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. 2023 Mar 9;8(11):9896–9903. doi: 10.1021/acsomega.2c06840

Crack Evolution Behaviors and Bursting Liability of Sandstone with Different Sizes: An Experimental Study

Yan Chen †,‡,§,∥,*, Guolong Zhang †, Baohua Guo †,∥, Rong Dou ⊥
PMCID: PMC10035014  PMID: 36969456

Abstract

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The size ranges of ore pillars play important roles in preventing the occurrences of rock burst phenomena. Due to the lack of research on the relationship between crack evolution behaviors and bursting liability of rocks with different sizes, uniaxial compression tests on sandstone with different height-to-diameter ratios (H/D) were conducted. The results showed that the mechanical parameters of sandstone have obvious size effects, in which both the peak strength and peak strain decrease with increases in the H/D. Moreover, the brittle index modified value (BIM) decreased, but the impact energy index increased gradually, which indicated the increase of bursting liability. With the increases in the BIM, the overall crack strain parameters increased, thereby indicating a positive correlation. With the increase in the impact energy index, the crack strain decreased and the bursting liability became higher. Although the axial crack closure stress and axial crack damage stress increased with the increases in BIM (indicating a positive correlation), the bursting liability became increasingly smaller. The crack stress decreased with the increases in the impact energy index, and the bursting liability became stronger. The findings of this study will potentially provide experimental references for furthering the current understanding of the mechanism of rock bursts in underground coal mines in China.

1. Introduction

In China, the major destructive elements of rock bursts pose serious threats to coal mine production processes. Rock bursts are dynamic phenomena characterized by sudden, sharp, and violent damages caused by the instantaneous release of the elastic properties of coal and rock mass around coal mine roadways or working faces. Rock bursts are often accompanied by instantaneous displacements and expulsion of coal and rock masses, accompanied with loud noise levels and air waves, which can cause casualties and damages to roadways in serious cases. Such bursts can even cause surface subsidence, leading to local earthquakes.1 For example, on October 20, 2018, a major rock burst accident occurred in the Shandong Longyun Coal Mine, which resulted in 21 deaths.2 Ore pillars are components in underground rock mass, which bear uniaxial compression loads. Structural failures in ore pillars can induce rock bursts due to the coalescence of cracks. In addition, the sizes of ore pillars have important influencing effects for the prevention of rock bursts. Therefore, research studies regarding crack evolution behaviors and the bursting liability of different sized rock specimens have important practical significance from the standpoint of safety in coal mines.

Fracturing is the failure process by which new surfaces in the form of cracks are formed or existing crack surfaces become extended in rock material.3 The main method used to identify material fracturing is to monitor whether new cracks become apparent. At the present time, the crack evolution behaviors of rock under compression are generally studied by using experimental or numerical simulations. Song et al.4 studied the effects of residual stress on creep crack initiation and growth behavior. The results revealed that higher compressive residual stress ahead of the crack tips delayed creep crack initiation and increased creep initiation life. Coal samples within original cracks were selected to conduct uniaxial compressive loading experiments by Kong et al.5 It was found that the original cracks had expanded and new cracks formed and propagated, resulting in fracturing of the samples. Zhao et al.6 studied the three-dimensional spatial evolution patterns of internal microcrack breeding, initiation, expansion, nucleation, and penetration of fractures, as well as the and instability processes of granite rock samples with different prefabricated cracks. Liu et al.7 selected four different strength types of rock samples in order to perform and analyze fracturing processes. The change characteristics of the number, angles, and lengths of the cracks during the process of rock failure were obtained. Also, based on CT images and slices, Duan et al.8 qualitatively studied the crack evolution of Longmaxi shale in the horizontal and vertical direction, and the distributions patterns of crack areas during loading were revealed. In another related study, Li et al.9 examined the real-time crack evolution processes of fractured shale samples under uniaxial compression. The results demonstrated that the crack evolution of fissured shale was progressive. Numerical simulations are a method commonly utilized to analyze the crack evolution behaviors of rock. For example, PFC2D has been used to build a variety of defective coal and rock specimens. The experimental results revealed that the mechanical properties, crack evolution characteristics, propagation forms of initial cracks, and the final crack distributions formed coal rock specimens with different types of fracture holes.10 In summary, numerous laboratory tests have shown that the crack evolution processes in the rock are one of the main causes of rock failure.

As an inherent property of coal and rock, bursting liability has been used to evaluate the rock burst tendency of underground coal mines or tunnels in China.11 At the present time, when exploiting deep high-gas coal seams with bursting liability, complicated rock burst disaster conditions exist due to the combined influences of stress and gas pressure.12 In deep hard-rock tunnel engineering processes, the promotion of large buckling deformations and the weakening of the strength of surrounding rock masses are two activities that induce rock bursts in tunnels.13 Zhang et al.14 discussed the bursting liability of dry and saturated water granite samples under uniaxial compression. The results suggested that water can significantly reduce the bursting liability of granite. In addition, in order to propose a new prediction method of rock burst without the influences of stress paths, Xu et al.15 proposed rock burst energy release rate as a new bursting liability index. That method was successfully verified by obtaining accurate predictions of the positions and intensity levels of rock burst events. Subsequently, in order to reasonably evaluate the bursting liability of coal in deep coal mining processes, a new modified bursting energy index, which considered the rebounding and damage effects of rock, was proposed by Zhang et al.16 Then, experimental tests confirmed that the modified bursting energy index increased linearly when the rock strength increased. Du et al.17 summarized the differences in mechanical parameters and bursting liability indices between coal and noncoal rocks and put forward a modified rock burst tendency classification criteria for noncoal rocks. Therefore, the research regarding the bursting liability of rock has made rich progress. However, the relationships between crack evolution behaviors and the bursting liability require further research in order to illustrate the effect of crack evolution on rock burst events.

In recent decades, the research regarding the size effect of rock or concrete has attracted extensive attention.18−21 For example, Hudson et al.22 conducted static uniaxial compression tests on marble of different sizes and concluded that the compressive strength of rock specimens with equal diameters decreased with increases of aspect ratio. You et al.23−25 pointed out that the uniaxial compressive strength of rock decreased with increases in length through the examination of size effect. It was proposed that the essential reason for the existence of rock size effect was the heterogeneity of the internal structures of rock material. Jin et al.26 studied the effects of strain rates on size effect and found that when the applied strain rate was more than the critical strain rate, the compressive strength increased with the increases in specimen size. Zhang et al.27 explored the size effect of rock dynamic fracture toughness. The results showed that the dynamic fracture toughness varied with the sizes of the specimens and the fracture process zone lengths, with the incubation times observed to increase with the increase in the sample size. Darbor et al.28 investigated the effects of specimen size on the strength levels of hard rock. The specimen size effect models were used thoroughly to analyze the laboratory data. In addition, Zhao et al.29 further analyzed the damage degrees of sandstone with different H/D values and found that the damage curves of sandstone exhibited nonlinear evolution. The larger the H/D, the higher the damage degree of the sandstone when the loads were equal. Triaxial compression tests of size effect on siltstone were conducted by Zhu et al.,30 in which the results showed that when the confining pressures were equal, the measured values of rock strength, Poisson’s ratio, and Young’s modulus tended to be stable when the H/D were exceeded. Masoumi et al.31 proposed a unified size-effect law, which correlated well with the ascending and descending uniaxial compressive strength trends. Shi et al.32 conducted uniaxial compression tests on fine sandstone samples with the same H/D and diameter distributions between 25 and 75 mm. It was concluded that the large sized samples had high compressive strength and larger amounts of pre-peak stored energy. However, after the peak, the samples exhibited brittle failure, rapid energy release, and strong impact tendency. Previously, the studies regarding the size effect of rocks have mainly focused on the basic mechanical parameters, such as strength and Young’s modulus. However, the relationships between size effect and crack evolution behaviors have not yet been investigated and require further study in order to reveal the mechanism of size effect on rock.

At the present time, few studies have been conducted regarding crack evolution behaviors and the bursting liability of rocks of different sizes. Therefore, a thorough examination of crack evolution behaviors and the bursting liability of rock will significantly improve the current understanding of the mechanism of rock burst events in underground coal mines. In this study, sandstone was taken as the research object for the purpose of conducting uniaxial compression tests. The crack evolution behaviors and burst liability parameters of sandstone samples were investigated, and the size effect on crack parameters and burst liability were investigated.

2. Method

2.1. Sample Preparation

The rock type used in this study was sandstone. The laboratory processing sequence of a standard rock sample is generally through rock drilling first in order to obtain the core. Then, a cutting machine is used to cut the appropriate length, and finally, a grinding machine is utilized to polish the two end faces. Therefore, when drilling a core, it is required that a certain bedding plane be drilled along so that the experimental samples maintain the same bedding structure. In this experiment, a rock core with a diameter of 50 mm was drilled using a rock core-taking machine. The core was cut into corresponding sizes using an automatic rock cutting machine, and the two ends of the samples were ground flat by an automatic rock double-sided stone grinder, in order to obtain the required experimental rock samples. However, the rock block was limited in size. Therefore, in order to make full use of the rock block, and the H/D values were set as 0.6, 1.2, 1.8, 2.0, 2.4, and 2.8.

2.2. Test Equipment

The test equipment utilized in this study was an RMT-150B electro-hydraulic servo rock mechanics testing machine developed by the Wuhan Rock and Soil Institute of the Chinese Academy of Sciences. The technical specifications were as follows: the Maximum vertical load is 1000 kN; maximum vertical stroke is 50 mm; force loading rates are in the range of 0.01–100 kN/s; displacement loading rates range between 0.0001 and 1.0 mm/s. The uniaxial compression in this study was set at a loading rate of 0.002 mm/s.

3. Results

3.1. Effects of H/D on Mechanical Parameters

In this study, uniaxial compression tests were carried out on sandstone samples with different H/D values. The stress–strain curves are shown in Figure 1, and the mechanical parameters are detailed in Table 1. In Table 1, H represents the height; D indicates the diameter; εp denotes the peak strain; Eave is the elastic modulus of the linear elastic segment; and UCS represents the uniaxial compression strength.

Figure 1.

Figure 1

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Stress–strain curves of sandstone with different height-diameter ratios, (a) sandstone specimens S1, S2, and S3; (b) sandstone specimens S4, S5, and S6.

Table 1. Mechanical Parameters of Sandstone with Different Sizes.

no. D (mm) H (mm) εp (10–3) Eave (GPa) UCS (MPa) H/D
S1–1 49.47 29.95 16.25 15.810 126.12 0.6
S1–2 49.66 30.89 13.24 17.589 120.52  
S2–1 49.92 60.69 8.707 19.723 85.41 1.2
S2–2 49.70 60.39 8.499 17.100 94.70  
S3–1 49.30 90.55 6.800 22.150 70.68 1.8
S3–2 49.39 91.09 5.712 24.367 89.14  
S4–1 49.51 99.18 4.716 26.594 80.29 2
S4–2 49.51 99.97 6.097 18.817 77.77  
S5–1 49.45 121.30 4.768 24.547 74.65 2.4
S5–2 49.55 119.93 4.502 24.671 71.75  
S6–1 49.71 141.25 4.212 23.941 69.69 2.8
S6–2 49.73 142.55 3.745 25.090 66.89  
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Based on the experimental results, the relationships between the mechanical parameters and H/D of the sandstone were obtained, as shown in Figure 2. In addition, as can be seen in Figure 2, with the increases in H/D, the UCS and peak strain levels of the rock samples displayed decreasing trends. For example, when the H/D was less than 2, the UCS and peak strain decreased faster. However, when H/D increased to 2, the rate of decrease tended to be flat. The results were consistent with Zhu et al.30 In regard to the average modulus, it was observed that with the increases in the H/D, the elastic modulus gradually increased, but the increasing rate then gradually slowed down. It was found that the mechanical parameters of the sandstone samples had an obvious size effect. Rock is a natural geological material containing such initial defects as joints, cracks, and microcracks. The UCS decreases with the H/D mainly due to the fact that larger specimens contain more initial defects. Meanwhile, small deformations can cause the failure of a specimen with larger H/D due to the coalescence of the cracks within the specimen. Theoretically speaking, the elastic modulus represents the deformation characteristics of the rock itself and its changes are related to the rock species and its properties. Brace33 found that the elastic modulus of unfractured rocks was slightly dependent on specimen size up to a meter. However, it has been found that under different experimental conditions, the measurement values of elastic modulus will change. It has been determined that this is mainly due to the changes in the relative ranges of stress field redistributions due to the end effects.30

Figure 2.

Figure 2

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Relationships between mechanical parameters and H/D of sandstone samples.

3.2. Effects of H/D on Crack Evolution Behaviors

The progressive failure of rock is an important process from “elasticity” to “damage” and then to “fracture”. It is necessary to fully understand the evolution law of cracks in rock during loading processes in order to accurately understand the progressive failure process of rock. Therefore, Martin and Chandler34 introduced the concept of crack strain for the purpose of calculating the sizes of the rock cracks during loading and to then analyze the progressive failure process of the rock. Crack strain can be used to quantitatively analyze the size and number of cracks in rock, which is defined as the axial and radial deformation caused by the initiation, propagation, and penetration of primary cracks and the initiation of new cracks under the action of external loads.34 Under uniaxial compression conditions, namely, a confining pressure of 0, the calculation formula of crack axial strain can be written as follows:

3.2. 1

where ε1c represents the crack axial strain, ε1 is the axial strain, σ1 denotes the axial stress, and E is the elastic modulus.

In this investigation, taking sandstone sample S6–2 as an example, Figure 3 shows the calculation diagram of the axial crack parameters, where Inline graphic denotes the axial crack closure stress, σcd is the axial crack damage stress, εcm indicates the crack axial closure strain, and εcf represents the crack axial peak strain. Subsequently, in order to study the law of crack propagation in sandstone during uniaxial compression, the difference between axial crack peak strain and the axial crack closure strain was calculated. The crack axial propagation strain Inline graphic was obtained as follows:

3.2. 2

Figure 3.

Figure 3

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Calculation diagram of crack parameter calculation.

The crack parameters of the sandstone samples with different H/D values under uniaxial compression were calculated and are shown in Table 2. Figure 4 shows the relationship between the crack parameters and H/D of the sandstone samples under uniaxial compression conditions.

Table 2. Crack Parameters of Sandstone with Different Height to Diameter Ratios.

no. εcm (10–3) εcf (10–3) εcp (10–3) σcc (MPa) σcd (MPa)
S1–1 4.794 8.617 3.823 42.89 94.79
S1–2 5.845 6.780 0.935 47.76 113.57
S2–1 3.271 4.376 1.105 42.44 78.19
S2–2 2.410 2.961 0.551 44.70 87.46
S3–1 2.524 2.948 0.424 41.57 73.34
S3–2 2.053 2.555 0.501 39.69 78.69
S4–1 1.646 1.701 0.055 37.62 78.98
S4–2 1.725 1.964 0.238 39.11 70.14
S5–1 1.635 1.727 0.093 36.00 71.59
S5–2 1.461 1.792 0.331 34.62 70.94
S6–1 0.968 1.288 0.320 32.13 55.22
S6–2 0.619 0.701 0.082 28.87 59.82
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Figure 4.

Figure 4

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Relationship between the height to diameter ratio and sandstone crack parameters. (a) Crack strain and (b) crack stress.

It can be seen in Figure 4a that the crack parameters tended to decrease with the increases of H/D. The crack axial closure strain reflects the initial crack closure in the rock. Figure 4a shows that the smaller the H/D, the more the initial crack is closed. Also, the larger the H/D, the less the initial crack is closed. It can be seen from the evolution law of the crack axial peak strain that the smaller the H/D, the smaller the crack axial peak strain. The crack axial peak strain reflects the number of cracks in the rock specimen at the point of failure. This study’s experimental results showed that the larger H/D is, the fewer cracks occurred during failure. Meanwhile, the smaller H/D, the more cracks were evident during the failure process. It was indicated that there was an inverse relationship between crack axial propagation strain and H/D. That is to say, the larger H/D, the less crack initiation and propagation occurred in the samples. In summary, the generation of small cracks led to the failure of the samples with larger H/D. Figure 4b shows that the axial crack stress, including the crack closure stress and crack damage stress, tended to decrease with the increases in H/D. In conclusion, for a sample with a larger H/D, smaller crack and stress could potentially induce failure. Meanwhile, a smaller H/D required more cracks and greater stress to produce failure. Therefore, in engineering practices, it should be assumed that ore pillars with smaller H/D are more stable.

The crack strain and crack stress both decrease with H/D mainly due to the smaller bearing capacity of larger specimens. The reduction of the crack stress of rock samples is due to the internal shear slip, and its macroscopic manifestation is that the rock samples produce unrecoverable crack propagation strain. It is indicated that the ore pillar with larger H/D can bear higher stress to avoid failure when the mining induced stress is small.

3.3. Effects of H/D on Bursting Liability

At the present time, the majority of coal mines in China are in the deep mining stage. The increases in ground stress results in major mining difficulties, such as rock bursts and coal and gas outbursts. Consequently, major economic losses have occurred. The deeper the mining depth, the greater the amount of ground stress exerted on ore pillars. Therefore, the greater the mining depth, the greater the pressure on the ore pillars, and the greater the possibility of destabilizing impact failure. Some researchers have suggested the use of small-sized pillars to avoid rock burst disasters.23 This section mainly details this study’s analysis of bursting liability parameters, which were used to establish the relationships between each parameter and H/D.

3.3.1. Introduction to the BIM

Aubertin et al.35 proposed such brittleness indexes as revised (brittle index modified value, BIM) transform. The specific algorithm of the BIM is as follows: the area between the pre-peak curve and the abscissa (as shown in Figure 5) is A2, which represents all the energy stored in the sample under uniaxial compression. Then, by taking the elastic modulus in the linear elastic stage as the slope and the area sandwiched between the line and the abscess through the peak point, A1 will represent the elastic energy stored in the sample, and A3 is the area covered by the post-peak curve, which reflects the deformation energy consumed after the peak, which can be called post-peak dissipated energy. The value of BIM can be expressed as follows:

3.3.1. 3
Figure 5.

Figure 5

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Schematic diagram of the calculation of burst liability parameters.

The shorter the yield stage of a rock sample, the closer A1 is to A2. When the elastic energy is approximated to the full energy, the value of BIM will be close to 1. The work done by all external forces on the sample is stored in the form of elastic energy. If the elastic energy is released instantaneously after the peak, rock bursts will occur. Therefore, it can be considered that the smaller BIM, the higher the bursting liability of the sample. Based on the aforementioned, Aubertin et al.35 divided the bursting liability in accordance with the value of BIM, as shown in Table 3.

Table 3. Evaluation of Bursting Liability Based on BIM.
BIM bursting liability
1.00–1.20 high
1.20–1.50 medium
>1.50 low
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3.3.2. Impact Energy Index, KE

Under uniaxial compression conditions, the ratio of the deformation energy A2 accumulated before the peak to the deformation energy A3 consumed after the peak is referred to as impact energy index of the total stress–strain curve of rock samples. It contains the entire process of the stress and strain changes of a sample, which can intuitively and comprehensively reflect the entire process of energy storage and energy consumption and reveal the physical nature of bursting liability as follows:

3.3.2. 4

For brittle rock, (for example KE < 1.0) no burst liability exists. However, when 1.0 ≤ KE < 2.0, the burst liability is considered to be weak, and 2.0 ≤ KE indicates a strong bursting liability.

In this study, according to eqs 3 and 4 and Figure 5, the bursting liability parameters of the examined sandstone with different H/D were calculated. The calculation results are detailed in Table 4.

Table 4. Burst Liability Parameters of the Examined Sandstone Specimens.
no. A1 A2 A3 BIM KE
S1–1 0.462 0.965 0.741 2.090 1.302
S1–2 0.413 0.532 0.899 1.287 0.591
S2–1 0.185 0.223 0.325 1.204 0.686
S2–2 0.201 0.346 0.553 1.725 0.625
S3–1 0.142 0.214 0.124 1.510 1.730
S3–2 0.163 0.224 0.322 1.376 0.698
S4–1 0.121 0.141 0.023 1.163 6.050
S4–2 0.161 0.199 0.128 1.238 1.554
S5–1 0.113 0.133 0.033 1.174 4.017
S5–2 0.104 0.126 0.043 1.206 2.941
S6–1 0.102 0.131 0.011 1.288 11.83
S6–2 0.089 0.099 0.003 1.107 29.89
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Figure 6 shows the relationships between H/D and the bursting liability parameters. As can be seen in Figure 6a, with the increases in H/D, the pre-peak elastic energy A1, pre-peak accumulated energy A2, and post-peak dissipated energy A3 all displayed decreasing trends. As detailed in Figure 6b, with the increases of H/D, the BIM value gradually decreased. In other words, the bursting liability gradually increased. With the increase of H/D, the impact energy index KE also gradually increased and the bursting liability increased. Therefore, according to the relationships between BIM and KE values, it was concluded that the bursting liability of the sandstone samples with different H/D values could be divided into three blocks (high, middle, and low), as shown in Figure 6b. The results were consistent with Shi et al.32 It has been determined that this is mainly due to the brittle failure of rock specimens with larger H/D. Brittleness is the characteristic that a material exhibits little or no plastic deformation before failure.36 Bursting liability is related with brittleness. The rock specimens with larger H/D are more likely to have larger brittleness due to the buckling failure of rock specimens.37 Therefore, it has significant importance to use ore pillars with small H/D to avoid rock burst in underground mining.

Figure 6.

Figure 6

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Relationship between the ratio of height to diameter and the impact liability parameter. (a) Strain energy density; (b) BIM and KE.

4. Discussion

4.1. Relationships between Crack Strain and Bursting Liability

Figure 7 presents the observed relationships between crack strain and bursting liability parameters. In Figure 7a, it can be seen that the BIM value of the sample is mainly located in the range of more than 1.2 and less than 1.5, which belongs to the category of medium bursting liability. In addition, as can be seen from the relationship diagram between crack strain and BIM in Figure 7a, with the increases in the BIM, the overall crack strain parameters also increased, thereby indicating a positive correlation. Therefore, the larger the crack parameter, the smaller the bursting liability. Furthermore, detailed in Figure 7b, it was observed that with the increases in the impact energy index, the crack strain decreased and the bursting liability became increasingly higher. The results clearly revealed that the smaller the crack strain, the stronger the bursting liability. Small crack parameter values meant that the rock specimens had stronger brittleness due to little plastic deformation before rock failure. In conclusion, there was determined to be a negative correlation between the crack strain values and bursting liability.

Figure 7.

Figure 7

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Relationships between crack strain and bursting liability parameters. (a) BIM; (b) KE.

4.2. Relationships between Crack Stress and Bursting Liability

Figure 8 presents this study’s findings regarding the relationships between the crack stress and bursting liability parameters. It can be seen in Figure 8a that the crack stress, namely, the axial crack closure stress (Inline graphic) and the axial crack damage stress (Inline graphic), had both increased with the increases in the BIM value, thereby indicating a positive correlation. Meanwhile, the bursting liability became increasingly smaller. As detailed in Figure 8b, it was observed that the crack stress decreased with the increases in the impact energy index, and the bursting liability became stronger. Small crack stress values meant that the rock specimens could easily induce crack closure and crack damage and failure with small cracks. Therefore, it was determined that the smaller the crack stress values, the stronger the bursting liability.

Figure 8.

Figure 8

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Relationship between crack stress and impact liability parameters. (a) BIM; (b) KE.

This study mainly focused on the rock burst phenomena of ore pillar in underground mining. The uniaxial compression tests were conducted to analyze the size effect on crack evolution behaviors and bursting liability of sandstone. However, in underground mining, various rocks (such as, basalt, mudstone, and other rocks) can be encountered. The results in this study were obtained from sandstone. Further, more kinds of rocks can be adopted to investigate the size effect on crack evolution behaviors and bursting liability. Meanwhile, there are multiple indices to reflect the bursting liability. In this study, the impact energy index was adopted to analyze bursting liability. In the next step, more bursting liability indices can be used to analyze the rock burst mechanism in underground mining. In a word, the relationships between crack strain, crack stress, impact energy index, BIM, and H/D were expected to provide theoretical reference on the rock burst mechanism of the ore pillar in underground mining. Also, the research results can provide the experimental reference on rock mechanics in rock engineering, such as tunnel engineering.

5. Conclusions

The UCS and peak strain values of the sandstone samples decreased with the increases in H/D, and the elastic modulus was inversely proportional to the H/D. With the increases in H/D, the pre-peak elastic energy, pre-peak accumulated energy, and post-peak dissipated energy had all decreased. Furthermore, the BIM value gradually decreased, but the impact energy index gradually increased, thereby indicating that the bursting liability had increased gradually.

The relationships between the crack parameters and the bursting liability parameters of the examined sandstone samples with different H/D were discussed and analyzed. It was found that with the increases in the BIM value, the overall crack strain parameters also increased, thereby indicating a positive correlation. Therefore, the larger the crack parameter, the smaller the bursting liability. Moreover, with the increases in the impact energy index, the crack strain values were observed to decrease, while the bursting liability became increasingly higher. It was found that both the axial crack closure stress and the axial crack damage stress increased with the increase in the BIM value, thereby suggesting a positive correlation. However, the bursting liability was observed to become increasingly smaller. In summary, the crack stress values decreased with the increases in the impact energy index, and the bursting liability became stronger.

Acknowledgments

Financial supports from the National Natural Science Foundation of China (51904092), the Fundamental Research Funds for the Universities of Henan Province (NSFRF210454, NSFRF230403), Young backbone teachers funding program of Henan Polytechnic University (2022XQG-01), the research fund of Henan Key Laboratory for Green and Efficient Mining & Comprehensive Utilization of Mineral Resources (KCF2202), and the research fund of Jiaozuo Road Traffic and Transportation Science and Technology research center (JRTT2023004, ZD2021002, YB2021001) are gratefully acknowledged.

The authors declare no competing financial interest.

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