## 3 Effect of the Particle Size Ratio on the Macroscopic Mechanical Behavior
### Effect of the Particle Size Ratio on the Models' Shear Mechanical Behavior  
Keeping the average particle size (the average of \(R_{max}\) and \(R_{min}\)) unchanged at 0.75 mm, set five groups of different particle size ranges and corresponding particle size ratios. The relevant particle size parameters are shown in Table 8. For each particle size ratio, four models numbered 01, 02, 03, and 04 with dimension of 100 mm \(\times\) 100 mm are generated. The schematic diagram of the generated models and the relationship between the particle amount and particle size ratio are shown in Figures 21 and 22, respectively. Direct shear test with different normal stress levels (0.5 MPa, 1.0 MPa, 1.5 MPa, 2.0 MPa, and 2.5 MPa) are carried out on the model. As can be seen in Figure 22, the particle gradation of other models is better, except for the particle size ratio of 1.0. This is caused by the fact that the particle size is then same when the particle size ratio is 1.0, and there are gaps inside the model after generation, which affects the model's integrity. From Figure 23, the particle amount decreases with an increasing particle size ratio, which can be expressed by a linear expression  
\[y=5284.51-192.87x \tag{5}\]  
where \(y\) is the particle amount, \(x\) is the particle size ratio, and the fitting coefficient \(R^{2}\) is 0.99. This indicates that the particle amount decreases linearly with the increase in the particle size ratio. \begin{table}
\begin{tabular}{c c c c} \hline
**Specimen ID** & **Particle Size Range (mm)** & \(R_{max}/R_{min}\) & **Normal Stress (MPa)** \\ \hline  & 0.75–0.75 & 1.0 & \\  & 0.6–0.9 & 1.5 & \\
01, 02, 03, 04 & 0.5–1.0 & 2.0 & 0.5, 1.0, 1.5, 2.0, 2.5 \\  & 0.429–1.0725 & 2.5 & \\  & 0.3–1.2 & 4.0 & \\ \hline \end{tabular}
\end{table} 
Table 8: Particle size ranges and test parameters. The average value of shear stiffness of four repeated specimens is calculated, which can be fitted by the nonlinear equation  
\[y_{s}=y_{0}+a\cdot e^{-x/t} \tag{6}\]  
where \(y_{s}\) is the shear stiffness and \(x\) represents the particle size ratio. \(y_{0}\), \(a\), and \(t\) all are fitting parameters. The variation curve of the average shear stiffness with the particle size ratio is shown in Figure 24. The obtained function parameters are shown in Table 9 and the variation in the fitted parameters with normal stress is shown in Figure 25.  
Figure 23: Relationship between the particle amount and particle size ratio. Figure 22: Schematic diagram of the generated models for the direct shear test. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter**Normal Stress** & **0.5 MPa** & **1.0 MPa** & **1.5 MPa** & **2.0 MPa** & **2.5 MPa** \\ \hline \(y_{0}\) & 2.55 & 3.24 & 4.02 & 4.91 & 5.70 \\ \(a\) & 5.66 & 5.74 & 5.81 & 5.96 & 6.09 \\ \(t\) & 0.62 & 0.66 & 0.70 & 0.65 & 0.70 \\ \(R^{2}\) (COD) & 0.994 & 0.983 & 0.982 & 0.971 & 0.987 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Fitting parameters of the average shear stiffness equation \(y_{s}=y_{0}+a\times e^{-x/t}\). It can be seen from Figure 24 that the average shear stiffness gradually decreases with the increase in the particle size ratio. With the increase in normal stress, the average shear stiffness decreases by 31.36%, 28.63%, 26.04%, 21.02%, and 21.18%, respectively. The average shear stiffness decreases first at a fast rate and then at a slow rate as the particle size ratio increases, and tends to level off when the particle size ratio is greater than 2.5. This is because the particle size is the same in the model with a particle size ratio of 1.0. They are better able to resist each other's relative motion, which allows for better conversion of particle contact stiffness to macroscopic shear stiffness. As the particle size ratio increases, the amount of small particles increases, and the small particles in the shear plane cannot resist the movement of other large particles well, so that the deformation increases and the corresponding shear stiffness decreases.