## 3 Results and Discussions
### Elastic Composites with Square Inclusions and Epoxy Host
The simulation results show that increasing the number of inclusions not only increased the bandwidth but resulted in an upward shift in the frequency range. The bandwidth was increased from 1316 Hz for one inclusion to 2678 Hz and 3915 Hz for four and nine inclusions. Interestingly, the LBF and UBF show a nonmonotonic relationship with the filling fraction, as shown in Figure 2c,d. This relationship causes the bandwidth of the bandgap to reach a maximum at a filling factor of 0.6. 3. Eigenmode analysis. To analyze what causes the LBF and UBF of the tungsten carbide epoxy phononic crystal to not follow the same trend with filling fractions, their deformation distributions of eigenmodes at the lowest and highest frequencies of the bandgap were calculated, as shown in Figure 3. The simulation results show that at the lower frequency near the bandgap, the localized elastic wave modes are excited and the inclusion/epoxy vibrations behave as mass-spring oscillators [39], as indicated in Figure 3b,c. The phononic crystal exhibits a bandgap at wavelengths less than gap width among inclusions and inclusion sizes [4]. The nonmonotonic shift of the LBF results from two different trend of inclusion sizes and the width among inclusions with increase in the filling fraction. Therefore, LBF has a minimal value when the filling fraction is near 0.4 for a different number of inclusions within a periodic unit cell, as shown in Figure 2c. Meanwhile, according to a previous study conducted by [PERSON] et al. [40], local resonances mainly depend on the resonator natural frequency and total mass density ratio. Therefore, the YMR does not have a pronounced effect on the LBF, as shown in Figure A1a. Figure 2: (**a**) The dispersion curves generated for a phononic crystal with four-square inclusions with FF = 0.5. (**b**) The dispersion curves generated for a phononic crystal with nine-square inclusions with FF = 0.5. (**c**–**e**) lower bound frequency (LBF), upper bound frequency (UBF), and bandwidth versus filling fraction for different unit cells with one-, four-, and nine-square inclusions. At the UBF near the bandgap, elastic Bragg scattering effects were observed between the square inclusion and matrix at a higher frequency, as shown in Figure 3d,e. The Bragg scattering phenomena were induced by destructive interferences when the wavelength of the elastic wave was comparable to the periodicity of the scatterers embedded in the host elastic medium [41; 42]. The first band gap was opened at wavelengths larger than the wavelength allowed by the Bragg mechanism, which is also supported by [PERSON]'s study [43]. Therefore, the observed wide broad bandgap in this study resulted from the coupling between two types of mechanisms [4]. The introduction of additional inclusions at a constant FF, as discussed in the previous section, reduced the periodicity of the inclusions, gap width among inclusions, and inclusion sizes. Thus, the reduction in these dimensions led to higher UBF and LBF, as shown in Figure 2c-e.  
Figure 3: (**a**) Dispersion relations of the phononic crystal with four-square inclusions. (**b–e**) Deformation distributions of eigenmodes in the phononic crystal, corresponding to four different points in the k-space near the bandgap.