This article deals with the problem of predicting the brittle fracture of rocks and similar materials, which can also include frozen sandy soils. Such materials, due to the diversity of their conditions of origin, are characterized by natural heterogeneity at the micro-, meso-, and macro-levels, which makes it difficult to develop sufficiently universal criteria for their strength. Despite a number of known models and criteria of strength and fracture, the search for such criteria remains an urgent problem. In this paper, using the energy approach to the mathematical modeling of mechanical systems, the fracture criterion is justified, which differs from the known criteria that do not require integration to calculate the strain energy $W_e$ and dissipation energy $W_d$. The well-known relation for the input energy $W=W_e+W_d$ is used. The object of the study was the ratio of $dW=d{W}_e+d{W}_d$. The main research question concerned what the ratio of $d{W}_e$ and $d{W}_d$ would be at the point of brittle failure. The search for an answer to the question led to the justification of a differential energy criterion for the failure of brittle materials on the descending branch of the full stress–strain curve. It was found that the point of predicted fracture is determined by the equality $\sigma =0.5 \epsilon E_{tangential}$ (if there is an inflection point on the ascending branch) or $\sigma =0.5 \epsilon E_{secant}$_secant. The main result of the work was ascertaining the differential strength and fracture criteria of brittle materials in the form of inequalities and equations, which were oriented for application in engineering calculations. Examples of application of the developed criteria are given; their consistency with the experimental data known from the literature confirmed.

This paper discusses modeling the behavior and prediction of fracture of brittle materials. Numerous publications show that progress in this area is characterized by the emergence of a number of new models that meet the requirements of the mining industry, construction and other engineering practices. The authors focus only on one class of models, paying special attention to the compromise between simplicity of solution and versatility of the model. A new version of the model is proposed, taking into account the advantages of previous models. We present a differential energy criterion for brittle fracture substantiated, according to which, fracture occurs at a certain ratio of dissipated and stored (elastic) energy. Fracture is considered as the end of the deformation process with a virtual transformation of the initial material almost without cracks into a real material with cracks. The highest and lowest elastic moduli are analytically determined, respectively, on the ascending and descending branches of the stress–strain curve. A graphical version of the algorithm for determining the brittle fracture point on the post-peak branch of the stress–strain curve is proposed. The modeling results are consistent with the experimental data known from the literature.

This article considers the effect of constant and variable Poisson’s ratio on cracking in concrete and rock specimens under uniaxial compression using mechanical systems modeling methods. The article presents an analysis of the data confirming the increase in Poisson’s ratio under specimen loading. A system of equations for modeling the effect of Poisson’s ratio on cracking under uniaxial compression is proposed. The comparison showed that the model with a constant Poisson’s ratio predicts a thickness of the surface layer with cracks that is underestimated by approximately 10%. In practice, this means that the model with a constant Poisson’s ratio underestimates the risk of failure. A technique for analyzing random deviations of Poisson’s ratio from the variable mathematical expectation is proposed. The comparison showed that the model with a variable Poisson’s ratio leads to results that are more cautious, i.e., it does not potentially overestimate the safety factor. The model predicts an increase in uniaxial compression strength when using external reinforcement. An equation is proposed for determining the required wall thickness of a conditional reinforcement shell depending on the axial compressive stress. The study contributes to understanding the potential vulnerability of load-bearing structures and makes a certain contribution to increasing their reliability.

This study investigates the application of the Pure Random Orthogonal Search (PROS) method, introduced in the literature in 2021, for approximating force and displacement measurement data obtained from rock specimen testing, using granite as a case study. The primary objective is to simplify the data approximation procedure and improve the accuracy of experimental data analysis by reducing the influence of subjective factors within a predefined protocol. The research focuses on determining the maximum value of the tangent modulus of elasticity during the pre-peak deformation stage of granite specimens under uniaxial compression. The study employs methods of mathematical modeling of rock mechanical behavior and experimental data analysis. To approximate the experimental data, a modified two-parameter S-curve equation is proposed. The optimal parameter values are determined using the PROS method, which reduces the problem to solving a two-dimensional objective function minimization task. The dimensionality of this optimization problem remains independent of the number of experimental data points, thereby enhancing computational efficiency. A systematic computational procedure is developed for the automated calculation of the approximating equation’s parameters and the determination of the maximum tangent modulus of elasticity. In the context of challenges associated with accurately measuring displacements using conventional testing machines, a numerical correction procedure is proposed and implemented to account for the compliance of the loading system. The results of the study are consistent with both the literature-reported experimental data and the data obtained in this work. The methodology and findings can be adapted for analyzing the properties of concrete as an artificial analog of natural rock materials.

This paper presents a model for predicting the thickness of the cracked surface layer in rock specimens subjected to axial compression with lateral pressure. The study focuses on radial cracks that are most prominently open on the lateral surface, manifesting as axial (or longitudinal) cracks on this surface. The research employs methods for analyzing mechanical and geotechnical systems. The proposed model determines the conditions under which damage to the surface layer decreases depending on Poisson’s ratio and lateral pressure. Radial cracks typically precede spalling, making it crucial to examine the conditions of their initiation and the potential for influencing their development, and this holds significant importance for mining and underground construction practices. A comparison indicated that the predicted values and those reported in the literature for the thickness of the damaged surface layer in rock pillars are within a similar range, varying from 44% to 70% of the initial cross-sectional radius. Although this study focuses on rock materials, the proposed model also offers potential applications in examining crack formation in cement and geopolymer concretes, serving as artificial analogs of rock, thereby helping to mitigate the risk of structural failure in building systems.