## 4 Conclusions  
This paper presents an analytical model and simulation results for predicting the thickness of the cracked surface layer in cylindrical rock specimens under uniaxial compression with lateral pressure. The simulation results are verified using experimental data from the literature. The verification showed that the developed method allows the thickness of the cracked layer to be predicted. The new modeling approach achieved good agreement between predicted and actual fracture layer thickness. The innovative nature of the approach lies in the fact that the surface layer of the rock at the model level functions as a virtual \"hoop\" or \"bandage\", resembling a shell under internal pressure. To determine the internal pressure, [PERSON]'s hypothesis is adapted, which expresses the horizontal pressure of the rock as a function of the vertical pressure through a nonlinear relationship including Poisson's ratio. There are no analogs of this approach to modeling cracks in the surface layer of cylindrical specimens under axial compression in the literature known to us. In addition, the novelty of the model is in the separation of the functions of the outer and inner surfaces of the mentioned virtual shell. Namely, the inner surface of this shell is subject to internal pressure (according to [PERSON]'s hypothesis [24; 25]), and cracks are initiated on the outer surface (as shown by well-known literature reviews of damage to rock pillars). The distance between the inner and outer surfaces is equal to the predicted thickness of the layer with potential cracks. The thickness of the layer with potential cracks is determined as the solution of the equilibrium equation. The developed model assumes that there are no initial cracks in the mentioned surface layer (natural microcracks in the rock are not taken into account). The condition for the initiation of new cracks in the surface layer is the growth of circumferential stresses to values equal to the tensile strength of the rock. The above-mentioned new aspects of rock behavior analysis led to the creation of a simple but accurate model, which made it possible to obtain new results. Examples of the use of the model are given; tables and graphs reflect the results of calculations for various initial data. The comparison showed that the predicted values and literature values for the thickness of the damaged surface layer in rock pillars are in a similar range, varying from 44% to 70% of the initial cross-sectional radius depending on the axial load, Poisson's ratio, and tensile strength of the rock. The model not only predicts the thickness of the specified cracked surface layer, but also allows us to determine the lateral pressure required to reduce the thickness of this layer and thereby increase the bearing capacity of the rock pillars. However, as the literature analysis showed, the problem of increasing the bearing capacity of rock pillars using ropes, steel mesh, and shotcrete to create lateral pressure requires additional research. Finally, despite the fact that all the data obtained were only for rock formations, applied studies of cracks in the surface layer are no less important for cement concrete and geopolymer concrete, as artificial analogs of rock formations, since cracks affect the durability and safety of buildings and other engineering structures of above-ground and underground construction. Therefore, this topic requires additional research to improve models for the more accurate prediction of crack formation and propagation in concrete. These same models will allow for the prediction of initial damage in the surface layer before it develops into macroscopic damage and eventually leads to the corrosion of reinforcement elements and concrete failure. Thus, the proposed model is a potentially useful tool in applied research on crack formation aspects. Conceptualization, V.S. and G.K.; methodology, V.S. and G.K.; software, V.S. and G.K.; validation, V.S. ; formal analysis, G.K.; investigation, V.S. and G.K.; resources V.S. ; data curation, V.S.