1 Introduction

Shallow colliery resources are gradually being exhausted, and resource exploitation requires continually deeper mining (Xie et al 2019, 2022; Gao et al 2021). In China, coal plays a pivotal role in the use of energy for the shortage of petroleum and natural gas (Bai et al 2018; Ju et al 2018). Exploitation of 1000 m deep mines has gradually become common practice. At such depths, coal and rock masses are under an environment featuring high initial stress, high geothermal and high fluid pressure (Zheng et al 2022; Zheng et al 2023a, b). As a result, the incidence of dynamic hazards represented by rock bursts and coal or gas outbursts is rapidly increasing (Chen et al 2021; Li et al 2021a, b). Dynamic disasters are usually manifested as the failure of coal or rock mass in a short time under an external loading caused by multiple mining-induced projects, such as fault slip, breakdown of roof and blasting (Feng et al 2018). Investigations on the mechanical behaviors of coal under dynamic loads are therefore essential to uncover the mechanisms of these dynamic disasters.

The Kolsky bar also called split Hopkinson pressure bar (SHPB) is a helpful experimental instrument to understand the dynamic mechanical behaviors of solid materials, including concretes, rocks and coals (Kolsky 1964; Xia and Yao 2015). Dynamic mechanical behaviors of coal have been investigated by many researchers using this technique. In these studies, the dynamic tensile strength (DTS) of coal is associated with water content (Zhao et al 2016) and bedding plane (Zhao et al 2014; Ai et al 2020; Shi et al 2022). Saturated coal specimens have higher DTS than dry coal specimens. The bedding angle effects the DTS of coal, but no linear relationship was well found between them at similar loading rate. Bedding plane also influences the dynamic fracture toughness (DFT) of coal (Wang and Yang 2017; Zhao et al 2017). The effects of bedding angles on the DFT decrease with the increase of impact velocity. The dynamic compressive strength (DCS) of coal is also affected by heat treatment (Yu et al 2017), water content (Gu et al 2019) and bedding plane (Liu et al 2015; Li et al 2021a, b). The results indicated that the DCS of coal at 20 °C is greater than 400 °C. The DCS of coal is negatively correlated with the water content. The DCS of coal along different bedding directions exhibit distinct variations, featuring significant anisotropy. The bedding effect of coal decreases with the increase of strain rate. However, all these studies utilized dynamic uniaxial compression without confinement or axial pressure before the dynamic loading pressure. In actual coal mining, the coal is under high in situ stresses. However, limited investigations have concerned the dynamic behavior of coal under pre-existing stresses. Wang et al (2019) conducted three-dimensional coupled static-dynamic loading experiments on water-saturated and natural coal specimens using SHPB with a confinement pressure of 4 or 8 MPa. However, for deep coal mining (usually larger than 400 m), the ground and mining-induced stress are much higher.

The strength of rocks or concretes is sensitive to hydrostatic pressure (Liu et al 2020; Zheng et al 2023a, b). The DCS of rocks was found to increase with the confinement pressure and the loading rate (Christensen et al 1972; Xu et al 2022). Wu et al (2016) conducted dynamic tensile testing on Laurentia granite treated with hydrostatic confinements, and the results indicated that the augmentation of the DTS declines with increase of hydrostatic confinement, which is in line with the effect seen for the static strength under confinement. Du et al (2018) studied the dynamic strength and failure mechanism of rocks under hydrostatic confinement in SHPB tests simulated by discrete element method (DEM), and found that the dynamic strength of rock under hydrostatic confinement increases with the strain rate, but the rate-sensitivity of strength is negatively related to the hydrostatic confining pressure.

In this study, a modified Kolsky bar system was invested to carry out dynamic compressive studies of coal under stresses corresponding to burial depths of 0, 200, 400, 600, and 800 m with different loading rates. The study results and correlations between the DCS and burial depth, and loading rate are discussed. A dynamic Mohr–Coulomb strength criterion about coal is established, considering the influence of burial depth.

2 Coal samples preparation

The samples used for tests were drilled from the longwall panel No. 24130 in the Pingdingshan No. 10 Coal Mine of Pingdingshan Mining Group, China. The depth of burial (DOB) of the coal bed is 1073 ~ 1173 m. In order to eliminate the influence of bedding on the properties of coal, the drilling direction is uniformly parallel to the bedding direction for drilling and sample preparation. The cores were then carefully sliced into discs and the diameter and thickness are 38.1 mm of 25 mm respectively. Both ends of the coal specimens were cautiously polished to achieve a surface roughness of 0.5% and straightness to ± 0.02 mm according to the International Society for Rock Mechanics and Rock Engineering (ISRM) suggested method (Zhou et al 2012). Five sets of coal samples were prepared for dynamic compressive tests.

The samples were analyzed by the electron probe microanalyzer (EPM), and the mineral map is shown in Fig. 1. The minerals of the coals were identified as mainly carbonaceous organics, followed by plagioclase, clay and Fe oxide. The mineral component of the coal samples is summarized in Table 1.

Fig. 1
figure 1

Coal specimen: a prepared specimens; b minerals map

Full size image
Table 1 Composition and content of coal samples
Full size table

3 Experimental setups

3.1 Testing facilities

Dynamic compressive tests of coal subjected to hydrostatic confinement were carried out by a modified Kolsky bar (Wu et al 2015, 2016; Xu et al 2019), the schematics of the system is illustrated in Fig. 2. The equipment comprises a series of bars and two hydraulic systems. The loading system consists of an incident bar (input bar), a transmitted bar (output bar) and a striker bar with lengths of 1800 mm, 1500 mm and 200 mm, respectively. All the bars have the same diameter of 38 mm and are made of high-strength steel with a density of 8100 kg/m3, Young’s modulus of 200 GPa, and P-wave velocity of 5107 m/s. Two hydraulic cylinders are used to provide the axial pressure and confining pressure, respectively. The average density of the overlying strata is about 2500 kg/m3. Using these systems, coal samples were loaded to hydrostatic pressures of 0, 5, 10, 15, and 20 MPa, simulating the stresses at burial depths of 0, 200, 400, 600, and 800 m, respectively. A shrink film is used to parcel the specimen to prevent oil from seeping into the specimen. By controlling the pressure of the high-pressure gas in the gas gun, the striker produces different impact speeds to achieve various loading rates. Dynamic mechanical experiments around the coal at various loading rates were then conducted.

Fig. 2
figure 2

Schematic of Kolsky bar system for dynamic compression tests

Full size image

During a test, the coal sample was fixed between the input and output bars, and the striker bar was driven by a gas gun to hit the input bar. The incoming wave propagates along the input bar until it impacts the coal sample, causing a reflected wave and a transmitted wave, and the strain gauges attached to the rod superficies are used to record these signals (Yao et al 2016). Then the forces P1 and P2 at both ends of the coal sample can be calculated respectively:

$$P_{1} \left( {\text{t}} \right) = EA\left( {\varepsilon_{i} \left( {\text{t}} \right) + \varepsilon_{r} \left( {\text{t}} \right)} \right)$$
(1)
$$P_{2} \left( {\text{t}} \right) = EA\varepsilon_{t} \left( {\text{t}} \right)$$
(2)

where Young’s modulus E is 200 GPa and A is the cross-section area of the rods. \(\varepsilon_{i}\),\(\varepsilon_{r}\), and \(\varepsilon_{t}\) are the incident, reflected, and transmitted strains on the bars, respectively.

Compressive stress \(\sigma\) and strain \(\varepsilon\) at different times during the tests can be calculated:

$$\sigma \left( {\text{t}} \right){ = }\frac{{{\text{EA}}}}{{2{\text{A}}_{{\text{S}}} }}\left( {\varepsilon_{{\text{i}}} \left( {\text{t}} \right) + \varepsilon_{{\text{r}}} \left( {\text{t}} \right) + \varepsilon_{{\text{t}}} \left( {\text{t}} \right)} \right)$$
(3)
$$\varepsilon \left( {\text{t}} \right){ = }\frac{{{\text{C}}_{{\text{B}}} }}{{{\text{L}}_{{\text{S}}} }}\int\limits_{0}^{{\text{t}}} {\left( {\varepsilon_{{\text{i}}} \left( {\text{t}} \right) - \varepsilon_{{\text{r}}} \left( {\text{t}} \right) - \varepsilon_{{\text{t}}} \left( {\text{t}} \right)} \right)} dt$$
(4)

where AS is the initial area of the sample, CB is the wave velocity in the elastic bar, and LS is the original length of the sample.

3.2 Dynamic force balance

For a valid test, the dynamic force balance is essential for samples during the dynamic loading process to eliminate the inertia effect (Zhou et al 2012). According to the ISRM suggested method (Zhou et al 2012), circle coppers with a diameter of 10 mm and thickness with 0.8 mm are utilized to shape the incident wave from rectangle to sinusoidal. Then, the P1 and P2 at each loading end of the specimen are obtained from Eq. (1), and shown in Fig. 3. From Fig. 3, these forces are almost equal for a typical dynamic experiment. This result indicates that the dynamic force equilibrium is achieved at both ends of the coal sample. The mismatch between forces P1 and P2 at early stage of the dynamic loading is caused by the axial pressure in the bars, which does not affect the dynamic force equilibrium near the peak loading.

Fig. 3
figure 3

Dynamic force balance for typical dynamic compression test. In, Re and Tr represent incident wave, reflected wave, and transmitted wave, respectively

Full size image

When the force balance is achieved, the dynamic compressive stress can be calculated by Eq. (3). A typical stress history is shown in Fig. 4. In this study, the loading rate is used to describe the rate dependence about the compression strength of coals. The slope of the linear region in the figure is determined as the loading rate (Zhou et al 2012). The loading rate \(\dot{\sigma }\) of the typical stress history determined by this methodology is 649 GPa/s (Fig. 4). The peak stress in Fig. 4 is the corresponding dynamic compressive strength of the coal sample.

Fig. 4
figure 4

Determination of loading rate for a typical dynamic test

Full size image

4 Results and discussion

4.1 Effects of loading rates and burial depths

The dynamic strengths of the coal specimens at different depths (0, 200, 400, 600, and 800 m) are determined by applying the corresponding hydrostatic confining pressure (0, 5, 10, 15, and 20 MPa) before the dynamic axial loading. At each confining pressure, different loading rates are applied to the samples. Using the methods mentioned above, the dynamic compression strength of samples at various depths and loading rates are summarized in Table 2 and Fig. 5.

Table 2 Dynamic strength of coal at various depths and loading rates
Full size table
Fig. 5
figure 5

Dynamic test results for different loading rates and depths

Full size image

From Fig. 5, the dynamic compressive strengths of coal are associated with the loading rate and the depth. The results demonstrate that the dynamic strength of specimen is higher at deeper depths at a given loading rate, and its growth rate decreased with increasing depth. Similarly, at a certain depth, the strength of coal increased with increasing the loading rate and showed obvious nonlinear growth.

The experimental results are fitted as follows:

$$\sigma_{{{\text{cd}}}} = \sigma_{0} \left( {1 + \alpha \ln \left( {1 + \frac{h}{{h_{0} }}} \right)} \right)\left( {1 + \beta \left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{n} } \right)$$
(5)

where \(\sigma_{{{\text{cd}}}}\) and \(\sigma_{0}\) are the dynamic and static strengths of coal, respectively, h is the coal burial depth, h0 is the reference depth (taken as 100 m), \(\dot{\sigma }\) is the experimental loading rate, \(\dot{\sigma }_{0}\) is the reference loading rate (taken as 1 GPa/s), and α, β, and n are fitting parameters. For this data set, the values of α, β, and n are 0.281, 0.0367, and 0.788, respectively.

Figures 6 and 7 show that the fitting curve well matches the experimental results, with a correlation coefficient of 0.986.

Fig. 6
figure 6

Dynamic compressive strength of coals under various loading rates at different depths

Full size image
Fig. 7
figure 7

Comparison of dynamic strength and fitting curve of coals at various depths and loading rates

Full size image

4.2 Dynamic failure characteristics

Selected coal samples after the dynamic tests are shown in Figs. 8 and 9. Coal samples without hydrostatic confinement broke into pieces (Fig. 8) (Gao et al 2020). In the presence of low loading rate, coal samples fragment into relatively large blocks; in the presence of high loading rate, the samples fragment into small granules. Motivated by the experimental results, the dynamic loading rate makes a great impact on the failure behaviors of the specimens.

Fig. 8
figure 8

Dynamic failure characteristics of coal samples without hydrostatic confinement (Gao et al 2020)

Full size image
Fig. 9
figure 9

Fracture characteristics of coal at various depths at similar loading rates

Full size image

When hydrostatic confinement is applied, the coal specimens tend to fragment into relatively larger blocks (Fig. 9). As the depth increases, the volume of the fragmented blocks gradually increases. At a depth of 800 m (with the hydrostatic pressure of 20 MPa), a single shear fracture is observed along the direction of the maximum shear stress. This phenomenon is in accordance with experimental results reported by previous researchers (Hokka et al 2016; Xu et al 2022).

5 Dynamic failure criterions

The Mohr–Coulomb strength criterion is common and valuable in rock mechanics (Nick 2013). The general expression is as follows:

$$\left| \tau \right| = C + \sigma \tan \varphi$$
(6)

where C and φ are the cohesive strength and the internal friction angle of the rock materials, respectively; τ and σ are the shear and normal stresses, respectively.

The research results of Zhao (2000), Huang et al (2012), Gong et al (2016), Si et al (2019), Xu et al (2019) and others have shown that the Mohr–Coulomb strength criterion is well suited for the dynamic conditions. Therefore, this study attempts to propose a dynamic Mohr–Coulomb strength criterion for coal considering different burial depths (i.e., confining stress).

The dynamic strength of coal under loading rates of 400–700 GPa/s in increments of 50 GPa/s is obtained by interpolating the dynamic strength results of coal under various depth stress conditions, and summarized in Table 3. The coal sample is subjected to hydrostatic pressure (i.e., \(\sigma_{{1{\text{s}}}} { = }\sigma_{2s} { = }\sigma_{3s}\)) before applying the dynamic axial pressure. Therefore, the maximum principal stress: \(\sigma_{1} { = }\sigma_{{{\text{cd}}}} + \sigma_{1s}\) The relationships between the maximum and minimum principal stresses σ1 and σ3 are given in Table 4.

Table 3 Dynamic strength of coal under corresponding loading rate at different depth stress states
Full size table
Table 4 The maximum principal stress σ1 and the minimum principal stress σ3 of sample under various loading rates
Full size table

Analysis of the experimental results showed that σ1 increases linearly with the σ3 in the loading rate range of 400–700 GPa/s (Fig. 10). The relationship can be summarized as follows:

$$\sigma_{1} { = }a\sigma_{3} + b$$
(7)
Fig. 10
figure 10

First principal stress of coal under various loading rates and confining pressures

Full size image

The expression between the principal stresses in the Mohr–Coulomb strength criterion is given by:

$$\sigma_{1} { = }\frac{1 + \sin \varphi }{{1 - \sin \varphi }}\sigma_{3} + 2{\text{c}} \cdot \frac{\cos \varphi }{{1 - \sin \varphi }}$$
(8)

According to Eqs. (7 and 8), the internal friction angle φ and cohesive strength c are given by:

$$\varphi\, { = }\,\arcsin \frac{a - 1}{{a + 1}}$$
(9)
$$c = \frac{b}{2\sqrt a }$$
(10)

where φ and c can be calculated under different loading rates according to Eqs. (9 and 10), respectively. The calculation results are given in Table 5. It can be found that c gradually increased when the loading rate changes from 400 to 700 GPa/s. The rate dependency of cohesion of coal is consistent with the observation of other rock like materials, such as sandstones (Huang et al 2011; Xu et al 2019). The φ also gradually increased with the loading rate in this study ranges (Table 5). This phenomenon is distinct from the results reported by Xu et al (2019), in which the φ of green sandstone seems to be rate independent. This discrepancy may be related to the difference of sandstone and coal used in this study.

Table 5 The maximum and minimum principal stresses
Full size table

Then, power relationships between c, φ, and the loading rate are found by fitting the data in Figs. 11 and 12:

$$c = 0.4269\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{0.5442}$$
(11)
$$\varphi { = }0.4262\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{0.5359}$$
(12)
Fig. 11
figure 11

Relationship between cohesive strength c and loading rates

Full size image
Fig. 12
figure 12

Relationship between internal friction angle φ and loading rates

Full size image

The maximum principal stress can be obtained by substituting Eqs. (11 and 12) into Eq. (8) for the Mohr–Coulomb criterion:

$$\sigma_{1} { = }\sigma_{{{\text{cd}}}} + \sigma_{3} = \frac{{1 + \sin \left( {0.4262\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{0.5359} } \right)}}{{1 - \sin \left( {0.4262\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{0.5359} } \right)}}\sigma_{3} + 2 \cdot \left( {0.4269\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{0.5442} } \right) \cdot \frac{{\cos \left( {0.4262\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{0.5359} } \right)}}{{1 - \sin \left( {0.4262\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{0.5359} } \right)}}$$
(13)

Under hydrostatic confining pressure, the initial stress of the sample is given by \(\sigma_{1s} { = }\sigma_{2s} { = }\sigma_{3s} { = }\sigma_{3}\) before the dynamic load is applied. Therefore, the measured dynamic strength is:

$$\sigma_{{{\text{cd}}}} { = }\sigma_{1} - \sigma_{{1{\text{s}}}} = \frac{1 + \sin \varphi }{{1 - \sin \varphi }}\sigma_{3} + 2{\text{c}} \cdot \frac{\cos \varphi }{{1 - \sin \varphi }} - \sigma_{3} = \frac{2\sin \varphi }{{1 - \sin \varphi }}\sigma_{3} + 2{\text{c}} \cdot \frac{\cos \varphi }{{1 - \sin \varphi }}$$
(14)

The dynamic strength of coal under hydrostatic pressure can then be obtained:

$$\sigma_{{{\text{cd}}}} { = }\frac{{2\sin \left( {0.4262\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{0.5359} } \right)}}{{1 - \sin \left( {0.4262\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{0.5359} } \right)}}\sigma_{3} + 2 \cdot \left( {0.4269\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{0.5442} } \right) \cdot \frac{{\cos \left( {0.4262\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{0.5359} } \right)}}{{1 - \sin \left( {0.4262\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{0.5359} } \right)}}$$
(15)

The dimensionless depth influence factor H/H0 is introduced to establish the dynamic strength criterion taking into account the role of burial depth. The confining pressure \(\sigma_{3}\) is related to depth by σ3 = f(H/H0). In this study, 800 m is taken as the characteristic depth value H0.

The formula for calculating the dynamic strength of coal considering the effect of depths can be simultaneously obtained:

$$\sigma_{{{\text{cd}}}} { = }\frac{{1 + \sin \left( {0.4262\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{0.5359} } \right)}}{{1 - \sin \left( {0.4262\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{0.5359} } \right)}} \cdot 0.025\left( {\frac{H}{{H_{0} }}} \right) + 2 \cdot \left( {0.4269\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{0.5442} } \right) \cdot \frac{{\cos \left( {0.4262\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{0.5359} } \right)}}{{1 - \sin \left( {0.4262\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{0.5359} } \right)}}$$
(16)

As illustrated in Fig. 13, theoretical curves for loading rates of 400 to 700 GPa/s are plotted using Eq. (16), which are consistent with the experimental results, and validate the proposed dynamic Mohr–Coulomb criterion according to the data in this study. The proposed dynamic Mohr–Coulomb strength criterion can be used to estimate the dynamic compressive strength of coal under different loading rates and burial depth 0–800 m. This criterion can provide a basis for a future dynamic constitutive model of coal that considers the effect of burial depth.

Fig. 13
figure 13

Comparison between dynamic Mohr–Coulomb criterion and experimental results

Full size image

6 Conclusions

In this study, dynamic compression experiments on coal specimens were conducted, and hydrostatic pressures of 0, 5, 10, 15, and 20 MPa were applied to simulate burial depths of 0, 200, 400, 600, and 800 m, respectively. The following main conclusions can be drawn:

The DCS of coal at various depths is rate dependent. The DCS increases significantly when the loading rate increases. Furthermore, the dynamic strength of coal exhibits a positive association with the burial depth at a given loading rate.

Under dynamic loading, the failure behaviors of coal are strongly correlated with the loading rate. After dynamic impact, coal without hydrostatic confinement broke into pieces. As the depth increases, the size of the fragments gradually increases.

Motivated by the experimental evidence, a dynamic Mohr–Coulomb criterion for coal under different depth stress conditions was established for the dynamic strength of coal at different burial depths.