Classical solutions for coupling analysis of unsymmetric magneto-electro-elastic composite laminated thin plates

For unsymmetric magnetic-electro-elastic (MEE) composite laminated thin plates, the coupling effects occur not only from mechanical, electric and magnetic interaction but also from lamination sequence and anisotropic properties. Stretching and bending deformation as well as electric and magnetic potentials will all be coupled each other. Thus, even a simple rectangular MEE laminated thin plate with simply supported edges under generalized loads, no explicit analytical solutions have been obtained in the literature. In this study to simplify the possible complicated mathematical expressions, the governing partial differential equations are written in matrix form. With simple matrix form expressions, we derive the explicit solutions for two different boundary conditions corresponding to the Navier's solution and Lévy's solution. The former is valid for a rectangular plate with all edges simply supported, and the latter is valid for one with two opposite edges simply supported and no restriction is set on the other two edges. Both explicit solutions are presented for the first time. To avoid the ill-conditioned matrices induced by the wide-ranged MEE material constants and the matrix exponential operation, some remarks on scaling and numerical calculation are provided. Moreover, the matrix form solutions cover all the reduced cases such as elastic and electro-elastic laminates with the difference only in matrix dimensions. For the purpose of verification and engineering application, a newly developed boundary element method is also introduced to compare with the derived classical solutions.