Study on the Dynamic Response Characteristics of Cylindrical Coal-Rock Samples under Dynamic Loads

To research the dynamic response characteristics of cylindrical coal-rock samples under impact loads, the impact of rigid bars on cylindrical coal-rock samples is simulated under different speed conditions, based on LS-DYNA software, and the dynamic distribution characteristics of the stress, strain, and energy of cylindrical coal-rock samples are analyzed..e results demonstrated the following: (1) the cylindrical coal-rock sample failed at the center first, and the damage developed downward along the axial direction. (2) .e critical effective stress and strain have an exponential function relationship with the velocity, and the critical time has a linear relationship with the velocity. (3).e energy change law of the cylindrical coal-rock sample is consistent with the destruction morphology. (4) .e axial stress peaks in the severe damage part have a linear relationship with the speed, the axial stress attenuates rapidly after passing the stress yield point, and the axial strain does not increase continuously. (5).e peaks stress and strain on the central axis and the radial line obey the power function distribution, the axial stress produces tensile stress in the axial propagation direction, and the axial stress and strain peaks at the same position are larger than those of the radial stress and strain peaks. .is research provides a reference for studying coal and rock dynamic disasters.


Introduction
e effect of the impact load on coal-rock sample mechanics is the basis of mining [1]. Dynamic disasters such as coal and gas outbursts and rock bursts occur suddenly, develop rapidly, are destructive, and sweep over a wide range. ey are likely to cause secondary accidents and pose a threat to the safety of a mine [2][3][4]. After a mine enters the deep mining stage, the geological conditions of the coal mine become more complicated [5,6]. To solve these problems, it is necessary to understand the law and mechanism of the deformation, failure, and fracture evolution of coal-rock masses [7]. Mining activities and coal-rock kinetics responses are the main sources of mining dynamic loads [8].
Coal and rock masses are mainly compressed and sheared under static loading [9][10][11][12][13][14]. e split-Hopkinson pressure bar (SHPB) is the main experimental device that is used to explore the mechanism of coal mass dynamic damage [15][16][17]. Feng et al. [18] conducted dynamic loading tests on coal using an SHPB system, and the mechanism of energy dissipation was discussed based on the fracture processes of coal under dynamic loads. Yin et al. [19] performed dynamic compression experiments on coal-rock using SHPBs and found that the peak strain decreased when the peak stress increased. Based on the SHPB experiment and RMT-150C test systems, Wang et al. [20] reported that the dynamic strength of 7D saturated coal specimens is lower than that of natural coal specimens under one-dimensional static-dynamic loading. Gong et al. [21] used the SHPB experiment to conclude that the dynamic compressive strength (the second peak stress) and dynamic peak strain (the second peak strain) of the coal-rock combined body have a strong loading rate effect and will generally increase linearly with the loading rate. Wang et al. [22] established a fracture model of a hydrous wing branch fracture under static-dynamic loading based on the SHPB and RMT-150 system tests. Yang et al. [23] used the SHPB to study the strength and microstructures of outburst-prone coal under compressive impact loading. Yin et al. [24] studied the dynamic behaviors of the strain and energy changes of gascontaining coal under the SHPB test, with the condition of controlling the initial gas pressure and the axial static preload. Kong et al. [25] analyzed the relationship among the dynamic strength, failure strain and axial static load, confining pressure, air pressure, and impact load, based on a split-Hopkinson pressure bar experimental system. Ai et al. [26] used SHPB experiments to study the crack propagation and dynamic mechanical properties of coal under high strain rate loading. Liao et al. [27] studied the tensile and compressive properties of rock based on SHPB experiments. Li et al. [28] demonstrated the strain, stress, displacement, vibration velocity, and energy distribution in all directions of cylindrical coal-rock samples in high-order P-waves. It can be seen from the above literature that SHPB experimental research has difficulty monitoring the dynamic change process of various parameters at the internal points of the sample.
Numerical simulation research has certain advantages compared with experimental research [29]. Hao et al. and Li et al. [30,31] studied the tensile and compressive properties of concrete materials based on the numerical simulation of the SHPB experiment. Li et al. and Zhu et al. [32][33][34] studied the mechanical properties of rock materials based on numerical simulations of SHPB experiments. Zhao et al. [35] applied LS-DYNA software to the constitutive model considering the dynamic compression and tensile failure for the analysis of crack propagation caused by coal blasting. Majidi et al. [36] and Zhai et al. [37] used LS-DYNA software to study the properties of concrete. Yuan et al. [38] simulated the rock SHPB experiment of the Holmquist-Johnson-Cook (HJC) model based on LS-DYNA software. Xie et al. [39] analyzed the numerical simulation with the failure form of coal-rock under dynamic loading based on the HJC model.
According to past studies in the available literature, the damage characteristics of coal-rock samples under dynamic loading have been insufficiently studied. Research on the damage mechanism of coal-rock samples under dynamic loading has mostly focused on the overall shape after failure and the overall stress, strain, energy dissipation, and compressive strength during the failure process. ere are few studies on the damage characteristics of the internal points of coal-rock samples. is paper took cylindrical coal-rock samples as the research object.
e cylindrical coal-rock sample impact experiment based on the HJC model was simulated by LS-DYNA software, and the failure characteristics and internal dynamic response characteristics of the cylindrical coal-rock samples during the failure process were analyzed. is research is helpful for deeply understanding the dynamic response characteristics of coal-rock samples under impact dynamic loads and provides a reference for studying coal and rock dynamic disasters.

HJC Constitutive Model and Modeling of Coal-Rock Sample
e units of quality, time, and length used in the LS_DYNA program are g, μs, and cm, respectively. Table 1 provides the  rigid body model parameters of the impact bar. e HJC dynamic constitutive model can be established in the LS-DYNA package [40]. e HJC model is capable of simulating the large deformation problems of concrete under high pressure and a high strain rate [41]. e equivalent yield strength is a function of the pressure, strain rate, and damage. e pressure is a function of the volumetric strain and contains the effects of permanent crushing. Damage accumulation is a function of the plastic volumetric strain, equivalent plastic strain, and pressure. e damage accumulation mainly contains the strength equation, damage evolution equation, and state equation [42]. e LS-DYNA package generates cracks in the structure through the failure of the unit. Table 2 provides the HJC model parameters of the coal-rock sample with reference to the HJC model definition and related literature [39,43,44]. e model was established with a cylindrical coal-rock sample with a diameter of 50 mm and height of 100 mm, an impact bar with a diameter of 60 mm and height of 200 mm, and a gasket with a diameter of 64 mm and height of 20 mm. To reduce the amount of simulation operation, actual onequarter symmetric size modeling was adopted. e vertical displacement constraint and the nonreflection boundary condition were set on the model symmetry plane. e cylindrical coal-rock samples adopted the HJC material model, and the impact bar and the gasket were established by the rigid body material model. e model was meshed by the hexagonal mapping method and adopted the three-dimensional solid element. e radial line, central axis, and edge axis of the cylindrical coal-rock samples on the model are shown in Figure 1.
In the simulation, the gasket was fixed, the cylindrical coal-rock sample was located on the gasket, which had a displacement constraint in the direction of the vertical gasket, and the impact bar impacted the cylindrical coal-rock sample at a certain speed. After several simulation tests, based on the SHPB experiment, representative speeds were selected for analysis. e simulated speeds of the impact bar

Damage of the Cylindrical Coal-Rock
Samples under Different Impact Speeds      e effective stress at the moment of failure of cylindrical coal-rock samples is called the critical effective stress, and the corresponding time and effective strain are called the critical time and the critical effective strain, respectively, under different speeds. Taking the speed as the independent variable and taking the critical effective stress and the critical time as the dependent variables, the fitting curves are shown in Figure 4. e critical effective stress of the cylindrical coal-rock samples increases with increasing speed, but the increasing amount gradually decreases and tends to a fixed value, indicating that the impact speed can change the critical effective stress of cylindrical coal-rock samples.
e critical time decreases linearly with increasing speed, indicating that the greater the velocity is, the faster the transfer is. e relationship between the critical effective stress and the impact velocity has an exponential function, and the critical time has a linear relationship with the impact velocity.
Taking the impact velocity as the independent variable and the critical effective strain as the dependent variable, the fitting curve is shown in Figure 5. e critical effective strain increases with increasing velocity, and the increase gradually decreases and tends to a fixed value, indicating that dynamic loading can change the effective strain of cylindrical coalrock sample damage. e relationships between the critical effective strain and the impact velocity have exhibited an exponential function. e critical effective stress and strain of the cylindrical coal-rock samples have the same trend as the change in velocity, which indicates that the stress and deformation of the cylindrical coal-rock samples are consistent. When the dynamic load is large, the effective stress and effective strain of the cylindrical coal-rock samples are also large, and the damage is also more serious.

Energy Time History.
e internal energy here is the internal energy converted from kinetic energy when the quarter coal-rock samples are subjected to a dynamic load. e total energy is the sum of the kinetic energy and internal energy obtained by the quarter coal-rock samples under the dynamic load. e cylindrical coal-mass sample has not been damaged at a speed of 0.125 m/s, and the energy-time history is shown in Figure 6. By impact loading, the kinetic energy of the cylindrical coal-rock sample instantaneously reaches the maximum value at the beginning, and the kinetic energy is almost converted into internal energy at the last moment. e total energy of the cylindrical coal-rock sample drops slightly during the impact. e cylindrical coal-rock sample is slightly damaged at speeds of 0.25 m/s, 0.5 m/s, 1 m/s, and 2 m/s, and the energy-time history at a speed of 2 m/s is shown in Figure 7. e speed and proportion of the conversion of kinetic energy into internal energy gradually decrease. e total energy decreases slightly with time.
e cylindrical coal-rock sample is seriously damaged at speeds of 4 m/s, 6 m/s, 8 m/s, and 10 m/s. e energy-time history at a velocity of 4 m/s is shown in Figure 8. After failure, only a small part of the kinetic energy is converted into internal energy, most of the energy is in the form of kinetic energy, the proportion of the internal energy is small, and the total energy decreases slightly with time.
Taking the impact velocity as the independent variable and the total energy peak and internal energy peak of the cylindrical coal-rock sample and kinetic energy of the impact bar as the dependent variables, the relationships are shown in Figure 9. e total energy peak increases exponentially with increasing speed. e internal energy peak of the cylindrical coal-rock sample first increases and then decreases with increasing velocity and reaches a maximum value when the impact velocity is 2 m/s. e cylindrical coal-rock sample is severely damaged if the impact velocity is greater than 2 m/s; by combining the relationship between internal energy and impact speed, we can get that the cylindrical coal-rock sample will be where E k , m, v, ρ, r, and h are defined as the kinetic energy, mass, speed, density, radius, and height of the impact bar, respectively. e parameters of the impact bar and the impact velocity are substituted into equation (1) to obtain the corresponding kinetic energy of the impact bar, as shown in Figure 9. After calculation, the kinetic energy of the impact bars at different impact speeds is approximately equal to the total energy obtained by the cylindrical coal-rock samples, and the fitting equations of the two are consistent.

Element Point Axial Stress-Strain Curve.
Six-element points were selected on the cylindrical coal-rock sample, and the axial stress-strain curve of each element point was analyzed.
e six-element points included the apex on the    When the impact speed is 0.125 m/s, the axial stressstrain curve of the six-element points tends to conform to the static-mechanical stress-strain curve. When the speeds are 0.25 m/s, 0.5 m/s, 1 m/s, 2 m/s, and 4 m/s, the axial stressstrain curve of the six-element points does not completely conform to the static-mechanical stress-strain curve change, and the axial stress-strain curve of the apex and the midpoint on the central axis quickly decays after passing the yield point. When the speeds are 6 m/s, 8 m/s, and 10 m/s, the axial stress-strain curves of the six-element points quickly decay after passing the yield point. In the part where the cylindrical coal-rock sample is severely damaged by impact loading, the axial stress attenuates rapidly after passing the stress yield point, and the axial strain does not continuously increase, which is different from the constant increase in the strain under static loading.
Taking the impact bar velocity as the independent variable and the peak value of the axial stress of the sixelement points as the dependent variable, the fitting curve is shown in Figure 11. e axial stress peaks of the apex on the central axis, the midpoint on the central axis, and the apex on the edge axis of the cylindrical coal-rock sample have a linear relationship with the impact speed. e damage of the coal-rock samples in the three parts is also obvious, and the peak values of the axial stress of these three element points are also relatively large.
e literature [45,46] uses an SHPB to obtain the stressstrain curve of coal samples at different strain rates. After the yield point, the stress rapidly decays, the axial strain does not continuously increase, and the peak stress increases linearly with increasing strain rate, which is consistent with the conclusions of the numerical simulation in this paper.

Discussion on the Axial and Radial Stress and Strain Distribution
When the impact speed is 10 m/s, the cylindrical coal-rock sample is the most severely damaged, which is a dynamic load damage and the axial stress distributions at 3 μs, 13 μs, 18 μs, 29 μs, and 1000 μs are shown in Figures 12(a)-12(e). e maximum axial stress is located in the center of the contact surface and develops along the central axis. e point of the maximum value of axial stress is the first point to be destroyed, and the failure point and the maximum axial stress are consistent.
On the central axis of the cylindrical coal-rock sample, fourteen element points are equidistantly selected from the top to the bottom. When the impact speed is 10 m/s, the axial stress time history of the fourteen element points is shown in Figure 13. e axial stress on the central axis develops from top to bottom, and the axial stress peak decreases  continuously. e axial stress is the compressive stress in the axial direction. When the axial stress at the apex of the coalrock sample exceeds the peak value, the axial stress behind the direction of propagation of the axial stress is smaller than that of the front. At the same moment, the cylindrical coal-rock sample is prone to damage in the axial stress propagation cross section where the axial stress becomes tensile stress. e distribution of the axial stress peak, radial stress peak, axial strain peak, and radial strain peak at fourteen element points on the central axis of the cylindrical coal-rock sample are shown in Figures 14 and 15, respectively. e axial stress peak, radial stress peak, axial strain peak, and radial strain peak distribution of the central axis are in accordance with the Farazdagiharris-type power function, and the correlation coefficient square (R 2 ) is greater than 0.98. e axial stress peak at the same position is greater than the radial stress peak, and the axial strain peak is greater than the radial strain peak. e axial stress and radial stress distribution on the cross section of the cylindrical coal-rock sample at a velocity of 10 m/s are shown in Figures 16(a) and 16(b). e maximum stress is at the center of the circle.
Ten element points are equidistantly selected from the center to the edge on the radial line of the contact surface of the cylindrical coal-rock sample. e axial stress time history of the ten element points at impact speed 10 m/s is shown in Figure 17. e axial stress on the radial line    Shock and Vibration increases almost simultaneously, and the axial stress peak from the center to the edge on the radial line is continuously reduced. e distribution of the axial stress peak, radial stress peak, axial strain peak, and radial strain peak at ten element points on the radial line of the contact surface of the cylindrical coal-rock sample is shown in Figures 18 and 19, respectively. e distribution of the axial stress peak, radial stress peak, and axial strain peak at the radial line conforms to the Farazdagiharris-type power function, and the correlation coefficient square (R 2 ) is greater than 0.94. e axial stress peak at the same position is greater than the radial   Shock and Vibration stress peak, and the axial strain peak is greater than the radial strain peak. e strain near the edge on the radial line increases due to the unconstrained edge and greater deformation under the same force.
For plastic deformation of coal and rock masses under stress, the impact stress wave is transmitted and reflected at the interface of the elastic and plastic coal bodies. e wave impedance of the plastic coal is smaller than the wave impedance of the elastic coal, and the wave velocity of the coal mass particle is expressed as where σ, ρc, ρc l , and ρc e are defined as the stress, wave impedance, wave impedance of the plastic coal, and wave impedance of the elastic coal, respectively. Because ρvc l < ρvc e , the stress in the plastic zone of the coal and rock mass is less than that in the unchanging elastic zone in the front; in other words, the stress behind the direction of propagation of the stress wave is smaller than that of the front. In the propagation direction of the compressive stress, the stress at the front of the elastic-plastic contact is greater than the stress at the rear. Relatively, tensile stress is generated on the elastic-plastic contact. Under the action of tensile stress, the coal and rock masses are prone to damage. e numerical simulation results show that the axial stress produces tensile stress in the axial propagation direction, which is consistent with the theoretical analysis.

Conclusion
Under impact loading, the effective stress of the cylindrical coal-rock sample moves from the edge of the contact surface to the center and reaches the maximum stress at this location. en, the coal-rock samples break at the center, and the damage develops downward along the axial direction.
(1) e critical effective stress and strain have an exponential function relationship with the impact velocity, and the critical time has a linear relationship with the impact velocity. (2) e kinetic energy of the impact bars at different impact speeds is approximately equal to the total energy obtained by the cylindrical coal-rock samples. e cylindrical coal-rock sample will be destroyed after the internal energy peak reaches a certain value under different impact speeds. e energy change law of the cylindrical coal-rock sample is consistent with the destruction morphology of the cylindrical coal-rock sample.
(3) e axial stress peaks in the severely damaged part have a linear relationship with the impact speed, the axial stress attenuates rapidly after passing the stress yield point, and the axial strain does not increase continuously. is is consistent with the conclusion of the SHPB dynamic experiment. (4) e peak distributions of the axial stress, axial strain, radial stress, and radial strain on the central axis and radial line are in accordance with the Farazdagiharris-type power function. e axial stress  produces tensile stress in the axial propagation direction. e axial stress peak and strain peak at the same position are greater than the radial stress peak and strain peak, respectively.

Data Availability
e primary data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.