# Failure Mechanism of a Coal-Rock Combined Body with Inclinations of Structural Planes and a Calculation Model for Impact Energy

Authors: Yi-Chao Zhao; Ming-Shi Gao; Yong-Liang He; Dong Xu
Journal: Advances in Civil Engineering
 (2019)

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## Abstract

A coal-rock (CR) combined body can be used to simulate structures of coal and rock strata, and its impact-induced failure characteristic conforms more close to engineering practice. Exploring the mechanical properties and impact energy in a CR combined body contributes to better predictions of rock bursts in coal mines. In the study, the mechanical properties of CR combined bodies with four different inclinations (0°, 15°, 30°, and 45°) of structural planes were measured, and also their failure mechanism was analysed. Based on the theory of particle mechanics, a calculation model for impact energy in a CR combined body with inclinations was established and then verified by using monitored acoustic emission (AE) data. The test results showed that inclination affected mechanical properties and failure characteristics of the CR combined body, i.e., the larger the inclination, the lower the strength and impact energy in the CR combined body and the lower the level of damage. The proposed calculation model for impact energy revealed the mechanical essence of energy accumulation and release of a CR combined body, providing a reference for investigating rock burst in coal mines.


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## Body

## 1. Introduction

Rock bursts in coal mines are a dynamic disaster induced by failure of coal and rock masses and release much energy [1, 2]. Mechanical properties of coal and rock masses are considered to be the basis for exploring rock bursts in coal mines [3, 4]. At present, the mechanical parameters and impact-induced failure characteristics of coal and rock are generally tested by applying stress on pure coal (PC) and pure rock (PR) specimens; however, mining spaces are integral structures composed of CR combined bodies. If only PC and PR are separately investigated, structural planes and interaction of coal and rock masses will be ignored, therefore, mechanical properties and failure process of PC and PR specimens fail to reflect the actual condition of failure [5]. Additionally, a CR combined body contains two materials, and the calculation method for the impact energy in the CR combined body certainly differs from that for PC and PR. Hence, exploring the impact-induced failure and energy evolution of a CR combined body is conducive to investigating the generation of rock bursts in coal mines.To explore the mechanical essence of accumulation and release of impact energies in a CR combined body, it is necessary to first investigate the mechanical properties and failure mechanism of CR combined specimens (Figure1). By measuring the mechanical properties of CR combined specimens, Liu et al. [6] investigated the effect of material strength on failure mode of specimens. Wang and Du [7] measured the mechanical properties of a CR combined body under unloading conditions and found that a CR combined body is more prone to deformation and failure during unloading. Du et al. [8] studied the mechanical properties of a CR combined body containing gas, and the results demonstrate that a CR combined body has a higher strength than PC and expands outwards upon unloading. Liu et al. [9] measured the mechanical properties of a CR combined body and established the corresponding damage constitutive model thereof. Huang and Liu [10] discussed various mechanical behaviours of CR combined specimens under different loading rates. Guo et al. [11] measured strengths of a CR combined body with various inclinations of its structural planes and described the failures therein. The aforementioned research is expected to provide a reference for measurement of the mechanical properties of a CR combined body in the laboratory.Figure 1
CR combined specimens (a) without inclination and (b) with inclinations.
(a)
(b)Two methods (laboratory monitoring and numerical simulation) are frequently applied in such research into the energy release from coal and rock masses [12–21]. By monitoring the acoustic emission (AE) energy in coal masses during loading and unloading, Wang et al. [22] found that energy release of coal masses is related to the fractures therein. Meng et al. [23] suggested that energy release is correlated with loads while being unrelated to loading rate by monitoring AE events under uniaxial load. By utilising PFC numerical simulation software, Lin et al. [24] investigated the crack propagation and energy release in coal masses under different lateral pressure coefficients. By using PFC numerical simulation software, Jin et al. [25] discussed the effect of initial cracks on energy release in rocks and found that the energy released from rock approximately rises with increasing angle of cracks to the loaded axis. Additionally, some scholars also investigated the energy release from a CR combined body. Lu et al. [26] discussed the energy release and vibrational characteristics of a CR combined body by using a microseismic monitoring instrument. The result showed that the vibrational signals recorded before impact-induced failure exhibit remarkable precursory characteristics indicative of a rock burst. Tan et al. [27] explored the effect of the homogeneity of a CR combined body on AE energy by PFC numerical simulation software. The result indicated that the stronger the homogeneity of coal and rock masses, the larger the rock burst tendency of a CR combined body. The aforementioned research indicates that structure of a CR combined body, loading mode, and material characteristics affect the failure and energy release of coal and rock.Above all, failure characteristics of a CR combined body are more consistent with engineering practice. Therefore, exploring the failure characteristics and impact energies of a CR combined body is conducive to revealing the evolution of a rock burst in coal mines. In the study, mechanical properties of CR combined bodies were tested, and its failure mechanism were analysed. Failure process of the specimens was monitored through the use of an AE monitoring system and high-speed photography [28, 29]. Furthermore, a calculation model for impact energies in a CR combined body with inclinations was proposed to reveal the mechanical essence of the energy release. The results can provide new ideas and methods for exploring and predicting rock bursts in coal mines.


## 2. Test Process

### 2.1. Preparation of Specimens

Coal mass and rock used in the test were taken from coal seams and roof strata, with strong rock burst tendency, in a coal mine in Inner Mongolia Autonomous Region, China. The fundamental mechanical properties of materials are listed in Table1. The structures of the CR combined body are displayed in Figure 2. The specimens measured 50 mm × 50 mm × 100 mm, and the thickness δ of each coal section was 20 mm. CR combined specimens were bonded with a resin-anchoring agent for bolts used in mines. The serial number of the specimens was represented by a combination of sample types (CR, PC, and PR refer to CR combined specimens, pure coal specimens, and pure rock specimens, respectively), inclinations α (i.e., 0–0°, 15–15°, 30–30°, and 45–45°) of a coal section, and sequence number (1, 2, or 3). According to the characterisation requirements for specimens recommended by the International Society for Rock Mechanics and Rock Engineering (ISRMRE), the specimens were ground. The unevenness of the two end faces of the shaped specimens was less than 0.05 mm, and the volumetric error was within ±2500 mm3.Table 1
Mechanical properties of coal and rock.
Material
Uniaxial compressive strength (MPa)
Shear strength (103 MPa)
Dynamic failure time (ms)
Elastic energy index
Impact energy index
Elastic modulus (103 MPa)
Poisson’s ratio
Coal
24.97
7.97
34
7.75
8.91
6.72
0.31
Rock
46.91
16.56
152
14.14
11.42
12.34
0.29Figure 2
Photograph of a CR combined specimen with inclinations of structural planes.


### 2.2. Testing and Verification

#### 2.2.1. Test Method

A uniaxial compression test was carried out on the CR combined specimens along the axial direction. The axial stress on the specimens is expressed as follows:(1)σ=PS,where σ, P, and S denote axial stress (MPa), the load (N) applied, and cross-sectional area (mm2) of the specimens, respectively. The axial strain on the coal section could be indirectly attained by measuring the amount of deformation of specimens and the strain on rock and expressed as follows [9]:(2)εc=Δl−εrlrlc,where εc, Δl, and εr represent the axial strain in a coal section, the amount (mm) of axial deformation of CR combined specimens, and axial strain in the rock, respectively; lc and lr denote the axial height lc=δ/cosαmm of the coal section and axial height (lx=l−lc) of the rock, respectively.By monitoring the AE energy in the loading process, the cumulative AE energy is given by(3)EAE=∑eAEi,where EAE refers to the cumulative AE energy (ms·mV) in specimens during the entire loading process and EAEi represents the AE energy (ms·mV) of a single fracture event.


#### 2.2.2. Equipment

An MTS-C64.106 electrohydraulic servo universal testing machine was used to apply load on the specimens; it had a maximum vertical load of 1000 kN and a displacement resolution of 0.2μm. The AE was monitored by employing a PCI-2 multichannel AE system. Moreover, the strain on the specimens was acquired by using an AKEMONDTSK-32-8C strain measuring instrument, with the resistance of all strain gauges being 120 Ω. The monitoring devices were connected as shown in Figure 3.Figure 3
Loading monitoring system. 1, AE monitoring system; 2, monitoring system for axial force; 3, strain monitoring system; 4, iron ore; 5, axial force sensor; 6, strain gauge; 7, AE sensor; 8, iron plate; 9, specimen; 10, upper pressurised cylinder; 11, steel prop to avoid eccentricity; 12, lower pressurised cylinder; and 13, base of the press machine.On condition that the strength of structural plane of coal and rock was too low, the specimens with a large inclinationα would slide along the structural plane before being damaged. As a result, mechanical properties of CR combined specimens could not be attained. To prevent incorrect test data induced by slippage and failure of structural planes, two iron plates were fixed on the two sides of the specimens by using four screws, and also four axial force sensors were separately installed on a screw, as shown in Figure 4. The sensor with the measurement range of 0 to 5 kN had a sensitivity of 5‰ (25 N). Before failure of a specimen, if the axial force on a single screw exceeded 50 N, the specimen was supposed to slide, resulting in an erroneous test result. The spacing between the specimen and iron plate was between 1 and 2 mm so that the specimen was not affected by iron plates, and the AE sensor can remain in close contact with the surface of the specimen after crossing the iron plates.Figure 4
Iron plates and force sensors for monitoring lateral sliding.


#### 2.2.3. Verification Methods

In the present study, load, strain, and AE energy were monitored. According to equation (1), the axial stress σ on the specimens was calculated, and then the maximum axial stress is taken as the strength σcr of the specimens, that is, σcr=maxσ. On this condition, the strengths of different specimens need to satisfy the following equation:(4)Vσi=σcri−σcr¯σcr¯×100%<10%,where VEi, σcri, and σcr¯ refer to the coefficient of standard deviation of the strength of specimens, the strength (MPa) of specimens and the mean strength (MPa) of a group of specimens, respectively.Eight AE sensors were fixed to the surfaces of all specimens. Afterwards, the cumulative AE energies in each channel were screened by using the following equation:(5)VEi=EAEi−EAE¯EAE¯×100%<10%,where VEi and EAEi represent the coefficient of standard deviation of cumulative AE energy and cumulative AE energy (ms·mV) of the ith channel during the entire loading process, respectively; EAE¯ denotes the mean (ms·mV) cumulative AE energy in all channels. When the coefficient VEi of the standard deviation of cumulative AE energy in a certain channel does not satisfy equation (5), the monitoring result of the channel was eliminated, and the coefficient VEi of standard deviation of the cumulative AE energies in the other channels was calculated. No less than six sets of data should be involved in calculation here to prove satisfaction of equation (5); otherwise, it is considered that the monitoring data of AE energy were subject to significant errors.


### 2.3. Test Procedure and Results

#### 2.3.1. Test Procedure

According to the recommendation of the International Society for Rock Mechanics and Stipulations in the Methods for Test, Monitoring, and Prevention of Rock Burst for the coal industry in China, a load-controlled method of testing was adopted. The specimens were subjected to the uniaxial loading test on the electrohydraulic servo universal testing machine, at the loading rate of 0.5 MPa/s (1.25 kN/s) and the frequency of data acquisition of 1000 Hz. By using the AE system, the AE energy in specimens was monitored, with the sampling frequency of 500 kHz, the threshold of 40 dB, and a operating frequency of 100–400 kHz. Moreover, the PDT, HDT, and HLT were set to 50, 200, and 300 μs, respectively. The sampling frequency of the strain measuring instrument was 2 Hz. In the test process, the failure process of specimens was recorded by using a NAC Memrecam HX-7s high-speed camera at a frequency of 1000 Hz.


#### 2.3.2. Test Results

The statistical parameters pertaining to the strength of the specimens are listed in Table2. The coefficients of standard deviation of the strength on different specimens all satisfy equation (4). The cumulative AE energies during the entire loading process are listed in Table 3, and the coefficients of standard deviation of cumulative AE energies in various specimens all satisfy equation (5). The strengths and cumulative AE energies in specimens with the same structures were consistent; this indicated that the initial cracks in rock and coal section had no significant effect on the test results, and the test method was thus verified as being effective.Table 2
Statistical results: strength of the specimens.
Serial number of specimens
σ
cr (MPa)
V
σ
i (%)
Serial number of specimens
σ
cr (MPa)
V
σ
i (%)
CR-0-1
32.1103
−4.21
CR-30-1
17.3016
−4.94
CR-0-2
33.1421
−1.13
CR-30-2
19.3541
6.34
CR-0-3
35.3115
5.34
CR-30-3
17.9453
−1.4
Mean
33.5213
Mean
18.2003
CR-15-1
25.9652
−2.91
CR-45-1
12.5735
−3.69
CR-15-2
28.2658
5.69
CR-45-2
13.6984
4.92
CR-15-3
26.0026
−2.77
CR-45-3
12.8954
−1.23
Mean
26.7445
Mean
13.0558
PC-1
26.1556
4.75
PR-1
48.0924
2.52
PC-2
24.5884
−1.53
PR-2
46.7412
−0.36
PC-3
24.1654
−3.22
PR-3
45.8905
−2.17
Mean
24.9698
Mean
46.908Table 3
Statistical data: cumulative AE energies in specimens during the entire loading process.
Serial number of specimens
E
AE (ms·mV)
V
E
i (%)
Serial number of specimens
E
AE (ms·mV)
V
E
i (%)
CR-0-1
151889
−5.7
CR-30-1
62905
−5.89
CR-0-2
158520
−1.59
CR-30-2
72354
8.25
CR-0-3
172820
7.29
CR-30-3
65258
−2.37
Mean
161076
Mean
66839
CR-15-1
111559
−7.87
CR-45-1
41604
−4.89
CR-15-2
131118
8.28
CR-45-2
47265
8.05
CR-15-3
120595
−0.41
CR-45-3
42365
−3.15
Mean
121091
Mean
43745
PC-1
112247
3.58
PR-1
316279
4.39
PC-2
108554
0.17
PR-2
307146
1.38
PC-3
104295
−3.76
PR-3
285494
−5.77
Mean
108365.33
Mean
302973


#### 2.3.3. Mechanical Properties

According to Table2, it can be seen that the strength of CR combined specimens ranged from 12.5735 to 35.3115 MPa. With increasing inclination of structural planes, the strength of CR combined specimens declined. The strength of CR combined specimens differed from those of PC and PR, and inclination could directly determine the change in strength of CR combined specimens.Figure5 shows the stress-strain curves of specimens with different structures. PC specimens and CR combined specimens (Figures 5(a) and 5(c)–5(f)) underwent a significant compaction stage in the initial loading process, while PR specimens (Figure 5(b)) exhibited linear stress-strain behaviour.Figure 5
Stress-strain curves of specimens with different structures: (a) PC specimens; (b) PR specimens; (c) CR combined specimens withα = 0°; (d) CR combined specimens with α = 15°; (e) CR combined specimens with α = 30°; (f) CR combined specimens with α = 45°.
(a)
(b)
(c)
(d)
(e)
(f)The loading of a CR combined body could be divided into three stages: a compaction stage, an elastic stage, and a failure stage. The stress-strain curve in the compaction stage is wavy or arcuate. In this stage, initial microfractures in coal masses are closed, bringing the structure of the CR combined body to the stage of dynamic adjustment. The stress-strain curve in the elastic stage is an inclined line. In this stage, the CR combined body has a stable structure and accumulates much elastic energy. The failure stage appears at, and after, reaching peak stress. The stress-strain curves of CR combined specimens are similar to those of PC and PR specimens, all showing no residual strength. Due to the strong rock burst tendency of such a coal mass and its surrounding rock, CR combined specimens were suddenly fractured and damaged when loaded to their ultimate strength [30].Parameters of deformation characteristics of each combined specimen are listed in Table4. With increasing inclination, the maximum strain of CR combined specimens decreased. When the inclination increased, the axial elastic modulus of the coal section decreased, while the axial elastic modulus and Poisson’s ratio of the rock section increased. In other words, the greater the inclination, the lower the stiffness of the coal mass in such combined specimens, while the stiffer the rock mass. Compared with CR combined specimens with an inclination of 0°, the coal section and rock section in CR combined specimens with an inclination of 45° exhibited the most significant changes in mechanical properties; the axial elastic modulus and Poisson’s ratio of the rock section separately increased by 7.74% and 45.45%, while the axial elastic modulus of coal section decreased by 18.03%; therefore, the greater the inclination, the more the mechanical properties of such a coal mass were weakened, which led to a lower stiffness of the CR combined body.Table 4
Deformation characteristics of CR combined specimens.
Serial number of different groups of specimens
Maximum axial strain in specimens/10−6
Variation (%)
Axial elastic modulus of coal section (GPa)
Variation (%)
Axial elastic modulus of rock section (GPa)
Variation (%)
Poisson’s ratio of rock section
Variation (%)
CR-0
2543
0
7.32
0
10.73
0
0.33
0
CR-15
2027
−20.29
6.85
−6.42
10.89
1.49
0.41
24.24
CR-30
1510
−40.62
6.2
−15.3
11.16
4.01
0.47
42.42
CR-45
1068
−58
6
−18.03
11.56
7.74
0.48
45.45


#### 2.3.4. Failure Characteristics

The corresponding relationships between AE energy and axial stress on specimens with different structures are illustrated in Figure6. PC, PR, and CR combined specimens all developed a small number of microfractures and also emitted low AE energies from the initial loading stage to that before peak axial stress. At peak axial stress, the cumulative AE energies in different specimens suddenly increased, and then the energies were synchronously released within a short time, thus exhibiting impact characteristics.Figure 6
Relationship between stress and AE energy: (a) PC specimens; (b) PR specimens; (c) CR combined specimens withα = 0°; (d) CR combined specimens with α = 15°; (e) CR combined specimens with α = 30°; (f) CR combined specimens with α = 45°.
(a)
(b)
(c)
(d)
(e)
(f)The impact-induced failure processes of different CR combined specimens are shown in Figure7. The figure displays the impact-induced failure conditions at peak axial stress and final failure stage. Coal and rock sections in CR combined specimens with inclinations of 0° and 15° developed interconnected fractures at peak axial stress. In contrast, rock sections in CR combined specimens with inclinations of 30°and 45°did not undergo failure. The general level of damage of the CR combined specimens declined with increasing inclination.Figure 7
Impact-induced failure process of CR combined specimens: (a) CR combined specimens withα = 0°; (b) CR combined specimens with α = 15°; (c) CR combined specimens with α = 30°; (d) CR combined specimens with α = 45°.
(a)
(b)
(c)
(d)


## 2.1. Preparation of Specimens

Coal mass and rock used in the test were taken from coal seams and roof strata, with strong rock burst tendency, in a coal mine in Inner Mongolia Autonomous Region, China. The fundamental mechanical properties of materials are listed in Table1. The structures of the CR combined body are displayed in Figure 2. The specimens measured 50 mm × 50 mm × 100 mm, and the thickness δ of each coal section was 20 mm. CR combined specimens were bonded with a resin-anchoring agent for bolts used in mines. The serial number of the specimens was represented by a combination of sample types (CR, PC, and PR refer to CR combined specimens, pure coal specimens, and pure rock specimens, respectively), inclinations α (i.e., 0–0°, 15–15°, 30–30°, and 45–45°) of a coal section, and sequence number (1, 2, or 3). According to the characterisation requirements for specimens recommended by the International Society for Rock Mechanics and Rock Engineering (ISRMRE), the specimens were ground. The unevenness of the two end faces of the shaped specimens was less than 0.05 mm, and the volumetric error was within ±2500 mm3.Table 1
Mechanical properties of coal and rock.
Material
Uniaxial compressive strength (MPa)
Shear strength (103 MPa)
Dynamic failure time (ms)
Elastic energy index
Impact energy index
Elastic modulus (103 MPa)
Poisson’s ratio
Coal
24.97
7.97
34
7.75
8.91
6.72
0.31
Rock
46.91
16.56
152
14.14
11.42
12.34
0.29Figure 2
Photograph of a CR combined specimen with inclinations of structural planes.


## 2.2. Testing and Verification

### 2.2.1. Test Method

A uniaxial compression test was carried out on the CR combined specimens along the axial direction. The axial stress on the specimens is expressed as follows:(1)σ=PS,where σ, P, and S denote axial stress (MPa), the load (N) applied, and cross-sectional area (mm2) of the specimens, respectively. The axial strain on the coal section could be indirectly attained by measuring the amount of deformation of specimens and the strain on rock and expressed as follows [9]:(2)εc=Δl−εrlrlc,where εc, Δl, and εr represent the axial strain in a coal section, the amount (mm) of axial deformation of CR combined specimens, and axial strain in the rock, respectively; lc and lr denote the axial height lc=δ/cosαmm of the coal section and axial height (lx=l−lc) of the rock, respectively.By monitoring the AE energy in the loading process, the cumulative AE energy is given by(3)EAE=∑eAEi,where EAE refers to the cumulative AE energy (ms·mV) in specimens during the entire loading process and EAEi represents the AE energy (ms·mV) of a single fracture event.


### 2.2.2. Equipment

An MTS-C64.106 electrohydraulic servo universal testing machine was used to apply load on the specimens; it had a maximum vertical load of 1000 kN and a displacement resolution of 0.2μm. The AE was monitored by employing a PCI-2 multichannel AE system. Moreover, the strain on the specimens was acquired by using an AKEMONDTSK-32-8C strain measuring instrument, with the resistance of all strain gauges being 120 Ω. The monitoring devices were connected as shown in Figure 3.Figure 3
Loading monitoring system. 1, AE monitoring system; 2, monitoring system for axial force; 3, strain monitoring system; 4, iron ore; 5, axial force sensor; 6, strain gauge; 7, AE sensor; 8, iron plate; 9, specimen; 10, upper pressurised cylinder; 11, steel prop to avoid eccentricity; 12, lower pressurised cylinder; and 13, base of the press machine.On condition that the strength of structural plane of coal and rock was too low, the specimens with a large inclinationα would slide along the structural plane before being damaged. As a result, mechanical properties of CR combined specimens could not be attained. To prevent incorrect test data induced by slippage and failure of structural planes, two iron plates were fixed on the two sides of the specimens by using four screws, and also four axial force sensors were separately installed on a screw, as shown in Figure 4. The sensor with the measurement range of 0 to 5 kN had a sensitivity of 5‰ (25 N). Before failure of a specimen, if the axial force on a single screw exceeded 50 N, the specimen was supposed to slide, resulting in an erroneous test result. The spacing between the specimen and iron plate was between 1 and 2 mm so that the specimen was not affected by iron plates, and the AE sensor can remain in close contact with the surface of the specimen after crossing the iron plates.Figure 4
Iron plates and force sensors for monitoring lateral sliding.


### 2.2.3. Verification Methods

In the present study, load, strain, and AE energy were monitored. According to equation (1), the axial stress σ on the specimens was calculated, and then the maximum axial stress is taken as the strength σcr of the specimens, that is, σcr=maxσ. On this condition, the strengths of different specimens need to satisfy the following equation:(4)Vσi=σcri−σcr¯σcr¯×100%<10%,where VEi, σcri, and σcr¯ refer to the coefficient of standard deviation of the strength of specimens, the strength (MPa) of specimens and the mean strength (MPa) of a group of specimens, respectively.Eight AE sensors were fixed to the surfaces of all specimens. Afterwards, the cumulative AE energies in each channel were screened by using the following equation:(5)VEi=EAEi−EAE¯EAE¯×100%<10%,where VEi and EAEi represent the coefficient of standard deviation of cumulative AE energy and cumulative AE energy (ms·mV) of the ith channel during the entire loading process, respectively; EAE¯ denotes the mean (ms·mV) cumulative AE energy in all channels. When the coefficient VEi of the standard deviation of cumulative AE energy in a certain channel does not satisfy equation (5), the monitoring result of the channel was eliminated, and the coefficient VEi of standard deviation of the cumulative AE energies in the other channels was calculated. No less than six sets of data should be involved in calculation here to prove satisfaction of equation (5); otherwise, it is considered that the monitoring data of AE energy were subject to significant errors.


## 2.2.1. Test Method

A uniaxial compression test was carried out on the CR combined specimens along the axial direction. The axial stress on the specimens is expressed as follows:(1)σ=PS,where σ, P, and S denote axial stress (MPa), the load (N) applied, and cross-sectional area (mm2) of the specimens, respectively. The axial strain on the coal section could be indirectly attained by measuring the amount of deformation of specimens and the strain on rock and expressed as follows [9]:(2)εc=Δl−εrlrlc,where εc, Δl, and εr represent the axial strain in a coal section, the amount (mm) of axial deformation of CR combined specimens, and axial strain in the rock, respectively; lc and lr denote the axial height lc=δ/cosαmm of the coal section and axial height (lx=l−lc) of the rock, respectively.By monitoring the AE energy in the loading process, the cumulative AE energy is given by(3)EAE=∑eAEi,where EAE refers to the cumulative AE energy (ms·mV) in specimens during the entire loading process and EAEi represents the AE energy (ms·mV) of a single fracture event.


## 2.2.2. Equipment

An MTS-C64.106 electrohydraulic servo universal testing machine was used to apply load on the specimens; it had a maximum vertical load of 1000 kN and a displacement resolution of 0.2μm. The AE was monitored by employing a PCI-2 multichannel AE system. Moreover, the strain on the specimens was acquired by using an AKEMONDTSK-32-8C strain measuring instrument, with the resistance of all strain gauges being 120 Ω. The monitoring devices were connected as shown in Figure 3.Figure 3
Loading monitoring system. 1, AE monitoring system; 2, monitoring system for axial force; 3, strain monitoring system; 4, iron ore; 5, axial force sensor; 6, strain gauge; 7, AE sensor; 8, iron plate; 9, specimen; 10, upper pressurised cylinder; 11, steel prop to avoid eccentricity; 12, lower pressurised cylinder; and 13, base of the press machine.On condition that the strength of structural plane of coal and rock was too low, the specimens with a large inclinationα would slide along the structural plane before being damaged. As a result, mechanical properties of CR combined specimens could not be attained. To prevent incorrect test data induced by slippage and failure of structural planes, two iron plates were fixed on the two sides of the specimens by using four screws, and also four axial force sensors were separately installed on a screw, as shown in Figure 4. The sensor with the measurement range of 0 to 5 kN had a sensitivity of 5‰ (25 N). Before failure of a specimen, if the axial force on a single screw exceeded 50 N, the specimen was supposed to slide, resulting in an erroneous test result. The spacing between the specimen and iron plate was between 1 and 2 mm so that the specimen was not affected by iron plates, and the AE sensor can remain in close contact with the surface of the specimen after crossing the iron plates.Figure 4
Iron plates and force sensors for monitoring lateral sliding.


## 2.2.3. Verification Methods

In the present study, load, strain, and AE energy were monitored. According to equation (1), the axial stress σ on the specimens was calculated, and then the maximum axial stress is taken as the strength σcr of the specimens, that is, σcr=maxσ. On this condition, the strengths of different specimens need to satisfy the following equation:(4)Vσi=σcri−σcr¯σcr¯×100%<10%,where VEi, σcri, and σcr¯ refer to the coefficient of standard deviation of the strength of specimens, the strength (MPa) of specimens and the mean strength (MPa) of a group of specimens, respectively.Eight AE sensors were fixed to the surfaces of all specimens. Afterwards, the cumulative AE energies in each channel were screened by using the following equation:(5)VEi=EAEi−EAE¯EAE¯×100%<10%,where VEi and EAEi represent the coefficient of standard deviation of cumulative AE energy and cumulative AE energy (ms·mV) of the ith channel during the entire loading process, respectively; EAE¯ denotes the mean (ms·mV) cumulative AE energy in all channels. When the coefficient VEi of the standard deviation of cumulative AE energy in a certain channel does not satisfy equation (5), the monitoring result of the channel was eliminated, and the coefficient VEi of standard deviation of the cumulative AE energies in the other channels was calculated. No less than six sets of data should be involved in calculation here to prove satisfaction of equation (5); otherwise, it is considered that the monitoring data of AE energy were subject to significant errors.


## 2.3. Test Procedure and Results

### 2.3.1. Test Procedure

According to the recommendation of the International Society for Rock Mechanics and Stipulations in the Methods for Test, Monitoring, and Prevention of Rock Burst for the coal industry in China, a load-controlled method of testing was adopted. The specimens were subjected to the uniaxial loading test on the electrohydraulic servo universal testing machine, at the loading rate of 0.5 MPa/s (1.25 kN/s) and the frequency of data acquisition of 1000 Hz. By using the AE system, the AE energy in specimens was monitored, with the sampling frequency of 500 kHz, the threshold of 40 dB, and a operating frequency of 100–400 kHz. Moreover, the PDT, HDT, and HLT were set to 50, 200, and 300 μs, respectively. The sampling frequency of the strain measuring instrument was 2 Hz. In the test process, the failure process of specimens was recorded by using a NAC Memrecam HX-7s high-speed camera at a frequency of 1000 Hz.


### 2.3.2. Test Results

The statistical parameters pertaining to the strength of the specimens are listed in Table2. The coefficients of standard deviation of the strength on different specimens all satisfy equation (4). The cumulative AE energies during the entire loading process are listed in Table 3, and the coefficients of standard deviation of cumulative AE energies in various specimens all satisfy equation (5). The strengths and cumulative AE energies in specimens with the same structures were consistent; this indicated that the initial cracks in rock and coal section had no significant effect on the test results, and the test method was thus verified as being effective.Table 2
Statistical results: strength of the specimens.
Serial number of specimens
σ
cr (MPa)
V
σ
i (%)
Serial number of specimens
σ
cr (MPa)
V
σ
i (%)
CR-0-1
32.1103
−4.21
CR-30-1
17.3016
−4.94
CR-0-2
33.1421
−1.13
CR-30-2
19.3541
6.34
CR-0-3
35.3115
5.34
CR-30-3
17.9453
−1.4
Mean
33.5213
Mean
18.2003
CR-15-1
25.9652
−2.91
CR-45-1
12.5735
−3.69
CR-15-2
28.2658
5.69
CR-45-2
13.6984
4.92
CR-15-3
26.0026
−2.77
CR-45-3
12.8954
−1.23
Mean
26.7445
Mean
13.0558
PC-1
26.1556
4.75
PR-1
48.0924
2.52
PC-2
24.5884
−1.53
PR-2
46.7412
−0.36
PC-3
24.1654
−3.22
PR-3
45.8905
−2.17
Mean
24.9698
Mean
46.908Table 3
Statistical data: cumulative AE energies in specimens during the entire loading process.
Serial number of specimens
E
AE (ms·mV)
V
E
i (%)
Serial number of specimens
E
AE (ms·mV)
V
E
i (%)
CR-0-1
151889
−5.7
CR-30-1
62905
−5.89
CR-0-2
158520
−1.59
CR-30-2
72354
8.25
CR-0-3
172820
7.29
CR-30-3
65258
−2.37
Mean
161076
Mean
66839
CR-15-1
111559
−7.87
CR-45-1
41604
−4.89
CR-15-2
131118
8.28
CR-45-2
47265
8.05
CR-15-3
120595
−0.41
CR-45-3
42365
−3.15
Mean
121091
Mean
43745
PC-1
112247
3.58
PR-1
316279
4.39
PC-2
108554
0.17
PR-2
307146
1.38
PC-3
104295
−3.76
PR-3
285494
−5.77
Mean
108365.33
Mean
302973


### 2.3.3. Mechanical Properties

According to Table2, it can be seen that the strength of CR combined specimens ranged from 12.5735 to 35.3115 MPa. With increasing inclination of structural planes, the strength of CR combined specimens declined. The strength of CR combined specimens differed from those of PC and PR, and inclination could directly determine the change in strength of CR combined specimens.Figure5 shows the stress-strain curves of specimens with different structures. PC specimens and CR combined specimens (Figures 5(a) and 5(c)–5(f)) underwent a significant compaction stage in the initial loading process, while PR specimens (Figure 5(b)) exhibited linear stress-strain behaviour.Figure 5
Stress-strain curves of specimens with different structures: (a) PC specimens; (b) PR specimens; (c) CR combined specimens withα = 0°; (d) CR combined specimens with α = 15°; (e) CR combined specimens with α = 30°; (f) CR combined specimens with α = 45°.
(a)
(b)
(c)
(d)
(e)
(f)The loading of a CR combined body could be divided into three stages: a compaction stage, an elastic stage, and a failure stage. The stress-strain curve in the compaction stage is wavy or arcuate. In this stage, initial microfractures in coal masses are closed, bringing the structure of the CR combined body to the stage of dynamic adjustment. The stress-strain curve in the elastic stage is an inclined line. In this stage, the CR combined body has a stable structure and accumulates much elastic energy. The failure stage appears at, and after, reaching peak stress. The stress-strain curves of CR combined specimens are similar to those of PC and PR specimens, all showing no residual strength. Due to the strong rock burst tendency of such a coal mass and its surrounding rock, CR combined specimens were suddenly fractured and damaged when loaded to their ultimate strength [30].Parameters of deformation characteristics of each combined specimen are listed in Table4. With increasing inclination, the maximum strain of CR combined specimens decreased. When the inclination increased, the axial elastic modulus of the coal section decreased, while the axial elastic modulus and Poisson’s ratio of the rock section increased. In other words, the greater the inclination, the lower the stiffness of the coal mass in such combined specimens, while the stiffer the rock mass. Compared with CR combined specimens with an inclination of 0°, the coal section and rock section in CR combined specimens with an inclination of 45° exhibited the most significant changes in mechanical properties; the axial elastic modulus and Poisson’s ratio of the rock section separately increased by 7.74% and 45.45%, while the axial elastic modulus of coal section decreased by 18.03%; therefore, the greater the inclination, the more the mechanical properties of such a coal mass were weakened, which led to a lower stiffness of the CR combined body.Table 4
Deformation characteristics of CR combined specimens.
Serial number of different groups of specimens
Maximum axial strain in specimens/10−6
Variation (%)
Axial elastic modulus of coal section (GPa)
Variation (%)
Axial elastic modulus of rock section (GPa)
Variation (%)
Poisson’s ratio of rock section
Variation (%)
CR-0
2543
0
7.32
0
10.73
0
0.33
0
CR-15
2027
−20.29
6.85
−6.42
10.89
1.49
0.41
24.24
CR-30
1510
−40.62
6.2
−15.3
11.16
4.01
0.47
42.42
CR-45
1068
−58
6
−18.03
11.56
7.74
0.48
45.45


### 2.3.4. Failure Characteristics

The corresponding relationships between AE energy and axial stress on specimens with different structures are illustrated in Figure6. PC, PR, and CR combined specimens all developed a small number of microfractures and also emitted low AE energies from the initial loading stage to that before peak axial stress. At peak axial stress, the cumulative AE energies in different specimens suddenly increased, and then the energies were synchronously released within a short time, thus exhibiting impact characteristics.Figure 6
Relationship between stress and AE energy: (a) PC specimens; (b) PR specimens; (c) CR combined specimens withα = 0°; (d) CR combined specimens with α = 15°; (e) CR combined specimens with α = 30°; (f) CR combined specimens with α = 45°.
(a)
(b)
(c)
(d)
(e)
(f)The impact-induced failure processes of different CR combined specimens are shown in Figure7. The figure displays the impact-induced failure conditions at peak axial stress and final failure stage. Coal and rock sections in CR combined specimens with inclinations of 0° and 15° developed interconnected fractures at peak axial stress. In contrast, rock sections in CR combined specimens with inclinations of 30°and 45°did not undergo failure. The general level of damage of the CR combined specimens declined with increasing inclination.Figure 7
Impact-induced failure process of CR combined specimens: (a) CR combined specimens withα = 0°; (b) CR combined specimens with α = 15°; (c) CR combined specimens with α = 30°; (d) CR combined specimens with α = 45°.
(a)
(b)
(c)
(d)


## 2.3.1. Test Procedure

According to the recommendation of the International Society for Rock Mechanics and Stipulations in the Methods for Test, Monitoring, and Prevention of Rock Burst for the coal industry in China, a load-controlled method of testing was adopted. The specimens were subjected to the uniaxial loading test on the electrohydraulic servo universal testing machine, at the loading rate of 0.5 MPa/s (1.25 kN/s) and the frequency of data acquisition of 1000 Hz. By using the AE system, the AE energy in specimens was monitored, with the sampling frequency of 500 kHz, the threshold of 40 dB, and a operating frequency of 100–400 kHz. Moreover, the PDT, HDT, and HLT were set to 50, 200, and 300 μs, respectively. The sampling frequency of the strain measuring instrument was 2 Hz. In the test process, the failure process of specimens was recorded by using a NAC Memrecam HX-7s high-speed camera at a frequency of 1000 Hz.


## 2.3.2. Test Results

The statistical parameters pertaining to the strength of the specimens are listed in Table2. The coefficients of standard deviation of the strength on different specimens all satisfy equation (4). The cumulative AE energies during the entire loading process are listed in Table 3, and the coefficients of standard deviation of cumulative AE energies in various specimens all satisfy equation (5). The strengths and cumulative AE energies in specimens with the same structures were consistent; this indicated that the initial cracks in rock and coal section had no significant effect on the test results, and the test method was thus verified as being effective.Table 2
Statistical results: strength of the specimens.
Serial number of specimens
σ
cr (MPa)
V
σ
i (%)
Serial number of specimens
σ
cr (MPa)
V
σ
i (%)
CR-0-1
32.1103
−4.21
CR-30-1
17.3016
−4.94
CR-0-2
33.1421
−1.13
CR-30-2
19.3541
6.34
CR-0-3
35.3115
5.34
CR-30-3
17.9453
−1.4
Mean
33.5213
Mean
18.2003
CR-15-1
25.9652
−2.91
CR-45-1
12.5735
−3.69
CR-15-2
28.2658
5.69
CR-45-2
13.6984
4.92
CR-15-3
26.0026
−2.77
CR-45-3
12.8954
−1.23
Mean
26.7445
Mean
13.0558
PC-1
26.1556
4.75
PR-1
48.0924
2.52
PC-2
24.5884
−1.53
PR-2
46.7412
−0.36
PC-3
24.1654
−3.22
PR-3
45.8905
−2.17
Mean
24.9698
Mean
46.908Table 3
Statistical data: cumulative AE energies in specimens during the entire loading process.
Serial number of specimens
E
AE (ms·mV)
V
E
i (%)
Serial number of specimens
E
AE (ms·mV)
V
E
i (%)
CR-0-1
151889
−5.7
CR-30-1
62905
−5.89
CR-0-2
158520
−1.59
CR-30-2
72354
8.25
CR-0-3
172820
7.29
CR-30-3
65258
−2.37
Mean
161076
Mean
66839
CR-15-1
111559
−7.87
CR-45-1
41604
−4.89
CR-15-2
131118
8.28
CR-45-2
47265
8.05
CR-15-3
120595
−0.41
CR-45-3
42365
−3.15
Mean
121091
Mean
43745
PC-1
112247
3.58
PR-1
316279
4.39
PC-2
108554
0.17
PR-2
307146
1.38
PC-3
104295
−3.76
PR-3
285494
−5.77
Mean
108365.33
Mean
302973


## 2.3.3. Mechanical Properties

According to Table2, it can be seen that the strength of CR combined specimens ranged from 12.5735 to 35.3115 MPa. With increasing inclination of structural planes, the strength of CR combined specimens declined. The strength of CR combined specimens differed from those of PC and PR, and inclination could directly determine the change in strength of CR combined specimens.Figure5 shows the stress-strain curves of specimens with different structures. PC specimens and CR combined specimens (Figures 5(a) and 5(c)–5(f)) underwent a significant compaction stage in the initial loading process, while PR specimens (Figure 5(b)) exhibited linear stress-strain behaviour.Figure 5
Stress-strain curves of specimens with different structures: (a) PC specimens; (b) PR specimens; (c) CR combined specimens withα = 0°; (d) CR combined specimens with α = 15°; (e) CR combined specimens with α = 30°; (f) CR combined specimens with α = 45°.
(a)
(b)
(c)
(d)
(e)
(f)The loading of a CR combined body could be divided into three stages: a compaction stage, an elastic stage, and a failure stage. The stress-strain curve in the compaction stage is wavy or arcuate. In this stage, initial microfractures in coal masses are closed, bringing the structure of the CR combined body to the stage of dynamic adjustment. The stress-strain curve in the elastic stage is an inclined line. In this stage, the CR combined body has a stable structure and accumulates much elastic energy. The failure stage appears at, and after, reaching peak stress. The stress-strain curves of CR combined specimens are similar to those of PC and PR specimens, all showing no residual strength. Due to the strong rock burst tendency of such a coal mass and its surrounding rock, CR combined specimens were suddenly fractured and damaged when loaded to their ultimate strength [30].Parameters of deformation characteristics of each combined specimen are listed in Table4. With increasing inclination, the maximum strain of CR combined specimens decreased. When the inclination increased, the axial elastic modulus of the coal section decreased, while the axial elastic modulus and Poisson’s ratio of the rock section increased. In other words, the greater the inclination, the lower the stiffness of the coal mass in such combined specimens, while the stiffer the rock mass. Compared with CR combined specimens with an inclination of 0°, the coal section and rock section in CR combined specimens with an inclination of 45° exhibited the most significant changes in mechanical properties; the axial elastic modulus and Poisson’s ratio of the rock section separately increased by 7.74% and 45.45%, while the axial elastic modulus of coal section decreased by 18.03%; therefore, the greater the inclination, the more the mechanical properties of such a coal mass were weakened, which led to a lower stiffness of the CR combined body.Table 4
Deformation characteristics of CR combined specimens.
Serial number of different groups of specimens
Maximum axial strain in specimens/10−6
Variation (%)
Axial elastic modulus of coal section (GPa)
Variation (%)
Axial elastic modulus of rock section (GPa)
Variation (%)
Poisson’s ratio of rock section
Variation (%)
CR-0
2543
0
7.32
0
10.73
0
0.33
0
CR-15
2027
−20.29
6.85
−6.42
10.89
1.49
0.41
24.24
CR-30
1510
−40.62
6.2
−15.3
11.16
4.01
0.47
42.42
CR-45
1068
−58
6
−18.03
11.56
7.74
0.48
45.45


## 2.3.4. Failure Characteristics

The corresponding relationships between AE energy and axial stress on specimens with different structures are illustrated in Figure6. PC, PR, and CR combined specimens all developed a small number of microfractures and also emitted low AE energies from the initial loading stage to that before peak axial stress. At peak axial stress, the cumulative AE energies in different specimens suddenly increased, and then the energies were synchronously released within a short time, thus exhibiting impact characteristics.Figure 6
Relationship between stress and AE energy: (a) PC specimens; (b) PR specimens; (c) CR combined specimens withα = 0°; (d) CR combined specimens with α = 15°; (e) CR combined specimens with α = 30°; (f) CR combined specimens with α = 45°.
(a)
(b)
(c)
(d)
(e)
(f)The impact-induced failure processes of different CR combined specimens are shown in Figure7. The figure displays the impact-induced failure conditions at peak axial stress and final failure stage. Coal and rock sections in CR combined specimens with inclinations of 0° and 15° developed interconnected fractures at peak axial stress. In contrast, rock sections in CR combined specimens with inclinations of 30°and 45°did not undergo failure. The general level of damage of the CR combined specimens declined with increasing inclination.Figure 7
Impact-induced failure process of CR combined specimens: (a) CR combined specimens withα = 0°; (b) CR combined specimens with α = 15°; (c) CR combined specimens with α = 30°; (d) CR combined specimens with α = 45°.
(a)
(b)
(c)
(d)


## 3. Failure Mechanism

### 3.1. Stress and Failure Mechanism

To explore the accumulation and release of impact energy in the combined body, it is necessary to investigate the stress conditions causing the failure of the CR combined body. As shown in Figures5(a) and 5(b), the stress-strain curves of PR specimens were quasilinear throughout the loading process. In contrast, although PC specimens were slightly compacted in the initial loading stage, the corresponding stress-strain curves in the middle and later loading stages were also quasilinear; therefore, when analysing the stress before the failure of CR combined specimens, the coal and rock sections were regarded as having a quasilinear constitutive relationship [31].The stress on CR combined specimens is shown in Figure8(a). The loading system applied axial stress σ along the specimens, and then the stress was transferred to the coal section through the rock section. A coordinate system was established along the structural planes and the direction normal to the structural planes; the stress on coal section is shown in Figure 8(b) (where σxc=σcosα and τxyc=σcosα). The deformation of each coal section was constrained by the rock section through structural planes under the constraint imposed by stress τs, as shown in Figure 8(c). Before failure, the strain in the coal section was consistent with that in the rock section along the structural planes, so(6)εyc=εyr,where εyc and εyr refer to strains on coal and rock section along the direction of the structural planes, respectively. According to elastic theory,(7)τs=σcosαEcμr−ErμcEcμr+Erμc−Ec−Er,where μc and μr denote Poisson’s ratios of coal and rock sections, respectively. Coal and rock sections were under three-dimensional stress states, and according to the Mohr–Coulomb strength criterion, the failure condition of the coal section is presented as follows:(8)σ1c=ξcσ3c+σbc,where σ1c denotes the maximum principal stress (MPa) on the coal section in a CR combined specimen; ξc represents the slope of the straight line of coal section based on the Mohr–Coulomb strength criterion; and σ3c refers to the minimum principal stress (MPa) on the coal section in CR combined specimens. Moreover, σ1c and σ3c could be calculated as follows:(9)σ1c=σxc+σyc2+σxc−σyc22+τxyc2,σ3c=σxc+σyc2−σxc−σyc22+τxyc2.Figure 8
Stress analysis of specimens. Stress on (a) specimens and (b) coal section; (c) constraint force of coal section.
(a)
(b)
(c)By substituting material parameters in Table1 into equations (8) and (9), the axial stress σ required for failure of the coal section in CR combined specimens could be attained as follows:(10)σcr−c=249.72.957100−81.11cos2α−5.411cosα.Similarly, the axial stressσ required for the failure of a rock section was calculated as follows:(11)σcr−r=938.23.006400−272.14cos2α−8.746cosα.Owing to the axial stress (σcr-c) triggering the failure of a coal section being lower than that (σcr-r) leading to failure of the rock section, the theoretical strength of CR combined specimens was calculated by using equation (10).


### 3.2. Verification

The strength of CR combined specimens was predicted by using equation (10), and the predicted and measured values are listed in Table 5. To evaluate the error between predicted and measured values, the relative error was introduced:(12)v=x−μxμx×100%,where v, x, and μx denote the relative error between predicted and measured values, measured and predicted values, respectively. The relative error of various specimens was between −4.94 and 6.34, implying that the aforementioned stress analysis on CR combined specimens conformed to actual conditions. According to equation (10) and the measured result, it can be found that the strength of a CR combined body was mainly determined by the inclination of the structural planes; the larger the inclination, the lower the strength of the specimens.Table 5
Predicted strengths and measured strengths of CR combined specimens.
Serial number of specimens
Predicted strength (MPa)
Measured strength (MPa)
Relative error (%)
Serial number of specimens
Predicted strength (MPa)
Measured strength (MPa)
Relative error (%)
CR-0-1
33.56
32.1103
−4.21
CR-30-1
18.07
17.3016
−4.94
CR-0-2
33.1421
−1.13
CR-30-2
19.3541
6.34
CR-0-3
35.3115
5.34
CR-30-3
17.9453
−1.4
Mean
33.5213
Mean
18.2003
CR-15-1
26.69
25.9652
−2.91
CR-45-1
13.16
12.5735
−3.69
CR-15-2
28.2658
5.69
CR-45-2
13.6984
4.92
CR-15-3
26.0026
−2.77
CR-45-3
12.8954
−1.23
Mean
26.7445
Mean
13.0558


## 3.1. Stress and Failure Mechanism

To explore the accumulation and release of impact energy in the combined body, it is necessary to investigate the stress conditions causing the failure of the CR combined body. As shown in Figures5(a) and 5(b), the stress-strain curves of PR specimens were quasilinear throughout the loading process. In contrast, although PC specimens were slightly compacted in the initial loading stage, the corresponding stress-strain curves in the middle and later loading stages were also quasilinear; therefore, when analysing the stress before the failure of CR combined specimens, the coal and rock sections were regarded as having a quasilinear constitutive relationship [31].The stress on CR combined specimens is shown in Figure8(a). The loading system applied axial stress σ along the specimens, and then the stress was transferred to the coal section through the rock section. A coordinate system was established along the structural planes and the direction normal to the structural planes; the stress on coal section is shown in Figure 8(b) (where σxc=σcosα and τxyc=σcosα). The deformation of each coal section was constrained by the rock section through structural planes under the constraint imposed by stress τs, as shown in Figure 8(c). Before failure, the strain in the coal section was consistent with that in the rock section along the structural planes, so(6)εyc=εyr,where εyc and εyr refer to strains on coal and rock section along the direction of the structural planes, respectively. According to elastic theory,(7)τs=σcosαEcμr−ErμcEcμr+Erμc−Ec−Er,where μc and μr denote Poisson’s ratios of coal and rock sections, respectively. Coal and rock sections were under three-dimensional stress states, and according to the Mohr–Coulomb strength criterion, the failure condition of the coal section is presented as follows:(8)σ1c=ξcσ3c+σbc,where σ1c denotes the maximum principal stress (MPa) on the coal section in a CR combined specimen; ξc represents the slope of the straight line of coal section based on the Mohr–Coulomb strength criterion; and σ3c refers to the minimum principal stress (MPa) on the coal section in CR combined specimens. Moreover, σ1c and σ3c could be calculated as follows:(9)σ1c=σxc+σyc2+σxc−σyc22+τxyc2,σ3c=σxc+σyc2−σxc−σyc22+τxyc2.Figure 8
Stress analysis of specimens. Stress on (a) specimens and (b) coal section; (c) constraint force of coal section.
(a)
(b)
(c)By substituting material parameters in Table1 into equations (8) and (9), the axial stress σ required for failure of the coal section in CR combined specimens could be attained as follows:(10)σcr−c=249.72.957100−81.11cos2α−5.411cosα.Similarly, the axial stressσ required for the failure of a rock section was calculated as follows:(11)σcr−r=938.23.006400−272.14cos2α−8.746cosα.Owing to the axial stress (σcr-c) triggering the failure of a coal section being lower than that (σcr-r) leading to failure of the rock section, the theoretical strength of CR combined specimens was calculated by using equation (10).


## 3.2. Verification

The strength of CR combined specimens was predicted by using equation (10), and the predicted and measured values are listed in Table 5. To evaluate the error between predicted and measured values, the relative error was introduced:(12)v=x−μxμx×100%,where v, x, and μx denote the relative error between predicted and measured values, measured and predicted values, respectively. The relative error of various specimens was between −4.94 and 6.34, implying that the aforementioned stress analysis on CR combined specimens conformed to actual conditions. According to equation (10) and the measured result, it can be found that the strength of a CR combined body was mainly determined by the inclination of the structural planes; the larger the inclination, the lower the strength of the specimens.Table 5
Predicted strengths and measured strengths of CR combined specimens.
Serial number of specimens
Predicted strength (MPa)
Measured strength (MPa)
Relative error (%)
Serial number of specimens
Predicted strength (MPa)
Measured strength (MPa)
Relative error (%)
CR-0-1
33.56
32.1103
−4.21
CR-30-1
18.07
17.3016
−4.94
CR-0-2
33.1421
−1.13
CR-30-2
19.3541
6.34
CR-0-3
35.3115
5.34
CR-30-3
17.9453
−1.4
Mean
33.5213
Mean
18.2003
CR-15-1
26.69
25.9652
−2.91
CR-45-1
13.16
12.5735
−3.69
CR-15-2
28.2658
5.69
CR-45-2
13.6984
4.92
CR-15-3
26.0026
−2.77
CR-45-3
12.8954
−1.23
Mean
26.7445
Mean
13.0558


## 4. Calculation Model for Impact Energy

### 4.1. Calculation Model

As shown in Figures6(c) to 6(f), energy release occurred throughout the loading process of CR combined specimens, and microfractures were constantly generated, therein AE energy exhibited a high discreteness, which indicated that the microfracturing degree of specimens also showed a high discreteness. To quantify the process of energy release in specimens, according to the theory of particle mechanics, it is supposed that the relationship between uniaxial strength (σ) and volume density (fσ) of particulate elements of coal and rock masses is expressed as follows [32–34]:(13)fσ=λe−λσ,where λ represents the strength-volume rate parameter of particle elements. It is supposed that particle elements exhibited a linear-elastic constitutive relationship, and their energy density is expressed as follows [35]:(14)eE=12σxεx+σyεy+σzεz+τxyγxy+τxzγxz+τyzγyz,where eE denotes the energy density (MPa) of particle elements. Before the impact-induced failure of CR combined specimens, with increasing load, cumulative energy released from coal or rock masses was governed by equation (15) on condition that the particulate element with uniaxial strength of σ was damaged:(15)E′=∫0σeEVfσdσ,where E′ represents the cumulative energy released (10−3 J) from coal or rock masses before impact-induced failure and V represents the volume (mm3) of the coal or rock masses. By substituting equation (13) into equation (15), the cumulative energy release before impact-induced failure of coal and rock sections is as follows:(16)E′=∫0σeEVλe−λσdσ.When impact-induced failure of CR combined specimens occurred, the impact energy (Eb) released from coal or rock masses was equivalent to the elastic energy accumulated in unbroken particle elements (with uniaxial strength larger than σb). Thus,(17)Eb=eEV1−∫0σbfσbdσb=eEVe−λσb.Hence, the calculation model for impact energy (Eber) in a CR combined body with inclinations of structural planes is expressed as follows:(18)Ebcr=eEcVce−λcσbc+eErVre−λrσbr,where Eber denotes the impact energy (10−3 J) of the CR combined body with inclinations of structural planes, eEc and eEr represent energy densities (MPa), Vc and Vr represent the volumes (mm3), while λc and λr denote the strength-volume rate parameters, of coal and rock masses, respectively, and σbc and σbr denote the uniaxial strength (MPa) of particle elements, of coal and rock masses, which were fractured immediately before impact-induced failure of the CR combined body, respectively.


### 4.2. Strength-Volume Rate Parameter

#### 4.2.1. AE Energy Conversion

AE energy (EAE) reflects the fact that the energy (Ee) released during fracturing of specimens could be given by(19)EAE=nEe,where EAE, n, and Ee represent AE energy (ms·mV), AE energy conversion coefficient (ms·mV/10−3 J) and energy (10−3 J) released from specimens, respectively. It can be seen from Figure 5(b) that, the stress-strain curve of PR specimens was approximately linear, and the energy Ee released from the PR specimens can be approximately equivalent to the elastic energy accumulated in the specimens. Thus, according to equation (14), the elastic energy accumulated in the PR specimens is expressed as follows:(20)Ee=Vσcr22Er.Thus, the cumulative energy released from PR specimens during the entire loading process was calculated by using equation (20). The cumulative energy (Ee) released from of different rock specimens and corresponding cumulative AE energy (EAE) are listed in Table 6. Through calculation (equation (19), the AE energy conversion coefficient (n) of different PR specimens ranged from 13.435 to 14.249 ms·mV/10−3 J, and its mean (13.796 ms·mV/10−3 J) was used as the conversion coefficient here.Table 6
Cumulative energy during the entire loading process and conversion coefficient.
Serial number of specimens
E
AE (ms·mV)
E
e (10−3 J)
n (ms·mV/10−3 J)
PR-1
316279
23541.2
13.4351
PR-2
303657
22157.1
13.7047
PR-3
327701
22998.7
14.2487
Mean
13.7962


#### 4.2.2. Strength-Volume Density Constant of Coal and Rock

In accordance with equation (16), the cumulative energy releases Ec′ and Er′ of PC and PR specimens before impact-induced failure were separately calculated as follows:(21)Ec′=∫0σeEcVλce−λcσdσ,(22)Er′=∫0σeErVλre−λrσdσ.According to equation (19), the cumulative AE energies of PC and PR specimens before impact-induced failure were separately obtained as follows:(23)Ec−AE′=n∫0σeEcVλce−λcσdσ,(24)Er−AE′=n∫0σeErVλre−λrσdσ.Under uniaxial loading, PC and PR specimens were only subjected to axial load. Based on equation (14), the energy densities of PC and PR specimens were separately shown as follows:(25)eEc=σ22E,eEr=σ22E.By separately fitting the curves of cumulative AE energies in PC and PR specimens before impact-induced failure by using equations (23) and (24), the strength-volume rate parameter λc and λr of coal and rock masses could be attained. The impact energies in specimens were equivalent to the energy released by all particle elements with uniaxial strength greater than σ, which is beyond the range of data-fitting adopted here. Owing to dynamic failure times of different specimens all being less than 200 ms, the fitting data range was deemed to be from the beginning of loading to 200 ms before reaching the peak stress. The fitting curves are shown in Figure 9.Figure 9
Fitting curves of cumulative AE energies in coal and rock masses before impact-induced failure: (a) PC specimens; (b) PR specimens.
(a)
(b)The fitting results of different specimens are listed in Table7: R2 for data from PC and PR specimens was between 0.94365 and 0.97692, and equations (21) and (22) can accurately describe the process of energy release of specimens before impact-induced failure. The strength-volume rate parameters λc and λr of coal and rock masses were separately valued as their corresponding means (0.02557 and 0.00156).Table 7
Fitting results of cumulative AE energies in coal and rock masses.
Serial number of specimens
λ
c
R
2
Serial number of specimens
λ
r
R
2
PC-1
0.02495
0.97841
PR-1
0.00158
0.94062
PC-2
0.02671
0.97105
PR-2
0.00149
0.94329
PC-3
0.02506
0.95187
PR-3
0.00161
0.95468
Mean
0.02557
0.96711
Mean
0.00156
0.94620


### 4.3. Prediction of Cumulative AE Energies before Impact-Induced Failure

#### 4.3.1. Fracturing Conditions and Energy Density of Particle Elements in CR Combined Specimens

Based on the analysis presented in Section3, coal and rock sections in CR combined specimens are under plane stress. Based on equations (6) ∼ (9), the failure condition of particle elements of coal sections with strength σbc in CR combined specimens is as follows:(26)σbcσ=ξc+1cosα4+m2−2m−3−cosαξc−1m+12σ.The failure condition of particle elements of rock sections with strengthσbr was obtained as follows:(27)σbrσ=ξr+1cosα4+m2+2m−3+cosαξr−1m−12σ,where m=Ecμr−Erμc/Ecμr+Erμc−Ec−Er. Equations (28) and (29) separately show the energy densities eEc and eEr of particle elements in coal and rock sections within CR combined specimens:(28)eEcσ=2+2μc−cos2α2μcm+12−2m2+12Ecσ2,(29)eErσ=2+2μr−cos2α2μrm−12−2m2+12Erσ2.


#### 4.3.2. Cumulative Energy before Impact-Induced Failure

The cumulative energy releases from coal and rock section in CR combined specimens with inclinations of structural planes before impact-induced failure are separately expressed as follows:(30)Ec′=∫0σbceEcVcλce−λcσbcdσbc,(31)Er′=∫0σbreErVrλre−λrσbrdσbr.The cumulative energy release (Ecr′) before impact-induced failure of CR combined specimens with different inclinations is the energy jointly released from coal and rock specimens, so the following equation could be obtained as follows:(32)Ecr′=Ec′+Er′.According to equation (19), the corresponding cumulative AE energy (Ecr−AE′) before impact is given by:(33)Ecr−AE′=nEc′+Er′.Parameters pertaining to the mechanical properties of coal and rock masses in Table1, the strength-volume density constant in Table 7, and equations (27) ∼ (33) were substituted into equations (30) and (31), and AE energy conversion coefficient n = 13.7962 was substituted into equation (32). Based thereon, the predicted curves of cumulative AE energy before impact-induced failure of CR combined specimens with different inclinations of structural planes can be obtained. The measured values of AE energies and measured and predicted curves of cumulative AE energies before impact-induced failure of CR combined specimens with different inclinations of structural planes are summarised in Figure 10. The AE energy exhibited a high discreteness, while the curve of cumulative AE energy showed strong regularity and was consistent with the predicted curve. The cumulative AE energies before impact-induced failure of CR combined specimens with different inclinations of structural planes were predicted, and the data were compared with the measured value (Table 8), with a relative error between −7.6 and 7.6.Figure 10
Curves of cumulative AE energies before failure of CR combined specimens (a) withα = 0°; (b) α = 15°; (c) α = 30°; (d) α = 45°.
(a)
(b)
(c)
(d)Table 8
Cumulative AE energy before failure of CR combined specimens.
Serial number of specimens
Predicted cumulative AE energy before impact-induced failure (ms·mV)
Measured cumulative AE energy before impact-induced failure (ms·mV)
Relative error
CR-0-1
8677
8875
2.3
CR-0-2
9444
9036
−4.3
CR-0-3
11181
10936
−2.2
Mean
9735
9616
CR-15-1
6324
5846
−7.6
CR-15-2
7917
7425
−6.2
CR-15-3
6348
6532
2.9
Mean
6840
6601
CR-30-1
3730
3801
1.9
CR-30-2
5003
4832
−3.4
CR-30-3
4106
4026
−1.9
Mean
4261
4220
CR-45-1
2980
2871
−3.7
CR-45-2
3723
3611
−3
CR-45-3
3183
3425
7.6
Mean
3287
3302


### 4.4. Prediction of Impact Energy

In accordance with equations (6) ∼ (9), when CR combined specimens were damaged, the minimum strengths of unfractured particle elements of coal and rock sections are separately given by(34)σbc=ξc+1cosα4+m2−2m−3−cosαξc−1m+12σcr,(35)σbr=ξr+1cosα4+m2+2m−3+cosαξr−1m−12σcr.The cumulative elastic energy densities in unfractured particle elements in coal and rock sections separately are given by(36)eEc=2+2μc−cos2α2μcm+12−2m2+12Ecσcr2,(37)eEr=2+2μr−cos2α2μrm−12−2m2+12Erσcr2.The AE impact energyEbcr−AE of CR combined specimens could be predicted by using the following equation(38)Ebcr−AE=neEcVce−λcσbc+eErVre−λrσbr.By substituting equations (34) ∼ (37) into (38), the predicted value of AE impact energy in CR combined specimens was obtained. The measured value of AE impact energy was equivalent to the cumulative AE energy within 200 ms before, and after, reaching peak stress. As shown in Table 9, compared with the predicted value, the relative error of the measured values of AE impact energies in different CR combined specimens ranged from −8.9 to 4.8. With increasing inclination, the impact energy in CR combined specimens decreased.Table 9
Measured values of AE impact energy in specimens and their relative errors.
Serial number of specimens
Predicted AE impact energy (ms·mV)
Measured AE impact energy (ms·mV)
Relative error (%)
Serial number of specimens
Predicted AE impact energy (ms·mV)
Measured AE impact energy (ms·mV)
Relative error (%)
CR-0-1
149120
143014
−4.1
CR-30-1
59406
59104
−0.5
CR-0-2
158076
149484
−5.4
CR-30-2
72918
67522
−7.4
CR-0-3
177639
161884
−8.9
CR-30-3
63519
61232
−3.6
Mean
161612
151461
Mean
65281
62619
CR-15-1
108598
105713
−2.7
CR-45-1
39346
38733
−1.6
CR-15-2
126963
123693
−2.6
CR-45-2
45893
43654
−4.9
CR-15-3
108887
114063
4.8
CR-45-3
41178
38940
−5.4
Mean
114816
114490
Mean
42139
40442


### 4.5. Analysis on Effects of Inclination on Impact Energy in a CR Combined Body

According to equations (34) and (35), when the inclination α was large, particle elements were easily fractured, which led to a small volume density of unfractured particle elements when the CR combined body was damaged. In addition, in accordance with equations (36) and (37), the energy density of particle elements in the CR combined body showed a quadratic positive correlation with the strength σcr of a CR combined body when failure occurred. Based on equations (10), the strength σcr decreased with the increase of inclination α. Therefore, the greater the inclination, the lower the strength of the CR combined body and the smaller the volume density and energy density of unfractured particle elements at failure. This results in less impact energy being released when the CR combined body was damaged.


## 4.1. Calculation Model

As shown in Figures6(c) to 6(f), energy release occurred throughout the loading process of CR combined specimens, and microfractures were constantly generated, therein AE energy exhibited a high discreteness, which indicated that the microfracturing degree of specimens also showed a high discreteness. To quantify the process of energy release in specimens, according to the theory of particle mechanics, it is supposed that the relationship between uniaxial strength (σ) and volume density (fσ) of particulate elements of coal and rock masses is expressed as follows [32–34]:(13)fσ=λe−λσ,where λ represents the strength-volume rate parameter of particle elements. It is supposed that particle elements exhibited a linear-elastic constitutive relationship, and their energy density is expressed as follows [35]:(14)eE=12σxεx+σyεy+σzεz+τxyγxy+τxzγxz+τyzγyz,where eE denotes the energy density (MPa) of particle elements. Before the impact-induced failure of CR combined specimens, with increasing load, cumulative energy released from coal or rock masses was governed by equation (15) on condition that the particulate element with uniaxial strength of σ was damaged:(15)E′=∫0σeEVfσdσ,where E′ represents the cumulative energy released (10−3 J) from coal or rock masses before impact-induced failure and V represents the volume (mm3) of the coal or rock masses. By substituting equation (13) into equation (15), the cumulative energy release before impact-induced failure of coal and rock sections is as follows:(16)E′=∫0σeEVλe−λσdσ.When impact-induced failure of CR combined specimens occurred, the impact energy (Eb) released from coal or rock masses was equivalent to the elastic energy accumulated in unbroken particle elements (with uniaxial strength larger than σb). Thus,(17)Eb=eEV1−∫0σbfσbdσb=eEVe−λσb.Hence, the calculation model for impact energy (Eber) in a CR combined body with inclinations of structural planes is expressed as follows:(18)Ebcr=eEcVce−λcσbc+eErVre−λrσbr,where Eber denotes the impact energy (10−3 J) of the CR combined body with inclinations of structural planes, eEc and eEr represent energy densities (MPa), Vc and Vr represent the volumes (mm3), while λc and λr denote the strength-volume rate parameters, of coal and rock masses, respectively, and σbc and σbr denote the uniaxial strength (MPa) of particle elements, of coal and rock masses, which were fractured immediately before impact-induced failure of the CR combined body, respectively.


## 4.2. Strength-Volume Rate Parameter

### 4.2.1. AE Energy Conversion

AE energy (EAE) reflects the fact that the energy (Ee) released during fracturing of specimens could be given by(19)EAE=nEe,where EAE, n, and Ee represent AE energy (ms·mV), AE energy conversion coefficient (ms·mV/10−3 J) and energy (10−3 J) released from specimens, respectively. It can be seen from Figure 5(b) that, the stress-strain curve of PR specimens was approximately linear, and the energy Ee released from the PR specimens can be approximately equivalent to the elastic energy accumulated in the specimens. Thus, according to equation (14), the elastic energy accumulated in the PR specimens is expressed as follows:(20)Ee=Vσcr22Er.Thus, the cumulative energy released from PR specimens during the entire loading process was calculated by using equation (20). The cumulative energy (Ee) released from of different rock specimens and corresponding cumulative AE energy (EAE) are listed in Table 6. Through calculation (equation (19), the AE energy conversion coefficient (n) of different PR specimens ranged from 13.435 to 14.249 ms·mV/10−3 J, and its mean (13.796 ms·mV/10−3 J) was used as the conversion coefficient here.Table 6
Cumulative energy during the entire loading process and conversion coefficient.
Serial number of specimens
E
AE (ms·mV)
E
e (10−3 J)
n (ms·mV/10−3 J)
PR-1
316279
23541.2
13.4351
PR-2
303657
22157.1
13.7047
PR-3
327701
22998.7
14.2487
Mean
13.7962


### 4.2.2. Strength-Volume Density Constant of Coal and Rock

In accordance with equation (16), the cumulative energy releases Ec′ and Er′ of PC and PR specimens before impact-induced failure were separately calculated as follows:(21)Ec′=∫0σeEcVλce−λcσdσ,(22)Er′=∫0σeErVλre−λrσdσ.According to equation (19), the cumulative AE energies of PC and PR specimens before impact-induced failure were separately obtained as follows:(23)Ec−AE′=n∫0σeEcVλce−λcσdσ,(24)Er−AE′=n∫0σeErVλre−λrσdσ.Under uniaxial loading, PC and PR specimens were only subjected to axial load. Based on equation (14), the energy densities of PC and PR specimens were separately shown as follows:(25)eEc=σ22E,eEr=σ22E.By separately fitting the curves of cumulative AE energies in PC and PR specimens before impact-induced failure by using equations (23) and (24), the strength-volume rate parameter λc and λr of coal and rock masses could be attained. The impact energies in specimens were equivalent to the energy released by all particle elements with uniaxial strength greater than σ, which is beyond the range of data-fitting adopted here. Owing to dynamic failure times of different specimens all being less than 200 ms, the fitting data range was deemed to be from the beginning of loading to 200 ms before reaching the peak stress. The fitting curves are shown in Figure 9.Figure 9
Fitting curves of cumulative AE energies in coal and rock masses before impact-induced failure: (a) PC specimens; (b) PR specimens.
(a)
(b)The fitting results of different specimens are listed in Table7: R2 for data from PC and PR specimens was between 0.94365 and 0.97692, and equations (21) and (22) can accurately describe the process of energy release of specimens before impact-induced failure. The strength-volume rate parameters λc and λr of coal and rock masses were separately valued as their corresponding means (0.02557 and 0.00156).Table 7
Fitting results of cumulative AE energies in coal and rock masses.
Serial number of specimens
λ
c
R
2
Serial number of specimens
λ
r
R
2
PC-1
0.02495
0.97841
PR-1
0.00158
0.94062
PC-2
0.02671
0.97105
PR-2
0.00149
0.94329
PC-3
0.02506
0.95187
PR-3
0.00161
0.95468
Mean
0.02557
0.96711
Mean
0.00156
0.94620


## 4.2.1. AE Energy Conversion

AE energy (EAE) reflects the fact that the energy (Ee) released during fracturing of specimens could be given by(19)EAE=nEe,where EAE, n, and Ee represent AE energy (ms·mV), AE energy conversion coefficient (ms·mV/10−3 J) and energy (10−3 J) released from specimens, respectively. It can be seen from Figure 5(b) that, the stress-strain curve of PR specimens was approximately linear, and the energy Ee released from the PR specimens can be approximately equivalent to the elastic energy accumulated in the specimens. Thus, according to equation (14), the elastic energy accumulated in the PR specimens is expressed as follows:(20)Ee=Vσcr22Er.Thus, the cumulative energy released from PR specimens during the entire loading process was calculated by using equation (20). The cumulative energy (Ee) released from of different rock specimens and corresponding cumulative AE energy (EAE) are listed in Table 6. Through calculation (equation (19), the AE energy conversion coefficient (n) of different PR specimens ranged from 13.435 to 14.249 ms·mV/10−3 J, and its mean (13.796 ms·mV/10−3 J) was used as the conversion coefficient here.Table 6
Cumulative energy during the entire loading process and conversion coefficient.
Serial number of specimens
E
AE (ms·mV)
E
e (10−3 J)
n (ms·mV/10−3 J)
PR-1
316279
23541.2
13.4351
PR-2
303657
22157.1
13.7047
PR-3
327701
22998.7
14.2487
Mean
13.7962


## 4.2.2. Strength-Volume Density Constant of Coal and Rock

In accordance with equation (16), the cumulative energy releases Ec′ and Er′ of PC and PR specimens before impact-induced failure were separately calculated as follows:(21)Ec′=∫0σeEcVλce−λcσdσ,(22)Er′=∫0σeErVλre−λrσdσ.According to equation (19), the cumulative AE energies of PC and PR specimens before impact-induced failure were separately obtained as follows:(23)Ec−AE′=n∫0σeEcVλce−λcσdσ,(24)Er−AE′=n∫0σeErVλre−λrσdσ.Under uniaxial loading, PC and PR specimens were only subjected to axial load. Based on equation (14), the energy densities of PC and PR specimens were separately shown as follows:(25)eEc=σ22E,eEr=σ22E.By separately fitting the curves of cumulative AE energies in PC and PR specimens before impact-induced failure by using equations (23) and (24), the strength-volume rate parameter λc and λr of coal and rock masses could be attained. The impact energies in specimens were equivalent to the energy released by all particle elements with uniaxial strength greater than σ, which is beyond the range of data-fitting adopted here. Owing to dynamic failure times of different specimens all being less than 200 ms, the fitting data range was deemed to be from the beginning of loading to 200 ms before reaching the peak stress. The fitting curves are shown in Figure 9.Figure 9
Fitting curves of cumulative AE energies in coal and rock masses before impact-induced failure: (a) PC specimens; (b) PR specimens.
(a)
(b)The fitting results of different specimens are listed in Table7: R2 for data from PC and PR specimens was between 0.94365 and 0.97692, and equations (21) and (22) can accurately describe the process of energy release of specimens before impact-induced failure. The strength-volume rate parameters λc and λr of coal and rock masses were separately valued as their corresponding means (0.02557 and 0.00156).Table 7
Fitting results of cumulative AE energies in coal and rock masses.
Serial number of specimens
λ
c
R
2
Serial number of specimens
λ
r
R
2
PC-1
0.02495
0.97841
PR-1
0.00158
0.94062
PC-2
0.02671
0.97105
PR-2
0.00149
0.94329
PC-3
0.02506
0.95187
PR-3
0.00161
0.95468
Mean
0.02557
0.96711
Mean
0.00156
0.94620


## 4.3. Prediction of Cumulative AE Energies before Impact-Induced Failure

### 4.3.1. Fracturing Conditions and Energy Density of Particle Elements in CR Combined Specimens

Based on the analysis presented in Section3, coal and rock sections in CR combined specimens are under plane stress. Based on equations (6) ∼ (9), the failure condition of particle elements of coal sections with strength σbc in CR combined specimens is as follows:(26)σbcσ=ξc+1cosα4+m2−2m−3−cosαξc−1m+12σ.The failure condition of particle elements of rock sections with strengthσbr was obtained as follows:(27)σbrσ=ξr+1cosα4+m2+2m−3+cosαξr−1m−12σ,where m=Ecμr−Erμc/Ecμr+Erμc−Ec−Er. Equations (28) and (29) separately show the energy densities eEc and eEr of particle elements in coal and rock sections within CR combined specimens:(28)eEcσ=2+2μc−cos2α2μcm+12−2m2+12Ecσ2,(29)eErσ=2+2μr−cos2α2μrm−12−2m2+12Erσ2.


### 4.3.2. Cumulative Energy before Impact-Induced Failure

The cumulative energy releases from coal and rock section in CR combined specimens with inclinations of structural planes before impact-induced failure are separately expressed as follows:(30)Ec′=∫0σbceEcVcλce−λcσbcdσbc,(31)Er′=∫0σbreErVrλre−λrσbrdσbr.The cumulative energy release (Ecr′) before impact-induced failure of CR combined specimens with different inclinations is the energy jointly released from coal and rock specimens, so the following equation could be obtained as follows:(32)Ecr′=Ec′+Er′.According to equation (19), the corresponding cumulative AE energy (Ecr−AE′) before impact is given by:(33)Ecr−AE′=nEc′+Er′.Parameters pertaining to the mechanical properties of coal and rock masses in Table1, the strength-volume density constant in Table 7, and equations (27) ∼ (33) were substituted into equations (30) and (31), and AE energy conversion coefficient n = 13.7962 was substituted into equation (32). Based thereon, the predicted curves of cumulative AE energy before impact-induced failure of CR combined specimens with different inclinations of structural planes can be obtained. The measured values of AE energies and measured and predicted curves of cumulative AE energies before impact-induced failure of CR combined specimens with different inclinations of structural planes are summarised in Figure 10. The AE energy exhibited a high discreteness, while the curve of cumulative AE energy showed strong regularity and was consistent with the predicted curve. The cumulative AE energies before impact-induced failure of CR combined specimens with different inclinations of structural planes were predicted, and the data were compared with the measured value (Table 8), with a relative error between −7.6 and 7.6.Figure 10
Curves of cumulative AE energies before failure of CR combined specimens (a) withα = 0°; (b) α = 15°; (c) α = 30°; (d) α = 45°.
(a)
(b)
(c)
(d)Table 8
Cumulative AE energy before failure of CR combined specimens.
Serial number of specimens
Predicted cumulative AE energy before impact-induced failure (ms·mV)
Measured cumulative AE energy before impact-induced failure (ms·mV)
Relative error
CR-0-1
8677
8875
2.3
CR-0-2
9444
9036
−4.3
CR-0-3
11181
10936
−2.2
Mean
9735
9616
CR-15-1
6324
5846
−7.6
CR-15-2
7917
7425
−6.2
CR-15-3
6348
6532
2.9
Mean
6840
6601
CR-30-1
3730
3801
1.9
CR-30-2
5003
4832
−3.4
CR-30-3
4106
4026
−1.9
Mean
4261
4220
CR-45-1
2980
2871
−3.7
CR-45-2
3723
3611
−3
CR-45-3
3183
3425
7.6
Mean
3287
3302


## 4.3.1. Fracturing Conditions and Energy Density of Particle Elements in CR Combined Specimens

Based on the analysis presented in Section3, coal and rock sections in CR combined specimens are under plane stress. Based on equations (6) ∼ (9), the failure condition of particle elements of coal sections with strength σbc in CR combined specimens is as follows:(26)σbcσ=ξc+1cosα4+m2−2m−3−cosαξc−1m+12σ.The failure condition of particle elements of rock sections with strengthσbr was obtained as follows:(27)σbrσ=ξr+1cosα4+m2+2m−3+cosαξr−1m−12σ,where m=Ecμr−Erμc/Ecμr+Erμc−Ec−Er. Equations (28) and (29) separately show the energy densities eEc and eEr of particle elements in coal and rock sections within CR combined specimens:(28)eEcσ=2+2μc−cos2α2μcm+12−2m2+12Ecσ2,(29)eErσ=2+2μr−cos2α2μrm−12−2m2+12Erσ2.


## 4.3.2. Cumulative Energy before Impact-Induced Failure

The cumulative energy releases from coal and rock section in CR combined specimens with inclinations of structural planes before impact-induced failure are separately expressed as follows:(30)Ec′=∫0σbceEcVcλce−λcσbcdσbc,(31)Er′=∫0σbreErVrλre−λrσbrdσbr.The cumulative energy release (Ecr′) before impact-induced failure of CR combined specimens with different inclinations is the energy jointly released from coal and rock specimens, so the following equation could be obtained as follows:(32)Ecr′=Ec′+Er′.According to equation (19), the corresponding cumulative AE energy (Ecr−AE′) before impact is given by:(33)Ecr−AE′=nEc′+Er′.Parameters pertaining to the mechanical properties of coal and rock masses in Table1, the strength-volume density constant in Table 7, and equations (27) ∼ (33) were substituted into equations (30) and (31), and AE energy conversion coefficient n = 13.7962 was substituted into equation (32). Based thereon, the predicted curves of cumulative AE energy before impact-induced failure of CR combined specimens with different inclinations of structural planes can be obtained. The measured values of AE energies and measured and predicted curves of cumulative AE energies before impact-induced failure of CR combined specimens with different inclinations of structural planes are summarised in Figure 10. The AE energy exhibited a high discreteness, while the curve of cumulative AE energy showed strong regularity and was consistent with the predicted curve. The cumulative AE energies before impact-induced failure of CR combined specimens with different inclinations of structural planes were predicted, and the data were compared with the measured value (Table 8), with a relative error between −7.6 and 7.6.Figure 10
Curves of cumulative AE energies before failure of CR combined specimens (a) withα = 0°; (b) α = 15°; (c) α = 30°; (d) α = 45°.
(a)
(b)
(c)
(d)Table 8
Cumulative AE energy before failure of CR combined specimens.
Serial number of specimens
Predicted cumulative AE energy before impact-induced failure (ms·mV)
Measured cumulative AE energy before impact-induced failure (ms·mV)
Relative error
CR-0-1
8677
8875
2.3
CR-0-2
9444
9036
−4.3
CR-0-3
11181
10936
−2.2
Mean
9735
9616
CR-15-1
6324
5846
−7.6
CR-15-2
7917
7425
−6.2
CR-15-3
6348
6532
2.9
Mean
6840
6601
CR-30-1
3730
3801
1.9
CR-30-2
5003
4832
−3.4
CR-30-3
4106
4026
−1.9
Mean
4261
4220
CR-45-1
2980
2871
−3.7
CR-45-2
3723
3611
−3
CR-45-3
3183
3425
7.6
Mean
3287
3302


## 4.4. Prediction of Impact Energy

In accordance with equations (6) ∼ (9), when CR combined specimens were damaged, the minimum strengths of unfractured particle elements of coal and rock sections are separately given by(34)σbc=ξc+1cosα4+m2−2m−3−cosαξc−1m+12σcr,(35)σbr=ξr+1cosα4+m2+2m−3+cosαξr−1m−12σcr.The cumulative elastic energy densities in unfractured particle elements in coal and rock sections separately are given by(36)eEc=2+2μc−cos2α2μcm+12−2m2+12Ecσcr2,(37)eEr=2+2μr−cos2α2μrm−12−2m2+12Erσcr2.The AE impact energyEbcr−AE of CR combined specimens could be predicted by using the following equation(38)Ebcr−AE=neEcVce−λcσbc+eErVre−λrσbr.By substituting equations (34) ∼ (37) into (38), the predicted value of AE impact energy in CR combined specimens was obtained. The measured value of AE impact energy was equivalent to the cumulative AE energy within 200 ms before, and after, reaching peak stress. As shown in Table 9, compared with the predicted value, the relative error of the measured values of AE impact energies in different CR combined specimens ranged from −8.9 to 4.8. With increasing inclination, the impact energy in CR combined specimens decreased.Table 9
Measured values of AE impact energy in specimens and their relative errors.
Serial number of specimens
Predicted AE impact energy (ms·mV)
Measured AE impact energy (ms·mV)
Relative error (%)
Serial number of specimens
Predicted AE impact energy (ms·mV)
Measured AE impact energy (ms·mV)
Relative error (%)
CR-0-1
149120
143014
−4.1
CR-30-1
59406
59104
−0.5
CR-0-2
158076
149484
−5.4
CR-30-2
72918
67522
−7.4
CR-0-3
177639
161884
−8.9
CR-30-3
63519
61232
−3.6
Mean
161612
151461
Mean
65281
62619
CR-15-1
108598
105713
−2.7
CR-45-1
39346
38733
−1.6
CR-15-2
126963
123693
−2.6
CR-45-2
45893
43654
−4.9
CR-15-3
108887
114063
4.8
CR-45-3
41178
38940
−5.4
Mean
114816
114490
Mean
42139
40442


## 4.5. Analysis on Effects of Inclination on Impact Energy in a CR Combined Body

According to equations (34) and (35), when the inclination α was large, particle elements were easily fractured, which led to a small volume density of unfractured particle elements when the CR combined body was damaged. In addition, in accordance with equations (36) and (37), the energy density of particle elements in the CR combined body showed a quadratic positive correlation with the strength σcr of a CR combined body when failure occurred. Based on equations (10), the strength σcr decreased with the increase of inclination α. Therefore, the greater the inclination, the lower the strength of the CR combined body and the smaller the volume density and energy density of unfractured particle elements at failure. This results in less impact energy being released when the CR combined body was damaged.


## 5. Conclusions

The mechanical properties of a CR combined body with different inclinations of its structural planes were tested, and its failure mechanism was analysed. On this basis, a calculation model for impact energy in a CR combined body was proposed, and its rationality was validated by using data from AE monitoring. The following conclusions could be drawn:(1)
Inclination significantly affected the mechanical properties of a CR combined body. With increasing inclination, the axial elastic modulus and Poisson’s ratio of rock section in a CR combined body increased, while the axial elastic modulus of such coal samples decreased.(2)
Compared with rock burst-prone coal and rock masses, a CR combined body also showed significant impact-induced failure characteristics. With increasing inclination, the failure mode of a CR combined body varied from one of overall failure to failure of only the coal samples. Therefore, the level of damage to specimens decreased.(3)
The calculation model for impact energy in a CR combined body with different inclinations of structural planes is given byEbcr=eEcVce−λcσbc+eErVre−λrσbr. Inclination exerted a significant influence on the impact energy in a CR combined body, that is, the greater the inclination, the lower the impact energy in the CR combined body.



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*Source: 3735427-2019-12-17.xml*