## 3 Derivation of the Theoretical Formula of the Surrounding Rock Loose Zone
### Theoretical Derivation of Work-Energy Relationship Method
#### 3.2.2 Based on Principles of Fracture Mechanics  
First, the total charge of a cycle is calculated with the volume formula, and then it is associated with the explosive energy per unit charge to obtain the total energy of each cycle blasting. Then, based on fracture mechanics principles, the work performed by the loose zone with a certain thickness of the surrounding rock is calculated, and the thickness of the loose zone is determined through the work-energy relationship. The formula for calculating the total charge is:  
\[N=nSL \tag{40}\]  
where \(N\) is the total charge of a blasting cycle (kg), \(n\) is the consumption of explosive required for blasting per cubic meter of rock mass (selected by looking up a table according to the rock firmness coefficient, kg/m\({}^{3}\)), \(S\) is the excavation section area (m\({}^{2}\)), and \(L\) is the planned cyclic footage (m). Then, the explosive energy consumed per excavation footage is:  
\[Q=qN=qnSL \tag{41}\]  
where \(q\) is the energy released when each kilogram of explosive explodes (kJ/kg), which can be found in the _Construction Manual of Shaft and Roadway Engineering_. The damage effect of blasting on the rock is mainly manifested in the formation of cracks in the surrounding rock; that is, the explosive works on the surrounding rock. Due to the stress wave generated by blasting at the boundary of the chamber, the rock is pulled tangentially to form a crack zone. For the crack formed by blasting, the plane wedge crack model (under the action of stress wave, the initial crack grid is formed in the rock mass around the blast hole; then, under the quasi-static action of the explosive gas, the initial crack is penetrated and further extended, and finally the macro fracture of the rock mass is completed. At this time, the rock crack presents a wedge-shaped state in the plane region, so the crack model is called the plane wedge crack model) is more suitable. At the same time, according to the principle of rock fracture mechanics, the cylindrical charge blasting of the blast hole can be regarded as a plane strain problem, and the energy release rate per unit length can be obtained [44]:  
\[G_{1}=\frac{\left(1-\mu^{2}\right)}{E}K_{2}^{2} \tag{42}\]  
Due to crack length expansion \(\Delta a\), the stress intensity factor at the crack end can be considered constant. Therefore, it can be determined that the work performed by the tangential stress during the expansion of the crack is:  
\[W=n_{1}\int_{r_{p}}^{R_{22}}\frac{\left(1-\mu^{2}\right)}{E}K_{2}^{2}dr \tag{43}\]  
where \(n_{1}\) is the number of radial main fractures, which is generally 8-12. When only the incident stress is considered to perform all the work, for an open crack, \(K_{2}=\sigma_{\lambda}\sqrt{2\pi R_{j}}\), which accounts for the effect of the blasting load on the crack propagation. \(R_{j}\) here is the integral of the distance from the blast hole center to the crack end. Combined with the above analysis, it can be seen that:  
\[W=\frac{n_{1}\pi\lambda^{2}P_{1}^{2}\gamma_{p}^{2}\left(1-\mu^{2}\right)}{E(1- \alpha)}\left[\left(\frac{R_{22}}{r_{p}}\right)^{2(1-\alpha)}-1\right] \tag{44}\]  
where \(\lambda=\mu/(1-\mu)\) is the lateral pressure coefficient, \(\alpha=2-\mu_{d}/(1-\mu_{d})\) is the pressure attenuation coefficient, \(P_{\lambda}=\rho_{m}D_{0}\rho_{0}D_{1}^{2}/2(D_{0}\rho_{m}+D_{1}\rho_{0})\) is the initial impact pressure of the hole wall, \(r_{p}\) is the blast hole radius, \(E\) is the elastic modulus of the surrounding rock, and \(R_{b}\) is the radius of the loose zone. The effect of blasting on the expansion of existing cracks is calculated from the blast hole. It is assumed that the proportion of the blasting energy to the energy forming the loose zone is \(\epsilon\), and the effective energy utilization rate \(e\) is 23-48%, according to previous research. Because the energy released by explosives is considered to perform work on the surrounding rock, only the expansion of the main crack is considered in this process.