1 Introduction

With the continuous development of coal resources in China, shallow resources are becoming scarcer and resources with a buried depth of 1000 m are now under consideration. With the increase of mining depth, deep coal masses are subject to a complex stress environment that includes high ground stress, high earth temperature, high karst water pressure, and mining disturbance (Gao et al. 2002). In order to avoid the distortion of the experimental results, the method recommended by ISRM is adopted, and C1100 copper alloy disc (diameter: 12.75 mm; thickness: 0.7 mm) was added as a shaper material to improve the square loading wave relative to the bevel loading wave (Culshaw 2015). Comparison of the loading wave before and after sha** is shown in Fig. 3. An improved SHPB test system with confining pressure was used in these experiments, as shown in Fig. 4 (Wu et al. Full size image

Fig. 4
figure 4

Improved split Hopkinson pressure bar system with confining pressure

Full size image

Based on the geological conditions on site, the average density of the overlying rock is determined to be 2500 kg/m3 (Li et al. 2023), and different depths are established to apply equivalent stress loads. To determine the dynamic strength of coal at different depths, the improved SHPB system was first used to load prepared coal samples to initial hydrostatic pressures of 0, 5, 10, 15, and 20 MPa to simulate original ground stress environments of the coal at depths of 0, 200, 400, 600, and 800 m, respectively. Dynamic mechanical testing of the coal at different impact rates under stress conditions at different depths was then carried out. Mechanical parameters, such as peak stress, peak strain, and dynamic strength of the coal under different loading rates, were analyzed according to the test results, and dynamic damage constitutive models of the coal at different depths were constructed and verified.

2.2 Determination of coal infinitesimal element strength considering rate effect

Coal has numerous original defects, such as micro fissures and joint structures. These original defects gradually propagate and coalesce under the action of external loads, until macroscopic fracture occurs. Damage mechanics is most commonly used to study damage evolution of rock materials. Determination of the infinitesimal element strength of coal is key to establishing the damage constitutive model. The damage constitutive model based on random distribution of infinitesimal element strength and the dynamic damage constitutive model based on the element model were analyzed, combined with comprehensive consideration of the properties of deep surrounding rock, dynamic deformation characteristics of the coal, and the stress environment at different depths, to determine the infinitesimal element strength of the coal considering the rate effect. Using the proposed dynamic strength criterion of coal considering the rate effect (Li et al. 2023), a method to measure the dynamic infinitesimal element strength of coal was established, as shown in Eq. 1:

$$F_{{\text{d}}} = \sigma^{\prime}_{1d} - \sigma^{\prime}_{3d} \frac{1 + \sin \varphi }{{1 - \sin \varphi }} - \frac{2c\cos \varphi }{{1 - \sin \varphi }}$$
(1)

where Fd is the dynamic infinitesimal element strength of coal, \(\dot{\sigma }\) is the loading rate, \(\dot{\sigma }_{0}\) is the reference loading rate, 1 GPa/s.

c is the cohesive forces, \(c = 0.4245\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{0.5451}\) (Li 2019),

φ is the angle of internal friction, \(\varphi = 0.4299\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{0.5360}\) (Li 2019),

\(\sigma^{\prime}_{{1{\text{d}}}}\), \(\sigma^{\prime}_{{3{\text{d}}}}\) is the dynamic effective stress of coal.

On the basis of Lemitre’s strain equivalence hypothesis, we assumed that damaged coal does not have bearing capacity; the entire load is borne by undamaged coal. Therefore, the constitutive relation of the damaged material only needs to replace stress in the constitutive relation of the original material by the effective stress, then:

$$\sigma_{id} = \sigma_{id}^{\prime } (1 - D_{d} )\quad (i = 1,2,3)$$
(2)

where \(\sigma_{id}\) is the dynamic macroscopic nominal stress of coal, \(\sigma_{2d} = \sigma_{3d} = \sigma_{3}\), and \(D_{d}\) is the dynamic damage variable of coal.

Assuming the stress–strain relationship of coal prior to damage obeys a linear elastic relationship, then:

$$\sigma_{1d} = E_{d} \varepsilon_{1d} \left( {1 - D_{d} } \right) + 2\mu \sigma_{3d}$$
(3)

where \(\varepsilon_{1d}\) is the axial strain, \(E_{d}\) is the dynamic elastic modulus of coal, and \(\mu\) is Poisson’s ratio of the coal.

The above relations can be substituted into Eq. (1) to determine a strong expression for the infinitesimal element of coal considering the rate effect:

$$\begin{aligned} F_{d} {\text{ }} & = \frac{{E_{d} \varepsilon _{{1d}} \left[ {\left( {\sigma _{{1d}} - \sigma _{{3d}} } \right) - \left( {\sigma _{{1d}} + \sigma _{{3d}} } \right)\sin \varphi } \right]}}{{\sigma _{{1d}} - 2\mu \sigma _{{3d}} }} - \frac{{2c\cos \varphi }}{{1 - \sin \varphi }}{\text{ }} \\ & = \frac{{E_{d} \varepsilon _{{1d}} \left[ {\left( {\sigma _{{1d}} - \sigma _{{3d}} } \right) - \left( {\sigma _{{1d}} + \sigma _{{3d}} } \right)\sin \left( {0.4299\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{{0.5360}} } \right)} \right]}}{{\left( {\sigma _{{1d}} - 2\mu \sigma _{{3d}} } \right)\left( {1 - \sin \left( {0.4299\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{{0.5360}} } \right)} \right)}} \\ & \quad - \frac{{2\left( {0.4245\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{{0.5451}} } \right)\cos \left( {0.4299\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{{0.5360}} } \right)}}{{\left( {1 - \sin \left( {0.4299\left( {\frac{{\dot{\sigma }}}{{\dot{\sigma }_{0} }}} \right)^{{0.5360}} } \right)} \right)}} \\ \end{aligned}$$
(4)

Mechanical parameters are obtained from dynamic tests that consider the occurrence depth of the coal, and thus this method to determine the dynamic infinitesimal element strength of coal is universal. Equation (4) applies to the determination of infinitesimal element strength in dynamic mechanical testing of coal at different depths.