# 3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions
## 3 3D Exact Shell Model for Hygro-Elastic Stress Analysis
This reconstruction can be defined as  
\[M^{j}(\bar{z}^{j})=a_{M}^{j}\bar{z}^{j}+b_{M}^{j}\,, \tag{55}\]  
in a \(j\)th mathematical layer, \(a_{M}^{j}\) and \(b_{M}^{j}\) are two constant coefficients. The first represents the slope of the moisture content profile inside a mathematical layer; the second the moisture content at the bottom. \(\bar{z}^{j}\) is a local thickness coordinate for each \(j\) mathematical layer, and it changes from \(0\) at the bottom of the considered \(j\) mathematical layer to the top value \(h^{j}\) of the same mathematical layer. The system of first-order differential equations in \(\bar{z}\) or \(z\) shown in Equation (54) is not homogeneous because the hygroscopic term \(\mathbf{L}^{*j}\mathbf{M}^{j}\) depends on \(\bar{z}^{j}\) or \(z^{j}\). In the case of a generic system of non-homogeneous first-order differential equations having an unknown \(G\times 1\) vector \(\mathbf{x}\), a \(G\times G\) \(|\mathbf{A}\) matrix containing constant coefficients, and a known function vector \(\mathbf{f}(t)=[f_{1}(t)\,...\,f_{G}(t)]^{T}\),we can write  
\[\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}t}=\mathbf{A}\mathbf{x}+\mathbf{f}(t)\,, \tag{56}\]  
a possible solution of the Equation (56) can be obtained via the exponential matrix method:  
\[\mathbf{x}(t)=\mathbf{e}^{\mathbf{A}(t-t_{0})}\mathbf{x}_{0}+\int_{t_{0}}^{t}\mathbf{e}^{ \mathbf{A}(t-s)}\mathbf{f}(s)ds\,. \tag{57}\]  
The known term in Equation (54) can be given in the following complete form:  
\[\mathbf{M}^{*j}=\mathbf{L}^{*j}\mathbf{M}^{j}= \begin{bmatrix}0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ -L_{1}^{*j}&0&0&0&0&0\\ -L_{2}^{*j}&0&0&0&0&0\\ -L_{4}^{*j}&-L_{3}^{*j}&0&0&0&0\end{bmatrix}\begin{bmatrix}a_{M}^{j}\bar{z}^{ j}+b_{M}^{j}\\ a_{M}^{j}\\ 0\\ 0\\ 0\\ 0\end{bmatrix} = \begin{bmatrix}0\\ 0\\ 0\\ -L_{1}^{*j}(a_{M}^{j}\bar{z}^{j}+b_{M}^{j})\\ -L_{M}^{*j}(a_{M}^{j}\bar{z}^{j}+b_{M}^{j})\\ -L_{4}^{*j}(a_{M}^{j}\bar{z}^{j}+b_{M}^{j})-L_{3}^{*j}a_{M}^{j}\end{bmatrix} \,. \tag{58}\]  
Therefore, Equation (54) can be rewritten as  
\[\mathbf{U}^{j^{\prime}}=\mathbf{A}^{*j}\mathbf{U}^{j}+\mathbf{M}^{*j}\,, \tag{59}\]  
\(\mathbf{M}^{*j}\) includes only linear and known functions in \(\bar{z}^{j}\) coordinate. The Equation (59) can be solved through the exponential matrix method:  
\[\mathbf{U}^{j}(\bar{z}^{j})=\mathbf{e}^{(\mathbf{A}^{*j}\bar{z}^{j})}\mathbf{U}^{j}(0)+\int _{0}^{\bar{z}^{j}}\mathbf{e}^{(\mathbf{A}^{*j}(\bar{z}^{j}-s))}\mathbf{M}^{*j}(s)ds\,. \tag{60}\]  
\(\mathbf{A}^{*j}=\mathbf{e}^{(\mathbf{A}^{*j}\bar{h}\bar{z}^{j})}\) and \(\mathbf{L}^{*sj}=\int_{0}^{\bar{h}^{j}}\mathbf{e}^{(\mathbf{A}^{*j}(\bar{h}^{j}-s))}\mathbf{ M}^{*j}(s)ds\) must be opportunely defined for each \(j\) layer having thickness \(h^{j}\). In this way, the displacement vector at the top of each \(j\) mathematical layer is calculated. Both terms are defined via the exponential matrix opportunely expanded and evaluated in each \(j\) mathematical layer with thickness \(h^{j}\):  
\[\mathbf{A}^{*j}=\mathrm{e}^{(\mathbf{A}^{*j}h^{j})}=\mathbf{I}+\mathbf{A}^{*j}h^{j}+\frac{{\mathbf{A }^{*j}}^{2}}{2!}{h^{j}}^{2}+\frac{{\mathbf{A}^{*j}}^{3}}{3!}{h^{j}}^{3}+\cdots+\frac{ {\mathbf{A}^{*j}}^{N}}{N!}{h^{j}}^{N}\,, \tag{61}\]  
\[\mathbf{L}^{*sj}= \int_{0}^{h^{j}}\mathrm{e}^{(\mathbf{A}^{*j}(h^{j}-s))}\mathbf{M}^{*j}(s) ds=\int_{0}^{h^{j}}\Big{(}\mathbf{I}+\mathbf{A}^{*j}(h^{j}-s)+\frac{{\mathbf{A}^{*j}}^{2}}{2!}(h^ {j}-s)^{2}+\frac{{\mathbf{A}^{*j}}^{3}}{3!}(h^{j}-s)^{3}+\] \[\cdots+\frac{{\mathbf{A}^{*j}}^{N}}{N!}(h^{j}-s)^{N}\Big{)}\mathbf{M}^{*j }(s)ds\,, \tag{62}\]  
where \(\mathbf{I}\) is the \(6\times 6\) identity matrix, and the integral given in Equation (60) can be calculated in each \(j\) layer having thickness \(h^{j}\) by using the exponential matrix and the same order \(N\) already shown in Equation (61).