## 5 Conclusions  
In this study, three different loading rates were designed, and cyclic loading tests were conducted under two paths of increasing and constant amplitude. The evolution of the dynamic shear modulus and damping ratio of silica rock under different strain amplitudes and loading rates was thoroughly investigated, and two damage models were subsequently established. The findings of this study demonstrate a complex relationship between the loading rate, damping ratio, and the mechanical behavior of rocks, which is crucial for understanding the response of rocks under various loading conditions. In particular, gaining a deeper understanding of the relationship between the loading rate and damping ratio is crucial for assessing the response of rocks under dynamic loads, such as earthquakes, and enhancing the safety of earthquake engineering. Future studies should continue to explore how multiple factors, including the loading rate, amplitude, and frequency, comprehensively influence the mechanical behavior of rocks to develop a more accurate rock mechanics model. This model can serve as a scientific basis for engineering design in earthquake engineering and related fields. The conclusions of this study are as follows:  
1. Under constant amplitude loading conditions, the shear modulus tended to increase during both loading and unloading as the amplitude interval increased. Conversely, the shear modulus in the transverse amplitude test exhibited a decreasing trend throughout the test phase. Additionally, the shear modulus ratios (Gr) of two neighboring cyclic groups exhibited a sudden decrease, suggesting that higher loading rates could result in larger Gr values.
2. The damping ratio of the diorite initially decreases and then increases with the number of cycles during the cycling process, a trend observed under both constant amplitude and increasing value loading paths. The mean values of the damping ratios for each cyclic group at the three different rates exhibited a stepwise decrease in response to the loading rate.
3. The strain-based damage model revealed that the rock damage was larger under low-rate loading conditions, suggesting that the loading rate had a significant effect on the damage accumulation process. The model fitting captures that the damage variables dynamically change with an increasing number of cycles following an S-shaped function, offering a quantitative description of the damage evolution process.
4. Drawing on the correlation between the calculated dissipated energy and the relative cyclic number, a new damage model was proposed in this study. The model accurately predicts the dissipated energy density damage accumulation process of silica rock under cyclic loading, offering an effective tool for assessing the damage of rocks under cyclic loading.  
Figure 12: Specim damage form diagram: (**a**) specimen X54; (**b**) specimen X48; (**c**) specimen X53; (**d**) specimen X43.  
**Author Contributions:** Conceptualization, [PERSON] ([PERSON]); methodology, C.L.; software, B.Z.; formal analysis, C.S.; resources, Z.T.; data curation, Y.Z.; writing--original draft preparation, [PERSON] ([PERSON]); writing--review and editing, [PERSON] ([PERSON]); funding acquisition, [PERSON] ([PERSON]). All authors have read and agreed to the published version of the manuscript.  
**Funding:** This article is funded by the National Natural Science Foundation of China (52174069), the Beijing Natural Science Foundation (8202033), and the Henan Key Laboratory for Green and Efficient Mining & Comprehensive Utilization of Mineral Resources (Henan Polytechnic University) (KCF2203).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
**Conflicts of Interest:** The authors declare no conflicts of interest.