## 5 Nonuniform Expansion Mechanism of Plastic Zone  
Mechanical Analysis of the Influence of Principal Stress Direction on Plastic Zone Expansion of Inclined Coal Seam Roadway  
The principal stress direction will have a certain angle with the coal (rock) layer due to the influence of the occurrence angle of the coal seam, which affects the distribution and expansion of the plastic zone of the surrounding rock. Based on the mechanical model in Section 2.2, from the theory of elastic-plastic mechanics, the stress fields of surrounding rock in inclined rock roadway can be expressed as Equations (2)-(4).  
\[\sigma_{r}=\frac{p_{z}}{2}\left\{\begin{array}{l}\left[\frac{3(\eta+1)\sin( \alpha-\gamma)}{2}+(\eta+1)\cos(\alpha-\gamma)\right]\left(1-\frac{\alpha^{2 }}{r^{2}}\right)-\\ \left[\frac{3(\eta-1)\sin(\alpha-\gamma)}{2}+(\eta-1)\cos(\alpha-\gamma) \right]\left(1+3\frac{\alpha^{4}}{r^{2}}\right)\cos 2\theta\end{array}\right\}, \tag{2}\]  
\[\sigma_{\theta}=\frac{p_{z}}{2}\left\{\begin{array}{l}\left[\frac{3(\eta+1) \sin(\alpha-\gamma)}{2}+(\eta+1)\cos(\alpha-\gamma)\right]\left(1-\frac{\alpha ^{2}}{r^{2}}\right)-\\ \left[\frac{3(\eta-1)\sin(\alpha-\gamma)}{2}+(\eta-1)\cos(\alpha-\gamma) \right]\left(1+3\frac{\alpha^{4}}{r^{2}}\right)\cos 2\theta\end{array}\right\}, \tag{3}\]  
\[\tau_{r\theta}=\frac{p_{z}}{2}\left\{\left[\frac{3(\eta-1)\sin(\alpha-\gamma) }{2}+(\eta-1)\cos(\alpha-\gamma)\right]\left(1+2\frac{\alpha^{2}}{r^{2}}-3 \frac{\alpha^{4}}{r^{4}}\right)\sin 2\theta\right\}, \tag{4}\]  
From the relevant mechanical calculation formula of elastic-plastic theory, the principal stress at any point in polar coordinates can be expressed as Equations (5) and (6).  
\[\sigma_{1}=\frac{\sigma_{r}+\sigma_{\theta}}{2}+\frac{1}{2}\sqrt{(\sigma_{r} -\sigma_{\theta})^{2}+4\tau_{r\theta}^{2}}, \tag{5}\]  
\[\sigma_{3}=\frac{\sigma_{r}+\sigma_{\theta}}{2}-\frac{1}{2}\sqrt{(\sigma_{r} -\sigma_{\theta})^{2}+4\tau_{r\theta}^{2}}, \tag{6}\]  
According to the MC criterion, the limit equilibrium condition can be expressed as the principal stress, as given in Equation (7).  
\[\sigma_{1}=2C\frac{\cos\varphi}{1-\sin\varphi}+\frac{1+\sin\varphi}{1-\sin \varphi}\sigma_{3}, \tag{7}\]  
According to Equations (2)-(7), and combined with the cosine quartic implicit equation of the principal stress direction deflection angle and the polar coordinate position angle of the plastic zone [28], the expression of the angle between the maximum principal stress and the coal seam and the polar coordinate position of the plastic zone can be obtained from Equation (8).  
\[\begin{array}{l}9(1-\eta)^{2}\big{(}\frac{\alpha}{r}\big{)}^{8}-12(1-\eta)^{ 2}\big{(}\frac{\alpha}{r}\big{)}^{6}+6\big{(}1-\eta^{2}\big{)}\big{(}\frac{ \alpha}{r}\big{)}^{6}\big{[}2\cos^{2}(\theta+\alpha-\gamma)-1\big{]}+10(1- \  
When calculating the plastic zone in heterogeneous layered rock strata, the redistribution of surrounding rock stress caused by the expansion deformation of the plastic zone was ignored. The relevant calculation program was written using Equation (5), and the parameters of each rock stratum were input for calculation and superposition. The equivalent calculation results of the plastic zone distribution under the different angle between the maximum principal stress and coal seam were obtained (Figure 12). In the calculation, the coal seam dip angle \(\alpha\) was taken as \(0^{\circ}\) (horizontal coal seam) and \(30^{\circ}\) (inclined coal seam), respectively.  
As shown in Figure 11, under different stress field conditions, the plastic zone expands nonuniformly under the action of coal seam dip angle and stress direction. When the angle is \(45^{\circ}\) (\(\pm 5^{\circ}\)), the degree of nonuniform expansion of the plastic zone is the largest (Figure 12(a-iv,b-iv)). The expansion degree of the plastic zone in the coal strata is affected by the lithology strength of the rock strata, and the expansion degree in the soft rock strata is higher than that in the hard rock strata.