## 3 Results and Discussions  
### Elastic Composites with Square Inclusions and Epoxy Host  
1. Single-square inclusion/epoxy host with different Young's moduli. Firstly, the unit cell composed of a single-square inclusion within a soft host was studied by varying its filling fraction (FF). The filling fraction was defined as \(\text{FF}=(a_{i})^{2}/(a_{o})^{2}\), where \(a_{i}\) is the side length of the square inclusion and \(a_{o}\) is the unit cell length, as shown in Figure 1b. The length \(a_{o}\) was kept constant at 1 m, and the \(a_{i}\) was varied from 0.316 to 0.948 m, corresponding to different FFs from 0.1 to 0.9. The effect of the stiffness of the inclusion in the matrix was studied by changing its Young's modulus ratio (YMR), defined as \(\text{YMR}=E_{i}/E_{o}\), where \(E_{i}\) is [PERSON]'s modulus of the inclusion material and \(E_{o}\) is [PERSON]'s modulus of the host material. The material considered here for the matrix is epoxy with \(E_{o}\) = 4.3438 GPa, and [PERSON]'s modulus of the inclusion \(E_{i}\) was varied to yield YMRs of 9, 90, and 300. Figure 1c show the dispersion curves of phononic crystals with the single-square inclusion at FF = 0.5, where the shaded region represents the bandgap and points A and B are located at the maximum and minimum frequencies of the bandgap, respectively. Figure 1d shows that in the case of YMR = 300, the bandwidth increases as the filling fraction increases, reaching a maximum of 3280 Hz at FF = 0.8, after which it starts to decrease with any further increase in FF. When YMR was 9, the elastic composite bandwidth showed a slight change with an increased filling factor value, which results from the lack of the significant contrast in the elastic stiffness of the inclusion to that of the matrix. In general, a higher YMR of the elastic composite has a broader bandgap, and its bandwidth increases and reaches a maximum point at a specific FF. Then, the bandwidth would decrease with increase in its FF. The relationship between bandwidth and filling fractions are strongly related owing to the corresponding changes in geometric dimensions in the periodic unit cell, which affects the resonance frequency of the elastic eigenmodes [26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38]. In Figure 1a,b of Appendix A, the lower bound frequency (LBF) and upper bound frequency (UBF) are plotted against the filling fraction of the square inclusion at YMR = 9, 90, and 300. The simulation result shows that the relationship between UBF and filling fraction followed a similar trend to that of the bandwidth in Figure 1d, which indicates that the variations in bandwidth are more dominated by UBF, and this difference between LBF and UBF would be discussed based on the eigenmode analysis in the part (iii) of Section 3.1. 2. Increasing number of square inclusions within a unit cell. To target a wider bandgap, in the following sections, the YMR of the inclusion/matrix was fixed at 90, corresponding to the ratio of tungsten carbide to the epoxy. The phononic crystals discussed above consisted only of one inclusion in each unit cell. Here, the number of square inclusions within the unit cell was increased to four and nine to investigate the effect of the inclusion number within the unit cell on the phononic band structure. The unit cells were designed as shown in the inset of Figure 2a,b, where the filling fraction was calculated as \(\text{FF}=4(a_{i})^{2}/(a_{o})^{2}\) in the case of four-square inclusions within the unit cell and \(\text{FF}=9(a_{i})^{2}/(a_{o})^{2}\) in the case of nine-square inclusions within the unit cell. The relationship between bandgap bandwidth and their filling fractions of three different unit cells was studied by changing \(a_{i}\) while fixing the \(a_{o}\) at 1 m.  
The dispersion curves of phononic crystals with the four- and nine-square inclusions are shown in Figure 2a,b, where the shaded region represents the bandgap.