## 2 Methodology and Results  
### Methodology  
This section introduces a simplified model to describe the potential mechanism of crack formation within the surface layer of a cylindrical specimen subjected to uniaxial compression combined with lateral pressure. In the proposed model, the uniform lateral pressure is proportional to the axial stress, represented as \(\sigma_{cnf}=s\sigma_{z}\), \(0\leq s\leq 1\). One form of failure under consideration (as illustrated in Figure 4a) involves the localization of radial and annular cracks within a layer of thickness t (Figure 5), leading to the separation of fragments from the core of the specimen. The presence of radial cracks within this surface layer is indicative of tensile annular stresses \(\sigma_{t}\). Consequently, the surface layer can be conceptualized as a virtual \"hoop\" or bandage, resembling a shell of wall thickness \(t\), subjected to internal pressure \(\sigma_{R}\) and external pressure \(\sigma_{cnf}\) (Figure 5). The axial stress (\(\sigma_{z}\)) is uniformly distributed across the areas of the upper and lower ends of the specimen. The proposed model assumes that the circumferential stresses within the surface layer are evenly distributed across vertical sections with an area of \(tdz\). This assumption minimally affects modeling accuracy under two conditions: (1) when the actual distribution of circumferential stresses is nearly uniform, and (2) when the thickness of the cracked surface layer is relatively small in comparison to the specimen radius. Subsequent comparisons with experimental data will help assess the validity and limitations of these simplifications. Figure 4: Longitudinal tensile cracks on the lateral surface and spalling of fragments during compression of cylindrical specimens: (**a**) sandstone (modified after [18]); (**b**) garnet amphibite. Stages of failure of the same specimen under uniaxial compression. \(D=H=50\) mm. Circumferential tensile stresses in the surface layer led to the formation of fragments 1, 2, and 3. Shear stresses caused the formation of fragments 4 and 5. The thickness of the layer, denoted as \(t\), is initially unknown, and determining its value constitutes the primary objective of this study. In the current model, it is assumed that the material properties of the surface layer are identical to those of the original specimen. While adhesion between the surface layer and the core of the specimen is acknowledged, this analysis focuses solely on radial cracks present within the surface layer. The adhesion between the surface layer and the core of the sample may be insufficient under certain conditions, but the analysis of these conditions is a subject for further research, beyond the scope of this paper. To proceed, we establish the equilibrium condition for a surface layer element by summing the projections of all forces along the \(x\)-axis. This condition ensures mechanical balance and serves as the basis for further analysis and modeling of stress distributions within the cracked surface layer:  
\[\sigma_{R}(R-t)d\varphi dz-2\sigma_{t}tdzsin\frac{d\varphi}{2}-\sigma_{cnf}Rd \varphi dz=0. \tag{3}\]  
This is due to the smallness of the angles \(\sin\frac{d\varphi}{2}\approx\frac{d\varphi}{2}\). \[\sigma_{R}(R-t)-\sigma_{t}t-\sigma_{cnf}R=0; \tag{4}\]  
\[\sigma_{t}=\frac{\sigma_{R}(R-t)-\sigma_{cnf}R}{t}. \tag{5}\]  
The model under consideration incorporates [PERSON]'s ratio, \(\
u\), within relation (6), which establishes a connection between the radial stress, \((\sigma_{R})\), and the vertical pressure, \((\sigma_{z})\), as per [PERSON]'s hypothesis [24; 25]. This relationship is pivotal in describing how lateral deformations influence stress distributions under uniaxial compression with lateral constraints:  
\[\sigma_{R}=n\sigma_{z};\ n=\frac{\
u}{1-\
u}. \tag{6}\]  
If \(0\leq\
u\leq 0.5\), then \(0\leq n\leq 1\).