## 3 Results
### Regularity of Particle Size Effect on Strength Behavior  
The definitions of friction angle (\(\varphi\)) and cohesion (\(c\)) in coarse-grained soils are primarily introduced for analytical convenience. However, it is challenging to strictly distinguish between these two parameters. Theoretically, cohesion (\(c\)) should be zero in non-cohesive soils, yet practical experiments often yield a non-zero value for \(c\). In this study, the observed non-zero \(c\) value (Table 2) for smooth glass beads does not represent true cohesion but rather results from inter-particle interactions such as sliding friction and interlocking. When the specimen is subjected to shear stress, the particles transition from a static state and must overcome their maximum static frictional force to initiate movement. This maximum static frictional force is macroscopically incorporated into the \(c\) value, which is defined as \"characterizing cohesion\".  
As previously discussed, the variation of shear strength behavior with particle size (\(d\)) for both small and large-sized granular materials indicates a consistent effect of particle size on the internal friction angle (\(\varphi\)) of cohesionless soils. The influence of particle size on strength is primarily reflected in the variation of \(\varphi\). Generally, \(\varphi\) exhibits a monotonically increasing trend with increasing \(d\), although there are intervals where this increase is minimal.  
To further investigate the influence of particle size on the internal friction angle, the relationship between \(\varphi\) and particle diameter (\(d\)) for both small and large-sized particles was plotted, as shown in Figure 14. This visualization allows for an intuitive comparison of how particle size affects \(\varphi\), thereby enhancing our understanding of the impact of particle size on the strength behavior of cohesionless soil.  
In the Figure 14, the internal friction angle (\(\varphi\)) of large-sized particle materials is generally significantly greater than that of small-sized particle materials. Moreover, both large and small-sized granular materials exhibit a gentle interval in \(\varphi\) values as particle size increases. The large-scale triaxial tests on large-sized particles were conducted to expand the particle size range of the study and to further compare the \(\varphi\) of small and large-sized particles, particularly focusing on the gentle zone discontinuity in the particle size increase process. The diameters of small and large-sized specimens are compared with the particle sizes in Table 5. It is evident that the diameter-to-diameter ratio of the insensitive particle size interval (4 mm-5 mm for small particles and 20 mm-25 mm for large particles) is nearly equal. Based on this observation, the gentle interval is defined as the \"insensitive particle size range\" for \(\varphi\) with respect to particle diameter \(d\), where the  
Figure 14: Relationship between \(\varphi\) and particle diameter (\(d\)) for both small-sized and large-sized particles.  
\(\varphi\) value remains almost unchanged as the particle diameter increases. Therefore, it can be concluded that the strength of particle materials generally increases monotonically with particle size. However, within certain particle size ranges (i.e., the insensitive particle size range), this increase is not significant, and \(\varphi\) shows insensitivity to changes in particle size. This finding is important for understanding the mechanical performance differences of materials with different particle sizes and has significant implications for material selection and design in practical engineering applications.