Stress analysis of crack characteristic of anthracite under dynamic loading-深地科学

Stress analysis of crack characteristic of anthracite under dynamic loading

Abstract

The coal dynamic characteristic stress identification under dynamic load is important for guiding underground mineral mining and predicting underground dynamic disasters. In this article, the dynamic compression test of anthracite under five strain rates is carried out, the evolution law of three kinds of crack characteristic stress is analyzed, and a prediction model of the crack characteristic stress threshold considering the strain rate effect is established. Then, the rationality of crack characteristic stress under dynamic loading is discussed from the damage evolution standpoint, and the crack extension response mechanism during dynamic compression of anthracite is discussed. The result shows that the crack characteristic stress threshold is significantly influenced by the strain rate. The three characteristic stress thresholds are positively correlated with the strain rate, but the ratios to the crest stress gradually decrease. The increase in the strain rate strongly contributes to the crack extension behavior of anthracite. In the crack unstable extension phase, because of the increase of the strain rate, anthracite shows more energy dissipation under the same deformation in association with the stress concentration effect and the dynamic strength enhancement effect. The crack propagation rate is increased, the crack propagation path of the section is more complex, and more severe damage occurs before the dynamic failure of anthracite, which leads to even more severe damage.

Highlights

The strain rate effect of anthracite crack characteristic stress is revealed.
A crack characteristic stress prediction model of anthracite considering the strain rate effect is established.
The response mechanism of anthracite crack damage evolution to the strain rate is discussed.

1 INTRODUCTION

Rock burst is a complex and dangerous disaster in mining engineering. Its main feature is that the coal mass suddenly releases huge energy, resulting in severe crushing and ejection of coal seams in underground gallery or goafs (Hosseini et al., 2023; Wang et al., 2023; Wu et al., 2019). Rock burst not only poses a serious threat to the safe production of mines but may also lead to casualties and equipment damage (Xie et al., 2020). Therefore, it has become a major challenge in mine engineering (Cao et al., 2016; Kan et al., 2022; Liu et al., 2015; Liu et al., 2024; Nie et al., 2024). An in-depth understanding of the rockburst mechanism can enable prevention and management of power hazards. However, the mechanism of rock burst is complex. At present, the academic community has not yet gained a complete understanding of the causes and mechanisms of rock burst. Many scholars generally believe that it is closely related to many factors such as the mine stress concentration, coal brittleness and strength characteristics, mining technology, and layout. Specifically, stresses within the coal block gradually accumulate owing to the redistribution of stresses caused by mining disturbances. The stress concentration effect gradually aggravates and the microcracks in the coal mass begin to expand and penetrate, eventually leading to the sudden destruction of the coal block. Therefore, the study of microfracture propagation is of significant theoretical relevance to explore the mechanism of rock burst.

The analysis of crack characteristic stress is helpful to study the crack propagation of underground rock masses. Crack characteristic stress refers to the stress characteristics during microfracture germination, crack propagation, and perforation. The study of crack characteristic stress can be traced back to the 1960s. Bieniawski et al. divided the loading curve of a rock mass before peak load into four processes, including the compaction process, the elastic process, the crack stable expansion process, and the crack unstable expansion process (Bieniawski, 1967a, 1967b, 1967c). The separation boundaries between each adjacent process correspond to crack closure stress σcc, crack initiation stress σci, and crack damage stress σcd (Bieniawski, 1967c). Subsequently, many scholars explored methods to determine the characteristic stress (Chen et al., 2012; Gao et al., 2018; Li et al., 2020; Zhou et al., 2023). At present, the methods for defining characteristic stresses largely fall into three categories. One is the axial strain method (including the axial strain stiffness method (Gao et al., 2018) and the axial plastic strain method (Zhou et al., 2023)). The second is the volume strain method, which combines axial and circumferential strains, including the volume stiffness method (Eberhardt et al., 1998) and the crack volume strain method (Li et al., 2020). In addition, combined with an increase in existing monitoring methods, geophysical methods like acoustic emission (AE) are increasingly being applied for the identification of characteristic stress. For example, according to the variation characteristics of monitoring parameters such as the ringing count (Zhao et al., 2013), energy (Moradian et al., 2016), and AE frequency (Chen et al., 2012) with the loading process, identification is carried out.

Similar to most rock masses, coal is a complex porous medium containing a multitude of natural microcracks and defects (Xia et al., 2014). Using the identification method of characteristic stress, many scholars have studied the microcrack propagation mechanism of coal. According to the phase properties of AE energy, Qijun Hao proposed a method to establish the characteristic stress threshold by using a damage evolution curve (Hao et al., 2020). Lin Gao used the volumetric strain method to analyze the energy evolution mechanism of coal (Gao et al., 2020). The influence of confining pressure on characteristic stress was investigated by Jianguo Ning through triaxial compression testing, revealing a proportional relationship between the crack initiation threshold and peak stress (Ning et al., 2018). The investigation conducted by Qiangling Yao revealed that the proportions of the stress threshold remained unaffected by variations in the water content (Yao et al., 2016).

Impact disturbance is one of the key factors that induces rock burst. The mechanical properties of coal show a significant strain rate effect under impact disturbance (Gong et al., 2020; Kong et al., 2022; Zhou et al., 2017). Flaws like microcracks may further expand and intensify under impact load, resulting in a damage evolution mechanism that is significantly different from that in static load conditions. However, the identification and assessment of the characteristic stress of coal are carried out under a quasi-static compression load, and there is little research on how to accurately identify the characteristic stress of coal crack under impact disturbance. Therefore, dynamic compression experiments were conducted on anthracite at different impact velocities. The characteristic stress threshold evolution law is investigated using axial plastic strain and axial strain stiffness methods. A prediction model of the characteristic stress threshold considering the strain rate effect is established. The applicability of the crack characteristic stress threshold is discussed from a dynamic damage perspective. Combined with the energy–damage–strain relationship in the unstable failure stage of crack, the strain rate effect of the coal damage evolution mechanism is analyzed. It provides a new perspective and reference for the investigation of rock burst.

2 TEST SCHEME AND METHODOLOGY

2.1 Experimental materials and procedures

The overall test flow is shown in Figure 1. In this paper, anthracite coal from Wuhai City, Inner Mongolia, was selected for research. First, the specimens were prepared according to the compression test protocol of the ISRM. The specimen for the static loading test is a standard cylinder measuring 50 mm in diameter and 100 mm in height, while the dynamic impact test specimen is a standard cylinder with a diameter of 50 mm and a height of 25 mm. To ensure consistency of the specific material properties in the experiment, density and wave velocity measurements were carried out on well-processed specimens. The average density of anthracite in this test was 1.632 g/cm³ and the average longitudinal wave speed was 1.846 km/s.

Details are in the caption following the image — Figure 1
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Experimental process.

Next, quasi-static compression tests were conducted on anthracite using a TAWA-2000 electro-hydraulic servo press in the Materials Mechanical Properties Testing Laboratory. The mechanical parameters of the specimens under static load were obtained as reference data for the test. The static uniaxial compression test was loaded according to the displacement mode at 0.2 mm/min, and the applied prestress was set at 1 MPa. Dynamic impact tests were performed using the Separate Hopkinson pressure bar (SHPB) test system of the impact dynamics laboratory of the Jiangsu Mechanics Experimental Center. The test system controls different impact rates by controlling the nitrogen pressure and the bullet position. Through the preimpact test, it was finally determined that five different impact velocities were selected in this test. In order to ensure the rigor and rationality of the test, it is necessary to achieve uniform deformation and stress balance of the sample during the test. Therefore, a rubber gasket is used as an auxiliary tool for stress waveform formation. The friction effect during the impact loading process was reduced by uniformly smearing vaseline on the longitudinal surface of the sample.

2.2 SHPB test theory

The mechanical data of the impact compression test were determined using the three-wave method (Yi et al., 2020):

\begin{matrix} σ (t) = \frac{E_{0} A_{0}}{2 A_{S}} [ε_{i} (t) + ε_{t} (t) + ε_{r} (t)], \\ ε (t) = \frac{A_{0}}{2 A_{S}} \int_{0}^{t} [ε_{i} (t) - ε_{t} (t) - ε_{r} (t)] d t \\ \dot{ε} (t) = \frac{C_{0}}{L_{S}} E_{0} [ε_{i} (t) - ε_{t} (t) - ε_{r} (t)], \end{matrix},

(1)

where the σ( t) is the sample stress and ε( t) is the axial strain. Similarly, ε i( t) is the incident strain, ε r( t) is the reflection strain, and ε t( t) is the transmission strain.

\dot{ε}

( t) is the strain rate. t is the loading time. E 0, A 0, and C 0 are the material parameters of the impact system. A s is the cross-sectional area of the sample and L s is the length of the sample.

In the dynamic impact test, it is necessary to verify the stress balance at both ends of the specimen. The strain balance equation is as follows:

ε_{i} (t) + ε_{r} (t) = ε_{t} (t) .

(2)

In this article, the strain signal obtained from the impact test under 0.2 MPa pressure is used for verification. It can be seen that during the impact process of the sample, the sum of the incident strain and the reflected strain basically coincides with the transmission strain shown in Figure 2. The basic requirements of the test are fulfilled.

2.3 Crack characteristic stress identification method

Zilong Zhou and colleagues introduced a novel method to evaluate crack stress (Zhou et al., 2023), as illustrated in Figure 3a. Assuming that the modulus of elasticity is constant until failure, the plastic strain is obtained from the difference of the principal strain and the elastic strain. The starting and ending points of the axial plastic strain curve equilibrium section correspond to σcc and σcd, respectively. In addition, some scholars have proposed a solution to identify crack initiation stress (Gao et al., 2018). This is illustrated in Figure 3b, and the end point of the relative equilibrium phase of axial strain stiffness is defined as σci.

3 RESULTS AND ANALYSIS

3.1 The characteristic stress variation law of anthracite under different strain rates

Table 1 presents the data of the crack characteristic stress of anthracite under different strain rates, and shows the mean value of each group of data for analysis. Considering the mean value of all the data and the analysis, as indicated in Figure 4, the characteristic stress thresholds of the samples tend to increase with the strain rate. Specifically, the sample peak strength ( $σ_{p}$ ) increases from 16.45 to 40.81 MPa when the strain rate develops varies from 10−4 to 130.17 s−1, showing a significant strain rate effect. σcc increases from 6.36 to 11.70 MPa, and its ratio to peak stress decreases from 38.60% to 28.98%. The crack closure process is shortened as the strain rate increases. σci varies from 11.24 to 19.26 MPa; the ratio of σci to peak stress decreases from 68.28% to 47.69%. The dynamic loading process entered the crack propagation stage relatively earlier. In addition, σcd increases from 13.26 to 28.35 MPa, and the ratio to σp decreases from 80.60% to 69.48%, which also indicates that the loading process enters the crack unstable propagation phase earlier.

Table 1. Characteristic stress data at five strain rates.

Number	$\dot{ε}$ (s−1)	$\bar{\dot{ε}}$ (s−1)	$σ_{p}$ (MPa)	$\bar{σ_{p}}$ (MPa)	$σ_{cc}$ (MPa)	$\bar{σ_{cc}}$ (MPa)	$\bar{σ_{cc} / σ_{p}}$ (%)	$σ_{ci}$ (MPa)	$\bar{σ_{ci}}$ (MPa)	$\bar{σ_{ci} / σ_{p}}$ (%)	(MPa)	(MPa)	(%)
J-1	10−4	10−4	15.88	16.45	6.32	6.36	38.66	11.27	11.24	68.28	13.39	13.26	80.60
J-2	10−4		16.46		6.20			10.68			12.58
J-3	10−4		17.01		6.56			11.76			13.82
D-1-1	50.07	51.29	19.18	20.06	7.29	7.55	37.66	10.98	11.69	58.23	15.36	15.98	79.67
D-1-2	50.96		20.49		7.57			12.07			16.17
D-1-3	52.85		20.52		7.79			12.01			16.41
D-2-1	70.76	71.05	23.48	24.52	8.76	9.09	37.10	13.62	14.07	57.43	18.94	18.73	76.41
D-2-2	71.08		24.85		9.20			14.25			18.05
D-2-3	71.32		25.22		9.32			14.35			19.20
D-3-1	89.65	90.88	27.89	28.37	9.73	9.91	34.91	14.45	14.71	51.84	21.09	21.22	74.81
D-3-2	90.12		27.91		9.64			14.42			20.89
D-3-3	92.88		29.32		10.35			15.26			21.69
D-4-1	108.99	109.95	34.97	35.56	11.62	11.66	32.79	17.83	17.85	50.19	25.83	26.11	73.43
D-4-2	110.03		35.68		11.78			17.79			26.05
D-4-3	110.84		36.03		11.58			17.92			26.45
D-5-1	128.12	130.17	39.91	40.81	11.71	11.70	28.98	19.20	19.26	47.69	28.15	28.35	69.48
D-5-2	130.74		40.34		11.53			19.25			27.84
D-5-3	131.66		42.18		11.85			19.33			29.05

In summary, dynamic loading at high strain rates significantly affects the anthracite crack extension behavior, which in turn has an important impact on its failure mechanism.

3.2 Characteristic stress threshold prediction model of anthracite

Different types of anthracite have different compressive strengths. Furthermore, the characteristic stresses are influenced by the strain rate (Zhao et al., 2021). In order to more fully characterize the changes in characteristic stresses at different strain rates and coal types, the characteristic stress prediction model is established as follows:

σ_{c} = f (σ_{sp}, \dot{ε}),

(3)

where σ sp represents the static compressive strength and σ c represents the general term of three characteristic stresses.

The dynamic compressive strength is usually higher than the static compressive strength (Fan et al., 2021), and the dynamic strength growth factor ( DIF) is used to reflect the change law of strength under dynamic load conditions (Li et al., 2023). The specific results are illustrated in Figure 5a. DIF shows accelerated growth as the strain rate increases. Therefore, the change in the characterization function of DIF with the strain rate is determined by fitting with a cubic polynomial function as follows:

\begin{array}{l} DIF = 1 - 3.42 \times 1 0^{- 3} \dot{ε} + 1.724 \times 1 0^{- 4} {\dot{ε}}^{2} - 4.466 \times 1 0^{- 7} {\dot{ε}}^{2}, \\ R^{2} = 0.9911 . \end{array}

(4)

The relationship between peak stress and three characteristic stresses is illustrated in Figure 5b. The characteristic stress shows a logarithmic or S-shaped growth trend as the peak stress increases. Therefore, the logistics function is used for fitting, and the correlation among the peak stress and the three crack characteristic stresses' fitting functions is as follows:

\begin{array}{l} σ_{cc} = 12.18 - 8.6 / {1 + exp [(σ_{dp} - 21.55) / 6.15]}, R^{2} = 0.98543, \\ σ_{ci} = 21.73 - 12.91 / {1 + exp [(σ_{dp} - 28.95) / 8.15]}, R^{2} = 0.97385, \\ σ_{cd} = 34.35 - 31.38 / {1 + exp [(σ_{dp} - 24.34) / 11.28]}, R^{2} = 0.99717, \end{array}

(5)

where σ dp represents the dynamic compressive strength.

In addition, the relationship between dynamic compressive strength and static compressive strength is as follows (Li et al., 2023):

σ_{dp} = DIF (\dot{ε}) σ_{sp} .

(6)

Therefore, the three characteristic stress threshold prediction models of anthracite obtained using Equations ( 3-6) are as follows:

\begin{array}{l} σ_{cc} = 12.18 - 8.6 / {1 + exp [(σ_{dp} - 21.55) / 6.15]}, \\ σ_{ci} = 21.73 - 12.91 / {1 + exp [(σ_{dp} - 28.95) / 8.15]}, \\ σ_{cd} = 34.35 - 31.38 / {1 + exp [(σ_{dp} - 24.34) / 11.28]}, \\ σ_{dp} = σ_{sp} (1 - 3.42 \times 10^{- 3} \dot{ε} + 1.724 \times 10^{- 4} {\dot{ε}}^{2} - 4.466 \times 10^{- 7} {\dot{ε}}^{2}) . \end{array}

(7)

4 DISCUSSION

In this article, three crack characteristic stresses were identified. It is necessary to reflect the rationality of the determination of characteristic stress. Therefore, this section attempts to discuss the rationality of the determination of the characteristic stress from the perspective of damage evolution. In addition, combined with the behavior of characteristic stress, the crack extension mechanism is further discussed.

4.1 The rationality of anthracite characteristic stress identification

Characteristic stress is a vital aspect for characterizing the dynamic damage evolution in anthracite. Therefore, the rationality of the selection of the characteristic stresses is verified by establishing a dynamic damage constitutive model. It is assumed that the microelement consists of damaged and cohesive bodies in parallel (Shan et al., 2003), as illustrated in Figure 6a. Based on the equivalent strain assumption, the Drucker–Prager failure criterion is adopted as the microelement strength, and the dynamic damage constitutive model is as follows:

σ = E ε exp [- {(\frac{ε}{ε_{\max}})}^{m} \frac{1}{m}] + η \frac{d ε}{d t},

(8)

D = 1 - exp [- {(\frac{ε}{ε_{\max}})}^{m} \frac{1}{m}],

(9)

where m is the shape parameter, E is the sample elastic modulus,

η

is the viscosity coefficient, and D is the degree of damage.

The fitting effect is illustrated in Figure 6b. It is clear that the fitted curves are basically consistent with the actual curves, the fitting effect is good, and the model has strong applicability. According to Equation (9), the damage evolution process of anthracite is presented in Figure 7. The characteristic strain corresponding to the characteristic stress obtained by the test results is shown in Figure 7.

Depending on the characteristic stress, the pre-peak loading is broken down into four phases. The first is the compacted stage. The original microcracks within the specimen gradually close and gradually transition to the elastic phase. No new cracks are formed in this stage. The damage values of the five groups of strain rates in Figure 7 are also basically zero, and there is no obvious damage. Second, after loading to the crack initiation stress threshold, nascent cracks begin to develop within the sample, and these cracks show stable expansion. The damage increases steadily. The damage values under five groups of strain rates also show linear growth characteristics. Then, the unstable failure stage follows, and numerous cracks are produced within the sample in a short duration and eventually result in large cracks. The damage values of the five groups of strain rates also show characteristics of accelerated growth. In summary, the characteristic stress determined can effectively and accurately depict the damage evolution throughout the loading stage.

4.2 Strain rate effect of the crack propagation mechanism in the crack unstable propagation stage

According to the sample damage evolution in Figure 7, considerable damage within the sample mainly arises from the unstable crack extension phase before the failure. An increase in the dynamic loading rate causes the specimen to rapidly enter the elastic phase before the original crack has fully closed. The inertial effect promotes the rapid propagation of stress waves within the sample. This in turn causes a significant stress concentration phenomenon at the crack tip, more small-sized cracks are formed, the cracks develop and expand earlier, and finally the unstable expansion stage arises early. The proportion of the crack unstable extension phase in the overall damage accumulation process gradually increases. This phenomenon is particularly significant at high strain rates, indicating that an increase in the dynamic loading rate strongly contributes to crack extension behavior. Dynamic loading also leads to the dynamic strength enhancement effect. Due to the combined effect of the stress concentration effect and the dynamic strength enhancement effect at the crack tip under dynamic loading, the crack propagation speed within anthracite is usually faster than that under static loading. In addition, the interaction of merging, intersection, and branching between adjacent cracks when the crack propagates within the specimen leads to a more intricate crack propagation path and a faster propagation rate, which consequently influences the overall fracture behavior.

Crack propagation is closely related to the energy dissipation mechanism (Li et al., 2014; Xie et al., 2009). Based on this, the mean values of dissipated energy density, strain, and damage degree at the crack unstable propagation stage are analyzed in Figure 8. The strain, dissipated energy density, and damage increase, and the growth rate also increases in the crack unstable extension phase. It is also known that an increase of the strain rate leads to a gradual increase in crack propagation.

To further analyze the influence of the strain rate on the damage-driving mechanism in the crack unstable extension phase, the following characterization parameters are defined:

X = U_{D} / (D_{p} - D_{cd}),

(10)

Y = U_{D} / (ε_{p} - ε_{cd}),

(11)

where U D represents the dissipation energy density during the crack unstable extension phase. D p and D cd are the damage values under the peak stress and crack damage stress, respectively. The parameter X represents the dissipated energy required for the specimen to produce the same damage and the parameter Y is the dissipated energy required for the specimen to produce the same strain.

The change of the parameters X and Y with the strain rate is shown in Figure 9. The sample produces the same size strain, and more energy dissipation occurs within the sample at high strain rates. With the same degree of damage, more energy dissipation is also generated within the high strain rate specimen. This also shows that in the crack unstable extension phase, the increase in internal crack formation at a high strain rate is more intense and severe, and the overall damage is greater.

Exploration of the fracture surface morphology is useful in the analysis of the damage features and fracture mechanism of anthracite. Therefore, the fracture surface morphology was measured and analyzed using the 3D profile measuring instrument. The specific scanning process and measurement results are shown in Figure 10. A relatively complete fracture interface is selected for scanning, and the roughness of the defined area of a fixed size is measured. Five defined areas of the same size are selected under each section. The S dr index is used to define the cross-section roughness, and S dr is as follows:

S_{dr} = \frac{1}{A} {\int \int}_{A} [\sqrt{1 + {(\frac{\partial z (x, y)}{\partial x})}^{2} + {(\frac{\partial z (x, y)}{\partial y})}^{2}} - 1] d x d y,

(12)

where S dr is the area ratio of the development interface. It represents how much the definition region has increased in size relative to the area of the definition region. A is the measurement region, z ( x, y) is the height value of the measured point, and x and y are the measurement point positions.

As depicted in Figure 10, the roughness of the section is estimated by determining the mean value of roughness of five measurements. Sdr increases as the strain rate increases, signifying a gradual increase in the specimen section roughness. From the three-dimensional cloud diagram of the fracture interface, it can be seen that the cracks have obvious branches and intersections along the dynamic loading direction. The crack path becomes more intricate. The more uneven the distribution of mineral particles on the fracture surface, the more severe the fluctuation of the fracture surface and the more complex the section morphology. The increase of the complexity and roughness of the fracture morphology proves that with an increase in the strain rate, the specimen undergoes more severe crack unstable propagation, resulting in huge damage before the specimen is destroyed, and finally makes specimen is completely damaged.

4.3 Discussion of the strain rate effect on damage evolution

It can be found from Figures 8 and 9 that the damage evolution of coal is affected by the strain rate. In the unstable crack propagation stage, with an increase in the strain rate, the damage evolution process becomes more intense. However, in the viscoelastic damage constitutive Equation (8), the viscous stress is obviously a variable related to the strain rate, but the influence of the strain rate is not directly reflected in the damage, and the influence of the strain rate on the damage evolution cannot be directly reflected. It can be seen from the damage expression (9) that the only parameter that changed is the shape parameter m. Therefore, it is speculated that the shape parameter is a characteristic parameter affected by the strain rate. Combined with Figures 9 and 10, it can be found that in the unstable crack propagation stage, the energy consumed to produce the same damage at different strain rates is different, which also shows that there is a certain correlation between damage and the strain rate from the side. This paper focuses on the identification of characteristic stress, and only a brief discussion is presented on the relationship between damage evolution and the strain rate.

5 CONCLUSIONS

In this article, the characteristic stress threshold evolution law of anthracite under different impact disturbance intensities is analyzed, and the prediction model of the characteristic stress threshold considering the strain rate effect is established. The applicability of characteristic stress is discussed. The principal findings are as follows:

1.
The crack characteristic stress is significantly affected by the strain rate. The three characteristic stress thresholds are positively correlated with a change in the strain rate, but the proportion of the three eigenstresses to the crest stress decreases. In addition, a prediction model of crack characteristic stress under dynamic load is established as follows:

$\begin{array}{l} σ_{cc} = 12.18 - 8.6 / {1 + exp [(σ_{dp} - 21.55) / 6.15]}, \\ σ_{ci} = 21.73 - 12.91 / {1 + exp [(σ_{dp} - 28.95) / 8.15]}, \\ σ_{cd} = 34.35 - 31.38 / {1 + exp [(σ_{dp} - 24.34) / 11.28]}, \\ σ_{dp} = σ_{sp} (1 - 3.42 \times 10^{- 3} \dot{ε} + 1.724 \times 10^{- 4} {\dot{ε}}^{2} - 4.466 \times 10^{- 7} {\dot{ε}}^{2}) . \end{array}$
2.
The crack characteristic stress determined yields a good description of the anthracite damage evolution process in the dynamic loading stage. Moreover, the increase of the strain rate contributed markedly to the anthracite crack propagation behavior. Under the combined action of the stress concentration effect at the crack tip and the dynamic strength enhancement effect, anthracite generates more energy dissipation under the same deformation, because of which the internal cracks undergo more severe and unstable expansion, resulting in huge damage. Finally, the crack propagation rate increases considerably, and the crack extension trail of the section becomes more complex, which in turn affects the overall fracture behavior of anthracite.

ACKNOWLEDGMENTS

This study was funded by the National Natural Science Foundation of China (No. 51934007, No. 12072363, No. 12372373, No. 52174091, and No. 52104234).

CONFLICT OF INTEREST STATEMENT

The authors declare no conflicts of interest.

Biographies

Yun Bai is a doctoral student at the State Key Laboratory of Intelligent Construction and Healthy Operation and Maintenance of Deep Underground Engineering, China University of Mining and Technology. His research interests mainly focus on the mechanism and early warning of underground dynamic disasters.
Feng Gao is a professor and doctoral supervisor at the State Key Laboratory of Intelligent Construction and Healthy Operation and Maintenance of Deep Underground Engineering, China University of Mining and Technology. His research work focuses on rock damage and fracture, fractal rock mechanics, deep rock engineering and disaster, multi-field coupling theory, and computational mechanics.

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