## 4 Results and Analysis
### Frequency Bandgaps of Particle Rotation Waves  
Numerous numerical and experimental results indicated that there are frequency bandgaps, in which no waves can propagate, in granular media because of the effect of microstructure. The presented micromorphic model can predict well the presence of frequency bandgaps. Figure 4 illustrates the presence of frequency bandgaps between particle rotation waves. For rotation longitudinal and transverse waves, there are two frequency bandgaps, respectively, i.e., one between acoustic and optical branches (e.g., RLB1) and another between two optical branches (e.g., RLB2). Generally, the width of RLB1 (or RTB1) is larger than that of RLB2 (or RTB2). There is also a frequency bandgap (RSB) between the two TSO branches. Besides, it is expected that the presented model can predict the existence of a bandgap (pink area) in which all the particle rotation waves cannot propagate in the granular media. This property of granular materials can be used to develop new materials for the field of energy absorption or vibration dissipation.  
Figure 5 shows the variation of frequency bandgaps with stiffness ratio \(\eta\). As shown in Figure 5, the contact stiffness ratio has a significant effect on the frequency bandgaps in both width and location (the frequency corresponding to the center of the bandgap). For the bandgaps of longitudinal waves, with the tangential to normal stiffness ratio increasing from 0 to 1, the asymptotic frequency of LA branch slightly increases, and the cutoff frequency of lower-frequency LO branch also increases, resulting in an increase in both the width and the location of bandgap RLB1. On the contrary, the width of bandgap RLB2 gradually decreases to 0, the center of which almost remains constant. The bandwidth and location of RTB1 between the TA branch and the lower-frequency TO branch significantly increase from 0 with regard to stiffness ratio. It is worth noting that the bandwidth of RTB1 becomes zero at the case of \(\eta=0\), where the lower-frequency TO branch disappears. The bandwidth of RTB2 gradually decreases to 0, while its location moves upwards as the stiffness ratio increases. The variation of RSB with stiffness ratio is similar to that of RTB2.  
Figure 6 shows the effect of fabric anisotropy on the frequency bandgaps of particle rotation waves. One can see that the anisotropic parameter \(a\) has a significant effect on the width of frequency bandgap of longitudinal waves, while it has little effect on the width of frequency bandgap of transverse and in-plane shear waves. Specifically, with the change in \(a\) from 0 to 1, the frequency of lower boundary of bandgap RLB1 continuously increases, while the upper boundary first increases slightly and then decreases, which leads to an overall decrease in the bandwidth of RLB1. Especially, it disappears when \(a=1.0\). The bandwidth of RLB2, however, continuously widens due to the decrease in lower boundary and the increase in upper boundary. With the increase in \(a\), the locations of bandgaps between transverse waves gradually increase, while the location of the bandgap between in-plane shear waves decreases.  
Figure 4: Frequency bandgap between the particle rotation waves: longitudinal waves (**left**), transverse waves (**middle**), and in-plane shear waves (**right**).  
Figure 5: Variation of frequency bandgaps with stiffness ratio \(\eta\): longitudinal waves (**left**), transverse waves (**middle**), and in-plane shear waves (**right**).