## 3 Solution Procedure
### FGM Layer  
We consider simply supported edge boundary conditions with the following relations:  
\[ \begin{array}{l}\sigma_{x}=0,v=0,w=0,\text{ and }T=0\;at\;x=0,a\\ \sigma_{y}=0,u=0,w=0,\text{ and }T=0\;at\;y=0,b\end{array}  \tag{18}\]  
These boundary conditions are identically satisfied by the following Fourier series expansions for the stresses, displacements, and temperature along the in-plane coordinates  
\[ \begin{array}{l}\hat{u}_{f}=\sum\limits_{n=1}^{\infty}\sum\limits_{n=1}^{ \infty}\hat{u}_{f^{\prime}}(z)\cos(p_{m}x)\sin(p_{n}y)\;\;\;\hat{v}_{f}=\sum \limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\hat{v}_{f^{\prime}}(z)\sin(p _{n}x)\cos(p_{n}y)\\ \hat{w}_{f}=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\hat{w}_{f^{ \prime}}(z)\sin(p_{m}x)\sin(p_{n}y)\;\;\;\hat{v}_{fz}=\sum\limits_{m=1}^{ \infty}\sum\limits_{n=1}^{\infty}\hat{v}^{\prime}_{fz}(z)\sin(p_{m}x)\sin(p_{ n}y)\\ \hat{t}_{fyz}=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\hat{v}^{ \prime}_{fyz}(z)\sin(p_{m}x)\cos(p_{n}y)\;\;\hat{t}_{fzz}=\sum\limits_{m=1}^{ \infty}\sum\limits_{n=1}^{\infty}\hat{v}^{\prime}_{fzz}(z)\cos(p_{m}x)\sin(p_{ n}y)\\ \hat{v}_{fx}=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\hat{v}_{fyz} (z)\sin(p_{m}x)\sin(p_{n}y)\;\;\hat{v}_{fy}=\sum\limits_{m=1}^{\infty}\sum \limits_{n=1}^{\infty}\hat{v}^{\prime}_{fy}(z)\sin(p_{m}x)\sin(p_{n}y)\\ \hat{t}_{fxy}=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\hat{v}^{ \prime}_{fxy}(z)\cos(p_{m}x)\cos(p_{n}y)\;\;\hat{T}_{f}=\sum\limits_{m=1}^{ \infty}\sum\limits_{n=1}^{\infty}\hat{T}_{f^{\prime}}(z)\sin(p_{m}x)\sin(p_{ n}y)\end{array}  \tag{19}\]  
with \(p_{m}=\frac{m\pi}{a}\), \(p_{n}=\frac{m\pi}{b}\). For convenience, non-dimensional quantities are introduced:  
\[ \begin{array}{l}\overline{T}=\frac{\alpha^{*}T}{p},\;\left(\overline{u}_{f}, \overline{v}_{f},\overline{w}_{f}\right)=\left(\hat{u}_{f}^{\prime},\hat{v}_{f ^{\prime}}^{\prime},\hat{w}_{f}^{\prime}\right)\frac{1}{p\hat{u}},\;\;\overline {E}=\frac{\hat{E}}{k^{k}}\\ \left(\overline{\sigma}_{fx},\overline{\sigma}_{fyy},\overline{\sigma}_{fz}, \overline{\tau}_{fzx},\overline{\tau}_{fzy},\overline{\tau}_{fxy}\right)= \left(\hat{\omega}^{\prime}_{fx},\hat{\sigma}^{\prime}_{fyz},\hat{\sigma}^{ \prime}_{fzx},\hat{\tau}^{\prime}_{fzx},\hat{\tau}^{\prime}_{fzy},\hat{\tau}^ {\prime}_{fxy}\right)\frac{1}{p\hat{u}^{k}}\\ \overline{x}=\underline{\bar{x}},\;\overline{y}=\underline{\bar{y}}\;,\; \overline{z}=\underline{\bar{z}}\;,\overline{p}_{m}=ap_{m},\;\;\overline{p}_{n}= bp_{n}\end{array}  \tag{20}\]  
Here \(k^{*}=1\;GPa\) and \(\alpha^{*}=10^{-6}/K\) are the scale factors, and \(p=\alpha^{*}T_{o}\). Using Equations (19) and (20) in Equation (12) results in the following non-dimensional state-space equation  
\[\frac{d}{d\overline{z}}\overline{\delta}_{f}=\overline{G}_{f}\overline{\delta} _{f}+\overline{B}_{f}\overline{T}_{f} \tag{21}\]  
where \(\overline{\delta}_{f}=\left[\overline{\sigma}_{fz},\overline{u}_{f},\overline{v }_{f},\overline{w}_{f},\overline{\tau}_{fxz},\overline{\tau}_{fyz}\right]^{T}\) is the state vector in the FGM core and \(\overline{G}_{f}\) and \(\overline{B}_{f}\) are a non-dimensional square matrix and a vector of coefficients (see Equations (A5) and (A6) in the Appendix A). The non-dimensional in-plane stresses can be computed from the non-dimensional state variables as \[ \begin{cases}\overline{\sigma}_{f_{X}}\\ \overline{\sigma}_{fy}\\ \overline{\tau}_{xy}\end{cases} =\left[\overline{F}\right] \begin{cases}\overline{ \sigma}_{fz}\\ \overline{u}_{f}\\ \overline{v}_{f}\\ \overline{T}_{f}\end{cases}  \tag{22}\]  
in which \(\overline{F}\) is a coefficient matrix (see Equation (A7) in the Appendix A). Since the components of the coefficient matrix \(\overline{G}_{f}\) depend on the thickness coordinate \(z\), it is not possible to solve Equation (21) analytically.