## 2 Materials and Methods
### The Blasting Rock Mass Throwing Model  
The underground blasting space was relatively confined, and the blasting operations in mine tunnels imposed multiple constraints on rock mass throwing. Research on the movement of blasted rock in underground environments was relatively scarce, with most studies focusing on rock mass movement in open-pit bench blasting. Many scholars had used the principle of spherical charge to approximate cylindrical charges as multiple spherical charges for the study of blasting issues. In this paper, based on the principle of spherical charge blasting, a model for underground blasting and rock throwing was constructed (Figure 4). In a two-dimensional cross-section of the blast hole, the radial lines emanating outward from the initiation point of the spherical charge divided the underground rock mass into multiple triangular finite elements. Assuming that the momentum imparted by the explosive energy was the same on different finite elements, according to the principle of spherical charge blasting, the movement velocity and direction of an individual element could be determined by the vector summation of the initial throw velocities imparted by \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Values** & **Parameters** & **Values** \\ \hline Compaction pressure PCO (GPa) & 6.0 & Residual surface parameter AN & 0.62 \\ \hline Damage parameter D1 & 0.04 & Reference compressive strain rate EOC & 3.0 \(\times\) 10\({}^{-5}\) \\ \hline Damage parameter D2 & 1.0 & Reference tensile strain rate EOT & 3.0 \(\times\) 10\({}^{-6}\) \\ \hline Tensile strain rate dependence exponent BETAT & 0.014 & Break compressive strain rate EC & 3.0 \(\times\) 10\({}^{-25}\) \\ \hline Tensile yield surface parameter GT\({}^{\star}\) & 0.7 & Break tensile strain rate ET & 3.0 \(\times\) 10\({}^{-25}\) \\ \hline Hugoniot polynomial coefficient A1 (GPa) & 65.78 & Failure surface parameter A & 2.40 \\ \hline Hugoniot polynomial coefficient A2 (GPa) & 84.19 & Failure surface parameter N & 0.823 \\ \hline Hugoniot polynomial coefficient A3 (GPa) & 22.28 & Lode angle dependence factor Q0 & 0.68 \\ \hline Parameter for polynomial EOS B0 & 1.28 & Lode angle dependence factor B & 0.0105 \\ \hline Parameter for polynomial EOS B1 & 1.28 & Shear modulus reduction factor XI & 0.5 \\ \hline Parameter for polynomial EOS T1 (GPa) & 65.78 & Minimum damaged residual strain EPM & 0.015 \\ \hline Parameter for polynomial EOS T2 & 0.0 & Gruneisen gamma GAMMA & 0.0 \\ \hline Porosity exponent NP & 3.0 & Initial porosity ALPHA & 1.0 \\ \hline Volumetric plastic strain fraction in tension PTF & 0.001 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._ \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Density/(kg\(\cdot\)m\({}^{-3}\))** & **Velocity of Donation/(m\(\cdot\)s\({}^{-1}\))** & **Pcj/GPa** & **A/GPa** & **B/GPa** & **R1** & **R1** & **\(\mathbf{\omega}\)** & **E0 (GPa)** \\ \hline
1320 & 6690 & 16 & 586 & 21.6 & 5.81 & 1.77 & 0.282 & 7.38 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Explosive material parameters. multiple spherical charges. The direction vector of the initial throw velocity acting on a specific element, generated by the explosive energy of several spherical charges, determined its movement direction. This method allowed for the determination of the blasting rock throwing velocity direction for any rock element within the underground rock mass. As shown in Figure 4, the underground blasting charge segment (BC) in this study is equivalently represented by N spherical charges, spaced equally along the axial direction. The mining rock mass in the stope is then divided into multiple triangular finite elements. The i-th rock mass element on the \(x\)-axis and the j-th rock mass element on the \(y\)-axis are denoted as \(\mathrm{M_{ij}}\), with an initial throw velocity of \(\mathrm{V_{ij}}\). The initial throw velocity generated by the n-th spherical charge at the rock mass element \(\mathrm{M_{ijn}}\) is denoted as \(\mathrm{V_{ijn}}\).