# 3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions
## 3 3D Exact Shell Model for Hygro-Elastic Stress Analysis
While the latter has already been discussed, for the displacement components it holds that  
\[u^{k}(\alpha,\beta,z) =U^{k}(z)cos(\bar{\alpha}\alpha)sin(\bar{\beta}\beta)\, \tag{31}\] \[v^{k}(\alpha,\beta,z) =V^{k}(z)sin(\bar{\alpha}\alpha)cos(\bar{\beta}\beta)\,\] (32) \[w^{k}(\alpha,\beta,z) =W^{k}(z)sin(\bar{\alpha}\alpha)sin(\bar{\beta}\beta)\, \tag{33}\]  
displacement amplitudes are indicated as \((U^{k}(z),V^{k}(z),W^{k}(z))\). As already described, \(m\) and \(n\) are the half-wave numbers, \(a\) and \(b\) the mid-surface dimensions, and \(\bar{\alpha}=\frac{m\pi}{a}\) and \(\bar{\beta}=\frac{n\pi}{b}\). The three-dimensional equilibrium equations for spherical shells having constant radii of curvature \(R_{\alpha}=R_{\beta}\) and a total number \(N_{L}\) of physical \(k\) (either classical or FGM) layers are  
\[H_{\beta}(z)\frac{\partial\sigma_{\alpha\alpha}^{k}}{\partial \alpha}+H_{\alpha}(z)\frac{\partial\sigma_{\alpha\beta}^{k}}{\partial\beta}+H_ {\alpha}(z)H_{\beta}(z)\frac{\partial\sigma_{\alpha z}^{k}}{\partial z}+( \frac{2H_{\beta}(z)}{R_{\alpha}}+\frac{H_{\alpha}(z)}{R_{\beta}})\sigma_{ \alpha zz}^{k}=0, \tag{34}\] \[\epsilon_{\beta\beta}^{k} =\epsilon_{\beta\beta m}^{k}-\epsilon_{\beta\beta M}^{k}=\frac{1 }{H_{\beta}(z)}\frac{\partial w^{k}}{\partial\beta}+\frac{w^{k}}{H_{\beta}(z) R_{\beta}}-\eta_{\beta}^{k}(z)\mathcal{M}^{k}\,\]0 (35) \[\epsilon_{\beta\beta}^{k} =\epsilon_{\beta\beta m}^{k}-\epsilon_{\beta\beta M}^{k}=\frac{1 }{H_{\beta}(z)}\frac{\partial w^{k}}{\partial\beta}+\frac{w^{k}}{H_{\beta}(z) R_{\beta}}-\eta_{\beta}^{k}(z)\mathcal{M}^{k}\,\]1  
Three-dimensional equilibrium equations for cylinders/cylindrical panels and plates are obtained from Equations (34)-(36) when one of the two radii of curvature is infinite and when both the radii of curvature are infinite, respectively. The constitutive equations are developed by considering the algebraic summation of mechanical elastic strains and hygroscopic strains:  
\[\epsilon_{\beta\beta}^{k} =\epsilon_{\beta\beta m}^{k}-\epsilon_{\beta\beta M}^{k}=\frac{1 }{H_{\beta}(z)}\frac{\partial w^{k}}{\partial\beta}+\frac{w^{k}}{H_{\beta}(z) R_{\beta}}-\eta_{\beta}^{k}(z)\mathcal{M}^{k}\,\]2  
\(\mathbf{\sigma}^{k}\) is the stress vector having a \(6\times 1\) dimension, \(\mathbf{C}^{k}(z)\)is the \(6\times 6\) elastic coefficient matrix and it could depend on the \(z\) coordinate in the case of a \(k\)th FGM layer. The strains are included in the constitutive equations by using the form shown in Equations (25)-(30). Closed-form solutions are possible when orthotropic angles are \(0^{\circ}\) or \(90^{\circ}\) to obtain \(C^{k}_{16}(z)=C^{k}_{26}(z)=C^{k}_{36}(z)=C^{k}_{45}(z)=0\) in the structural reference system (\(\alpha\), \(\beta\), \(z\)). Therefore,  
\[\mathbf{C}^{k}(z)= \begin{bmatrix}C^{k}_{11}(z)&C^{k}_{12}(z)&C^{k}_{13}(z)&0&0&0\\ C^{k}_{12}(z)&C^{k}_{22}(z)&C^{k}_{23}(z)&0&0&0\\ C^{k}_{13}(z)&C^{k}_{23}(z)&C^{k}_{33}(z)&0&0&0\\ 0&0&0&C^{k}_{44}(z)&0&0\\ 0&0&0&0&C^{k}_{55}(z)&0\\ 0&0&0&0&0&C^{k}_{66}(z)\end{bmatrix} \;.