## 2 Effect of the Average Particle Size on the Model's Macroscopic Mechanical Behavior
### Effect of the Average Particle Size on the Model's Shear Mechanical Behavior  
By changing the random factor, four different numerical models with same particles are generated, which are equivalent to four specimens with the same mechanical properties but different particle spatial arrangements. Four groups of samples numbered 01, 02, 03, and 04 with dimensions of 100 mm \(\times\) 100 mm are complete rock mass cube samples. After applying different normal stresses (0.5 MPa, 1.0 MPa, 1.5 MPa, 2.0 MPa, and 2.5 MPa) to the four groups of models with different average particle sizes, direct shear tests are carried out. The particle size ratio of model is kept at 1.5, and five different particle size ranges are set, as shown in Table 5.  
The generated numerical model is shown in Figure 9, and the particle amount corresponding to each \(L/d_{\text{ave}}\) is shown in Figure 10. It can be seen that the amount of particles and the increase speed of the particle amount increase with the increase in \(L/d_{\text{ave}}\). The quadratic function is used for fitting and the equation is as follows:  
\[y=28.5-2.24x+1.16x^{2} \tag{3}\]  
Figure 9: Schematic diagram of the generated model used for the shear test. \begin{table}
\begin{tabular}{c c c c} \hline
**Specimen ID** & **Particle Size Range (\(R_{min}\)–\(R_{max}\)) (mm)** & \(L/d_{\text{ave}}\) & **Normal Stress (MPa)** \\ \hline  & 4.0–6.0 & 10 & \\  & 2.0–3.0 & 20 & 0.5, 1.0, 1.5, \\
01, 02, 03, 04 & 0.8–1.2 & 50 & 2.0, 2.5 \\  & 0.6–0.9 & 67 & \\  & 0.5–0.75 & 80 & \\ \hline \end{tabular}
\end{table} 
Table 5: Particle size ranges and relevant test parameters. where \(y\) is the number of particles, \(x\) is the corresponding \(L/d_{\text{ave}}\), and the fitting determination coefficient R\({}^{2}\) is 0.99994, indicating that the relationship of the particle amount in the cube model with \(L/d_{\text{ave}}\) can be predicted by fitting the quadratic polynomial equation. For models under a normal stress of 2.0 MPa, the relationship between their shear stiffness and \(L/d_{\text{ave}}\) is plotted in Figure 11. The average shear stiffness of the four groups of models with different normal stresses and \(L/d_{\text{ave}}\) are shown in Table 6. It can be seen from Figure 11 that under a normal stress of 2.0 MPa, the shear stiffness gradually increases with the increase in \(L/d_{\text{ave}}\), and the average shear stiffness increases from 3.98 MPa/mm to 5.56 MPa/mm, with a growth rate of 39.7%. The increasing trend of shear stiffness slows down with the increase in \(L/d_{\text{ave}}\). When \(L/d_{\text{ave}}\) is greater than 67, the shear stiffness basically no longer increases, indicating that the influence of \(L/d_{\text{ave}}\) on shear stiffness is limited. When \(L/d_{\text{ave}}\) reaches a certain value, the shear stiffness will no longer increase. This is because under the premise of a fixed model size, when \(L/d_{\text{ave}}\) is low, the size of the particles is larger, resulting in a higher porosity and a lower shear stiffness under shear loading. With the increase in \(L/d_{\text{ave}}\) and the decrease in the particle size, the internal void ratio decreases, resulting in an increase in shear stiffness. However, the internal void ratio cannot be reduced indefinitely, and so the increasing rate of shear stiffness also gradually decreases, and, finally, it can only increase slightly or even stop increasing with the increase in \(L/d_{\text{ave}}\). It can be seen from Table 5 that when \(L/d_{\text{ave}}\) increases from 10 to 80, the shear stiffness also increases, and the increasing ratio of average shear stiffness ranges from 30% to 50%.