and energy conservation theory, Tu et al. [11] established a slope instability criterion using a strength-reduction method based on energy conversion. Peng et al. [12] studied the relationship between crack angle and energy inside the loaded rock, and the results showed that angle was positively correlated with energy storage capacity. Wang et al. [13] used numerical simulation software to calculate the energy storage of surrounding rock, expounded the relationship between rock failure and elastic energy, and confirmed that the simulated strain energy analysis method could be used for rockburst prediction. Wu et al. [14] studied the energy evolution of rock under different loading modes. Wang et al. [15] conducted a uniaxial cyclic charge test on the lower dry and saturated sandstone, and analyzed the strength and deformation characteristics of the rock, as well as the change and distribution of energy under the dry and saturated state. Jia et al. [16] studied the energy variation law of rock mass at different depths during mining, and the results showed that the increase of the depth of rock mass would lead to the increase of all types of energy inside it. Deng et al. [17] carried out dynamic uniaxial compression tests on rock samples under impact velocity and analyzed the energy dissipation law in the dynamic failure process of rock. Zhang et al. [18] studied the influence of confining pressure on the change of energy inside the rock in the triaxial test, and the results showed that the increase of confining pressure could improve the efficiency of energy accumulation to a certain extent. Munoz et al. [19,20] defined a new rock brittleness index to elucidate the energy accumulation and release of rock failure. Song et al. [21] performed uniaxial cyclic loading and unloading tests on coal and rock, collected electromagnetic radiation signals released in the test process, and established the correlation between electromagnetic radiation and dissipated energy.

The above research has greatly enriched the application of energy theory in rocks, but most scholars focus on intact rocks and have little research on rock discontinuity, and in the cyclic loading and unloading test, often with unloading in the next cycle to 0 MPa, by changing the upper limit of cyclic loading for different stress amplitudes. The lower limit of cyclic load is often not zero, and therefore in the present work cement mortar was used to represent the rock, and a CSS-1950 biaxial rheological testing machine was adopted to conduct graded cyclic loading and unloading tests on Barton's standard profile line discontinuities of different joint roughness coefficients (JRCs). The energy evolution of samples in the process of failure was revealed, which can provide a theoretical basis for studying rock damage and failure mechanisms from the perspective of energy.

#### **2. Test Equipment and Specimens**

#### *2.1. Test Equipment*

The CSS-1950 (Model of testing machine, CSS is creep shear strength) rock biaxial rheological testing machine was used in this test (Figure 1a), which is manufactured by the Changchun Institute of Testing Machines in Changchun, China. The test machine includes vertical and horizontal loading systems with maximum vertical and horizontal compression loads of 500 and 300 kN, respectively. Two linear variable displacement transducers (LVDT) were used with a measurement range of 0–10 mm and accuracy of 0.0001 mm to monitor vertical and horizontal deformation, as shown in Figure 1b. The test machine adopts servo-motor control, a pressure system for screw pressure, load-rate control, and a continuous working time of more than 1000 h.

**Figure 1.** (**a**) CSS-1950 rock biaxial rheological testing machine; (**b**) specimen and monitoring system. **Figure 1.** (**a**) CSS-1950 rock biaxial rheological testing machine; (**b**) specimen and monitoring system.

#### *2.2. Sample Preparation*

*2.2. Sample Preparation*  Due to the random composition and surface morphology of natural rock mass, it is difficult to prepare relatively uniform samples using natural rock mass, and it is impossible to quantify the roughness of the rock discontinuity, which results in difficulty when comparing test results; therefore, because cement mortar has good uniformity, it is often used as a kind of rock material to simulate rock discontinuities [22,23]. Therefore, in the present work, cement mortar was used as a similar material to prepare Barton's standard Due to the random composition and surface morphology of natural rock mass, it is difficult to prepare relatively uniform samples using natural rock mass, and it is impossible to quantify the roughness of the rock discontinuity, which results in difficulty when comparing test results; therefore, because cement mortar has good uniformity, it is often used as a kind of rock material to simulate rock discontinuities [22,23]. Therefore, in the present work, cement mortar was used as a similar material to prepare Barton's standard profile line discontinuities with different JRCs for testing.

profile line discontinuities with different JRCs for testing. In this test, the cement mortar samples named 1#,4#, 8#, and 10# were respectively adopted to represent the features of structural planes with different roughness. Their Barton's standard profile line features are shown in Figure 2b. For the convenience of analysis, the JRC of the sample was taken as the intermediate value, that is, the cement mortar In this test, the cement mortar samples named 1# , 4# , 8# , and 10# were respectively adopted to represent the features of structural planes with different roughness. Their Barton's standard profile line features are shown in Figure 2b. For the convenience of analysis, the JRC of the sample was taken as the intermediate value, that is, the cement mortar samples named 1# , 4# , 8# , and 10# were, respectively, 1, 7, 15, and 19 herein. *Materials* **2022**, *15*, x FOR PEER REVIEW 4 of 16

stored in the standard curing chamber at the temperature and humidity of (20 ± 1) °C and 95%, respectively, for 28 days before testing. **Figure 2.** (**a**) Steel mold; (**b**) Barton's standard profile lines; (**c**) sample. **Figure 2.** (**a**) Steel mold; (**b**) Barton's standard profile lines; (**c**) sample.

According to the test requirements, a total of 29 specimens were prepared, including 5 intact samples and 24 rock discontinuity samples (the cement mortar samples named 1#,4#, 8#, and 10# were prepared in 6 pieces each). In total, the intact samples were used for the uniaxial compression test, and the rock discontinuity samples were used for the direct shear test and graded cyclic shear test. **3. Test Procedure**  *3.1. Uniaxial Compression Test*  To determine the normal stress value of shear test, uniaxial compression tests were conducted on five complete cement mortar specimens at a loading rate of 0.4 kN/s. The The steel mold used in this test was made by referring to Barton's standard profiles and using high precision (0.1 µm) computer control [24], as shown in Figure 2a. According to need, the upper and lower parts of the sample were prepared, and the two parts were combined to form a complete rock discontinuity sample (Figure 2c). Each specimen measured 100 × 100 × 100 mm.

test results are shown in Table 1. Here, 10%, 20%, and 30% of the average compressive strengths were taken as the normal stresses of subsequent shear tests, which were 2.17, 4.35, and 6.52 MPa, respectively. **Table 1.** Uniaxial compressive strength of each sample (*σ*c is compressive strength). **Sample Peak Stress/kN** *σ***c/MPa**  The sample material was Portland cement with a compressive strength of 32.5 MPa, standard sand and water, and the mixing ratio was sand/cement/water of 4:2:1. After mixing evenly, the mold was filled. After filling, the sample was removed after the cement mortar was formed. After removal, the samples were placed in the curing chamber and stored in the standard curing chamber at the temperature and humidity of (20 ± 1) ◦C and 95%, respectively, for 28 days before testing.

1 228.47 22.85 2 233.44 23.34 3 177.69 17.77 4 248.37 24.84 5 198.55 19.86 According to the test requirements, a total of 29 specimens were prepared, including 5 intact samples and 24 rock discontinuity samples (the cement mortar samples named 1# , 4 # , 8# , and 10# were prepared in 6 pieces each). In total, the intact samples were used for the uniaxial compression test, and the rock discontinuity samples were used for the direct shear test and graded cyclic shear test.

Average value 217.30 21.73

**Sample Peak Stress/kN τc/MPa**  1–2.17 7.5 1.5 1–4.35 15.2 3.0 1–6.52 18.5 3.7 4–2.17 10.1 2.0 4–4.35 15.2 3.0 4–6.52 22.5 4.5 8–2.17 12.2 2.4 8–4.35 20.0 4.0 8–6.52 25.0 5.0 10–2.17 12.5 2.5

To obtain the shear strength of rock discontinuities, the cement mortar samples named 1#,4#, 8#, and 10# (JRC = 1, 7, 15, 19) were selected and shear tests were conducted at a rate of 0.2 kN/s under the normal stress of 2.17, 4.35, and 6.52 MPa, respectively, until failure. The shear strength obtained was used as the basis for the classification of shear load grades in subsequent cyclic shear tests, and Barton's standard profile line number and normal stress were used to number the test results. For example, the test results of rock discontinuity No. 1 under the normal stress of 2.17 MPa were denoted "1–2.17", and

**Table 2.** Shear strength of each sample (*τ*c is shear strength).

*3.2. Direct Shear Test*

the test data are listed in Table 2.

#### **3. Test Procedure**

#### *3.1. Uniaxial Compression Test*

To determine the normal stress value of shear test, uniaxial compression tests were conducted on five complete cement mortar specimens at a loading rate of 0.4 kN/s. The test results are shown in Table 1. Here, 10%, 20%, and 30% of the average compressive strengths were taken as the normal stresses of subsequent shear tests, which were 2.17, 4.35, and 6.52 MPa, respectively.

**Table 1.** Uniaxial compressive strength of each sample (*σ*c is compressive strength).

#### *3.2. Direct Shear Test*

To obtain the shear strength of rock discontinuities, the cement mortar samples named 1 # , 4# , 8# , and 10# (JRC = 1, 7, 15, 19) were selected and shear tests were conducted at a rate of 0.2 kN/s under the normal stress of 2.17, 4.35, and 6.52 MPa, respectively, until failure. The shear strength obtained was used as the basis for the classification of shear load grades in subsequent cyclic shear tests, and Barton's standard profile line number and normal stress were used to number the test results. For example, the test results of rock discontinuity No. 1 under the normal stress of 2.17 MPa were denoted "1–2.17", and the test data are listed in Table 2.

**Table 2.** Shear strength of each sample (*τ*c is shear strength).

#### *3.3. Graded Cyclic Shear Tests*

The cement mortar samples named 1# , 4# , 8# , and 10# (JRC = 1, 7, 15, 19) were selected to conduct cyclic shear tests under normal stresses of 2.17, 4.35, and 6.52 MPa, respectively. The loading of samples is shown in Figure 3a. The normal stress was first added to a predetermined value, and the shear stress was applied after the normal deformation was stabilized. The shear stress was divided into multiple stages. The upper limit of the shear stress at the first stage was 30% of the shear strength, and each stage increased the stress by 10% of the shear strength. Under the same level of loading, the cyclic amplitude of shear stress was 10% of the shear strength. Under the same level of loading, 10 cycles were carried out with a loading rate of 0.2 kN/s. Continuous loading was conducted until failure. The actual loading stress is listed in Table 3.

10–4.35 22.1 4.4 10–6.52 27.5 5.5

The cement mortar samples named 1#,4#, 8#, and 10# (JRC = 1, 7, 15, 19) were selected to conduct cyclic shear tests under normal stresses of 2.17, 4.35, and 6.52 MPa, respectively. The loading of samples is shown in Figure 3a. The normal stress was first added to a predetermined value, and the shear stress was applied after the normal deformation was stabilized. The shear stress was divided into multiple stages. The upper limit of the shear stress at the first stage was 30% of the shear strength, and each stage increased the stress by 10% of the shear strength. Under the same level of loading, the cyclic amplitude of shear stress was 10% of the shear strength. Under the same level of loading, 10 cycles were carried out with a loading rate of 0.2 kN/s. Continuous loading was conducted until fail-

**Figure 3.** (**a**) Sample loading diagram; (**b**) schematic diagram of test loading path (*σ* is the normal stress and *τ* is the shear stress). **Figure 3.** (**a**) Sample loading diagram; (**b**) schematic diagram of test loading path (*σ* is the normal stress and *τ* is the shear stress).

**Table 3.** Loading stress table. **Table 3.** Loading stress table.

*3.3. Graded Cyclic Shear Tests*

ure. The actual loading stress is listed in Table 3.

#### 10–4.35 1.32 0.44 **4. Results and Discussion**

#### 10–6.52 1.65 0.55 *4.1. Deformation Characteristics of Specimen during Failure*

**4. Results and Discussion**  *4.1. Deformation Characteristics of Specimen during Failure*  The purpose of this research was to reveal the energy evolution of rock discontinuity The purpose of this research was to reveal the energy evolution of rock discontinuity under cyclic loading. Due to the limitation of word count, only one group of test data was studied, and the stress–strain relationship of other samples is similar, so it will be repeated here.

under cyclic loading. Due to the limitation of word count, only one group of test data was Figure 4 displays the whole process of the stress and displacement curve for sample 4–6.52. As shown in the figure, during each cycle, the unloading curve does not coincide with the original loading curve, and the unloading curve can form a completely closed annular area with the reloading curve, namely, the hysteresis loop [25]. The reason is that rock materials are not ideal elastomers, and there are a large number of internal defects such as pores and micro-cracks, which lead to the closure of the original cracks and the initiation of new cracks in the test process [26], resulting in some irreversible damage to the sample. The appearance of the hysteresis loop is not only an experimental phenomenon, but also a manifestation of energy dissipation, and the area of the hysteresis loop can be characterized as the energy dissipated by crack closure, expansion, and through-cracking failure of the loaded sample; the larger the area of the hysteresis loop, the more energy consumed and the more severe the damage to the sample.

The rock discontinuity sample is different from the intact rock sample, and in the first loading process of each grading cycle stage, the upper and lower parts of the rock discontinuity sample usually slip, resulting in large residual deformation. With the continuous increase of stress, this sliding deformation gradually increases, and the deformation curve of the sample is gradually sparse, indicating that the internal damage of the sample is increasingly aggravated with the increase of stress, and its ability to resist shear is gradually

decreased, resulting in the gradual increase of irreversible deformation. However, under the same cyclic stress, the stress–strain curve changes from thinning to dense, and the hysteresis loop area decreases gradually, indicating that the specimen gradually compacts with the increase of cyclic number and the hardening degree increases. ally decreased, resulting in the gradual increase of irreversible deformation. However, under the same cyclic stress, the stress–strain curve changes from thinning to dense, and the hysteresis loop area decreases gradually, indicating that the specimen gradually compacts with the increase of cyclic number and the hardening degree increases.

studied, and the stress–strain relationship of other samples is similar, so it will be repeated

Figure 4 displays the whole process of the stress and displacement curve for sample 4–6.52. As shown in the figure, during each cycle, the unloading curve does not coincide with the original loading curve, and the unloading curve can form a completely closed annular area with the reloading curve, namely, the hysteresis loop [25]. The reason is that rock materials are not ideal elastomers, and there are a large number of internal defects such as pores and micro-cracks, which lead to the closure of the original cracks and the initiation of new cracks in the test process [26], resulting in some irreversible damage to the sample. The appearance of the hysteresis loop is not only an experimental phenomenon, but also a manifestation of energy dissipation, and the area of the hysteresis loop can be characterized as the energy dissipated by crack closure, expansion, and through-cracking failure of the loaded sample; the larger the area of the hysteresis loop, the more energy

The rock discontinuity sample is different from the intact rock sample, and in the first loading process of each grading cycle stage, the upper and lower parts of the rock discontinuity sample usually slip, resulting in large residual deformation. With the continuous increase of stress, this sliding deformation gradually increases, and the deformation curve of the sample is gradually sparse, indicating that the internal damage of the sample is increasingly aggravated with the increase of stress, and its ability to resist shear is gradu-

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consumed and the more severe the damage to the sample.

here.

**Figure 4.** Stress–strain curves of specimen 4–6.52. **Figure 4.** Stress–strain curves of specimen 4–6.52.

#### *4.2. Energy Density Calculation Method 4.2. Energy Density Calculation Method*

For the loaded sample, it often goes through the extrusion compaction stage, the stable development stage, and the unstable development stage of the crack before the final failure, and these processes are often accompanied by energy input, accumulation, dissipation, and release. The energy transformation of samples from deformation to failure is a dynamic process, which is represented by the transformation and balance among input energy, elastic energy, and dissipated energy. According to the first law of thermodynam-For the loaded sample, it often goes through the extrusion compaction stage, the stable development stage, and the unstable development stage of the crack before the final failure, and these processes are often accompanied by energy input, accumulation, dissipation, and release. The energy transformation of samples from deformation to failure is a dynamic process, which is represented by the transformation and balance among input energy, elastic energy, and dissipated energy. According to the first law of thermodynamics, the energy of substances in a thermodynamic system can be transformed and transferred, and the total amount of energy remains unchanged in the process of transformation and transfer [6]. The work performed by the surrounding to the rock mass can produce energy input into the rock mass, causing reversible deformation and irreversible deformation of the rock mass. The reversible deformation accumulates in the form of elastic energy. Irreversible deformation dissipates energy in the form of plastic deformation energy, internal friction of rock mass, and thermal radiation [27]. The sample in the test is deformed by external force loading and is a closed system. Therefore, the energy exchange between the specimen and the outside world is not considered [28,29]. According to the law of the conservation of energy,

$$
\mathcal{U} = \mathcal{U}\_{\mathcal{e}} + \mathcal{U}\_{\mathcal{d}} \tag{1}
$$

where *U* is the total energy input from the outside; *U*<sup>e</sup> denotes the elastic energy; and *U*<sup>d</sup> is the dissipated energy, that is, the sum of the damage energy and plastic strain energy in rock mass.

At any time in the deformation process of the sample, there is a specific energy state corresponding to it [30], which is a function of stress, strain, and time. According to the stress–strain curve characteristics of samples, the elastic energy density and dissipation energy density of rock mass under cyclic loading were calculated. The relationship between the elastic energy density *u*<sup>e</sup> and dissipation energy density *u*<sup>d</sup> per unit of volume in the loading and unloading curve of samples under certain stress levels is shown in Figure 5. The total energy density *u* input by the outside world is the area formed by *OBAE*. The

elastic energy density *u*<sup>e</sup> denotes the area formed by *ACDE*. The dissipative energy density *u*<sup>d</sup> is the total energy density *u* minus the elastic energy density *u*e, which is the area formed by *OBACD*. The *u*<sup>e</sup> and *u*<sup>d</sup> are calculated as follows: energy density *u*d is the total energy density *u* minus the elastic energy density *u*e, which is the area formed by *OBACD*. The *u*e and *u*d are calculated as follows: 2 *ε <sup>ε</sup> u = <sup>σ</sup>d<sup>ε</sup>* (2)

e

d

ics, the energy of substances in a thermodynamic system can be transformed and transferred, and the total amount of energy remains unchanged in the process of transformation and transfer [6]. The work performed by the surrounding to the rock mass can produce energy input into the rock mass, causing reversible deformation and irreversible deformation of the rock mass. The reversible deformation accumulates in the form of elastic energy. Irreversible deformation dissipates energy in the form of plastic deformation energy, internal friction of rock mass, and thermal radiation [27]. The sample in the test is deformed by external force loading and is a closed system. Therefore, the energy exchange between the specimen and the outside world is not considered [28,29]. According to the

where *U* is the total energy input from the outside; *U*e denotes the elastic energy; and *U*<sup>d</sup> is the dissipated energy, that is, the sum of the damage energy and plastic strain energy

At any time in the deformation process of the sample, there is a specific energy state corresponding to it [30], which is a function of stress, strain, and time. According to the stress–strain curve characteristics of samples, the elastic energy density and dissipation energy density of rock mass under cyclic loading were calculated. The relationship between the elastic energy density *u*e and dissipation energy density *u*d per unit of volume in the loading and unloading curve of samples under certain stress levels is shown in Figure 5. The total energy density *u* input by the outside world is the area formed by *OBAE*. The elastic energy density *u*e denotes the area formed by *ACDE*. The dissipative

$$\mathfrak{u}\_{\mathfrak{e}} = \int\_{\mathfrak{e}\_1}^{\mathfrak{e}\_2} \mathfrak{e} d\mathfrak{e} \tag{2}$$

$$\mathfrak{e}\_{\mathfrak{e}\_1} \qquad \mathfrak{e}\_{\mathfrak{e}\_2}$$

*U =U +U* e d (1)

$$u\_d = \int\_{\varepsilon\_0}^{\varepsilon\_2} \sigma d\varepsilon - \int\_{\varepsilon\_1}^{\varepsilon\_2} \sigma d\varepsilon \tag{3}$$

where *σ* is the stress at any point in the stress–strain curve; *ε* denotes the strain corresponding to *σ*; *ε*<sup>0</sup> refers to the strain corresponding to the initial stress *σ*<sup>0</sup> of the loading curve; *ε*<sup>1</sup> is the strain corresponding to the lower limit stress *σ*<sup>0</sup> of the unloading curve; and the strain corresponding to the loading of the upper limit stress *σ*<sup>1</sup> is represented by *ε*2. sponding to *σ*; *ε*0 refers to the strain corresponding to the initial stress *σ*0 of the loading curve; *ε*1 is the strain corresponding to the lower limit stress *σ*0 of the unloading curve; and the strain corresponding to the loading of the upper limit stress *σ*1 is represented by *ε*2.

*Materials* **2022**, *15*, x FOR PEER REVIEW 7 of 16

law of the conservation of energy,

in rock mass.

**Figure 5. Figure 5.** Schematic diagram of energy density calculation. Schematic diagram of energy density calculation.

#### *4.3. Energy Evolution Process in Samples*

According to the above, during the first loading of each stage of the graded cycle, rock discontinuities usually produce large residual deformation, which makes it difficult for the unloading curve to form a hysteresis loop with the reloading curve, and the error in the calculation of the energy density is large; therefore, the second cycle was explored as if it were the first cycle. According to Equations (2) and (3), the energy density of the sample in each cycle of loading and unloading can be calculated. Taking specimen 4–6.52 as an example, the relationship between the energy density and cycle times under various cyclic stresses is shown in Figure 6.

As illustrated in Figure 6, the external input energy shows an overall downward trend with the increase in the number of cycles, but its change process is different under different cyclic stresses. When the stress is small (1.35 to 2.25 MPa), the external input energy decreases gradually with increasing number of cycles, and the decrease is significant in the first few cycles, small in the later stage, then tends to be stable. When the stress is large (2.70 to 4.95 MPa), the external input energy decreases rapidly with the increase in the number of cycles, and then decreases again after a small increase.

With the increase in the number of cycles, the elastic energy density shows an overall upward trend, and increases significantly in the early stage, and gradually stabilizes in the later stage, while the dissipated energy density gradually decreases, which is consistent with the phenomenon that the area of the hysteresis loop in the stress–strain curve decreases with increasing number of cycles. The results show that in the cyclic loading and unloading process associated with the same stress level, the micro-cracks inside the sample gradually close, the hardening of the sample increases, and the energy consumed by the friction inside the sample decreases.

*4.3. Energy Evolution Process in Samples*

cyclic stresses is shown in Figure 6.

by the friction inside the sample decreases.

According to the above, during the first loading of each stage of the graded cycle, rock discontinuities usually produce large residual deformation, which makes it difficult for the unloading curve to form a hysteresis loop with the reloading curve, and the error in the calculation of the energy density is large; therefore, the second cycle was explored as if it were the first cycle. According to Equations (2) and (3), the energy density of the sample in each cycle of loading and unloading can be calculated. Taking specimen 4–6.52 as an example, the relationship between the energy density and cycle times under various

As illustrated in Figure 6, the external input energy shows an overall downward trend with the increase in the number of cycles, but its change process is different under different cyclic stresses. When the stress is small (1.35 to 2.25 MPa), the external input energy decreases gradually with increasing number of cycles, and the decrease is significant in the first few cycles, small in the later stage, then tends to be stable. When the stress is large (2.70 to 4.95 MPa), the external input energy decreases rapidly with the increase

With the increase in the number of cycles, the elastic energy density shows an overall upward trend, and increases significantly in the early stage, and gradually stabilizes in the later stage, while the dissipated energy density gradually decreases, which is consistent with the phenomenon that the area of the hysteresis loop in the stress–strain curve decreases with increasing number of cycles. The results show that in the cyclic loading and unloading process associated with the same stress level, the micro-cracks inside the sample gradually close, the hardening of the sample increases, and the energy consumed

With the continuous increase of cyclic stress, the dissipated energy density of each stage gradually exceeds the elastic energy density and gradually occupies a dominant po-

The energy evolution of the other samples at each stress is akin to that of specimen

sition, indicating that the internal damage to the sample is gradually intensified.

4–6.52, which is not repeated in this paper due to limited space.

in the number of cycles, and then decreases again after a small increase.

**Figure 6.** Variations in energy density of specimen 4–6.52 with the number of cycles *N* under various cyclic stresses: (**a**) 1.35 MPa; (**b**) 1.80 MPa; (**c**) 2.25 MPa; (**d**) 2.70 MPa; (**e**) 3.15 MPa; (**f**) 3.60 MPa; (**g**) 4.05 MPa; (**h**) 4.50 MPa; (**i**) 4.95 MPa. **Figure 6.** Variations in energy density of specimen 4–6.52 with the number of cycles *N* under various cyclic stresses: (**a**) 1.35 MPa; (**b**) 1.80 MPa; (**c**) 2.25 MPa; (**d**) 2.70 MPa; (**e**) 3.15 MPa; (**f**) 3.60 MPa; (**g**) 4.05 MPa; (**h**) 4.50 MPa; (**i**) 4.95 MPa.

According to the actual failure strength of the sample, the cyclic stress is normalized, and the energy density of the last nine cycles under all levels of cyclic stress is averaged to obtain the relationship between the average energy density and the stress level (the With the continuous increase of cyclic stress, the dissipated energy density of each stage gradually exceeds the elastic energy density and gradually occupies a dominant position, indicating that the internal damage to the sample is gradually intensified.

ratio of the upper limit of cyclic stress at all levels to the actual failure strength of the sample, as shown in Table 4), as shown in Figure 7. With the increase in stress, the energy The energy evolution of the other samples at each stress is akin to that of specimen 4–6.52, which is not repeated in this paper due to limited space.

absorbed by the sample from the outside firstly decreases greatly, and then decreases significantly. When the stress reaches a certain value, it shows a significant upward trend. The elastic energy density presents a non-linear decreasing trend with the increase of stress, and the rate of change decreases gradually thereafter. With the increase in stress, the dissipated energy density decreases slightly at first, then increases slowly at a low rate, and increases significantly as the specimen approaches failure. At the initial stage of fractional loading, the three energies all decreased. This phenomenon is due to the existence of many pores and cracks in the sample itself. In the pro-According to the actual failure strength of the sample, the cyclic stress is normalized, and the energy density of the last nine cycles under all levels of cyclic stress is averaged to obtain the relationship between the average energy density and the stress level (the ratio of the upper limit of cyclic stress at all levels to the actual failure strength of the sample, as shown in Table 4), as shown in Figure 7. With the increase in stress, the energy absorbed by the sample from the outside firstly decreases greatly, and then decreases significantly. When the stress reaches a certain value, it shows a significant upward trend. The elastic energy density presents a non-linear decreasing trend with the increase of stress, and the

cess of first-stage cyclic loading and unloading, the closure and friction between these pores and cracks need to absorb more energy, and the absorbed energy is mainly stored

inside the sample have closed, and the degree of compaction is significantly improved, resulting in a decrease in the energy of all parts. With the further increase in stress, the primary cracks in the sample begin to expand, and new cracks are constantly initiated, leading to the decline of the ability of the sample to accumulate elastic energy. The energy consumed by internal friction and plastic failure increases gradually, which implies that the elastic energy density decreases continuously, while the dissipated energy density in-

**Sample Stress Level**

1–2.17 0.28, 0.37, 0.46, 0.55, 0.64, 0.74, 0.83, 0.92 1–4.35 0.27, 0.36, 0.46, 0.55, 0.64, 0.73, 0.82, 0.91

4–2.17 0.39, 0.52, 0.65, 0.79, 0.92 4–4.35 0.31, 0.41, 0.52, 0.62, 0.72, 0.83, 0.93 4–6.52 0.26, 0.35, 0.44, 0.53, 0.62, 0.71, 0.79, 0.88, 0.97 8–2.17 0.29, 0.39, 0.48, 0.58, 0.68, 0.78, 0.87, 0.97 8–4.35 0.31, 0.41, 0.51, 0.61, 0.71, 0.82, 0.92

1–6.52 0.20, 0.27, 0.34, 0.41, 0.48, 0.55, 0.62, 0.69, 0.75, 0.82, 0.89

creases gradually.

**Table 4.** Stress level.

rate of change decreases gradually thereafter. With the increase in stress, the dissipated energy density decreases slightly at first, then increases slowly at a low rate, and increases significantly as the specimen approaches failure.

**Table 4.** Stress level.

**Figure 7.** Relationship between energy density and stress level of samples under different working conditions: (**a**) *u*; (**b**) *u*e; (**c**) *u*d. **Figure 7.** Relationship between energy density and stress level of samples under different working conditions: (**a**) *u*; (**b**) *u*e; (**c**) *u*d.

*4.4. Energy Distribution in Samples*  In the closed test system, the energy input to rock samples by the testing machine is mainly transformed into elastic energy and dissipated energy, which will affect the defor-At the initial stage of fractional loading, the three energies all decreased. This phenomenon is due to the existence of many pores and cracks in the sample itself. In the

mation and failure of rock samples. The relationship between the proportion division of

sample show a non-linear trend, and the elastic energy presents a downward trend as a whole, while the dissipated energy shows the opposite trend, and the rates of change of

At the initial stage of fractional loading, the internal pores of the sample are compressed and compacted, and the proportion of elastic energy is much higher than that of dissipated energy, indicating that most of the energy input from the outside is converted into elastic energy and stored in the sample. The energy consumed by internal crack closure and friction slip is small. With the increase in stress, the elastic energy ratio decreases step-by-step at a low rate, while the dissipated energy ratio changes in the opposite way.

both gradually increase.

process of first-stage cyclic loading and unloading, the closure and friction between these pores and cracks need to absorb more energy, and the absorbed energy is mainly stored in the sample in the form of elastic energy. Thus, the energy of each part of the sample is high in the first stage, and after entering the next stage of the cycle, most of the pores inside the sample have closed, and the degree of compaction is significantly improved, resulting in a decrease in the energy of all parts. With the further increase in stress, the primary cracks in the sample begin to expand, and new cracks are constantly initiated, leading to the decline of the ability of the sample to accumulate elastic energy. The energy consumed by internal friction and plastic failure increases gradually, which implies that the elastic energy density decreases continuously, while the dissipated energy density increases gradually. *Materials* **2022**, *15*, x FOR PEER REVIEW 11 of 16

#### *4.4. Energy Distribution in Samples*

In the closed test system, the energy input to rock samples by the testing machine is mainly transformed into elastic energy and dissipated energy, which will affect the deformation and failure of rock samples. The relationship between the proportion division of types of energy in samples and stress level is shown in Figure 8. With the constant increase in stress, the proportion of elastic energy and the proportion of dissipated energy in the sample show a non-linear trend, and the elastic energy presents a downward trend as a whole, while the dissipated energy shows the opposite trend, and the rates of change of both gradually increase. At this stage, although most of the pores inside the sample are closed, the stress concentration leads to the expansion and initiation of micro-cracks, increasing the dissipated energy ratio, but most of the energy from the external input is still accumulated in the sample. When the ratio of dissipated energy and elastic energy approach each other, the connection of micro-cracks in the sample and the formation and unstable expansion of macrocracks lead to the dissipation of most of the external input energy. The elastic energy accumulated in the sample begins to release, showing that the proportion of dissipated energy decreases rapidly.

**Figure 8.** Relationship between energy ratio and stress level under different working conditions: (**a**) *u*e; (**b**) *u*d. **Figure 8.** Relationship between energy ratio and stress level under different working conditions: (**a**) *u*e; (**b**) *u*d.

*4.5. Energy Criterion for Rock Discontinuity Failure*  For an ideal material, any form of energy applied to it can all be converted into releasable elastic strain energy within the material, and during the deformation process, its internal structure does not suffer damage, and the absorption and storage of elastic energy will not dissipate. However, for rock materials, the friction of pores or micro-cracks inside the specimen in the compaction stage, and the expansion of micro-cracks and connection in the elastic and plastic stages are all accompanied by energy dissipation. The occurrence and accumulation of irreversible deformation are the direct causes of specimen failure [31], and the dissipative energy can indirectly reflect the irreversible deformation generated in the specimen. The accumulation of dissipative energy will facilitate the sample in its gradual transformation from the initial stable state to an unstable state, and then to another stable state (the strength therein being the residual strength after the main fractures have inter-connected and split into multiple rock blocks) through the reorganization At the initial stage of fractional loading, the internal pores of the sample are compressed and compacted, and the proportion of elastic energy is much higher than that of dissipated energy, indicating that most of the energy input from the outside is converted into elastic energy and stored in the sample. The energy consumed by internal crack closure and friction slip is small. With the increase in stress, the elastic energy ratio decreases step-by-step at a low rate, while the dissipated energy ratio changes in the opposite way. At this stage, although most of the pores inside the sample are closed, the stress concentration leads to the expansion and initiation of micro-cracks, increasing the dissipated energy ratio, but most of the energy from the external input is still accumulated in the sample. When the ratio of dissipated energy and elastic energy approach each other, the connection of micro-cracks in the sample and the formation and unstable expansion of macro-cracks lead to the dissipation of most of the external input energy. The elastic energy accumulated in the sample begins to release, showing that the proportion of dissipated energy decreases rapidly.

of the internal structure of the sample. The change from a stable state to an unstable state entails the process of internal damage accumulation to unstable failure, as well as the process of internal energy transformation. Therefore, the energy dissipation ratio *K* (the ratio of the dissipated energy density to the elastic energy density of the loaded rock sample)

> d e

When *K* < 1, it can be considered that the internal structure of the sample is in a relatively stable state, and the damage to it is small; when *K* = 1, it can be considered that the loaded rock sample reaches its energy storage limit, is in a critical state, and is about to

(4)

<sup>=</sup> *<sup>u</sup> <sup>K</sup> u*

of the sample. *K* is given by

#### *4.5. Energy Criterion for Rock Discontinuity Failure*

For an ideal material, any form of energy applied to it can all be converted into releasable elastic strain energy within the material, and during the deformation process, its internal structure does not suffer damage, and the absorption and storage of elastic energy will not dissipate. However, for rock materials, the friction of pores or micro-cracks inside the specimen in the compaction stage, and the expansion of micro-cracks and connection in the elastic and plastic stages are all accompanied by energy dissipation. The occurrence and accumulation of irreversible deformation are the direct causes of specimen failure [31], and the dissipative energy can indirectly reflect the irreversible deformation generated in the specimen. The accumulation of dissipative energy will facilitate the sample in its gradual transformation from the initial stable state to an unstable state, and then to another stable state (the strength therein being the residual strength after the main fractures have inter-connected and split into multiple rock blocks) through the reorganization of the internal structure of the sample. The change from a stable state to an unstable state entails the process of internal damage accumulation to unstable failure, as well as the process of internal energy transformation. Therefore, the energy dissipation ratio *K* (the ratio of the dissipated energy density to the elastic energy density of the loaded rock sample) is used to characterize the deterioration of the sample, and also indirectly reflect the state of the sample. *K* is given by

$$K = \frac{\mu\_d}{\mu\_\varepsilon} \tag{4}$$

When *K* < 1, it can be considered that the internal structure of the sample is in a relatively stable state, and the damage to it is small; when *K* = 1, it can be considered that the loaded rock sample reaches its energy storage limit, is in a critical state, and is about to enter the unstable development stage; when *K* > 1, it can be considered that the rock sample is in an unstable state.

Figure 9 shows the stress level–energy dissipation ratio deformation of samples under different working conditions and Figure 10 demonstrates the energy consumption ratio of each sample that varies with the stress level.

According to the sample energy dissipation ratio and shear deformation seen in Figure 9, the failure process of the sample can be roughly divided into three stages. In the early stage of fractional loading, the sample is in the extrusion compaction stage (the first stage), and the internal pores and cracks of the sample are closed in this stage. The hardening of the sample is significantly improved, resulting in the irreversible deformation of the rock discontinuity which decreases significantly, and the energy dissipation ratio remains quasi-constant. Then the sample enters the stage of steady crack propagation (the second stage), and the internal damage intensifies, and the dissipated energy increases continuously due to the increase of internal friction and plastic deformation. The sample gradually reaches its energy storage limit (*K* = 1), which shows that the deformation of the structural plane and energy dissipation ratio increase linearly with the increasing stress. With the further increase of the stress level, the sample enters the unstable crack development stage (the third stage). The connection and penetration of cracks results in the worsening of the plastic failure of the sample, the rapid increase of dissipative energy, and the energy accumulated in the sample begins to release. When the failure is near, both the deformation and energy consumption ratio of the rock discontinuity change dramatically, indicating that a large amount of plastic failure occurs in the sample. The stored energy in the sample is released instantly, leading to the rapid loss of bearing capacity and instability failure.

Figure 10 illustrates that the energy dissipation ratio of all samples has roughly the same variation with stress, showing slow growth in the early stage, and a significant increase in the growth rate after reaching the energy storage limit (*K* = 1). However, there are significant differences in the stress on all samples when reaching the energy storage limit.

Instability and failure of specimens occurred after reaching the energy storage limit and are the root cause of a rapid release of elastic energy. Energy storage limits of the samples at the corresponding stress level can describe the samples' accumulated elastic

*Materials* **2022**, *15*, x FOR PEER REVIEW 12 of 16

tio of each sample that varies with the stress level.

ple is in an unstable state.

energy capacity: the higher the stress level, the more the sample storage limits the strength or stiffness, and instability and failure are less likely. in the sample is released instantly, leading to the rapid loss of bearing capacity and instability failure.

enter the unstable development stage; when *K* > 1, it can be considered that the rock sam-

Figure 9 shows the stress level–energy dissipation ratio deformation of samples under different working conditions and Figure 10 demonstrates the energy consumption ra-

According to the sample energy dissipation ratio and shear deformation seen in Figure 9, the failure process of the sample can be roughly divided into three stages. In the early stage of fractional loading, the sample is in the extrusion compaction stage (the first stage), and the internal pores and cracks of the sample are closed in this stage. The hardening of the sample is significantly improved, resulting in the irreversible deformation of the rock discontinuity which decreases significantly, and the energy dissipation ratio remains quasi-constant. Then the sample enters the stage of steady crack propagation (the second stage), and the internal damage intensifies, and the dissipated energy increases continuously due to the increase of internal friction and plastic deformation. The sample gradually reaches its energy storage limit (*K* = 1), which shows that the deformation of the structural plane and energy dissipation ratio increase linearly with the increasing stress. With the further increase of the stress level, the sample enters the unstable crack development stage (the third stage). The connection and penetration of cracks results in the worsening of the plastic failure of the sample, the rapid increase of dissipative energy, and the energy accumulated in the sample begins to release. When the failure is near, both the deformation and energy consumption ratio of the rock discontinuity change dramatically, indicating that a large amount of plastic failure occurs in the sample. The stored energy

**Figure 9.** Stress level–energy dissipation ratio deformation of samples under different working conditions (Δ*D* is the total deformation of 10 cycles under different stress levels): (**a**) 1–2.17; (**b**) 1–4.35; (**c**) 1–6.52; (**d**) 4–2.17; (**e**) 4–4.35; (**f**) 4–6.52; (**g**) 8–2.17; (**h**) 8–4.35; (**i**) 8–6.52; (**j**) 10–2.17; (**k**) 10–4.35; (l) 10–6.52. **Figure 9.** Stress level–energy dissipation ratio deformation of samples under different working conditions (∆*D* is the total deformation of 10 cycles under different stress levels): (**a**) 1–2.17; (**b**) 1–4.35; (**c**) 1–6.52; (**d**) 4–2.17; (**e**) 4–4.35; (**f**) 4–6.52; (**g**) 8–2.17; (**h**) 8–4.35; (**i**) 8–6.52; (**j**) 10–2.17; (**k**) 10–4.35; (**l**) 10–6.52.

Figure 10 illustrates that the energy dissipation ratio of all samples has roughly the same variation with stress, showing slow growth in the early stage, and a significant increase in the growth rate after reaching the energy storage limit (*K* = 1). However, there are significant differences in the stress on all samples when reaching the energy storage limit. Instability and failure of specimens occurred after reaching the energy storage limit and are the root cause of a rapid release of elastic energy. Energy storage limits of the samples at the corresponding stress level can describe the samples' accumulated elastic To explore the influence of roughness on the energy storage capacity of a specimen, the stress corresponding to each sample at *K* = 1 in Figure 10 was taken as the ultimate stress associated with its energy storage, and the results under the action of three different normal stresses under the same roughness were averaged to determine the variations in the ultimate stress of the energy storage of the sample with the roughness (Figure 11). The energy storage limiting stress for the specimen shows a positive correlation with the increase of JRC; that is, the rougher the rock discontinuity, the greater its energy storage limit and stiffness. According to the linear fitting results in the figure, the energy storage

energy capacity: the higher the stress level, the more the sample storage limits the strength

the stress corresponding to each sample at *K* = 1 in Figure 10 was taken as the ultimate stress associated with its energy storage, and the results under the action of three different normal stresses under the same roughness were averaged to determine the variations in the ultimate stress of the energy storage of the sample with the roughness (Figure 11). The energy storage limiting stress for the specimen shows a positive correlation with the increase of JRC; that is, the rougher the rock discontinuity, the greater its energy storage limit and stiffness. According to the linear fitting results in the figure, the energy storage capacity of the rock discontinuity sample increases by about 0.75% when the value of JRC

**Figure 10.** The energy consumption ratio of each sample varies with the stress level.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Stress level

increases by 1.

 1-2.17 1-4.35 1-6.52 4-2.17 4-4.35 4-6.52 8-2.17 8-4.35 8-6.52 10-2.17 10-4.35 10-6.52 *K*=1

0.0

0.5

1.0

1.5

2.0

*K*

2.5

3.0

3.5

*Materials* **2022**, *15*, x FOR PEER REVIEW 13 of 16

*K* Δ*D*

(**j**) (**k**) (**l**)

Stress level

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0

I II

or stiffness, and instability and failure are less likely.

0.00

0.05

Δ*D*(mm)

0.5

1.0

*K*=1

*K*

1.5

0.10

0.15

10–6.52.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0

Stress level

*K* Δ*D*

I II III

0.5 1.0 1.5 2.0 2.5 3.0

*K*=1

*K*

limit.

capacity of the rock discontinuity sample increases by about 0.75% when the value of JRC increases by 1. limit and stiffness. According to the linear fitting results in the figure, the energy storage capacity of the rock discontinuity sample increases by about 0.75% when the value of JRC increases by 1.

**Figure 9.** Stress level–energy dissipation ratio deformation of samples under different working conditions (Δ*D* is the total deformation of 10 cycles under different stress levels): (**a**) 1–2.17; (**b**) 1–4.35; (**c**) 1–6.52; (**d**) 4–2.17; (**e**) 4–4.35; (**f**) 4–6.52; (**g**) 8–2.17; (**h**) 8–4.35; (**i**) 8–6.52; (**j**) 10–2.17; (**k**) 10–4.35; (l)

0.00

0.05

Δ*D*

0.5 1.0 1.5 2.0 2.5

*K*=1

*K*

(mm)

0.10

III

0.15

Figure 10 illustrates that the energy dissipation ratio of all samples has roughly the same variation with stress, showing slow growth in the early stage, and a significant increase in the growth rate after reaching the energy storage limit (*K* = 1). However, there are significant differences in the stress on all samples when reaching the energy storage

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0

Stress level

0.00

0.05

0.10

Δ*D*(mm)

0.15

0.20

*K* Δ*D*

I II III

Instability and failure of specimens occurred after reaching the energy storage limit and are the root cause of a rapid release of elastic energy. Energy storage limits of the samples at the corresponding stress level can describe the samples' accumulated elastic energy capacity: the higher the stress level, the more the sample storage limits the strength

To explore the influence of roughness on the energy storage capacity of a specimen, the stress corresponding to each sample at *K* = 1 in Figure 10 was taken as the ultimate stress associated with its energy storage, and the results under the action of three different normal stresses under the same roughness were averaged to determine the variations in the ultimate stress of the energy storage of the sample with the roughness (Figure 11). The energy storage limiting stress for the specimen shows a positive correlation with the increase of JRC; that is, the rougher the rock discontinuity, the greater its energy storage

**Figure 10.** The energy consumption ratio of each sample varies with the stress level. **Figure 10.** The energy consumption ratio of each sample varies with the stress level. *Materials* **2022**, *15*, x FOR PEER REVIEW 14 of 16

**Figure 11.** Variations in stress at the energy storage limit of samples with JRC. **Figure 11.** Variations in stress at the energy storage limit of samples with JRC.

#### **5. Conclusions 5. Conclusions**

The following conclusions can be drawn following hierarchical cyclic loading and unloading tests and energy calculation and analysis of specimens containing discontinui-The following conclusions can be drawn following hierarchical cyclic loading and unloading tests and energy calculation and analysis of specimens containing discontinuities:

ple under load. In the initial stage of loading, the sample is in the extrusion and com-

ratio increases linearly (albeit at a low rate). Before the specimen is about to fail, the change rate is accelerated, and then a sudden change occurs, indicating that the rapid release of energy is the fundamental reason for the failure of the rock discontinuity. (5) The increase of the roughness of rock discontinuity samples will improve their energy storage capacity to a certain extent: the higher the JRC of the rock discontinuity,

**Author Contributions:** writing—original draft preparation, conceptualization, W.Z.; writing—review and editing, methodology, L.G., Z.W.; data curation, supervision, J.M., H.L., H.Z. All authors

**Funding:** This work was financially supported by the National Natural Science Foundation of China (Grant No. 42002266), the Fundamental Research Funds for the Central Universities (Grant

have read and agreed to the published version of the manuscript.

**Institutional Review Board Statement:** Not applicable.

the greater the energy storage limit and stiffness of the specimen.

No.30922010918) and the Chinese Postdoctoral Science Foundation (2020M673654).

and compaction stage, and the energy dissipation ratio remains unchanged. Then, the fracture develops steadily, the damage in the sample intensifies, and the energy dissipation ratio increases linearly (albeit at a low rate). Before the specimen is about to fail, the change rate is accelerated, and then a sudden change occurs, indicating that the rapid release of energy is the fundamental reason for the failure of the rock discontinuity.

(5) The increase of the roughness of rock discontinuity samples will improve their energy storage capacity to a certain extent: the higher the JRC of the rock discontinuity, the greater the energy storage limit and stiffness of the specimen.

**Author Contributions:** Writing—original draft preparation, conceptualization, W.Z.; writing—review and editing, methodology, L.G., Z.W.; data curation, supervision, J.M., H.L., H.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was financially supported by the National Natural Science Foundation of China (Grant No. 42002266), the Fundamental Research Funds for the Central Universities (Grant No.30922010918) and the Chinese Postdoctoral Science Foundation (2020M673654).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Informed consent was obtained from all subjects involved in the study.

**Data Availability Statement:** The data used to support the findings of this study are included within the article.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

## *Article* **Comparative Study on the Seepage Characteristics of Gas-Containing Briquette and Raw Coal in Complete Stress–Strain Process**

**Ke Ding <sup>1</sup> , Lianguo Wang 1,\*, Zhaolin Li 2,3,\* , Jiaxing Guo <sup>1</sup> , Bo Ren <sup>1</sup> , Chongyang Jiang <sup>1</sup> and Shuai Wang <sup>1</sup>**

**Abstract:** In this study, triaxial compression and seepage tests were conducted on briquette and raw coal samples using a coal rock mechanics-seepage triaxial test system (TAWD-2000) to obtain the complete stress–strain curves of the two samples under certain conditions. On this basis, the different damage forms of the two coal samples and the effect of their deformation and damage on their permeability were analyzed from the perspective of fine-scale damage mechanics. Moreover, the sensitivity of permeability to external variables and the suddenness of coal and gas outbursts were discussed. The results show that the compressive strength of raw coal is 27.1 MPa and the compressive strength of briquette is 17.3 MPa, the complete stress–strain curves of the two coal samples can be divided into four stages and show a good correspondence to the permeability–axial strain curves. Since briquette and raw coal have different structural properties, they present different damage mechanisms under load, thus showing great diversity in the permeability-axial strain curve, especially in the damage stage. The deformation affects the seepage characteristics of briquette mainly in the latter two stages, while it affects raw coal throughout the test. The four stages of the complete stress–strain seepage test of raw coal can well explain the four stages of coal and gas outburst process, i.e., preparation, initiation, development, and termination. Hence, the law of coal permeability to gas variation can be utilized for the coal and gas outburst prediction and forecast. The research results are valuable for exploring the real law of gas migration in coal seams.

**Keywords:** coal and gas outburst; gas-containing coal; complete stress–strain process; permeability; solid-gas coupling

#### **1. Introduction**

Energy, a main driver of the economy, plays a pivotal role in national economic development [1]. Along with the increasing amount and depth of coal mining, problems such as high ground stress and high gas pressure emerge one after another. In the underground mining process in coal mines, the coal body is deformed and damaged by the mining, accompanied by changes in gas permeability characteristics, which is a major cause of gas dynamic damages such as gas gushing from the working face and coal and gas outbursts [2–6]. Therefore, for an effective reduction of the gas-induced adverse effects in underground coal mining, it is of great theoretical significance and practical value to study the law of gas seepage during coal deformation and understand the coal and gas outburst mechanism.

Many scholars believe that briquette shares a similar variation law with raw coal, despite their significant difference in true relative density and apparent relative density (their porosities differ about 4 times and their pore volumes differ 4–10 times). Besides, briquette is easy and efficient to be processed into coal samples. Therefore, briquette can be

**Citation:** Ding, K.; Wang, L.; Li, Z.; Guo, J.; Ren, B.; Jiang, C.; Wang, S. Comparative Study on the Seepage Characteristics of Gas-Containing Briquette and Raw Coal in Complete Stress–Strain Process. *Materials* **2022**, *15*, 6205. https://doi.org/10.3390/ ma15186205

Academic Editor: Saeed Chehreh Chelgani

Received: 26 July 2022 Accepted: 30 August 2022 Published: 7 September 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

used as a research object on the general law of gas-containing coal, and the yielded law is applicable to coal seams [7]. At present, researches have been conducted on both briquette and raw coal. By performing comparative tests on the permeabilities of gas-containing briquette and raw coal, Ge et al. [8] found that using briquette samples in place of raw coal samples in the laboratory can only obtain a rough variation law. Pang et al. [9] studied the deformation of coal after gas adsorption during gas pressure change using a self-developed experimental device and discussed the effect of gas on coal based on the existing studies. They concluded that the adsorbed gas would cause expansion and deformation of coal, which weakens its strength and increase its brittleness, so that it becomes more prone to sudden instability damage. Gan et al. [10] investigated the effect of gas pressure on the coal permeability to gas characteristics of coal rock materials in the complete stress–strain process. The results show that at a constant confining pressure, the complete stress–strain curve corresponds well to the seepage rate-axial strain curve, and raising the gas pressure within a certain range can enhance the permeability of coal. Sun et al. [11] conducted triaxial compression and seepage tests on briquette and raw coal using a self-developed gravitational device and found a good correspondence between their complete stress–strain curves and seepage rate-axial strain curves. Zhang et al. [12] explored the deformation characteristics and compressive strengths of briquette and raw coal under triaxial stress conditions based on the results of gas-containing triaxial tests, and concluded that although the two coal samples are similar in the above two properties, they differ significantly in the permeability characteristics. Katarzyna et al. [13] used Brazilian tests to evaluate the work of disintegration of rock resulting from the stresses produced by gas present in its porous structure. However, these research results did not provide a comprehensive comparative analysis on the permeability characteristics of briquette and raw coal. In fact, it is still necessary to further explore the differences in the permeability characteristics between the two during the complete stress–strain process and determine which one is more in line with the actual situation [14–27].

In this paper, gas-containing complete stress–strain tests were conducted on raw coal and briquette by using a coal rock mechanics-seepage triaxial test system (TAWD-2000) to study the similarity and difference in seepage characteristics of the two coal samples during the deformation process. The research results are expected to provide reference for the experimental study using briquette in place of raw coal and serve as an experimental basis for further exploration on the real law of gas migration and the mechanism of coal and gas outbursts in coal seams.

#### **2. Test Process**

#### *2.1. Test Materials*

The test coal samples were taken from the No. 8<sup>2</sup> coal seam of Zouzhuang Coal Mine of Huabei Mining Stock Corporation, China. The average buried depth of the coal seam is 853.7 m, as shown in Figure 1. This coal mine is bounded by the F22 fault and the Dual-Stack fault in the east, the first limestone cropline at the top of the Carboniferous Taiyuan Formation in the south, the Nanping fault in the west, and the #27 exploration line in the north. The No. 8<sup>2</sup> coal seam is 2.48 m thick on average. The coal body, which belongs to semi-bright coal due to its weak glassy luster, is black, pulverized-fragmented, and has developed endogenous fractures. Soft and fragile, this coal seam is typically prone to coal and gas outbursts.

**Figure 1.** Diagram of the sampling site. **Figure 1.** Diagram of the sampling site.

The preparation of raw coal samples is a difficult task: The raw coal blocks taken from the site were sealed with plastic film and transported directly to the mechanics experiment center of the China University of Mining and Technology where they were cored and processed under the condition that the stratification was perpendicular to the processing axis. According to the test platform and test specifications, the prepared raw coal samples were cylindrical with a height of 95–102 mm, a diameter of about 50 mm, and a parallelism below ±0.05 mm between the upper and lower end surfaces, and they The preparation of raw coal samples is a difficult task: The raw coal blocks taken from the site were sealed with plastic film and transported directly to the mechanics experiment center of the China University of Mining and Technology where they were cored and processed under the condition that the stratification was perpendicular to the processing axis. According to the test platform and test specifications, the prepared raw coal samples were cylindrical with a height of 95–102 mm, a diameter of about 50 mm, and a parallelism below ±0.05 mm between the upper and lower end surfaces, and they were sealed and stored before the test.

were sealed and stored before the test. In contrast, the preparation of briquette samples is easier: The raw coal blocks were polished into 40–60 mesh pulverized coal and mixed well with a little water and binder. Then, a certain amount of the mixture was weighed with a balance by experience and put into the mold. After being shaped by a pressure of 200 kN for 20 min, they were made In contrast, the preparation of briquette samples is easier: The raw coal blocks were polished into 40–60 mesh pulverized coal and mixed well with a little water and binder. Then, a certain amount of the mixture was weighed with a balance by experience and put into the mold. After being shaped by a pressure of 200 kN for 20 min, they were made into cylinders with a height of 100 mm, a diameter of 50 mm, and a parallelism of 0.02 mm between the upper and lower end surfaces.

into cylinders with a height of 100 mm, a diameter of 50 mm, and a parallelism of 0.02 mm between the upper and lower end surfaces. Comparing the two samples, it is clear that the briquette sample had a standard size, a smooth surface and uniform texture, while the raw coal sample had primary damages such Comparing the two samples, it is clear that the briquette sample had a standard size, a smooth surface and uniform texture, while the raw coal sample had primary damages such as vertical and horizontal fractures, cracks, and holes. Therefore, the raw coal samples with similar density, cracks, and longitudinal wave velocities were selected for the test.

#### as vertical and horizontal fractures, cracks, and holes. Therefore, the raw coal samples with similar density, cracks, and longitudinal wave velocities were selected for the test. *2.2. Test Device and Principle*

*2.2. Test Device and Principle*  The coal rock mechanics-seepage triaxial test system (TAWD-2000) of the China University of Mining and Technology was used for the test (Figure 2). The system, which mainly consists of a pressure host system, a pressure and temperature control system, and a microcomputer operating system, can determine the permeabilities of coal rock under different pressure conditions. The maximum working pressures of confining pressure, The coal rock mechanics-seepage triaxial test system (TAWD-2000) of the China University of Mining and Technology was used for the test (Figure 2). The system, which mainly consists of a pressure host system, a pressure and temperature control system, and a microcomputer operating system, can determine the permeabilities of coal rock under different pressure conditions. The maximum working pressures of confining pressure, injection pressure, and axial pressure are 70 MPa, 70 MPa, and 800 MPa, respectively, with the pressures fluctuating within 0.5% in 48 h. The test was conducted at a constant temperature of 25 ◦C with CH<sup>4</sup> as the seepage medium.

injection pressure, and axial pressure are 70 MPa, 70 MPa, and 800 MPa, respectively, with the pressures fluctuating within 0.5% in 48 h. The test was conducted at a constant

temperature of 25 °C with CH4 as the seepage medium.

mm; CH4 μ

**Figure 2.** TAWD-2000 coal rock mechanics-seepage triaxial test system. **Figure 2.** TAWD-2000 coal rock mechanics-seepage triaxial test system.

Coal is a porous medium in which the seepage characteristics of gas depend on the number, size, and connectivity of pores and the pressure at both ends of pores in the flow direction. Hence, the permeability characteristics of the sample can be reflected by the relation curve between permeability and other related physical quantities. The principle of the coal permeability measurement test is shown in Figure 3. According to the principle, the steady-state method was adopted in the permeability test: First, different gas pressures were applied to the two ends of a coal sample with a constant pressure difference, so that a certain pressure gradient was formed in the coal sample to promote the flow of gas Coal is a porous medium in which the seepage characteristics of gas depend on the number, size, and connectivity of pores and the pressure at both ends of pores in the flow direction. Hence, the permeability characteristics of the sample can be reflected by the relation curve between permeability and other related physical quantities. The principle of the coal permeability measurement test is shown in Figure 3. According to the principle, the steady-state method was adopted in the permeability test: First, different gas pressures were applied to the two ends of a coal sample with a constant pressure difference, so that a certain pressure gradient was formed in the coal sample to promote the flow of gas through the fractures. Meanwhile, the gas flow was measured. When the flow within the coal sample developed into a steady-state flow, the amount of gas flowing through the sample over a while was recorded. The recorded amount can be substituted into the governing Equation (1) to calculate the permeability of the coal sample [19].

$$K = \frac{2\eta\_0 Q I\_{\text{coal}} \mu\_{\text{CH}\_4}}{A \left(p\_1^2 - p\_2^2\right)}\tag{1}$$

*<sup>A</sup> p p* , (1)

coal sample developed into a steady-state flow, the amount of gas flowing through the sample over a while was recorded. The recorded amount can be substituted into the governing Equation (1) to calculate the permeability of the coal sample [19]. where *K* is the permeability, 10−<sup>15</sup> m<sup>2</sup> ; *p*<sup>0</sup> is the atmospheric pressure, 0.1 MPa; *Q* is the gas flow through the coal sample, cm3/s; *L*coal is the standard length of the coal sample, mm; *µ*CH<sup>4</sup> is the gas dynamic viscosity coefficient, MPa·s; *A* is the cross sectional area of the coal sample, mm<sup>2</sup> ; *p*<sup>1</sup> is the inlet pressure, MPa; and *p*<sup>2</sup> is the outlet pressure, MPa.

<sup>=</sup> <sup>−</sup>

the coal sample, mm2; <sup>1</sup> *<sup>p</sup>* is the inlet pressure, MPa; and <sup>2</sup> *<sup>p</sup>* is the outlet pressure, MPa.

2

*p QL <sup>K</sup>*

( ) <sup>4</sup> 0 CH 2 2

is the gas dynamic viscosity coefficient, MPa·s; *A* is the cross sectional area of

2

μ

coal

1

**Figure 3.** Schematic diagram of the principle of the permeability determination test. **Figure 3.** Schematic diagram of the principle of the permeability determination test.

#### *2.3. Test Procedure 2.3. Test Procedure*

In this test, to investigate the changes in gas seepage velocities of briquette and raw coal samples in the complete stress–strain process, a comparative test was carried out according to the geological situation of the No. 82 coal seam in Zouzhuang Coal Mine at a gas pressure of 1.2 MPa and a confining pressure of 4 MPa. The specific procedure and precautions are as follows: (1) The sample, whose cling film was unwrapped before the test, was fixed with the upper and lower ventilative plates by a thin heat shrinkable film. Then, the fixed In this test, to investigate the changes in gas seepage velocities of briquette and raw coal samples in the complete stress–strain process, a comparative test was carried out according to the geological situation of the No. 8<sup>2</sup> coal seam in Zouzhuang Coal Mine at a gas pressure of 1.2 MPa and a confining pressure of 4 MPa. The specific procedure and precautions are as follows:

*3.1. Comparative Analysis on the Complete Stress–Strain Curves*

attributed to the macroscopic cracks penetrating the sample.

**Figure 5.** Complete stress–strain curves of briquette and raw coal.

**3. Test Results and Analysis**

In the triaxial compression test, the complete stress–strain curves of briquette and raw coal show similar variation trends (Figure 5). Both of them can be divided into four development stages: initial compaction stage, elastic deformation stage, plastic deformation

slight upsweep, which results from the compaction of pores and fractures inside the coal sample. In the elastic deformation stage, the stress and strain are linearly correlated with each other, and the elastic modulus becomes constant, following the Hoek–Brown criterion. In the plastic deformation stage, when the axial stress reaches the yield strength, internal damage occurs inside the coal sample, leading to a reduction in the sample's load-carrying capacity. At this time, the elastic modulus decreases, and the stress–strain curve is no longer linear and curves downward, which is caused by the continuously developing internal damage and new fractures in the sample. In the instability damage stage, after reaching the strength limit, the axial stress begins to decrease with the increase of the strain, which is

**Figure 4.** Sealed samples. **Figure 4.** Sealed samples.

#### **3. Test Results and Analysis 3. Test Results and Analysis**

**Figure 4.** Sealed samples.

#### *3.1. Comparative Analysis on the Complete Stress–Strain Curves 3.1. Comparative Analysis on the Complete Stress–Strain Curves*

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In the triaxial compression test, the complete stress–strain curves of briquette and raw coal show similar variation trends (Figure 5). Both of them can be divided into four development stages: initial compaction stage, elastic deformation stage, plastic deformation stage, and instability damage stage. In the initial compaction stage, the elastic modulus increases with the increase of axial stress and strain, and the stress–strain curve shows a slight upsweep, which results from the compaction of pores and fractures inside the coal sample. In the elastic deformation stage, the stress and strain are linearly correlated with each other, and the elastic modulus becomes constant, following the Hoek–Brown criterion. In the plastic deformation stage, when the axial stress reaches the yield strength, internal damage occurs inside the coal sample, leading to a reduction in the sample's load-carrying capacity. At this time, the elastic modulus decreases, and the stress–strain curve is no longer linear and curves downward, which is caused by the continuously developing internal damage and new fractures in the sample. In the instability damage stage, after reaching the strength limit, the axial stress begins to decrease with the increase of the strain, which is attributed to the macroscopic cracks penetrating the sample. In the triaxial compression test, the complete stress–strain curves of briquette and raw coal show similar variation trends (Figure 5). Both of them can be divided into four development stages: initial compaction stage, elastic deformation stage, plastic deformation stage, and instability damage stage. In the initial compaction stage, the elastic modulus increases with the increase of axial stress and strain, and the stress–strain curve shows a slight upsweep, which results from the compaction of pores and fractures inside the coal sample. In the elastic deformation stage, the stress and strain are linearly correlated with each other, and the elastic modulus becomes constant, following the Hoek–Brown criterion. In the plastic deformation stage, when the axial stress reaches the yield strength, internal damage occurs inside the coal sample, leading to a reduction in the sample's load-carrying capacity. At this time, the elastic modulus decreases, and the stress–strain curve is no longer linear and curves downward, which is caused by the continuously developing internal damage and new fractures in the sample. In the instability damage stage, after reaching the strength limit, the axial stress begins to decrease with the increase of the strain, which is attributed to the macroscopic cracks penetrating the sample.

**Figure 5. Figure 5.**Complete stress–strain curves of briquette and raw coal. Complete stress–strain curves of briquette and raw coal.

The comparative analysis suggests many differences between the test results of the two samples. First, briquette experiences more severe transverse deformation and axial deformation than raw coal, its transverse deformation being twice that of raw coal and its axial deformation being three times that of raw coal. Second, briquette has a lower compressive strength than raw coal, as the compressive strength of raw coal is 27.1 MPa and the compressive strength of briquette is 17.3 MPa, the former being 63.7% of the latter. The elastic modulus of briquette obtained from the test is 0.517 GPa, the elastic modulus of raw coal is 2.306 GPa, the Poisson's ratio of briquette is 0.22, and the Poisson's ratio of raw coal is 0.16. Third, the two coal samples exhibit obvious differences in the post-peak deformation and damage stage. Specifically, after the peak, briquette presents a strain softening phenomenon and a gentle decrease in axial pressure, while raw coal shows a stress drop and significant and sharp change in axial pressure, just like the instantaneously and rapidly occurring coal and gas outburst on-site.

The factor that most directly determines coal permeability to gas in coal seams is the development degree of pores and fractures. In the laboratory, both briquette and raw coal have certain primary micropores and microfractures, which are referred to as primary damages [20]. The lower initial permeability of raw coal indicates its much slighter primary damages than briquette. However, as the applied load continues to grow, these primary damages will further develop, extend, and finally connect with each other, leading to a change in permeability of the coal sample.

Based on the changes in strains and permeabilities of briquette and raw coal during the loading stage, the complete stress–strain curves and permeability-strain curves of the two samples were obtained (Figure 6). A correspondence can be found between the complete stress–strain curves and the permeability-strain curves of briquette and raw coal, but the two coal samples have different gas permeability variation laws due to their different damage forms:

#### *3.2. Relation between Permeability and Axial Stress*

Based on the data of strains and permeabilities of briquette and raw coal in the loading stage, the permeability-axial stress curves of the two samples were obtained (Figure 7). When the axial stress is below the yield stress, the permeabilities of raw coal and briquette show essentially the same variation trend, that is, they decline with the increase of axial pressure and reach the minimum at the yield stress point. The difference is that the permeability of raw coal diminishes more gently than that of briquette. This shows that briquette is loose and soft, with a large number of voids and a large compressible space; although raw coal has primary fractures, its initial permeability is low and cannot be enhanced obviously by compression.

permeability grows only gently.

permeability grows only gently.

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generated by the local damage. These large instability-induced fractures enable the gas to pass through smoothly and promote the permeability of raw coal rapidly. However, as briquette only develops based on shear damage, its bearing capacity

generated by the local damage. These large instability-induced fractures enable the gas to pass through smoothly and promote the permeability of raw coal rapidly. However, as briquette only develops based on shear damage, its bearing capacity begins to decline, and its internal structure disenables a sudden stress drop, so its

**Figure 6.** Axial pressure-permeability-strain curves of briquette and raw coal. (**a**) Briquette; (**b**) raw **Figure 6.** Axial pressure-permeability-strain curves of briquette and raw coal. (**a**) Briquette; (**b**) raw coal. and cannot be enhanced obviously by compression.

**Figure 7.** Axial pressure-permeability curves of briquette and raw coal. (**a**) Briquette; (**b**) raw coal. **Figure 7.** Axial pressure-permeability curves of briquette and raw coal. (**a**) Briquette; (**b**) raw coal.

*K/*10-15 m2 When the axial pressure exceeds the yield stress, the permeabilities of both coal samples begin to grow, but in obviously different variation trends. In the plastic deformation stage (BC section), briquette exhibits only a smooth rise in permeability, When the axial pressure exceeds the yield stress, the permeabilities of both coal samples begin to grow, but in obviously different variation trends. In the plastic deformation stage (BC section), briquette exhibits only a smooth rise in permeability, while raw coal shows a steep rise due to the fracture seepage caused by the development of primary fractures and the generation of new fractures.

(**a**) (**b**) **Figure 7.** Axial pressure-permeability curves of briquette and raw coal. (**a**) Briquette; (**b**) raw coal. while raw coal shows a steep rise due to the fracture seepage caused by the development of primary fractures and the generation of new fractures. The peak stress point indicates that the sample reaches its maximum bearing capacity. At this time, the fractures accumulated before the peak reach a critical number, and the sample is on the verge of complete damage, which is a turning point for permeability. The instability damage stage (CD section) reflects the post-peak permeability variation trend. From Figure 7a, it is observed that after the peak stress, raw coal exhibits a rapid stress drop and a steep permeability rise, and this phenomenon indicates that its main fracture emerges and extends suddenly, so the damage is sudden. After raw coal is damaged, its gas pressure gradient soars, thus raising the risk of coal and gas outbursts. From Figure 7b, it can be seen that, unlike raw coal, briquette does not feature suddenness in its parameters, that is, the two samples differ essentially in terms of damage modes and seepage characteristics.

while raw coal shows a steep rise due to the fracture seepage caused by the development

of primary fractures and the generation of new fractures.

When the axial pressure exceeds the yield stress, the permeabilities of both coal

#### **4. Discussion** In this test, the change in axial pressure affects the permeability of the coal sample at

of damage modes and seepage characteristics.

#### *4.1. Sensitivity Analysis on Gas Permeabilities of Briquette and Raw Coal* all stages. The entire variation of axial pressure was normalized to analyze the sensitivity

*4.1. Sensitivity Analysis on Gas Permeabilities of Briquette and Raw Coal*

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In this test, the change in axial pressure affects the permeability of the coal sample at all stages. The entire variation of axial pressure was normalized to analyze the sensitivity of coal permeability to axial pressure in each stage. The axial pressure and permeability exhibit different trends throughout the process and the absolute values of their variations in all stages were summed as 1 for normalization. The axial pressure and permeability gradient curves of the two coal samples are given in Figure 8. of coal permeability to axial pressure in each stage. The axial pressure and permeability exhibit different trends throughout the process and the absolute values of their variations in all stages were summed as 1 for normalization. The axial pressure and permeability gradient curves of the two coal samples are given in Figure 8.

The peak stress point indicates that the sample reaches its maximum bearing

capacity. At this time, the fractures accumulated before the peak reach a critical number, and the sample is on the verge of complete damage, which is a turning point for permeability. The instability damage stage (CD section) reflects the post-peak permeability variation trend. From Figure 7a, it is observed that after the peak stress, raw coal exhibits a rapid stress drop and a steep permeability rise, and this phenomenon indicates that its main fracture emerges and extends suddenly, so the damage is sudden. After raw coal is damaged, its gas pressure gradient soars, thus raising the risk of coal and gas outbursts. From Figure 7b, it can be seen that, unlike raw coal, briquette does not feature suddenness in its parameters, that is, the two samples differ essentially in terms

**Figure 8.** Axial pressure-permeability gradient curves of briquette and raw coal. **Figure 8.** Axial pressure-permeability gradient curves of briquette and raw coal.

For briquette, its permeability and axial pressure gradients have the same variation trend, both being the largest (55% and 63% respectively) in the elastic deformation stage (AB section) and relatively small in the other stages. This shows that, under triaxial For briquette, its permeability and axial pressure gradients have the same variation trend, both being the largest (55% and 63% respectively) in the elastic deformation stage (AB section) and relatively small in the other stages. This shows that, under triaxial compression, the permeability of briquette is significantly affected by the axial stress and is the most sensitive to axial pressure in the elastic deformation stage.

compression, the permeability of briquette is significantly affected by the axial stress and is the most sensitive to axial pressure in the elastic deformation stage. In contrast, for raw coal, the variation laws of permeability and axial pressure are relatively inconsistent. Nevertheless, in the plastic deformation stage (BC section) and In contrast, for raw coal, the variation laws of permeability and axial pressure are relatively inconsistent. Nevertheless, in the plastic deformation stage (BC section) and instability damage stage (CD section), the variation laws of the two parameters are roughly consistent, which indicates a relatively high sensitivity of its permeability to axial pressure in these two stages.

#### instability damage stage (CD section), the variation laws of the two parameters are *4.2. Analysis of the Suddenness of Coal and Gas Outbursts*

roughly consistent, which indicates a relatively high sensitivity of its permeability to axial pressure in these two stages. *4.2. Analysis of the Suddenness of Coal and Gas Outbursts*  Generally, the occurrence of coal and gas outbursts can be divided into four stages: Generally, the occurrence of coal and gas outbursts can be divided into four stages: preparation, initiation, development, and termination. However, some scholars believe that initiation and termination are only two mutation points, while only preparation and development are continuous processes [21]. The seepage tests on briquette and raw coal under triaxial compression can provide new insights into the mechanism of coal and gas outbursts.

preparation, initiation, development, and termination. However, some scholars believe that initiation and termination are only two mutation points, while only preparation and development are continuous processes [21]. The seepage tests on briquette and raw coal Based on the complete stress–strain test and seepage test on raw coal, it is found that the initial compaction stage and elastic deformation stage of the stress–strain-seepage velocity variation belong to the preparation stage of coal and gas outbursts. In this stage, with the rise of axial pressure, the increasing elastic energy from the elastic deformation blocks the gas flow channel, and thus the gas internal energy mounts up obviously. These changes create conditions for the initiation of an outburst. In the plastic deformation stage, plastic deformation occurs because of stable extension of continuously distributed microfractures. At this time, the stress of the sample reaches a limited equilibrium state.

In the instability damage stage, the stress drop is a mutation point of outburst initiation. Under the action of axial pressure, the limited equilibrium state of stress is damaged into an instability state. At this time, the sudden release of elastic energy and gas internal energy accumulated in the solid medium of the coal body leads to sudden coal damage, which enables a continuous outburst until the coal body completely loses its bearing capacity. When the subsequent elastic energy and gas internal energy are below the surrounding constraining force, the outburst terminates.

Coal and gas outburst is a kind of dynamic disaster that can be intensely completed within a short time. The difficulty in predicting it lies in its suddenness. As can be concluded from the test, after reaching the peak, the stress of raw coal drops sharply within a short time, accompanied by an increase in gas pressure gradient and a surge of seepage velocity, which is quite close to the occurrence of coal and gas outbursts on site, while the changes in briquette are quite different from the real outburst. Such a result shows that the suddenness of coal and gas outburst is determined by the sudden coal damage under the joint action of in situ stress (axial pressure and confining pressure) and gas pressure, which cannot be accurately reflected by briquette.

#### **5. Conclusions**

**Author Contributions:** Conceptualization, K.D.; Formal analysis, Z.L. and C.J.; Funding acquisition, Z.L. and J.G.; Investigation, K.D. and Z.L.; Methodology, L.W.; Software, B.R.; Validation, S.W.; Visualization, J.G.; Writing – original draft, K.D.; Writing – review & editing, Z.L. and J.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This study was supported by Natural Science Foundation of Jiangsu Province, China (Grant No. BK20200628).

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data used to support the findings of this study are included within the article.

**Acknowledgments:** This study was supported by Natural Science Foundation of Jiangsu Province, China (Grant No. BK20200628) and the support is gratefully acknowledged.

**Conflicts of Interest:** The authors declare that they have no conflict of interest regarding the publication of this work.

#### **References**

## *Article* **Mechanical Properties and Failure Mechanism of Anchored Bedding Rock Material under Impact Loading**

**Yunhao Wu <sup>1</sup> , Xuesheng Liu 1,2,\* , Yunliang Tan 1,2,\*, Qing Ma <sup>1</sup> , Deyuan Fan <sup>1</sup> , Mingjie Yang <sup>3</sup> , Xin Wang <sup>1</sup> and Guoqing Li <sup>1</sup>**

**Abstract:** In view of the problem that anchored bedding rock material is prone to instability and failure under impact loading in the process of deep coal mining, and taking the lower roadway of a deep 2424 coal working face in the Suncun coal mine as the engineering background, a mechanical model of anchored bedding rock material was established, and the instability criterion of compression and shear failure of anchored bedding rock material was obtained. Then, the separated Hopkinson pressure bar was used to carry out an impact-loading test on the anchored bedding rock material, and the dynamic mechanical properties of the rock with different anchoring modes and bolt bedding angles were studied; the evolution law of the strain field of the anchored bedding rock material was also obtained. The results show the following: (1) The bolt support could effectively improve the dynamic load strength and dynamic elastic modulus of the rock material with anchorage bedding, the degree of improvement increased with the increase in the angle of the bolt bedding, and the full anchorage effect was much higher than the end anchorage effect was. (2) The bolt bedding angle and anchorage mode greatly influenced crack development and displacement characteristics. After an impact, the bedding rock material had obvious shear displacement along the bedding direction, and obvious macroscopic cracks were produced in the bedding plane. The research results offer theoretical guidance to and have reference significance for deep roadway anchorage support engineering.

**Keywords:** anchored bedding rock material; impact loading; instability mechanism; complex stress environment; shear failure

#### **1. Introduction**

Deep coal resources are an important guarantee as China's main energy sources [1–3]. Statistics show that coal resources with a depth of more than 600 m account for about 73% of the total coal, while coal resources with a depth of more than 1000 m account for about 53%. Deep mining is becoming the new normal in coal-resource development [4,5].

In the process of deep coal mining, the surrounding rocks of deep roadways bear large initial ground stress, and the superposition of supporting pressure leads to stress concentration, which places the surrounding rock in a state of high strength compression for a long time [6–8]. In addition, the mining process is often accompanied by the periodic fracturing and the lateral collapse of the overlying strata, fault slips, blasting, and other activities [9,10]. The impact loading generated by these activities causes strong disturbance to the surrounding rock of the roadway [11–14]. When the static load and impact loading of the surrounding rock in the mining space exceed the critical load of the impact failure of coal and rock material, the surrounding rock becomes unstable, resulting in roof collapses, hydraulic support compression frames, partial sidewalls, and other rock burst

**Citation:** Wu, Y.; Liu, X.; Tan, Y.; Ma, Q.; Fan, D.; Yang, M.; Wang, X.; Li, G. Mechanical Properties and Failure Mechanism of Anchored Bedding Rock Material under Impact Loading. *Materials* **2022**, *15*, 6560. https:// doi.org/10.3390/ma15196560

Academic Editor: Tomasz Sadowski

Received: 29 August 2022 Accepted: 19 September 2022 Published: 21 September 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

accidents [15–18]. The complex geological environment of the surrounding rock is also an important factor that threatens the stability of the surrounding rock of roadways [19,20]. As layered composite rock material is widely distributed in underground roadways, and the bedding plane of the rock stratum is its natural weak surface, the existence of bedding weakens the antideformation ability of the rock material and reduces the stability of the surrounding rock [21–25]. Therefore, the layered surrounding rock of deep coal mining roadways is prone to instability and failure under impact loading, and supporting it is very difficult [26,27], which has become one of the main bottlenecks restricting the safety of deep coal mining.

Aiming at the failure problem of anchorage structures under impact loading, Wu Yongzheng [28,29] analyzed the dynamic load response characteristics of surrounding anchorage rock under impact loading, and studied the dynamic response law of anchorage rock under lateral impact load, and the influence of impact energy and bolt mechanical properties on rock material mechanical behavior. Mu Zonglong [30] analyzed the failure conditions of roadway surrounding rock under static load, and dynamic and static loads, and experimentally studied the failure conditions of dynamic load on surrounding roadway rock. Wang Zhengyi [31] analyzed the dynamic effect of longitudinal waves, established the criterion of rock burst of anchoring roadway support structures, and formulated the corresponding preventive measures. Wang Aiwen et al. [32] found that there is an obvious time difference effect between the vibration of the bolt and the surrounding rock under impact loads, which leads to the asynchronous vibration of the bolt and the surrounding rock, resulting in the dynamic shear of the anchorage agent. Qiu Penggqi [33] believed that improving the antisliding characteristics and coordinated deformation ability of a rock, the anchoring agent and bolt can effectively prolong the anti-impact aging time of anchoring rock and reduce the impact of dynamic load on the supporting structure of surrounding anchoring rock. Skrzypkowski K [34,35] believed that the places of particular exposure to shear stresses are faults and layered roof layers; between them, there are surfaces of reduced cohesion. Jiao Jiankang [36] analyzed the dynamic load response characteristics and impact failure evolution process of roadway anchorage bearing structures under dynamic load disturbances, and put forward the impact-failure criterion, and the criterion of anchorage bearing structures under dynamic load disturbance and the control technology of surrounding roadway anchorage rock under dynamic load disturbance.

These studies revealed the failure mechanism of anchorage structures under impact loading to a certain extent, and put forward the corresponding control technology of anchorage rocks of rock burst roadways, but failed to fully reveal the overall instability mechanism of anchorage bodies. There is little research on dynamic instability of anchorage bodies under complex stratum conditions, which fails to solve the problem of the instability and failure of anchored bedding rock material that widely exists in engineering geology.

Therefore, the lower roadway of 2424 coal working face in Suncun Coal Mine was taken as the engineering background, the mechanical model of the anchored bedding rock material is established, the compression and shear instability criterion of the anchored bedding rock material is put forward, and the split Hopkinson pressure bar test device was used to carry out an impact loading test on the anchored bedding rock material and reveal the influence law and instability mechanism of the angle between different bolts and bedding and the anchoring mode on the mechanical properties of the bedding rock material.

#### **2. Stress Analysis and Instability Mechanism of Anchored Bedding Rock Material**

#### *2.1. Engineering Background*

Taking the lower roadway of 2424 coal working face in the Suncun coal mine as an example, the coal seam of the −800 m level 2424 coal working face was four coals, and the rock column diagram is shown in Figure 1.

*2.1. Engineering Background*

*2.1. Engineering Background*

2.

the rock column diagram is shown in Figure 1.

the rock column diagram is shown in Figure 1.

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**Figure 1.** 2424 Coal working face rock histogram. **Figure 1.** 2424 Coal working face rock histogram. **Coal 4 Siltstone** 2.4 7.2 Coal seam structure Simplicity. Gray, containing fossil plants, containing a

The average thickness of the coal seam was 2.4 m, and the dip angle was 26.5◦ . The direct top was siltstone and sandstone, thickness was about 5.8 m, bedding development was relatively hard, and compressive strength was 21.2 MPa. The basic roof was mainly sandstone with about 11.2 m thick and hard bedding. The buried depth of the lower roadway in the 2424 coal working face was 1254 m, and the total length was 1132 m. The section was trapezoidal, and the net width was 4 m. The height of the left side was 3 m, and the height of the right side was 4 m. The roof was an arc section, as shown in Figure 2. **Sandstone Mudstone Sandstone** 10.7 8.0 4.3 Sandstone, grayish white, hard, thick bedding. Black, fragile, stratiform, containing fossil plants. Sandstone, grayish white, medium, hard. **Figure 1.** 2424 Coal working face rock histogram.

small amount of mudstone.

**2. Stress Analysis and Instability Mechanism of Anchored Bedding Rock Material**

**2. Stress Analysis and Instability Mechanism of Anchored Bedding Rock Material**

Taking the lower roadway of 2424 coal working face in the Suncun coal mine as an example, the coal seam of the −800 m level 2424 coal working face was four coals, and

The average thickness of the coal seam was 2.4 m, and the dip angle was 26.5°. The direct top was siltstone and sandstone, thickness was about 5.8 m, bedding development was relatively hard, and compressive strength was 21.2 MPa. The basic roof was mainly sandstone with about 11.2 m thick and hard bedding. The buried depth of the lower roadway in the 2424 coal working face was 1254 m, and the total length was 1132 m. The section was trapezoidal, and the net width was 4 m. The height of the left side was 3 m, and the height of the right side was 4 m. The roof was an arc section, as shown in Figure

Taking the lower roadway of 2424 coal working face in the Suncun coal mine as an example, the coal seam of the −800 m level 2424 coal working face was four coals, and

The average thickness of the coal seam was 2.4 m, and the dip angle was 26.5°. The

**Figure 2.** Schematic diagram of roadway support. **Figure 2.** Schematic diagram of roadway support.

The mining process of the coal working face uses a Φ22 × 2400 mm HRB 600 resin anchored anchor bolt + diamond mesh + W steel belt + tray support. The anchor bolt anchorage length was 1.2 m. The three middle rows of anchor bolts were installed on the vertical roof (at an angle of 14◦ with the vertical line), and the outermost anchor bolts were installed at an angle of 20◦ with the vertical line.

This was adopted by two gangs, namely, Φ22-2200 mm MSGLD-600(x) equal strength threaded steel bolt + diamond mesh + W steel strip + high-strength tray support. The lowest row of bolts on the two sides of the roadway was 500 mm from the roadway floor and inclined downward at an angle of 30◦ with the horizontal line. The lowest row of bolts on the two sides of the roadway was 500 mm from the roadway roof and inclined upward at an angle of 30◦ with the horizontal line. cium, and water). *2.2. Stress Analysis of the Anchored Rock Material* The dynamic action in the mining process mainly comes from mining activity and the stress response of surrounding rock to the mining activity. The mining process of a coal seam is always accompanied by the movement of the overlying strata. The high

The mining process of the coal working face uses a Ф22 × 2400 mm HRB 600 resin anchored anchor bolt + diamond mesh + W steel belt + tray support. The anchor bolt anchorage length was 1.2 m. The three middle rows of anchor bolts were installed on the vertical roof (at an angle of 14° with the vertical line), and the outermost anchor bolts

This was adopted by two gangs, namely, Ф22‐2200 mm MSGLD‐600(x) equal strength threaded steel bolt + diamond mesh + W steel strip + high‐strength tray sup‐ port. The lowest row of bolts on the two sides of the roadway was 500 mm from the roadway floor and inclined downward at an angle of 30° with the horizontal line. The lowest row of bolts on the two sides of the roadway was 500 mm from the roadway roof

A half‐length anchor was adopted for the two sides and the roof. The anchoring length was 1.4 m. Fast‐setting resin was selected as the anchoring agent, composed of two kinds of materials, namely, resin cement (cement containing resin, dolomite pow‐ der, silica, and accelerator) and a curing agent (containing benzoyl peroxide, light cal‐

A half-length anchor was adopted for the two sides and the roof. The anchoring length was 1.4 m. Fast-setting resin was selected as the anchoring agent, composed of two kinds of materials, namely, resin cement (cement containing resin, dolomite powder, silica, and accelerator) and a curing agent (containing benzoyl peroxide, light calcium, and water). stress waves generated by the roof breaking in the movement of the overlying strata form mutual interference and superposition in the coal and rock, and have a strong dis‐turbance effect on the surrounding rock of a roadway. Taking the lower drift of coal working face 2424 of the Suncun coal mine as an ex‐

#### *2.2. Stress Analysis of the Anchored Rock Material* ample, when the layered composite surrounding rock roof and two sides of the deep mining roadway were supported by bolts, the anchored surrounding rock was subject to

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were installed at an angle of 20° with the vertical line.

and inclined upward at an angle of 30° with the horizontal line.

The dynamic action in the mining process mainly comes from mining activity and the stress response of surrounding rock to the mining activity. The mining process of a coal seam is always accompanied by the movement of the overlying strata. The high stress waves generated by the roof breaking in the movement of the overlying strata form mutual interference and superposition in the coal and rock, and have a strong disturbance effect on the surrounding rock of a roadway. high ground stress and impact loading, which led to the surrounding anchored layered rock very easily losing stability and breaking the ring, as shown in Figure 3a. In order to study the stress condition and instability mechanism of the anchored surrounding rock at the bedding development under impact loading, the mechanical model of the anchor under impact loading was established with the roof anchor as the research object, as shown in Figure 3b. The inclination angle of the anchor rod was *θ*1,

Taking the lower drift of coal working face 2424 of the Suncun coal mine as an example, when the layered composite surrounding rock roof and two sides of the deep mining roadway were supported by bolts, the anchored surrounding rock was subject to high ground stress and impact loading, which led to the surrounding anchored layered rock very easily losing stability and breaking the ring, as shown in Figure 3a. and bedding dip angle was *α*. The anchor body was subjected to vertical stress of sur‐ rounding rock *σ*v, and horizontal stress *σ*<sup>h</sup> was also affected by the dynamic load dis‐ turbance caused by roof fracture instability. The dynamic load was *σ*d, which included angle with anchor rod *θ*2.

**Figure 3.** Stress environment of the surrounding rock and mechanical model of anchored bedding rock material. (**a**) Anchorage structure and stress environment of layered composite surrounding rock in Inclined Mining Roadway; (**b**) Mechanical model of anchorage bedding rock mass. **Figure 3.** Stress environment of the surrounding rock and mechanical model of anchored bedding rock material. (**a**) Anchorage structure and stress environment of layered composite surrounding rock in Inclined Mining Roadway; (**b**) Mechanical model of anchorage bedding rock mass.

In order to study the stress condition and instability mechanism of the anchored surrounding rock at the bedding development under impact loading, the mechanical model of the anchor under impact loading was established with the roof anchor as the research object, as shown in Figure 3b. The inclination angle of the anchor rod was *θ*1, and bedding dip angle was *α*. The anchor body was subjected to vertical stress of surrounding rock *σ*v, and horizontal stress *σ*<sup>h</sup> was also affected by the dynamic load disturbance caused by roof fracture instability. The dynamic load was *σ*d, which included angle with anchor rod *θ*2.

A dynamic load generated by roof fracture instability mainly propagates in the form of stress waves in the rock structure. As the surrounding rock in the actual project is heterogeneous and anisotropic material, it is difficult to accurately analyze it. Therefore, the following were assumed: <sup>1</sup> The surrounding rock of the roadway is an ideal homogeneous elastic–plastic body; <sup>2</sup> at a certain distance from the source, the stress waves propagating in the rock can be regarded as plane waves [37]. The shear wave effect of the stress waves was not considered, and the dynamic load (*σ*p) generated by longitudinal waves (P) P can be expressed as follows:

$$
\sigma\_\mathbf{P} = \rho \mathbb{C}\_\mathbf{P} v\_\mathbf{P} \tag{1}
$$

where *C*<sup>p</sup> is the propagation velocity of P waves, m/s; *v*<sup>p</sup> is the peak vibration velocity of the particle at the boundary of the dynamic load source, m/s; and *ρ* is the density of rock mass, kg/m<sup>3</sup> .

The research shows that the propagation of stress waves in a coal rock medium presents a power function attenuation law [38], and the attenuation law of particle vibration velocity can be expressed as follows:

$$
\upsilon = \upsilon\_{\rm P} L^{-\eta} \tag{2}
$$

where *v* is the peak vibration velocity of the particle at the propagation of the stress wave, m/s; *L* is the distance from the boundary of a moving source, m; *η* is the peak velocity attenuation coefficient related to the propagation medium; attenuation coefficients *η* of the intact rock mass, intact coal body, and fractured coal body are [38] 0.5755, 0.867, 1.0655, respectively.

Combining Formulas (1) and (2), the dynamic load caused by stress-wave propagation to the surface of the anchorage structure is:

$$
\sigma\_{\mathbf{d}} = \rho \mathbf{C}\_{\mathbf{P}} v\_{\mathbf{P}} L^{-\eta} \tag{3}
$$

At this time, stress at any point O on the anchorage body can be characterized along the inclined direction (x axis) and the vertical direction (y axis) of the bolt as follows:

$$\begin{cases} \sigma\_1 = \sigma\_\text{x} = \sigma\_\text{v} \cdot \sin \theta\_1 + \rho \mathbf{C}\_\text{p} v\_\text{p} L^{-\eta} \cdot \cos \theta\_2\\ \sigma\_3 = \sigma\_\text{y} = \sigma\_\text{v} \cdot \cos \theta\_1 + \rho \mathbf{C}\_\text{p} v\_\text{p} L^{-\eta} \cdot \sin \theta\_2 \end{cases} \tag{4}$$

According to the Mohr–Coulomb criterion, because of *σ*<sup>3</sup> > 0, when *σ*<sup>1</sup> = *σ<sup>v</sup>* · sin *θ*<sup>1</sup> + *ρC*p*v*p*L* −*η* · cos *θ*<sup>2</sup> ≥ *σ*<sup>c</sup> (*σ*<sup>c</sup> is uniaxial compressive strength of anchorage), compression fracture occurs in the surrounding rock of an anchor solid.

#### *2.3. Shear Failure Mechanism of Anchorage Rock Material*

Since the dynamic load propagation process is transferred from sandstone to siltstone, and the propagation velocity of stress waves in different strata is different, shear dislocation occurs on the joint surface, and shear failure occurs on the anchorage body. In order to analyze this problem, the instantaneous shear model of anchor solid bedding surface under dynamic load was established, as shown in Figure 4. Assuming that the transverse displacement of the structural plane was *U*1, the longitudinal displacement was *U*2, and *U*<sup>2</sup> = *U*<sup>1</sup> tan*ψ*, *ψ* is the shear angle of the structural plane.

Then, the lateral deformation of the bolt is *u*1, and the axial deformation is *u*2. According to the geometric relationship:

$$\begin{aligned} \mu\_1 &= \mathcal{U}\_1 \sin(\mathfrak{a} + \theta\_1) / \cos \mathfrak{a} \\ \mu\_2 &= \mathcal{U}\_2 / \sin(\mathfrak{a} + \theta\_1) \end{aligned} \tag{5}$$

The shear strength of the anchored bedding rock material is generally composed of four parts: the shear strength of the bedding plane itself *τ<sup>j</sup>* , the shear strength contributed by the 'pin' action of the bolt body *τbd*, and the shear strength contributed by the normal and tangential components of the axial load of the bolt along the joint surface *τbd*, *τbs* [39]. Therefore, shear strength *τbd* of the anchored bedding rock material can be expressed as follows:

$$
\tau\_{bj} = \tau\_j + \tau\_{bd} + \tau\_{bt} + \tau\_{bs} \tag{6}
$$

**Figure 4.** Instantaneous shear model of anchor solid bedding plane. **Figure 4.** Instantaneous shear model of anchor solid bedding plane.

Then, the lateral deformation of the bolt is *u*1, and the axial deformation is *u*2. Ac‐ cording to the geometric relationship: According to the Coulomb–Mohr criterion, the shear strength *τ<sup>j</sup>* of the bedding plane itself is:

$$\tau\_{\dot{j}} = \sigma\_{\dot{j}} \tan \left( \psi + \varphi' \right) + c\_{\dot{j}} \tag{7}$$

 11 1 22 1 sin cos sin *u U α θ / α u U/ α θ* (5) where *σ<sup>j</sup>* is the normal stress of the structural plane, *c<sup>j</sup>* is the structural surface adhesion, and *ϕ* 0 is the structural surface friction angle.

The shear strength of the anchored bedding rock material is generally composed of *τbd*, *τbd*, and *τbd* can be calculated with the following formula:

$$\begin{cases} \quad \pi\_{bd} = \pi\_b \cdot \eta \left[ \sin(a + \theta\_1) / \cos a \right] \\ \quad \pi\_{bt} = \sigma\_b \cdot \sin(a + \theta\_1) \cdot \eta \cdot \tan \phi' \\ \quad \pi\_{bs} = \sigma\_b \cdot \cos(a + \theta\_1) \cdot \eta \end{cases} \tag{8}$$

follows: *bj j bd bt bs τ = ττ τ τ* (6) where *σ<sup>b</sup>* is axial stress of the bolt, *τ<sup>b</sup>* is the average shear stress on the cross-section of the bolt, and *η* is the ratio of the cross-sectional area of the bolt to the area of the bedding plane containing a single bolt. *Materials* **2022**, *15*, x FOR PEER REVIEW 7 of 18

Therefore, shear strength *τbd* of the anchored bedding rock material can be expressed as

According to the Coulomb–Mohr criterion, the shear strength *τ<sup>j</sup>* of the bedding plane itself is: In order to obtain the axial stress of the bolt, a bolt element was taken for stress analysis, as shown in Figure 5.

*bt*

*b*

cos *bs b τ σ θη α* **Figure 5.** Mechanical model of a bolt element. **Figure 5.** Mechanical model of a bolt element.

tained:

ness, 1 2

terial, respectively).

expressed as follows:

chor.

*K K <sup>K</sup> K K*

1 2

From the above analysis:

*N* (*x* = 0) = *N*0, *N* (*x* = *l*) = 0, and the solution is:

Then the axial stress *σ<sup>b</sup>* of bolt is:

where *σ<sup>b</sup>* is axial stress of the bolt, *τ<sup>b</sup>* is the average shear stress on the cross‐section of the According to the unit force balance, the differential equation is established: According to the unit force balance, the differential equation is established:

$$N(\mathbf{x}) - \pi D \tau(\mathbf{x}) d\mathbf{x} = N(\mathbf{x}) + d N(\mathbf{x}) \tag{9}$$

(*K*<sup>1</sup> and *K*<sup>2</sup> are shear stiffness of slurry and shear stiffness of rock ma‐

In order to obtain the axial stress of the bolt, a bolt element was taken for stress analysis, as shown in Figure 5. where *N*(*x*) is the axial force of the bolt at position *x* from the bedding plane, *D* is the bolt diameter, and *τ*(*x*) is the interfacial shear stress at *x* position from the bedding where *N*(*x*) is the axial force of the bolt at position *x* from the bedding plane, *D* is the bolt diameter, and *τ*(*x*) is the interfacial shear stress at *x* position from the bedding plane.

1 1

plane. Through the linear relationship between load and displacement, and the stress equilibrium state of bolt, the equation of interfacial shear stress with displacement is ob‐ Through the linear relationship between load and displacement, and the stress equilibrium state of bolt, the equation of interfacial shear stress with displacement is obtained:

> *dN x πDKu x*

Then, the axial strain *ε*(*x*) of the bolt at a distance from the bedding plane *x* can be

*<sup>x</sup> du x N x <sup>σ</sup>*

where *E* is the elastic modulus of the anchor, and *A* is the cross‐sectional area of the an‐

 <sup>2</sup> <sup>2</sup> 0

When the bedding plane is damaged, it is assumed that the axial force of the bolt at the bedding plane is pretightening force *N*0, and the axial force at the distance from the joint plane *u*<sup>2</sup> (axial deformation of the bolt) is 0. Therefore, the boundary conditions are

<sup>2</sup>

*K*

*EA*

*u x*

2

*u*

*K*

*EA*

*dN x K N x dx EA*

$$
\pi(\mathbf{x}) = K u(\mathbf{x}) \tag{10}
$$

*τ x Ku x* (10)

*dx* (11)

*dx E EA* (12)

(13)

(14)

(8)

Combining Formulas (9) and (10), we can obtain the following:

*ε x*

*Nx N*

0

sinh

sin

where *u*(*x*) is the axial normal strain of the bolt; and *K* is the comprehensive shear stiffness, *K* = *K*1*K*<sup>2</sup> *K*1+*K*<sup>2</sup> (*K*<sup>1</sup> and *K*<sup>2</sup> are shear stiffness of slurry and shear stiffness of rock material, respectively).

Combining Formulas (9) and (10), we can obtain the following:

$$\frac{dN(\mathbf{x})}{d\mathbf{x}} = \pi DKu(\mathbf{x}) \tag{11}$$

Then, the axial strain *ε*(*x*) of the bolt at a distance from the bedding plane *x* can be expressed as follows:

$$
\varepsilon(\mathbf{x}) = \frac{d\mu(\mathbf{x})}{d\mathbf{x}} = \frac{\sigma\_\mathbf{x}}{E} = \frac{\mathbf{N}(\mathbf{x})}{EA} \tag{12}
$$

where *E* is the elastic modulus of the anchor, and *A* is the cross-sectional area of the anchor. From the above analysis:

$$\frac{d^2N(\mathbf{x})}{d\mathbf{x}^2} - \frac{\mathbf{K} \cdot \mathbf{N}(\mathbf{x})}{EA} = \mathbf{0} \tag{13}$$

When the bedding plane is damaged, it is assumed that the axial force of the bolt at the bedding plane is pretightening force *N*0, and the axial force at the distance from the joint plane *u*<sup>2</sup> (axial deformation of the bolt) is 0. Therefore, the boundary conditions are *N* (*x* = 0) = *N*0, *N* (*x* = *l*) = 0, and the solution is:

$$N(\mathbf{x}) = N\_0 \frac{\sinh\left[\sqrt{\frac{K}{EA}}(\mu\_2 - \mathbf{x})\right]}{\sin\left(\mu\_2 \sqrt{\frac{K}{EA}}\right)}\tag{14}$$

Then the axial stress *σ<sup>b</sup>* of bolt is:

$$\sigma\_b = \frac{N\_0 \sinh\left[\sqrt{\frac{4K}{E\pi D^2}}(u\_2 - \chi)\right]}{\frac{\pi D^2}{4}\sin\left(u\_2\sqrt{\frac{4K}{E\pi D^2}}\right)}\tag{15}$$

Before the failure of the bedding anchor solid, the anchor and rock near the bedding plane are in an elastic state. According to the principle of elastic mechanics, the lateral deformation of the anchor satisfies:

$$\frac{d^4\mu\_1}{dy^4} + \frac{Dk}{EI}\mu\_1 = 0\tag{16}$$

where *k* = 300 *σ*c/*D*, the rock reaction coefficient; and *EI* is the anchor bending stiffness.

Analysis shows that, when *y* = 0, the bending moment is 0; when *y*→∞, the bending moment is 0, and the shear force is 0:

$$\begin{cases} M\_{y=0} = -EI\frac{d^2u\_1}{dy^2} = 0\\ M\_{y\to\infty} = -EI\frac{d^2u\_1}{dy^2} = 0\\ \tau\_{y\to\infty} = -EI\frac{d^3u\_1}{dy^3} = 0 \end{cases} \tag{17}$$

From these boundary conditions, the relationship between the lateral displacement of the bolt and shear stress *τ<sup>b</sup>* in the shear-slip process of the bedding plane can be obtained:

$$\pi\_b = \frac{8EI\mu\_1(y)}{pD^2 \left(\frac{4EI}{kD}\right)^{\frac{3}{4}}} \tag{18}$$

The reaction force provided by the rock in the elastic state is *p* = *ku*1. When the compressive stress is too large, the rock produces plastic failure and reaches the ultimate reaction, *p* = *nσ*c. At this time, the lateral elastic displacement limit of the bolt end is *u*<sup>1</sup> = *nD*/300. *n* is a coefficient dependent on the internal friction angle of rock, and its value is 2–5 according to the degree of rock hardness.

Comprehensive Formulas (6)–(8), (15) and (18) can be substituted into numerical values to obtain the shear strength at any point (*x*, *y*) of the rock material with anchor bedding:

$$\begin{split} \tau\_{bj}(\mathbf{x},\boldsymbol{y}) &= \sigma\_{\hat{\boldsymbol{\eta}}} \tan(\boldsymbol{\psi} + \boldsymbol{\varrho}^{\prime}) + \frac{N\_{0} \sin \mathbb{h} \left[ \sqrt{\frac{4\mathcal{K}}{E \pi D^{2}}} (\boldsymbol{u}\_{2} - \boldsymbol{x}) \right]}{\frac{\pi D^{2}}{4} \sin \big[ \boldsymbol{u}\_{2} \sqrt{\frac{4\mathcal{K}}{E \pi D^{2}}} \big]} \cdot \eta \left[ \sin(\boldsymbol{a} + \theta\_{1}) / \cos \boldsymbol{a} \right] + \\ &\frac{8EI \boldsymbol{u}\_{1}(\boldsymbol{y}) \eta}{p D^{2} \left(\frac{4E}{kD}\right)^{\frac{3}{4}}} \cdot \left[ \sin(\boldsymbol{a} + \theta\_{1}) \cdot \tan \boldsymbol{\varrho}^{\prime} + \cos(\boldsymbol{a} + \theta\_{1}) \right] \end{split} \tag{19}$$

When *σ*<sup>d</sup> sin(*θ*<sup>1</sup> + *α*) + *σ<sup>v</sup>* sin *α* = *ρC*p*v*p*L* −*η* sin(*θ*<sup>1</sup> + *α*) + *σ<sup>v</sup>* sin *α* ≥ *τbj*max, the bedding plane shear failure occurs in the anchored bedding rock material under dynamic load disturbance, resulting in anchoring failure.

On the basis of the stress analysis of anchored bedding rock material, the instability criterion of anchored bedding rock material in a complex stress environment can be obtained:

$$\begin{cases} \ \sigma\_{\upsilon} \cdot \sin \theta\_1 + \rho \mathbf{C}\_{\text{P}} \upsilon\_{\text{P}} L^{-\eta} \cdot \cos \theta\_2 \ge \sigma\_{\upsilon} \text{ Compression Fourier} \\ \ \rho \mathbf{C}\_{\text{P}} \upsilon\_{\text{P}} L^{-\eta} \sin(\theta\_1 + a) + \sigma\_{\upsilon} \sin a \ge \tau\_{b \text{max}} \text{ Shear Fourier} \end{cases} \tag{20}$$

where *σ*<sup>c</sup> is the uniaxial compressive strength of rock mass.

According to this formula, the parameters of the bedding anchor solid can be obtained with field measurements, and physical and mechanical tests, and the instability of the anchored bedding rock material can be judged:

#### **3. Anchored Bedding Rock Material Impact Loading Test**

#### *3.1. Sample Preparation*

According to the instability criterion of anchored bedding rock material, under the condition of a constant surrounding rock stress environment, the angle between bolt and bedding, and the physical and mechanical properties of anchorage body are important factors affecting the instability of anchored bedding rock material. In order to further study the mechanical properties and instability mechanism of anchored bedding rock material, the impact loading test of anchored bedding rock material was carried out.

According to the stress conditions of the anchor solid, as shown in Figure 3b, assuming that the dynamic load of the rock fracture acted vertically on the anchor, the anchorage bedding rock specimen was produced as shown in Figure 6, where the angle between the anchor and the bedding plane was *θ*, *θ* = *α* + *θ*1.

Test 1:

tained:

**Figure 6. Figure 6.** Schematic diagram of inclined bedding surrounding rock anchorage specimen. Schematic diagram of inclined bedding surrounding rock anchorage specimen. tion angles were produced. The siltstone and sandstone specimens were bonded with epoxy resin, and specimens of Ф50 × *h*50 mm were produced. In order to study the intion angles were The siltstone and sandstone specimens were bonded with epoxy resin, and specimens of Ф50 × *h*50 mm were produced. In order to study the intion angles were produced. The siltstone and sandstone specimens were bonded with epoxy resin, and specimens of Ф50 × *h*50 mm were produced. In order to study the in-Group A was the control group and the full-length anchorage. The bolt bedding angle Group A was the control group and the full-length anchorage. The bolt bedding angle Group A was the control group and the full-length anchorage. The bolt bedding angle