# 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
### Energy Conversion in Collisions  
This study analyzed the energy conversion of collision in the perfect model and the effect of IRDT on rock fragmentation. The perfect model for analyzing collisions had two types of central and non-central collisions. As shown in Figure 3, two balls, A and B, were used, with masses \(m_{A}\) and \(m_{B}\), respectively, and both had relative velocities \(v\). Assume that two balls have inelastic properties and rough surfaces. For a central collision (Figure 3a), the direction of relative velocity at the time of collision coincides with the line joining the centers of the two balls. At the time of the collision, the axial forces \(F_{NA}\) and \(F_{NB}\) were generated, such as \(F_{NA}=F_{NB}=F_{N}\). The axial force \(F_{NA}\) caused the velocity of ball A to decrease \(V_{A}\), and the axial force \(F_{NB}\) increased the velocity of ball B to \(V_{B}\). It is noteworthy that during a collision, the axial force \(F_{NA}\) causes the velocity of ball A to decrease initially and may reduce to zero before increasing in the opposite direction, but it does not exceed the velocity before the collision. In an inelastic collision, the kinetic energy of ball A decreases while that of ball B increases, but the total kinetic energy is not conserved before and after the collision. This is because the axial forces \(F_{NA}\) and \(F_{NB}\) cause non-elastic deformation of balls A and B, which is converted into internal energy of the balls and dissipated. Damage occurs when the ball deformation cannot withstand the internal energy gained. According to the law of conservation of momentum, the energy loss during a central collision can be obtained as follows [19] (pp. 26-29):  
\[E_{k}=\frac{1}{2}mv^{2}\Big{(}1-e^{2}\Big{)} \tag{2}\]  
where \(E_{k}\) is the energy loss, kJ; \(m\) is relative mass \(m=m_{A}m_{B}/(m_{A}+m_{B})\), kg; \(v\) is the relative velocity of A and B at the time of the collision, m/s; \(e\) is the recovery coefficient, which is related to the nature of the material, and for inelastic material, \(0<e<1\). According to Equation (2), the larger the relative velocity of the ball in an inelastic central collision, the greater the energy loss and the more severe the damage to the ball. For non-central collision (Figure 3b), the angle between the direction of relative velocity in a collision and the line joining the centers of the two balls is \(\theta(0<\theta<\pi/2)\), and the horizontal and vertical components of the relative velocity are \(v_{x}\) and \(v_{y}\), respectively, where \(v_{x}=vcos\theta\) and \(v_{y}=vsin\theta\). Furthermore, the energy loss at the time of the collision was analyzed in the horizontal and vertical directions. The horizontal direction was similar to the central collision model. The collision generated the axial forces \(F_{NA}\) and \(F_{NB}\), where the axial force \(F_{NA}\) decreased the horizontal velocity of ball A to \(V_{Ax}\). The axial force \(F_{NB}\) increased the horizontal velocity of ball B to  
Figure 3: Schematic diagram of an inelastic collision: (**a**) a central collision; (**b**) a non-central collision. \(V_{Bx}\). Meanwhile, part of the energy was converted into the internal energy of the ball and loss. The energy loss can be derived as follows:  
\[E_{kx}=\frac{1}{2}m{v_{x}}^{2}\Big{(}1-e^{2}\Big{)}=\frac{1}{2}m(vcos\theta)^{2} \Big{(}1-e^{2}\Big{)} \tag{3}\]  
In the vertical direction, the collision produced frictional forces \(F_{fA}\) and \(F_{fB}\) for balls A and B, and \(F_{fA}=F_{fB}\) The frictional force \(F_{fA}\) reduced the velocity of ball A in the vertical direction to \(V_{Ay}\), and \(F_{fB}\) increased the velocity of ball B in the vertical direction to \(V_{By}\). Meanwhile, the frictional forces \(F_{fA}\) and \(F_{fB}\) caused balls A and B to produce rotational angular velocities \(\omega_{A}\) and \(\omega_{B}\), respectively. After a collision in the y-direction, part of the kinetic energy was converted into thermal and angular kinetic energies, which were not conducive to ball destruction [3] (pp.