Magneto-electro-elastic 3D coupling in free vibrations of layered plates

A three-dimensional (3D) analytical formulation is proposed to put together magnetic, electric and elastic fields to analyze the vibration modes of simply-supported layered piezo-electro-magnetic plates. The present 3D model allows analyses for layered smart plates in both open-circuit and closed-circuit configurations. The second-order differential equations written in the mixed curvilinear reference system govern the magneto-electro-elastic free vibration problem for multilayered plates. This set consists of the 3D equations of motion and the 3D divergence equations for the magnetic induction and electric displacement. Navier harmonic forms in the planar directions and the exponential matrix method in the transversal direction of the plate are applied to solve the second-order differential equations in terms of displacements. For these reasons, simply-supported boundary conditions are considered. Imposition of interlaminar continuity conditions on primary variables (displacements, magnetic potential, electric potential), and some secondary variables (transverse normal and transverse shear stresses, transverse normal magnetic induction/electric displacement) allows the implementation of the layer-wise approach. Assessments for both load boundary configurations are proposed in the results section to validate the present 3D approach. 3D electro-elastic and 3D magneto-elastic coupling validations are performed separately considering different models from the open literature. A new benchmark involving a full magneto-electro-elastic coupling for multilayered plates is presented considering both load boundary configurations for different thickness ratios. For this benchmark, circular frequency values and related vibration modes through the transverse direction in terms of displacements, magnetic and electric potential, transverse normal magnetic induction/electric displacement are shown to visualize the magneto-electro-elastic coupling and material and thickness layer effects. The present formulation has been entirely implemented in an academic Matlab (R2024a) code developed by the authors. In this paper, for the first time, the second-order differential equations governing the magneto-electro-elastic problem for the free vibration analysis of plates has been solved considering the mixed mode of harmonic forms and exponential matrix. The exponential matrix permits computing the secondary variable of the problem (stresses, electric displacement components and magnetic induction components) exactly, directly from constitutive and geometrical equations. In addition, the very simple and elegant formulation permits having a code with very low computational costs. The present manuscript aims to fill the void in open literature regarding reference 3D solutions for the free vibration analysis of magneto-electro-elastic plates.