## 3 Results  
The results proposed in this section are given in order to identify the optimum for combined values of the \(M\) number of mathematical layers and the \(N\) order of expansion for the exponential matrix. The introduction of the mathematical layers is necessary to obtain constant coefficients in the equilibrium equations by approximating the FGM properties and the curvature terms for shells. Four different geometries are proposed. The first case considers a square plate, and geometry is shown in Figure 3. The in-plane dimensions are \(a=b=1\) m. Two different thickness ratios have been taken into account: \(a/h=20\) for the thinner plate and \(a/h=4\) for the thicker plate. Figure 4 shows the second geometry for the cylinder. The radius of curvature is infinite in \(\beta\)-direction (\(R_{\beta}=\infty\)) and it is finite in \(\alpha\) direction (\(R_{a}=10\) m). The \(\alpha\) dimension of its middle surface is equal to \(a=2\pi R_{a}\), while the dimension in \(\beta\) direction is \(b=20\) m. The considered thickness ratios are \(R_{\alpha}/h=10\) and \(R_{\alpha}/h=4\). Both the plate and the cylinder consider the same material configuration, a single FGM layer with a metallic and a ceramic phase. The Young modulus changes across the thickness with the following law:  
\[E(z)=E_{m}+(E_{c}-E_{m})V_{c}. \tag{40}\]  
\(E_{m}\) is the Young modulus for the metallic phase and \(E_{c}\) is the Young modulus for the ceramic phase. \(E_{m}=70\) GPa and \(E_{c}=380\) GPa are considered. \(V_{c}\) is the volume fraction of the ceramic phase, and it is defined as:  
\[V_{c}=\left(\frac{2z+h}{2h}\right)^{x}\quad\text{where}\quad-\frac{h}{2}\leq z \leq\frac{h}{2}\,. \tag{41}\]\(V_{c}\) is zero at the bottom of the structure (\(z=-\frac{h}{2}\)), which means fully metallic, and it is equal to 1 at the top of the structure (\(z=\frac{h}{2}\)), which means fully ceramic. \(\kappa\) is a parameter, which changes the volume fraction law across the thickness. It may be equal to 0.5, 1 and 2. In the first benchmark, \(\kappa=0.5\) is considered; in the second benchmark, a linear evolution of \(V_{c}\) is chosen with \(\kappa=1\). Figure 5 shows the evolution of the elastic properties through the thickness and the influence of the parameter \(\kappa\). The Poisson ratio is constant through the thickness (\(\
u_{m}=\
u_{c}=0.3\)). This feature makes the Young modulus be the only quantity that changes through the thickness. The third geometry is presented in Figure 6 and it is a cylindrical shell panel. The only finite radius of curvature is the one in \(\alpha\) direction, and it is equal to \(R_{\alpha}=10\) m. The following in-plane dimensions are taken into account: \(a=\frac{\pi}{3}R_{\alpha}\) and \(b=20\) m. Figure 7 introduces the fourth and last geometry. A spherical shell panel is considered, having the same radii of curvature in both directions (\(R_{\alpha}=R_{\beta}=10\) m). In addition, the in-plane dimensions are the same in the two directions and they are expressed in terms of \(R_{\alpha}\) as \(a=b=\frac{\pi}{3}R_{\alpha}\). Both the third and the fourth cases consider a sandwich configuration, with bottom and top isotropic skins and an FGM core. The thickness of the bottom skin is \(h_{1}=0.15h\), and it is full metallic with Young modulus \(E_{m}=70\) GPa. The top skin is full ceramic with the Young modulus \(E_{c}=200\) GPa. The top skin has the same thickness of the bottom skin (\(l_{3}=0.15h\)). The FGM core has thickness \(h_{2}=0.7h\); the Young modulus of this layer changes according to the same law presented in Equation (40). It changes from a fully metallic phase on the bottom to a fully ceramic phase on the top. As the mechanical properties only change in the range \(-0.7\frac{h}{2}\leq z\leq 0.7\frac{h}{2}\), the thickness of the core (\(h_{2}=0.7h\)) must replace the total thickness \(h\) in Equation (41). The parameter \(\kappa\) has been chosen equal to 2 in the case of the cylindrical shell and equal to 1 for the spherical panel.