## 3 Effect of the Particle Size Ratio on the Macroscopic Mechanical Behavior
### Effect of the Particle Size Ratio on the Uniaxial Compressive Strength  
The uniaxial compressive strength of specimens with different particle size ratios are extracted, and the average of the shear strengths of the four groups of specimens at the same particle size ratio is calculated, which is shown in Figure 17. The dispersion of uniaxial compressive strength corresponding to each particle size ratio is analyzed, and the strength standard deviation is shown in Figure 18.  
Figure 16: Failure modes of models with different particle size ratios.  
Figure 17: Variation in the uniaxial compressive strength with particle size ratio.  
Figure 18: Standard deviation of the uniaxial compressive strength with the particle size ratio.  
As can be seen in Figure 16, the average uniaxial compressive strength decreases gradually with the increase in the particle size ratio. When the particle size ratio ranges from 1.0 to 4.0, the average compressive strength decreases from 10.16 MPa to 7.21 MPa, with a decline amplitude of 29.10%. The average uniaxial compressive strength decreases faster when the particle size ratio is less than 2.0. When the particle size ratio is 1.0, the uniaxial compressive strength difference between specimens with different IDs is large, and the compressive strength range is around 3 MPa. For specimens with a particle size ratio greater than 1.5, the strength of specimens with different IDs is all roughly close to each other, and the range is within 1 MPa.  
From Figure 18, except for the standard deviation that is relatively high when the particle size ratio is 1.0, the standard deviations corresponding to other particle size ratio are maintained at a low level. When the particle size ratio ranges from 1.0 to 4.0, the standard deviation decreases from 1.27 to 0.32, with a decrease ratio of 75.15%. This indicates that when the particle size ratio is 1.0, not only is the uniaxial compressive strength higher but also the dispersion between the uniaxial compressive strength of specimens with different IDs is larger. When the particle size ratio increases to 1.5, the dispersion decreases significantly. This is because when the particle size ratio is 1.0, the space between the particles will be larger. Different particle spatial arrangements will cause different structures of the models, resulting in a large fluctuation of the uniaxial compressive strength and increased dispersion.  
As shown in Figure 19, the black thick line represents the compressive force, and the red line is the tensile force. When the particle size ratio is 1.0, the particles are arranged in a regular way, and the path of force chain is clear and neat, so that the compressive stress can be transmitted from the upper loading plate to the lower loading plate. Additionally, the tensile stress is small because of the clear path of force transmission. On the contrary, for specimen with particle size ratio of 2.0, due to the inconsistent size and irregular arrangement of particles, the path of force transmission is complicated, which results in a compressive stress that cannot be well transmitted to the lower loading plate, and more tensile stresses appear in the particles. The above phenomena finally lead to the largest compressive strength of model with a particle size ratio of 1.0.  
Figure 19: Force chain of models with particle size ratios of 1.0 versus 2.0.