# Correlation Analysis of Macro-Micro Parameters of Sandstone Based on PFC3D
## 3 Results and Discussion  
### Orthogonal Experiment Results  
Based on the orthogonal experimental design, 27 sets of numerical simulation analyses were conducted with the improved PBM. Through simulations of uniaxial compression, uniaxial tension, and triaxial compression, seven macroscopic parameters corresponding to each curve were extracted. These parameters included the elastic modulus (\(E\)), Poisson's ratio (\(v\)), uniaxial compressive strength (\(\sigma_{t}\)), internal friction angle (\(\varphi\)), cohesion (\(c\)), ratio of crack damage stress to uniaxial compressive strength (\(\sigma_{cal}/\sigma_{c}\)), and compressive-tensile strength ratio (\(\sigma_{c}/\sigma_{t}\)). The results of the numerical simulation are presented in Table 5.  
The results indicate that the configuration of different microscopic parameters significantly impacts the macroscopic performance of the sandstone model. The elastic modulus (\(E\)) ranges from 21.534 GPa to 36.037 GPa, the uniaxial compressive strength (\(\sigma_{c}\)) varies from 13.204 MPa to 141.244 MPa, the internal friction angle (\(\varphi\)) spans from 33.218\({}^{\circ}\)to 44.663\({}^{\circ}\), and the cohesion (\(c\)) ranges from 2.635 MPa to 12.294 MPa. The compressive-tensile strength ratio (\(\sigma_{c}/\sigma_{t}\)) fluctuates between 5.773 and 14.977, demonstrating that the improved PBM model effectively simulates the compressive-tensile strength ratio characteristics of sandstone, thereby resolving the issue of the low compressive-tensile strength ratio in the traditional PBM model. Figure 3: Determination of macro parameters. tion, \(0.5\leq|\,r\,|<0.7\) indicating a moderate correlation, \(0.3\leq|\,r\,|<0.5\) indicating a weak correlation, and \(|\,r\,|<0.3\) indicating essentially no correlation. As shown in the figure, the effects of the various microscopic parameters on the macroscopic parameters are as follows:  
(1) The bonding effective modulus (r = 0.876) exerts the most significant influence on the elastic modulus \(E\), followed by the packing ratio \(R_{f}\) (r = \(-0.416\)). The elastic modulus \(E\) increases with the rise in the bonding effective modulus but decreases as \(R_{f}\) increases. This indicates that variations in the bonding effective modulus directly affect the sample's stiffness, whereas \(R_{f}\) impacts stiffness by modifying the ratio of strong to weak inter-particle contacts. The other microscopic parameters with an \(|\,r\,|\) less than 0.3 have a negligible effect on \(E\) and are omitted from further discussion. (2) The impact of various microscopic parameters on [PERSON]'s ratio \(v\), from greatest to least, is as follows: filler ratio \(R_{f}\) (r = 0.612), tangential stiffness ratio \(k\) (r = 0.549), and effective bond modulus (r = \(-0.484\)). [PERSON]'s ratio \(v\) increases with increases in \(R_{f}\) and \(k\) but decreases with the increase in the effective bond modulus. This indicates that changes in \(R_{f}\) directly affect the sample's deformation capacity. An increase in \(k\) and the effective bond modulus enhances the inter-particle bonding stiffness, which suppresses relative particle movement, thereby limiting deformation and resulting in a decrease or increase in [PERSON]'s ratio. (3) The uniaxial compressive strength \(\sigma_{c}\) is primarily influenced by the moment distribution coefficient \(\overline{\beta}\) (r = \(-0.477\)), the packing ratio \(R_{f}\) (r = \(-0.434\)), the tensile strength \(\overline{\sigma}_{t}\)(r = 0.414), \(\overline{c}/\overline{\sigma}_{t}\) (r = 0.386), and the strength ratio \(R_{\sigma}\) (r = 0.342). The negative correlation between \(\overline{\beta}\) and \(R_{f}\) indicates that an increase in either leads to a decrease in \(\sigma_{c}\). \(\overline{\beta}\) affects the overall stability of the specimen through uneven moment distribution, while \(R_{f}\) reduces the compressive performance by increasing the proportion of weak contact.