## 2 Theory and Formulation
### Hamilton's Principle  
In this study, the Hamilton's principle was employed to derive the governing equations of motion  
\[\int_{t_{1}}^{t_{2}}{(\delta T-\delta U+\delta W_{1}-\delta W_{2})dt}=0 \tag{21}\]  
in which \(T\) and \(U\) stand for the kinetic energy and strain energy, respectively, and \(W_{1}\) and \(W_{2}\) denote the work done by the thermal load and the elastic substrate, respectively. Taking the first variation of the strain energy yields  
\[\delta U=\frac{1}{2}\int\limits_{V}{\sigma_{ij}\delta\varepsilon_{ij}dV}= \int\limits_{R_{1}}^{R_{2}}{ \begin{array}{c}\left\{\frac{\partial N_{RR}}{ \partial R}+\frac{N_{\theta\theta}}{R}\right\}\delta u\\ +\Big{(}\frac{\partial M_{RR}}{\partial R}+\frac{M_{\theta\theta}}{R}-c_{1} \frac{\partial P_{RR}}{\partial R}-\frac{c_{1}}{R}P_{\theta\theta}+(Q_{RZ}-3c _{1}S_{RZ})\Big{)}\delta\phi\\ +\Big{(}-c_{1}\frac{\partial^{2}P_{RR}}{\partial R^{2}}-\frac{c_{1}}{R}\frac{ \partial P_{\theta\theta}}{\partial R}+\Big{(}\frac{\partial Q_{RZ}}{\partial R }-3c_{1}\frac{\partial S_{RZ}}{\partial R}\Big{)}\Big{)}\delta w\end{array}  \right\}dR \tag{22}\]  
The resultants of the force and moment can be expressed as  
\[\{N_{RR},M_{RR},P_{RR}\}=\int_{z}{\sigma_{RR}\big{[}1,z,z^{3}\big{]}dz} \tag{23}\]  
\[\{N_{\theta\theta},M_{\theta\theta},P_{\theta\theta}\}=\int_{z}{\sigma_{ \theta\theta}\big{[}1,z,z^{3}\big{]}dz} \tag{24}\]  
\[\{Q_{Rz},S_{Rz}\}=\int_{z}{\sigma_{Rz}\big{[}1,z^{2}\big{]}dz} \tag{25}\]  
The variation of the work induced by thermal gradient is given by  
\[\delta W_{1}=\int\limits_{R_{1}}^{R_{2}}{\left[N^{T}\frac{\partial w}{ \partial x}\frac{\partial\delta w}{\partial x}\right]dR} \tag{26}\]  
with the force resultant of the thermal loading \(N^{T}\)  
\[N^{T}=\int_{-h/2}^{h/2}{(Q_{11}\alpha_{11}+Q_{12}\alpha_{22})\ (T-T_{0})dz} \tag{27}\]  
Note that three different patterns were considered for the temperature gradient across the thickness: Uniform temperature rise (UTR), power form temperature rise (PTR), and sinusoidal temperature rise (STR) [74]. As linear temperature rises are unrealistic in real applications, we studied two other conditions to finally compare the difference to the first scenario. \[UTR:\ \Delta T=T-T_{0} \tag{28}\]  
\[PTR:\ T=T_{0}+\Delta T\Big{(}\frac{1}{2}+\frac{z}{h}\Big{)}^{\beta} \tag{29}\]\[STR:T=T_{0}+\Delta T\bigg{(}1-\cos\frac{\pi}{2}\bigg{(}\frac{1}{2}+\frac{z}{h} \bigg{)}\bigg{)}^{\beta} \tag{30}\]  
while the applied work by the elastic substrate can be expressed as  
\[\delta w_{2}=\int\limits_{R_{1}}^{R_{2}}\left(-K_{w}w\delta w+K_{p}\bigg{[} \bigg{(}\frac{\partial w}{\partial x}\bigg{)}\bigg{(}\frac{\partial}{\partial x }\delta w\bigg{)}\bigg{]}\right)dR \tag{31}\]  
with \(K_{p}\) and \(K_{w}\) being the Pasternak and Winkler coefficients of the substrate, respectively. The kinetic energy and its first variation can be written as  
\[T=\int\limits_{A}\frac{1}{2}\rho\bigg{[}\bigg{(}\frac{\partial U}{\partial t} \bigg{)}^{2}+\bigg{(}\frac{\partial V}{\partial t}\bigg{)}^{2}+\bigg{(}\frac{ \partial W}{\partial t}\bigg{)}^{2}\bigg{]}dR\text{d}Z \tag{32}\]  
\[\{N_{\theta\theta},M_{\theta\theta},P_{\theta\theta}\}=\int_{z}{\sigma_{ \theta\theta}\big{[}1,z,z^{3}\big{]}dz} \tag{24}\]0  
\[\delta T=\int_{R_{1}}^{R_{2}}\left[