Dynamic characteristics and crack evolution laws of coal and rock under split Hopkinson pressure bar impact loading

The dynamic mechanical properties and crack evolution characteristics of coal and rock during split Hopkinson pressure bar (SHPB) impact failure are important contents for analysis. In previous studies, the coal and rock specimens used have usually been independent and not closely correlated. In addition, quantitative characterization and analysis methods for coal and rock cracks are immature, and more information has not been fully revealed. The aims of this paper are to comprehensively explore both the dynamic mechanical properties and crack evolution characteristics of coal and rock during impact failure. First, experimental specimens are prepared from coal seam, direct roof rock strata and direct floor rock strata in the same area to highlight the correlations between test pieces. Second, a dynamic strain gauge and high-speed (HS) camera are adopted to reflect the stress wave signal and crack evolution. Then, based on digital image correlation (DIC) technology and the mass screening method, the evolution laws of surface cracks during crushing and the distribution characteristics of sample fragments after crushing are studied from the perspective of fractal, and finally compared with those of the simulation analysis. The results are as follows. (1) The coal and rock samples from the same area have both consistency and differences. The dynamic mechanical properties of coal and rock are affected by the impact velocity and the physical properties of the specimen. Higher impact speeds and densities lead to the more obvious brittleness of the specimen when destroyed. Conversely, the sample shows more plasticity and ductile yield. (2) The self-similarity is significantly manifested in the evolution of surface cracks during impact and the distribution characteristics of fragments after impact. The box dimension and quality screening dimension are applicable to quantitatively characterize the evolution process and results of coal and rock fractures. (3) The simulation results based on the Holmquist–Johnson–Cook (HJC) and Riedel–Hiermaier–Thoma (RHT) constitutive models agree well with the experimental results, and the RHT constitutive model is more consistent. This study may contribute to a more comprehensive understanding of the dynamic characteristics and crack evolution laws of coal and rock under impact loading and provide references for further research and discussion.


Introduction
To date, an increasing number of mines are entering the state of deep mining [1]. Unlike shallow mining, deep coal and rock masses always exist in the special mechanical environment of three highs and one disturbance, i.e. high ground stress, high ground temperature, high karst water pressure and mining disturbance [2], resulting in a more complex mechanism. Mining disturbance mainly refers to blasting, crosscut coal uncovering, rapid mining and other production operations [3], with a relatively high strain rate, which is more likely to cause rapid changes in stress distribution and transient instability of coal and rock structure [4]. Statistics show that 96.5% [5] of outburst accidents and 93.3% [6] of rockburst are caused by external mining disturbances. More seriously, with the extension of the mining level, the consequences of coal and rock dynamic disasters induced by mining disturbance are becoming increasingly serious.
Studies on material failure and instability in the range of high strain rates (10 1 -10 4 ) usually adopt the split Hopkinson pressure bar (SHPB) device [7]. By taking the dynamic compression of coal and rock as an example, the International Society for Rock Mechanics (ISRM) has proposed dynamic test methods of uniaxial compression [8] and triaxial compression [9]. Many scholars have made targeted improvements to SHPBs based on their needs. Li et al [10] and Gong et al [11] developed active confining pressure and axial pressure devices based on a conventional SHPB device and realized dynamic and static combined loading in a preliminary sense. Yin et al [12] added a temperature control device to SHPB and then analyzed the dynamic mechanical properties of rock under the coupling of temperature and pressure. However, in the above devices, the most commonly used device is still the uniaxial compression SHPB system, especially in the experiment of observing the crack evolution law during the impact compression failure of coal and rock.
Quantitative characterizations of the dynamic mechanics and fracture processes of coal and rock during SHPB impact failure are critical for evaluating the stability and failure processes of the system and are challenging subjects. Although traditional contact measurement technologies, such as strain gauges, characterize the dynamic mechanical properties of coal and rock through the stress−strain rate curve, they are still indirect reflection methods. This method cannot provide intuitive and sufficient information to fully reflect the evolution of cracks, nor is it conducive to an effective understanding of the failure mechanisms of coal and rock [7]. In this case, many optical testing technologies, such as laser measurement [13], molar technology [14], photoelectric technology [15], holographic interferometry [16], caustics [17], x-ray micro CT [18] and infrared thermal imaging technology [19], are applied to the study of coal and rock impact processes. With the introduction and development of high-speed (HS) photography [20] and digital image correlation (DIC) technology [21], HS-DIC technology, as a high-speed, high-resolution, noncontact and full-field observation technology, has been widely used in the field of coal and rock impact dynamics.
HS photos provide a synchronous relationship between crack initiation time and stress history [7], while DIC is an optical method for addressing the image differences before and after deformation [22]. For crack images of coal and rock, scholars have adopted different in-depth processing methods. Zhang and Zhao [23] integrated a dynamic stress−strain curve, dynamic horizontal strain fields and a normalized dynamic compressive strength function to verify the effectiveness of HS-DIC technology in rock crack analysis during uniaxial impact. Gao et al [22] applied HS-DIC technology to fracture measurements in notched semi-circular bend (NSCB) tests of rocks and extracted the crack tip positions and dynamic stress intensity factors. Ai et al [24,25] used HS-DIC technology to study the crack propagation of coal and rock under the impact of SHPB. Germanovich et al [26], Dyskin et al [27], Lu et al [28], Fu et al [29], Li et al [30] and Zhang et al [31] discussed the influences of preset defects on rock crack growth with the help of DIC technology. Furthermore, to overcome the heterogeneous and opaque features of coal and rock, HS-DIC has been preliminarily applied in the crack failure of 3D printing samples [32][33][34][35][36]. However, although the above achievements provide some methods for analyzing cracks using HS-DIC technology, the accuracy and quantitative characterization of surface crack parameters still need to be improved. In this case, it is particularly necessary to use fractal theory to study the surface cracks of heterogeneous materials, such as coal and rock [37]. Li et al [38] and Ma et al [39] carried out fractal investigations on the surface crack evolution characteristics of coal and rock failure and achieved preliminary quantitative results.
Considering that the coal and rock samples prepared in the SHPB impact tests are mostly cylinders, it is difficult to accurately measure the circumferential deformation field and crack field under impact compression with one camera [23]. In addition, crack propagation in the specimen cannot be observed, which greatly limits the application of HS-DIC. Therefore, many scholars adopt the mass screening method for analyzing the fragments of coal and rock after impact damage. Hou et al [40] carried out screening tests on slate, sandstone and granite fragments after SHPB impact and discussed the impact load and rock characteristics on the final results. Li et al [41] used the SHPB device to conduct impact tests on artificially prepared coal, rock and combined bodies and studied the mass fractal dimension of the fragments using the screening method.
We must emphasize that HS-DIC technology effectively characterizes the surface progressive fracture processes of coal and rock during impact, and the mass screening rule focuses on the distribution characteristics of fragments after impact. Both are closely related to fractal theory and have a temporal inheritance relationship. The unification of the two is helpful for comprehensively reflecting the dynamic characteristics and failure processes of coal and rock under impact conditions. To date, there are few studies on the comprehensive applications of surface crack and fragment characteristics to evaluate the impact damage degrees of coal and rock. In addition, as typical heterogeneous and anisotropic media, coal and rock specimens under high-velocity impact often show typical discontinuity and nonlinearity in the failure process [42]. Relative to the laboratory test, numerical simulation reveals the overall process and internal mechanism of sample failure from the detail level, which is crucial to the study of coal and rock impact [43]. Currently, common numerical methods including finite element method (FEM) [44], extended finite element method (XFEM) [45], numerical manifold method (NMM) [46], and phase-field method [47][48][49], are employed to simulate crack growth and failure. Notably, both the FEM and the phase-field method are simulation methods for crack propagation under the finite element framework, and have wide applications in the simulation of brittle materials such as rock [50,51], coal [52,53] and concrete [54][55][56].
In this paper, coal and rock samples obtained at the same location are subjected to the SHPB uniaxial impact failure test; the damage evolution process of the specimen is comprehensively revealed from the two aspects of surface crack and fragment characteristics. Specifically, coal and rock samples are prepared from the coal seams, direct roof and direct bottom strata of the east fifth north wing heading face of the Sihe Coal Mine, which is seriously affected by coal and rock dynamic disasters. Then, based on HS-DIC technology and the mass screening method, the evolution laws of surface cracks during crushing and distribution characteristics of sample fragments after crushing are studied from the perspective of fractals, and the correlation between the two is explored to comprehensively reflect the impact damage processes of coal and rock. Finally, the Holmquist-Johnson-Cook (HJC) and Riedel-Hiermaier-Thoma (RHT) dynamic damage simulation models are established to reproduce the crack evolution process and verify the test results. The research is expected to further reveal the failure mechanisms of coal and rock materials under impact loading, and provide support for prewarning of coal and rock dynamic disasters.

Material introduction and specimen preparation
As illustrated in figure 1(a), the experimental specimens were taken from the Sihe Coal Mine of Jinneng Holding Equipment Manufacturing Group Co., Ltd, which was seriously affected by coal and rock dynamic disasters. The east five north wing heading roadway in the Sihe Mine was a double tunneling working face, and the sampling point was located at the end of the No. 1 heading face (shown in figure 1(b)). According to the stratigraphic logging records of the boreholes near the sampling point shown in figure 1(c), coal seam #3 was in the Shanxi Group of the Lower Permian System, with silty mudstone on the direct roof and siltstone on the direct floor. To ensure the consistency of the coal and rock media, a hydraulic drilling rig was used to core the coal seams, direct roof strata and direct floor strata in the same area. The field sampling diagram of roof rock is shown in figure 1(d).
To minimize the inertial effect of the test piece and meet the internal stress homogenization assumption, by referring to the relevant research results [57][58][59][60], the samples were made into cylinders with dimensions of 50 mm diameter and 30 mm height. According to the suggested method for dynamic tests [61], the upper and lower surfaces of the specimen were precisely polished to ensure that the nonparallelism of the two ends of the cylinder was less than 0.05 mm, and the nonperpendicularity of the circumference and end face was less than 0.25 • . As shown in figure 2, the samples were processed into eight cylindrical coal seam, roof rock stratum and floor rock stratum specimens, defined as C1-8, RR1-8 and FR1-8, respectively.  Figure 3 illustrates the SHPB system developed by China University of Mining and Technology (Beijing). As illustrated in figure 3(a), the system consists of a dynamic system, bar system and data measurement system. (1) The dynamic system, i.e. the stress wave generator, is initiated by high-pressure nitrogen. The gases with different pressures drive the bullet to impact the incident bar and finally make the stress wave act on the specimen along the bar system. (2) The bar system, considering the large particle size and local anisotropy of coal and rock media, adopts an incident bar and transmission bar with a relatively large diameter (φ = 50 mm) and a length of 3000 mm and 2500 mm, respectively. During the test, the bar system should be at the same horizontal position to ensure the integrity of the test waveform and the accuracy of the data. (3) The data measurement system includes velocity measurement, strain measurement and HS photography subsystems, which are relatively complex. The velocity measurement subsystem consists of a parallel light source and a velocity tester. The photoelectric method is used to measure the velocity. That is, the instantaneous velocity of the impact bar is determined by the distance between two light sources and the truncation time difference. The velocity  tester is connected to the data acquisition system of the SHPB test, and the system automatically obtains the initial velocity of the impact bar and records its value. The strain measurement subsystem includes a semiconductor strain gauge, super dynamic strain gauge, waveform collector and data acquisition system. Two strain gauges, MC-AF-120, with a sensitivity of (110 ± 5)%, are pasted on the input bar and output bar to record the electrical signals during the test. The output signal of the strain gauge is collected by the LK2107A super dynamic strain gauge, displayed and stored on the DPO72004C oscillograph recorder produced by Tektronix Company, USA. The oscillograph recorder is connected to a computer to read, store and process signals.
Notably, to accurately depict the failure forms and crack evolution laws of coal and rock during SHPB impact, the HS photography subsystem is highlighted in figure 3(b). The main body of this subsystem is the GX-3 HS camera produced by NAC Company, Japan. The acquisition resolution is 1280 × 1024 pixels, with a pixel size of 21.7 µm, and the shooting frequency is 10 000 frames/second. The video data of the specimen under the impact of dynamic loading from beginning to end are recorded. Due to the particularity of coal and rock medium materials, the VL300W constant fill light produced by Godox Company, Shenzhen, China, is used in the experiment. The colour temperature adjustment range is 5400-5800 K, and the maximum illumination in the standard lampshade state is 77 000 LUXm −1 . The light has a strong spotlight effect and colour fidelity to display more details during coal and rock damage and crack evolution. In addition, to effectively collect the specimen fragments, a plexiglass sleeve with a thickness of 2 mm is set outside the sample. The angles and positions of the sleeve, fill light and camera are adjusted many times to minimize the impact on the acquisition of speckle photos.

Principles of the SHPB test.
In stress wave propagation theory, the specimen in the SHPB impact experiment produces one-dimensional (1D) strain states, which is different from the 1D stress states of the tested materials in the low strain rate experiment [7]. Specifically, in the 1D strain states, the bullet is emitted from the chamber and hits the incident bar. The stress pulse is transmitted from the incident bar to the contact surface between the bar and the sample as a compression wave. Due to the significant difference in wave impedance values, the reflected tensile wave is generated at the interface, accompanied by a transmission wave. Similarly, reflection and transmission occur at the interface between the sample and the transmission bar. After multiple reflections and transmissions, stress−strain equilibrium is achieved at the two end faces of the sample. The bars selected in this experiment are homogeneous materials with the same cross-sectional area as the coal and rock specimens. The incident wave, transmission wave and transmission wave signals recorded by the strain gauge are combined with the 1D stress wave theory and the three assumptions of the SHPB test to simplify the three-wave formula, thus obtaining the variations in the stress, strain and strain rate values of coal and rock samples with time [62]: whereε(t), ε(t) and σ(t) are the loaded strain rate, loaded strain and loaded stress for the specimen, respectively; ε r (t) and ε t (t) correspond to the reflected wave strain and transmitted wave strain, respectively; E 0 and C 0 are the elastic modulus and wave velocity of the bar, respectively, and these two values are determined when the bar is determined; and L is the length of the specimen.

Box dimension calculation model.
Many studies indicate that within a certain scale range, the surface cracks of brittle heterogeneous materials, such as coal and rock, show fractal characteristics to some extent, and fractal theory is applicable to quantitatively study parameters, such as crack morphology [63][64][65][66]. Among the common fractal dimension representation methods, the box dimension is the most widely used. The method has high practicability and accuracy and meets the requirements of computational efficiency and dynamic characteristics [67].
The principle of the box dimension [68,69] is to cover coal and rock cracks with boxes composed of squares (boundary length of δ), and the number N δ of nonempty boxes containing relevant crack pixels is counted. Obviously, there are significant differences in the number of N δ values under different box sizes δ [70]. By changing δ, the statistical expression of fractal dimension D is as follows: where a is the prefactor in the fractal dimension scaling rule. For a fixed crack image, a is a constant. If equation (4) is expressed in logarithmic form, the new expression is as follows: According to equation (5), the slope of ln N δ relative to ln δ is obtained by linear regression, and the result is the fractal dimension D of the coal and rock surface crack image.
To ensure the accuracy of the calculation results, it is necessary to preprocess the image to eliminate the artefacts, noise and other noncrack elements in the original image recorded by the HS camera and reveal the real characteristics of cracks in the image. In this paper, Ratsnake annotation software [71] with high accuracy and resolution is used to manually extract cracks. Then, the crack sketch results are input into the FracLac plug-in on ImageJ software for box fractal dimension analysis. The flow chart is shown in figure 4. Ratsnake software and the FracLac plug-in are integrated to realize quantitative descriptions of surface cracks in coal and rock samples. In particular, ImageJ software evaluates whether the crack sketch is a binary image. If it is a binary image, the FracLac plug-in is manually opened for analysis. Otherwise, the image is binarized by setting a threshold. In Fraclac, the parameters of the control panel to analyze the cracks are set in the region of interest (ROI). Finally, the box fractal dimension is obtained by image scanning and the statistical results of equation (5).

2.2.4.
Mass-frequency relationship model. In fact, fractal theory is used to describe the crack propagation laws of coal and rock during failure and to analyze the block size distribution characteristics after failure [72,73]. Since the number of broken blocks is difficult to count, a screening test is carried out. To realize the quantitative expression of the degree of coal and rock fragmentation, Mandelbrot [74] and Zhao et al [75] established a mass-frequency relationship model based on the screening method, as shown in equation (6): where ε is the pore size of the splitting sieve; σ is the average scale; M(ε) is the mass of the broken body of coal and rock with a diameter less than ε; M is the total mass of the broken body of coal and rock; and D is the fractal dimension of mass. Obviously, the relationship between M(ε) and M is expressed by the cumulative mass percentage of each particle size under the sieve. By taking the logarithms of the left and right sides of equation (6) and sorting them out, we obtain the following equation: According to equation (7), in the double logarithmic coordinate system composed of lg[M(ε)/M] and lg (ε/σ), we use the least square method to fit the data and then subtract the slope of the straight line from 3, and the calculation result is the mass fractal dimension D.

Test results of physical and mechanical parameters.
To facilitate comparative analysis, two specimens were selected from each group of coal and rock samples for the following experiments: (1) the uniaxial compression failure test based on an electrohydraulic servo testing machine and (2) the porosity measurement experiment based on the mercury intrusion method. In this process, some basic physical and mechanical parameters, such as the uniaxial compression coefficient, elastic modulus, Poisson's ratio and porosity, were measured. In addition, to reflect the compactness of the sample, a ZBL-U510 nonmetallic ultrasonic detector produced by Beijing Zhibo Lian Technology Co., Ltd was selected to conduct a single-emission and single-receiving acoustic transmission test on the remaining specimens [76] with reference to the test method specified in ASTM D 2845 [77]. The arithmetic mean of the longitudinal wave velocity was calculated after multiple tests. The results are summarized in table 1.
Relative to the results of the ultrasonic longitudinal wave velocity test of coal, roof rock and floor rock specimens in the Sihe Mine shown in table 1, although the data have some dispersion, the longitudinal wave velocity in each group is basically similar; this phenomenon reflects the high homogeneity characteristics of the collected and prepared samples, providing scientific support for subsequent analysis. In addition, by comparing the longitudinal wave velocities of different samples, the values of rock samples are significantly greater than those of coal samples, and the values of floor rock samples are generally slightly greater than those of roof rock samples. By comprehensive comparisons of the above physical and mechanical parameters, although coal and rock are both elastoplastic materials, the former is significantly denser and more prone to brittleness. In comparison, the plasticity and ductility yield of coal are more prominent.

Dynamic stress equilibrium.
After testing the ultrasonic longitudinal wave velocities of coal and rock samples, we conduct 18 groups of impact load tests. The specific experimental parameters are shown in table 1 in section 3.1.1. The impact speed ranges from 4 m s −1 -14 m s −1 , and each 2 m s −1 increase is set as a speed step, which is divided into six levels. After each SHPB test, to ensure the accuracy of the results, the stress equilibrium at both sides of the rock specimen requires careful checking. Figure 5 shows the typical dynamic stress equilibrium states, which are obtained from the floor rock sample (FR4) with an impact velocity of 8.885 m s −1 . According to the SHPB test principle, the stresses at both ends of the sample are calculated from the test signals on the elastic bar. Specifically, the superposition of the incident wave and reflected wave reflects the stress at the incident end of the sample; the transmitted wave reflects the stress at the transmission end. Figure 5 shows that the curves at both ends are basically coincident, which indicates that the specimen is in a state of stress equilibrium during the dynamic loading process.

Time history curve of stress wave and signal denoising.
In the SHPB impact damage experiment of coal and rock, semiconductor strain gauges are arranged on the input bar and output bar to capture the incident wave, reflected wave and transmitted wave. Then, the signals are stored through a super dynamic strain gauge, waveform collector and data acquisition system. As space is limited, by taking the impact test with a speed level of 6 m s −1 as an example, the time history curve of the stress waveform is displayed in figure 6(a). The figure shows that each waveform roughly presents a half-sine curve, and the dispersion of the waveform and the resulting wave head oscillation are effectively suppressed. However, the original waveform signals in figure 6(a) contain a large amount of noise interference, which is particularly evident in the transmitted wave. This phenomenon is caused by external environmental impact and Pochhammer-Chree (PC) oscillation [78]. Due to high noise, short duration and rapid mutation, the waveform signal generated by the impact dynamic load is a nonstationary signal, which can be denoised by the Hilbert-Huang transformation (HHT) method [79,80]. The denoised  signal is shown in figure 6(b). Relative to figure 6(a), the signal amplitudes before and after filtering have little changes; however, high-frequency noise is effectively eliminated, and the signal-to-noise ratio is significantly improved. Similarly, 18 groups of stress waves are filtered and denoised, and the time history curves of incident-reflected waves and transmitted waves are shown in figure 7. Figures 7(a) and (b) show that with increasing impact speed, the stress wave amplitudes of coal and rock impact is increased, the time to the peak is advanced, and the overall duration of the stress wave is gradually shortened. Most notably, in figure 7(a), when the impact velocity is low, negative values appear at the tail end of the reflected wave. The reason for this phenomenon is that the specimen is not destroyed completely under incident wave loading conditions, and a large amount of strain energy is stored in the medium instead. These strain energies are released in the unloading phase of the incident wave and rebound to the bar as compression waves. In contrast, when the impact velocity is high, the coal and rock samples are destroyed instantaneously under the action of the incident wave, and the compression rebound in the reflected wave disappears.
In addition, at similar impact velocities, the stress waves of each coal and rock specimens have high coincidence. The peak values of the incident wave and transmission wave of the coal samples are lower than those of the rock samples; however, the reflected waves are higher. Similarly, when the test sample is coal, the time to the peak is often relatively slow. According to a comparison of the ultrasonic longitudinal wave velocity test results in table 1, denser samples have faster stress wave propagation speeds. In conclusion, the time history curve of the stress wave is affected by both the impact velocity and the physical properties of the sample.  the strain rate with time at different speeds. When the impact speed is low, the strain rate curve slowly increases to the peak point and then gradually decreases. With increasing impact velocity, the slope value of the strain rate curve increases rapidly, the end position of the curve gradually approaches the peak point, and there is a short fluctuation in the peak section. Figure 8(b) shows the time−strain curves. With the increase in impact speed, the times for initial strain and failure instability characteristics of coal and rock materials are significantly shortened. At similar impact velocities, the jumping points of the strain curves of rock samples are earlier than those of coal samples, and the cut-off time of the strain curves of roof rock samples are earlier than those of coal samples and floor rock samples in most cases. The reasons for this phenomenon are as follows. (1) Although rock and coal are both elastic−plastic complexes, the former is more inclined to show its brittleness, while the latter has more prominent plasticity and ductility.
(2) The crack derivation and damage failure characteristics of roof rock samples are obviously faster than those of floor rock samples, which needs to be verified by subsequent test results. Figure 8(c) shows the relationships between strain and stress at different impact velocities. When the impact speed is small, the initial slope of each sample curve has little difference. After entering the plastic stage, the curve slowly drops from the peak stress, and vibration and fluctuation occur. With the increase in impact, the initial rates of coal and rock samples gradually coincide, and each curve first drops and then rebounds after reaching the first peak, showing a saddle shape, which is obviously a manifestation of strain strengthening. At a higher strain rate, the sample reaches the stress peak at a smaller strain, and its brittleness is more obvious.

Process of crack propagation and specimen failure
3.2.1. Recognition and feature extraction of surface cracks. To date, many scholars have used HS cameras to record the evolution laws of surface cracks and explore the failure processes of coal and rock masses. Image processing is a prerequisite for crack analysis, especially for coal and rock samples with low contrast and robustness between cracks and specimens. The floor rock sample (FR1) is taken as an example, and its impact velocity is 4.668 m s −1 . FR1 takes 29 ms from the contact between the incident bar and the sample to the ejection of rock fragments. The HS camera shoots at a frequency of 10 000 frames s −1 and extracts pictures every 2 ms, obtaining 15 pictures. Ratsnake is utilized to annotate the cracks in each image, and the analysis results are shown in figure 9. Figure 9 shows the crack evolution process of the FR1 sample under the impact load of the SHPB, as follows: (1) The crack first occurs in the middle of the sample, and the main radius direction of the crack equivalent ellipse [24] is basically 45 • oblique to the impact loading direction of the axis.
(2) The crack rapidly extends to both ends of the sample, and another parallel main crack appears in other areas. (3) Both main cracks continue to expand, accompanied by the simultaneous generation of minor cracks. Relative to the main crack, the propagation direction of the minor crack has great randomness. (4) As the impact continues, the main cracks and the minor cracks expand synchronously, and the area and pixel ratios of the cracks increase. By the 29-frame shooting time, fragments are ejected, and the identification and analysis of surface cracks are terminated.

Crack quantification and box dimension calculation.
Based on the Ratsnake surface crack identification results, according to the box dimension calculation model, the FracLac plug-in is used to operate and process the crack patterns in the SHPB impact failure of each coal and rock specimens to quantitatively depict the evolution law of each sample from crack generation to fragment ejection. According to the flow chart shown in figure 4, a total of 75 coal seam sample fractal dimensions, 60 roof rock fractal dimensions, and 73 floor rock fractal dimensions are obtained through linear regression and average value calculation. The above results are summarized, and the box dimension progress curves of coal and rock samples are drawn, as shown in figure 10.  1) For impact failure at a specific speed, the box dimension of the coal and rock specimens shows the laws of overall increase and local oscillation. The reason for the overall increase is obvious: the box dimension itself is a quantitative reflection of the whole process, from crack expansion to the overall failure of the specimen with impact. The local oscillation fully reflects the complexity, heterogeneity, and anisotropy characteristics of coal and rock materials, and the crack may still be closed during the overall growth. (2) Generally, the growth rate of the box dimension is relatively fast at the initial stage, while the subsequent increasing trend is slowed. Rapid crack initiation occurs immediately when the incident bar hits the specimen. Relative to the initial incident wave, the damage levels of the reflected tensile wave and subsequent superimposed wave on the specimen are weakened. (3) With increasing impact velocity, the slope of the box dimension curves of surface cracks in coal and rock specimens increase synchronously; however, the duration is continuously shortened. The results show that a higher impact velocity promotes the damage process, significantly accelerates crack evolution, and greatly advances the ejection times of coal and rock fragments.
Additionally, as shown in figure 10(d), we have purposely compared the box dimension differences of coal and rock samples. First, the variation trend of the box dimension of the roof rock sample is similar to that of the floor rock sample. Second, under a similar impact velocity, the rock sample oscillation degree is more intense than that of the coal sample. Third, the numbers of crack evolution images of roof rock samples captured by a high frame rate camera are relatively less than those of coal and floor rock samples. The curve durations of most roof rocks are relatively short, which is a prominent reflection of the rapid transition of the sample from the initial derivations of cracks to the ejection state of fragments. Finally, when the impact velocity is low, the box dimensions of the rock samples are larger than those of the coal samples. When the impact speed is high, the opposite is true. The results indicate that for the prepared test pieces, the rock samples show more apparent heterogeneity and impact resistance levels (i.e. the ability to resist damage) than the coal samples. In other words, as elastic−plastic complexes, most rock samples show brittle failure under the action of incident waves. During the impact failure of coal samples, the plastic characteristics are more obvious.
To quantitatively characterize the crack evolution law of the specimen, this paper adopts the research method of first using an HS camera to obtain the impact failure processes of coal and rock and then using the box dimension to conduct fractal analysis. There are three defects in this method. (1) The sketches of cracks are manually marked by Ratsnake. The cracks require a heavy workload, and the crack-filling situation and the fragment ejection cut-off criteria are subjective.
(2) The failure observation surface of the specimen is bent. The plane picture is used to characterize the crack evolution law on the circumference face of the coal and rock mass. The normal direction of the cracks may have an angle with the analysis plane, producing corresponding result errors [23]. (3) The coal and rock samples are prepared as three-dimensional cylinders. As limited by the experimental conditions and equipment, we analyze the plane illuminated by the high-power projector, leading to the failure fully expressing the evolution of some cracks, which is the main reason for the error in figure 10(d). To compensate for the impacts of the above drawbacks on the analysis results, we select sample sieves with different apertures to fully screen the coal and rock fragments after impact and discuss impact damage from another perspective. Figure 11 shows the fragment morphologies of coal and rock specimens collected under different strain rates after impact. The coal and rock specimens under impact loading have strong strain rate sensitivities [81]. Specifically, when the impact velocity is low, the coal and rock samples still maintain certain degrees of integrity, and most of them are in a massive dispersion state. With increasing impact speed, the damage degrees of coal and rock increase. For the broken body, the volume is significantly reduced, and the quantity is correspondingly increased. The failure form of coal and rock changes from fragmentation to pulverization [82].

Fractal characteristics of crushing products.
With reference to relevant standards [83], sample sieves with 9.50 mm, 4.75 mm, 2.36 mm, 1.18 mm, 0.60 mm, 0.30 mm, 0.15 mm, and 0.075 mm apertures are selected for combination, and the coal and rock fracture sieves after SHPB impact damage are divided into eight grades. A high-precision balance is used to weigh the broken blocks of coal and rock of each grade. The statistical results are summarized in table 2.
As a note, since the coal sample (C1) is broken into three pieces after impact, which has no screening significance, the value is discarded. By referring to equation (7), the double logarithmic expressions of the impact samples are established, and the regression analysis is carried out. The results are shown in figure 12.
In table 2 and figure 12, there is a good logarithmic correlation between the cumulative mass percentage of the particle size under the sieve and the diameter of the sample sieve. The fractal dimension fluctuates between 1.1601 and 2.6029, and the fitting coefficient is relatively high, indicating that the    coal and rock fragmentation block has obvious self-similarity and fractal characteristics. In general, the slope of the fitting curve decreases with increasing impact velocity and strain rate. Accordingly, the fractal dimension increases, indicating that the fragmentation of coal and rock blocks gradually intensifies after impact. In addition, the fractal dimension of the roof rock sample under similar impact speed is larger than that of the floor rock sample; the block is broken more severely. Combined with the ultrasonic test results in table 1, the floor rock samples are generally denser than the roof rock samples, and they are also easier to preserve their structural integrity at similar impact speeds. The similar rules can be mutually corroborated with figures 10(d) and 11.

Correlation characteristics of the box dimension and mass dimension.
The box fractal dimension is used to characterize the surface progressive fracture processes of coal and rock during impact, and the mass fractal dimension is employed to describe the distribution characteristics of fragments after impact. Obviously, exploring the correlation between these two fractal dimensions helps to understand the processes of coal and rock masses impact failure. The box fractal dimension is obtained from the analysis of specific crack images by HS-DIC technology, and there are significant differences in crack images at different times. For each coal and rock sample, we select the last frame image before fragment ejection and calculate its box fractal dimension. This value is compared with the mass fractal dimension, and the relationship between them and impact velocity is established, as shown in figure 13. Figure 13(a) shows that both the box fractal dimension and mass fractal dimension are positively correlated with the impact velocity. From the perspective of energy, when the SHPB device is used for the impact test, the energy absorbed by the coal and rock specimen is mainly converted into breakage energy for crack propagation and the generation of a new fracture surface. With increasing impact speed, the breakage energy increases synchronously, producing the following results. (1) On the surface of the sample, the evolution of cracks is significantly promoted, and the number of cracks is greatly increased, increasing the box dimension. (2) In the interior of the sample, the breakage energy converts into the surface energy of the newly broken fragments and intensifies the collision and friction between the fragments, further reducing the fragmentation of the broken body and finally leading to a synchronous increase in the mass dimension. In addition, in figure 13(a), the two curves of the box fractal dimension and mass fractal dimension are approximately parallel in some sections, indicating that there is a certain correlation between them. Therefore, we establish a scatter diagram of the box fractal dimension and mass fractal dimension under the same impact speed and carried out data fitting. The result is shown in figure 13(b). There is a positive linear correlation between the box fractal dimension and the mass fractal dimension. Specifically, the box fractal dimension and the mass fractal dimension are both quantitative characterizations of the impact failure of the specimen. As the box dimension increases, the mass dimension increases synchronously. In figure 13(b), the box fractal dimension is a planar description of the circumference face crack of the cylinder, which is a two-dimensional calculation of the surface crack. In contrast, the mass fractal dimension focuses on the failure patterns of the internal and external fragments of the specimen, which is a three-dimensional interpretation of the overall failure. The unification of the two is helpful for comprehensively reflecting the dynamic characteristics and failure processes of coal and rock under impact conditions. Notably, the two dimensions are different expressions of coal and rock damage, and there is a certain randomness of crack growth, which is the main reason for the large difference in the correlation coefficient (R 2 value) of the coal and rock material fitting curve in figure 13(b).

Numerical simulation of coal and rock impact failure
Earlier in the article, we systematically described the failure processes of coal and rock masses based on experimental results. However, the dynamic characteristics and crack evolution laws of coal and rock specimens under impact loads are extremely complex and are significantly different from those under quasistatic loads. In this case, the impact responses of heterogeneous materials, such as coal and rock, are usually determined by comparing laboratory tests with numerical simulations [84]. In the finite element simulation analysis of nonlinear dynamics, such as the HS impact of coal and rock, the LS-DYNA program launched by ANSYS, Inc. and Livermore Software Technology Corporation (LSTC) is widely used [85]. In this section, based on two dynamic damage constitutive models, HJC [86] and RHT [87], we have utilized the LS-DYNA program to simulate the impacts of coal and rock, compared the simulated crack evolution process with the experimental test results, revealed the overall process and internal mechanism of sample failure from the level of detail, and evaluated the impact of the constitutive model on the simulation results of coal and rock impact. These parameters are classified into five categories: basic material parameters, material strength parameters, material damage parameters, material pressure parameters, and software parameters. According to the physical and mechanical properties of the coal and rock used in the experiment and by referring to the relevant literature [88][89][90][91], the HJC constitutive model parameters are determined as shown in table 3 (unit: g-cm-us system). In addition, the ADD_EROSION keyword has been added to the k-file to add a tensile damage failure criterion, i.e. MXEPS = 0.0005. Based on the HJC model, Riedel et al proposed the RHT model [87] and made several improvements [92,93]. The model is used for concrete materials. The elastic limit surface equation, failure limit surface equation and residual strength limit surface equation related to pressure are embedded in the model to describe the evolution laws of the initial yield strength, peak yield strength, and postpeak residual strengths of rock mass materials, respectively [94]. RHT is divided into three stages: the linear elastic phase, the linear strengthening phase, and the damage softening phase. The linear strengthening phase is an advantage of the RHT model, which is characterized by the P − α state equation of porous materials; it highlights the strain-hardening effect of coal and rock media [95]. In contrast, the polynomial state equation of the dense material is used to describe the damage softening stage [96].
The RHT model exists in MAT No.272 material of LS-DYNA and contains 34 parameters. These parameters are summarized as relevant material mechanics parameters, material basic strength parameters, failure limit surface parameters, elastic limit surface parameters, linear strength segment parameters, residual strength limit surface parameters, damage evolution parameters, damage softening effect parameters, etc. Notably, the RHT model was developed as an enhancement to the Johnson and Holmquist (JH) concrete model [97]. Although the AUTODYN model has been widely applied to simulate the damage evolution process in concrete, it is rarely used to model coal and rock materials. In addition, the RHT model has not been widely employed in LS-DYNA [98]. To apply the RHT model in LS-DYNA to simulate the damage evolution processes of coal and rock, the mechanical parameters must be determined based on existing mechanical tests [99][100][101]. For some parameters that cannot be established temporarily, we refer to the default values in relevant manuals [102]. The results are shown in table 4 (unit: g-cm-us system).

Simulation results and discussion
With reference to the SHPB test system (figure 3), the simulation model is selected according to the dimensions of the test, as shown in figure 14. The lengths of the incident bar and transmission bar are 3000 mm and 2500 mm, respectively, and their diameters are both 50 mm. The material is 35CrMn steel with a density of 7800 kg m −3 and an elastic modulus of 206 GPa. In the model, a solid 164 eight-node hexahedron element is selected for grid generation. The axial directions of the bullet, incident bar, sample and transmission bar are divided into 44, 375, 50, and 312 parts, respectively. SET_SEGMENT is defined at the rear end face of the transmission bar to set the boundary condition. The contact type between the bullet and bar is set to ERODING_SURFACE_TO_SURFACE, and the contact type between the bar and the test piece is set to AUTOMATIC_SURFACE_To_ SURFACE, ignoring friction between contact surfaces. A penalty function is set to reduce the hourglass effect in the contact algorithm. The HJC and RHT constitutive models are selected to simulate the failure processes of coal and rock specimens under SHPB impact, and the simulation results are compared with the experimental results.
To verify the accuracy of the simulation results, it is necessary to compare the dynamic response characteristics obtained based on the HJC and RHT constitutive models with the experimental results. Due to space limitations, we take the timestress curves at the impact velocity level of 6 m s −1 as an example. The data collected in the experiment is shown in figure 6(b). On this basis, the incident wave, reflected wave and transmission wave curves obtained by simulation are added. The results are shown in figure 15. The figure shows that whether based on the HJC or RHT constitutive model, the simulation results are similar to the measured results, which shows that it is feasible to use LS-DYNA software to conduct SHPB simulation of coal and rock materials; the selected constitutive model accurately reflects the dynamic response characteristics of coal and rock masses under impact conditions. Specifically, the measured curves of the incident-reflected stress waves of each coal and rock sample are in good agreement with the simulated curve. At the peak value of the incident waves, the measured results are similar to the simulation results. Correspondingly, the measured peak values of reflected waves are significantly lower than the simulated values, which is obviously caused by the energy attenuation and loss of stress waves during propagation. In addition, due to the heterogeneity of coal and rock materials, the measured curves of transmission waves are often slightly more complex than the simulated curves, and their value has certain volatility. Overall, the measured value and the simulated value of the stress wave are slightly different by the same order of magnitude, and the simulation results are reasonable.
According to the consistency comparison of stress waveforms, we verify the accuracy of the LS-DYNA simulation results. Moreover, the damage process of the specimen is reproduced by numerical simulations. By taking the roof rock sample (RR1) as an example, the experimental and simulation comparison diagram of the specimen failure process is shown in figure 16(a). The impact velocity of the bullet is 4.377 m s −1 . The bullet takes 29 ms from the contact between the incident bar and the sample to the ejection of rock fragments. A total of eight test pictures are captured every 4 ms. The HJC and RHT constitutive models are used to describe the failure rule of the specimen under SHPB impact; the front view of the simulation results is intercepted, in which the interception time and angle are consistent with the experiment. The experimental diagrams show that when the sample is impacted, the microcracks are derived from the middle of the sample and extend along the axial direction to the contact between the two ends of the sample and the bar, finally forming several splitting cracks that run throughout the sample. With time, the cracks gradually expand and develop from the surface to the interior. Furthermore, several cracks are connected with each other, which makes the crack distribution on the surface of the sample more complex. Relative to the experimental results, the simulations based on the HJC and RHT constitutive models show similar crack evolution laws. Fracture first occurs in the middle of the specimen, then the cracks develop along the axial direction to the end face, and finally, Notably, in the numerical simulation, the failure causes of the surface and internal microelements of the specimen are different. The surface microelement is basically in the compression state. There is a natural free surface on one end face, and its failure reason is mainly excessive strain. In contrast, the internal microelement is restrained by the peripheral microelements due to their deformation; thus, the strain is not large and they are prone to local tension. A few microelements are destroyed first because the tensile stress reaches the limit value.
When an internal microelement is damaged, the surrounding microelements form a free surface due to the constraint disappearance, leading to the consequences of strain increase and final failure, forming a damage accumulation phenomenon at the crack tip. Figures 16(b)-(d) show a comparison of the research results of different impact speeds and samples. The crack evolution law is indeed affected by both the impact energy and the physical properties of the test piece. This point has been fully explained previously and is not repeated here. In the LS-DYNA simulation, with increasing impact velocity, the numbers of damaged microelements increase significantly, the sizes of microelements decrease correspondingly, and the damage degrees of the samples increase accordingly.   Furthermore, the number of points involved in the initial failure increases, and the disorder of crack propagation further increases. The failure form of coal and rock changes from 'crack gradually penetrated' to 'instant fracturing crushing'. In addition, by comparing the simulation results of HJC and RHT, the latter shows a higher agreement with the experimental results. The reason for this phenomenon is that the RHT is proposed based on the HJC. RHT introduces the third invariant of the deviatoric stress tensor and effectively distinguishes the tension and compression meridians. Relative to HJC, RHT is more sensitive to impact and better describes the mechanical properties and damage characteristics of coal and rock materials under dynamic loading.
Since the coal and rock masses are heterogeneous and opaque anisotropic materials, the initiation and propagation of cracks are three-dimensional, leading to the deviation of surface cracks in the comparison between the experimental and simulation results. In addition, there are typically two ways for simulation of the fracture process and failure prediction of coal and rock materials under dynamic test, i.e. mesh-based simulation methods and mesh-less simulation methods [25]. In this article, LS-DYNA, a meshbased method, is used to simulate coal and rock impact. When large deformations occur, it is challenging to maintain the nonsingularity and continuity of the mesh. Accordingly, a series of mesh-less simulation methods such as smoothed particle hydrodynamics(SPH) [103][104][105][106], cracking particle model (CPM) [107,108], and reproducing kernel particle method (RKPM) [109,110] are proposed. Due to the separation of constraints from meshes and elements, meshless methods are particularly suitable for studying oversized deformation and crack growth such as impact. In future work, we attempt to explain the evolution laws of coal and rock fractures from the perspectives of three-dimensional and meshless simulation to improve the comprehensiveness of the conclusions.

Conclusions
In this study, coal and rock samples obtained at the same location are taken as the research objects, and the dynamic mechanical properties and crack evolution laws under impact loading are systematically analyzed using indoor experiments and numerical simulation methods. The following conclusions are drawn: (1) The dynamic mechanical properties of coal and rock are affected by the impact velocity and the physical properties of the medium during SHPB impact. The prepared experimental samples have both consistency and differences. At similar impact velocities, the denser the sample is, the faster the stress wave propagates. Although rock and coal are both elastic−plastic complexes, the former is more inclined to show its brittleness; the latter plasticity and ductility yield is more prominent. With increasing impact velocity, the strain strengthening is gradually obvious, and the brittleness of the sample is more typical. (2) The surface cracks in the processes of coal and rock impact failure have obvious self-similarity, and the box dimension is applicable to quantitatively describe the crack evolution law. In general, the box dimensions of the coal and rock specimens show the regularity of overall increase and local oscillation, and the degree of oscillation of the rock specimen is more intense than that of the coal specimen. The growth rate of the box dimension is relatively fast at the initial stage; the subsequent increasing trend slows. When the impact velocity is low, the box dimensions of the rock samples are larger than those of the coal samples. When the impact speed is high, the opposite phenomenon is true. (3) The fragment distributions of coal and rock after impact damage have good fractal laws. There is a good logarithmic correlation between the cumulative mass percentage of the particle size under the sieve and the diameter of the sample sieve. The fractal dimension fluctuates between 1.1601 and 2.6029, and the fitting coefficient is relatively high. (4) Both the box fractal dimension and mass fractal dimension are quantitative characterizations of the impact failure of the specimen, and there is a certain positive correlation between them and the impact speed. With increasing impact speed, more energy is used for crack propagation and the generation of a new fracture surface, resulting in a synchronous increase in the box fractal dimension and mass fractal dimension. These two dimensions are different expressions of coal and rock failure, and their unification is helpful for comprehensively reflecting the failure processes of coal and rock under impact conditions. (5) Based on the HJC and RHT constitutive models, LS-DYNA simulates the crack evolution process under impact load. The simulation results are in good agreement with the experimental results. Relative to HJC, the simulation based on the RHT constitutive relation better describes the mechanical and damage characteristics of coal and rock materials under dynamic loading.

Data availability statement
The data cannot be made publicly available upon publication because no suitable repository exists for hosting data in this field of study. The data that support the findings of this study are available upon reasonable request from the authors.