## 2 Engineering Background
### Geological Conditions
#### 3.1.1 Mechanical Model of a Nonuniform Stress Field in a Circular Roadway  
Based on the current research on the elastic-plastic deformation of the hole, the damage range of the roadway is quantified by mechanical-mathematical methods, which adopt the circular hole plane from the strain model. The stress field under the mining influence is simplified to a planar axisymmetric, nonuniform hydrostatic pressure condition. A bidirectional, nonuniform pressure circular roadway model is established, as shown in Figure 3.  
Figure 3: Computational model of the surrounding rock plastic zone in a bidirectional, nonuniform stress circular roadway. Figure 2: Stratigraphic column of the 22-116 working face. In the model, the radius of the circular roadway is a, the coordinates of any point in the polar coordinate system are (\(r\), \(\theta\)), the nonuniform stress field around the circular roadway is represented using the regional principal stress field, \(\mathrm{P_{1}}\) is the regional maximum principal stress, and \(\mathrm{P_{3}}\) is the regional minimum principal stress. A bidirectional, nonuniform stress condition is called a deviatoric stress environment, which uses the ratio \(\eta\) of \(\mathrm{P_{1}}\) to \(\mathrm{P_{3}}\) to reflect the degree of nonuniform stress distribution, as shown in Equation (1). \[\eta=\frac{\mathrm{P_{1}}}{\mathrm{P_{3}}}>1, \tag{1}\]  
1.2 Evolution of the Shape of the Surrounding Rock Plastic Zone in the Deviatoric Stress Environment  
In the literature, the bidirectional, nonuniform stress circular roadway model is analyzed, which combines the stress expression at a point in the roadway surrounding rock using polar coordinates in elastic mechanics and the Mohr-Coulomb damage criterion to derive the implicit equation for the boundary of the plastic zone in the circular roadway in a nonuniform stress field [23]. The plastic zone boundary location in the roadway can be calculated for the given roadway specifics and surrounding rock stress conditions, as shown in Equation (2). \[f(R,\theta)=a\times F(P_{1},P_{3},\eta,\mathrm{C},\varphi), \tag{2}\]  
Equation (2) shows that the surrounding rock plastic zone boundary in a circular roadway in a nonuniform stress field is mainly related to the roadway radius a, the regional principal stress environment (\(\mathrm{P_{1}}\), \(\mathrm{P_{3}}\), and \(\eta\)), and the rock lithology (\(\varphi\): internal friction angle, \(\mathrm{C}\): cohesion). For the same lithology, the degree of nonuniformity in the stress field distribution (\(\eta\)) is the critical factor for determining the damage form and size of the roadway surrounding rock in a partial stress environment [24]. The following parameters are used in building the FLAC\({}^{3\mathrm{D}}\) numerical simulation model: the circular roadway has a radius of 3 m, the mechanical properties of the surrounding rock are \(\mathrm{C}\) = 3 MPa and \(\varphi\) = 25\({}^{\circ}\), and the minimum principal stress \(\mathrm{P_{3}}\) is 20 MPa. Under these conditions, Equation (2) was prepared using the MAPLE2020 mathematical software to calculate the roadway plastic zone boundary with a change in the degree of the nonuniform stress distribution (\(\eta\)). The evolution of the size and shape of the plastic zone in the surrounding rock in the roadway is shown in Figure 4.  
With an increase in \(\eta\), the surrounding rock plastic zone shape from the FLAC\({}^{3\mathrm{D}}\) simulation and MAPLE calculation indicates the development pattern of the circular-elliptical-butterfly shaped, and the maximum boundary range of the plastic zone increases with the change in shape. When \(\eta\) = 1, the roadway is in a bidirectional equal stress condition, the plastic zone is not large, it is uniformly distributed in the surroundings of the roadway, and the form is a regular circular shape. With an increase in \(\eta\), the plastic zone begins to expand above the roof and below the floor in the roadway.