# Correlation Analysis of Macro-Micro Parameters of Sandstone Based on PFC3D
## 3 Results and Discussion
### Orthogonal Experiment Results
Changes in \(\overline{\sigma}_{t}\), \(\overline{c}/\overline{\sigma}_{t}\), and \(R_{\sigma}\) impact the inter-particle bond strength and homogeneity, thereby influencing the specimen's compressive performance. (4) The distribution coefficient \(\overline{\beta}\) (r = \(-0.597\)) exerts the most significant influence on the internal friction angle \(\varphi\), followed by the packing ratio \(R_{f}\) (r = \(-0.495\)), the friction coefficient \(\mu\) (r = 0.408), and the effective bond modulus \(\overline{E}_{c}\) (r = 0.342). \(\varphi\) increases with the rise in \(\mu\) but decreases with the increase in \(\overline{\beta}\) and \(R_{f}\). This indicates that variations in \(\mu\) directly impact inter-particle friction, whereas changes in \(\overline{\beta}\) and \(R_{f}\) affect the effective inter-particle contact, thereby influencing friction. (5) The factors influencing cohesion \(c\), ranked from most to least significant, are tensile strength \(\overline{\sigma}_{t}\) (r = 0.502), effective bond modulus \(\overline{E}_{c}\) (r = \(-0.415\)), cohesion/tensile strength  
Figure 4: Pearson correlation coefficient heatmap. \(\overline{c}/\overline{\sigma}_{t}\) (r = 0.386), packing ratio \(R_{f}\) (r = \(-\)0.368), strength ratio \(R_{r}\) (r = 0.364), and friction angle \(\overline{\phi}\) (r = \(-\)0.341). \(R_{r}\) is positively correlated with \(c\), indicating that an increase in these factors enhances inter-particle bonding, thereby increasing \(c\). Conversely, \(R_{f}\) and \(\overline{\phi}\) are negatively correlated with \(c\), suggesting that a decrease in the effective inter-particle contact reduces \(c\). Generally, an increase in bond strength enhances inter-particle bonding, but excessively strong bonding restricts particle movement, thereby increasing the specimen's deformation resistance and leading to a decrease in \(c\). (6) The critical deviatoric stress ratio is primarily influenced by the tensile strength \(\overline{\sigma}_{t}\)(r = \(-\)0.488), packing ratio \(R_{f}\) (r = 0.453), the ratio of cohesion to tensile strength \(\overline{c}/\overline{\sigma}_{t}\) (r = \(-\)0.439), the moment distribution coefficient \(\overline{\beta}\) (r = 0.365), and the friction angle \(\overline{\phi}\) (r = 0.361). This ratio decreases as \(\overline{\sigma}_{t}\) and \(\overline{c}/\overline{\sigma}_{t}\) increase and rises with the increase in \(R_{f}\), \(\overline{\beta}\), and \(\overline{\phi}\). This suggests that variations in \(\overline{\sigma}_{t}\) and \(\overline{c}/\overline{\sigma}_{t}\) directly impact the expansion of microcracks, whereas \(R_{f}\), \(\overline{\beta}\), and \(\overline{\phi}\) limit crack propagation by decreasing particle contact and enhancing friction, thereby elevating the ratio. (7) The friction angle \(\overline{\phi}\) (r = 0.569) exerts the most significant influence on \(\sigma_{c}/\sigma_{t}\), followed by the moment distribution coefficient \(\overline{\beta}\) (r = \(-\)0.486), the strength ratio \(R_{r}\) (r = \(-\)0.435), and the packing ratio \(R_{f}\) (r = 0.417). The positive correlation between \(\overline{\phi}\) and \(R_{f}\) indicates that increasing both parameters elevates the compression-tension ratio. \(\overline{\phi}\) impacts this ratio by modifying the inter-particle friction force, whereas \(R_{f}\) affects it by enhancing the material's deformation capacity. Conversely, the negative correlation between \(\overline{\beta}\) and \(R_{r}\) suggests that their increase leads to a reduction in the compression-tension ratio. An increase in \(\overline{\beta}\) causes local weakening in the sample, while a rise in \(R_{r}\) results in a more uniform strength distribution, thereby decreasing the compression-tension ratio.