# The Stability and Failure of Deep Underground Structures at Potash Mining Deposits
## 2 Materials and Methods
### Complex Limit State Criterion for Rock Masses  
In [15], it is shown that to evaluate the limit state of the rock masses in the vicinity of underground structures, it is efficient to use the complex limit state criterion, which is a specific system of failure criteria. The application of any single criterion in the case of rock masses may be a controversial decision even if the criterion is well-know and well-validated, since various common criteria can give completely opposite assessment of critical zones on the rock mass [15; 20]. The application of this criteria depends on the type of the stress-strain state at a particular point of the rock mass. Mathematically the complex limit state criterion can be written as (3). As follows from (3), the complex limit state criterion includes the most common and effective criteria used in the practice of geotechnical engineering [20]. These common criteria are applied for each particular point of the rock mass depending on the value of the Nadai-Lode parameter [21], which defines the type of the stress-strain state. More details about complex limit state criterion can be found in the work [15]; in this article only the main principle is briefly described. In addition to that, the system of failure criteria used in (3) can be expanded if specific data of mechanical properties of the rock mass are available. In particular, these data should include the information on specific parameters used for other failure criteria used in geomechanics and geotechnics.  
\[\left\{\begin{aligned} &[\sigma_{3}\leq\ \sigma_{c},\ \varepsilon_{3}\leq \varepsilon_{c}]\ \text{when}\ \mu\in(0.5;1]\\ &[\sigma_{1}\leq\ \sigma_{t},\ \varepsilon_{1}\leq\varepsilon_{t}] \ \text{when}\ \mu\in(-0.5;-1]\end{aligned}\right. \tag{3}\]  
Here, and below, \(\sigma_{1}\) and \(\sigma_{3}\) are the maximum and minimum principal stresses; \(\sigma_{c}\) and \(\sigma_{t}\) are the ultimate compressive and tensile strengths; \(\varepsilon_{1}\) and \(\varepsilon_{3}\) are the maximum and minimum principal strains; \(\varepsilon_{c}\), \(\varepsilon_{t}\) and \(\varepsilon_{s}\) are the ultimate compressive, tensile, and shear strains; \(\mathbb{I}_{1},\mathbb{I}_{2}\) are the first and the second invariants of the stress tensor, \(\varphi\) is inner friction angle, \(\mathbb{C}\) is cohesion; \(\alpha=\frac{2\sin\varphi}{\sqrt{3}(3-\sin\varphi)}\); \(s=\frac{6\cos\varphi}{\sqrt{3}(3-\sin\varphi)}\), \(\lambda=\frac{1+\sin\varphi}{1-\sin\varphi}=\frac{\sigma_{c}}{\sigma_{t}}\), and \(\mu\) is the Nadai-Lode coefficient.  
The Nadai-Lode coefficient, \(\mu\), is determined by the following expression [15; 21]:  
\[\mu=\frac{2(\sigma_{2}-\sigma_{3})}{\sigma_{1}-\sigma_{3}}-1, \tag{4}\]  
The values of the Nadai-Lode coefficient always belong to the interval \([-1,\ 1]\). If the value of this coefficient is in the interval \([-1,\ -0.5)\), then the stress state at this point corresponds to the generalized tensile state. The values in the interval \([-0.5,\ 0.5]\) correspond to the generalized shear state, and the values in the interval \((0.5,\ 1]\) correspond to generalized compressive state. Here, brackets mean that the value is not included in the interval and braces mean the value is included in the interval.  
Note that the majority of common geomechanical criteria, like Coulomb-Mohr and Drucker-Prager predict the failure of the rocks due to the shear stress [20]. However, generalized shear stress state is not the only case especially at great mining depths. This will be illustrated in the next section.