## 2 Theoretical Analysis of the Plastic Zone under a Non-Isobaric Stress Field
### Theoretical Analysis of Optimization Orientation of Roadway Surrounding Rock
#### 2.2.1 Optimal Orientation Theory Analysis of the \(\sigma_{\rm HZ}\)-Dominant Stress Field  
In the \(\sigma_{\rm HZ}\) stress field, \(\sigma_{\rm H}>\sigma_{\rm Z}>\sigma_{\rm h}\). It can be seen from Equation (6) that with the increase in the axial angle \(\alpha\), \(\cos 2\alpha\) monotonically decreases and the stress \(S^{\prime}_{\rm xx}\) of the  
Figure 4: The stress analysis chart of the roadway after an axial rotation at \(\alpha\) angle.  
corresponding vertical roadway side monotonically increases. In the process of increasing, at one point, angle \(S^{\prime}_{xx}=\sigma_{Z}\) and then \(S^{\prime}_{xx}\) continues to increase to become equal to the axial stress \(\sigma_{\rm H}\). In the \(\sigma_{\rm HZ}\)-dominant stress field, there are four stress environments: (1) If the difference between \(\mid\sigma_{\rm H}-\sigma_{Z}\mid\) and \(\mid\sigma_{Z}-\sigma_{\rm h}\mid\) is small, it shows that the difference between \(S^{\prime}_{xx}\) and \(\sigma_{Z}\) is not significant during the rotation of the roadway, and the ratio of high confining pressure is not formed, which is not favorable to the formation of a butterfly plastic zone. (2) If the difference between \(\mid\sigma_{\rm H}-\sigma_{Z}\mid\) and \(\mid\sigma_{Z}-\sigma_{\rm h}\mid\) is large, the difference between \(S^{\prime}_{xx}\) and \(\sigma_{Z}\) is large, at \(0^{\circ}\) and \(90^{\circ}\). The roadway is in a high-stress-ratio environment and the stress environment is in a high-low-high state in the axial rotation process. A certain angle range in the rotation process is conducive to avoiding the formation of a butterfly plastic zone. (3) If the difference between \(\mid\sigma_{\rm H}-\sigma_{Z}\mid\) is large and the difference between \(\mid\sigma_{Z}-\sigma_{\rm h}\mid\) is small, the difference between \(S^{\prime}_{xx}\) and \(\sigma_{Z}\) is small at \(0^{\circ}\), and the roadway is in a low-stress environment. With the increase in the distance to \(\sigma_{\rm H}\), the stress difference gradually increases, and a deviatoric stress area begins to form. (4) If the difference between \(\mid\sigma_{\rm H}-\sigma_{Z}\mid\) is small and the difference between \(\mid\sigma_{Z}-\sigma_{\rm h}\mid\) is large, the difference between \(S^{\prime}_{xx}\) and \(\sigma_{Z}\) is large at \(0^{\circ}\), and there is a high-stress difference, making it easy for a butterfly plastic zone to develop. With the increase in \(\alpha\), the deviatoric stress difference gradually decreases. In short, in the \(\sigma_{\rm HZ}\)-dominant stress field, there is an angle \(\alpha\), which makes \(S^{\prime}_{xx}\) equal to \(\sigma_{Z}\) and minimizes the deviatoric stress difference.  
Take \(\eta=\frac{S^{\prime}_{xx}}{\sigma_{Z}}=1\) into Equation (6) to obtain Equation (9):  
\[\alpha=\frac{1}{2}\arccos\left(\frac{2\sigma_{Z}-\sigma_{\rm h}-\sigma_{\rm H} }{\sigma_{\rm h}-\sigma_{\rm H}}\right) \tag{9}\]  
This \(\alpha\) is the optimal angle deduced from the butterfly failure theory of roadway surrounding rock to avoid the formation of a butterfly plastic zone. The adjacent range of this angle is the optimal orientation. In this range, the plastic zone is elliptical or quasi-circular, which is conducive to the stability of the roadway.