# 3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions
## 4 Results
### New Benchmarks
The results of all the following benchmarks consider \(N=3\) and \(M=300\) as an a priori prerequisite for results accuracy. The first benchmark studies a square plate with a single FGM layer and simply-supported sides. The plate has \(a=b=10\) m as in-plane dimensions but comes with several and different thicknesses, which allow the effect of this geometrical parameter to be evaluated. In fact, the thickness ratio goes from \(a/h=2\) to \(a/h=100\), thus ranging from very thick to very thin plates. In this benchmark, the volume fraction \(V_{c}\) of the ceramic phase evolves linearly: \(p=1\) is set in the material law defined through Equation (83). This relation also implies a fully ceramic top surface and a fully metallic bottom one. The moisture content is imposed on the top and the bottom external surfaces; it has a harmonic form on both with amplitudes \(\mathcal{M}_{t}=1.0\%\) and \(\mathcal{M}_{b}=0.0\%\), on top and bottom, respectively. The harmonic form of the moisture content has \(m=n=1\) as half-wave numbers in \(\alpha\) and \(\beta\) directions, respectively. The elastic and hygroscopic properties of the metallic and ceramic phases are the same as those introduced at the beginning of this section. The FGM nature of the layer makes its mechanical properties evolve through the thickness direction; Figure 3a,b, respectively, shows how the volume fraction \(V_{c}\) and the bulk modulus \(K\) evolve with respect to non-dimensional thickness coordinate \(\tilde{z}/h\). Note that \(K\) is not linear as \(V_{c}\) due to Equation (85). Table 5 and Figures 4 report an extract of the main results. The results in tabular form give the amplitude of some variables of the problem; they reflect the three different ways of evaluating the moisture content. 3D(\(\mathcal{M}_{a}\)) implies the assumed linear moisture content profile; 3D(\(\mathcal{M}_{c},1D\)) relies on the 1D version of the Fick moisture diffusion equation; finally, 3D(\(\mathcal{M}_{c},3D\)) relies on a 3D solution of the moisture diffusion problem. The prefix 3D underlines that the elastic part of the solution is three-dimensional in all three models. Such an analysis allows grasping the differences between the three approaches. The 3D(\(\mathcal{M}_{c},3D\)) shell model considers both the mechanical/hygrometric properties evolution through z and the three-dimensional nature of the problem. As discussed in the previous section, it always delivers an accurate result. Table 5 underlines that the 3D(\(\mathcal{M}_{c},1D\)) model results get closer to those of the 3D(\(\mathcal{M}_{c},3D\)) model as the thickness decreases. It considers how the mechanical/hygrometric properties evolve through z but disregards the moisture diffusion through alpha and beta direction, which have a negligible weight in thin structures. The results of the 3D(\(\mathcal{M}_{a}\)) model are always unreliable, as they are built on a moisture content profile that is far from the actual scenario in a layer embedding an FGM. Figure 3c,d further facilitates understanding these concepts; it compares the three moisture content profiles for a thick and a thin plate. In thick structures, the difference between the three profiles is very pronounced: the three-dimensionality of the problem and the mechanical/hygrometric properties variability in the thickness direction make the 3D profile differ from the linear assumption. Even the 1D profile differs from the linearity: the hygrometric properties vary through z, reflecting on the moisture content at different thickness coordinates. These concepts also apply to thin structures; however, the three-dimensionality of the problem is insignificant, and the 3D and 1D profiles coincide. Figure 4 shows the complete profile of the tree displacement components, two stresses, and a strain. Note that all the quantities evolve with continuity, which is essential as it demonstrates both the graded elastic/hygrometric properties and the correct introduction of the continuity conditions.