## 2 Theoretical Analysis of the Plastic Zone under a Non-Isobaric Stress Field  
### General Form of the Plastic Zone in Rock Surrounding a Circular Roadway  
In coal mining, roadway layout, stope strata control and movement, roadway strata behavior control, and the coal mine dynamic pressure phenomenon are closely related to the crustal stress field. The size and direction of original rock stress largely affect roadway deformation and failure. Therefore, it is important to study the initial stress field environment of a roadway to analyze the internal stress change of a rock mass during excavation and design reasonable support. The schematic diagram of roadway surrounding rock under a three-dimensional stress environment is shown in Figure 1.  
In Figure 1, \(\sigma_{\rm H}\) is the maximum horizontal principal stress, parallel to the roadway axis; \(\sigma_{\rm h}\) is the minimum horizontal principal stress, perpendicular to the roadway side; and \(\sigma_{\rm Z}\) is the vertical stress. In the existing research, the actual roadway model can be simplified into the axisymmetric plane strain circular hole problem in elastoplastic mechanics. The mechanical model is shown in Figure 2, where \(P_{1}\) is the principal stress of the side of the vertical roadway, \(P_{3}\) is the direct lead stress, a is the radius of the roadway, r and \(\theta\) are the polar coordinates of any point in the coordinates, and \(\eta\) is the confining pressure ratio (i.e., the ratio of the main stress of two sides of the vertical roadway to the vertical stress of the roadway). The expression is shown in Equation (1):  
\[\frac{|P_{1}|}{|P_{3}|}=\eta\geq 1 \tag{1}\]  
Through the stress analysis of a circular roadway under a non-uniform stress field, the stress calculation formula of any point around the roadway under non-uniform pressure is substituted into the Mohr-Coulomb criterion, the 8-order implicit equation of the plastic zone boundary of the surrounding rock of a circular roadway under a non-uniform stress field is obtained, as shown in Equation (2), and the concept of the butterfly plastic zone is proposed [19]. \[f\Big{(}\frac{a}{r}\Big{)}=K_{1}\Big{(}\frac{a}{r}\Big{)}^{8}+K_{2}\Big{(} \frac{a}{r}\Big{)}^{6}+K_{3}\Big{(}\frac{a}{r}\Big{)}^{4}+K_{4}\Big{(}\frac{a }{r}\Big{)}^{2}+K_{5}=0 \tag{2}\]  
Figure 1: Schematic diagram of the stress field environment in the roadway envelope. Figure 2: Mechanical calculation model and plastic zone evolution model of a circular roadway under a non-uniform stress field. where  
\[K_{1} =9(1-\eta)^{2}\] \[K_{2} =-12(1-\eta)^{2}-6\big{(}1-\eta^{2}\big{)}\cos 2\theta\] \[K_{3} =2(1-\eta)^{2}\big{[}\cos^{2}2\theta(5+2\sin^{2}\varphi)-\sin^{2}2 \theta\big{]}+(1+\eta)^{2}+4\big{(}1-\eta^{2}\big{)}\cos 2\theta\] \[K_{4} =-4(1-\eta)^{2}\cos 4\theta-2\big{(}1-\eta^{2}\big{)}\cos 2 \theta\big{(}1+2\sin^{2}\varphi\big{)}+\tfrac{4}{\gamma H}(1-\eta)\cos 2\theta \sin 2\varphi C\] \[K_{5} =(1-\eta)^{2}-\sin^{2}\varphi\Big{(}1+\eta+\tfrac{2C\cos\varphi}{ \gamma H\sin\varphi}\Big{)}^{2}\]  
where \(C\) is for coal and rock medium cohesion, \(a\) is the internal friction angle, \(r\) is the roadway radius, \(f\) is the radius of the plastic zone, and \(K\) is for the polar coordinates of any point around the roadway surrounding rock. When \(\eta=1\), Equation (2) is the same as the famous Kastner formula. The plastic zone morphology of circular roadway surrounding rock under different \(\eta\) values under the same mechanical properties as those of roadway surrounding rock is obtained through Equation (2) and visual programing calculation, as shown in \[\frac{|P_{1}|}{|P_{3}|}=\eta\geq 1 \tag{1}\]0 In the diagram, the plastic zone in the rock surrounding the circular roadway has three forms: circular, elliptical, and butterfly [20]. When the roadway plastic zone is circular and elliptical, the failure of the surrounding rock of the roadway is mostly uniform, the rock in the failure zone is not concentrated in a certain area, and the surrounding rock of the roadway is relatively stable.