## 3 Numerical Simulation
### Model Calibration and Validation
The F values and associated \(p\)-values are shown in Figure 10. We can accept the hypothesis that the property is sensitive to the factor if the corresponding \(p\)-value is less than \(\alpha\); otherwise, the property and the factor are relatively independent variates. The following correlations are evident in Figure 10:  
1. The unconfined compressive strength is sensitive to \(\sigma_{c}\), as well as \(c\) and \(\phi\);
2. The tensile strength is highly sensitive to \(\sigma_{c}\) and can be affected by \(\kappa^{*}\);
3. The elastic modulus is highly sensitive to \(E^{*}\) and \(\kappa^{*}\);
4. Poisson's ratio is sensitive to \(\kappa^{*}\) and will be affected by \(c\) and \(\phi\). During the model calibration process, some tests were applied with different random seeds to check the model's reliability. The results showed that the calculated coefficients of variation for the simulated macroproperties were less than 1%. The above observations indicate certain correlations between the micromechanical parameters and macroproperties. However, this paper does not intend to establish a quantitative prediction model of the macroproperties of flat-jointed material. The parametric study was primarily conducted to \begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Value** \\ \hline \(E^{*}\) (GPa) & 6.0, 8.0, 12.0 \\ \(\kappa^{*}\) & 2.0, 2.5, 3.0 \\ \(\mu\) & 0.3, 0.5, 0.8 \\ \(\sigma_{c}\) (MPa) & 3.0, 5.0, 8.0 \\ \(c\) (MPa) & 20.0, 30.0, 60.0 \\ \(\phi\) (\({}^{\circ}\)) & 25.0, 30.0, 35.0 \\ \hline \end{tabular}
\end{table} 
Table 2: Tested values of micromechanical parameters. Figure 9: Initial microstructure of flat-joint contact. optimize the calibration process. A set of best-fit parameters were obtained based on the parametric study, as listed in Table 3. Note that in this study, \(E^{*}=E^{*}_{hmm}\), \(\kappa^{*}=\kappa^{*}_{hmm}\), and \(\mu=\mu_{hmm}\). The test results were compared with those of the experiments. The force-displacement curves are shown in Figure 11. The result for the experiment was the average curve of five repeated tests. The displacement during the initial stage of the splitting test was relatively large. A possible reason for this was that the contact area was relatively small and the steel bar used for loading may not have been absolutely flat; it was difficult to ensure complete contact between a specimen and the loading equipment; thus, the initial stiffness was relatively small. However, the slope and peak value of the numerical model corresponded with those of the experiment very well. Figure 10: Results of a multifactor ANOVA (\(\alpha=0.05\)): (**a**) unconfined compressive strength; (**b**) tensile strength; (**c**) elastic modulus; and (**d**) [PERSON]’s ratio. \begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Value** \\ \hline \(\rho_{b}\) (kg/m\({}^{3}\)) & 2650 \\ \(N_{r}\) & 1 \\ \(N_{a}\) & 3 \\ \(C_{\lambda}\) & 0 \\ \(\lambda_{v}\) & 1 \\ \(g_{i}\) (m) & \(2.0\times 10^{-4}\) \\ \(\phi_{B}\) & 1.0 \\ \(E^{*}\) (GPa) & 8.0 \\ \(\kappa^{*}\) & 3.0 \\ \(\mu\) & 0.577 \\ \(\sigma_{c}\) (MPa) & 4.0 \\ \(c\) (MPa) & 23.0 \\ \(\phi\) (\({}^{\circ}\)) & 30.0 \\ \hline \end{tabular}
\end{table} 
Table 3: Tested values of micromechanical parameters. The results show that the numerical model was well fitted with the mechanical properties of high-strength gypsum we used and could reflect its basic mechanical characteristics. The microparameters were well calibrated and were suitable for the study of particle breakage under ball-plane contact conditions. The physical characteristics for numerical material are shown in Table 4  
The shape and size of the numerical model were consistent with the spherical gypsum particles in the experimental test. The diameter of the model was 50 mm. The minimum diameter of elementary particles was 1.0 mm, and the ratio of maximum to minimum size was 1.6. The ratio of model diameter to median particle size (\(L/d\)) was 38.5, and the effect on the simulation is generally very small, according to [PERSON] [45].