# A Thorough Investigation of the Dynamic Properties of Granite under Cyclic Loading
## 4 Results and Discussion
### Dynamic Elastic Modulus and Dynamic Damping Ratio of Specimens
#### 4.2.2 Effect of Loading Frequency  
Figure 11 presents the variations in the dynamic elastic modulus of the specimens. Figure 11a illustrates the influence of the dynamic elastic modulus of the specimens concerning the frequency, dynamic stress amplitude, and number of cycles. It reveals that the dynamic elastic modulus of the specimens decreased with the increasing number of cycles, while it increased with the increasing frequency and dynamic stress amplitude. At frequencies ranging from 1 to 10 Hz, the dynamic elastic modulus of the specimens exhibits a similar trend at the same dynamic stress amplitude, with relatively small fluctuations across the different number of cycles. However, at the frequency of 20 Hz, the dynamic elastic modulus of the specimens experienced a notable increase, with considerable fluctuations observed for each number of cycles. Specifically, the dynamic elastic modulus of the specimen increased by approximately 7% from 1 Hz to 5 Hz, by about 28% from 1 Hz to 10 Hz, and significantly rose by around 80% from 1 Hz to 20 Hz. To portray more effectively the trend of the dynamic elastic modulus with varying frequencies, we depict the variation of the average dynamic elastic modulus with increasing frequency at different dynamic stress amplitudes using black curves in Figure 11b, with the data points corresponding to the black y-axis on the left. As the frequency increases, the dynamic elastic modulus of the specimen demonstrates a rising trend, and we conjecture that the relationship between them follows a quadratic polynomial pattern. The blue curve, acquired through data normalization, adequately represents the trend of the average dynamic elastic modulus of the specimens, with the data points corresponding to the blue y-axis on the right. Furthermore,  
Figure 10: Dynamic damping ratio of specimens with varying confining pressure: (**a**) the dynamic damping ratio of the specimens; (**b**) the average dynamic damping ratio of the specimens. it can be observed that the rate of growth of the dynamic damping ratio of the specimen decelerates as the frequency increases, especially at high dynamic stress amplitude. During the frequency loading, the particles generated inside the specimens infiltrated into the microscopic cracks, thereby inducing the closure of these cracks [54]. Additionally, the viscosity of the fluid between the mineral particles inside the specimens exhibited an increase with the rise in loading frequency. Consequently, these combined effects enhanced the specimens' resistance to deformation, resulting in an increase in the dynamic elastic modulus. The dynamic damping ratio of the specimens is presented in Figure 12. Figure 12a illustrates the impact of frequency, dynamic stress amplitude, and number of cycles on the dynamic damping ratio of the specimens. The dynamic damping ratio decreases with an increase in the number of cycles, rises with an increase in frequency, and declines with an increase in dynamic stress amplitude. Notably, at frequencies of 1-10 Hz, the dynamic damping ratio undergoes minor variations across the number of cycles. However, at a frequency of 20 Hz, the dynamic damping ratio exhibits a substantial increase, and its variation range becomes more pronounced as the number of cycles progress. To elucidate the dynamic damping ratio trend with different frequencies, the average dynamic damping ratio of the specimens with increasing frequency at various dynamic stress amplitudes is depicted using black curves in Figure 12b. A positive correlation between the dynamic damping ratio and frequency is observed. We hypothesize a power function relationship between them. The nearly overlapping black curves suggest a consistent growth rate of the dynamic damping ratio among the specimens. The experimental findings align with those of [PERSON] et al. [55], where it was noted that the accumulation of fine damage within the rock increased with elevated loading frequency, indicating enhanced energy dissipation inside the rock.