# Experimental and Numerical Investigations on the Slate Shearing Mechanical Behavior
## 4 Numerical Simulations  
### Numerical Model and Calibration  
In numerical simulations, uncertainties primarily stem from model parameter selection, boundary condition simplifications, rock heterogeneity, and numerical errors. Model parameters (such as particle diameter, contact stiffness, and bond strength) are typically set based on experimental or empirical data, but these parameters may exhibit significant variability in actual rocks, affecting the accuracy of the simulation results. The simplification of boundary conditions and the neglect of the complex internal structure of rocks (such as heterogeneity and anisotropy) can also lead to discrepancies between the simulation results and real-world conditions.  
The numerical simulation based on the discrete element method provides a robust supplement to indoor testing. A two-dimensional numerical model (50 mm \(\times\) 50 mm) was used to simulate the shear test. The minimum particle size was set at 0.20 mm, and the maximum to minimum particle radius ratio was 1.5, ensuring a uniform distribution. Through a trial-and-error method, the numerical simulation and experimental results are generally consistent. It cannot be entirely consistent due to numerical simulations typically assume that rock materials are homogeneous and isotropic, whereas actual rocks exhibit heterogeneity and anisotropy. In the simulations, particle shapes are idealized, while the geometrical structure of real rock particles is more complex. Bonding models assume that the bonding strength and fracture behavior between particles are relatively simple, but the microstructure of real rocks is more complex, especially during crack propagation and failure. Simplified bonding models cannot fully capture the mechanical responses observed in experiments (Figure 10). The determined micro-parameters are outlined in Table 2. With these micro-parameters, direct shear tests under various bedding dip angles were designed (Figure 11), where the bedding plane is marked in green. Eight rigid walls were used to simulate the upper and lower shear boxes.  
Figure 9: (**a**) AE count rate and (**b**) energy release rate of slate (\(\beta=0^{\circ}\), \(\sigma_{\rm n}=20\) MPa) with time elapse; relationship between mean value of maximum (**c**) AE count rate and (**d**) AE energy release rate and \(\beta\).  
The numerical model is somewhat idealized in the modeling process and can only analyze layered rocks through laboratory experiments, which presents certain limitations. A comparison of experimental results and numerical simulation results shows that, although there are some differences, the overall mechanical features, such as strength, stress drop, rate, failure characteristics, and failure modes of the numerical model, are similar to those of the experimental results. According to [34; 35; 36], these results are considered to have a certain degree of validity.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Description** & **Parameters** & **Value** \\ \hline \multirow{6}{*}{Ball} & \(r_{\text{min}}\) (mm) & 0.25 \\  & \(r_{\text{min}}/r_{\text{max}}\) & 1.5 \\  & Porosity ratio & 0.15 \\  & \(E\) (GPa) & 1 \\  & \(\rho\) (g/cm\({}^{3}\)) & 2661 \\  & Damp & 0.7 \\  & \(k_{n}/k_{\text{s}}\) & 1.5 \\ \hline \multirow{6}{*}{Parallel bond} & Pb\_ten (MPa) & 46 \\  & Pb\_coh (MPa) & 49 \\ \cline{1-1}  & \(E\) (GPa) & 2 \\ \cline{1-1}  & \(k_{n}/k_{\text{s}}\) & 1.3 \\ \cline{1-1}  & \(\text{Fric}\) & 0.8 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Mesoscopic parameters of numerical model.  
Figure 11: Numerical models of direct shear test for slate with different \(\beta\).  
Figure 10: Stress–strain curves of uniaxial compression and the corresponding numerical tests of slate.