## 3 Results
### Initial Failure Mode  
According to the stress analysis, the direction of the geo-stress and the lateral pressure coefficient can significantly affect the stress distribution around the opening within the rock mass, which in turn may impact the stability and failure characteristics of the opening. This study introduces the Discrete Element Method (DEM) numerical simulation to reveal the influence of stress conditions on the failure characteristics of the opening. In this study,  
Figure 5: Variation curves of maximum and minimum hoop stress at the opening boundary with \(\beta\) under different \(\lambda\): (**a**) maximum hoop stress; (**b**) minimum hoop stress. the PFC2D 6.0 software was used for numerical simulation of the mechanical behavior of samples containing openings. PFC2D software is a DEM simulation tool used to model the behavior of rock and soil materials. It is widely applied in the field of rock mechanics engineering to analyze the mechanical behavior, stability, and failure patterns of rock mass [24]. The linear parallel bond contact model was employed as the constitutive model for rock mass in the numerical modeling. This model effectively describes the mechanical behavior of rock or rock-like materials and has been widely applied in numerical simulations of rock mechanical failure [25]. Sandstone is a common rock found in deep underground mines, which is used as the reference material in the numerical simulation, and its mesoscale parameters are calibrated through uniaxial compression tests and are presented in Table 2.  
The geometry of the numerical model is shown in Figure 6, with the opening shape consistent with that used in the analytical calculation. To facilitate comparison between the analytical and numerical results, the opening area is set to 1 m\({}^{2}\), and the numerical simulation scheme is identical to the theoretical calculation. To minimize boundary effects, the overall model size is set to 7 m \(\times\) 7 m. When the lateral pressure coefficient \(\lambda\) = 0, the two side walls are set to non-active. When \(\lambda\) > 0, the side walls are set to servo control mode, with their stress equal to the product of axial stress and \(\lambda\), which changes with the axial stress. Due to the fact that the dimensions within the sample are much smaller than the overall sample size, the overall mechanical properties of the sample are minimally affected by the opening and cannot reflect the impact of stress direction on the stability of the opening. Alternatively, the study provides the distribution of initial failures at the opening boundary in some typical cases, which is presented in Figure 7. To better illustrate the damage of the surrounding rock, Figure 7 only presents the failure distribution within the central square region of the models. In the figure, the red areas represent fractures formed by contact failures between two contacting balls. \begin{table}
\begin{tabular}{c c c c c c c} \hline \(\rho\) & \(\mu\) & \(E_{\epsilon}\) & \(k^{*}\) & \(\overline{E}_{\epsilon}\) & \(\overline{k}^{*}\) & \(\overline{\epsilon}\) & \(\overline{\sigma}_{t}\) \\ \hline
2.70 g/cm\({}^{3}\) & 0.577 & 3.2 GPa & 1.0 & 3.2 GPa & 1.0 & 21.1 MPa & 13.2 MPa \\ \hline \(\rho\): density; \(\mu\): friction coefficient; \(E_{\epsilon}\): effective modulus; \(k^{*}\): normal-to-shear stiffness ratio; \(\overline{E}_{\epsilon}\): bond pressure modulus; \(\overline{k}^{*}\): bond normal-to-shear stiffness ratio; \(\overline{\epsilon}\): cohesion; \(\overline{\sigma}_{t}\): bond tensile strength. & & & \\ \end{tabular}
\end{table} 
Table 2: Rock parameters for the PFC2D numerical model. Figure 6: Geometry of rock mass containing an opening in the DEM numerical model. Figure 7: Initial failure distribution in the surrounding rock around the opening in typical cases. As shown in Figure 7, the initial failure distribution characteristics of the surrounding rock well validate the results of the stress analysis. In all cases, initial failures occur in areas of compressive or tensile stress concentration.