## 5 Conclusions  
In this paper, a large triaxial numerical simulation test under different densities is carried out with rockfill materials as the research object, and the deformation and damage laws of rockfill materials in the initial, phase transition, peak, and critical states are analyzed from macroscopic and microscopic points of view. The following conclusions are drawn:  
1. The stress-strain curve characteristics of rockfill materials can be divided into softening and hardening types. The softening type curve shows different mechanical properties in the initial, phase transition, peak, and critical states, resulting in significant shear deformation. The hardening type curve in the loading process of deviatoric stress continues to increase, and there is no obvious distinction between the phase transition, peak, and critical states; at the same time, more significant shear shrinkage deformation is produced.
2. The softening curve in the fine view shows that the displacement of the upper and lower ends of the shell unit on the flexible boundary increases with increasing loading, and the displacement of the middle ends decreases; however, the rotational deformation of the particles in the middle part of the specimen increases, and there is an \"X\"-type shear band. The strength of the force chain tends to increase and subsequently decrease. The coordination number is closely related to the deformation of the specimen, and it tends to increase first and subsequently decrease. The coordination number is closely related to the deformation of the specimen, showing a tendency to increase and then decrease, which corresponds to the macroscopic deformation of the specimen of shear shrinkage and then shear expansion.
3. According to the fine view of the hardening curve, with the loading of the flexible boundary on the shell unit, the softening type curve is similar to the performance of the upper and lower ends of the displacement and is large in the middle of the displacement, but the overall displacement is small. Similarly, for the specimens in the middle of the rotational deformation and the emergence of more cleavage zones rather than shear zones, the strength of the force chain continues to increase, and the coordination number of the first increases and then tends to stabilize, corresponding to macroscopic shear shrinkage deformation of the specimen.  
The authors confirm their contributions to the paper as follows: study conception and design: Y.C. and L.Z.; data collection: Y.C. and R.Z.; analysis and interpretation of results: Y.C. and L.Z.; draft manuscript preparation: Y.C. and C.S. All authors have read and agreed to the published version of the manuscript.  
The work was supported by the National Natural Science Foundation of China (Grants Nos. 52109136, 52379100), Outstanding Youth Science Fund Project of Xinjiang Uygur Autonomous Region of China (Grants Nos. 2022D01E45), Key R & D Tasks of Xinjiang Uygur Autonomous Region of China (Grants Nos. 2022B03024-3), The Belt and Road Special Foundation of the State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering (Grants Nos. 2020492211) and Postgraduate research project of Xinjiang Key Laboratory of Water Conservancy Engineering Safety and Water Disaster Prevention in 2022 of China (Grants Nos. ZDSYS-YJS-2022-16). The authors express their appreciation for the financial assistance.  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
**Acknowledgments:** The authors would like to express their appreciation to the editor and reviewers for their valuable suggestions, which greatly improved the presentation of this paper.  
**Conflicts of Interest:** The authors declare no conflicts of interest.