1 INTRODUCTION

The mechanical behavior of progressive failure induced by external compressive loadings of brittle rocks has significance for the design and safety evaluation of deep underground engineering (e.g., energy exploitation and storage, deep repository for waste nuclear fuel, tunneling engineering, deep laboratory, etc.) (Cai et al., 2023; Kang et al., 2023; Shen et al., 2024). Many microcracks are present in brittle rocks. The initiation, growth, nucleation, and coalescence of microcracks have a great influence on the mechanical behaviors of brittle rocks. Thus, study of the effects of microcrack characteristics on the compressive mechanical properties during progressive failure is significant for deep engineering applications.

Many studies have focused on the microcrack growth characteristics using the technologies of acoustic emissions (Browning et al., 2017; Zhai et al., 2020), scanning electron microscope (Tao et al., 2020), and computed tomography (Ma et al., 2016; Wang, Li, et al., 2019). Furthermore, to qualitatively analyze the initial crack geometry effects, the characteristics of crack initiation, growth, interaction, and coalescence were studied and stress–strain curves were constructed by prefabricating minor cracks in brittle solids (Figure 1) (Lee & Jeon, 2011; Liu et al., 2018; Morgan et al., 2013; Wang, Dyskin, et al., 2019; Wong et al., 2004; Zhou et al., 2019). However, in previous studies, the initial crack friction was not considered in brittle solids in which minor cracks were prefabricated. The initial crack friction was further studied by prefabricating minor cracks with initial crack friction (Park & Bobet, 2010), and initial crack friction strengthens the mechanical behaviors in brittle solids.

The new wing crack growth is formed when the external loadings trigger the sliding of the initial microcrack, and the growth direction of the new wing crack is parallel to the direction of the principal compressive stress in Figure 1. Based on this phenomenon, the expression of the stress intensity factor at the tips of wing cracks was proposed (Ashby & Hallam, 1986; Ashby & Sammis, 1990), the correlation of applied stress and wing crack growth was studied (Brantut et al., 2012), and deformation induced by the wing crack growth was analyzed (Li et al., 2018, 2019) subjected to the remote principal compressive stresses of brittle rocks. Furthermore, the microcrack growth models were presented in brittle rocks under remote general compressive stresses and shear stresses (Golshani et al., 2006; Yuan et al., 2013), the wing crack growth direction was still assumed to be parallel to the maximum compressive stress direction (Yuan et al., 2013), and the macroscopic stress–strain relationship was analyzed.

However, shear stress often exists around the rock in practical engineering. With the internal crack growth and external stress state variation of rocks, the direction of maximum principal compressive stress changes, which causes the newly generated wing crack growth direction to change. Under the general stress state (σy, σx, τxy) with compressive and shear stresses, the shear stress effects on the progressive compressive failure considering the evolution of the wing crack growth direction are rarely studied. This study will propose a micro–macro method to study this problem.

2 DESCRIPTION OF THE ANALYTICAL METHOD

An analytical method is proposed to predict the variation law of the growth direction of the tip of the wing crack under general stress states (σy, σx, τxy) during progressive failure in brittle rocks with initial microcracks in Figure 2a. With the variation of externally applied stress, the initiation, growth, nucleation, and coalescence of internal microcracks cause the progressive failure of brittle rocks. This study mainly focuses on the progressive failure of brittle rocks subjected to the changing compressive stress of y-direction σy. Furthermore, the proposed model in Figure 2 is a homogenization model; the mechanical properties caused by characteristics of randomly distributed microcracks cannot be analyzed and are described by the conventional method of microcrack geometry parameters.

Note: The positive direction of angle θ rotates anticlockwise from the x-direction to the direction of normal stress. The tensile stress is positive and the shear stress rotating clockwise in the rock element is positive.

The stress–strain relationship in Equation (22) describes the wing crack growth-induced macroscopic deformation. The stress–strain relationship before wing crack growth is assumed to be linear in this study. Furthermore, for the parameters in the above-mentioned model, the friction coefficient μ can be measured by a friction test, fracture toughness KIC can be measured by a three-point bending test, the initial damage Do can be determined by a Scanning Electron Microscope test, and the average parameters of initial crack size a and angle φi can be approximately determined by comparing the theoretical and experimental stress–strain curve repeatedly.

According to Equations (13) and (22), the y-direction stress σy varies with the growth of wing crack l or the increment of y-direction strain εy under given shear stress τxy and x-direction stress σx under the progressive compressive failure of brittle rocks. Based on Equation (5), angle θ from the direction of maximum principal compressive stress σ1 to the y-direction of compressive stress σy also varies with the growth of the wing crack or the increment of y-direction strain. Thus, based on the experimental phenomenon of the growth direction of the wing crack being parallel to the direction of the maximum principal compressive stress (Ashby & Sammis, 1990) in Figure 2d, it is predicted that the growth direction of the wing crack should also be changing with the growth of the wing crack or the increment of y-direction strain under the general stress states (σy, σx, τxy). The specified variation law of the growth direction of the tip of the wing crack under the general stress states during progressive failure in brittle rocks will be analyzed below.

3 RESULTS AND DISCUSSION

3.1 Shear stress effect

Figure 3a shows the shear stress effect on the relationship between y-direction compressive stress and strain. The y-direction compressive stress increases to a peak value, and then decreases to failure with the increment of the y-direction strain. Figure 3b shows the shear stress effect on the relationship between y-direction compressive stress and wing crack growth, which corresponds to the macroscopic phenomenon of Figure 3a. Furthermore, the peak compressive stress of the y-direction increases with an increase in shear stress under the given initial crack angle φi = 45°. The initial crack angle strongly influences the variation of the y-direction peak compressive stress relating to shear stress. The initial crack angle effect will be discussed in detail below.

Due to the limitations of the test apparatus, the stress–strain curve with the shear stress effect in Figure 3a has hardly been experimentally considered to date. However, the specified relationship between the principal compressive stress and strain without shear stress calculated by Equation (22) can be determined by a compression experiment. The comparison of the theoretical and experimental (Zhao et al., 2014) relationship between y-direction compressive stress and y-direction strain of granite is shown in Figure 4. The variation tendency of the theoretical and experimental curve between y-direction compressive stress and y-direction strain is identical. There are some differences in the numerical value between theory and experiment. The reason for this difference may be that (1) the theoretical model is an average method, and nonuniformity of rock cannot be achieved, (2) the geometric parameters (e.g., size and angle) of all microcracks are assumed to be an average value, and (3) the model parameters are determined independently by different tests, and errors of model parameters arise.

In Equation (5), angle θ between the direction of maximum principal compressive stress σ1 and the y-direction of compressive stress σy is close to the y-direction stress, x-direction stress, and shear stress. Figure 3 shows that the y-direction stress changes with the y-direction strain increment and wing crack growth. Thus, angle θ between the direction of maximum principal compressive stress σ1 and the y-direction of compressive stress σy also changes with the wing crack growth. Figure 5a shows the variation of angle θ with the wing crack growth under different shear stresses. For negative shear stress, angle θ is positively decreased to a minimum value, and then is positively increased with the wing crack growth. For positive shear stress, angle θ is negatively decreased to a minimum value, and then is negatively increased with the wing crack growth. Figure 5b shows the variation of angle θ with shear stress under a given wing crack length. Angle θ is positively increased with the negative increment of shear stress, and is negatively increased with the positive increment of shear stress under a given wing crack length. The changing angle θ reflects directly the evolution of direction wing crack growth in Figure 6.

In general, the shear stress strongly influences the evolution of direction wing crack growth and the mechanical properties of the progressive compressive failure in brittle rocks with initial microcracks.

3.2 Initial crack angle effect

Figure 7a shows the initial crack angle effect on the relationship between y-direction peak compressive stress and shear stress. For initial crack angle φi = 25 and 31°, the y-direction peak compressive stress decreases with the algebraic increment of shear stress. For initial crack angle φi = 45 and 51°, the y-direction peak compressive stress increases with the algebraic increment of shear stress. There is a critical value of the initial crack angle that makes the opposite variation tendency of the y-direction peak stress with shear stress in Figure 7a. This critical value φic of the initial crack angle (i.e., 34° or so) can be determined by the relation between y-direction peak compressive stress and the initial crack angle under different shear stresses in Figure 7b. Furthermore, this critical initial crack angle is close to the friction coefficient between the initial crack surface, and this critical angle decreases with the increment of the friction coefficient of initial crack in Figure 7c. This critical initial crack angle shows that the applied y-direction peak compressive stress strongly depends on the coupling effect of the initial crack angle, shear stress, and the initial crack friction coefficient. In Figure 7, the y-direction peak compressive stress decreases first and then increases with increment of the initial crack angle, which is identical to the experimental and numerical results (Vergara et al., 2016; Zhang et al., 2024) of the pre-existing minor flaw inclination effect in Figure 8.

Figure 9 shows the initial crack angle effect on the relationship between angle θ and wing crack growth under negative and positive shear stresses. Based on Figure 9, the initial crack angle effect on the relationship between angle θ and shear stress is shown in Figure 10a. Angle θ increases positively with the negative increment of shear stress, and increases negatively with the positive increment of shear stress under a given initial crack angle. Figure 10b shows the effect of the initial crack angle on angle θ under different wing crack lengths. Angle θ increases positively first and then decreases with the increment of the initial crack angle at l = 0.2–1.8 mm. Angle θ always positively decreased with the increment of the initial crack angle at l = 2.2 mm.

4 CONCLUSIONS

CONFLICT OF INTEREST STATEMENT

The authors declare no conflict of interest.