## 3 Effect of the Particle Size Ratio on the Macroscopic Mechanical Behavior
### Effect of the Particle Size Ratio on the Models' Uniaxial Compression Mechanical Behavior  
The generated models' particle size parameters are shown in Table 7, and the generated models are shown in Figure 13. By changing the random factor, four groups of specimens with different particle spatial arrangements and identical contact properties are generated for repeated experiments. The relationship between the particle amount and particle size ratio is shown in Figure 14.  
From Figure 13, it can be found that, except for the model with particle size ratio of 1.0, the particle gradation of other models is well graded and can be densely bonded together. When the particle size ratio is 1.0, the particle size is same, which creates large voids during model generation and affects the integrity of the model. Therefore, in most of the models  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Specimen ID** & **Particle Size Range** & **Particle Size Ratio** \\  & **(\(R_{min}\)-\(R_{max}\)) (mm)** & \(R_{\text{mas}}\)/\(R_{\text{min}}\) \\ \hline  & 0.75–0.75 & 1.0 \\  & 0.6–0.9 & 1.5 \\
01, 02, 03, 04 & 0.5–1.0 & 2.0 \\  & 0.429–1.0725 & 2.5 \\  & 0.3–1.2 & 4.0 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Particle size range and corresponding particle size ratio.  
Figure 14: Relationship between the particle amount and particle size ratio.  
Figure 13: Schematic diagram of uniaxial compression model generation.  
that do not specify the particle gradation, the particle size ratio is generally from 1.5 to 2.0, according to the empirical value.  
From Figure 14, it can be seen that the particle amount decreases linearly with an increasing particle size ratio. This is because increasing the particle size ratio without changing the average particle radius will increase the maximum particle radius. Large particles take up more space in the model, resulting in a decrease in the number of particles.  
A linear equation is used to fit the variation in the particle number with the particle size ratio, and the result is as follows:  
\[y=2637.66-89.75x \tag{4}\]  
where \(y\) is the number of particles and \(x\) is the particle size ratio.  
The stress-strain curves of model number 01 with different particle size ratios in the uniaxial compression test are shown in Figure 15. The peak strength decreases gradually with the increase in the particle size ratio, but the decline amplitude becomes small when the particle size ratio reaches 2.0, and the strength is basically the same for models with particle size ratios of 2.0, 2.5, and 4.0. In addition, the slope of the elastic stage also decreases gradually with the increase in the particle size ratio. It should be noted that some stress curves will increase after some point decreases. This is because an increase in the loading force causes the appearance of cracks and the release of stress in rock, but does not destroy the bearing capacity of the rock.  
Figure 16 shows the schematic diagram of final failure characteristics for model number 01 with different particle size ratios, where the blue represents particles and the red means microcracks. At a particle size ratio of 1.0, the microcracks propagate along the gaps generated by the particles to the interior of the model. The failure mode of models with other four particle size ratios are similar to those of the laboratory test. This is because when the particle size ratio is 1.0, the model particles are regular in size. When uniaxial loading is applied on the specimen, the particle force transmission path is clear and the particle size is consistent, which can better constrain the particle displacement and timely alter the contact stiffness between particles, resulting in a larger elastic modulus and compressive strength.  
Figure 15: Stress–strain curves of models with different particle size ratios.