## 4 Analysis of Simulation Results  
### Stress Variation  
The hoop stresses are higher than the radial stresses in an opening subjected to dynamic loading. Furthermore, the local concentration of hoop stress results in surrounding rock failure. Figure 10 shows the evolution of cracks around the hole. The hoop stresses can be monitored by measuring circles arranged around the hole, as shown in Figure 11. The duration of the incident/reflected stress waves can be estimated by monitoring the variation in the x-direction velocity of point A in Figure 11, as shown in Figure 12.  
Figure 10: Evolution of cracks around the hole.  
Figure 9: Final failure mode around the hole.  
The duration from crack generation to the final failure of the specimen was 220-600 us. Figure 13 shows the temporal and spatial variations in the hoop stresses around the hole in the period. According to Figure 11, the x-direction velocity of point A was consistent with the incident direction during 220-360 us. This indicates that the incident stress wave reached the hole for the first time, thereby causing hoop stress concentration at the edge of the hole. The compressive stress was concentrated at the angles of 30-150 and 210-330 with peak stress of 30 MPa. Meanwhile, the tensile stress was concentrated at angles of 150-210, 0-30 and 330-360 with peak stress of 2 MPa. At 320 us, a crack started to initiate at 180 from the hole edge (Figure 10). The tangential tensile stress exceeded the tensile strength of the specimen, which resulted in the generation of cracks in this area. The tensile crack was first generated at an angle of 180 because the hoop tensile stress exceeded the tensile strength of the specimen at 320 us. There were a few cracks in the compression stress concentration area. This indicated that the compressive hoop stress did not attain its compressive strength. Correspondingly, the number of cracks increased abruptly (Figure 12) during 320-360 us. This stage mainly manifested the generation and expansion of horizontal cracks. In the period 360-456 us, the x-direction velocity of point A was opposite to the incident direction owing to the reflection of the stress wave transmitted to the transmission bar. Accordingly, the tensile stress concentration started to gather and form at the angles of 90 and 270 around the hole, and cracks began to be generated in this area. Therefore, the cracks started to rapidly add again (see Figure 10) during 396-456 us. This stage is the generation and propagation of vertical cracks. During 456-545 us (Figure 12), the stress wave was reflected again when it arrived at the incident bar. However, the stress wave amplitude reduced significantly owing to most of the energy consumption for crack propagation and friction loss. As shown in Figure 12, the tensile stress concentration was transferred to the area at the angles of 0-60 and 270-330. However, the cracks increased less owing to the low strength of the stress wave.  
Figure 11: Layout of monitoring point and measuring circles.  
Figure 12: X–direction velocity of monitoring point and crack number.  
To conclude, the formation of horizontal and vertical macrocracks in the specimen is a result of the tensile hoop stress concentration around the hole caused by multiple reflections of the incident stress wave. For the first incidence, the compressive stress and tensile stresses were concentrated in the areas of 90\({}^{\circ}\) and 270\({}^{\circ}\) and in the areas of 0\({}^{\circ}\) and 180\({}^{\circ}\), respectively. Subsequently, the tensile stress concentration area was transferred to the vertical direction owing to reflection. This resulted in tensile failure in different areas. The stress wave strength decreased gradually after multiple reflections, crack growth, and friction damping consumption, and the failure of the specimen stopped.