## 3 Numerical Model
### Principle of Numerical Simulation Method  
Considering the transport of a caved body, the saving method is used to control the ground pressure through collapse, and the damage form of typical caved mining surrounding rock is shown in Figure 2. The simulation should not only consider the subduing, softening, and failure of the surrounding rock area but should also consider the impact of the surrounding rock on an expansive bulk filling goat after failure. Additionally, it should have a supporting effect on the surrounding rock in the empty area, that is, the influence of caving body migration. Therefore, under the condition of considering the influence of caved body transport, when performing rock stability simulation analysis, instead of using large deformation calculation, the caved area is considered as Eulerian description; the Mohr-Coulomb ideal plasticity criterion is calculated for the caved body; and the strain-softening criterion is calculated for the surrounding rock around the caved body. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Modulus of Elasticity** & **Poisson’s Ratio** & **Internal Friction Angle** & **Compressive Strength** & **Tensile Strength** & **Cohesion** \\ \hline
65.12 MPa & 0.3 & 35\({}^{\circ}\) & 106.44 MPa & 8.87 MPa & 25 kPa \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical parameters of rock mass. (1) The Mohr-Coulomb criterion is implemented using the following steps:  
1. The elastic principal structure is used to calculate \(\sigma_{\text{x}}\), \(\sigma_{\text{y}}\), and \(\sigma_{\text{xy}}\) for the new step. The principal stresses \(\sigma_{1}\) and \(\sigma_{3}\) and their directions are calculated. \(\sigma_{1}<\sigma_{3}\), i.e., \(\sigma_{1}<\sigma_{3}\) is the maximum principal stress (if the pull is positive). 2. The shear and tensile damage criteria for rocks are shown in Equations (1) and (2), respectively, and \(\sigma_{1}\) and \(\sigma_{3}\) are substituted to determine whether damage occurs. \[\text{f}_{\text{s}}=\sigma_{1}-\sigma_{3}\text{N}_{\varphi}+2\text{c}\sqrt{ \text{N}_{\varphi}}\] (1) \[\text{f}_{\text{t}}=\sigma_{3}-\sigma_{\text{t}})0\] (2) \[\text{N}_{\varphi}=\frac{1+\sin\varphi}{1-\sin\varphi}\] (3) where c is the cohesive force; \(\varphi\) is the angle of internal friction; and \(\sigma_{\text{t}}\) is the tensile strength. 3. If shear damage occurs, then the principal stress is recalculated according to Equations (4) and (5) and \[\sigma_{1}^{\text{N}}=\sigma_{1}-\lambda_{\text{s}}(\text{a}_{1}-\text{a}_{2 }\text{N}_{\psi})\] (4) \[\sigma_{3}^{\text{N}}=\sigma_{3}-\lambda_{\text{s}}(\text{a}_{2}-\text{a}_{1 }\text{N}_{\psi})\] (5) Among them, \[\lambda_{\text{s}}=\frac{\text{f}_{\text{s}}(\sigma_{1}+\sigma_{3})}{\text {a}_{1}-\text{a}_{2}\text{N}_{\psi}-(\text{a}_{2}-\text{a}_{1}\text{N}_{\psi })\text{N}_{\psi}}\] (6) \[\text{a}_{1}=\text{K}+\frac{3}{4}\sigma\] (7) \[\text{a}_{2}=\text{K}-\frac{2}{3}\sigma\] (8)  
Figure 2: Schematic diagram of saving in saving mining. \[\text{f}_{\text{s}}=\sigma_{1}-\sigma_{3}\text{N}_{\varphi}+2\text{c}\sqrt{ \text{N}_{\varphi}}\]0  
where \(\psi\) is the shear expansion angle, and for the associative flow law, we have \(\psi=\varphi\). 4. If tensile damage occurs, the principal stresses are recalculated according to Equations (10) and (11). \[\text{f}_{\text{s}}=\sigma_{1}-\sigma_{3}\text{N}_{\varphi}+2\text{c}\sqrt{ \text{N}_{\varphi}}\]1 (10) \[\text{f}_{\text{s}}=\sigma_{1}-\sigma_{3}\text{N}_{\varphi}+2\text{c}\sqrt{ \text{N}_{\varphi}}\]2 (11) Among them, \[\text{f}_{\text{s}}=\sigma_{1}-\sigma_{3}\text{N}_{\varphi}+2\text{c}\sqrt{ \text{N}_{\varphi}}\]3 (12)
5. \(\sigma_{\mathrm{x}}\), \(\sigma_{\mathrm{y}}\) and \(\sigma_{\mathrm{xy}}\) are recalculated based on the maximum and minimum principal stresses as well as the principal stress directions.