# Bridging Macro-Microscopic Parameters: Correspondence and Calibration Approaches
## 3 Fundamental Parameters in Particle Mechanics
### Micro Parameters  
To determine the normal stiffness \(k_{n}\) and tangential stiffness \(k_{s}\) of the particles based on the macroscopic mechanical parameters of the sand sample, the compression modulus \(E_{s}\) was used as a macroscopic mechanical parameter to characterize the elastic deformation capability of the sample. Therefore, the macro-microscopic compression modulus relationship shown in Equation (8) was established. \[E_{s}=f(k_{n},k_{s},\eta_{1},\eta_{2},\eta_{3},\rho) \tag{8}\]  
where \(k_{n}\) represents the normal stiffness of the particles, \(k_{s}\) represents the tangential stiffness of the particles, \(\eta_{1}\) represents the friction coefficient between particles, \(\eta_{2}\) represents the friction coefficient between particles and the wall, \(\eta_{3}\) represents the rolling friction coefficient between particles and the wall, and \(\rho\) represents the particle density. \(\eta_{1}\), \(\eta_{2}\), and \(\eta_{3}\) can be obtained from previous experiments, thereby allowing further derivation of Equation (9). \[E_{s}=f(k_{n},k_{s}) \tag{9}\]  
To further investigate the relationship between particle contact stiffness and the macroscopic compression modulus, uniaxial compression numerical simulations were conducted using the parameters listed in Table 2. The particle size and grading are consistent with those of the original test sand (Table 1). However, to mitigate the issue of excessively small particles resulting in small time steps, particles smaller than 1 mm were resized to 1 mm in diameter, with a uniform distribution applied across each diameter segment. Figure 5: Uniaxial compression tests data. The model size is designed to be five times the size of the uniaxial compression test apparatus (Figure 6). Larger model sizes allow for more particles in the model, leading to more stable simulation results. The model is surrounded by rigid walls forming the boundaries. The side walls provide lateral constraint for uniaxial compression tests, and the upper wall applies pressure using servo principles. A \"facet\" simulation approach is used to observe the influence of \(k_{n}\) and \(k_{s}\) on particle compression modulus \(E_{s}\). By fixing \(k_{n}\) and \(k_{s}\) separately and continuously changing the other parameter, local simulation points reflect the simulation surface to some extent. Firstly, in the variable \(k_{n}\) type simulation, with \(k_{s}\) set to \(1\times 10^{8}\), \(5\times 10^{8}\), and \(1\times 10^{9}\), \(k_{n}\) is initially set to \(1\times 10^{7}\), then incremented by \(1\times 10^{7}\) until it reaches \(1\times 10^{8}\), and subsequently incremented by \(1\times 10^{8}\) until \(k_{n}\) reaches \(1\times 10^{9}\). The influence of \(k_{n}\) on \(E_{s}\) under different \(k_{s}\) values is observed (refer to Figure 7). The specific simulation process involves initially setting \(k_{n}\) and \(k_{s}\), followed by preloading the stacked particles with a normal force of 50 kPa to eliminate gaps or ineffective contacts between particles. Subsequently, the applied stress is continuously increased, with compression measurements recorded every 50 kPa until reaching 500 kPa. This completes one cycle of uniaxial compression numerical simulation, and finally, the compression modulus of the sand sample is calculated using the formula \(E_{s}=\frac{\Delta\sigma}{\Delta\xi}\). \begin{table}
\begin{tabular}{l c c} \hline
**Parameters** & **Value** & **Unit** \\ \hline Particle density & 2210 & kg/m\({}^{3}\) \\ Interparticle friction & 0.762 & / \\ Cohesion & 0 & kPa \\ Particle normal stiffness & - & N/m \\ Particle tangential stiffness & - & N/m \\ Wall normal stiffness & \(1\times 10^{8}\) & N/m \\ Wall tangential stiffness & \(1\times 10^{8}\) & N/m \\ Particle–wall friction & 0.424 & / \\ Particle–wall rolling & 0.306 & / \\ friction & 0.05 & / \\ Local damping & 0.2 & / \\ Viscous damping & 0.2 & / \\ \hline \end{tabular} Note: The “-” means calibration is required.  
\end{table} 
Table 2: Parameters used for numerical simulation of 1–5 mm sample.