## 2 Research Background
### Determination of Mechanical Parameters of Surrounding Rock
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Lithology**} & **Density** & **Elastic** & **Normal** & **Tangential** & **Cohesive** & **Internal** & **Tensile** \\  & **(kg\(\cdot\)m\({}^{-3}\))** & **Modulus** & **Strength** & **Strength** & **Force (MPa)** & **Friction** & **Strength** \\  & **(GPa)** & **(GPa/m)** & **(GPa/m)** & **(GPa/m)** & **Angle (\({}^{\circ}\))** & **(MPa)** \\ \hline No. 3 Coal & 1400 & 2.6 & 112 & 44.8 & 1.25 & 20 & 0.64 \\ Mudstone & 2510 & 5.4 & 228 & 91 & 9.34 & 27 & 1.99 \\ No. 4 Coal & 1400 & 2.7 & 113 & 45.1 & 1.2 & 19 & 0.59 \\ Silstone & 2550 & 9.7 & 351.2 & 140 & 7.1 & 31 & 2.15 \\ Calcareous & 2800 & 10.4 & 464.4 & 185.7 & 16.08 & 36 & 8.05 \\ Limestone & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistical table of the results of the mechanical experiment on the coal rock body. Figure 3: Coal rock body mechanical properties are part of the standard rock sample test damage map. According to the theory of elastic mechanics, the polar coordinate expression for the stress solution at any point of the circular hole enclosure is  
\[\left\{ \begin{array}{l}\sigma_{r}=\frac{P_{Y}+P_{X}}{2}(1-\frac{\tau_{0}^{2}}{ \tau_{0}^{2}})+\frac{P_{Y}-P_{X}}{2}(1-4\frac{\tau_{0}^{2}}{\tau_{0}^{2}}+3\frac {\tau_{0}^{4}}{\tau^{4}})\cos 2\theta\\ \sigma_{\theta}=\frac{P_{Y}+P_{X}}{2}(1+\frac{\tau_{0}^{2}}{\tau^{2}})-\frac{P _{Y}-P_{X}}{2}(1+3\frac{\tau_{0}^{4}}{\tau^{4}})\cos 2\theta\\ \tau_{\tau\theta}=\frac{P_{Y}-P_{X}}{2}(1+2\frac{\tau_{0}^{2}}{\tau^{2}}-3 \frac{\tau_{0}^{4}}{\tau^{4}})\sin 2\theta\end{array} \right., \tag{1}\]  
where \(\sigma_{r}\) is the radial stress at any point outside the circular borehole, \(\sigma_{\theta}\) is the circumferential stress, \(\tau_{\theta\theta}\) is the shear stress, \((r,\theta)\) is the polar coordinates of any point, \(r_{0}\) is the radius of the roadway, \(P_{Y}\) is the vertical stress, and \(P_{X}\) is the horizontal stress. There is also the conversion relation between the stress expression in elastic mechanics in the right-angle coordinate system and the stress expression in the polar coordinate system there:  
\[\left\{ \begin{array}{l}\sigma_{x}=\frac{\sigma_{r}+\sigma_{\theta}}{\sigma_{ x}}+\frac{\sigma_{r}-\sigma_{\theta}}{2}\cos 2\theta-\tau_{\theta}\sin 2\theta\\ \sigma_{x}=\frac{\sigma_{r}^{2}-\sigma_{\theta}}{2}\sin 2\theta+\tau_{\theta} \cos 2\theta\end{array} \right., \tag{2}\]  
The expressions for the maximum and minimum principal stresses at any point in elastic mechanics are  
\[\left\{ \begin{array}{l}\sigma_{1}=\frac{\sigma_{x}+\sigma_{y}}{2}+\frac{1}{ 2}\sqrt{\left(\sigma_{x}-\sigma_{y}\right)^{2}+4\tau_{xy}}\\ \sigma_{3}=\frac{\sigma_{x}+\sigma_{y}}{2}-\frac{1}{2}\sqrt{\left(\sigma_{x}- \sigma_{y}\right)^{2}+4\tau_{xy}}\end{array} \right., \tag{3}\]  
where \(\sigma_{1}\) is the maximum principal stress and \(\sigma_{3}\) is the minimum principal stress. The polar coordinate expressions for the maximum and minimum principal stresses at any point within the surrounding rock can be obtained by combining Equations (2) and (3):  
\[\left\{ \begin{array}{l}\sigma_{1}=\frac{\sigma_{r}+\sigma_{\theta}}{2}+ \sqrt{\left(\frac{\sigma_{r}-\sigma_{\theta}}{2}\right)^{2}+\left(\tau_{ \theta}\right)^{2}}\\ \sigma_{3}=\frac{\sigma_{r}+\sigma_{\theta}}{2}-\sqrt{\left(\frac{\sigma_{r}- \sigma_{\theta}}{2}\right)^{2}+\left(\tau_{\theta}\right)^{2}}\end{array}