# Bridging Macro-Microscopic Parameters: Correspondence and Calibration Approaches
## 3 Fundamental Parameters in Particle Mechanics
### Micro Parameters
Figure 6: Numerical simulation of uniaxial compression tests. (**a**) Numerical simulation model, with (**b**) numerical simulation process. For the variable \(k_{n}\) type simulation, the influence of the three \(k_{s}\) values on \(E_{s}\) is minimal before \(k_{n}\) reaches \(1\times 10^{8}\). When \(k_{n}\) exceeds \(1\times 10^{8}\), the influence of \(k_{s}\) on \(E_{s}\) gradually increases. Under the three \(k_{s}\) conditions, \(k_{n}\) and \(E_{s}\) are positively correlated, indicating that \(\frac{\partial f(k_{n},k_{s})}{\partial k_{n}}\) is positive, and an increase in \(k_{n}\) amplifies the effect of \(k_{s}\) on \(E_{s}\), i.e., \(\frac{\partial^{2}f(k_{s},k_{s})}{\partial k_{n}\partial k_{n}}\) is greater than 0. Another approach is the variable \(k_{s}\) type simulation, which is the inverse of the previous method. The influence of \(k_{s}\) on \(E_{s}\) under different \(k_{n}\) values is observed (refer to Figure 8). Figure 8: Variable \(k_{s}\) type simulation. Figure 7: Variable \(k_{n}\) type simulation. From the perspective of \(k_{s}\), the curve can be divided into two segments below and above \(1\times 10^{8}\). For \(k_{s}\) = \(1\times 10^{8}\), the curve can be approximated as two straight lines. When \(k_{n}\) is \(1\times 10^{8}\), \(k_{s}\) above \(1\times 10^{8}\) can be approximated as a horizontal line, indicating that the change in \(k_{s}\) has a minimal impact on \(E_{s}\) when \(k_{n}\) is less than \(1\times 10^{8}\). When \(k_{n}\) exceeds \(1\times 10^{8}\), \(k_{s}\) positively correlates with \(E_{s}\), i.e., \(\frac{\partial f(k_{n},k_{s})}{\partial k_{s}}\) is greater than \(0\), and the slope for \(k_{s}\) less than \(1\times 10^{8}\) is greater than \(k_{s}\) greater than \(1\times 10^{8}\). Additionally, the increase in \(k_{s}\) enhances the impact of \(k_{n}\) on \(E_{s}\), indicating \(\frac{\partial^{2}f(k_{n},k_{s})}{\partial k_{n}\partial k_{s}}\) is greater than \(0\). From the above analysis, both \(k_{n}\) and \(k_{s}\) are positively correlated with \(E_{s}\). Additionally, \(\frac{\partial^{2}f(k_{n},k_{s})}{\partial k_{n}\partial k_{s}}\), \(\frac{\partial^{2}f(k_{n},k_{s})}{\partial k_{n}\partial k_{n}}\) are greater than \(0\), while \(\frac{\partial^{2}f(k_{n},k_{s})}{\partial k_{s}\partial k_{s}}\) is close to \(0\). Notably, \(k_{n}\) exerts a stronger influence on \(E_{s}\) compared to \(k_{s}\). Therefore, it is reasonable to assume that \(E_{s}\) behaves as a linearly increasing function of the coupling between \(k_{n}\) and \(k_{s}\). Thus, Equation (9) can be further represented as follows:  
\[E_{S}=e+a\times k_{n}+b\times k_{s}+c\times{k_{n}}^{2}+d\times k_{n}\times k_ {s} \tag{10}\]  
As per  
consistent. However, it is worth noting that the resulting error remains within an acceptable range. Fitting the data through MATLAB facilitates obtaining Figure 10 and Equation (12). \[E_{S}=\frac{987.6\times k_{n}\times 10^{7}+7.4\times k_{s}\times 10^{7}-3.7\times{k_{n}} ^{2}+5\times k_{n}\times k_{s}}{10^{14}} \tag{12}\]  
Upon observing Equation (12), it is apparent that the parameter b is relatively smaller than a, suggesting that \(k_{n}\) has a greater linear impact on \(E_{s}\) compared to \(k_{s}\).