# 3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions
## 3 3D Exact Shell Model for Hygro-Elastic Stress Analysis
\tag{38}\]  
The explicit form of the constitutive equations develops thanks to the introduction of the geometrical Equations (25)-(30) into the constitutive Equation (37):  
\[\sigma^{k}_{\alpha\alpha} =\frac{C^{k}_{11}(z)}{H_{\alpha}(z)}u^{k}_{\mathcal{A}}+\frac{C^ {k}_{11}(z)}{H_{\alpha}(z)R_{\alpha}}w^{k}+\frac{C^{k}_{12}(z)}{H_{\beta}(z)}v ^{k}_{\mathcal{,}\beta}+\frac{C^{k}_{12}(z)}{H_{\beta}(z)R_{\beta}}w^{k}+C^{k }_{13}(z)w^{k}_{\mathcal{,}}-\xi^{k}_{\alpha}(z)\mathcal{M}^{k}\;, \tag{39}\] \[\sigma^{k}_{\beta\beta} =\frac{C^{k}_{12}(z)}{H_{\alpha}(z)}u^{k}_{\mathcal{,}}+\frac{C^ {k}_{12}(z)}{H_{\alpha}(z)R_{\alpha}}w^{k}+\frac{C^{k}_{22}(z)}{H_{\beta}(z)}v ^{k}_{\mathcal{,}\beta}+\frac{C^{k}_{22}(z)}{H_{\beta}(z)R_{\beta}}w^{k}+C^{k }_{23}(z)w^{k}_{\mathcal{,}}-\xi^{k}_{\beta}(z)\mathcal{M}^{k}\;,\] (40) \[\sigma^{k}_{zz} =\frac{C^{k}_{13}(z)}{H_{\alpha}(z)}u^{k}_{\mathcal{,}}+\frac{C^ {k}_{13}(z)}{H_{\alpha}(z)R_{\alpha}}w^{k}+\frac{C^{k}_{23}(z)}{H_{\beta}(z)}v ^{k}_{\mathcal{,}}+\frac{C^{k}_{23}(z)}{H_{\beta}(z)R_{\beta}}w^{k}+C^{k}_{33} (z)w^{k}_{\mathcal{,}}-\xi^{k}_{z}(z)\mathcal{M}^{k}\;,\] (41) \[\sigma^{k}_{\beta z} =\frac{C^{k}_{44}(z)}{H_{\beta}(z)}w^{k}_{\beta}+C^{k}_{44}(z)v^{ k}_{\mathcal{,}}-\frac{C^{k}_{44}(z)}{H_{\beta}(z)R_{\beta}}v^{k}\;,\] (42) \[\sigma^{k}_{\alpha z} =\frac{C^{k}_{55}(z)}{H_{\alpha}(z)}w^{k}_{\mathcal{,}}+C^{k}_{55 }(z)u^{k}_{\mathcal{,}}-\frac{C^{k}_{55}(z)}{H_{\alpha}(z)R_{\alpha}}u^{k}\;,\] (43) \[\sigma^{k}_{\alpha\beta} =\frac{C^{k}_{66}(z)}{H_{\alpha}(z)}v^{k}_{\mathcal{,}}+\frac{C^ {k}_{66}(z)}{H_{\beta}(z)}u^{k}_{\mathcal{,}}\;, \tag{44}\]  
partial derivatives \((\frac{\partial}{\partial\alpha})\), \((\frac{\partial}{\partial\beta})\), and \((\frac{\partial}{\partial z})\) are indicated in Equations (39)-(44) through subscripts (\(\alpha\)), (\(\beta\)), and (\(z\)), respectively. Terms \(\xi^{k}_{\alpha}(z)\), \(\xi^{k}_{\beta}(z)\), and \(\xi^{k}_{z}(z)\) designate the hygomechanical coupling coefficients in the structural reference system (\(\alpha\), \(\beta\), \(z\)), and they could depend on \(z\) in the case of FGM layers:  
\[\xi^{k}_{\alpha}(z) =C^{k}_{11}(z)\eta^{k}_{\alpha}(z)+C^{k}_{12}(z)\eta^{k}_{\beta}(z )+C^{k}_{13}(z)\eta^{k}_{z}(z)\;, \tag{45}\] \[\xi^{k}_{\beta}(z) =C^{k}_{12}(z)\eta^{k}_{\alpha}(z)+C^{k}_{22}(z)\eta^{k}_{\beta}(z )+C^{k}_{23}(z)\eta^{k}_{z}(z)\;,\] (46) \[\xi^{k}_{z}(z) =C^{k}_{13}(z)\eta^{k}_{\alpha}(z)+C^{k}_{23}(z)\eta^{k}_{\beta}(z )+C^{k}_{33}(z)\eta^{k}_{z}(z)\;. \tag{47}\]  
The final system is obtained thanks to the substitution of the harmonic form equations for displacements and moisture content Equations (1)-(31) and the modified constitutive relations Equations (39)-(44) into the three-dimensional equilibrium Equations (34)-(36) developed for spherical shells:  
\[A_{1}^{j}U^{j}+A_{2}^{j}V^{j}+A_{3}^{j}W^{j}+A_{4}^{j}U_{z}^{j}+A_ {5}^{j}W_{z}^{j}+A_{6}^{j}U_{zz}^{j}+L_{1}^{j}M^{j}=0\;, \tag{48}\] \[\sigma^{k}_{\beta\beta} =\frac{C^{k}_{12}(z)}{H_{\alpha}(z)}u^{k}_{\mathcal{,}}+\frac{C^ {k}_{12}(z)}{H_{\alpha}(z)R_{\alpha}}w^{k}+\frac{C^{k}_{22}(z)}{H_{\beta}(z)}v ^{k}_{\mathcal{,}\beta}+\frac{C^{k}_{22}(z)}{H_{\beta}(z)R_{\beta}}w^{k}+C^{k }_{23}(z)w^{k}_{\mathcal{,}}-\xi^{k}_{\beta}(z)\mathcal{M}^{k}\;,\]0 (49) \[\sigma^{k}_{\beta\beta} =\frac{C^{k}_{12}(z)}{H_{\alpha}(z)}u^{k}_{\mathcal{,}}+\frac{C^ {k}_{12}(z)}{H_{\alpha}(z)R_{\alpha}}w^{k}+\frac{C^{k}_{22}(z)}{H_{\beta}(z)}v ^{k}_{\mathcal{,}\beta}+\frac{C^{k}_{22}(z)}{H_{\beta}(z)R_{\beta}}w^{k}+C^{k }_{23}(z)w^{k}_{\mathcal{,}}-\xi^{k}_{\beta}(z)\mathcal{M}^{k}\;,\]1  
The above equations have no constant coefficients because the parametric coefficients \(H_{\alpha}\) and \(H_{\beta}\) depend on \(z\) in the case of shell geometries and/or elastic and hygrometric coefficients depend on \(z\) in FGM layers. For these reasons, each \(k\) physical layer is divided into a suitable number of mathematical layers indicated with a new index \(j\) which changes from 1 to the global number of mathematical layers \(G\).