Burst Failure Characteristics and Energy Evolution Law of Coal with Prefabricated Cracks at Different Angles

In order to study the infuence of fssures on the burst tendency of coal, the test and numerical simulation of the burst tendency of coal with diferent burst angles were carried out. Te evolution law of the burst tendency index of coal under the infuence of burst angle was analyzed, and the mechanism of energy storage and release of coal under the infuence of fssure angle was revealed. Te results show that compared with the specimens without prefabricated cracks, the uniaxial compressive strength of the specimens with 0 ° cracks is reduced by 48.4%, the dynamic failure time is increased by 279.4%, the burst energy index is reduced by 54%, and the burst energy velocity index is reduced by 87.9%. After that, with the increase of prefabricated crack angle, the uniaxial compressive strength of coal increases gradually, the dynamic failure time decreases gradually, the burst energy index increases gradually, and the burst energy velocity index increases gradually. Tat is to say, the larger the crack angle contained in the coal body, the stronger the burst tendency of the coal body, but it is still lower than that of the complete coal body. With the increase of prefabricated crack angle, the proportion of prepeak elastic energy of coal body increases, the less energy dissipation in the whole loading process of coal body, and the faster energy release rate during failure. Te research results can provide some theoretical support for the prevention and control of rock burst disaster.


Introduction
Coal is a typical heterogeneous material that contains a large number of joints, cracks, or structural planes inside.Te instability and failure of coal mine roadways are caused by the initiation, expansion, and connection of internal cracks in the coal body [1][2][3][4].With the gradual deepening of coal mining, the joint cracks of the roadway surrounding rock are more developed, loose, and broken, and the deformation of the roadway is aggravated [5][6][7][8][9].Te fracture of coal can be regarded as a process of energy conversion and transmission, and its fnal failure is a state instability phenomenon driven by energy.Many scholars believe that the essential mechanism of coal dynamic failure and rock burst can be revealed from the perspective of energy [10][11][12][13][14]. Terefore, it is of great engineering signifcance to study the burst failure and energy evolution of cracked coal.
At present, scholars at home and abroad have systematically studied the infuence of crack dip angle, number and length on the strength, and deformation and failure characteristics of coal mass through laboratory tests and numerical simulation [15][16][17][18][19].
Te whole process from loading deformation to destruction of coal body is accompanied by the accumulation and transformation of energy.Energy is the essential factor leading to the destruction of the coal bodies [19][20][21][22][23][24][25][26].Many scholars have carried out a lot of research on energy and rock failure, but there are few studies on the energy of cracked coal, which still needs further research.
Previous studies have focused on the infuence of crack angle, crack number, and crack length on the mechanical properties of coal and rock mass.Terefore, on the basis of previous studies, this paper fxed the crack length and crack number, prefabricated four kinds of coal with diferent crack angles, and carried out the impact tendency test of coal with diferent crack angles to study the impact failure characteristics of coal [27,28].Trough the indoor test and numerical simulation of coal body with diferent fracture angles, the energy evolution process of coal body is analyzed and the energy evolution law of coal body is studied.

Indoor Experiment and Result Analysis
2.1.Specimen Preparation and Loading Scheme.Te size of the coal in the test is 50 mm in diameter and 100 mm in height.Te prefabricated crack is located in the center of the coal body.Te crack size is 20 mm in length and 2 mm in width, and the crack penetrates the coal body.In order to reduce the individual diference of the sample, the longitudinal wave velocity of the coal sample was tested before the prefabricated crack, and the coal sample with the wave velocity between 1200 and 1500 m/s was selected for the prefabricated crack.According to the angle of the prefabricated crack, it is divided into fve groups of tests, namely, no crack group, 0 °crack group, 30 °crack group, 60 °crack group, and 90 °crack group.Each group is tested on three specimens.Te specifc specimen preparation is shown in Figure 1.
Te test loading system adopts an RLJW-2000 rock testing machine, axial compression is applied at a stress loading rate of 0.5 mm/min, and the load-displacement data of the sample are collected synchronously until the sample is destroyed.

Measurement of Burst Tendency.
Te burst tendency of coal refers to the ability and inherent properties of coal to accumulate energy and produce burst damage, which is a necessary condition for rock burst.Terefore, the determination of coal burst tendency is an important part of the prevention and control of rock burst, and the burst tendency index can be used to evaluate the risk of coal burst.In the national standard GB/T 25217.2-2010determination of rock burst monitoring and prevention methods Part 2: classifcation of coal burst tendency and determination method of index, four indexes are given to measure the strength of burst tendency.At the same time, many researchers have also proposed many indexes to measure the burst tendency [29,30].
In this test, four parameters in the national standard, namely, dynamic failure time D T , uniaxial compressive strength σ c , burst energy index K E , and burst energy velocity index W ST proposed by authoritative scholars [28], were selected to measure the burst tendency of coal body, which can be divided into no burst tendency, weak burst tendency, and strong burst tendency.
Te uniaxial compressive strength can refect the ultimate bearing capacity of coal samples.Te higher the uniaxial compressive strength, the stronger the bearing capacity of coal samples before failure, the more energy accumulated inside under the external force, and the greater the kinetic energy converted and released when the peak strength is reached.Te dynamic failure time is used to measure the degree of burst tendency, which is the transient duration time of coal from the beginning to the end of the peak strength, as shown in Figure 2 [29].Te duration of the failure process is a comprehensive refection of the dynamic characteristics of energy accumulation and dissipation, refecting the speed of postpeak failure of coal samples.
Te whole process of the stress-strain curve of coal contains rich information about burst tendency, which intuitively and comprehensively refects the whole process from energy storage to energy consumption.It is of great signifcance to reveal the physical nature of burst tendency and analyze other burst tendency indexes.Te burst energy index is calculated by dividing the prepeak total input energy by the postpeak failure dissipation energy in the stress-strain curve, as shown in Figure 3 [29].Te calculation formula is as follows: In the formula, W p is the total input energy before the peak, and its value is the area enclosed by the prepeak curve and the coordinate axis; W f is the postpeak damage dissipation energy, and its value is the area enclosed by the postpeak curve and the coordinate axis.
Te residual energy released per unit time during the failure process of coal samples represents the amount of elastic energy converted into kinetic energy per unit time, which also refects the degree of burst tendency of coal.Te burst energy velocity index is the ratio of the burst energy index to the dynamic failure time.Te specifc calculation formula is as follows: Te physical meaning of the index characterizes the ratio of energy accumulation and release during the compression process of coal samples per unit time and refects the burst release ability of energy during the compression failure process of coal samples.
Te determination methods of four burst tendency indexes are listed in Table 1.

Te Final Failure Form of Coal.
Figure 4 shows the fnal failure form of the coal specimen after uniaxial compression loading.It can be seen from Figure 4 that the damage degree 2 Shock and Vibration of the 0 °crack specimen is the smallest, and only a small number of fragments fall of along the crack tip.For the 30 °crack specimen, some coal blocks are thrown along the crack area, and the whole crack area of the 60 °crack specimen is basically completely destroyed, while the 90 °crack specimen has large pieces of coal body spalling and throwing, and the degree of damage is the largest.

Uniaxial Compressive Strength σ c
. Te stress-time curve of coal containing prefabricated cracks with diferent angles is shown in Figure 5.When the crack angle is 0 °, 30 °, and 60 °, the postpeak curve is a step-down curve; the new cracks in the specimen are slowly generated and expanded, and the specimen is slowly destroyed.When the prefabricated crack angle increases to 90 °, the step-down characteristic of the postpeak curve of the specimen weakens, showing rapid failure and enhanced brittle failure characteristics.
Te statistics of uniaxial compressive strength of prefabricated cracks at diferent angles are shown in Table 2 and Figure 6.About the uniaxial compressive strength, compared with the specimens without prefabricated cracks, the uniaxial compressive strength of the coal is reduced by 48.4% when the crack angle is 0 °, 44.9% when the crack angle is 30 °, and 32.8% when the crack angle is 60 °.Te coal body with three crack angles shows a weak burst tendency; when the crack angle is 90 °, the uniaxial compressive strength is reduced by 2%.In this case, it still shows a strong burst tendency.With the increase of prefabricated crack angle, the measured value of uniaxial compressive strength of coal specimens also increases, and a power function growth relationship is presented.
Since the coal is limited in the loading direction, there is free expansion space in the other two directions, so the microcracks are mainly tensile cracks parallel to the loading direction.In the mutual expansion and penetration of small tensile cracks, the prefabricated cracks play the role of bridging, so that the cracks generated inside the coal body with the increase of load can interact more easily, and then, the ability of the coal sample to withstand the load is reduced.Terefore, with the increase of the dip angle, the number of tensile cracks that can be connected by the prefabricated cracks of the same length along the loading direction is less and less, so the uniaxial compressive strength of the coal body is on the rise.Especially when the crack dip angle is 90 °, the prefabricated crack has little efect on the uniaxial compressive strength of the coal body.

Dynamic Failure Time D T .
Te dynamic failure time of coal obtained from the stress-time curve is shown in Table 3 and Figure 7. Compared with the coal specimen without prefabricated cracks, the dynamic failure time increases by 279.4% when the crack angle is 0 °, 236.7% when the crack angle is 30 °, 171.8% when the crack angle is 60 °, and 15.9% when the crack angle is 90 °.According to the standard, there is no burst tendency.With the increase of crack angle, the measured value of dynamic failure time gradually decreases and fnally shows a trend of power function decrease.Te smaller the crack angle, the longer the dynamic failure time of the specimen.Tis is because with the increase of the crack angle, the projection length of the crack length in the vertical loading direction is smaller, so that the number of cracks in the coal body parallel to the loading direction is reduced, the damage degree of the coal body is reduced, and fnally, the dynamic failure time of the coal body is reduced.
2.6.Burst Energy Index K E .Trough the analysis of stressstrain curve data, the fnal calculated coal burst energy index is shown in Table 4 and Figure 8.Compared with the coal specimen without prefabricated cracks, the burst energy index is reduced by 54% when the crack angle is 0 °, 42.2% when the crack angle is 30 °, 24.4% when the crack angle is 60 °, and 11.9% when the crack angle is 90 °.As the crack angle increases, the measured value of the burst energy index gradually increases, showing a linear growth trend.Te smaller the crack angle, the smaller the burst energy index of the coal, and the less severe the damage.Te results also verify the infuence principle of fracture angle on uniaxial compressive strength and dynamic failure time of coal.Shock and Vibration 3 2.7.Burst Energy Velocity Index W ST .According to the average value of burst energy index and dynamic failure time and combined with formula (2), the fnal burst energy velocity index of coal is shown in Table 5 and Figure 9.
According to the standard, the coal without prefabricated cracks has weak burst tendency.Compared with the coal without prefabricated cracks, the burst energy velocity index decreases by 87.9% when the crack angle is 0 °, 82.6% when the crack angle is 30 °, and 72.2% when the crack angle is 60 °.Te coal with these three abovementioned crack angles shows no burst tendency.When the crack angle is 90 °, the burst energy velocity index is reduced by 23.9%, and this crack angle still shows a weak burst tendency.With the increase of crack angle, the measured value of burst energy velocity index increases gradually, showing a trend of power function growth.Te smaller the crack angle, the smaller the burst energy velocity index of coal, and the less the elastic energy converted into kinetic energy per unit time.In this simulation, the parallel bond model is used to simulate the mechanical behavior of coal.Te parameters are compared by the trial and error method and indoor test.Te stress-strain curve of the complete specimen is shown in Figure 10.Te specifc parameters of the model are listed in Table 6.

Numerical Simulation and Result Analysis
In PFC, rock crack and macroparameters are closely related to the size of particles, which is called the size efect.With increasing of L/R (i.e. the ratio of specimen size to     Shock and Vibration particle radius), the infuence of the size efect decreases.Based on works by Zhao et al., when the value of L/R is greater than 120, the size of particles has little infuence on macro parameters of the specimen [31].So in this paper, in order to reduce the size efect, the radius of the particles was set 0.4∼0.5 mm, and L/R was 220.
In the simulation, the coal is prefabricated with cracks at diferent angles, and then, uniaxial loading is carried out to destroy the coal.Te changes of stress, elastic energy, dissipation energy, and input energy in the whole process are recorded, and the energy evolution law of the coal body is analyzed.

Failure Form of Specimen.
After uniaxial loading to failure, the crack expansion of coal under diferent crack angles is shown in Figure 11.From the diagram, it can be seen that after uniaxial loading, the tip of the prefabricated crack will be destroyed frst.With the increase of the angle of the prefabricated crack, the generated crack gradually expands to the diagonal direction.When the angle of the prefabricated crack is small, the damage range is basically near the prefabricated crack.Te larger the angle of the prefabricated crack, the larger the damage range, which is basically similar to the indoor test results.

Te Energy Calculation Method.
In PFC2D simulation, the strain energy accumulated in the specimen includes two parts.One is the contact strain energy E c str stored at all contacts, and the other is the parallel-bond strain energy E pb str stored in parallel bonds.Te strain energy can be expressed as where In Equations ( 4) and ( 5), F n i , F s i , and M s i are the normal force, shear force, and the moment in the parallel bond i, respectively; F n i and F s i are the normal force and shear force in the contact i, respectively; A i and I i are the area and inertia moment of the bond cross section, respectively; N c is the number of contacts; and N pb is the number of parallel bonds.

Energy Evolution Process of Coal with Diferent Crack
Angles.During the loading process of the coal, the stress and energy evolution curve of the whole process can show the loading and failure process inside the coal during the whole process.Te energy evolution law in the process of coal loading and failure is analyzed, respectively, as shown in Figure 12.
It can be seen from Figure 9 that as the prefabricated crack angle increases, the strain at failure gradually increases.When the prefabricated crack angle is 0 °, the strain is 0.148%, and when it is 90 °, the strain is 0.18%.Te greater the angle of the prefabricated crack, the greater the uniaxial compressive strength and the greater the peak elastic energy; the larger the prefabricated crack angle is, the faster the release rate of stress and elastic energy in the postpeak failure stage is, the faster the dissipation energy increases, and the more severe the damage is.Compared with the noncracked coal, the uniaxial compressive strength and elastic energy peak of the cracked coal will change, resulting in the change of the maximum energy storage of the coal.Terefore, the uniaxial compressive strength and elastic energy peak of the nonprefabricated crack and diferent prefabricated crack angles are compared and analyzed.
Figure 13 shows the uniaxial compressive strength curve at diferent crack angles.Te uniaxial compressive strength of coal without prefabricated cracks is 18.38 MPa.It decreases by 31.23% when the prefabricated crack angle is 0 °,     16.81% when the prefabricated crack angle is 60 °, 9.96% when the prefabricated crack angle is 75 °, and 6.69% when the prefabricated crack angle is 90 °.Te greater the prefabricated crack angle of the coal body, the greater the uniaxial compressive strength increases in a power function, and the smaller the reduction is compared with the nonprefabricated crack, which is basically the same as the test results.Figure 14 shows the peak curve of elastic energy at diferent crack angles.When there is no prefabricated crack, the peak value of elastic energy of coal is 93.67 J.It is reduced by 49.72% when the prefabricated crack angle is 0 °, 47.91% when the prefabricated crack angle is 15 °, 43.39% when the prefabricated crack angle is 30 °, 45.5% when the prefabricated crack angle is 45 °, 35.35% when the prefabricated crack angle is 60 °, 27.15% at 75 °, and 18.46% at 90 °.Te larger the prefabricated crack angle of the coal body, the peak value of the elastic energy increases in a power function, but it is still lower than the peak value of the elastic energy of the coal body without prefabricated cracks.

Energy Dissipation and Release Law of Coal with Diferent
Crack Angles.During the whole loading process of the coal, there will be some energy dissipation.When the peak point is reached, the diference between the input energy and the elastic energy represents the energy dissipation during the whole loading process.Terefore, the elastic energy ratio (the ratio of the elastic energy peak to the input energy) is defned to analyze the energy dissipation law during the loading process of the coal.

Shock and Vibration
Figure 15 shows the elastic energy ratio curve at diferent crack angles.When there is no prefabricated crack, the proportion of elastic energy of coal is 89.35%.It is reduced by 24.83% when the prefabricated crack angle is 0 °, 18.75% when the prefabricated crack angle is 15 °, 8.24% when the prefabricated crack angle is 30 °, 9.68% when the prefabricated crack angle is 45 °, 4.85% when the prefabricated crack angle is 60 °, 2.18% at 75 °, and 2.64% at 90 °.Te larger the prefabricated crack angle of coal body is, the larger the proportion of elastic energy is, showing a power function.Compared with no prefabricated crack, the smaller the reduction range is, the less the energy dissipation in the whole loading process of coal body is.
In the loading process of the whole coal body, the proportion of elastic energy can refect the energy storage and dissipation characteristics inside the coal body.Te postpeak energy release rate (i.e., failure energy release rate) can refect the speed of energy release after coal failure.Te slower the postpeak energy release rate is, the lower the burst risk of postpeak failure is. Figure 16 shows the energy release rate curves of coal with diferent crack angles after failure.For the postpeak failure stage, when the prefabricated crack angle is 0 °, the elastic energy release amount of each 1% strain is 143.1 J, and the elastic energy release amount of each 1% strain increases by 8.04% at 15 °.At 30 °, the elastic energy release amount of each 1% strain increases by 126.62%.At

3. 1 .
Simulation Scheme.In order to study the energy evolution law of coal with prefabricated cracks at diferent angles, PFC numerical simulation software is used to establish a numerical model similar to the laboratory test.Te model size is 50 mm × 100 mm, and the prefabricated crack angles are 0 °, 15 °, 30 °, 45 °, 60 °, 75 °, and 90 °.Te crack length is 20 mm, and the width is 2 mm.Te crack is located at the center of the model.

Figure 6 :
Figure 6: Te relationship curve between crack angle and uniaxial compressive strength.

Figure 7 :
Figure 7: Relationship curve between crack angle and dynamic failure time.

Figure 8 :
Figure 8: Relationship curve between crack angle and bursting energy index.

Figure 9 :Figure 10 :
Figure 9: Relationship curve between crack angle and burst energy velocity index.

Table 2 :
Uniaxial compressive strength of coal under diferent crack angles.

Table 3 :
Dynamic failure time of coal under diferent crack angles.

Table 4 :
Bursting energy index of coal under diferent crack angles.
3.5.Energy Storage Law of Coal with Diferent Crack Angles.

Table 5 :
Burst energy velocity index of coal under diferent crack angles.

Table 6 :
Micromechanical parameters of coal samples.85% when the prefabricated crack angle is 15 °, 25.79% when the prefabricated crack angle is 30 °, 26.33% when the prefabricated crack angle is 45 °,