# 3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions
## 4 Results
### New Benchmarks
The transverse stress \(\sigma_{zz}\) and the transverse shear strain \(\gamma_{\beta z}\) satisfy the external mechanical boundary conditions: it equals 0 at both the top and the bottom surfaces as no external load acts on them. Figure 3: First benchmark: one-layered FGM (\(p=1\)) square plate with an imposed moisture content on the top and bottom surfaces. The figures show the volume fraction of the ceramic phase, the bulk 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**) Bulk modulus K. (**c**) Moisture content profile of the \(a/h=2\) plate. (**d**) Moisture content profile of the \(a/h=100\) plate. Figure 4: First benchmark: one-layered FGM square plate with an imposed moisture content on the top and bottom surfaces. The results are calculated for a moderately thick (\(a/h=4\)) structure via the 3D(\(\mathcal{M}_{c},3D\)) model. **(a)** Amplitude of \(u\) displacement component. **(b)** Amplitude of \(v\) displacement component. **(c)** Amplitude of \(w\) displacement component. **(d)** Amplitude of \(\sigma_{zz}\) stress component. **(e)** Amplitude of \(\sigma_{aa}\) stress component. **(f)** Amplitude of \(\gamma_{\beta z}\) strain component. The second benchmark focuses on a closed cylinder, featuring a single FGM layer and simply-supported sides. The dimensions of the reference mid-surface, \(a=2\pi R_{\rm{a}}\) and \(b=30\) m, are a function of the radii of curvature of the shell, one of which is infinite: \(R_{\rm{a}}=10\) m and \(R_{\beta}=\infty\). Different thicknesses have been considered; the thickness ratio \(R_{\rm{a}}/h\) is expressed with respect to \(R_{\rm{a}}\) and ranges from 2 to 100 also in this second case study. The material volume fraction of the ceramic phase is a quadratic function of the thickness coordinate; the material law defined through Equation (83) consider \(p=2\). Given that a single layer is considered, the cylinder is metallic in the inner surface and ceramic in the outer. The moisture content is imposed on the outer external surface, \(\mathcal{M}_{t}=1.0\%\), and on the inner one, \(\mathcal{M}_{b}=0.0\%\). The half wave numbers of both the harmonic forms are the same, \(m=2\) and \(n=1\). The elastic and hygroscopic properties of both the phases introduced previously also apply here. Figure 5a,b, respectively, shows the volume fraction \(V_{c}\) and moisture diffusion coefficient \(\mathcal{D}\) vs. the non-dimensional thickness coordinate \(\hat{z}/h\). \(V_{c}\) follows a power-law of order \(p=2\), \(\mathcal{D}\) follows Equation (85). Table 6 and Figure 6 summarize an extract of the main results. This second benchmark also reports three different sets of results: the elastic model is the same (prefix 3D), but the moisture content profile follows the different approaches. This leads to models 3D(\(\mathcal{M}_{a}\)), in which the moisture content is a priori assumed, and 3D(\(\mathcal{M}_{c},1D\))-3D(\(\mathcal{M}_{c},3D\)), in which the profile is calculated following a monodimensional or three-dimensional approach. This analysis allows highlighting the distinctions between the three methods. The last one is the only model in which no assumptions are made concerning the three-dimensionality of the problem as the moisture content amplitude derives from [PERSON]'s law of diffusion. The results coming from 3D(\(\mathcal{M}_{a}\)) are always wrong because the moisture content evaluation is inaccurate. The differences between 3D(\(\mathcal{M}_{c},1D\)) and 3D(\(\mathcal{M}_{c},3D\)) are less pronounced if compared with the previous benchmark and decrease with the thickness.