# The Stability and Failure of Deep Underground Structures at Potash Mining Deposits
## 2 Materials and Methods
### Problem Statement, Material Properties, and Mathematical Models of Rock Behavior  
Consider a stability problem for single mining excavation located in salt rock mass at a moderate depth, H = 600 m, and great depth, H = 1200 m. Typical cross-sections for underground tunnels and mining excavations at potash mining deposits are shown below in Figure 1. The dimensions of the cross-sections were chosen according to the real characteristics of mining excavations at potash deposits of the Republic of Belarus. The problems below are solved in the 3D elasto-plastic statement; however, the results demonstrate the solution in transverse cross-sections of the model for convenience. In addition to that, for 2D problems in rock mechanics coefficient, (4) is not always applicable; therefore, the 3D problem statement is a crucial peculiarity of the presented case studies.  
Mechanical properties of salt rock mass are presented in Tables 1 and 2 below. These properties are obtained as a result of standard lab tests of rock samples at various mining depths. In this article we purposely do not describe the exact methodology of lab tests because it is well known, and the main aim of this article is to demonstrate the efficiency proposed coupled numerical algorithm but not to solve a particular problem for the particular cross-section and material properties. The proposed algorithm can be used to solve similar problems for any other cross-section, material properties, and even other geological structure of the rock mass.  
In this article the main research method is numerical simulation. FEM and DEM are used to predict stability and failure of considered geotechnical systems. Well-known mathematical models of mechanical behavior of salt rock mass used for the FEM simulation includes Hooke's law for isotropic media (5) and the Coulomb-Mohr model (6) [15; 20]  
\[\sigma_{ij}\leq\frac{E}{1+\
u}\epsilon_{ij}+\frac{\
u E}{(1+\
u)(1-2\
u)} \epsilon_{ij}\delta_{ij} \tag{5}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Geological Layer** & **Density, kg/m** & **Youngs’s** & **Poisson’s Ratio** & **Inner Friction** & **Cohesion, MPa** \\  & & **Modulus, MPa** & **Poisson’s Ratio** & **Angle, deg.** & **Cohesion, MPa** \\ \hline Rock salt at 600 m & 2300 & 1750 & 0.28 & 69 & 2.74 \\ \hline Rock salt at 1200 m & 2400 & 2280 & 0.35 & 63 & 4.2 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Effective mechanical properties of potassium rock salt strata.  
Figure 1: Typical cross-sections of mining excavations for potash mining deposits. (**a**) Filleded square cross-section, (**b**) Arched cross-section.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Geological Layer** & **Ultimate Tensile Strength, MPa** & **Ultimate Compressive Strength, MPa** & **Ultimate Compressive Strain, \%** & **Ultimate Tensile Strain, \%** & **Ultimate Tensile Strength, MPa** \\ \hline Rock salt at 600 m & \(1\) & 30 & \(1.8\) & 0.22 & \(1\) \\ \hline Rock salt at 1200 m & 1.2 & 32 & 2 & 0.25 & 1.2 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Effective strength limits of potassium rock salt strata.  
\[\tau_{n}\leq\sigma_{n}\tan\varphi+C, \tag{6}\]  
where \(\sigma_{ij}\)  and \(\varepsilon_{ij}\) are stress and strain tensors, respectively, \(\sigma_{n}\) and \(\tau_{n}\) are normal and shear stress, respectively, \(\delta_{ij}\) is the Kronecker delta, \(E\) is [PERSON]'s modulus, and \(\
u\) is [PERSON]'s ratio.  
Thus, evaluation of the limit state zones is performed using the complex limit state criterion [15] based in the results of FEM simulation. After the limit state zones are established, the DEM simulation is conducted in this zone. DEM simulation is described in more detail in Section 3.2.