## 5 Conclusions  
In this paper, a micromechanical model incorporating the effect of microstructure is developed for transversely isotropic granular media and then applied to study the propagational characteristics of particle rotation waves, i.e., dispersion characteristics and frequency bandgaps. From the point of particle kinematics, we take the particle translation and rotation as independent variables and ignore the particle displacement at contact due to particle rotation. Hence, the total deformation energy of granular media can be expressed as the sum of microscale deformation energy, which can be easily calculated by introducing contact constitutive laws, over all contacts. At macroscale level, the relative deformation tensors are defined to describe the local fluctuation in deformations. The macroscale deformation energy can be assumed to be the product of macroscale deformations and their corresponding stress measures. The governing equations and corresponding boundary conditions can be obtained based on [PERSON]'s principle. Subsequently, based on micro-macro energy conservation, the constitutive relations are derived through transferring the summation into an integral and introducing the contact fabric tensor. Then, the dispersion relations and bandgap features of particle rotation waves in transversely isotropic granular media are analyzed, and finally a detailed parametric analysis of the dispersion relations and frequency bandgaps is carried out. The main conclusions in this paper are summarized as follows:  
1. The present model can predict 12 particle rotation waves, including 3 rotation longitudinal waves, 6 rotation transverse waves, and 3 rotation in-plane shear waves in the transversely isotropic granular media. These waves can reflect eight kinds of dispersion relations. 2. The effect of the change in fabric on the dispersion relation of particle rotation waves can be mainly attributed to the effect of equivalent stiffness on frequency. With the increase in \(a\), the frequencies of rotation longitudinal and transverse waves increase, except for the lower-frequency LO wave, while all rotation in-plane shear waves decrease. 3. The present model can predict the presence of bandgaps between particle rotation waves. With the increase in stiffness ratio from 0 to 1, the bandwidths of RLB1 and RTB1 gradually increase, while that of other bandgaps decrease to 0. 4. The degree of anisotropy has significant effect on the width of frequency bandgap of longitudinal waves, while has little effect on the width of frequency bandgap of transverse and in-plane shear waves. 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. Writing--original draft preparation and investigation, P.Y. ; conceptualization, methodology, supervision and funding acquisition, Y.L. ; writing--review and editing, M.S. and Z.C. All authors have read and agreed to the published version of the manuscript. This research was funded by Beijing Natural Science Foundation, grant number 8222020. Figure 6: Variation of frequency bandgaps with anisotropic parameter \(a\): longitudinal waves (**left**), transverse waves (**middle**), and in-plane shear waves (**right**). **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** The data used to support the findings in this paper are available from the corresponding author upon request. **Acknowledgments:** The authors would like to express gratitude to [PERSON] for his valuable advice and help in formula derivation. **Conflicts of Interest:** The authors declare no conflict of interest. **Abbreviations**