## 2 Theoretical Analysis of the Plastic Zone under a Non-Isobaric Stress Field
### Theoretical Analysis of Optimization Orientation of Roadway Surrounding Rock
#### 2.2.3 Optimal Orientation Theory Analysis of the \(\sigma_{\rm H}\)-Dominant Stress Field  
In the \(\sigma_{\rm H}\) stress field, \(\sigma_{\rm H}>\sigma_{\rm h}>\sigma_{Z}\). It can be seen from Equation (6) that with the increase in the axial angle \(\alpha\) of the roadway, \(\cos 2\alpha\) decreases monotonically and the stress \(S^{\prime}_{xx}\) corresponding to the vertical roadway sidewall increases monotonically. As angle \(\alpha\) increases, \(S^{\prime}_{\rm xx}\) transits from \(\sigma_{\rm h}\) to \(\sigma_{\rm H}\). In the \(\sigma_{\rm HZ}\)-dominant stress field, there are the following four stress environments: (1) If the difference between \(\mid\sigma_{\rm H}-\sigma_{\rm h}\mid\) and \(\mid\sigma_{\rm h}\mid\) - \(\sigma_{\rm Z}\mid\) is small, in the process of roadway rotation, the stress difference between \(S^{\prime}_{\rm xx}\) and \(\sigma_{\rm Z}\) is small. With the increase in the axial angle \(\alpha\) of the roadway, the deviatoric stress difference of the surrounding rock of the roadway gradually increases but the high confining pressure ratio environment is not formed. (2) If the difference between \(\mid\sigma_{\rm H}-\sigma_{\rm h}\mid\) and \(\mid\sigma_{\rm h}-\sigma_{\rm Z}\mid\) is large, when the roadway is at \(0^{\circ}\), the difference between \(S^{\prime}_{\rm xx}\) and \(\sigma_{\rm Z}\) is large and the deviatoric stress difference continues to increase with the increase in angle \(\alpha\), rapidly increasing the risk of butterfly malignant plastic zone formation. (3) If the difference between \(\mid\sigma_{\rm H}-\sigma_{\rm h}\mid\) is large and the difference between \(\mid\sigma_{\rm h}-\sigma_{\rm Z}\mid\) is small, the difference in the stress of roadway orientation in the low-angle range is small and the butterfly risk coefficient increases with the increase in the roadway angle. (4) If the difference between \(\mid\sigma_{\rm H}-\sigma_{\rm h}\mid\) is small and the difference between \(\mid\sigma_{\rm h}-\sigma_{\rm Z}\mid\) is large, then there is a high-deviatoric-stress environment in the roadway at \(0^{\circ}\). With the increase in the angle, the butterfly risk will continue to increase as in situations (2) and (3). In short, in the \(\sigma_{\rm H}\)-type stress field, with an increase in the roadway angle, when the angle range is large, the high confining pressure ratio environment is easier to form, and the probability of butterfly risk also increases greatly. Morphological and Size Characteristics of Theoretical Plastic Zones with Different Orientations under Three Dominant Stress Fields  
According to the above three different dominant stress fields, the plastic zone boundary in Equation (1) under the condition of a non-uniform stress field is used to theoretically analyze the distribution characteristics and geometric forms of the plastic zone of a surrounding rock in the 0-90\({}^{\circ}\) range under the three stress fields. Under three dominant stress fields, each stress field gives a set of stress values, and the shape and size of the plastic zone are preliminarily determined by the equation. In a \(\sigma_{\rm HZ}\)-dominant stress field, when \(\sigma_{\rm H}=20\) MPa, \(\sigma_{\rm Z}=8\) MPa, and \(\sigma_{\rm h}=7\) MPa, the shape of the plastic zone is shown in Table 1a. With the increase in the roadway angle \(\alpha\), the plastic zone morphology of the roadway surrounding rock transits from ellipse to quasi-circular to butterfly. In addition, the maximum sizes of the roadway plastic zone from \(0^{\circ}\) to \(90^{\circ}\) are 0.47, 0.46, 0.5, 0.68, 0.85, 1.06, 1.5, 2.15, 2.74, and 2.98 m, respectively, and the difference between the maximum plastic zone size and the minimum plastic zone size is 2.52 m.