## 2 Effect of the Average Particle Size on the Model's Macroscopic Mechanical Behavior
### Effect of the Average Particle Size on the Elastic Modulus  
The elastic modulus of each specimen and the relationship between the average modulus and \(L/d_{\text{ave}}\) is shown in Figure 7. The larger the \(L/d_{\text{ave}}\), the bigger the elastic modulus. When \(L/d_{\text{ave}}\) increases from 10 to 100, the average elastic modulus increases from 1.29 GPa to 1.93 GPa by 49.6. However, the increase rate of the elastic modulus decreases with the increase in \(L/d_{\text{ave}}\), and the elastic modulus decreases greatly when \(L/d_{\text{ave}}\) is greater than 75. In addition, the larger the \(L/d_{\text{ave}}\) is, the smaller the difference between the elastic modulus of specimens with different IDs. It means that the denser the particles, the smaller the void inside the model, resulting in an increase in the elastic modulus. However, the increase in the elastic modulus is finite. When \(L/d_{\text{ave}}\) is larger, the increase rate of the elastic modulus becomes smaller. Figure 8 shows the variation in the elastic modulus variance with \(L/d_{\text{ave}}\). It is clear that the larger the \(L/d_{\text{ave}}\), the smaller the variance in the elastic modulus, and it suddenly decreases to a lower level when \(L/d_{\text{ave}}\) is 33. After that, the variance in the elastic modulus remains at a low level and gradually decreases. The variance in the elastic modulus decreases from the initial 0.057 to 0.0001, indicating that an increase in \(L/d_{\text{ave}}\) will make the specimens gradually tend to be identical, and the influence caused by the particles' spatial arrangement is gradually weakened. It can be seen in Table 4 that when \(L/d_{\text{ave}}\) is 33, the percentage of range of the average elastic modulus drops to 5%, and when \(L/d_{\text{ave}}\) is 75, it is already lower than 5%, indicating that the larger the \(L/d_{\text{ave}}\), the smaller the error of the numerical simulation result. Figure 6: Relationship between the uniaxial compressive strength variance and \(L/d_{\text{ave}}\). \begin{table}
\begin{tabular}{c c c} \hline \multirow{2}{*}{\(\mathbf{L/d_{\text{ave}}}\)} & **Compressive Strength** & **Percentage of the Range of** \\  & **Range (MPa)** & **Average Strength** \\ \hline
10 & 1.99 & 26.5\% \\
33 & 1.72 & 19.3\% \\
50 & 1.51 & 15.1\% \\
75 & 1.23 & 10.9\% \\
100 & 0.55 & 4.5\% \\ \hline \end{tabular}
\end{table} 
Table 3: Range analysis of compressive strength. Based on the above analysis, an increase in the particle density leads to an increase in the uniaxial compressive strength and elastic modulus, but the increase speed gradually decreases. This indicates that the contribution of the particle density to the mechanical parameters is limited when the particle density reaches a certain level. The denser the particles, the more the model tends to be uniform and homogeneous, the smaller the dispersion of the test results, and the influence caused by the spatial arrangement of particles gradually decreases. The denser the particles, the more identical the model tends to be. The dispersion of the simulation results become smaller, and the influence caused by the spatial arrangement of particles gradually decreases. In addition, the compressive \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{\(L/d_{\text{ave}}\)} & **Elastic Modulus** & **Percentage of the Range of the Average** \\  & **Range (GPa)** & **Elastic Modulus** \\ \hline
10 & 0.30 & 23.3\% \\
33 & 0.08 & 5.3\% \\
50 & 0.08 & 4.9\% \\
75 & 0.04 & 2.2\% \\
100 & 0.01 & 0.5\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Range of elastic moduli corresponding to different \(L/d_{\text{ave}}\). Figure 8: Variation in the elastic modulus variance with \(L/d_{\text{ave}}\). Figure 7: Variation in the elastic modulus with \(L/d_{\text{ave}}\). strength is stable when \(L/d_{\text{ave}}\) is 75, and the corresponding range drops to 10%. In terms of the elastic modulus, when \(L/d_{\text{ave}}\) is 33, the simulation results are relatively stable, and the relevant range drops to 5%.