## 2 Governing Equations  
Let us describe the mathematical problem and its governing equations for a long FG cylindrical panel with an inner and outer piezoelectric layer. The laminated panel features an opening angle \(\theta_{0}\), an inner radius \(R_{i}\), and an outer radius \(R_{o}\), whereby \(h_{c}\) is the FG layer thickness, \(h_{p}\) is the piezoelectric layer thickness, and for a total thickness \(h=h_{c}+2h_{p}\) (see Figure 1). A cylindrical coordinate system \((r,\,\theta,\,z)\) is chosen to identify the material points of the laminated panel in the undeformed configuration, as visible in Figure 1. The FG core layer is assumed to change gradually throughout the radial direction, namely, the mechanical properties vary continuously and smoothly from a ceramic-rich material (at the inner surface) to a metal-rich material (at the outer surface) according to a power law distribution. Thus, the effective material property \(P\) of the core is defined, accordingly, as  
\[P(r)=P_{c}+(P_{m}-P_{c})\left(\frac{r-R_{i}}{R_{o}-R_{i}}\right)^{n} \tag{1}\]  
where \(n\) is a positive real number associated with the power law index (or the material property gradient index). Long panels follow the plane strain assumptions, as employed thereinafter. Accordingly, the linear constitutive relations for the core (i.e., the FG layer) and the piezoelectric layers take the following form, respectively [30]:  
\[\left\{ \begin{array}{c}\sigma_{rr}^{c}\\ \sigma_{\theta\theta}^{c}\\ \sigma_{r\theta}^{c}\end{array} \right\}=\left[ \begin{array}{ccc}C_{11}^{c} &C_{12}^{c}&0\\ C_{12}^{c}&C_{22}^{c}&0\\ 0&0&C_{55}^{c}\end{array} \right]\left\{\begin{array}{c}\left( \begin{array}[] {c}\frac{\partial\ell^{c}}{\partial\ell}-\alpha^{c}\left(\Delta T\right)^{c} \\ \frac{1}{T}\left(\frac{\partial\ell^{c}}{\partial\ell}+u^{c}\right)-\alpha^{c} \left(\Delta T\right)^{c}\\ \frac{1}{T}\left(\frac{\partial\ell^{c}}{\partial\ell}-v^{c}\right)+\frac{ \partial\ell^{c}}{\partial\ell}\end{array} \right)\right\}, \tag{2a}\]  
Figure 1: (a) and (b): Geometry of the FG cylindrical panels embedded in piezoelectric layers. \[\left\{ \begin{array}{l}\sigma_{rr}^{p}\\ \sigma_{\theta\theta}^{p}\\ \sigma_{r\theta}^{p}\end{array} \right\}=\left[ \begin{array}{ccc}C_{11}^{p}&C_{ 12}^{p}&0\\ C_{12}^{p}&C_{22}^{p}&0\\ 0&0&C_{55}^{p}\end{array} \right]\left\{ \begin{array}{l}\frac{\partial d^{p} }{\partial t}-\alpha^{p}\left(\Delta T\right)^{p}\\ \frac{1}{r}\left(\frac{\partial d^{p}}{\partial\theta}+u^{p}\right)-\alpha^{p} \left(\Delta T\right)^{p}\\ \frac{1}{r}\left(\frac{\partial u^{p}}{\partial\theta}-v^{p}\right)+\frac{ \partial^{p}}{\partial\theta^{r}}\end{array} \right\}-\left\{ \begin{array}{l}0 \\ 0\\ \epsilon_{4}E_{\theta}\end{array} \right\}, \tag{2b}\]  
\[\left\{ \begin{array}{l}D_{r}\\ D_{\theta}\end{array} \right\}=\left[ \begin{array}{ccc}e_{3}&e_{2}&0\\ 0&0&e_{4}\end{array} \right]\left\{ \begin{array}{l}\frac{\partial d^{c}}{ \partial t}\\ \frac{1}{r}\left(\frac{\partial d^{c}}{\partial\theta}+u^{c}\right)\\ \frac{1}{r}\left(\frac{\partial d^{c}}{\partial\theta}-v^{r}\right)+\frac{ \partial^{p}}{\partial\theta^{r}}\end{array} \right\}+\left[ \begin{array}{cc} K_{33}&0\\ 0&K_{22}\end{array} \right]\left\{ \begin{array}{l}E_{r}\\ E_{\theta}\end{array} \right\}+\left\{ \begin{array}{l}P_{3}(\Delta T)^{c}\\ 0\end{array}