## 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
The relationship between the vertical component and the incident stress is as follows:  
\[\sigma_{\perp}=\sigma_{\lambda}\cos\alpha \tag{25}\]  
where \(\alpha\) is the angle of oblique incidence of the blasting stress wave, and the size is determined by the following formula:  
\[\cos\alpha=\frac{L_{1}}{\sqrt{L_{1}^{2}+L_{2}^{2}}} \tag{26}\]  
From the vertical component of the incident stress \(\sigma_{\perp}\), the tensile stress \(\sigma_{\theta\perp}\) is derived as follows:  
\[\sigma_{\theta\perp}=\frac{\mu_{d}}{1-\mu_{d}}\sigma_{\perp} \tag{27}\]  
The derived tensile stress \(\sigma_{\theta\perp}\) makes the crack in the roadway sidewall expand. According to the theory of rock fracture mechanics, for an open crack, when the crack is stable, under the action of the blasting stress wave, the stress field at any point near the top of the crack \((r,\theta)\) is [42]:  
\[ \begin{cases}\sigma_{x}\\ \sigma_{y}\\ \sigma_{z}\end{cases} =\frac{K_{1}}{\sqrt{2\pi r}}\cos\frac{\theta}{2} \begin{cases} 1-\sin\frac{\theta}{2}-\sin\frac{3\theta}{2}\\ 1+\sin\frac{\theta}{2}+\sin\frac{3\theta}{2}\\ \sin\frac{\theta}{2}\sin\frac{3\theta}{2}\end{cases}  \tag{28}\]  
From the three-dimensional stress field expression, the maximum principal stress expression can be calculated as follows:  
\[\sigma_{1}=\frac{K_{1}}{\sqrt{2\pi r}}\cos\frac{\theta}{2}\Big{(}1+\sin\frac{ \theta}{2}\Big{)} \tag{29}\]where \(K_{1}\) is the dynamic stress intensity factor, which is \(K_{1}=\sigma_{\phi\perp}\sqrt{2\pi R}\) for an open crack, and \(R\) is the half length of the crack. The strength factor of the surrounding rock will decrease continuously due to the damage to the surrounding rock caused by blasting. In the chamber excavation, the crack propagation can be judged by comparing the strength factor with the dynamic fracture toughness \(K_{1D}\) of the surrounding rock. Cracking means that the maximum tensile stress of the concentrated stress field generated by the blasting stress at the crack tip exceeded the dynamic tensile strength of the damaged rock, and the critical state is:  
\[\sigma_{1}=(1-D)\sigma_{D} \tag{30}\]  
Combined with the above analysis, we can obtain the following:  
\[\frac{K_{1}}{\sqrt{2\pi r}}\cos\frac{\theta}{2}\left(1+\sin\frac{\theta}{2} \right)=(1-D)\sigma_{D} \tag{31}\]  
By solving for the crack length, it can be determined that the length of the blasting propagation crack is as follows:  
\[r=\frac{L_{1\parallel}}{2}\left[\frac{\sigma_{\phi_{\perp}}}{(1-D)\sigma_{D} }\cos\frac{\theta}{2}\left(1+\sin\frac{\theta}{2}\right)\right]^{2} \tag{32}\]  
When \(\theta=0\), the maximum length of the crack propagation can be deduced. From this, the maximum propagation length \(r_{\max}\) of the blasting crack under the far zone of the blasting stress wave can be calculated as follows:  
\[r_{\max}=\frac{L_{1\parallel}}{2}\left[\frac{\sigma_{\phi_{\perp}}}{(1-D) \sigma_{D}}\right]^{2} \tag{33}\]  
where \(L_{b1}=R_{b1}-r_{t}\) is the thickness of the first-stage stress release ring, and \(\sigma_{D}\) is the dynamic tensile strength of the rock. Therefore, in the process of roadway blasting excavation, under the action of a blasting load, the maximum length of crack propagation of the stress loose zone under the action of the blasting load, \(r_{\max}\), is equal to the expanded thickness of the blasting-expansion loose zone \(L_{b2}\):  
\[L_{b2}=r_{\max}=\frac{L_{b1}}{2}\left[\frac{\sigma_{\phi_{\perp}}}{(1-D)\sigma _{D}}\right]^{2} \tag{34}\]  
According to the above analysis, the final loose zone thickness \(L_{b}\) is the combination of the stress loose zone thickness \(L_{b1}\) and the blasting-expansion loose zone thickness \(L_{b2}\). Here, influence coefficients \(m\) and \(n\) are introduced, which are related to the actual surrounding rock conditions of the project.