# 3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions
## 2 Fick Moisture Diffusion Equation
### 1D Fick Equation  
The three-dimensional problem of defining the moisture content field can be simplified when the structure under investigation is thin enough; such condition is expressed through sufficiently high thickness ratios. By recalling Equation (4) and the harmonic form for the moisture content field, the three components of the moisture content flux inside a \(k\)-th layer are  
\[g_{1}^{k}=\mathcal{D}_{1}^{\,*k}(z)\bar{\alpha}M^{k}(z)cos(\bar{ \alpha}\alpha)sin(\bar{\beta}\beta)\, \tag{17}\] \[g_{2}^{k}=\mathcal{D}_{2}^{\,*k}(z)\bar{\beta}M^{k}(z)sin(\bar{ \alpha}\alpha)cos(\bar{\beta}\beta)\,\] (18) \[g_{3}^{k}=\mathcal{D}_{3}^{\,*k}(z)M_{z}^{k}(z)sin(\bar{\alpha} \alpha)sin(\bar{\beta}\beta). \tag{19}\]  
As already seen, the diffusion coefficients are, in general, a function of the thickness coordinate as they are not constant inside FGM layers. The relative weight of the moisture content fluxes \(g_{1}\) and \(g_{2}\) decreases if compared to \(g_{3}\) the thinner is the shell. When this happens, the first two components can be disregarded; Equation (19) now becomes  
\[g_{3}(z)= \left(\mathcal{D}_{3}^{\,*}\frac{\partial M}{\partial z}\right)= const. \tag{20}\]  
As already seen in the previous section, there is no practical difference between \(\mathcal{D}_{3}^{\,*}\) and \(\mathcal{D}_{3}\) because the third coordinate \(3\equiv z\) is rectilinear, and its parametric coefficient \(H_{2}\) equals 1. Equation (20) implies that \(g_{3}(z)\) is actually constant along with \(z\); it can be specialized for a generic \(j\)-th mathematical layer. The coefficient \(\mathcal{D}_{3}^{\,*}\) does not change inside it; this implies that the derivative of the moisture content to \(z\) is constant: the moisture content is linear inside each mathematical layer. Consequently Equation (20) can be simplified and rewritten in an algebraic form:  
\[g_{3}^{j}=-\mathcal{D}_{3}^{\,*j}\frac{\partial M^{j}}{\partial z}=-\frac{ \mathcal{D}_{3}^{\,*j}}{h^{j}}(M_{t}^{j}-M_{b}^{j})\,. \tag{21}\]For the term \(\mathcal{D}_{3}^{sj}/h^{j}\), it is possible to exploit the analogy with the electrical conductance, as already done in [21] where a sort of thermal conductance was defined: the layer diffuses more, becoming thinner, increasing \(\mathcal{D}_{3}^{*}\). This analogy is beneficial, as it enables defining an equivalent moisture content diffusion resistance for the overall structure:  
\[R_{z_{\alpha l}}=\sum_{j=1}^{G}\frac{h^{j}}{\mathcal{D}_{3}^{sj}}\,. \tag{22}\]  
The moisture flux in the thickness direction can now be easily quantified, as the dissertation demonstrated it is constant and quantified the equivalent diffusion resistance. Defined \(M_{t}\) the moisture content amplitude on the top external surface and \(M_{b}\) that on the bottom, \(g_{3}\) is  
\[g_{3}=-\frac{1}{R_{z_{\alpha l}}}(M_{t}-M_{b})=const. \tag{23}\]  
As \(g_{3}\) is constant across all the layers, the moisture content at any interface (and even at any \(z\) coordinate) is easily obtained:  
\[g_{3}^{j}=-\mathcal{D}_{3}^{sj}\frac{(M_{t}^{j}-M_{b}^{j})}{h^{j}}=g_{3}^{j+1} =-\mathcal{D}_{3}^{sj+1}\frac{(M_{t}^{j+1}-M_{b}^{j+1})}{h^{j+1}}=g_{3}=const.\,, \tag{24}\]  
The coefficient \(\mathcal{D}_{3}^{sj}\) changes in the thickness direction when FGM layers are considered in the stacking sequence. As a consequence, the moisture flux can keep constant only if the slope of the moisture content profile modifies accordingly. In this perspective, the effect of the material is considered (which means the stacking sequence and the FGM law), but the impact of the thickness is disregarded. The 3D hygro-elastic model including this hygrometric profile is referred to as 3D(\(\mathcal{M}_{c}\),1D).