## 3 Testing Results
### Damping Ratio Analysis  
In the process of cyclic loading and unloading, the sillimanite specimen has already shown obvious local damage, resulting in irrecoverable macroscopic cracks. This may promote the expansion of these cracks during further cyclic loading and unloading, leading to a decreased damping ratio as crack expansion diminishes the rock's overall energy absorption capacity [34; 35]. As the number of cyclic loadings increases, the damping ratio of silica rocks gradually decreases, attributed to the progressive expansion and coalescence of microscopic cracks and fissures within the rocks. This process results in a gradual reduction in the rocks' stiffness and strength, subsequently leading to a decrease in the  
Figure 7: Shear modulus versus number of cycles: (**a**) evolutionary pattern of shear modulus for specimen X54; (**b**) evolutionary pattern of shear modulus for specimen X48; (**c**) evolutionary pattern of shear modulus for specimen X53; (**d**) evolutionary pattern of shear modulus for specimen X43. damping ratio. After reaching a certain number of cyclic loadings, the damping ratio of skarn tends to stabilize. At this stage, the cracks and microstructure within the rock become relatively stable, and the development and coalescence rate of the cracks slow, resulting in slower changes to the damping ratio. Prolonged cyclic loading may lead to continual expansion [36] and coalescence of cracks within the rock, drastically reducing its strength and stiffness, which typically leads to a sharp decline in the damping ratio as the rock nears a state of damage, and this decline may coincide with the rock's destruction. Under cyclic loading, the damping ratio of the rock will follow a 'concave' pattern, decreasing and then gradually increasing, which reflects the rock's progression from compaction and elastic deformation to crack expansion and destruction. Figure 8 illustrates the relationship between the damping ratio and the number of cycles for the four specimens, and the comparison indicates that the damping ratio of silica exhibits a trend to first decrease and then increase with the increase in the number of cycles throughout the cycling stage. As illustrated in Figure 8a, the damping ratio initially decreases in the first five-cycle stage, and with the increase in the loading rate to 4 kN/s, the damping ratio experiences a sharp decrease and then stabilizes, and the damping ratio experiences a discontinuous decrease when the rate reaches 6 kN/s. Observations of the damping ratio change in the subsequent unused cycle group indicate that the rate-dependent behavior is consistent with that of the first cycle group, and this behavior is also observed in the other three specimens. The red bar graph in Figure 8 represents the mean damping ratio at each rate, whether at constant or increasing amplitude. The mean damping ratio at the three rates in each cycle group shows a stepwise decrease, suggesting that the loading rate significantly influences the damping ratio; specifically, higher rates result in lower damping ratios and correspondingly lower energy dissipation. In summary, the trends in the damping ratios of silica rocks under cyclic loading correlate with internal damage and crack development, as well as the loading rate. These trends are crucial for understanding and predicting the long-term stability of rocks under dynamic loading. Table 2 displays the fitting equation for the mean damping ratio of a single sample at a consistent rate. The fitting curve represents a logarithmic function, and the correlation coefficients listed in the table suggest a high degree of fit between them.