## 3 Derivation of the Theoretical Formula of the Surrounding Rock Loose Zone
### Theoretical Derivation of the Stress Method
#### 3.1.2 Derivation of Range of Blasting-Expansion Loose Zone  
In the process of crack propagation in the stress loose zone caused by the explosion, the open crack is the main form of crack propagation, and the analysis of the range of the blasting loose zone caused by the analysis of the crack propagation in the stress loose zone is mainly based on the analysis of the blasting stress. The stress wave generated by blasting can be divided into a vertical stress wave and a parallel stress wave [38; 39; 40], as shown in Figure 4.  
The incident stress wave of blasting is mainly related to the nature of the explosive and charging mode. After the blasting in the blast hole, the shock wave is aroused in the rock near the blast hole. The initial impact pressure of the hole wall is approximately calculated based on acoustic approximation theory [41], as follows:  
\[P_{\lambda}=\frac{\rho_{0}D_{1}^{2}}{4}\times\frac{2\rho_{m}D_{0}}{D_{0}\rho_ {m}+D_{1}\rho_{0}}=\frac{\rho_{m}D_{0}\rho_{0}D_{1}^{2}}{2(D_{0}\rho_{m}+D_{1 }\rho_{0})} \tag{20}\]  
As the propagation distance of the stress wave increases, the peak value of the stress decreases continuously. At the same time, in the action area of the stress wave, the radial and tangential stress of the cylindrical stress wave in the rock have the following relationship:  
\[\sigma_{\theta}=\frac{\mu_{d}}{1-\mu_{d}}\sigma_{r} \tag{21}\]  
If the rock body is regarded as a uniform elastic body, the stress at any point in the rock where the wave front goes can be expressed as follows:  
\[\sigma_{r}=P_{\lambda}\overline{r}^{-\kappa} \tag{22}\]  
Figure 4: Effect of blasting incident stress wave: (**a**) Stress of chamber section; (**b**) incident stress wave front. \(F_{1}\) is the chamber section, \(F_{2}\) is the incident stress wave front, \(\sigma_{\lambda}\) is the incident stress, \(\sigma_{\perp}\) is the vertical component of incident stress, and \(\sigma_{\parallel}\) is the parallel component of incident stress. where \(\sigma_{\theta}\) and \(\sigma_{r}\) are the tangential stress and radial stress of the rock, \(\overline{r}=r_{OA}/r_{0}\) is the contrast distance, \(P_{\lambda}\) is the initial impact pressure of the hole wall, \(\alpha=2-\mu_{d}/(1-\mu_{d})\) is the pressure attenuation index, and \(\mu_{d}\) is the dynamic [PERSON]'s ratio of the rock, which is generally taken as \(\mu_{d}=0.8\mu\). The incident stress of blasting can be calculated using the above formulas. The specific expression is as follows:  
\[\sigma_{\lambda}=\sigma_{r}=\frac{\rho_{m}D_{0}\rho_{0}D_{1}^{2}}{2(D_{0} \rho_{m}+D_{1}\rho_{0})}\cdot\left(\frac{r_{0}}{r_{OA}}\right)^{2-\frac{\mu_{d} }{1-\mu_{d}}} \tag{23}\]  
where \(\rho_{m}\) is the density of the rock medium, \(D_{1}\) is the wave velocity of the shock wave (detonation wave velocity) in the rock at the hole wall, \(\rho_{0}\) is the density of the explosive, \(D_{0}\) is the velocity of the elastic wave propagating in the explosion (initial velocity of the shock wave), \(r_{0}\) is the distance in the direction of the cylindrical charge (blast hole radius), \(r_{OA}\) is the distance in the incident direction of the stress wave, \(r_{OA}=\sqrt{L_{1}^{2}+L_{2}^{2}}\), \(L_{1}\) is the horizontal distance between the center of the cylindrical charge and the top of the roadway, \(L_{2}\) is the vertical distance between the center of the cylindrical charge and the outer contour of the loose zone at the top of the roadway, and \(\mu_{d}\) is the dynamic Poisson's ratio of the rock, which is generally taken as \(\mu_{d}=0.8\mu\). Considering the influence of porous blasting on the crack propagation and extension as well as the influence of porous blasting on the incident stress, the incident stress is calculated by the following formula:  
\[\sigma_{\lambda r}=(n)^{C}\sigma_{\lambda} \tag{24}\]  
where \(n\) is the number of blast holes and \(C\) is the influence coefficient. What causes crack propagation in surrounding rock is the derived tensile stress of the vertical component of the incident stress.