# The Stability and Failure of Deep Underground Structures at Potash Mining Deposits
## 3 Results
### FEM Simulation of the Rock Mass State in the Vicinity of Single Mining Excavations
Thus, the main factors that have the determining influence on the formation of the limit state zones in the rock mass at great depths are the effective mechanical properties of the rock strata and the type of stress state. Stability of Block-Fractured Rock Mass in the Vicinity of Single Mining Excavations (DEM Simulation)  
In this section we consider the further simulation of the rock mass behavior in the vicinity of an underground excavation, after the block structure formation. As mentioned previously, this simulation is conducted using DEM. To date, the applications of DEM in problems in geomechanics are quite common [14; 22; 23]. According to the definition of the limit state given in Section 1, the formation of block structure in the rock mass with underground structures is possible primarily in the areas of the limit state, evaluated by the complex limit state criterion (3). As is well-known, the block structure is formed by an induced system of detachment and cracks. Both of these types of cracks may be present in selected areas of the limit state at the rock mass (Figures 2 and 3). Therefore, the introduction of block elements into the calculation algorithm is carried out in the whole limit state zone. To determine the geometry of block structure formation in the rock strata, we propose to use the Lade criterion [20; 24]:  
\[ \begin{array}{c}\left(\frac{\mathrm{I}_{1}^{3}}{\mathrm{I}_{3}}-27\right) \left(\frac{\mathrm{I}_{1}}{\mathrm{P}_{\mathrm{a}}}\right)^{\mathrm{m}}\leq \mathrm{\eta}_{1},\\ \mathrm{I}_{1}=\overline{\mathrm{\sigma}_{1}}+\overline{\mathrm{\sigma}_{2}}+ \overline{\mathrm{\sigma}_{3}},\ \mathrm{I}_{2}=\frac{(\overline{\mathrm{\sigma}_{1}}- \overline{\mathrm{\sigma}_{2}})^{2}-(\overline{\mathrm{\sigma}_{2}}- \overline{\mathrm{\sigma}_{1}})^{2}-(\overline{\mathrm{\sigma}_{3}}-\overline {\mathrm{\sigma}_{2}})^{2}}{\overline{\mathrm{\sigma}_{1}}},\mathrm{I}_{3}= \overline{\mathrm{\sigma}_{1}\mathrm{\sigma}_{2}\mathrm{\sigma}_{3}},\\ \overline{\mathrm{\sigma}_{1}}=\sigma_{1}+\mathrm{\alpha}\mathrm{p}_{\mathrm{a }},\ \ \ \ \ \ \ \ \ \ \ \ \overline{\mathrm{\sigma}_{2}}=\mathrm{\sigma}_{2}+\mathrm{\alpha}\mathrm{p}_{ \mathrm{a}},\ \ \ \ \ \ \ \ \ \ \ \ \overline{\mathrm{\sigma}_{3}}=\mathrm{\sigma}_{3}+\mathrm{\alpha}\mathrm{p}_{ \mathrm{a}}.\end{array}  \tag{7}\]  
where (4) \(\alpha,\mathrm{m},\mathrm{and}\)\(\mathrm{\eta}_{1}\) are the experimentally defined parameters of rocks [24]. This criterion is not so widely used in geomechanics for the determination of the limit state zones; however, it can be very effective in the determination of the block structure in the rock mass. This is especially important when the limit state zones are already defined, but the scheme of the block fracture is not clear. In such areas of the rock mass, cohesion and inner friction become very small in the vicinity of yield surfaces or discontinuity zones where the block structure occurs. As known from [20; 24], the Lade criterion gives the accurate results for the rocks and geomaterials with relatively law inner friction. Therefore, it can be used as proposed above. The results of conducted FEM simulations approve that criterion (7) allows to predict the crack propagation and discontinuity formulation in the rock mass more accurately than well-known crack-predictive criteria such as the Griffith criterion [25]. This fact is illustrated by Figures 5 and 6. Those figures show that according to the Lade criterion (7), the formulation of block structure in the vicinity of rectangular excavation occurs at the entire outline, while in the vicinity of arch-shaped excavation the block structure formulates  
Figure 4: Limit states of the rock mass in the vicinity of mining excavations of different cross-sections at great mining depth (H = 1200 m) according to complex limit state criterion (3). Red color—limit state zones. at the bottom and sides of the excavation outline. According to the classic Griffiths criterion, in turn, the block structure formulates only at the bottom of the arch-shaped excavation.