## 2 Theoretical Analysis of Pre-Splitting Blasting
### Propagation Law of Explosion Stress Wave  
When the stress wave propagates in the rock mass, its magnitude decreases with the increase in propagation distance, and the nature and shape of the wave also change accordingly. According to the different properties, shapes and action results of the wave, the propagation process of the shock wave can be divided into three action regions, as shown in Figure 2, where R is the radius of the charge. The peak pressure of the shock wave is much larger than the dynamic compressive strength of the rock in the range of three to seven times the radius of the explosive source, which causes plastic deformation of the rock and consumes most of the energy. This range is the shock wave area. After the stress wave passes through the area, the energy is greatly depleted, and it attenuates into a stress wave without a steep wave peak. The state parameters on the wave front become relatively flat, which is called the stress wave action area or the compression stress wave action area. After the stress wave passes through the area, its strength and energy are further attenuated and evolved into seismic waves. Its effect can only cause elastic vibration of rock particles, but it cannot cause rock damage. The time when rock particles leave the static state is equal to the time when they return to the static state.  
Figure 1: Diagram of rock blasting mechanism.  
Figure 2: Diagram of stress wave propagation.  
After the explosion acts on the rock mass, the explosion stress wave propagates outward from the explosion source. The distance relationship between the peak stress and the explosion source can be expressed as [20; 21]:  
\[\sigma_{\rm{rmax}}=\frac{p_{1}}{\overline{r}^{a}} \tag{1}\]  
In the formula, \(p_{1}\)--radial peak stress of rock element and \(\alpha\)--the attenuation coefficient of stress wave.  
For radial peak stress, uncoupled charge is used, and \(p_{1}\) is expressed as [22]:  
\[p_{1}=\frac{1}{8}\rho_{a}D_{a}^{2}\left(\frac{r_{a}}{r_{b}}\right)^{6}m \tag{2}\]  
In the formula, \(\rho_{a}\)--density of explosives, \(D_{a}\)--explosive detonation velocity, \(r_{a}\), \(r_{b}\) --charge radius and hole radius, and \(m\)--pressure increase coefficient, generally \(m\) = 8-11.  
\[\sigma_{\rm{\theta max}}=w\sigma_{\rm{rmax}} \tag{3}\]  
The value of \(w\) is related to the distance between the stress wave front and the explosion source and the Poisson's ratio u of the rock. When the stress wave front is close to the explosion source, \(w=1\), and when the stress wave front is far from the explosion source, the coefficient \(w\) approaches \(w=\mu/(1-\mu)\)[23; 24].  
According to \(\alpha=2-\frac{\mu}{1-\mu}\) and simultaneous Formula (1), we can obtain:  
\[p_{1}=\sigma_{\rm{rmax}}\overline{r}^{(2-\frac{\mu}{1-\mu})} \tag{4}\]  
According to Formulas (2)-(4):  
\[\sigma_{\rm{rmax}}=\frac{\rho_{a}D_{a}^{2}\left(\frac{r_{a}}{r_{b}}\right)^{6} m}{8\overline{r}^{(2-\frac{\mu}{1-\mu})}} \tag{5}\]  
According to Formulas (1)-(5):  
\[\sigma_{\rm{\theta max}}=w\frac{\rho_{a}D_{a}^{2}\left(\frac{r_{a}}{r_{b}} \right)^{6}m}{8\overline{r}^{(2-\frac{\mu}{1-\mu})}} \tag{6}\]  
According to \(w=\mu/(1-\mu)\) and Formula (6), we can obtain:  
\[\sigma_{\rm{\theta max}}=\frac{\mu}{(1-\mu)}.\frac{\rho_{a}D_{a}^{2}\left( \frac{r_{a}}{r_{b}}\right)^{6}m}{8\overline{r}^{(2-\frac{\mu}{1-\mu})}} \tag{7}\]  
According to isentropic correlation theory, isentropic exponent initial stress is introduced:  
\[p_{0}=\frac{1}{2(K+1)}\rho_{a}{D_{a}}^{2} \tag{8}\]  
Combining Formulas (7) and (8), the expressions of radial compressive stress and tangential tensile stress considering the initial stress of isentropic index can be obtained:  
\[\sigma_{\rm{rmax}}=\frac{(K+1)p_{0}\left(\frac{r_{a}}{r_{b}}\right)^{6}m}{4 \overline{r}^{(2-\frac{\mu}{1-\mu})}} \tag{9}\]\[\sigma_{\theta\text{max}}=\frac{\mu}{(1-\mu)}.\frac{(K+1)p_{0}\left(\frac{r_{a}}{r_{ b}}\right)^{6}m}{4\pi^{(2-\frac{\mu}{1-\mu})}} \tag{10}\]  
Formula: adiabatic isentropic index of \(K\)-explosive.