# Bridging Macro-Microscopic Parameters: Correspondence and Calibration Approaches
## 4 Bridging Macro-Micro Parameters  
### Calibration Approaches  
Based on the preceding discussion, a microscopic parameter calibration method for sand is proposed, as follows:  
1. Obtain the fundamental coefficients required for the simulation. The parameters used in this paper are listed in Table 2, and the compression modulus \(E_{s}\) of the sample is obtained through uniaxial compression experiments. 2. Observe the influence of \(k_{n}\) and \(k_{s}\) on the \(E_{s}\) through the \"facet\" simulation approach. 3. Parameters for Equation (10) are fitted to obtain an empirical formula for \(E_{s}\) with respect to \(k_{n}\) and \(k_{s}\). 4. Direct shear experiments are conducted to obtain the shear stress-strain curve of the specimen. 5. Utilizing \(E_{s}\), together with the constraint \(k_{n}/k_{s}\) ranges from 1 to 3/2 and Equation (10), microscopic parameters \(k_{n}\) and \(k_{s}\) are jointly determined. 6. Numerical simulations of direct shear tests are performed. The true shear cutting line of the shear stress-strain curve and the simulated cutting line are compared. If the error is within 10%, the calibration is considered successful; otherwise, the process returns to step five for further calibration. Direct shear tests were conducted on the sample described earlier, with relevant parameters listed in Table 1. During the direct shear tests, the sand samples were subjected to shear under normal stresses of 100 kPa, 200 kPa, and 300 kPa, with a shear rate of 0.8 mm/min (Figure 11). Shear stress increases with increasing shear displacement, but the  
Figure 10: Three\(-\)dimensional relationship fitting between \(E_{s}\) and \(k_{n}\), \(k_{s}\) variables. rate of increase gradually decreases. At a pressure of 100 kPa, the shear stress reaches its peak value when the shear displacement reaches 1 mm, with the final peak value being 70 kPa. Similarly, at pressures of 200 kPa and 300 kPa, the shear displacements at which the shear stress approaches peak values are 3 mm and 4 mm, respectively, with peak stresses of 120 kPa and 270 kPa. It can be observed that increasing the pressure significantly enhances the shear strength of the sand sample, requiring longer shear distances to reach the maximum shear stress. Comparatively, [PERSON] and [PERSON] (1979) and [PERSON] (1949) are closer to [PERSON] and [PERSON] (1998), indicating that the ratio of \(k_{n}/k_{s}\) is between 3/2 and 1 [30; 31; 32]. Combining the \(E_{s}\) value obtained from the uniaxial compression experiment using Equation (12), we set \(k_{n}\) and \(k_{s}\) to be 2.65 \(\times\) 10\({}^{8}\) N/m and 2 \(\times\) 10\({}^{8}\) kPa N/m, respectively. Based on this, numerical simulations of direct shear experiments under 100 kPa were conducted. The sandy soil sample parameters and the simulation device used were consistent with those of the uniaxial compression simulation (Figure 6). A normal stress of 100 kPa was applied, and the upper wall was sheared at a rate of 1 mm/min, with the pressure stress maintained using servo principles throughout the process. The shear stress-shear displacement curve of the sand sample was obtained, and the simulated curve was compared with the actual curve (Figure 12). During the elastic stage of shearing (shear displacement less than 0.5 mm), the simulated values were in basic agreement with the experimental values. In the shear displacement range of 0.5 mm to 3 mm, the experimental values were slightly higher than the simulated values. This is attributed to the irregularities in the experimental particles compared to the simulated circular particles, allowing for a greater interlocking force among the particles during the initial displacement, resulting in a faster initial increase in shear stress in the experimental process than in the simulated process [33]. In the plastic stage of shearing (shear displacement greater than 3 mm), the simulated values also showed a high degree of agreement with the experimental values.