## 3 Results
In the thinner case (\(R_{a}/h=10\)), only \(M=10\) mathematical layers are necessary to stabilize the stress value (when it is coupled with a third-order of expansion for the exponential matrix). On the other side, the same order of expansion \(N=3\) needs at least \(M=160\) mathematical layers to obtain the convergence of the displacement \(\tilde{v}\) because a thicker shell is analyzed. Figure 10 confirms the trend shown in Table 2 for the displacement \(\tilde{v}\) of the thicker cylinder; the results move close to the exact value as the number \(M\) mathematical layers is increased. This result is achieved with only a third expansion order for the exponential matrix (\(N=3\)). \[E(z)=E_{m}+(E_{c}-E_{m})V_{c}. \tag{40}\]1 Case 2, one-layered FGM (\(\kappa=1\)) cylinder with thickness ratio \(R_{a}/h=4\). Displacement \(\tilde{v}(a/2,0,0)\) vs. \(N\) order of expansion of the exponential matrix for different \(M\) numbers of global mathematical layers. Reference solution in Brischetto [23] is here indicated as REF. \[E(z)=E_{m}+(E_{c}-E_{m})V_{c}. \tag{40}\]2 Case 1, one-layered FGM (\(\kappa=0.5\)) plate with thickness ratio \(a/h=20\). Stress \(\tilde{\sigma}_{\tilde{p}z}(a/2,0,0)\) vs. \(N\) order of expansion of the exponential matrix for different \(M\) numbers of global mathematical layers. Reference solution in Brischetto [23] is here indicated as REF. The results obtained for the sandwich cylindrical panel are shown in Table 3; the stress \(\sigma_{az}\) for the thinner shell (\(R_{\alpha}/h=10\)) and the displacement \(\bar{u}\) for the thicker geometry (\(R_{\alpha}/h=4\)), are presented. The material property law follows a second-order evolution inside the core, while constant elastic properties through the thickness are considered for both the external skins. This case is interesting as it shows that the impact of different factors follows a behavior, which is a priori unpredictable. In the presented case, the concerned stress reaches the convergence with \(M=160\) mathematical layers, starting from an order of expansion \(N\) for the exponential matrix equal to 3. This result can be easily explained because the geometry and the lamination scheme are harder to be described than those presented in the previous case (cylinder). On the contrary, the displacement \(\bar{u}\), which is a direct output for the proposed solution, converges using only \(M=40\) with the same order of expansion (\(N=3\)), even if a thicker plate is considered. The convergence of the stress \(\sigma_{az}\) with increasing values of M is shown in \[E(z)=E_{m}+(E_{c}-E_{m})V_{c}. \tag{40}\]3  
The results for the last considered geometry are presented in Table 4 where a sandwich spherical shell panel is analyzed. The through-the-thickness law for the elastic properties is simpler because the volume fraction in the FGM core follows a linear evolution with respect to \(z\) (\(\kappa=1\)). The stress \(\sigma_{az}\) is considered for the thicker structure. The second-order expansion combined with \(M=40\) guarantees the convergence. \(N=3\) expansion order allows the number of mathematical layers to be further reduced until \(M=20\). The thicker case (\(R_{\alpha}/h=4\)) also shows that the number of mathematical layers can be strongly reduced when the order of expansion is increased. Passing from \(N=2\) to \(N=3\), the mathematical layers can be reduced from \(M=200\) to \(M=80\).