# Thickness-Stretch Vibration of an Infinite Piezoelectric Plate with Flexoelectricity
## 2 Mathematical Modeling
### One-Dimensional Model for Thickness-Stretch Vibration
\begin{array}{l}A_{1}\big{[}\alpha\cosh(\alpha h)-k_{1}\alpha^{3}\cosh( \alpha h)\big{]}-A_{2}\big{[}\alpha\sinh(\alpha h)-k_{1}\alpha^{3}\sinh( \alpha h)\big{]}\\ +A_{3}\big{[}\beta\cos(\beta h)+k_{1}\beta^{3}\cos(\beta h)\big{]}+A_{4} \big{[}\beta\sin(\beta h)+k_{1}\beta^{3}\sin(\beta h)\big{]}=0\end{array} \] (24b) \[ \begin{array}{l}A_{1}\big{[}\alpha\sinh(\alpha h)+k_{2}\alpha^{2}\sinh( \alpha h)\big{]}+A_{2}\big{[}\alpha\cosh(\alpha h)+k_{2}\alpha^{2}\cosh( \alpha h)\big{]}\\ +A_{3}\big{[}\beta\cos(\beta h)-k_{2}\beta^{2}\sin(\beta h)\big{]}-A_{4} \big{[}\beta\sin(\beta h)+k_{2}\beta^{2}\cos(\beta h)\big{]}=0\end{array} \] (24c) \[ \begin{array}{l}A_{1}\big{[}\alpha\cosh(\alpha h)-k_{2}\alpha^{2}\sinh( \alpha h)\big{]}-A_{2}\big{[}\alpha\sinh(\alpha h)-k_{2}\alpha^{2}\cosh( \alpha h)\big{]}\\ +A_{3}\big{[}\beta\cos(\beta h)+k_{2}\beta^{2}\sin(\beta h)\big{]}+A_{4} \big{[}\beta\sin(\beta h)-k_{2}\beta^{2}\cos(\beta h)\big{]}=0\end{array}  \tag{24d}\]  
where  
\[k_{1}=\frac{\mu_{3333}^{2}}{c_{33}\varepsilon_{33}+e_{33}^{2}},\ k_{2}=\frac{ \mu_{3333}}{e_{33}}\]  
For nontrivial solutions of the coefficients \(A_{1}\), \(A_{2}\), \(A_{3}\), and \(A_{4}\), the determinant of Equation (24) must be zero. This will lead to the frequency equation of the given problem. * Short circuit condition (SC) For short circuit conditions, the boundary conditions are given below. \[T_{33}-\tau_{333,3}=0,\ \ \ \tau_{333}=0\ \ \ \ \ (x=\pm h)\] (25a) \[\varphi(x_{3})=\pm\,\varphi_{0},\ \ \ \ (x_{3}=\pm h)\] (25b)  
From the third equation of Equation (25), the potential condition leads to the following two equations. \[\varphi(h)=\frac{e_{33}u_{3}(h)+\mu_{3333}u_{3,3}(h)}{\varepsilon_{33}}+B_{1}h +B_{2}=\varphi_{0} \tag{26a}\] \[\varphi(-h)=\frac{e_{33}u_{3}(-h)+\mu_{3333}u_{3,3}(-h)}{\varepsilon_{33}}-B_{1}h +B_{2}=-\varphi_{0} \tag{26b}\]  
Then, \(B_{1}\) is obtained by the subtraction of the above two equations. \[B_{1}=\frac{\varphi_{0}}{h}-\frac{1}{2\varepsilon_{33}h}[\varepsilon_{33}(u_{3 }(h)-u_{3}(-h))+\mu_{3333}(u_{3,3}(h)-u_{3,3}(-h))] \tag{27}\]Similarly, the other two boundary conditions result in the following four equations at two surfaces. \[\left(c_{33}+\frac{e_{33}^{2}}{\varepsilon_{33}}\right)u_{3,3}(h)+e_{33}B_{1}- \frac{u_{3333}^{2}u_{3,333}(h)}{\varepsilon_{33}}=0 \tag{28a}\] \[\left(c_{33}+\frac{e_{33}^{2}}{\varepsilon_{33}}\right)u_{3,3}(-h)+e_{33}B_{1}- \frac{u_{3333}^{2}u_{3,333}(-h)}{\varepsilon_{33}}=0\] (28b) \[\frac{e_{33}\mu_{3333}u_{3,3}(h)+u_{3333}^{2}u_{3,33}(h)}{\varepsilon_{33}}+B_{1 }\mu_{3333}=0\] (28c) \[\frac{e_{33}\mu_{3333}u_{3,3}(-h)+u_{3333}^{2}u_{3,33}(-h)}{ \varepsilon_{33}}+B_{1}\mu_{3333}=0 \tag{28d}\]  
Substituting the displacement function and its derivatives into Equation (28) gives the following equations, with \(\varphi_{0}=0\) for free vibration. \[ \begin{array}{l}\left[\alpha\cosh(\alpha h)-k_{3}e_{33}\text{sinh}(\alpha h )-k_{1}\alpha^{3}\cosh(\alpha h)\right]A_{1}\\ +\left[\alpha\sinh(\alpha h)-k_{3}\mu_{3333}\text{sinh}(\alpha h)-k_{1}\alpha^ {3}\text{sinh}(\alpha h)\right]A_{2}\\ +\left[\beta\cos(\beta h)-k_{3}e_{33}\sin(\beta h)+k_{1}\beta^{3}\cos(\beta h) \right]A_{3}\\ +\left[-\beta\sin(\beta h)+k_{3}\mu_{3333}\beta\sin(\beta h)-k_{1}\beta^{3} \sin(\beta h)\right]A_{4}=0\end{array}  \tag{29a}\] \[ \begin{array}{l}\left[\alpha\cosh(\alpha h)-k_{3}e_{33}\text{sinh}(\alpha h )-k_{1}\alpha^{3}\cosh(\alpha h)\right]A_{1}\\ +\left[-\alpha\text{sinh}(\alpha h)-k_{3}\mu_{3333}\text{cosh}(\alpha h)+k_{1} \alpha^{3}\text{sinh}(\alpha h)\right]A_{2}\\ +\left[\beta\cos(\beta h)-k_{3}e_{33}\sin(\beta h)+k_{1}\beta^{3}\cos(\beta h )\right]A_{3}\\ +\left[\beta\sin(\beta h)+k_{3}\mu_{3333}\beta\sin(\beta h)+k_{1}\beta^{3} \sin(\beta h)\right]A_{4}=0\end{array} \] (29b) \[