Experiment is an important research method in the field of geotechnical engineering (Kang et al., 2022; Lin S. Q. et al., 2022; Wang et al., 2022; Zhao et al., 2023). In this study, the common granite was selected as the research object. The granite samples obtained after processing have no obvious joints and cracks, and have a very good compositional homogeneity. Its shape is a thin cuboid with length, width and thickness equal to 160, 80, and 10 mm respectively. In addition, there is a circular through hole with a diameter of 5 mm in the center of the sample, an oblique straight pre-crack passes through the hole, and their centers coincide with each other. The length and width of the pre-crack are 10 and 1 mm respectively. The pre-crack inclination angles α are 0°, 15°, 30°, 45°, 60°, 75°, and 90° respectively. The left side of Figure 1 shows a typical sample, its pre-crack inclination angle α is equal to 45°.

In order to study the influence of crack stop holes on the failure behavior of granite samples, for some samples with pre-crack inclination angle equal to 45°, a crack stop hole was drilled near each end of the pre-crack for comparative analysis. The function of the crack stop hole is to prevent further propagation of cracks in areas where stress is easily concentrated (Deng et al., 2022). Nevertheless, the role of crack stop hole in this study is to change the crack growth path, therefore, it is arranged near the route of wing crack propagation, as shown in Figure 1. The vertical distance d ₁ from the center of the crack stop hole to the tip of the pre-crack is 10 mm, while the corresponding horizontal distance d ₂ from their center to the corresponding pre-crack tip is 0, 5, and 10 mm respectively, and the radius of the crack stop hole is 2 mm. Figure 1 shows two typical crack stop holes with a horizontal distance of 5 mm and a vertical distance of 10 mm.

Uniaxial compression test was carried out by MTS816 rock mechanics testing system, as shown in Figure 2. During the test, the axial stress was applied to samples by displacement control, and the loading rate was 0.002 mm/s. The system can automatically acquire parameters such as time, axial load and displacement, etc., and the collection interval was set to 0.1 s.

Energy evolution plays an important role in the analysis of mechanical characteristics, there are mainly four types of energy changes in the process of rock failure under external load, namely, energy input, energy accumulation, energy dissipation and energy release. According to the laws of thermodynamics, based on the assumption that there is no heat exchange with the external environment during loading, and ignoring the energy released through thermal radiation, heat exchange and other forms, the total energy input from the outside is completely converted into elastic strain energy and dissipated energy inside the sample (Peng et al., 2020). Therefore, the energy of the pre-cracked granite sample under uniaxial compression is calculated as follows (Liu et al., 2020; Qiao et al., 2022). U = U e + U d (1) where U is the total energy input from the outside, MJ·mm⁻³; U _e is the elastic strain energy, MJ·mm⁻³; U _d is the dissipated energy, MJ·mm⁻³.

In the principal stress space, the total energy and elastic strain energy absorbed by sample element can be calculated by the following formula (Liu et al., 2020): U = ∫ 0 ε 1 σ 1 d ε 1 + ∫ 0 ε 2 σ 2 d ε 2 + ∫ 0 ε 3 σ 3 d ε 3 (2) U e = 1 2 σ 1 ε 1 e + 1 2 σ 2 ε 2 e + 1 2 σ 3 ε 3 e (3) ε i e = 1 E i σ i − μ i σ j + σ k (4) where σ i , σ j , σ k ( i , j , k =1, 2, 3) are the principal stresses in each direction; ε i , ε i e are the total strain and elastic strain in each principal stress direction; μ i , E i are Poisson’s ratio and elastic modulus during the unloading stage, which can be replaced by μ 0 , E 0 at elastic stage (Xie et al., 2008).

In this study, it should be noted here that the elastic modulus of pre-cracked granite samples should be called apparent stiffness. Under uniaxial compression, the circumferential stress is 0 MPa, and only the axial stress does work. Therefore, according to Eqs 2–4, the energy can be calculated as follows (Liu et al., 2020; Qiao et al., 2022): U = ∫ 0 ε i σ 1 d ε 1 = ∑ i = 1 n 1 2 σ 1 i + σ 1 i − 1 ε 1 i + ε 1 i − 1 (5) U e = 1 2 σ 1 ε 1 e = U e = 1 2 σ 1 σ 1 E 0 = σ 1 2 2 E 0 (6) where σ 1 i , ε 1 i are the stress and strain values corresponding to each point on the stress-strain curve.

According to Eqs 1, 5, 6, the dissipated energy of the sample is the difference between the total energy and the elastic strain energy (Peng et al., 2020). U d = U − U e = ∑ i = 1 n 1 2 σ 1 i + σ 1 i − 1 ε 1 i + ε 1 i − 1 − σ 1 2 2 E 0 (7)

It can be seen from Figure 3A that the trend of stress-strain curves of samples with different pre-crack inclination angles is basically consistent with that of samples without cracks, and they all go through four stages, namely, micro-crack compaction stage, elastic deformation stage, unstable fracture development stage and post failure stage. In addition, the peak strength of the pre-cracked samples is always lower than that of the sample without pre-cracks, indicating that pre-crack reduces the peak strength of the granite sample, and make it more vulnerable to damage.

As shown in Figure 3B, for granite samples with pre-crack inclination angle of 45°, crack stop hole has a certain influence on the stress-strain curve. When the horizontal distance d ₂ between the center of the crack stop hole and the tip of the pre-crack is equal to 0 mm, the peak strength of the sample is slightly lower than that of the one without crack stop hole. However, when the horizontal distance d ₂ is equal to 5 and 10 mm, the peak strength is increased slightly. Moreover, the overall strain decreases under the influence of crack stop hole, indicating that the crack stop hole reduces the ductility of the sample before failure. Although there are differences in the position of crack stop holes, the distance between curves is closer than that of the sample without crack stop holes, and the slope of the stress-strain curve is larger than that of the sample without crack stop holes, indicating that crack stop holes increase the apparent stiffness of granite samples. Finally, in Figures 3A, B, the stress-strain curve drops rapidly after the peak strength, reflecting the brittle fracture characteristics of granite.

Figure 4 shows the peak strength of different pre-cracked granite samples, it can be seen that the peak strength increases with the increase of the inclination angle, as shown in Figure 4A. When the inclination angle of the pre-crack is 0°, the peak strength of the sample reaches the lowest value equal to 49.22 MPa, and when the inclination angle rises to 90°, the peak strength increases to 151.14 MPa. According to fracture mechanics theory and Figure 5, the stress field near the crack tip in polar coordinate system can be obtained (Zhang et al., 2020), as shown in Eq. 8. σ r = 1 2 2 π r K Ⅰ 3 − cos ⁡ θ cos θ 2 + K Ⅱ 3 ⁡ cos ⁡ θ − 1 sin θ 2 σ θ = 1 2 2 π r cos θ 2 K Ⅰ 1 + cos ⁡ θ − 3 K Ⅱ ⁡ sin ⁡ θ τ r θ = 1 2 2 π r cos θ 2 K Ⅰ ⁡ sin ⁡ θ + K Ⅱ 3 ⁡ cos ⁡ θ − 1 (8) where σ _r is the radial stress; σ _θ is the circumferential stress; τ _rθ is the shear stress; θ is the angle in polar coordinate system; r is the distance from the crack tip; K _I and K _II are the stress intensity factors of type I and II cracks, respectively.

Under uniaxial compression, the pre-cracked granite sample is only subjected to axial stress. Therefore, for an open crack, K _I and K _II are determined by Eq. 9. K Ⅰ = σ 1 π a sin ⁡ 2 ⁡ β K Ⅱ = σ 1 π a sin ⁡ β ⁡ cos ⁡ β (9) where a is half the length of the long axis of the crack and equal to the distance from the crack tip to the crack center; β is the angle between crack strike and the direction of axial stress, as shown in Figure 1B.

According to Eq. 9, when β is equal to 90°, that is, the pre-crack inclination angle α (see Figure 1) is 0°, in this case, K _I obtains the maximum value and K _II = 0, therefore, it only shows the characteristics of type I crack. Considering that the tensile strength of rocks is far lower than the shear strength and compressive strength, the granite sample with pre-crack inclination angle of 0° does have the lowest peak strength. As pre-crack inclination angle increases, K _I gradually decreases, K _II is no longer zero, and the sample shows the characteristics of both type I and II cracks. Moreover, the width of the pre-crack in this study is 1 mm, according to the results obtained by Zhang et al. (2020), when the crack width is greater than 0.8 mm, the peak strength decreases with the increase of β, and the minimum value is obtained when β = 90°. When β = 0°, the pre-crack inclination angle α is 90°, both K _I and K _II are equal to zero, indicating that there is no stress concentration at the crack tip, in this case, the pre-cracked granite sample obtains the highest value of peak strength. Combined with the experimental results in this paper, it can be seen that the theoretical results mentioned above have been verified again.

For pre-cracked samples with crack stop holes, the peak strength changes with the horizontal distance d ₂ between the crack stop hole and the crack tip from 0 to 10 mm, as shown in Figures 3B, 4B. In detail, the peak strength is 80.68 MPa when the horizontal distance d ₂ is 0 mm, which is smaller than the peak strength of the sample without crack stop holes (87.45 MPa). The peak strength increases to 105.33 MPa when the horizontal distance d ₂ is equal to 5 mm, and decreases to 101.02 MPa when d ₂ is equal to 10 mm. It is estimated that the crack stop hole changes the law of crack propagation, resulting in changes in stress distribution. Therefore, there is a reasonable location of the crack stop hole to obtain the extreme value of the peak strength of the pre-cracked granite sample. In summary, the arrangement of the crack stop hole plays an important role in improving the stability of the structure of cracked rocks.

The peak strength of pre-cracked granite samples under uniaxial compression is positively and linearly correlated with the inclination angle of the pre-crack, and the fitting equation is as follows: σ p = 1.04 α + 45.54 R 2 = 0.92 (10) where σ p represents the peak strength of the pre-cracked granite samples, MPa; α is the pre-crack inclination angle, °, as shown in Figure 1B.

The ratio of the peak strength of the pre-cracked sample to the peak strength of the intact sample is defined as strength weakening coefficient to characterize the weakening degree of the peak strength of granite caused by pre-crack inclination angle, as shown in Figure 4A. When the pre-crack inclination angle is 0°, the peak strength of the sample is only 26% of that of the intact sample. When the pre-crack inclination angle increases from 0° to 60°, the strength weakening coefficient increases from 0.26 to 0.46, indicating that the peak strength of the pre-cracked granite sample increases relatively slowly. However, when the pre-crack inclination angle increases from 60° to 90°, the peak strength increases relatively quickly, for example, the peak strength is 78.73% of that of intact sample when the inclination angle is 90°.

The slope of the straight line segment of the stress-strain curve is defined as the elastic modulus of the rock. However, considering that the elastic modulus is only related to the performance of the rock, such as lithology, composition, particle size, water content and pore size, therefore, the so-called elastic modulus here should actually be regarded as apparent stiffness. According to the stress-strain curve in Figure 3, the apparent stiffness of different pre-cracked samples can be obtained, as shown in Figure 6. It can be seen that the apparent stiffness increases linearly with the increase of the pre-crack inclination angle. The linear fitting equation is as follows: E t = 0.21 α + 17.29 R 2 = 0.98 (11) where E t is the apparent stiffness of different pre-cracked granite samples, MPa.

The failure mode of the sample is a key feature reflecting its failure mechanism (Li et al., 2022). Generally speaking, the failure of the sample starts from the initial cracking, and then the crack propagates until it penetrates the sample. In this study, as the pre-crack inclination angle increases, the characteristics of tensile failure gradually weaken, while the characteristics of shear failure strengthen, and shear failure plays a dominant role during this process, as shown in Figure 7A, this characteristic can be clearly seen as the inclination angle increases from 0° to 90°. For example, when α = 0°, a tensile crack initially forms and propagates on the upper edges of the central circular through hole, however, as the pre-crack gradually closed during the loading process, the wing crack appears at the pre-crack tip, whose growth rate is gradually faster than that of the initial tensile crack, and finally the tensile failure mode is dominant, and the tensile crack still penetrates the upper part of the sample. Moreover, the number of cracks caused by shear increases with the increase of the pre-crack inclination angle, including wing cracks and secondary cracks, especially when α = 90°, the shear failure characteristics are more significant.

It can be seen from Figure 7B that, the crack propagation of samples is significantly affected by crack stop holes. The number of cracks after sample failure is relatively minimum when the horizontal distance d ₂ is equal to 0 mm, is maximum when d ₂ = 5 mm, and ranks second when d ₂ = 10 mm. This is closely related to the variation of peak strength of granite samples, in general, the more crack propagation, the higher the peak strength.

According to Eqs 1–3, the total energy, elastic strain energy and dissipation energy of different granite samples during uniaxial compression can be calculated respectively, the results are depicted in Figures 8, 9. It can be seen from Figure 8A that the change of total energy of different samples is roughly similar, that is, the total energy absorbed by each sample increases non-linearly. Except when α = 0°, the total energy increases with the increase of the pre-crack inclination angle, indicating that the larger the pre-crack inclination angle, the more energy the sample consumes during the failure process, which reflects the compression resistance of the sample macroscopically. In Figure 8B, the change of elastic strain energy is similar to the change of stress-strain curve, showing a non-linear increase at first and then a rapid decline, that is, the elastic strain energy is released rapidly after the peak strength, resulting in rapid failure of rock samples, especially when the pre-crack inclination angle is relatively large. In addition, similar to the total energy, the elastic strain energy increases with the increase of the pre-crack inclination angle, except when α = 0°, indicating that the energy storage capacity of the sample increases.

In Figure 8C, dissipated energy increases slowly and linearly at first, and then rapidly and non-linearly after reaching the peak strength. The reason why the dissipated energy increases slowly before the peak strength is that most of the energy is stored as elastic strain energy. After the peak strength, the elastic strain energy is released rapidly, leading to a rapid increase in dissipated energy. It should be noted that the data of elastic strain energy and dissipated energy in the residual stage cannot be collected due to the rapid failure of the sample after the peak strength.

It can be seen from Figures 9A, B that, under the influence of the crack stop hole, the total energy and elastic strain energy stored in the sample increase under the same strain. In addition, compared with the sample without crack stop hole, the total energy-strain curve and elastic strain energy-strain curve of the sample are closer to each other respectively, and the crack stop hole reduces the total strain. When d ₂ = 0 mm, the total energy and elastic strain energy are smaller than those without crack stop holes. When d ₂ = 5 mm and 10 mm, both types of energy are greater than those without crack stop holes. The results show that if the horizontal distance d ₂ is relatively small, the elastic strain energy of the sample will be reduced, therefore, the reasonable arrangement of crack stop holes can improve the energy storage capacity of the sample.

When the peak strength is reached, the elastic strain energy reaches the maximum value. If the loading continues, the elastic strain energy is released in the form of dissipated energy, resulting in a rapid growth of dissipated energy after the peak strength, as shown in Figure 9C.

The various energies of different samples at peak strength are shown in Table 1. It can be seen that the elastic strain energy of intact sample at peak strength accounts for 92.43%, while the dissipated energy accounts for 7.57%. For samples with pre-crack inclination angles of 0°, 15°, 30°, 45°, 60°, 75°, 90°, elastic strain energy accounts for 64.87%, 75.71%, 70.03%, 85.64%, 64.81%, 83.47%, 86.42% respectively, and dissipated energy accounts for 35.13%, 24.29%, 29.97%, 14.36%, 35.19%, 16.53%, 13.58% respectively. For the sample with crack stop holes, the elastic strain energy accounts for 78.69%, 92.39%, 89.51% respectively, and the dissipated energy accounts for 21.31%, 7.61%, 10.49% respectively. It shows that the elastic strain energy at peak strength is far more than the dissipated energy, in addition, the storage capacity of elastic strain energy of the intact sample is stronger than that of the pre-cracked samples.

The pre-crack significantly reduces the total energy absorbed by the sample, which may be due to the reduction of peak strength caused by the pre-crack, thus reducing its energy storage capacity. The total energy, elastic strain energy and their proportion can be increased within a limited range by crack stop holes, and both kinds of energy are ultimately dominated by the integrity of the sample. According to Table 1, the elastic strain energy and dissipated energy of each sample at peak strength are shown in Figure 10. Generally speaking, the elastic strain energy increases non-linearly with the increase of pre-crack inclination angle, and can be expressed by Eq. 12. The dissipated energy fluctuates between 0.032 and 0.073, and its distribution can be expressed mainly by Eq. 13. It should be noted that Eq. 13 does not consider the values of points where the pre-crack inclination angle is 30° and 60° in Figure 10, otherwise the correlation coefficient will fall to a relatively low value. The reason may be that the dissipation energy is very small at peak strength, and its slight fluctuation can easily lead to a larger change in the correlation coefficient. In addition, the dispersion of granite samples increases the difference of dissipated energy. U e = − 0.00164 α + 4.26 α 2 * 10 − 5 + 0.116 R 2 = 0.93 (12) U d = − 0.00134 α + 1.501 α 2 * 10 − 5 + 0.0547 R 2 = 0.79 (13)

The mechanical behavior of rock sample under uniaxial compression can be reflected by energy dissipation (Lin H. et al., 2022). The ratio of dissipated energy to total energy is defined as the dissipated energy rate η _e, as shown in Eq. 14, which reflects the relationship between the dissipated energy and the total energy during loading. η e = U e U (14)

The variation of dissipated energy rate of each sample with axial strain is shown in Figure 11. The dissipated energy rate increases first and then decreases during the microcracks compaction stage. The reason may be that the internal microcracks of the sample are compacted at the initial stage, and the energy is released as dissipated energy. After that, if the loading continues, strain hardening gradually takes a leading position, and the energy is mainly stored in the form of elastic strain energy. When approaching the peak strength, the decline of dissipated energy rate slows down, and then the dissipated energy rate gradually increases, which may be caused by the accelerated growth of microcracks. After the peak strength, the elastic strain energy is released rapidly due to the macroscopic failure of the sample, resulting in a sharp rise in the dissipated energy rate.

As shown in Figure 11B, all curves show rising characteristics at the initial stage, however, the dissipated energy rate of the sample without crack stop hole (α = 45°) keeps stable before ε = 0.002, and begins to decline after ε is greater than 0.002, indicating that the internal microcracks of the sample without crack stop hole are relatively more developed, so its dissipated energy rate remains at a higher value. On the other hand, the crack stop hole inhibits the crack growth from the initial stage of loading, thus reducing the dissipated energy rate of samples. In addition, the difference in the location of the crack stop hole leads to the difference of the dissipated energy rate.