## 3 Numerical Result of Bandgap Characteristics
### Effect of the Initial Tensile Stress on the Bandgap Characteristics  
Initial tensile stress occurs when the CCPM is stretched. Figure 10a,b show the changes in the bandgap diagrams with and without considering initial stress corresponding to the tensile displacement of the CCPM of 0 mm, 5 mm, 10 mm, 15 mm, 20 mm, 25 mm, and 30 mm from left to right. By comparing Figure 10a,b, it is indicated that initial tensile stress has obvious great effects on the bandgap characteristics. Figure 9: Specific bandgap variations: (**a**) first initial bandgap; (**b**) second additional bandgap. \begin{table}
\begin{tabular}{c c} \hline \hline
**Loading** & **Loading** \\
**Displacement** & **10 mm** \\ \hline
**Bandgap results** & **Bandgap results** \\
**obtained from** & **obtained from** \\
**reference [11]** & **Reference** \\ \hline
**Bandgap results** & **Bandgap results** \\
**obtained from** & **obtained from** \\
**the FEM in this** & **Reference** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: A comparison with equivalent research [11]. As shown in Figure 10b, the tensile loading displacement increases from 0 mm to 30 mm. The first initial bandgap (colored in yellow) begins to contract and move towards a low frequency. And the second additional bandgap (colored in red) appears and disappears again. However, in Figure 10a, as the initial tensile stress increases, the first initial bandgap (colored in yellow) begins to contract and move upwards. And the second additional bandgap (colored in red) and the third bandgap (colored in green) are opened gradually. The third additional bandgap has not appeared in Figure 10b. Based on Figure 10, the third additional bandgap (colored in green) appears and begins to widen when the tensile displacement exceeds 20 mm (the average axial average initial stress reaches 49,406 Pa). Compared to Figure 10b, the initial stress can close the second additional bandgap in advance, as shown in Figure 10a. Moreover, the first initial bandgap is closed at the tensile displacement of 30 mm (the average axial initial stress reaches 96,261 Pa). The detailed bandgap distribution of CCPM under different tensile displacements is illustrated in Figure 11. The width of the first initial bandgap gradually decreases as the tensile displacement increases [see Figure 11a]. Considering initial stress, the beginning and ending frequencies of the first initial bandgap both move towards high frequencies, as shown in Figure 11a. And the bandgap closes when the tensile displacement is 30 mm (the average axial initial stress reaches 96,261 Pa). Next, Figure 11b investigates the changes in the second additional bandgap. It begins to appear with an increase in tensile displacement. The width of the bandgap increases from 0 Hz to a maximum of 11 Hz and decreases again without initial stress. However, when the initial stress is taken into account, its bandgap width is significantly reduced at the stages of individual deformations. Both the beginning and ending frequencies shift up. As is seen, the second additional bandgap will be closed when the tensile displacement is 20 mm (average axial initial stress is 49,406 Pa). Figure 11: Specific bandgap variations: (**a**) first initial bandgap; (**b**) second additional bandgap. Figure 10: The bandgap of the CCPM at different tensile displacements: (**a**) with initial stress; (**b**) without initial stress. The variation in the center frequency of the first initial bandgap with displacement is plotted [see Figure 12] to demonstrate the dependence of initial stress on the position of bandgaps. As can be seen from Figure 12, considering the initial stress during compression causes the center frequency of the bandgap of the structure to drop, while considering the initial stress during stretching is the opposite. This is consistent with the results reported by [PERSON] [39]. The changing trend of center frequency about tensile is the opposite. With a continuous increase in tensile displacement, the center frequency of the bandgap with initial stress increases more.