## 2 Establishment of Numerical Model with Non-Parallel Overlapping Flaws
### Model Generation and Verification  
In this study, the numerical simulation experiment was based on the laboratory experiment conducted by [PERSON] et al. [22]. The samples with a size of 152 mm \(\times\) 76 mm were prepared for the static uniaxial compression test. In PFC2D, to reduce the interference of irrelevant factors in numerical simulation, the same random number was selected in all simulations, and the standard model with the same size as the laboratory experiment was established. There were 23,552 particles with a uniform distribution in each intact numerical model, and the particle size ranged from 0.3 mm to 0.5 mm. The sample model with non-parallel overlapping flaws was obtained by deleting particles in the specific area. The flaw length 2a was 12.7 mm, the flaw width 2b was 1.27 mm, the rock bridge length L was 12.7 mm, and the flaw inclination angle \(\beta\) was set as \(15^{\circ}\), \(30^{\circ}\), \(45^{\circ}\) and \(60^{\circ}\) respectively, as shown in Figure 2.  
The loading rate of 0.05 m/s was adopted, the specimen was loaded in a strain-controlled manner by specifying the speeds of the upper and lower walls. Local damping was used and the damping constant was set to 0.7. The axial stress in the numerical model was measured by the average value of the total force applied by the particles to the wall divided by the cross-sectional area of the specimen. After the establishment of the PFC2D numerical model, the micro-parameters of particles and particle bonding should be determined, and its mechanical properties should be consistent with the macro-mechanical response of laboratory physical experiments. At present, researchers mainly use the \"trial and error method\" to determine the meso parameters [49; 50; 51]. Firstly, the macro-elastic modulus of rock-like sample is determined by the stiffness ratio and meso modulus of the contact model. For a fixed sample model and a given stiffness ratio, the meso modulus can be calibrated by tests. Then, the calibrated meso parameters are fixed, and the bonding parameters are adjusted to match the strength of rock-like sample [33]. Finally, the meso parameters are fine-tuned to match the numerical simulation requirements during the numerical calculation. In this study, by repeatedly adjusting the meso parameters of the defect-free sample model, the macro-mechanical parameters were consistent with the laboratory experiments of [PERSON] et al. [22]. Table 1 shows the meso parameters of the sample model, and Table 2 shows the comparison of macro-mechanical parameters. Based on the calibrated model parameters, a simulation experiment was carried out on the sample with non-parallel overlapping flaws. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Value** & **Remark** \\ \hline Contact bond stiffness ratio & 1.2 & \\ Contact bond modulus (GPa) & 3.76 & \\ Friction coefficient & 0.35 & \\ Parallel bond stiffness ratio & 1.2 & \\ Parallel bond modulus (GPa) & 3.76 & \\ Parallel bond tension strength, mean (MPa) & 17.2 & Normal distribution \\ Parallel bond tension strength, standard deviation (MPa) & 2.06 & \\ Parallel bond cohesion, mean (MPa) & 28.0 & Normal distribution \\ Parallel bond cohesion, standard deviation (MPa) & 3.36 & \\ Parallel bond friction angle (\({}^{\circ}\)) & 30 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Meso parameters of rock-like materials in PFC2D model. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type** & **Density \(\rho\)/(g/cm\({}^{3}\))** & **Young’s Modulus E/GPa** & **Uniaxial Compressive Strength \(\sigma\)/MPa** \\ \hline Experimental results & 2.2 & 7.50 & 46.35 \\ Simulation results & 2.2 & 7.47 & 46.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of laboratory test results and numerical simulation results. Figure 2: Intact numerical model and a numerical model with non-parallel overlapping flaws. Table 3 shows the results of laboratory physical test and simulation results for samples with non-parallel overlapping flaws.