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As one of the most catastrophic dynamic hazards in underground coal mines, coal bursts have been a major safety concern around the world for many years. Although the coal bursts can occur in all cases of hard to soft coal if the right stress environment is created, the occurrence of coal bursts is closely related to the intrinsic mechanical properties of coal, such as the bursting proneness. In this study, a total of 27 coal specimens are selected in the open literature studies to obtain a group of fundament data, such as the mechanical parameters, four bursting proneness indices, stress-strain curves, and their geological conditions where the specimens were taken. The relationship between bursting proneness indices and the cohesion of the coal specimens is established by numerically fitting the stress-strain curves and theoretically deduction. By taking into account the coal heterogeneity, eight probability distribution functions are employed to assignment nonuniform cohesion to the numerical model and to study the influence of heterogeneity on bursting proneness. The results reveal that the coal cohesion, which combines the common advantages of the four proneness indices, can be used as bursting proneness index. In the research of heterogeneity, the coal bursting proneness will decrease with the increasing of cohesion scatter degree. The larger the cohesion scatter degree increase is, the lower the bursting proneness will be. The failure of coal specimen is more and more severe with the decrease of cohesion scatter degree. In addition, this paper provides two methods for assigning heterogeneous parameters to the numerical model. The contours of shear strain rate and plastic state between homogeneous and heterogeneous coal specimens are compared to study the failure types of coal specimens and to reveal the mechanism of violent failure in coal bursts.

As one of the most catastrophic dynamic hazards in an underground coal mine, coal bursts have been a major safety concern around the world for many years [

There are a large number of factors associated with the occurrence of coal bursts. However, although the coal bursts can occur in all cases of hard to soft coal if the right stress environment is created, the occurrence of coal bursts is closely related to the intrinsic mechanical properties of coal, such as the bursting proneness [_{c}), elastic strain energy (_{ET}), bursting energy (_{E}), and dynamic fracturing duration (_{t}) [_{ET}) to evaluate the ability of coal to burst [_{E}) be used as the main index in assessing the burst prone [_{ET}) to evaluate high-potential rockburst in a tunnel with overburden depth greater than 400 m [

As Kidybinski stated, the bursting proneness was the natural ability of coal to store and release elastic strain energy. Therefore, the bursting proneness of coal is an issue of distribution between energy storage and release. To gain in-depth understanding of the relationship between the bursting proneness and coal bursts, apart from these four indices, numerous studies have been carried out to obtain novel bursting proneness indices based on the investigation of energy storage and release, e.g., surplus energy [

In fact, the bursting proneness of coal is closely related to the internal composition and microstructure of coal [

According to the review of literature studies about the bursting proneness indices and relationship between bursting proneness and microstructure of material, it can be seen that the current studies of bursting proneness are mainly focused on the proneness index definition and application. However, there is limited information available in the open literature about the relationship between the indices _{c}, _{ET}, _{E}, and _{t}. In fact, there is a common mechanical parameter among the four indices, which is of great significance to assessing the bursting proneness. In addition, although the effectiveness of material composition and microstructure on the bursting proneness has been investigated for many years, there is little information about the numerical method for simulating the distribution of heterogeneous parameters, failure types of heterogeneous coal specimens, and its influence on the mechanism of violent failure in coal bursts. In this study, a total of 27 coal specimens are selected in the open literature studies to obtain a group of fundament data, such as the mechanical parameters, four bursting proneness indices, stress-strain curves, and their geological conditions where the specimens were taken. The relationship between bursting proneness indices and the cohesion of the coal specimens is established by numerically simulation and theoretically deduction. The reason that cohesion can be used as an index to assess the bursting proneness is analyzed based on the in-depth understanding on the common advantages of the four proneness indices. By taking into account the coal heterogeneity, the nonuniform distribution of cohesion in coal specimens is mapped by eight commonly used probability distribution functions to study the influence of heterogeneity on bursting proneness. In addition, the internal and external factors contributing to coal bursts are analyzed to reveal the mechanism of occurrence of coal bursts.

Bursting proneness is an inherent property of coal. It is used to evaluate the risk of coal bumps by assessing the energy accumulation capability of coal. According to the current standard in China, bursting proneness is classified into three levels: high, low, and none [_{ET}), the bursting energy index (_{E}), the duration of dynamic fracture (_{t}), and uniaxial compressive strength (_{c}) [

As shown in Figure _{c}) is the peak stress of stress and strain curve (Figure _{ET}) (Figure _{E}) (Figure _{t}) (Figure _{t} represents the speed of accumulated energy release, the smaller the value of _{t} is, the higher the bursting proneness of coal will be. Table _{c}, _{ET}, _{E}, and _{t}.

Four indices obtained from the uniaxial compressive test and employed to classify the bursting proneness of coal: (a) uniaxial compressive strength (_{c}); (b) elastic strain energy index (_{ET}); (c) bursting energy index (_{E}); (d) duration of dynamic fracture (_{t}).

Classification of the bursting proneness of coal and indices according to current standard in China [

Indices | No bursting proneness | Low bursting proneness | High bursting proneness |
---|---|---|---|

Elastic energy index (_{ET}) | _{ET} < 2 | 2 ≤ _{ET} < 5 | _{ET} ≥ 5 |

Bursting energy index (_{E}) | _{E} < 1.5 | 1.5 ≤ _{E} < 5 | _{E} ≥ 5 |

Dynamic rupture period ( | 50 < | ||

Uniaxial compressive strength (_{c}) | _{c} < 7 | 7≤_{c} < 14 | _{c} ≥ 14 |

A total of 27 coal specimens are taken from the open literature to establish a group of fundament data of the stress-strain curves and bursting proneness indices of specimens [_{c}. The elastic strain energy index (_{ET}), the bursting energy index (_{E}), and the duration of dynamic fracture (_{t}) of these coal specimens are also listed. The comprehensive results of bursting proneness are determined according to the current standard in Table

Coal specimens and values of bursting proneness indices accompanied by geological conditions of the sites from which the specimens were taken.

Specimens | Bursting proneness | Coal mine | Location | Geological conditions | Reference | ||||
---|---|---|---|---|---|---|---|---|---|

_{c} | _{ET} | _{E} | _{t} | Results | |||||

No. 1 | 27.26 | 9.36 | 12.02 | 12.11 | High | Zhaogu | Henan, China | Large amounts of faults and subsidiary fold structures | Su et al. [ |

No. 2 | 25.13 | 8.12 | 7.49 | 23.75 | High | Tangshan | Hebei, China | Existence of primary folds and large amounts of island longwall mining face | Pan et al. [ |

No. 3 | 24.58 | 7.52 | 7.79 | 31.56 | High | Tangshan | Hebei, China | Existence of primary folds and large amounts of island longwall mining face | Pan et al. [ |

No. 4 | 22.70 | 7.70 | 9.82 | 45.78 | High | Mentougou | Beijing, China | Existence of thrust faults and overturned rock strata. Fold structures such as syncline and anticline existed alternately | Okubo et al. [ |

No. 5 | 21.24 | 7.84 | 5.98 | 29.51 | High | Tangshan | Hebei, China | Existence of primary folds and large amounts of island longwall mining face | Pan et al. [ |

No. 6 | 21.08 | 7.25 | 13.44 | 22.70 | High | Mentougou | Beijing, China | Existence of thrust faults and overturned rock strata. Fold structures such as syncline and anticline existed alternately | Okubo et al. [ |

No. 7 | 20.68 | 7.51 | 6.91 | 56.27 | High | Sanjiaohe | Shanxi, China | A series of developed folds and subsidiary faults existence | Li et al. [ |

No. 8 | 20.39 | 6.12 | 13.26 | 51.68 | High | Pingdingshan | Henan, China | A series of faults and folds existence | Su et al. [ |

No. 9 | 20.33 | 7.02 | 4.93 | 29.87 | High | Mentougou | Beijing, China | Existence of thrust faults and overturned rock strata. Fold structures such as syncline and anticline existed alternately | Okubo et al. [ |

No. 10 | 20.24 | 6.97 | 5.80 | 28.78 | High | Pingdingshan | Henan, China | Large amounts of faults and subsidiary fold structures | Su et al. [ |

No. 11 | 19.53 | 7.53 | 5.19 | 49.79 | High | Zhaogu | Henan, China | Large amounts of faults and subsidiary fold structures | Su et al. [ |

No. 12 | 18.65 | 6.91 | 4.26 | 45.41 | High | Zhangcun | Shanxi, China | Coal seams existed in a monocline fold structure | Su et al. [ |

No. 13 | 16.71 | 6.34 | 3.35 | 38.88 | High | Qianqiu | Henan, China | Large synclines, reverse fault structures, and extreme thick roof strata | Pan et al. [ |

No. 14 | 16.65 | 6.29 | 5.12 | 42.32 | High | Chengjiao | Henan, China | Wide and gentle slope fold structures accompanied by a number of faults | Su et al. [ |

No. 15 | 16.09 | 5.85 | 3.59 | 25.32 | High | Qianqiu | Henan, China | Large synclines, reverse fault structures, and extreme thick roof strata | Pan et al. [ |

No. 16 | 15.82 | 5.45 | 2.87 | 50.88 | High | Qianqiu | Henan, China | Large synclines, reverse fault structures, and extreme thick roof strata | Pan et al. [ |

No. 17 | 14.88 | 5.83 | 4.62 | 58.19 | High | Zhangcun | Shanxi, China | Coal seams existed in a monocline fold structure | Su et al. [ |

No. 18 | 12.40 | 4.60 | 2.40 | 167.23 | Low | Pingdingshan | Henan, China | A series of faults and folds existence | Su et al. [ |

No. 19 | 11.46 | 4.78 | 3.59 | 26.45 | Low | Chengjiao | Henan, China | Wide and gentle slope fold structures accompanied by a number of faults | Su et al. [ |

No. 20 | 11.03 | 4.03 | 3.98 | 187.85 | Low | Jixi | Heilongjiang, China | Large amounts of fault structures | Zhang et al. [ |

No. 21 | 10.41 | 2.73 | 2.01 | 267.32 | Low | Pingdingshan | Henan, China | A series of faults and folds existence | Su et al. [ |

No. 22 | 9.08 | 3.51 | 2.26 | 368.00 | Low | Chengjiao | Henan, China | Wide and gentle slope fold structures accompanied by a number of faults | Su et al. [ |

No. 23 | 8.67 | 2.97 | 2.62 | 110.45 | Low | Pingdingshan | Henan, China | A series of faults and folds existence | Su et al. [ |

No. 24 | 6.87 | 2.48 | 0.64 | 548.20 | None | Pingdingshan | Henan, China | A series of faults and folds existence | Su et al. [ |

No. 25 | 6.64 | 1.99 | 1.88 | 517.06 | None | Chengjiao | Henan, China | Wide and gentle slope fold structures accompanied by a number of faults | Su et al. [ |

No. 26 | 6.07 | 0.89 | 1.29 | 514.40 | None | Chengjiao | Henan, China | Wide and gentle slope fold structures accompanied by a number of faults | Su et al. [ |

No. 27 | 4.98 | 1.02 | 1.23 | 426.13 | None | Chengjiao | Henan, China | Wide and gentle slope fold structures accompanied by a number of faults | Su et al. [ |

The stress-strain curves of these 27 coal specimens in the uniaxial compressive tests were also obtained, as shown in Figure

The stress-strain curves of 27 coal specimens grouped based on coal mines where they are taken: (a) Zhaogu coal mine; (b) Tangshan coal mine; (c) Mentougou coal mine; (d) Sanjiaohe coal mine; (e) Pingdingshan coal mine; (f) Zhangcun coal mine; (g) Qianqiu coal mine; (h) Chengjiao coal mine; (i) Jixi coal mine.

Although the stress-strain curves and bursting proneness are gained in the uniaxial compressive tests, it is impossible to obtain other important parameters of the specimens such as Young’s modulus, cohesion, friction angle, and tensile strength through the same tests. In this study, the 27 stress-strain curves will be numerically fitted to obtain these parameters.

According to the ISRM-suggested methods for rock mechanics, a cylinder with diameter of 50 mm and height of 100 mm is built in ANSYS (Analysis Systems). This ANSYS model is then transferred into FLAC^{3D} (Fast Lagrangian Analysis of Continua in 3-Dimensions) code as the numerical model for simulation, as shown in Figure ^{3D} mesh elements with size of 2.2 mm × 1.4 mm × 2.5 mm are presented.

Numerical model of coal specimen and the uniaxial compressive boundary condition.

The failure criterion is the strain-softening model which is based on the Mohr–Coulomb model with nonassociated shear and associated tension flow rules. In this model, the cohesion, friction, and tensile strength may soften after the onset of plastic yield by a user-defined piecewise linear function. Since Young’s modulus, Poisson’s ratio, cohesion, friction angle, and tensile strength of the 27 coal specimens were not given in the cited literature studies, the initial mechanical parameters are empirically determined according to the UCS, as listed in Table

Empirical parameters initially used in the numerical simulation [

UCS (MPa) | Young’s modulus (GPa) | Poisson’s ratio | Cohesion (MPa) | Friction angle (°) | Tensile strength (MPa) |
---|---|---|---|---|---|

4.90–10.0 | 0.29–2.45 | 0.1–0.30 | 0.98–9.81 | 19–40 | 0.24–5.79 |

9.81–15.7 | 2.45–6.37 | 0.1–0.30 | 1.96–3.92 | 28–35 | 1.47–2.45 |

19.6–49.0 | 8.83–22.60 | 0.1–0.35 | 3.92–5.88 | 35–45 | 1.96–9.81 |

A coordinate system is selected with the ^{−5} mm/step is applied in the

In this study, the numerical simulation is performed according to the following procedure:

Step 1: numerical fitting of the stress-strain curves of the 27 coal specimens with uniform distribution of mechanical parameters.

Step 2: to obtain the mechanical parameters of the 27 coal specimens, e.g., Young’s modulus, Poisson’s ratio, cohesion, friction angle, tensile strength, and uniaxial compressive strength.

Step 3: study on heterogeneity by probability distribution.

Step 4: nonuniform parameter assignment to the numerical model.

Step 5: study of the influence of nonuniform distribution of mechanical parameters on UCS of coal specimen.

Step 6: study of coal bursting proneness and failure.

Figure ^{2}, and the distribution interval is (0, 1). The larger the ^{2} is, the better the fitting degree is:^{2} is further derived, and the results are as follows:_{i} is the real observation value, the stress value of the test curve is substituted in the calculation,

Numerical fitted stress-strain and unload curves of the 27 coal specimens by the uniaxial compressive test: (a) No .1 to No. 9 coal specimens. (b) No. 10 to No. 18 coal specimens. (c) No. 19 to No. 27 coal specimens.

It is revealed that numerical fitted stress-strain curves can be used to study the mechanical parameters of coal specimens as most of fitting coefficients ^{2} between test and numerical results are greater than 98%. To calculate the dissipated plastic energy before 80–90% of the coal peak strength, Figure _{C}, _{ET}, and _{E} indexes, and low bursting proneness according to _{t} indexes; No. 16 and No. 17 coal are determined as bursting proneness according to _{C} and _{ET} indexes, and low impact tendency according to _{E} and _{t} indexes. There are still similar problems in the data in the table, which are not listed here one by one. Therefore, the existing identification index has the problem of inconsistent identification results.

The mechanical parameters determined by numerically fitting the stress-strain curve in the uniaxial compressive test.

Specimens | Young’s modulus (GPa) | Poisson’s ratio | UCS (MPa) | Cohesion | Friction angle | Tensile strength | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Original value (MPa) | Softening rate (%) | Residual value (MPa) | Original value (°) | Softening rate (%) | Residual value (°) | Original value (MPa) | Softening rate (%) | Residual value (MPa) | ||||

No. 1 | 3.69 | 0.22 | 27.26 | 6.82 | 0.20 | 1.00 | 37 | 0.20 | 35 | 2.87 | 0.20 | 0 |

No. 2 | 3.50 | 0.22 | 25.13 | 6.20 | 5.20 | 0.30 | 37 | 5.20 | 35 | 2.89 | 5.20 | 0 |

No. 3 | 3.65 | 0.22 | 24.58 | 6.07 | 2.80 | 0.50 | 37 | 2.80 | 35 | 2.43 | 2.80 | 0 |

No. 4 | 3.58 | 0.22 | 22.70 | 5.58 | 1.80 | 1.50 | 37 | 1.80 | 35 | 2.43 | 1.80 | 0 |

No. 5 | 3.93 | 0.22 | 21.24 | 5.25 | 4.15 | 0.30 | 37 | 4.15 | 33 | 2.16 | 4.15 | 0 |

No. 6 | 3.57 | 0.22 | 21.08 | 4.73 | 2.10 | 0.51 | 41 | 2.10 | 39 | 2.18 | 2.10 | 0 |

No. 7 | 3.82 | 0.23 | 20.68 | 5.10 | 7.00 | 0.50 | 37 | 7.00 | 35 | 2.21 | 7.00 | 0 |

No. 8 | 3.15 | 0.22 | 20.39 | 5.05 | 4.70 | 0.50 | 37 | 4.70 | 35 | 2.20 | 4.70 | 0 |

No. 9 | 3.81 | 0.22 | 20.33 | 5.02 | 2.15 | 0.50 | 37 | 2.15 | 35 | 2.10 | 2.15 | 0 |

No. 10 | 3.08 | 0.22 | 20.24 | 4.98 | 4.05 | 1.50 | 37 | 4.05 | 35 | 2.03 | 4.05 | 0 |

No. 11 | 3.22 | 0.22 | 19.53 | 4.81 | 1.20 | 0.50 | 38 | 1.20 | 35 | 2.07 | 1.20 | 0 |

No. 12 | 3.27 | 0.25 | 18.65 | 4.82 | 5.00 | 0.30 | 35 | 5.00 | 31 | 1.91 | 5.00 | 0 |

No. 13 | 3.02 | 0.22 | 16.71 | 4.02 | 3.50 | 0.50 | 37 | 3.50 | 35 | 1.78 | 3.50 | 0 |

No. 14 | 2.86 | 0.22 | 16.65 | 4.10 | 4.20 | 0.10 | 37 | 4.20 | 35 | 2.02 | 4.20 | 0 |

No. 15 | 2.47 | 0.22 | 16.09 | 3.97 | 4.32 | 0.50 | 37 | 4.32 | 35 | 1.77 | 4.32 | 0 |

No. 16 | 2.90 | 0.22 | 15.82 | 3.89 | 5.80 | 0.28 | 37 | 5.80 | 35 | 1.53 | 5.80 | 0 |

No. 17 | 3.24 | 0.25 | 14.88 | 3.85 | 2.40 | 0.30 | 35 | 2.40 | 31 | 1.54 | 2.40 | 0 |

No. 18 | 2.56 | 0.23 | 12.40 | 3.03 | 3.50 | 0.50 | 35 | 3.50 | 32 | 1.24 | 3.50 | 0 |

No. 19 | 2.33 | 0.28 | 11.46 | 3.63 | 5.70 | 0.10 | 25 | 5.70 | 22 | 1.18 | 5.70 | 0 |

No. 20 | 1.87 | 0.24 | 11.03 | 3.10 | 5.00 | 0.60 | 31 | 5.00 | 27 | 1.14 | 5.00 | 0 |

No. 21 | 1.83 | 0.24 | 10.41 | 2.79 | 7.50 | 0.15 | 33 | 7.50 | 29 | 1.01 | 7.50 | 0 |

No. 22 | 1.78 | 0.27 | 9.08 | 2.20 | 1.90 | 0.80 | 35 | 1.90 | 31 | 0.93 | 1.90 | 0 |

No. 23 | 1.34 | 0.25 | 8.67 | 2.39 | 4.50 | 0.30 | 32 | 4.50 | 28 | 0.89 | 4.50 | 0 |

No. 24 | 2.02 | 0.22 | 6.87 | 1.66 | 2.50 | 0.20 | 37 | 2.50 | 35 | 0.75 | 2.50 | 0 |

No. 25 | 1.78 | 0.22 | 6.64 | 1.47 | 3.20 | 0.01 | 41 | 3.20 | 39 | 0.70 | 3.20 | 0 |

No. 26 | 1.77 | 0.22 | 6.07 | 1.46 | 4.00 | 0.01 | 37 | 4.00 | 35 | 0.67 | 4.00 | 0 |

No. 27 | 1.16 | 0.22 | 4.98 | 1.19 | 3.55 | 0.10 | 37 | 3.55 | 35 | 0.55 | 3.55 | 0 |

As shown in Figure

The functional equations of stress-strain curve before peak strength, the unload curve before 80–90% of peak strength, and the curve after peak strength for No. 12 coal specimen.

In this study, the functional equations of these three curves can be assumed as_{1}, _{2}, _{3}, _{2}, and _{3} are the slope and intercept of these three linear curves. By calculating the fitted stress-strain curves, the values of slope and intercept for the 27 coal specimens are listed in Table

Parameters of _{1}, _{2}, and _{3} of stress-strain curve before peak strength, unload curve before 80–90% of peak strength, and curve after peak strength of the 27 coal specimens.

Specimens | _{1} | _{2} | _{3} | UCS (MPa) |
---|---|---|---|---|

No. 1 | 3.69 | 4.80 | 79.02 | 27.26 |

No. 2 | 3.50 | 4.24 | 73.12 | 25.13 |

No. 3 | 3.65 | 4.43 | 70.35 | 24.58 |

No. 4 | 3.58 | 6.67 | 55.33 | 22.70 |

No. 5 | 3.93 | 4.43 | 50.13 | 21.24 |

No. 6 | 3.57 | 4.32 | 35.16 | 21.08 |

No. 7 | 3.82 | 4.53 | 30.65 | 20.68 |

No. 8 | 3.15 | 3.66 | 35.93 | 20.39 |

No. 9 | 3.81 | 4.58 | 28.33 | 20.33 |

No. 10 | 3.08 | 3.73 | 27.63 | 20.24 |

No. 11 | 3.22 | 3.86 | 22.37 | 19.53 |

No. 12 | 3.27 | 3.74 | 21.66 | 18.65 |

No. 13 | 3.02 | 3.73 | 20.98 | 16.71 |

No. 14 | 2.86 | 3.73 | 14.99 | 16.65 |

No. 15 | 2.47 | 2.89 | 20.27 | 16.09 |

No. 16 | 2.90 | 3.43 | 18.17 | 15.82 |

No. 17 | 3.24 | 3.79 | 15.09 | 14.88 |

No. 18 | 2.56 | 3.51 | 14.20 | 12.40 |

No. 19 | 2.33 | 3.00 | 13.72 | 11.46 |

No. 20 | 1.87 | 2.34 | 10.52 | 11.03 |

No. 21 | 1.83 | 2.50 | 5.02 | 10.41 |

No. 22 | 1.78 | 2.46 | 3.37 | 9.08 |

No. 23 | 1.34 | 2.32 | 5.53 | 8.67 |

No. 24 | 2.02 | 2.98 | 1.56 | 6.87 |

No. 25 | 1.78 | 2.84 | 1.80 | 6.64 |

No. 26 | 1.77 | 2.62 | 2.49 | 6.07 |

No. 27 | 1.16 | 2.46 | 1.97 | 4.98 |

Figure _{1}, _{2}, _{3}, and the UCS. It is revealed that the _{1} and _{2} approximately increase linearly with the increase of the UCS while an exponential relation between _{3} and UCS can be assumed. Therefore, the relational equation between the parameters of _{1}, _{2}, _{3}, and UCS can be assumed as_{1}, _{2}, _{3}, _{1}, _{2}, and _{3} are constant coefficients of these three equations; _{c} is the UCS of coal specimens.

The variation of _{1}, _{2}, and _{3} versus the UCS of 27 coal specimens.

According to the coefficients of curves in Figure

According to the numerical parameters of these coal specimens in Table

The relationship between bursting proneness indices and cohesion: (a) Index _{c} (UCS). (b) Index _{ET}. (c) Index _{E} (d) Index _{t} and postpeak strain.

Figure _{c} and cohesion. It is revealed that the index _{c} linearly increases with the increase of cohesion. According to the theory of Mohr–Coulomb criterion, the UCS or index _{c} is related to cohesion and friction angle by

According to equation (_{c} can be obtained. Theoretical and numerical results are in good agreement, as shown in Figure

Figure _{ET} and cohesion. It can be seen from the numerical results that the index _{ET} approximately linearly increases with the increase of cohesion.

According to the current standard in China, the index _{ET} is calculated as a ratio of the elastic energy accumulated and the dissipated plastic energy before 80–90% of the coal peak strength [

Substituting equations (_{ET} is calculated as

Substitution of equation (_{ET} and cohesion as

Therefore, if the friction angle _{ET} and cohesion holds. This theoretical relation is identical with the numerical results, as shown in Figure

Figure _{E} and cohesion. It can be seen that the index _{E} increases with the increase of cohesion. The index _{E} gradually increases as the cohesion increases from 1.5 to 3.5 MPa, and then it sharply increases when the cohesion is greater than 3.5–4.0 MPa.

According to the current standard in China, the bursting energy index (_{E}) (Figure

Substituting equations (_{E} is calculated as

Substitution of equation (_{E} and cohesion as

If the friction angle _{E} and cohesion is obtained. This theoretical relation is consistent with the numerical results, as shown in Figure

Figure _{t} and cohesion. It can be seen that the index _{t} decreases with the increase of cohesion. The index _{t} sharply decreases when the cohesion increases from 1.5 to 3.5 MPa while it gradually decreases when the cohesion is over 3.5–4.0 MPa.

According to Figure _{t}) are equivalent to illustrate the span from peak strength to complete failure of coal specimen. Therefore, the dynamic fracturing duration (_{t}) can be replaced by the postpeak strain in this part to study the relation between index _{t} and cohesion. The postpeak strain can be calculated as

Substitution of equation (

If the friction angle

Coal heterogeneity is a major factor to influence its mechanical behaviors, such as the UCS, shear strength, failure behavior, crack evolution, and bursting proneness [

In this study, the influence of nonuniform distribution of cohesion on the coal strength (UCS) is mainly analyzed. Eight types of probability distribution functions are employed to assignment cohesion to the numerical model. Table

Eight probability density functions employed to simulate the heterogeneous distribution of cohesions in the numerical model.

Distribution | Probability density function ( | Mathematical expectation | Variance | Standard deviation |
---|---|---|---|---|

Weibull | ||||

Normal | ||||

Rayleigh | ||||

Chi-square | ||||

Student’s | ||||

Exponent | ||||

Cauchy | — | — | — | |

Fisher |

The No. 12 coal specimen is taken again as an example in this part. The cohesion of this specimen is 4.82 MPa under homogeneous conditions, as listed in Table

Values of standard deviation of the eight probability distributions under eight cohesion scatter intervals.

Distribution | Cohesion intervals (MPa) | |||||||
---|---|---|---|---|---|---|---|---|

0.82–8.82 | 1.32–8.32 | 1.82–7.82 | 2.32–7.32 | 2.82–6.82 | 3.32–6.32 | 3.82–5.82 | 4.32–5.32 | |

Weibull | 2.78 | 2.39 | 2.02 | 1.66 | 1.31 | 0.98 | 0.64 | 0.32 |

Normal | 2.79 | 2.41 | 2.05 | 1.69 | 1.34 | 0.99 | 0.66 | 0.33 |

Rayleigh | 2.57 | 2.24 | 1.91 | 1.58 | 1.26 | 0.94 | 0.62 | 0.31 |

Chi-square | 2.64 | 2.32 | 2.00 | 1.66 | 1.33 | 0.99 | 0.66 | 0.33 |

Student’s | 2.77 | 2.41 | 2.06 | 1.70 | 1.35 | 1.00 | 0.66 | 0.33 |

Exponent | 2.27 | 2.02 | 1.75 | 1.47 | 1.19 | 0.89 | 0.60 | 0.30 |

Cauchy | 2.56 | 2.33 | 2.03 | 1.70 | 1.35 | 1.01 | 0.67 | 0.33 |

Fisher | 2.38 | 2.18 | 1.91 | 1.62 | 1.31 | 0.99 | 0.66 | 0.33 |

Since the coal specimen is an axisymmetric cylinder which is obtained by laboratory drilling, the strength of the specimen center is generally greater than that of the specimen edge. Therefore, nonuniform cohesion will be assigned to the numerical model from the center to edge of cross section according to the above eight distribution functions. Figure

Nonuniform cohesion assignment results in coal specimen under eight cohesion scatter intervals: (a) cohesion interval of 0.82–8.82 MPa, (b) cohesion interval of 1.32–8.32 MPa, (c) cohesion interval of 1.82–7.82 MPa, (d) cohesion interval of 2.32–7.32 MPa, (e) cohesion interval of 2.82–6.82 MPa, (f) cohesion interval of 3.32–6.32 MPa, (g) cohesion interval of 3.82–5.82 MPa, and (h) cohesion interval of 4.32–5.32 MPa.

Since the UCS is one of the indices to evaluate the bursting proneness of coal, it is selected to study the bursting proneness in this part. Figure

Stress-strain curves of No. 12 coal specimen under nonuniform cohesion distribution in the model with eight different scatter intervals: (a) cohesion interval of 0.82–8.82 MPa, (b) cohesion interval of 1.32–8.32 MPa, (c) cohesion interval of 1.82–7.82 MPa, (d) cohesion interval of 2.32–7.32 MPa, (e) cohesion interval of 2.82–6.82 MPa, (f) cohesion interval of 3.32–6.32 MPa, (g) cohesion interval of 3.82–5.82 MPa, and (h) cohesion interval of 4.32–5.32 MPa.

Variation of UCS of coal specimen versus the standard deviation.

In addition, as shown in Figure

In this study, shear strain rate (SSR) is employed to study the failure characteristics of coal specimens [^{3D} [^{3D} element variable in which the maximum SSR will return at each calculation step. The contours of SSR represent the most likely failure zone in which both the tensile and shear failure may occur. Figure

Eight groups of stress-strain curves of coal specimen with nonuniform cohesion assignment by Weibull probability distribution.

Comparison between the tests images and numerical results in terms of failure characteristics of coal specimens with bursting proneness: (a) high bursting proneness; (b) low bursting proneness.

It can be revealed that the current indices used to identify bursting proneness have their own advantages and limitations. The index _{c} can be directly used to determine bursting proneness. However, the released strain energy and the accumulated strain energy in the compressive test are not considered in this index. Although both _{E} and _{ET} are used to assess the bursting proneness with respect to the strain energy, _{ET} focuses on the capacity of coal to absorb external inputs of energy before it achieves peak strength, while _{E} considers not only the accumulated elastic energy before peak strength but also the energy released after the peak strength. The index _{t} can reflect the speed of energy release, but it is not easy to be captured in the compressive test.

Su studied the influence of crystal size of material on the bursting proneness and concluded that the shear deformation associated with cohesion will influence the bursting proneness [

According to the numerical and theoretical results in Figure _{c} and _{ET} and the cohesion (see Figures _{E} and _{t} have gradual and sharp variation stages with the variation of strain. The power functional equation between these two indices and cohesion can reflect this nonlinear characteristic (see Figures

Due to the inconsistency of identification results in the existing bursting proneness indexes, each identification index has certain limitations. Therefore, a new identification index is needed. According to the data given in Tables

Cohesion used to identify the bursting proneness of coal.

Indices | No bursting proneness | Low bursting proneness | High bursting proneness |
---|---|---|---|

Cohesion ( | 1.5 ≤ |

Figure ^{3D} when the cohesion is uniformly distributed in coal specimens. The contours of SSR gradually transform from two zones to one zone when the cohesion changes from high to low. The shear strain zones distribute across the coal specimens when reaching the postpeak strength. In addition, the SSR of high bursting proneness coal is greater than that for the low or none bursting proneness coals. Since the bursting proneness can be identified by cohesion, the subfigures in Figure

Contours of SSR and plastic zones calculated by FLAC^{3D} for coal specimens with different cohesion: (a) contours of SSR; (b) plastic zones.

The appearance of contours of SSR does not necessarily mean that the shear failure will occur along the zones. The contours of plastic state after the peak strength are calculated to further study the failure characteristics. As shown in Figure

As shown in Figure _{0} is set as 4.82 MPa for the cohesion of this specimen. Figure

Random assignment of cohesion in the numerical model by Weibull function with different parameter

Figure

Stress-strain curves of No. 12 coal specimen under random assignment of cohesion by Weibull function.

Plastic strain contour of No. 12 coal specimen under random assignment of cohesion by Weibull function.

By comparing the stress-strain curves of Figures

Comparison of model failure pattern between the method of parameter assignment from center to edge and random assignment: (a) assignment from center to edge; (b) random assignment.

Figure

As discussed previously in this paper, coal bursts are closely related to the intrinsic bursting proneness of coal [

In conclusion, a coal burst is the interactional results of bursting proneness, unstable geological structure, extreme thick roof strata, and island longwall mining face. In addition, due to the greater cohesion between the particles in coal specimens, coal with high bursting proneness provides an ideal internal environment for energy accumulation. To weaken this environment, the technologies including coal seam softening by water injection, borehole drilling in front of mining face, and advanced blasting in coal seam are wildly applied in coal mines to eliminate the bursting proneness [

Relationship between the coal bursting proneness and coal homogeneity and heterogeneity are studied in this paper. The main conclusions are summarized as follows:

By numerically fitting the stress-strain curves of 27 coal specimens, the relationship between bursting proneness indices and cohesion of coal is statically established and the functional equations between them are theoretically obtained. It is suggested that the proneness indices _{c} and _{ET} linearly increase with the increase of cohesion while the index _{E} and cohesion have a positive power function relation. However, the negative power function between the index _{t} and cohesion is established.

The linear functional relation between the indices of _{c} and _{ET} and the cohesion suggests that the cohesion of coal can be directly used to reflect the elastic strain energy accumulation before coal peak strength. In addition, because of the power functional equation between cohesion and _{E} and _{t}, the coal cohesion could also be used to illustrate the nonlinear process of strain energy release after peak strength. Therefore, coal cohesion combines the common advantages of the four indices and it can be used as an index to identify the bursting proneness. It is suggested that coal is of high bursting proneness when the cohesion is greater than 4.0 MPa and the low bursting proneness is warranted when the cohesion is greater than 1.5 MPa.

By taking into account the coal heterogeneity, eight probability distribution functions are employed to assign the cohesion to the numerical model and to study the influence of heterogeneity on the bursting proneness. With the increasing of cohesion scatter degree, the UCS decrement of heterogeneous coal specimen assigned by Weibull and Normal distributions was smaller than that by the other six distributions. The UCS of heterogeneous coal specimen with Fisher distribution decreases fastest among the other distributions with the increasing of cohesion scatter degree. The coal bursting proneness will decrease with the increasing of cohesion scatter degree. The larger the cohesion scatter degree increase is, the lower the bursting proneness will be. The failure of coal specimen is more and more severe with the decrease of cohesion scatter degree.

There are many methods to assign heterogeneous parameters to the numerical models. This study provides two methods including the assignment from center to edge and random assignment. The selection of different method determines the different failure characteristics of the numerical model. By employing the method of assignment from center to edge, the edges of the model begin to fail and the failure zones gradually develop from edge to center until the model is completely destroyed. The selection of random assignment may simulate the characteristics in which failure zones appear first at the center of the numerical model.

The underground coal seam with high bursting proneness provides an ideal internal environment for energy accumulation. The high bursting proneness may be just an internal factor inducing coal burst accidents while the geological conditions are the significant external factors contributing to the occurrence of coal bursts. Therefore, the coal bursts are the interactional results of high bursting proneness, unstable geological structure including thrust faults and overturned folds, extreme thick roof strata, and island longwall panel mining.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This research was financially supported by the National Natural Science Foundation of China (41872205), Beijing Natural Science Foundation (8202041), Yue Qi Young Scholar Project, China University of Mining & Technology, Beijing (2018QN13), and the Fundamental Research Funds for the Central Universities (2021YJSLJ10).