## 2 Effect of the Average Particle Size on the Model's Macroscopic Mechanical Behavior
### Discussion  
The increases in uniaxial compressive strength and elastic modulus with decreasing particle size (increasing \(L/d_{\text{ave}}\)) can be attributed to the denser packing of particles within the model. As the particle size decreases, the number of particles increases, leading to a more uniform and homogeneous distribution of stress within the model. This results in a higher compressive strength and elastic modulus. The underlying mechanism can be explained by the enhanced inter-particle interactions and bonding strength due to the higher surface area-to-volume ratio of smaller particles. Additionally, the denser packing reduces the void ratio within the model, resulting in a more homogeneous microstructure where stress is more evenly distributed. Consequently, the model exhibits higher stiffness and strength. However, the rate of increase in these properties decreases as \(L/d_{\text{ave}}\) exceeds 50, suggesting that beyond a certain level of particle density, the contribution to the mechanical properties becomes limited due to the saturation of inter-particle interactions and the diminishing effect of further reducing the void ratio.  
The observed decreases in the dispersion of compressive strength, elastic modulus, and shear stiffness with increasing \(L/d_{\text{ave}}\) indicate that the influence of the particles' spatial arrangement diminishes as the particle size decreases. Smaller particles can more effectively fill the voids within the model, reducing the variability in mechanical behavior caused by differences in the particle arrangement. This leads to a more uniform stress distribution, resulting in a lower dispersion of the mechanical properties. As the particle size decreases, the model becomes less sensitive to the spatial arrangement of particles, which can adapt to the applied stress more uniformly, reducing the impact of local variations in the particle distribution. When \(L/d_{\text{ave}}\) reaches 75, the compressive strength stabilizes, and the corresponding range drops to 10%. For the elastic modulus, the simulation results become relatively stable when \(L/d_{\text{ave}}\) is 33, with the relevant range dropping to 5%. These findings suggest that setting \(L/d_{\text{ave}}\) greater than 50 can ensure more reliable numerical simulation results.  
The increase in shear stiffness with decreasing particle size can be explained by the reduction in the internal void ratio within the model. As the particle size decreases, the model becomes denser, leading to higher shear stiffness. The underlying mechanism involves the enhanced interlocking of smaller particles, which provides greater resistance to shear deformation. Additionally, the denser packing of smaller particles enhances the microstructural stability of the model, reducing the deformation under shear loading. However, the rate of increase in shear stiffness slows down as \(L/d_{\text{ave}}\) exceeds 67, indicating that the effect of the particle size on shear stiffness is limited. This is consistent with the observation that the internal void ratio cannot be reduced indefinitely, and thus the increase in shear stiffness is finite.