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Twenty-five uniaxial compression tests were performed to determine stress at onset of dilation, referred to herein as “the crack damage stress,” in heterogeneous dolomites and limestones. A simplified model for crack damage stress (σcd) is developed here using porosity, elastic modulus, Poisson's ratio and three empirical coefficients. The model sh...
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... dry bulk density (r), initial porosity (n), and elastic constants (elastic modulus, E, and Poisson's ratio, n) of the tested rocks are summarized in Table 1. The porosity (n) was calculated from measured values of dry bulk density (r) and specific gravity of solids, and ranged between 5.4% and 29%. ...
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... This can be explained by the strong relationship existing between the rock strength and its structure, which these two predictors surely represent. In fact, V p is notoriously affected by the rock texture (Starzec 1999;Al-Shayea 2004;Martínez-Martínez et al. 2011, and is function of the porosity, which represents a weakening feature from the mechanical point of view (e.g., Vernik et al. 1993;Al-Harthi et al. 1999;Palchik and Hatzor 2002;Pappalardo and Mineo 2022). On the other hand, the lower effect of input parameters on the rock deformation ( Fig. 18a) suggests that this latter also relies on further variables beyond the structure. ...
This study aims to make a unique contribution to the existing body of knowledge about rock strength and deformation parameters and crack stress thresholds through intelligent and statistical approaches applied to a database comprising various rock types (i.e., sedimentary, igneous, and metamorphic rocks). The database contains physical–mechanical and ultrasonic parameters. Six distinct machine learning (ML) algorithms— artificial neural network (ANN), random forest (RF), decision tree (DT), K-nearest neighbor (KNN), support vector regression (SVR), and bagging regressor (BR)— along with the conventional linear regression techniques, were employed to develop predictive models. These models estimate uniaxial compressive strength (σc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{\text{c}}$$\end{document}) and Tangent Young’s modulus (Et\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${E}_{\text{t}}$$\end{document}) based on bulk density (ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho$$\end{document}) and P-wave ultrasonic velocity (Vp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}_{\text{p}}$$\end{document}). Furthermore, they predict crack stress thresholds (i.e., crack closure stress σcc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{\text{cc}}$$\end{document}, crack initiation stress σci\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{\text{ci}}$$\end{document}, and crack damage stress σcd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{\text{cd}}$$\end{document}) as a function of σc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{\text{c}}$$\end{document}, Et\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${E}_{\text{t}}$$\end{document}, ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho$$\end{document}, Vp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}_{\text{p}}$$\end{document}, axial strain at failure (ε1f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varepsilon }_{1{\text{f}}}$$\end{document}), and lateral strain at failure (ε3f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varepsilon }_{3{\text{f}}}$$\end{document}). Various performance indices were utilized to evaluate and compare the performance of these models. The results indicated that the RF method outperformed other ML-based and linear regression-based approaches in predicting the output parameters. Additionally, the multiple parametric sensitivity analysis (MPSA) was carried out to determine the significance of input parameters in predicting the output variables. This analysis revealed that Vp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}_{\text{p}}$$\end{document} and ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho$$\end{document} have the highest and lowest impact on predicting σc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{\text{c}}$$\end{document} and Et\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${E}_{\text{t}}$$\end{document}, respectively. On the other hand, σc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{\text{c}}$$\end{document} was identified as the most influential parameter in predicting σci\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{\text{ci}}$$\end{document} and σcd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{\text{cd}}$$\end{document}, while parameters ε3f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varepsilon }_{3{\text{f}}}$$\end{document} and Vp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}_{\text{p}}$$\end{document} showed the least impact on the foregoing outputs, respectively. This is while ε1f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varepsilon }_{1{\text{f}}}$$\end{document} and ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho$$\end{document} were, respectively, found as the most important and least important factors in predicting σcc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{\text{cc}}$$\end{document}. Finally, to facilitate easy access to the prediction results and enhance the practicality of the proposed RF model, a graphical user interface (GUI) was developed, which enables the practical application of the most performing developed prediction model.
... Thus, the crack damage stress can be detected by identifying the axial stress at which the axial stress-volumetric strain curve reverses; a method which is often employed for unsaturated rocks under uniaxial compression (e.g. Martin, 1997;Palchik and Hatzor, 2002;Xue et al., 2014;Taheri et al., 2020) or true triaxial loading conditions (Gao et al., 2020;Zhang et al., 2023). Especially for rocks in which no reversal of the axial stress-volumetric strain can be observed prior to failure, the stress 30 threshold determination method using axial and lateral crack strains is proposed to determine the crack damage stress (Mo et al., 2024). ...
To detect the crack damage stress also known as onset of dilatancy in fully saturated rocks, we propose a new procedure which combines an innovative measurement technique using pore pressure diffusion with the well known technique of finding the pore pressure maximum. A precise determination of the crack damage stress is required to establish parameter dependencies and ultimately to develop a constitutive equation for the crack damage stress, which is of significant interest e.g. for the long-term safety analysis of repositories for radioactive waste. The new technique monitors the true axial strain as indicator for the crack damage stress during a pore pressure diffusion test. In addition to the crack damage stress, this new true axial strain method simultaneously yields pore pressure diffusion coefficients, thereby maximising the information gain. The true axial strain method was developed based on a multi-cycle, long-term experiment of one sample of Passwang Marl, but it can be applied to other types of rocks, which is demonstrated on a Bunter Sandstone.
... BI CD vs. BI 2 for different rocks, including granite (red), shale (grey), limestone (orange), diorite (dark blue), siltstone (green), sandstone (blue), coal (yellow), marble (black) and quartzites (brown)(Xue et al. 2014;Cai et al. 2004;Ghasemi et al. 2020;Palchik and Hatzor 2002;Zhao et al. 2013;Modiriasari et al. 2017;Gatelier et al. 2002;Palchik 2010;Wu et al. 2021;Taheri et al. 2020; Song et al. 2020a, b;Wang et al. 2016a, b) Fig. 22The comparisons between the predicted tensile strength for shale with different bedding plane inclinations using the empirical relationships and the experimental values(Ma et al. 2018) ...
Shale anisotropy characteristics have great effects on the mechanical behaviour of the rock. Understanding shale anisotropic behaviour is one of the key interests to several geo-engineering fields, including tunnel, nuclear waste disposal and hydraulic fracturing. This research adopted the finite discrete element method (FDEM) to create anisotropic shale models in ABAQUS. The FDEM models were calibrated using the mechanical values obtained from published laboratory tests on Longmaxi shale. The results show that the anisotropic features of shale significantly affect the brittleness and fracturing mechanism at the micro-crack level. The total fracture number in shale under the Uniaxial Compressive Strength (UCS) test is not only related to the brittleness of shale. It is also strongly dependent on the structure of the shale, which is sensitive to shale anisotropy. Two new brittleness indices, BIf and BICD, have been proposed in this paper. The expression for BIf directly incorporates the number of fractures formed inside of the rock, which provides a more accurate frac-ability using this brittleness index. It can be used to calculate the frac-ability of rocks in projects where there are concerns about fractures after excavation. Meanwhile, BICD links brittleness to the CD/UCS ratio in shale for the first time. BICD is easy to obtain in comparison to other brittleness indices because it is based on the Uniaxial Compressive Strength test only. In addition, it has been shown there is a relationship between tensile strength and the crack damage strength in shale. Based on this, an empirical relationship has been proposed to predict the tensile strength based on the Uniaxial Compressive Strength test.
... The material properties of intact blocks are defined by Young's modulus, Poisson's ratio, and density, and are summarized in Table B1. The properties of intact blocks in the sedimentary sequence were determined according to typical values for limestones and dolomites in the region (Palchik and Hatzor, 2002), and the properties of the three basement blocks were adopted from representative values for granites (Jaeger et al., 2009). We note that the simulations simplify the natural lithological and geometrical complexity. ...
... Last but not least, ci is also important for strength criteria and constitutive models. More concretely, it can be used to estimate the parameter mi of the Hoek-Brown (H-B) criterion (Cai 2010), the uniaxial compressive strength (UCS) (Nicksiar and Martin 2013) and parameters in some constitutive models (Cieślik 2007;Palchik and Hatzor 2002;Peng et al. 2015;Wen et al. 2018). ...
The crack initiation stress threshold ( $${\sigma }_{\mathrm{ci}}$$ σ ci ) is an essential parameter in the brittle failure process of rocks. In this paper, a volumetric strain response method (VSRM) is proposed to determine the $${\sigma }_{\mathrm{ci}}$$ σ ci based on two new concepts, i.e., the dilatancy resistance state index ( $${\delta }_{\mathrm{ci}}$$ δ ci ) and the maximum value of the dilatancy resistance state index difference ( $$\left|{\Delta \delta }_{\mathrm{ci}}\right|$$ Δ δ ci ), which represent the state of dilatancy resistance of the rock and the shear sliding resistance capacity of the crack-like pores during the compressive period, respectively. The deviatoric stress corresponding to the maximum $$\left|{\Delta \delta }_{\mathrm{ci}}\right|$$ Δ δ ci is taken as the $${\sigma }_{\mathrm{ci}}$$ σ ci . We then examine the feasibility and validity of the VSRM using the experimental results. The results from the VSRM are also compared with those calculated by other strain-based methods, including the volumetric strain method (VSM), crack volumetric strain method (CVSM), lateral strain method (LSM) and lateral strain response method (LSRM). Compared with the other methods, the VSRM is effective and reduces subjectivity when determining the $${\sigma }_{\mathrm{ci}}$$ σ ci . Finally, with the help of the proposed VSRM, influences from chemical corrosion and confining stress on the $${\sigma }_{\mathrm{ci}}$$ σ ci and $${\Delta \delta }_{\mathrm{ci}}$$ Δ δ ci of the carbonate rock are analyzed. This study provides a subjective and practical method for determining $${\sigma }_{\mathrm{ci}}$$ σ ci . Moreover, it sheds light on the effects of confinement and chemical corrosion on $${\sigma }_{\mathrm{ci}}$$ σ ci .
... The rocks of Israeli origin show pores in limestone and dolomite concentrating stresses. According to Palchick and Hatzor [10], influencing the resistance of the rock, where the Young's Modulus and the resistance stress increase as the porosity decreases and with an increase in the incidence of pores, both the Young's modulus and the resistance decrease, therefore, the compressive strength is influenced by porosity [11,12], being an important parameter and agreeing with the modeling of geotechnical predictions for the texture coefficient of this type of rocks [13], these parameters control the physical and mechanical response of oolithic carbonates [14,15], controlling by the distribution of microporosity within the ooids,whose characteristics establish their mechanical resistance conditions such as grain size, internal structure and composition of the mineral present [16,17]. Other studies carried out the analysis of petrographic and petrophysical data of different limestone lithofacies and their dolomitized equivalents within a carbonate succession with slope of a fault block, relating the distribution of fractures with the textural and mechanical properties of limestone lithofacies [18,19]. ...
There is a fundamental interest in studying travertine rocks, and this is to understand their structure, their geomechanical behavior and other particularities in order to guarantee their proper use in different engineering and architectural applications, and thus, evaluate the sustainability of the travertines, natural resources, the stability of slopes, the preservation of cultural heritage and the mitigation of possible anthropic risks. Travertine has petrological and mechanical properties similar to carbonates from oil fields such as those found in El Presal-Brazil, which currently contain the largest hydrocarbon reserves in the world. Given the impossibility of obtaining rock samples from this deposit to carry out the study, rocks similar to these were used. The present study specifically used samples of Lapis tiburtinus rocks, coming from the west of the city of Tivoli in Italy and these showed resistance to uniaxial and triaxial compression, and showed mechanical resistance due to increased porosity and brittleness. The investigation carried out an analysis of the geomechanical behavior travertine through an experimental program, which includes a petrological, structural, and mechanical characterization. It was determined the travertine is mainly composed of micrite and spastic calcite without the presence of grains or allochemical cements and presents high porosity of the fenetral and vulgar type. Macro and micropores were found to be chaotically distributed in the rock and have low connectivity, which demonstrates the complexity and heterogeneity of the porous structure of Roman travertine. Uniaxial and triaxial compressive strength tests were also carried out, observing a decrease in its mechanical strength due to the increase in porosity, presenting a property of brittleness in its behavior. The results were consistent and valid for this type of rock compared to other studies; determining that there is a correct and adequate operation of the triaxial cell used in the mechanical resistance tests.
... Andersson et al. [8] and Martin and Christiansson [9] proposed that the in situ spalling strength can be estimated using the rock crack initiation stress measured in a laboratory uniaxial compression test. In relevant studies [10][11][12][13], initiation stress and damage stress are two of the mechanical parameters of rocks in rock constitutive equations. ...
The deformation and failure process of rocks is a gradual process. The purpose of this study was to examine the characteristics of rocks in different stages through a cyclic loading experiment. The experiment was carried out based on the MTS815 rock mechanics test system combined with acoustic emission monitoring equipment to study the typical characteristics of two kinds of granite in the stages of crack closure, linear elastic deformation, crack initiation and stable crack growth, along with crack damage and unstable crack growth. The results showed that there were significant differences in the characteristics of the strain response, energy evolution, and acoustic emission of the two granites in the different stages. Although the microstructure and mineral elements of the two granites are different, the characteristics of the two granites in the same stage were similar, indicating that the stage characteristics of brittle rocks in the failure process may be widespread and have significant similarities.
... These values at dry and saturated conditions can be measured in the laboratory. (Rybacki et al., 2015), tuff (Avar et al., 2003), dolomite (Palchik and Hatzor, 2002), and sandstone (Palchik, 1999); (b) Different kinds of modulus for clean sandstone (Han et al., 1986;Yu et al., 2016); E, G, K, and λ are Young's modulus, shear modulus, bulk modulus, and Lame parameter, respectively. Symbols in the figures are measurements, and lines are fittings. ...
... Many instances demonstrate how arbitrary the method used to calculate the Young's modulus is. The research focuses on the effects of fissures or voids [17], porosity, mineral assemblage, water content, and permeability [10,[25][26][27][28], as well as the correlations between the Young's modulus and other physical parameters [25][26][27][28][29]. However, they pay li le a ention to determining the Young's modulus or any other stiffness characteristics throughout the loading process from the beginning to the failure phase. ...
... Many instances demonstrate how arbitrary the method used to calculate the Young's modulus is. The research focuses on the effects of fissures or voids [17], porosity, mineral assemblage, water content, and permeability [10,[25][26][27][28], as well as the correlations between the Young's modulus and other physical parameters [25][26][27][28][29]. However, they pay li le a ention to determining the Young's modulus or any other stiffness characteristics throughout the loading process from the beginning to the failure phase. ...
Any rock mechanics' design inherently involves determining the deformation characteristics of the rock material. The purpose of this study is to offer equations for calculating the values of bulk modulus (K), elasticity modulus (E), and rigidity modulus (G) throughout the loading of the sample until failure. Also, the Poisson's ratio, which is characterized from the stress-strain curve, has a significant effect on the rigidity and bulk moduli. The results of a uniaxial compressive (UCS) test on granitic rocks from the Morágy (Hungary) radioactive waste reservoir site were gathered and examined for this purpose. The fluctuation of E, G, and K has been the subject of new linear and nonlinear connections. The proposed equations are parabolic in all of the scenarios for the Young's modulus and shear modulus, the study indicates. Furthermore, the suggested equations for the bulk modulus in the secant, average, and tangent instances are also nonlinear. Moreover, we achieved correlations with a high determination factor for E, G, and K in three different scenarios: secant, tangent, and average. It is particularly intriguing to observe that the elastic stiffness parameters exhibit strong correlation in the results.
... The greater is the number of new cracks generated in the rock, the greater is the possibility of their coalescence with each other, resulting a the decrease in r cd /r p . Compared with the cementation between mineral particles, the effect of porosity on the strength of sandstone is Acta Geotechnica more significant [50]. The porosity of red sandstone used in this study is 5.38%, which is significantly higher than 2.27% for basalt and 1.04% for granite. ...
The crack closure stress σcc, crack initiation stress σci and crack damage stress σcd are important mechanical parameters to describe the brittle failure mechanism of hard rock. To investigate the strain rate effect on the crack stress threshold and to reveal the mechanical mechanism of hard rock showing different strength and deformation characteristics at various strain rates, uniaxial compression tests at strain rates of 10–5/s, 5 × 10–5/s, 10–4/s, 5 × 10–4/s and 10–3/s were performed on granite, sandstone and basalt. The research results show that the influence of the strain rate on the damage stress level of the three types of rock is significantly different. With the increase in the strain rate, the normalized crack damage stress (σcd/σp) and the normalized crack initiation stress (σci/σp) of granite show a step-by-step downwards trend. The σcd/σp of sandstone shows a step-by-step downwards trend while σci/σp and σcc/σp steadily decrease with increasing strain rate. The normalized stress thresholds (σcd/σp and σci/σp) of basalt remain constant with increasing strain rate. Based on the characteristics of the rock mineral structure, the internal mechanism of the variation law of the stress threshold was revealed. The effect of the strain rate on the strain characteristics of granite and sandstone is mainly reflected in the unstable crack propagation stage. The larger is the strain rate, the more obvious is the volumetric dilatation. Finally, the guiding significance of the research results for engineering excavation was discussed.
