## 3 Theoretical Formulations
### Equations of Motion
Therefore, by substitution of Equations (17) and (19) into Equation (16), by integrating the result by parts, and equating the coefficients of \(\delta u\), \(\delta w\), and \(\delta\phi\) to zero separately, we get the following nonlinear equations of motion:  
\[\frac{\partial N_{xx}}{\partial x}=m_{0}\ddot{u}+m_{1}\ddot{\phi}-c_{1}m_{3} \biggl{(}\ddot{\phi}+\frac{\partial\ddot{w}}{\partial x}\biggr{)}, \tag{21}\]  
\[\frac{\partial M_{xx}}{\partial x}-c_{1}\frac{\partial P_{xx}}{\partial x}-Q_{x }+c_{2}R_{x}=m_{1}\ddot{u}+m_{2}\ddot{\phi}-c_{1}m_{3}\ddot{u}-c_{1}m_{4} \biggl{(}2\ddot{\phi}+\frac{\partial\ddot{w}}{\partial x}\biggr{)}+c_{1}^{2}m_{6 }\biggl{(}\ddot{\phi}+\frac{\partial\ddot{w}}{\partial x}\biggr{)}, \tag{22}\]  
\[\frac{\partial}{\partial x}\biggl{(}N_{xx}\frac{\partial w}{\partial x}\biggr{)} +c_{1}\frac{\partial^{2}P_{xx}}{\partial x^{2}}+\frac{\partial Q_{x}}{\partial x }-c_{2}\frac{\partial R_{x}}{\partial x}=m_{0}\ddot{w}+c_{1}m_{3}\frac{ \partial\ddot{u}}{\partial x}+c_{1}m_{4}\frac{\partial\ddot{\phi}}{\partial x}-c _{1}^{2}m_{6}\biggl{(}\frac{\partial\ddot{\phi}}{\partial x}+\frac{\partial^{2} \ddot{w}}{\partial x^{2}}\biggr{)}. \tag{23}\]Thus, we define the stress resultants in terms of the displacement and rotation components of the sandwich beam as  
\[N_{xx}=A_{11}\!\left(\frac{\partial u}{\partial x}+\frac{1}{2}\!\left(\frac{ \partial w}{\partial x}\right)^{\!2}\right)+B_{11}\!\left(\frac{\partial\phi}{ \partial x}\right)-c_{1}E_{11}\!\left(\frac{\partial\phi}{\partial x}+\frac{ \partial^{2}w}{\partial x^{2}}\right), \tag{24}\]  
\[M_{xx}=B_{11}\!\left(\frac{\partial u}{\partial x}+\frac{1}{2}\!\left(\frac{ \partial w}{\partial x}\right)^{\!2}\right)+D_{11}\!\left(\frac{\partial\phi} {\partial x}\right)-c_{1}F_{11}\!\left(\frac{\partial\phi}{\partial x}+\frac{ \partial^{2}w}{\partial x^{2}}\right), \tag{25}\]  
\[P_{xx}=E_{11}\!\left(\frac{\partial u}{\partial x}+\frac{1}{2}\!\left(\frac{ \partial w}{\partial x}\right)^{\!2}\right)+F_{11}\!\left(\frac{\partial\phi} {\partial x}\right)-c_{1}H_{11}\!\left(\frac{\partial\phi}{\partial x}+\frac {\partial^{2}w}{\partial x^{2}}\right), \tag{26}\]  
\[Q_{x}=(A_{55}-c_{2}D_{55})\!\left(\phi+\frac{\partial w}{\partial x}\right), \tag{27}\]  
\[R_{x}=(D_{55}-c_{2}F_{55})\!\left(\phi+\frac{\partial w}{\partial x}\right) \tag{28}\]  
where  
\[(A_{11},B_{11},D_{11},E_{11},F_{11},H_{11})=\sum_{k=1}^{N}b\int_{h_{k}}^{h_{k+ 1}}Q_{11}^{(k)}\!\left(1,z,z^{2},z^{3},z^{4},z^{6}\right)\!dz, \tag{29}\]  
\[\frac{\partial N_{xx}}{\partial x}=m_{0}\ddot{u}+m_{1}\ddot{\phi}-c_{1}m_{3} \biggl{(}\ddot{\phi}+\frac{\partial\ddot{w}}{\partial x}\biggr{)}, \tag{21}\]0  
and  
\[\frac{\partial N_{xx}}{\partial x}=m_{0}\ddot{u}+m_{1}\ddot{\phi}-c_{1}m_{3} \biggl{(}\ddot{\phi}+\frac{\partial\ddot{w}}{\partial x}\biggr{)}, \tag{21}\]1  
In view of Equations (21)-(28), the nonlinear partial differential equations of motion of FG GPL-reinforced sandwich beams can be written as  
\[ \begin{array}{ll}A_{11}\!\left(\frac{\partial^{2}u}{\partial x^{2}}+\frac{ \partial^{2}w}{\partial x^{2}}\frac{\partial w}{\partial x}\right)&+\!\left(B _{11}-c_{1}E_{11}\right)\frac{\partial^{2}\phi}{\partial x^{2}}-c_{1}E_{11} \frac{\partial^{2}w}{\partial x^{2}}\\ &=m_{0}\frac{\partial^{2}u}{\partial x^{2}}+(m_{1}-c_{1}m_{3})\frac{\partial^ {2}\phi}{\partial x^{2}}-c_{1}m_{3}\frac{\partial^{2}w}{\partial x\partial x ^{2}},\end{array}  \tag{32}\]  
\[