## 2 Materials and Methods  
### Numerical Experiments of Uniaxial Compressive on Layered Rock  
The PFC2D 5.0 program is based on particle flow discrete elements and is widely used to simulate the generation and propagation of cracks in media. Simulation is performed by PFC (2D) 5.0, and the macroscopic mechanical parameters of the calculated model can be simulated by simply setting the meso-mechanical parameters of the particles. The PFC program provides two particle bonding models, contact bond (CB) junction and parallel bond (PB), and the parallel bond model can be used to simulate the adhesion of cementitious material between two adjacent particles; when the normal or tangential stress exceeds the corresponding parallel bond strength, tensile microcracks or shear microcracks and parallel bond failure will occur. Figure 1 shows the loading conditions of the layered rock specimens. \(\alpha\) is the bedding dip. The numerical model with PFC2D setting different bedding inclination dips (0\({}^{\circ}\) to 180\({}^{\circ}\), one every 30\({}^{\circ}\)) is used for numerical simulation, the model size is 50 mm \(\times\) 100 mm, and the displacement loading method is used for simulation (Figure 2). In this paper, the axial displacement loading rate is set to 0.0005 m/s, and the calculation termination condition is as follows: stop the calculation when the residual strength of the model reaches 60% of the peak strength. In order to better reflect the mechanical properties of rock, the parallel bond model is used as the contact model between particles, and the smooth joint model is used between bedding planes. The initial parameters of the model are derived from the research results of [PERSON] [39] and [PERSON] [18], and then adjusted according to the macroscopic mechanical parameters and failure modes. The final meso-mechanical parameters of the numerical calculation model are shown in Table 1.  
Figure 1: Numerical model loading conditions. Figure 2: Numerical model of bedding rock. (**a**) The bedding dip is 0\({}^{\circ}\); (**b**) the bedding dip is 30\({}^{\circ}\); (**c**) the bedding dip is 60\({}^{\circ}\); (**d**) the bedding dip is 90\({}^{\circ}\); (**e**) the bedding dip is120\({}^{\circ}\); (**f**) the bedding dip is 150\({}^{\circ}\). [PERSON] [19] proposed the single weak surface principle, which assumed that the transversely isotropic rock material consists of a rock medium and a set of parallel bedding weak surfaces. The failure of rock medium and weak bedding follows the Mohr-Coulomb criterion [40], but the cohesion and internal friction dip of rock medium and weak bedding are not equal. Therefore, the breaking criterion is formulated as follows:  
For the destruction of rock materials,  
\[\tau=c+\sigma\tan\varphi \tag{1}\]  
For the destruction of bedding surfaces,  
\[\tau_{\theta}=c_{w}+\sigma_{\theta}\tan\varphi_{w} \tag{2}\]  
where \(c\) and \(\varphi\) are the cohesive force and internal friction dips of the rock medium, respectively, and \(c_{w}\) and \(\varphi_{w}\) are the cohesion and internal friction dips of the bedding surface. \(\tau\) and \(\sigma\) are tangential stress and normal stress, respectively. \(\tau_{\theta}\) and \(\sigma_{\theta}\) are the tangential stress and normal stress when the dip of the bedding plane is \(\theta\), respectively. As shown in Figure 3, there is a weak surface in the rock, and the dip between this weak surface and the minimum principal stress is \(\beta\). From the single weak surface theory, it can be seen that the conditions for failure along the weak surface are  
\[\sigma_{1}=\sigma_{3}+\frac{2(\varsigma_{w}+\sigma_{3}\tan\varphi_{w})}{(1- \tan\varphi_{w}\cot\beta)\sin 2\beta} \tag{3}\]