## 3 Equilibrium and Stability Equations of FG Plate Placed on Elastic Foundation
 \tag{7}\]  
In order to use the principles of minimum potential energy, it is necessary to define strain energy \(\rm U_{s}\), the potential energy of the elastic foundation \(\rm U_{e}\) and the total potential energy \(\Pi\):  
\[\rm U_{s}=\int\limits_{-h/2}^{h/2}\int\limits_{A}\Big{(}\sigma_{xx}[\epsilon_{ xx}-\alpha(z)T(z)]+\sigma_{y}\Big{[}\epsilon_{yy}-\alpha(z)T(z)\Big{]}+\sigma_{xx} \epsilon_{zz}+\tau_{xy}\gamma_{xy}+\tau_{xz}\gamma_{xz}+\tau_{yz}\gamma_{yz} \Big{)}dAdz, \tag{8}\]  
\[\rm U_{e}=\frac{1}{2}\int\limits_{A}\Bigg{\{}k_{0}w^{2}+k_{1}\Bigg{[}\Bigg{(} \frac{\partial w}{\partial x}\Bigg{)}^{2}+\Bigg{(}\frac{\partial w}{\partial x }\Bigg{)}^{2}\Bigg{]}\Bigg{\}}dA, \tag{9}\]  
\[\Pi=\rm U_{s}+U_{e}. \tag{10}\]  
By applying the principles of minimum potential energy:  
\[ \begin{array}{l}\delta\Pi=\delta(U_{s}+U_{e})=\delta U_{s}+\delta U_{e}= \int\limits_{A}(N_{xx}\delta\epsilon_{xx}^{(0)}+N_{yy}\delta\epsilon_{yy}^{(0)}+N_ {xy}\delta\gamma_{xy}^{(0)}+M_{xx}\delta\rm{k}_{xx}^{(0)}+M_{yy}\delta\rm{k}_{ yy}^{(0)}+M_{xy}\delta\rm{k}_{xy}^{(0)}\\ +P_{xx}\delta\rm{k}_{xx}^{(1)}+P_{yy}\delta\rm{k}_{yy}^{(1)}+P_{xy}\delta\rm{k}_{ xy}^{(1)}+R_{x}\delta\rm{k}_{xx}^{(2)}+R_{y}\delta\rm{k}_{yz}^{(2)}dA+\int \limits_{A}\Big{\{}k_{0}w\delta w\delta w+k_{1}\Big{(}\frac{\partial w}{ \partial x}\frac{\partial w}{\partial x}+\frac{\partial w}{\partial y}\frac{ \partial w}{\partial y}\Big{)}\Big{\}}dA=0,\end{array}  \tag{11}\]  
equilibrium equations become:  
\[ \begin{array}{l}\delta u_{0}:\frac{\partial N_{xx}}{\partial x}+\frac{ \partial N_{yy}}{\partial y}=0,\quad\delta v_{0}:\frac{\partial N_{yy}}{\partial y }+\frac{\partial N_{yy}}{\partial x}=0,\\ \delta w_{0}:\frac{\partial M_{xx}}{\partial x^{2}}+\frac{\partial^{2}M_{xy}}{ \partial x\partial y}+\frac{\partial^{2}M_{yy}}{\partial y^{2}}+\frac{\partial^{2}M_{ yx}}{\partial x^{2}}+2N_{xx}\frac{\partial^{2}w_{0}}{\partial x\partial y}+N_{yy}\frac{ \partial^{2}w_{0}}{\partial y^{2}}-k_{0}w_{0}+k_{1}\Big{(}\frac{\partial^{2}w_{0} }{\partial x^{2}}+\frac{\partial^{2}w_{0}}{\partial y^{2}}\Big{)}=0,\\ \delta\rm{k}_{x}:\frac{\partial P_{xx}}{\partial x}+\frac{\partial P_{yy}}{\partial y }-R_{x}=0,\quad\delta\delta y:\frac{\partial P_{yy}}{\partial x}+\frac{\partial P _{yy}}{\partial y}-R_{y}=0.\end{array}  \tag{12}\]  
Based on the equilibrium Equation (12) and using the displacement components \(\rm u_{0},\,v_{0},\,w_{0},\,\theta_{\rm x0}\) and \(\rm\theta_{y0}\), the stability equation could be defined. The displacement components of the next stable configuration are:  
\[ \begin{array}{l}\rm u=u_{0}+u_{1},\,\,\,\,v=v_{0}+v_{1},\,\,\,w=w_{0}+w_{1},\\ \theta_{\rm x}=\theta_{\rm x0}+\theta_{\rm x1},\,\,\theta_{\rm y}=\theta_{ \rm y0}+\theta_{\rm y1},\end{array}  \tag{13}\]  
where \(\rm u_{1},\,v_{1},\,w_{1},\,\FG plate and that it changes only in the direction of \(z\)-axis, the stability equation can be obtained by substituting the Equation (13) into the Equation (12):  
\[ \begin{array}{l}\delta u_{0}:\frac{\partial N_{\rm xx}^{1}}{\partial x}+\frac{ \partial N_{\rm yy}^{1}}{\partial y}=0,\quad\delta v_{0}:\frac{\partial N_{ \rm yy}^{1}}{\partial y}+\frac{\partial N_{\rm yy}^{1}}{\partial x}=0,\\ \delta w_{0}:\frac{\partial^{2}N_{\rm xy}^{1}}{\partial x^{2}}+\frac{\partial^ {2}N_{\rm xy}^{1}}{\partial x\partial y}+\frac{\partial^{2}N_{\rm xy}^{1}}{ \partial y^{2}}+N_{\rm xx}^{0}\frac{\partial^{2}w_{0}}{\partial x^{2}}+2N_{ \rm xy}^{0}\frac{\partial^{2}w_{0}}{\partial x\partial y}+N_{\rm yy}^{0}\frac {\partial^{2}w_{0}}{\partial y^{2}}-k_{0}w_{1}+k_{\rm I}\Big{(}\frac{ \partial^{2}w_{1}}{\partial x^{2}}+\frac{\partial^{2}w_{1}}{\partial y^{2}} \Big{)}=0,\\ \delta\alpha:\frac{\partial^{2}N_{\rm xy}^{1}}{\partial x}+\frac{\partial^{2} N_{\rm xy}^{1}}{\partial y}-R_{\rm x}^{1}=0,\quad\delta y:\frac{\partial^{2}N_{ \rm xy}^{1}}{\partial x}+\frac{\partial^{2}N_{\rm xy}^{1}}{\partial y}-R_{\rm y }^{1}=0.\end{array}