## 3 Solution Methodology
### Boundary Discontinues Fourier Series Method
The above step generates an additional 4 n unknown boundary Fourier coefficients (\(\overline{a}_{n}\), \(\overline{b}_{n}\), \(\overline{c}_{n}\), \(\overline{d}_{n}\)) generated from the differentiation procedure. Boundary Fourier coefficients are defined in Appendix A. The substitution of the assumed displacement functions and their appropriately obtained derivatives into the governing PDE's yields 5 mn + 2 m + 2 n linear algebraic equations for equating the coefficients of \(\cos(\alpha\xi_{1})\sin(\beta\xi_{2})\), \(\sin(\beta\xi_{2})\), \(\sin(\alpha\xi_{1})\), etc., as shown below:  
\[ \begin{array}{l}\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\cos( \alpha\xi_{1})\sin(\beta\xi_{2})\{\left(-\text{A}_{11}\alpha^{2}-\text{A}_{66} \beta^{2}-f_{1}+f_{2}\beta^{2}\right)U_{mn}\\ \qquad+(-f_{2}-\text{A}_{12}\alpha\beta-\text{A}_{66}\alpha\beta)V_{mn}+(g_{ 1}\alpha+f_{1}\alpha)W_{mn}\}\\ \qquad+\left(-\text{B}_{11}\alpha^{2}+f_{1}-\frac{f_{2}\beta^{2}}{c}\right) X_{mn}+(-c\text{D}_{66}\alpha\beta)Y_{mn}+\text{A}_{11}b_{n}+(g_{1}+f_{1})d_{n}\\ \qquad=0\end{array}  \tag{31}\]  
\[ \begin{array}{l}\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\sin( \alpha\xi_{1})\cos(\beta\xi_{2})\{\left(-f_{2}\alpha\beta-\text{A}_{12}\alpha \beta-\text{A}_{66}\alpha\beta\right)U_{mn}\\ \qquad+\left(-\text{A}_{66}\alpha^{2}-\text{A}_{22}\beta^{2}-f_{5}+f_{2} \alpha^{2}\right)V_{mn}+f_{6}W_{mn}+(c\text{D}_{66}\alpha\beta)X_{mn}\\ \qquad+f_{7}Y_{mn}\}=0\end{array}  \tag{32}\]  
\[ \begin{array}{l}\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\sin( \alpha\xi_{1})\sin(\beta\xi_{2})\{\left(f_{3}\alpha\right)U_{mn}+(f_{6})V_{mn }+\left(f_{8}-\frac{f_{6}}{K_{9}}-f_{10}\right)W_{mn}+(f_{11})X_{mn}\\ \qquad+(f_{12})Y_{mn}-\text{A}_{55}\alpha b_{n}\}=\sum\limits_{m=1}^{\infty} \sum\limits_{n=1}^{\infty}q_{mn}\sin(\alpha\xi_{1})\sin(\beta\xi_{2})\\ \qquad\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\cos(\alpha\xi_{1} )\sin(\beta\xi_{2})\{\left(-\text{B}_{11}\alpha^{2}+c\text{D}_{66}\beta^{2}+ f_{1}\right)U_{mn}+\left(\frac{-f_{2}\alpha\beta}{c}\right)V_{mn}\\ \qquad+(f_{13}\alpha-\text{A}_{55}\alpha)W_{mn}\}+(g_{2})X_{mn}+(-\text{D}_{12 }\alpha\beta-\text{D}_{66}\alpha\beta)Y_{mn}+\text{B}_{11}b_{n}\\ \qquad+(f_{13}-\text{A}_{55})d_{n}=0\end{array}  \tag{33}\]  
\[ \begin{array}{l}\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\sin( \alpha\xi_{1})\cos(\beta\xi_{2})\{\left(c\text{D}_{66}\alpha\beta\right)U_{mn }+\left(-c\text{D}_{66}\alpha^{2}-\text{B}_{22}\beta^{2}+f_{5}R_{y}\right)V_{mn }+(f_{14}\beta\\ \qquad-\text{A}_{44}\beta)W_{mn}+(-\text{D}_{12}\alpha\beta-\text{D}_{66} \alpha\beta)X_{mn}+f_{15}Y_{mn}\}=0\end{array}  \tag{35}\]  
\[ \begin{array}{l}\sum\limits_{n=1}^{\infty}\sin(\beta\xi_{2})\biggl{\{}U_{0n }f_{4}+\frac{\text{A}_{11}}{2}a_{n}+\frac{f_{3}}{2}c_{n}-\frac{c\text{D}_{66} \beta^{2}}{2}X_{0n}\biggr{\}}=0\end{array}  \tag{36}\]  
\[ \begin{array}{l}\sum\limits_{n=1}^{\infty}\sin(\beta\xi_{2})\biggl{\{}\frac{ \text{D}_{66}\beta^{2}}{2}U_{0n}-\frac{\text{D}_{66}\beta^{2}}{2}X_{0n}+\left( \frac{-\text{A}_{55}}{2}+\frac{\text{B}_{11}}{2R_{x}}\right)c_{n}+\frac{\text {B}_{11}}{2}a_{n}\biggr{\}}=0\end{array}  \tag{37}\]  
\[ \begin{array}{l}\sum\limits_{m=1}^{\infty}\sin(\alpha\xi_{1})\Bigl{\{}V_{m0} \Bigl{(}\text{D}_{66}c^{2}-\text{A}_{66}\Bigr{)}+c\text{D}_{66}Y_{m0}\Bigr{\}} =0\end{array}  \tag{38}\]  
\[ \begin{array}{l}\sum\limits_{m=1}^{\infty}\sin(\alpha\xi_{1})\bigl{\{}V_{m0} (-c\text{D}_{66})-\text{D}_{66}Y_{m0}\bigr{\}}=0\end{array}  \tag{39}\]  
The constants of Equations (31)-(39) can be found in Appendix B. The remaining equations are supplied by the geometric and boundary conditions given in Equation (20). \(u_{1}\) and \(\text{Q}_{1}\) are forced to vanish at the edges, \(\xi_{1}=0\) and a. Satisfying these geometric boundary conditions generates an additional 4 n linear algebraic equations.