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CivilComp Proceedings
ISSN 17593433 CCP: 93
PROCEEDINGS OF THE TENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru and M.L. Romero
Paper 73
Analytical Solution for the Transient Response of Symmetric Magnetoelectric Laminated Beams C. Orlando, A. Milazzo and A. Alaimo
Department of Structural, Aerospace and Geotechnical Engineering, University of Palermo, Italy C. Orlando, A. Milazzo, A. Alaimo, "Analytical Solution for the Transient Response of Symmetric Magnetoelectric Laminated Beams", in B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru, M.L. Romero, (Editors), "Proceedings of the Tenth International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 73, 2010. doi:10.4203/ccp.93.73
Keywords: analytical solution, magnetoelectric composites, forced vibration, laminated beam.
Summary
In this paper an analytical model for magnetoelectric laminated beams is presented to study the response to mechanical and electromagnetic 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 magnetoelectric behavior can be considered quasistatic with respect to the mechanical one, since electromagnetic 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 meanline transverse displacement and the cross sectional rotation functions.
The constitutive equations for each layer are written under the hypothesis of monoaxial stress state and by neglecting the electric and magnetic fields components transverse to the thickness direction. First of all the Gauss' laws for electrostatic and magnetostatic are integrated in the thickness direction, highlighting that the electric and magnetic potentials depend on the first and second derivative of the crosssectional rotation and of the meanline transverse displacement, respectively. The electromagnetic continuity condition at the interface between adjacent layers and electromagnetic 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 magnetoelectric inputs have to be treated as timevarying 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. References
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