## 4 Conclusions  
This study employs the Particle Flow Code (PFC) to systematically investigate the influence of particle size parameters on the macroscopic mechanical behavior of rock models under uniaxial compression and direct shear. The findings provide novel insights into the relationships between the particle size, particle size ratio, and the mechanical properties of rock models. The conclusions drawn from this research are summarized as follows:  
1. Effect of the particle size on the mechanical properties--The results demonstrate that decreasing the particle size (increasing \(L/d_{\text{ave}}\)) leads to a denser packing of particles within the model, thereby enhancing the mechanical properties. Specifically, the uniaxial compressive strength increases by 62.7% and the elastic modulus increases by 49.6% as \(L/d_{\text{ave}}\) increases from 10 to 100. The shear stiffness also increases by approximately 40% as \(L/d_{\text{ave}}\) increases from 10 to 80.
2. Reduction in the dispersion of mechanical properties--The dispersion of the compressive strength, elastic modulus, and shear stiffness decreases significantly with increasing \(L/d_{\text{ave}}\). When \(L/d_{\text{ave}}\) increases from 10 to 100, the dispersion of compressive strength decreases by 88.51%, the dispersion of the elastic modulus decreases by 99.82%, and the dispersion of shear stiffness decreases by 71.87%. This indicates that smaller particle sizes result in more uniform and reliable numerical simulation results. Setting \(L/d_{\text{ave}}\) greater than 50 is recommended to ensure the reliability of the numerical simulations.
3. Effect of the particle size ratio on the mechanical properties--Increasing the particle size ratio enhances the particle gradation but reduces the uniaxial compressive strength and elastic modulus. When the particle size ratio increases from 1.0 to 4.0, the uniaxial compressive strength decreases by 29.10% and the elastic modulus decreases by 38.89%. However, the dispersion of mechanical properties is significantly reduced when the particle size ratio is greater than 1.5. This suggests that a particle size ratio greater than 1.5 should be used to ensure the accuracy and reliability of the numerical simulations.  
Conceptualization, C.Y., X.F. and H.L.; methodology, C.Y., S.L. and H.L.; software, X.D., Z.L. and H.L.; formal analysis, S.L.; investigation, S.L. and X.D.; resources, H.L. and X.F.; data curation, C.Y., Z.L. and H.L.; writing--original draft, S.L. and X.D.; writing--review and editing, C.Y., S.L. and H.L. All authors have read and agreed to the published version of the manuscript.  
This paper received funding from Projects 42277175 supported by the National Natural Science Foundation of China; Project 2023 JJ30657 supported by the Hunan Provincial Natural Science Foundation of China; the Guizhou Provincial Major Scientific and Technological Program (2023-425); and Yunnan Province Science and Technology Plan Project 202205 AD160063. The authors wish to acknowledge this support.  
Not applicable.  
The data used to support the findings of this study are available from the corresponding author upon request.  
Authors [PERSON] and [PERSON] were employed by the company Yunnan Diqing Non-Ferrous Metals Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.