## 5 Conclusions  
Based on the concept of the control variable method, ideal spherical glass beads were employed as the experimental material to systematically investigate the effect of particle size-a single internal variable-on the strength characteristics of cohesionless soil. Additionally, the intrinsic mechanisms underlying the influence of particle size variations on strength behavior were thoroughly analyzed. The findings provide a theoretical basis for optimizing the selection of particle sizes in fill materials for engineering applications such as tunnels and deep foundations in cohesionless soil regions. Furthermore, this study offers new insights and methods for the optimization of soil materials in underground engineering. The specific conclusions are as follows:  
1. Particle size significantly influences the mechanical properties of cohesionless soil. For a given confining pressure, the peak shear stress (\(q_{t}\)) of the specimen increases with particle size. This trend is observed for both small-sized (2 mm-6 mm) and large-sized (10 mm-30 mm) granular particles, with the growth rate of \(q_{t}\) being higher under high confining pressures (300 kPa and 400 kPa) compared to low confining pressures (100 kPa and 200 kPa).
2. The internal friction angle (\(\varphi\)) increases monotonically with particle diameter (\(d\)) for both small-sized and large-sized granular materials. However, there exists an insen  
Figure 18: Schematic diagram of occlusion areas for particles of different sizes.  
sitive particle size range where the increase in \(\varphi\) is minimal: 4 mm-5 mm for small particles and 20 mm-25 mm for large particles. Additionally, large-sized granular materials exhibit significantly higher strength than small-sized granular materials.
3. For large-sized granular particles, the particle breakage rate (\(B_{\mathrm{r}}\)) increases with confining pressure. Moreover, \(B_{\mathrm{r}}\) initially increases and then decreases with increasing particle diameter (\(d\)).
4. The insensitive particle size ranges in the increase of internal friction angle (\(\varphi\)) with particle size reflect the macroscopic manifestation of the combined effects of two mesoscopic structural parameters: the number of particle contact points and the micro-contact area between particles. These parameters reach an extremum, resulting in a plateau in the overall strength as particle size increases.  
This study has limitations due to the lack of micro-scale testing methods and equipment during triaxial tests, preventing observation of particle information during shearing. We hope to conduct more systematic research in the future when conditions permit.  
L.Z.: Conceptualization, Writing and Editing; F.L.: Conceptualization, Review and Supervision, Resources, Funding acquisition; Z.W.: Review and Editing, Visualization Figures, Funding acquisition; N.L.: Methodology, Investigation Supervision and Data Curation; H.L.: Writing--Review and Funding acquisition; B.W.: Writing--Review and Editing. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the National Natural Science Foundation of China (No. 12072260) and the National Natural Science Foundation of China (No. 52278370) and the Project for Enhancing Composite Foundation Treatment Construction Technology by Anhui Construction Engineering San Jian Group Co., Ltd. (RD2024-XB-03).  
Not applicable.  
The data presented in this study are available from the corresponding author upon reasonable request.  
The authors declare no conflicts of interest.