## 1 Introduction  
Functionally Graded Materials (FGMs) have changing properties across the volume [1]. This change in the properties is induced and described by a certain spatial function. FGMs may be obtained changing the porosity, the microstructure or the composition of the bulk material [2]. A first distinction may be done between the continuous-graded structures and the stepwise-graded structures. In the present paper, continuous-graded structures, using changing composition, are considered. A metallic and a ceramic phase are present and their related volume fractions continuously vary through the thickness direction. In the same way as the volume fraction, the mechanical and thermal properties also vary. FGMs with metallic and ceramic phases are commonly employed in aerospace field as structural components in high-temperature environments [3]. The main advantage given by continuous-graded materials is related to the possibility to achieve specific properties that common materials cannot guarantee. They do not introduce the common layers and related interfaces as in multilayered structures. This feature leads to the absence of stress discontinuities that are often at the base of delamination phenomena in classical multilayered structures. In view of the complexity of this problem, the static and dynamic analyses of structures embedding FGMs were conducted by several authors, which proposed two- (2D) and three-dimensional (3D) models for basic elements such as plates and shells. Both 3D and 2D models always need to be validated in order to confirm their capability in displacement, stress, strain and vibration evaluations. 3D exact solutions are the only tools capable of accurately evaluating the punctual behavior of these elements embedding FGM layers. These solution types are possible only for certain load/boundary conditions and for specific assumptions for the displacement components. A closed form of the equilibrium equations can be solved taking into account how the material properties change in the volume. Square and rectangular plates are the most common geometries in the literature. Several authors proposed exact solutions for the static and dynamic analysis of these elements when functionally graded layers are included. The bending of functionally graded rectangular plates was studied by [PERSON] [4] using the solution found by [PERSON] in [5] for the equilibrium equations of inhomogeneous isotropic materials. An exponential law for the volume fraction of both the phases was considered. Several results for displacements were presented. The power series method was used by [PERSON] and [PERSON] [6] to solve the equilibrium equations for simply supported functionally graded rectangular plates. [PERSON] and [PERSON] [7] studied the 3D static exact solution of functionally graded plates discussing the effects of different thickness ratios on displacements and stresses. [PERSON] and [PERSON] [8] studied the problem of free vibrations of FGM annular plates on elastic foundations considering different kinds of material property laws across the thickness. The equations of motion were solved using a semi-analytical approach based on series solution and on the differential quadrature method. Solutions for sandwich structures with one or more FGM layers were also proposed in the literature. The effects of different functionally graded material laws on the structural response of simply supported rectangular plates were studied by [PERSON] and [PERSON] in [9], the electroelastic coupling was also analyzed. [PERSON] et al. [10] investigated the free vibrations of a simply supported homogeneous rectangular plate coated by a functionally graded skin with an exponential law through the thickness for the elastic modulus and the mass density. Rectangular sandwich plates, with homogeneous skins and FGM core or with FGM skins and homogeneous core, were studied by [PERSON] et al. in [11]. The fundamental frequencies were found using the Ritz method. The solutions found for structures with curvatures are less numerous.