## 2 Methodology and Results
### Methodology
In the model under consideration, the vertical pressure, represented as \((\sigma_{z}=F/A_{0},\ A_{0}=\pi R^{2})\), determines the lateral pressure using a proportionality coefficient \(s\):  
\[\sigma_{cnf}=s\sigma_{z};\ 0\leq s\leq 1. \tag{7}\]  
The user specifies the proportionality coefficient when analyzing the effect of lateral pressure on the bearing capacity of a column or pillar made of materials such as concrete or rock. Substituting relations (6) and (7) into Equation (5) leads to Equation (8). Figure 5: Stresses in the surface layer. The user specifies the proportionality coefficient \(s\) when analyzing the effect of lateral pressure on the bearing capacity of a column or pillar composed of materials such as concrete or rock. By substituting relations (6) and (7) into Equation (5), we derive Equation (8):  
\[\sigma_{t}=\sigma_{z}\frac{n(R-t)-sR}{t}. \tag{8}\]  
Radial cracks will not form if the circumferential stress remains below a threshold value equal to the tensile strength of the rock, denoted as \(\sigma_{t,\ peak}\). In the limiting case, when \(\sigma_{t}=\sigma_{t,\ peak}\), this condition, combined with Equation (8), provides an estimate for the thickness of the fractured layer, as expressed in Equation (9):  
\[t=R\frac{\sigma_{z}(n-s)}{n\sigma_{z}+\sigma_{t,\ peak}}\ ;\ s\leq n. \tag{9}\]  
Equation (9) demonstrates that the circumferential stress is directly proportional to the radius, indicating that the maximum values of circumferential stresses--and consequently the localization of radial cracks--occur within the surface layer of the specimen, as evidenced by experimental observations (refer to Figure 4). Equation (10) defines a dimensionless characteristic representing the thickness of this layer:  
\[\frac{t}{R}=\frac{1-\frac{s}{n}}{1+\frac{\sigma_{t,\ peak}}{n\sigma_{z}}};\ s\leq n. \tag{10}\]  
In this analysis, the accuracy of the modeling may be influenced by the following limitations. The proposed model focuses on predicting the state of the material near the mid-height of the cylindrical specimen. This is because in this zone, edge effects are minimal, leading to the development of tensile circumferential stresses that increase with higher friction between the loading plates and the specimen ends [26]. Additionally, the hourglass fracture pattern (refer to Figure 4) highlights that the circumferential stresses and the thickness of the surface layer containing radial cracks reach their maximum approximately at the specimen's mid-height. Moreover, since both tensile and compressive stresses are present in the same specimen, a critical question arises: do these stresses reach their peak values simultaneously? Experimental evidence indicates that tensile cracks under uniaxial compression develop prior to the formation of shear cracks [2; 17], as depicted in \[\sigma_{R}(R-t)d\varphi dz-2\sigma_{t}tdzsin\frac{d\varphi}{2}-\sigma_{cnf}Rd \varphi dz=0. \tag{3}\]0 This phenomenon is attributed to the rock's lower tensile strength compared to its compressive strength. The sequence of failure (initial tension followed by shearing) is also indirectly represented in the stress-strain curve (Figure 2). After surpassing the inflection point 4, increased nonlinearity emerges and grows, likely attributed to the formation of radial tensile cracks. Tensile cracks initially form within the weakest particles of the material at a stress range of approximately 40% to 60% of \(\sigma_{peak}\). Subsequently, when stress levels increase to approximately 70-90% or even 100% of \(\sigma_{peak}\), these cracks grow unstably and coalesce. During this  
\[\sigma_{R}(R-t)d\varphi dz-2\sigma_{t}tdzsin\frac{d\varphi}{2}-\sigma_{cnf}Rd \varphi dz=0. \tag{3}\]1 The failure of a gabbro-dolerite specimen from the Khauki-5 deposit, Republic of Karelia, Russian Federation. The shear crack runs through a zone with a discontinuous failure pattern, breaking the specimen in a cone-shaped way. stage, the formation of shear microcracks within weak particle contacts in heterogeneous rock may also occur.