# 3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions
## 4 Results
### New Benchmarks
\(p=0.5\) is the coefficient for the volume fraction law across the FGM core, which defines how the elastic and hygrometric properties evolve in the thickness direction. The elastic and hygroscopic properties of both the phases already introduced in the assessments also apply here. The radii of curvature are coherent with those proposed in the previous benchmark, \(R_{\alpha}=10\) m and \(R_{\beta}=\infty\). The dimension of the reference mid-surface in \(\alpha\) direction is a function of the radius of curvature \(R_{\alpha}\) and equals \(a=\frac{\pi}{3}R_{\alpha}\); the dimension in the remaining direction \(\beta\) is fixed and equals \(b=30\). \(m=2\) and \(n=0\) have been chosen as half-wave numbers for the harmonic form of the moisture content imposed at the bottom and the top of the shell. The amplitude of the external fields discussed so far are as follows: the moisture content amplitude is \(\mathcal{M}_{t}=1.0\%\) on the top and \(\mathcal{M}_{b}=0.0\%\) on the bottom. This third case study also considers different thickness ratios to evaluate the effects of this parameter; as in the previous case, it is expressed with respect to \(R_{\alpha}\) and ranges from 2 to 100. The volume fraction \(V_{c}\) of the ceramic phase runs from 0 to 1 inside the core; it equals 0 inside the bottom skin as it is fully metallic, 1 inside the top coat as it is fully ceramic. This is visible in Figure 7a,b, showing the volume fraction and the shear modulus along with the thickness coordinate z; the shear modulus of the top skin coincides with that of the ceramic; the shear modulus of the bottom skin coincides with that of the metal. The amplitudes of some variables are reported for all the thickness ratios and at different thickness coordinates in Table 7; Figure 8 explores six variables and shows their trend through \(z\). The Figures rely on the 3D calculated moisture content profile; the table also reports the results obtained through \(\mathcal{M}_{a}\) and \(\mathcal{M}_{c},1D\). The differences between the three models can be already seen at the moisture content level and directly \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{R_{\alpha}/\hbar}\) & \(\mathbf{2}\) & \(\mathbf{4}\) & \(\mathbf{10}\) & \(\mathbf{50}\) & \(\mathbf{100}\) \\ \hline \multicolumn{6}{c}{\(\mathcal{M}\) at \((\alpha=a/2,\beta=b/2;z=h/2)[-]\)} \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 6: Second benchmark: closed cylinder with a single FGM layer subject to an external moisture content applied to the top and bottom surfaces. The results of all the 3D models consider \(N=3\) and \(NL=300\). reflect the mechanical quantities. Figure 7c,d compares the three moisture content profiles for a thick and a thin cylindrical shell panel and confirms that the differences are sharp not only at a specific thickness coordinate, but throughout all the thickness. The 3D(\(\mathcal{M}_{c},1D\)) model results get closer to those of the 3D(\(\mathcal{M}_{c},3D\)) model as the thickness decreases, and this is clear from the results of Table 7. Figure 8 gives the profile of the tree displacement components, two stresses, and a strain. As in the previous cases, all the quantities evolve with continuity: the mechanical properties are introduced into the model with continuity. The transverse stress \(\sigma_{zz}\) satisfies the external mechanical boundary conditions: it equals 0 at both the top and the bottom surfaces as no external load acts on them. Figure 7: Third benchmark: cylindrical sandwich shell panel featuring an FGM (\(p=0.5\)) core with an imposed moisture content on the top and bottom surfaces. The figures show the volume fraction of the ceramic phase, the shear modulus, and the moisture content profiles for a thick and a thin structure through their thickness. (**a**) Volume fraction of the ceramic phase \(V_{c}\). (**b**) Shear modulus \(\mu\). (**c**) Moisture content profile of the \(R_{\mathrm{g}}/h=2\) shell. (**d**) Moisture content profile of the \(R_{\mathrm{g}}/h=100\) shell.