## An analytical solution of direction evolution of crack growth during progressive failure in brittle rocks

## Abstract

Microcrack growth during progressive compressive failure in brittle rocks strongly influences the safety of deep underground engineering. The external shear stress *τ**xy* on brittle rocks greatly affects microcrack growth and progressive failure. However, the theoretical mechanism of the growth direction evolution of the newly generated wing crack during progressive failure has rarely been studied. A novel analytical method is proposed to evaluate the shear stress effect on the progressive compressive failure and microcrack growth direction in brittle rocks. This model consists of the wing crack growth model under the principal compressive stresses, the direction correlation of the general stress, the principal stress and the initial microcrack inclination, and the relationship between the wing crack length and strain. The shear stress effect on the relationship between *y*-direction stress and wing crack growth and the relationship between *y*-direction stress and *y*-direction strain are analyzed. The shear stress effect on the wing crack growth direction during the progressive compressive failure is determined. The initial crack angle effect on the *y*-direction peak stress and the wing crack growth direction during the progressive compressive failure considering shear stress is also discussed. A crucial conclusion is that the direction of wing crack growth has a U-shaped variation with the growth of the wing crack. The rationality of the analytical results is verified by an experiment and from numerical results. The study results provide theoretical support for the evaluation of the safety and stability of surrounding rocks in deep underground engineering.

## Highlights

An analytical solution of the shear stress effect on progressive compressive failure is proposed in brittle rocks.

The growth direction of the wing crack during progressive compressive failure is determined.

The coupled effect of shear stress and initial crack inclination on the rock strength is analyzed.

## 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 (*σ**y*, *σ**x*, *τ**xy*) 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 (*σ**y*, *σ**x*, *τ**xy*) 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 *σ**y*. 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.

*σ*

*y*,

*σ*

*x*,

*τ*

*xy*) and |

*σ*

*y*| >|

*σ*

*x*|, where

*σ*

*y*and

*σ*

*x*are the compressive stresses and

*τ*

*xy*is the shear stresses. The shear stress and normal stress acting on the inclined plane of this model in Figure 2b are

*θ*is the angle between the

*y*-direction of compressive stress

*σ*

*y*and the plane inclination or the angle between the

*x*-direction of compressive stress

*σ*

*x*and the direction of normal stress

*σ*n.

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.

*θ*, the maximum value of normal stress

*σ*n acting on the inclination plane of the brittle rock model in Figure 2c can be calculated, and the principal compressive stress (

*σ*1,

*σ*3) and |

*σ*1 | > |

*σ*3| can be determined as

*θ*when the normal stress

*σ*n becomes the principal compressive stress

*σ*3 on the inclined plane, that is, angle

*θ*between the direction of maximum principal compressive stress

*σ*1 and the

*y*-direction of compressive stress

*σ*

*y*, is solved as

*y*-direction of compressive stress

*σ*

*y*and the initial microcrack inclination in brittle rock is defined by

*φ*i in Figure 2a. The angle between the direction of the maximum principal compressive stress

*σ*1 and the initial microcrack inclination is defined by

*φ*in Figure 2d. The correlation of three angles above is:

*σ*1,

*σ*3), the microcrack growth model of brittle rock is shown in Figure 2d (Ashby & Sammis, 1990). The growth direction of the newly formed wing crack is parallel to the direction of the maximum principal compressive stress

*σ*1. The stress intensity factor of the mode-I crack (i.e.,

*K*I) in this model is (Ashby & Sammis, 1990; Li et al., 2018):

*μ*is the friction coefficient of the initial crack,

*a*is the size of the initial crack,

*l*is the extended length of the wing crack,

*β*is the correction factor,

*α*is the cosine of angle

*φ*,

*N*V is the initial crack number in unit volume, and initial damage

*D*o =

*N*V

*a*3.

*F*w is a wedging force acting on the initial crack plane,

*σ*i 3 is the internal stress acting between wing cracks, and

*τ*c and

*σ*nc are the shear stress and the normal stress on the initial crack plane, respectively. The limitation length for coalescence of two adjacent wing cracks can be solved as

*K*I =

*K*IC, where

*K*IC is the fracture toughness. Therefore, combining Equations ( 3-7), a correlation of the general stress state (

*σ*

*y*,

*σ*

*x*,

*τ*

*xy*) and wing crack length

*l*is obtained as:

*σ*1 can be derived (Li et al., 2018) as

*m*,

*ε*o are constants of the material.

*y*-direction strain

*ε*

*y*and the strain

*ε*1 along the direction of the maximum principal compressive stress

*σ*1 is

*θ*. The

*y*-direction strain

*ε*

*y*under the general stress state in Figure 1a is approximately expressed as

*ε*

*y*=

*ε*1/cos

*θ.*Then, combining Equation ( 5), the

*y*-direction strain

*ε*

*y*is expressed as

*σ*

*y*,

*σ*

*x*,

*τ*

*xy*) and

*y*-direction strain

*εy*is obtained as

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 *K*IC can be measured by a three-point bending test, the initial damage *D*o can be determined by a Scanning Electron Microscope test, and the average parameters of initial crack size *a* and angle *φ*i can be approximately determined by comparing the theoretical and experimental stress–strain curve repeatedly.

According to Equations (13) and (22), the *y*-direction stress *σ**y* varies with the growth of wing crack *l* or the increment of *y*-direction strain *ε**y* under given shear stress *τ**xy* and *x*-direction stress *σ**x* under the progressive compressive failure of brittle rocks. Based on Equation (5), angle *θ* from the direction of maximum principal compressive stress *σ*1 to the *y*-direction of compressive stress *σ**y* 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 (*σ**y*, *σ**x*, *τ**xy*). 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 *φ*i = 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 *σ*1 and the *y*-direction of compressive stress *σ**y* 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 *σ*1 and the *y*-direction of compressive stress *σ**y* 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 *φ*i = 25 and 31°, the *y*-direction peak compressive stress decreases with the algebraic increment of shear stress. For initial crack angle *φ*i = 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 *φ*ic 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

*θ*) in brittle rocks. The major conclusions are as follows:

1.

For negative shear stress, angle

*θ*positively decreases to a minimum value, and then positively increases with wing crack growth. For positive shear stress, angle*θ*negatively decreases to a minimum value, and then negatively increases with the wing crack growth. Angle*θ*shows a positive increase with the negative increment of shear stress and a negative increase with the positive increment of shear stress under a given wing crack length.2.

For initial crack angle

*φ*i = 25° and 31°, the*y*-direction peak compressive stress decreases with the algebraic increment of shear stress. For initial crack angle*φ*i = 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. 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 the initial crack.

## ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (Grant nos. 51708016 and 12172036), the R&D program of Beijing Municipal Education Commission (Grant no. KM202110016014), the Government of Perm Krai, research project (Grant nos. СED-26-08-08-28 and С-26/628), and the Graduate Innovation Program of Beijing University of Civil Engineering and Architecture (Grant no. PG2024035).

## CONFLICT OF INTEREST STATEMENT

The authors declare no conflict of interest.

## Biography

Xiaozhao Li is an associate professor at Beijing University of Civil Engineering and Architecture. He received his PhD degree in geotechnical engineering from Xi'an University of Architecture and Technology in 2016. His research interest is focused on the theory of micro–macro fracture mechanics of brittle rocks. He has published two Chinese and English monographs and more than forty research papers including thirty SCI- and EI-indexed papers as the first author. He has presided over 1 program of the National Natural Science Foundation of China and 1 program of the Postdoctoral Science Foundation of China. He received the First Prize in Natural Science of the Chinese Society of Rock Mechanics and Engineering in 2023 and the Shaanxi Province Excellent Doctoral Dissertation Award in 2018. He is a member of the Engineering Safety and Protection Branch and the Deep Earth Space Exploration and Development Branch of the Chinese Society of Rock Mechanics and Engineering.