# 3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions
## 2 Fick Moisture Diffusion Equation
### 3D Fick Equation  
By analogy with the heat flux \(\mathbf{q}\), suppose to define a moisture flux \(\mathbf{g}\). Then, the differential equation of moisture diffusion into a homogeneous solid, in the absence of chemical reactions and under steady-state conditions (\(\frac{\partial\mathcal{M}}{\partial t}=0\)) reads:  
\[\
abla\cdot\mathbf{g}(u_{1},u_{2},u_{3})=0\,. \tag{2}\]  
Equation (2) is written in a general orthogonal curvilinear coordinate system \((u_{1},u_{2},u_{3})\); however, in these conditions, the divergence of the moisture flux takes the following expression:  
\[\
abla\cdot\mathbf{g}=\frac{1}{a}\bigg{[}\frac{\partial}{\partial u_{1}}\bigg{(} \frac{a}{a_{1}}g_{1}\bigg{)}+\frac{\partial}{\partial u_{2}}\bigg{(}\frac{a} {a_{2}}g_{2}\bigg{)}+\frac{\partial}{\partial u_{3}}\bigg{(}\frac{a}{a_{3}}g _{3}\bigg{)}\bigg{]}\,, \tag{3}\]  
\(g_{1}\), \(g_{2}\), \(g_{3}\) express the components of the 3D flux in directions 1, 2, and 3; their explicit form is  
\[g_{i}=-\mathcal{D}_{i}\frac{1}{a_{i}}\frac{\partial\mathcal{M}}{\partial u_{ i}}\,, \tag{4}\]  
\(\mathcal{D}_{i}\) is the diffusion coefficient along with direction \(i\); \(a_{1}\), \(a_{2}\), and \(a_{3}\) are the so-called scale factors, with \(a\) the product of the three factors (\(a=a_{1}\,a_{2}\,a_{3}\)). The problem takes place in a mixed curvilinear orthogonal coordinate system (\(\alpha\), \(\beta\), \(z\)); [PERSON] [32] discussed that, in this context, Equation (3) might be rewritten as follows:  
\[\frac{1}{H_{a}H_{\beta}}\bigg{[}\frac{\partial}{\partial\alpha}\bigg{(}\frac {H_{a}H_{\beta}}{H_{\alpha}}\mathcal{D}_{1}\frac{1}{H_{\alpha}}\frac{ \partial\mathcal{M}}{\partial\alpha}\bigg{)}+\frac{\partial}{\partial\beta} \bigg{(}\frac{H_{a}H_{\beta}}{H_{\beta}}\mathcal{D}_{2}\frac{1}{H_{\beta}} \frac{\partial\mathcal{M}}{\partial\beta}\bigg{)}\bigg{]}+\mathcal{D}_{3} \frac{\partial^{2}\mathcal{M}}{\partial z^{2}}=0\,\,, \tag{5}\]  
Figure 2: Geometrical data and coordinate system for cylindrical and spherical shell panels. The figure also shows the stacking sequence used in Benchmarks 3 and 4. FGM layers distinguish from classical isotropic and orthotropic layers as the diffusion coefficients \(\mathcal{D}_{1}(z)\), \(\mathcal{D}_{2}(z)\), and \(\mathcal{D}_{3}(z)\) are a function of the thickness coordinate \(z\). Despite the lamination scheme, \(H_{a}(z)\) and \(H_{\beta}(z)\) are two parametric coefficients, defined as follows:  
\[H_{a}(z) =\left(1+\frac{z}{R_{\alpha}}\right)=H_{a}(\bar{z})=\left(1+\frac{ \bar{z}-h/2}{R_{\alpha}}\right)\,, \tag{6}\] \[H_{\beta}(z) =\left(1+\frac{z}{R_{\beta}}\right)=H_{\beta}(\bar{z})=\left(1+ \frac{\bar{z}-h/2}{R_{\beta}}\right)\,. \tag{7}\]  
When considering shells with constant radii of curvature, \(R_{\alpha}\) and \(R_{\beta}\), they are a linear function of the thickness coordinate \(z\), which varies from \(-h/2\) to \(+h/2\) or \(\bar{z}\), which varies from \(0\) to \(h\). \(h\) is the global thickness. The thickness coordinate \(z\) (or \(\bar{z}\)) is rectilinear; for consistency, a further coefficient might be defined: \(H_{z}=1\). Equation (5) points out two main blocks; the second block has a simpler formulation as the third coordinate \(z\) is rectilinear. Actually, this point is further confirmed by the work of [PERSON] [70], which showed that  
\[a_{1}=H_{\alpha},\ a_{2}=H_{\beta},\ a_{3}=H_{z}=1\,. \tag{8}\]  
The coefficients of Equation (5) are not constant even inside a \(k\)-th physical layer. This issue is due to the parametric coefficients \(H_{a}\) and \(H_{\beta}\), a function of the thickness coordinate \(z\), and to \(\mathcal{D}_{1}\), \(\mathcal{D}_{2}\), and \(\mathcal{D}_{3}\) which are not constant in FGM layers. A possible solution consists in dividing each physical layer into sufficiently thin mathematical (fictitious) layers. There, calculating the parametric and the diffusion coefficients at its middle allows moving the differential operators to the moisture content only.