## 2 Research Methods
### Effectiveness Verification of Numerical Model
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Microscopic Parameter** & \multicolumn{5}{c}{**Numerical Value**} \\ \hline Mineralogical composition & Sapphire & Plagioclase & Potassium feldspar & \multirow{2}{*}{Mics} & Fine mineral \\ Percentage by volume/\% & 29.4 & 42.2 & 21.9 & 2.6 & 4.0 \\ Mineral Radius Minimum \(R_{\text{G}}\)/mm & 1.8 & 1.4 & 1.6 & 1.2 & 0.65 \\ Mineral radius maximum to minimum ratio \(r_{\text{G}}\) & & & 2.0 & & \\ \hline Basic unit & Sapphire & Plagioclase & Potassium feldspar & \multirow{2}{*}{Mics} & Fine mineral \\ Radius minimum \(R_{\text{P}}\)/mm & & & 0.65 & & \\ Ratio of maximum to minimum radius \(r_{\text{P}}\) & & & 2.0 & & \\ Density \(\rho_{\text{P}}\)/kg/m\({}^{3}\) & 2600 & 2650 & 3000 & 1700 & 2400 \\ Modulus of elasticity \(E_{\text{P}}\)/GPa & 62.0 & 47.0 & 32.0 & 27.0 & 22.0 \\ Rigidity ratio \(k_{\text{n-P}}\)/\(k_{\text{s-P}}\) & 1.4 & 1.6 & 1.8 & 2.0 & 2.2 \\ Friction factor \(\mu_{\text{P}}\) & 0.25 & 0.30 & 0.40 & 0.50 & 0.60 \\ \hline \hline \end{tabular}
\begin{tabular}{c c c c c} \hline \hline
**Intercrystalline contact** & Quartz inner & Plagioclase & Potassium feldspar & \multirow{2}{*}{Mica} & Inside the fine \\ Modulus of elasticity \(E_{\text{c-tra}}\)/GPa & 62.0 & 47.0 & 32.0 & 27.0 & 22.0 \\ Rigidity ratio \(k_{\text{n-tra}}\)/\(k_{\text{s-tra}}\) & 1.4 & 1.6 & 1.8 & 2.0 & 2.2 \\ Angle of internal friction \(\phi_{\text{tra}}\)/\({}^{\circ}\) & 12 & 14 & 16 & 18 & 22 \\ Bonding strength \(c_{\text{tra}}\)/MPa & 366.0 & 306.0 & 286.0 & 246.0 & 106.0 \\ Tensile strength \(\sigma_{\text{tra}}\)/MPa & 183.0 & 153.0 & 143.0 & 123.0 & 83.0 \\ \hline \hline \end{tabular}
\begin{tabular}{c c c c} \hline \hline
**Intergranular contact** & Between identical minerals & \multicolumn{3}{c}{Between different minerals} \\ Parallel stiffness ratio \(k_{\text{pb-n-ter-s}}\)/\(k_{\text{pb-s-ter-s}}\) & 2.6 & 2.8 & \\ Linear stiffness ratio \(k_{\text{c-n-ter-s}}\)/\(k_{\text{c-s-ter-s}}\) & 2.6 & 2.8 & \\ Friction factor \(\mu_{\text{ter-s}}\) & 0.7 & 0.8 & \\ Angle of internal friction \(\phi_{\text{per-s}}\)/\({}^{\circ}\) & 26.0 & 28.0 & \\ Parallel modulus of elasticity \(E_{\text{pb-ter-s}}\)/GPa & 2.5 & 2.2 & \\ Linear modulus of elasticity \(E_{\text{c-ter-s}}\)/GPa & 2.5 & 2.2 & \\ Bonding strength \(c_{\text{ter-s}}\)/MPa & 54.0 & 44.0 & \\ Tensile strength \(\sigma_{\text{ter-s}}\)/MPa & 27.0 & 22.0 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Micro-parameters of the numerical sample. By comparing and analyzing the uniaxial compression test results of the sample under laboratory testing and numerical simulation, the effectiveness of the micro-parameter calibration of the numerical model is verified. Figure 5a shows a comparison of the load-displacement curves from the uniaxial compression tests. The comparison indicates a strong consistency between the load-displacement results of the laboratory tests and the numerical simulations, demonstrating the reliability of the numerical model's micro-parameter calibration. In order to investigate the fracture process of the specimen in depth, in this study, the following six characteristic moments were selected during the loading process: the peak load moments of 20%, 40%, 60%, 80%, and 100% before the peak, and the peak load moments of 40% after the peak. These moments are denoted as \(t_{20}\), \(t_{40}\), \(t_{60}\), \(t_{80}\), \(t_{100}\), and \(t_{140}\), respectively. The load values at these six characteristic moments are extracted for comparison, as shown in Figure 5b. The figure demonstrates that the load levels of the experimental and simulation results are almost identical at the same characteristic time. Specifically, at peak load time \(t_{100}\), the load values for the experimental and simulation results are 320.55 kN and 315.12 kN, respectively, with an error of only 3.00%.