Three-dimensional analytical solution for functionally graded magneto–electro-elastic circular plates subjected to uniform load

The problem of a functionally graded, transversely isotropic, magneto–electro-elastic circular plate acted on by a uniform load is considered. The displacements and electric potential are represented by appropriate polynomials in the radial coordinate, of which the coefficients depends on the thickness coordinate, and are called the generalized displacement functions. The governing equations as well as the boundary conditions for these generalized displacement functions are derived from the original equations of equilibrium for axisymmetric problems and the boundary conditions on the upper and lower surfaces of the plate. Explicit expressions are then obtained through a step-by-step integration scheme, with five integral constants determinable from the boundary conditions at the cylindrical surface in the Saint Venant’s sense. The analytical solution is suited to arbitrary variations of material properties along the thickness direction, and can be readily degenerated into those for homogeneous plates. A particular circular plate, with some material constants being the exponential functions of the thickness coordinate, is finally considered for illustration.