Modeling the Conditions of Occurrence and State of Radial Cracks in Rock Specimens Under Axial Compression with Lateral Pressure  
[PERSON]  
1 Institute of Geology, KarRC RAS, Pushkinskaya St., 11, 185610 Petrozavodsk, Russia 11 Institute of Forestry, Mining and Construction Sciences, Petrozavodska State University, Lenin Str. and cyclic loading are similar (Figure 1). This suggests that, under certain conditions, the cracking patterns of rocks in the surface layer under monotonic and cyclic loading will be similar (but not identical). This paper focuses on the behavior of a cylindrical specimen during the pre-peak stage of monotonic deformation. The pre-peak branch of the stress-strain curve (Figure 2) illustrates four distinct stages of brittle material behavior: the closure of initial cracks (stages 0-1, representing 10% to 15% of \(\sigma_{peak}\)); nearly linear deformation (stages 1-2, covering approximately 15% to 70-90% of \(\sigma_{peak}\)); the initiation and stable growth of new cracks (stages 2-4, ranging from \(\sim\)40-60% to \(\sim\)70-90% of \(\sigma_{peak}\)); and unstable crack growth (stages 2-3, from \(\sim\)70-90% to 100% of \(\sigma_{peak}\)) [2]. During the nearly linear deformation stage, an inflection point, denoted as point 4, emerges on the graph, at which the tangential modulus of elasticity (\(E\)) attains its maximum value. If the equation expressing the dependence \(\sigma(\varepsilon)\), is known, then the coordinates of the inflection point can be found from the condition \(\frac{d^{2}\sigma}{d\varepsilon^{2}}=0\); a number of such equations are known, and form the basis of many models of uniaxial compression of rocks [6; 7; 8; 9; 10] and similar materials [11; 12]. An acceptable option is Equation (1) [13], which is a product of the long-term evolution of the Furamura model and is a variant of the Blagojevich model [11]. If the equation describing the dependence of stress (\(\sigma\)) on strain (\(\varepsilon\)) is known, the coordinates of the inflection point can be determined by satisfying the condition \(\frac{d^{2}\sigma}{d\varepsilon^{2}}=0\). Various such equations have been established, forming the basis of numerous models for the uniaxial compression of rocks [8; 9; 10; 11; 12] and analogous materials [13; 14]. A viable approach is represented by the following Equation (1) [15], which is the result of an  
Figure 1: Gneiss granite specimens, testing machine, and load–displacement plot (explanations in text). Figure 2: Dependence of axial strain on stress for a cylindrical specimen of garnet amphibolic. \(D=H=50\) mm; \(\sigma_{peak}=75\) MPa. extended development of the Furamura model and serves as a variant of the Blagojevich model [13]. \[\sigma=\sigma_{peak}\Bigg{(}\frac{\varepsilon}{\varepsilon_{peak}}e^{\frac{1}{ 1}(1-\frac{\sigma^{p}}{p_{peak}^{n}})}\Bigg{)}^{B}. \tag{1}\]  
Since \(\sigma=F/A_{0}\) and \(\varepsilon=u/L_{0}\), where \(L_{0}\) is the initial height of the specimen and \(u\) is the change in its length, Equation (1) can be reformulated in terms of force (\(F\)) and displacement (\(u\)) while retaining the same model parameters \(n\) and \(B\):  
\[F=F_{peak}\Bigg{(}\frac{u}{u_{peak}}e^{\frac{1}{1}(1-\frac{u^{p}}{p_{peak}^{n} })}\Bigg{)}^{B}. \tag{2}\]  
The strain \(\varepsilon_{peak}\) and the stress \(\sigma_{peak}\) at the peak point are determined experimentally. The dimensionless model parameters \(n\) and \(B\) must be calibrated by fitting the theoretical curve (Equation (1)) to experimental data using methods such as least squares, as in [13]. Alternatively, applying a random search method [16] offers a more straightforward and generalized approach. Preliminary analyses and test calculations have demonstrated that the values of parameters \(n\) and \(B\) in Equation (1) vary based on the rock's physical and mechanical properties, but they generally change within narrow intervals for a given rock type. This is likely because Equation (1) does not explicitly incorporate material constants, such as the modulus of elasticity.