Keywords: analytical solution, magnetoelectric composites, forced vibration, laminated beam.

In this paper an analytical model for magnetoelectric laminated beams is presented to study the response to mechanical and electro-magnetic time varying loads. The model is based on the Timoshenko's beam theory to take into account the effect of the shear strain in the dynamic behavior of the magnetoelectric beam. By assuming that no density charge and no density current act on the analyzed domain and that the magneto-electric behavior can be considered quasi-static with respect to the mechanical one, since electro-magnetic wave propagation velocity is several orders of magnitude higher than the propagation velocity of elastic waves, the electric and magnetic fields are modeled by means of scalar electric and magnetic potential functions. On the other hand, the kinematical variables involved in the analysis are the beam mean-line transverse displacement and the cross sectional rotation functions.

The constitutive equations for each layer are written under the hypothesis of mono-axial stress state and by neglecting the electric and magnetic fields components transverse to the thickness direction. First of all the Gauss' laws for electro-static and magneto-static are integrated in the thickness direction, highlighting that the electric and magnetic potentials depend on the first and second derivative of the cross-sectional rotation and of the mean-line transverse displacement, respectively. The electro-magnetic continuity condition at the interface between adjacent layers and electro-magnetic boundary conditions on the beam top and bottom surfaces are imposed allowing the computation of the potentials through the thickness distributions in closed form.

To obtain the potential distribution along the length of the beam the mechanical problem must be solved. By virtue of the constitutive relationships and the aforementioned assumptions, the equations of motion are written for an equivalent elastic beam, highlighting the contributions of the piezoelectric, piezomagnetic and magnetoelectric couplings to the beam equivalent bending stiffness. Moreover the model highlights that the magneto-electric inputs have to be treated as time-varying mechanical boundary conditions and as an external equivalent bending moment per unit length. Following reference [1] the equations of motion are then solved by means of the method of separation of variables and by using the approach of Mindlin and Goodman [2], which takes full advantage of the property of orthogonality of the beam modes of vibration.

The model is validated by comparing finite element results with the present ones for a piezoelectric laminated beam undergoing both mechanical and electric loads. Last, the proposed solution is employed to study a magnetoelectric beam used as a strain sensing device analyzing the influence of the stacking sequence on the elastic, electric and magnetic responses.

1

A. Milazzo, C. Orlando, A. Alaimo, "An analytical solution for the magneto-electro-elastic bimorph beam forced vibrations problem", Smart Materials and Structures, 18(8), 2009. doi:10.1088/0964-1726/18/8/085012

2

R.D. Mindlin, L.E. Goodman, "Beam vibrations with time-dependent boundary conditions", ASME Journal of Applied Mechanics, 17, 377-380, 1950.

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