## 4 Analysis of Simulation Results
### Variation of Force Chain  
In PFC, the specimen consists of many rigid particles bonded by deformable and destructible contact. Furthermore, the failure of the specimen is manifested by contact fracture. Tensile cracks form when the normal stress \(\sigma\) of the contact exceeds the normal strength limit \(\sigma_{c}\). Shear cracks are generated when the shear stress \(\tau\) is larger than the shear strength limit \(\tau_{c}\). Tension crack: \(\lfloor\sigma\rfloor\geq\sigma_{c}\), \(\sigma<0\);  
Shear crack: \(\lfloor\tau\rfloor\geq\tau_{c}\). Thus, the failure of the specimen was directly caused by an increase or decrease in the contact force in the different areas. The simulation focused on the process of stress concentration and damage around the hole caused by the incident stress wave. Therefore, the normal and shear force chains in an annular area around the hole with a thickness of 0.01 m at different times were extracted. The monitoring area is shown in Figure 11.  
The normal-force evolution of the contact in the annular region is shown in Figure 14. Tensile cracks occurred only when the normal force was negative. Before the arrival of the incident stress wave, the normal force of contact was a compressive chain with a value closer to zero (Figure 14a). This was because the internal particles of the specimen after the initial formation were relatively loose with weak interactions. Subsequently, the particles began to compress each other, and the normal force increased accordingly with the effect of the stress wave at 260 \(\upmu\)s once the front end of the stress wave arrived. The compressive stress converged from the incident end at the positions of 0\({}^{\circ}\) to 90\({}^{\circ}\) and 270\({}^{\circ}\) during 270-280 ms. During 290-300 \(\upmu\)s, the compressive force at the positions of 90\({}^{\circ}\) and 270\({}^{\circ}\) gradually attained the peak (red force chain). However, no crack was generated because the compressive stress did not exceed the peak strength of the specimen. During 300-320 \(\upmu\)s, the tensile force began to increase centrally in the region of 180\({}^{\circ}\) and attained a peak at 320 \(\upmu\)s. Correspondingly, tensile cracks appeared and grew in the horizontal direction during 330-340 \(\upmu\)s. At 400 \(\upmu\)s, the stress wave was reflected after being transmitted to the transmission bar. The corresponding tensile stress area was transferred to the areas  
Figure 13: Temporal and spatial variation in hoop stress. Note: positive and negative values represent tensile stress and compressive stress, respectively. of 90\({}^{\circ}\) and 270\({}^{\circ}\). Thereafter, vertical tensile cracks began to appear. Finally, macrofractures were formed in the horizontal and vertical directions at 440 \(\upmu\)s.  
Similarly, the shear force distribution of the contact in the annular area followed clear laws before and after the incidence of the stress wave. The shear force was low before the stress wave incidence (Figure 15a). It began to increase at 260 \(\upmu\)s owing to the arrival of the stress wave at the front end. During 260-310 \(\upmu\)s, the stress wave was transmitted from the  
Figure 14: Evolution of normal force chain. right to the left in the specimen. Furthermore, the shear force transferred from 0\({}^{\circ}\) to 90\({}^{\circ}\) and 270\({}^{\circ}\) and attained the peak value. Similar to the distribution of the compressive stress peak, the shear force peak was concentrated in the regions of 90\({}^{\circ}\) and 270\({}^{\circ}\). No shear crack appeared because the shear force did not exceed the shear strength limit. At 400 us, the stress wave attained the left end of the sample and formed a tensile wave. A decrease in the normal compressive stress in the specimen caused a reduction in the shear force. Finally, the shear force in the entire annular region decreased significantly and was close to zero at 440 us. However, the shear force in the regions of 0\({}^{\circ}\) and 180\({}^{\circ}\) remained constant at zero.