptions of quasi-static electric and magnetic fields and that the electric and magnetic transverse components are prevalent with respect to the in-plane ones. The electro-magnetic state is first condensed to kinematical quantities only and the mechanical model is then written for a shear deformable Timoshenko beam [2] including the effects of electromagnetic stacking sequence and electromagnetic loads on both the equivalent stiffness properties and the equivalent mechanical boundary conditions. Polynomial shape functions are then selected for the beam mean-line axial and transverse displacements and the cross-sectional rotation, in such a way the approximated kinematical field satisfies the homogeneous form of the problem governing equations. It is found that these shape functions depend upon to parameters which are representative of the staking sequence through the equivalent magneto-electro-elastic stiffness coefficients of the beam. The weak form of the governing equations are then obtained by integrating over the element length the equation of motion of the beam opportunely multiplied by the virtual mean-line axial and transverse displacements and by the virtual cross-sectional rotation. The kinematical quantities are then expressed in terms of virtual and actual nodal variables by means of the proposed shape functions. By so doing, the definitions of the element mass and stiffness matrices and of the equivalent force vector are straightforwardly obtained. It is found that the electro-magnetic boundary conditions are transferred to the discrete finite element representation as work-equivalent nodal forces. The 6x6 stiffness matrix obtained is found to present an inherent axial-bending coupling which vanishes when the stacking sequence is symmetric with respect to the mid-line from the elastic, electric and magnetic points of views. Lastly, numerical results present in the literature, as well as plane stress finite element simulation carried out using commercial available finite element codes, have been used to validate the proposed finite element formulation. Magneto-electric frame structures have also been analysed to show the versatility of the finite element developed.

1

M. Fiebig, "Revival of the magnetoelectric effect", Journal of Physics D: Applied Physics, 38, R123-152, 2005. doi:10.1088/0022-3727/38/8/R01

2

S. Timoshenko, "Vibration problems in engineering", D. Van Nostrand Company, Inc., Princeton, 1955.

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