## 4 Numerical Simulation and Results Analysis
### Numerical Results Analysis  
Figures 14 and 15 show the evolution of mises stress in the medium containing the unfilled joint with an angle of 90\({}^{\circ}\) and 45\({}^{\circ}\), respectively. It can be seen that the unfilled joint severely obstructed the propagation of blast waves. When the blast wave encountered the joint, a strong reflected wave was generated and altered the stress field between the borehole and the joint, and thus significantly influenced the propagation behavior of the main crack. In addition, the results also show that the stress concentration at the tip of vertical joint attenuated rapidly with the spreading of the diffraction wave, whereas the stress concentration at the far tip of the oblique joint sustained much longer compared with the stress concentration at the end of the vertical joint. This also certified that the wing crack at the far end of the oblique joint may extend much longer than that at the end of the vertical joint. Figure 14: _Cont._  
Figure 13: The typical mesh of the numerical model around the borehole and the unfilled joint: (**a**) typical mesh around the borehole; (**b**) typical mesh around the unfilled joint with 45\({}^{\circ}\). Figures 16-18 show the stress distribution characteristic around the vertical joint for the time of 30 \(\upmu\)s, 40 \(\upmu\)s and 70 \(\upmu\)s, respectively. S11, S22 and S12 represent the normal stress in the x direction, normal stress in the y direction and in-plane shear stress, respectively. It can be seen that when the  
Figure 14: Medium with a vertical unfilled joint: (**a**) 10 \(\upmu\)s; (**b**) 20 \(\upmu\)s; (**c**) 30 \(\upmu\)s; (**d**) 40 \(\upmu\)s; (**e**) 50 \(\upmu\)s; (**f**) 60 \(\upmu\)s.  
Figure 15: Medium with oblique unfilled joint: (**a**) 10 \(\upmu\)s; (**b**) 20 \(\upmu\)s; (**c**) 30 \(\upmu\)s; (**d**) 40 \(\upmu\)s; (**e**) 50 \(\upmu\)s; (**f**) 60 \(\upmu\)s.  
blast wave arrived the end of the vertical joint at the time of 30 us, the normal stresses in both the x direction and y direction are negative, which implies that the wing crack is not expected to generate at this time. However, after a while, the normal stresses around the end of the vertical joint become tensile in both the x direction and y direction when a positive sign of blast wave encountered the joint. In addition, the in-plane shear stress become more obvious around the vertical joint compared with the normal stresses at the time of 70 us. This indicates that the mode II crack failure dominates the initiation of the wing crack at the tip of the vertical joint. This is also coincident with the experiment results, in which the propagation direction of wing crack is almost perpendicular to the vertical joint. Figure 16: The stress distribution characteristic around the vertical unfilled joint for the time of 30 μs: **(a)** S11 (normal stress in the x direction), **(b)** S22 (normal stress in the y direction), **(c)** S12 (in-plane shear stress). Figure 17: The stress distribution characteristic around the vertical unfilled joint for the time of 40 μs: **(a)** S11 (normal stress in the x direction), **(b)** S22 (normal stress in the y direction), **(c)** S12 (in-plane shear stress). Figures 19 and 20 show the stress distribution around the oblique joint for the time of 30 \(\upmu\)s and 55 \(\upmu\)s, respectively. For the near end of the oblique joint, a significant stress concentration is formed with the diffraction of the blast wave. For the far end of the oblique joint, a significant stress concentration is formed by the superposition between the reflected wave and the blast wave. Figure 19: The stress distribution characteristic around the oblique unfilled joint for the time of 30 \(\upmu\)s: (**a**) S11 (normal stress in the x direction), (**b**) S22 (normal stress in the y direction), (**c**) S12 (in-plane shear stress).