## 4 Results and Discussion  
### Stress-Strain Evolution  
In this section, we visualize the stress-strain distribution within the computational domain. Looking ahead, we note that this subsection presents the results for a 40 m pillar, which has been chosen as a compromise between the coal recovery ratio and the safety factor. The distributions are further used in the numerical analysis of the stability of the rocks adjacent to the openings. The stability evaluation is closely related to the stress distribution analysis in the vicinity of the protective pillar. Notably, the stress change \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Rock** & \(\rho\)**,\(g\)/\(cm^{3}\)** & \(K\)**,\(GP\)a** & \(\mu\)**,\(GP\)a** & \(\sigma\)**\(U\)**ics\_MPa** & \(\sigma\)**\(U\)**ITS\_MPa** & \(\chi_{0}\)**MPa** & \(\alpha_{1}\) & \(\alpha_{2}\) & \(\Lambda_{0}\) \\ \hline Sandstone & 2.54 & 12.22 & 10.43 & 22.1 & 2.82 & 3.45 & 1.34 & 0.84 & 0.08 \\ \hline Siltstone & 2.55 & 13.18 & 9.17 & 13.92 & 1.11 & 3.03 & 1.32 & 0.81 & 0.1 \\ \hline Carb.mudst. & 1.93 & 11.76 & 7.56 & 11.73 & 1.5 & 2.1 & 1.34 & 0.9 & 0.549 \\ \hline Coal & 1.29 & 3.29 & 2.275 & 2.616 & 1.1 & 0.9 & 0.7 & 0.7 & 0.11 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical-mechanical properties of rocks used in modeling. Figure 3: Dynamics of seismic activity due to panel 1 stope advancement in study site over \(\approx\) 70 days. Arrow points out the largest rock burst. is even more important than a single static value. For this reason, we first record the values of normal stresses along the vertical line drawn through the model center (line 1-1 in Figure 2a) and along the line associated with the dip direction of seam 3 (line 2-2 in Figure 2a). Figure 4a-c illustrate the stress distributions. It can be concluded that even in the initial state (prior to the mining operations) there is stress amplification associated with the roadways. The latter yields a deviation from the monotonically increasing vertical stress tensor component \(\sigma_{yy}\) in Figure 4a and the characteristic stress pattern in the coal seam horizon (Figure 4b,c). Figure 4c illustrates the vertical stress distribution in the seam plane. Since we do not account for tectonic stresses, two other horizontal components of the stress tensor, namely, \(\sigma_{zz}\) and \(\sigma_{xx}\), are related to the vertical stress through Poisson's ratio. Below, we compare further stress states along these lines with the initial states. Let us trace the stages of the inelastic strain distribution in the course of the gob increase in panel 1. This step of the numerical modeling is necessary to validate the model. To do so, we compare the steps of the roof caving with the field data. If the misfit error between the model estimation and the field data is smaller than 20%, we accept the model and use the results of the model in the further analysis. Figure 5 illustrates the stages of the development of inelastic deformation in the roof of the coal seam after the first collapse. The corresponding gob length is given in the figure caption. To clearly represent the inelastic strain distribution, we rotated the model by 180\({}^{\circ}\) in contrast to the view given in Figure 4. The mining direction is illustrated by a black arrow in Figure 5a. Judging by the distribution of the inelastic strain in Figure 5a, strong non-homogeneity and asymmetry can be observed. The corresponding gob length  
Figure 4: Distribution of normal stresses along: the vertical line drawn through the center of the model (**a**), the line drawn through the center along the dip direction of seam 3 (**b**), and the initial vertical stress distribution in seam plane (**c**). is \(\approx\)130 m. Further stages of inelastic deformation and damage development in the roof are illustrated in Figure 5b-d. The corresponding gob lengths are \(\approx\)180 m and 200 m. Figure 5d illustrates the final distribution of the inelastic strain in the roof, corresponding to the surface subsidence state.