# 3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions
## 3 3D Exact Shell Model for Hygro-Elastic Stress Analysis  
The 3D equilibrium equations for shells are written in the orthogonal mixed curvilinear coordinate system \((\alpha,\beta,z)\) shown in Figures 1 and 2. These equations are modified using 3D constitutive equations for Functionally Graded Materials (FGMs) and general geometrical relations for shells in \((\alpha,\beta,z)\) coordinates. Therefore, the system includes three differential equations of second order in \(z\), and the related coefficients are not constant because of the radii of curvature and elastic coefficients for FGMs. A reasonable number of mathematical layers is introduced to obtain constant-coefficient equations; redoubling the number of variables allows reducing the order of the differential equations. Simply-supported sides and harmonic variables allow the analytical calculation of the partial derivatives in \(\alpha\) and \(\beta\) directions. The final system, including the moisture content profile, has only first-order partial derivatives in \(z\), and the exponential matrix method allows determining both general and particular solutions. The investigated multilayered shells and plates subjected to a moisture content \(\mathcal{M}(\alpha,\beta,z)\) at the external surfaces have \(k\) classical/composite layers and/or functionally graded material layers. Stains defined in an orthogonal mixed curvilinear reference system (\(\alpha\),\(\beta\),\(z\)) are the algebraic summation of mechanical elastic parts (subscript \(m\)) and hygroscopic parts (subscript \(\mathcal{M}\)). The \(6\times 1\) vector (\(\epsilon_{\alpha a}^{k}\), \(\epsilon_{\beta\beta}^{k}\), \(\epsilon_{zz}^{k}\), \(\gamma_{\beta z}^{k}\), \(\gamma_{\alpha z}^{k}\), \(\gamma_{\alpha\beta}^{k}\)) for each \(k\) layer is defined as  
\[\epsilon_{\alpha a}^{k} =\epsilon_{\alpha am}^{k}-\epsilon_{\alpha a,\mathcal{M}}^{k}= \frac{1}{H_{\alpha}(z)}\frac{\partial u^{k}}{\partial\alpha}+\frac{w^{k}}{H_{ \alpha}(z)R_{\alpha}}-\eta_{\alpha}^{k}(z)\mathcal{M}^{k}\, \tag{25}\] \[\epsilon_{\beta\beta}^{k} =\epsilon_{\beta\beta m}^{k}-\epsilon_{\beta\beta M}^{k}=\frac{1 }{H_{\beta}(z)}\frac{\partial w^{k}}{\partial\beta}+\frac{w^{k}}{H_{\beta}(z) R_{\beta}}-\eta_{\beta}^{k}(z)\mathcal{M}^{k}\,\] (26) \[\epsilon_{zz}^{k} =\epsilon_{zzm}^{k}-\epsilon_{zz\mathcal{M}}^{k}=\frac{\partial w ^{k}}{\partial z}-\eta_{z}^{k}(z)\mathcal{M}^{k}\,\] (27) \[\gamma_{\beta z}^{k} =\gamma_{\beta zm}^{k}=\frac{1}{H_{\beta}(z)}\frac{\partial w^{k} }{\partial\beta}+\frac{\partial v^{k}}{\partial z}-\frac{v^{k}}{H_{\beta}(z) R_{\beta}}\,\] (28) \[\gamma_{\alpha z}^{k} =\gamma_{\alpha zm}^{k}=\frac{1}{H_{\alpha}(z)}\frac{\partial w ^{k}}{\partial\alpha}+\frac{\partial u^{k}}{\partial z}-\frac{u^{k}}{H_{\alpha }(z)R_{\alpha}}\,\] (29) \[\gamma_{\alpha\beta}^{k} =\gamma_{\alpha\beta m}^{k}=\frac{1}{H_{\alpha}(z)}\frac{\partial v ^{k}}{\partial\alpha}+\frac{1}{H_{\beta}(z)}\frac{\partial u^{k}}{\partial \beta}\, \tag{30}\]  
In the hygro-elastic strains, the three displacement components \(u^{k}\), \(v^{k}\), and \(w^{k}\) and the scalar moisture content \(\mathcal{M}^{k}\) are defined in the reference system (\(\alpha\), \(\beta\), \(z\)). Moisture expansion coefficients \(\eta_{\alpha}^{k}(z)\), \(\eta_{\beta}^{k}(z)\) and \(\eta_{z}^{k}(z)\) in the \(k\) physical layer could depend on the \(z\) coordinate in the case of Functionally Graded Material (FGM) layers; they are defined in the structural reference system (\(\alpha\), \(\beta\), \(z\)) starting from the moisture expansion coefficients \(\eta_{1}^{k}(z)\), \(\eta_{2}^{k}(z)\) and \(\eta_{3}^{k}(z)\) in the material reference system (1, 2, 3). The partial derivatives are defined via the symbol \(\partial\). As anticipated, an essential hypothesis for exact closed-form solutions is the harmonic forms for all the variables: displacement components and moisture content.