## 3 Discussion  
### Consistency with Examples from Previous Studies  
As previously mentioned, the nature of rock failure under uniaxial compression is significantly influenced by friction at force contacts. The model of axial compression with maximum friction corresponds well to the behavior of rock columns (pillars), which are an essential component in underground mining. Pillars consist of unmined sections of ore that are retained to support the roof and overburden [30, 31, 32]. The damage patterns observed in these structures (illustrated in Figure 11) [33] are consistent with those documented in laboratory conditions, as seen in Figure 4.  
Figure 10: Dependence of the relative layer thickness (Equation (10)) on the Poisson’s ratio and the ratio of circumferential and axial stresses for three scenarios of proportional lateral pressure (\(\sigma_{cnf}=s\sigma_{z};s=0;\,0.05;\,0.1\)). Values where \(t/R\sim 1\) indicate a completely destroyed specimen, as exemplified in Figure 3.  
The comparison of calculated results obtained using Equations (6), (8), and (10) and depicted in Figure 7 and Appendix A with experimental data from [33] (Figure 11) demonstrates that the relative thickness of the fractured surface layer in actual rock columns lies within the range of 0.44 to 0.70. The predicted average values of this layer also fall within this range. Thus, the developed model accurately represents the process of surface layer damage due to radial cracks while maintaining a satisfactory level of prediction accuracy. The Effect of a Sharp Decrease in the Degree of Failure of the Surface Layer Depending on the Poisson's Ratio and Lateral Pressure  
The impact of lateral pressure on the mechanical state of pillars and their bearing capacity and failure predictions is discussed in [31; 32]. According to [31], there is a threshold value of the confinement that causes a significant reduction in the failure rate. Analyzing this phenomenon has both scientific and practical implications. The sharp (high-gradient) reduction in cracking of the surface layer under a specific lateral pressure is a primary focus of this study within the proposed model framework. Equation (10) elucidates and predicts the observed sharp reduction in surface layer damage as a function of Poisson's ratio and lateral pressure. Specifically, Equations (6) and (9) indicate that the thickness of the cracked layer becomes zero when \(s=n\), i.e., if \(\sigma_{cnf}=\sigma_{R}=n\sigma_{z}=\frac{v}{1-v}\sigma_{z}\). Figure 12 illustrates the variation in relative thickness of the damaged layer in relation to lateral pressure and Poisson's ratio. A confining pressure \(\sigma_{cnf}=\frac{v}{1-v}\sigma_{z}\) can be induced through external (bandage) reinforcement. Practical applications involving external reinforcement using steel and polymer ropes and meshes are discussed in [31]. Reinforcement contributes to an increase in lateral pressure as transverse deformation escalates. This effect is captured in the proposed model, automatically simulating the proportional increase in lateral pressure relative to Poisson's ratio and axial compressive stress through Equations (6) and (9). As previously mentioned, lateral pressure diminishes the thickness of the layer damaged by radial cracks (as shown in Equation (10)). This effect is influenced by both the pressure magnitude and the Poisson's ratio. Analysis using the proposed model and exam  
Figure 12: Changes in relative thickness of the fractured layer as a function of confining pressure and Poisson’s ratio (for \(s\leq n\)). ples in Appendix A predict a rapid decrease in the thickness of the cracked surface layer when the lateral pressure \(\sigma_{cnf}=s\sigma_{z}\) and the proportionality coefficient sss falls within the range \(0.9n\) to \(n\), where \(n=\surd/(1-\
u)\) and \(\
u\) denotes the Poisson's ratio of the rock. This prediction aligns with the aforementioned study [31].