# Effect of the Inter-Ring Delay Time on Rock Fragmentation: Field Tests at the Underground Mine
## 3 Theoretical Analysis of IRDT in Tunnel Excavation Blasting
### Collision of Fragment  
Based on the energy conversion theory in the collision process and blasting process, a tunnel excavation blasting model (Figure 6) was established to analyze the effect of IRDT on rock fragmentation. \(\theta\) is the angle between the blast hole and the free surface; R1, R2, and R3 represent the cut hole, stopping hole, and contour hole, respectively; the TRDT for R1, R2, and R3 are the same; S1 represents the space of unknown fragments after the detonation of R1, and S2 represents the new compensating space. The tunnel section is the free surface, the shared face between S2 and R1 forms the new free surface, while the shared face between S1 and R1 forms the part free surface. Additionally, assuming the same time of free surface formation and the same velocity of rock movement after each blast means that the fragments have the same kinetic energy. Figure 4: Rock damage caused by explosive blast. Figure 5: Explosive load and rock interaction processes: (**a**) experience \(t_{1}+t_{2}\) time to form a free surface; (**b**) experience \(t_{1}+t_{2}+t_{3}\) time to form a blasting funnel, where \(t_{3}\) is the rock throwing time; (**c**) burst rock is completely thrown. As indicated by collision theory, the energy loss during collisions depends on the magnitude and direction of the relative velocity. Therefore, in the analysis of the tunnel excavation blasting process, the influence of collisions among the same-ring blasted rocks on rock fragmentation was considered. Additionally, a comparative analysis was carried out between the effects of rear-ring-blasted rocks impacting front-ring-blasted rocks on rock fragmentation. The IRDT is denoted as \(\Delta t_{1}\). As shown in Figure 6a, the blasting took \(\Delta t_{1}\) time. The rock adjacent to the S1 region along the partial free surface is thrown after R2 detonates, whereas the rock at S2 is thrown towards a new free surface. The collision between rocks from the front and rear rings occurs in the S1 region, with a relative velocity of \(\sqrt{2}\,V\) and an angle of \(45^{\circ}\) with respect to the horizontal direction. The relative velocity in the horizontal direction is \(V\). The collision between rocks of the same ring occurs at the S2 region, and the relative velocity in the horizontal direction is \(2 Vsin\theta\). As indicated in Section 3.2, the degree of rock fragmentation resulting from a collision depends on the magnitude of the relative velocity in the horizontal direction. Thus, when \(0<\theta<\pi/6\), \(V>2 Vsin\theta\), the degree of rock fragmentation at S1 is better than that at S2. When \(\pi/6<\theta\leq\pi/2\), \(V>2 Vsin\theta\), the degree of rock fragmentation at S2 is better than that at S1. Based on the actual field conditions, the case of \(0<\theta<\pi/6\) is not considered, and only the situation of \(\pi/6<\theta\leq\pi/2\) is discussed, i.e., the degree of rock fragmentation at S2 is better than that at S1. When the delay time increases, the IRDT is denoted as \(\Delta t_{2}(\Delta t_{2}>\Delta t_{1})\). As shown in Figure 6b, the blast experience at \(\Delta t_{2}\) time, the kinetic energy dissipated in area S2 after R2 is detonated is greater than that in area S1. This is because more energy is used for collision at S2, improving the fragmentation of the blasted rock and reducing its throw distance. As the delay time \(\Delta t_{2}\) increases compared to \(\Delta t_{1}\), the S1 region gradually decreases, and the S2 region gradually increases. With the increase in the IRDT, more energy is used for rock fragmentation after R2 is detonated. Meanwhile, when the delay time continues to increase, take the IRDT as \(\Delta t_{3}\ (\Delta t_{3}>\Delta t_{2})\). As shown in Figure 6c, after a period of \(2\Delta t_{3}\), R3 starts to detonate.