## 5 Discussion
### The Influence of Stress Wave Amplitude  
It is established that mine tremors with higher energy can straightforwardly induce dynamic disasters compared with those with lower energy. Thus, the amplitude of the stress wave has a significant effect on the dynamic stress response of a roadway. The failure mode and stress redistribution were studied by maintaining the frequency of the stress wave at 5000 Hz and setting its peak velocity to 2, 4, 6, 8, 10, and 12 m/s (see Figure 19). The final failure modes of the specimens at different impact velocities are illustrated in Figure 20. No cracks appeared at a velocity of 2 m/s. As the velocity increased to 4 m/s, tensile failure occurred in both horizontal and vertical directions. As the velocity increased to 6 m/s, shear failure began to occur at the angles of approximately 120\({}^{\circ}\) and 240\({}^{\circ}\) at the upper left and lower left corners around the hole. These continued to expand and converge into macro-cracks as the velocity increased to 8 m/s. When the velocity was 10 m/s, shear failure began to increase at the angles of approximately 60\({}^{\circ}\) and 300\({}^{\circ}\) in the  
Figure 18: Stress concentration factor under different frequencies. (**a**) DSCF around hole, (**b**) maximum stress concentration factor. upper right and lower right corners. X-shape shear failure occurred in the specimen when the velocity increased to 12 m/s. It can be concluded that the failure mode of the hole gradually transitioned from horizontal and vertical tensile failure to X-shape shear failure centered at the angles of 60\({}^{\circ}\), 120\({}^{\circ}\), 240\({}^{\circ}\), and 300\({}^{\circ}\) with an increase in impact velocity. Figure 19: Incident wave with different amplitudes. Figure 20: Failure mode around the hole under different amplitudes. The distribution of DSCF at different impact velocities is shown in Figure 21. With an increase in the impact velocity, the DSCF around the hole showed a decreasing trend (Figure 21a), the MCSCF decreased gradually, and the MTSCF first increased and then decreased. In addition, the peak value of compressive stress was essentially concentrated at the angles of 90\({}^{\circ}\) and 270\({}^{\circ}\) (i.e., in the vertical direction) under a relatively lower velocity of 2-6 m/s. As the velocity increased to 8-12 m/s, the peak value of compressive stress began to deflect to varying degrees from the vertical direction to both sides. This induced the formation of X-shape shear failure. Meanwhile, the tensile stress peak remained concentrated in the directions of 0\({}^{\circ}\) and 180\({}^{\circ}\) without significant variations. A comparison of Figures 20 and 21a reveals that the area of the compressive stress peak concentration approximately corresponds to the area of shear fracture propagation. With the increase in impact velocity, the distribution of DSCF gradually approaches the butterfly-type distribution, which corresponds to the final X-shape shear failure. To summarize, the evolution of the hoop stress and failure mode around the hole can be determined. With an increase in impact velocity, the compression shear stress and  
Figure 21: Variation in stress concentration factor under different amplitudes. (**a**) DSCF around the holes under different amplitudes, (**b**) maximum stress concentration factor for different amplitudes. tensile stress were concentrated in the vertical and horizontal areas, respectively, of the hole subjected to the instantaneous incidence of compressive stress waves. Because the tensile strength of coal is significantly lower than its compressive strength, horizontal tensile failures were generated first under a lower impact velocity. As the impact velocity increased further, the maximum compressive stress gradually attained its compressive strength and began to deflect from the vertical direction to both sides. The hoop stress around the hole gradually approached a butterfly-type distribution. This caused the occurrence of X-shape shear failure.