## 2 Methodology
### Piezoelectric Constitutive Equation of IEPCM  
The direction 1, 2, 3 of IEPCM is set as parallel to the \(z\)-axis, \(y\)-axis, and \(x\)-axis, respectively. Since the interdigitated finger electrode alternately appears on the same surface with the positive and negative electrodes (as shown in Figure 1), it makes the polarization centerline of the left half part electrode points to the negative direction of the \(x\)-axis, and the polarization centerline of other part points to the positive. Here, first, the polarized direction pointing to the positive part of the \(x\)-axis is considered. The piezoelectric equations related with the stress (**T**) and electric displacement (**D**) under the unconstrained boundary conditions and short-circuit state are constructed supposed the strain **S** and the electric field intensity **E** as independent variables. The constitutive relation between **D**, **T** matrix and **S**, **E** matrix are shown in Equations (5), (6-1) and (6-2), respectively. Second, if the constant values in Table 2 are taken as opposite ones, the corresponding piezoelectric equations with the polarized direction points to the negative part of the \(x\)-axis can also be constructed.  
\[\left\{\begin{array}{l}\mathbf{T}=\mathbf{c}^{E}\mathbf{S}-\mathbf{e} \mathbf{E}\\ \mathbf{D}=\mathbf{e}_{i}\mathbf{S}+\varepsilon^{\ast}\mathbf{E}\end{array}\right. \tag{5}\]  
\[\left[\begin{array}{l}T_{1}\\ T_{2}\\ T_{3}\\ T_{4}\\ T_{5}\\ T_{6}\end{array}\right]=\left[\begin{array}{cccccc}c_{33}^{E}&c_{33}^{E}&c_{3 3}^{E}&0&0&0\\ c_{33}^{E}&c_{33}^{E}&c_{33}^{E}&0&0&0\\ c_{33}^{E}&c_{33}^{E}&c_{33}^{E}&0&0&0\\ 0&0&0&c_{66}^{E}&0&0\\ 0&0&0&0&c_{44}^{E}&0\\ 0&0&0&0&0&c_{44}^{E}\\ \end{array}\right]\left[\begin{array}{cccc}S_{1}\\ S_{2}\\ S_{3}\\ S_{4}\\ S_{5}\\ S_{6}\end{array}\right]-\left[\begin{array}{cccc}e_{33}&0&0\\ e_{31}&0&0\\ e_{31}&0&0\\ 0&0&0\\ 0&0&e_{15}\\ \end{array}\right]\left[\begin{array}{cccc}E_{1}\\ E_{2}\\ E_{3}\\ \end{array}\right]\] (6-1)  
\begin{table}
\begin{tabular}{l l l l l l l l l l l l} \hline \hline
**Item** & **Value** & **Item** & **Value** & **Item** & **Value** & **Item** & **Value** & **Item** & **Value** & **Item** & **Value** \\ \hline \(C_{11}\) & 103.7 & & & & & & & & & (_Symmetry_) & \\ \(C_{12}\) & 67.0 & \(C_{22}\) & 119.1 & & & & & & & & \\ \(C_{13}\) & 67.0 & \(C_{23}\) & 70.2 & \(C_{33}\) & 119.1 & & & & & & \\ \(C_{14}\) & 0 & \(C_{24}\) & 0 & \(C_{34}\) & 0 & \(C_{44}\) & 23.5 & & & & \\ \(C_{15}\) & 0 & \(C_{25}\) & 0 & \(C_{35}\) & 0 & \(C_{45}\) & 0 & \(C_{55}\) & 28.0 & & \\ \(C_{16}\) & 0 & \(C_{26}\) & 0 & \(C_{36}\) & 0 & \(C_{46}\) & 0 & \(C_{56}\) & 0 & \(C_{66}\) & 23.5 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Elastic stiffness coefficient matrix in short circuit state of IEPCM **c** (GPa).  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**E-Field Direction** & **Stress Direction** & **X Axis** & **Y Axis** & **Z Axis** \\ \hline \(X\) & \(e_{11}\) (15.1) & \(e_{12}\) (0) & \(e_{13}\) (0) \\ \(Y\) & \(e_{21}\) (\(-5.2\)) & \(e_{22}\) (0) & \(e_{23}\) (0) \\ \(Z\) & \(e_{31}\) (\(-5.2\)) & \(e_{32}\) (0) & \(e_{33}\) (0) \\ \(XY\) & \(e_{41}\) (0) & \(e_{42}\) (12.7) & \(e_{43}\) (0) \\ \(YZ\) & \(e_{51}\) (0) & \(e_{52}\) (0) & \(\[\left[\begin{array}{c}D_{1}\\ D_{2}\\ D_{3}\end{array}\right]=\left[\begin{array}{cccccc}e_{33}&e_{31}&e_{31}&0&0&0\\ 0&0&0&0&e_{15}\\ 0&0&0&0&e_{15}&0\end{array}\right]\left[\begin{array}{c}S_{1}\\ S_{2}\\ S_{3}\\ S_{4}\\ S_{5}\\ S_{6}\end{array}\right]+\left[\begin{array}{ccc}\varepsilon_{33}^{S}&0&0\\ 0&\varepsilon_{11}^{S}&0\\ 0&\varepsilon_{11}^{S}\end{array}\right]\left[\begin{array}{c}E_{1}\\ E_{2}\\ E_{3}\end{array}\right]\] (6-2)