Study on Dynamic Parameters and Energy Dissipation Characteristics of Coal Samples under Dynamic Load and Temperature

Abstract

Coal and rock dynamic disasters such as rock burst and outburst seriously threaten the sustainable development of the coal mining industry, which are intimately correlated with the nonlinear dynamic response process of the deep coal and rock mass. This study conducts coal dynamic experiments under vibration load from room temperature to 60 °C by using the split Hopkinson bar (SHPB) with a temperature real-time control system and analyzes the variation in stress and strain and the energy dissipation characteristics of coal during the dynamic load process. The expression equation of dissipated energy of coal at different scales is established, and the judgment conditions of the macroscopic mechanical behavior of coal are analyzed theoretically. The stress curves show a multi-stress peak phenomenon when the coal samples are subjected to different temperatures and dynamic loads, and the coal’s dynamic stress and temperature show a polynomial fitting relationship at different stages. When the coal sample is subjected to temperature and dynamic load, the macroscopic changes in incident energy, reflected energy, and dissipated energy are consistent; that is, various energies gradually increase to a fixed value and tend to stabilize with the time of stress wave action. The transmission energy exhibits a rising trend in correlation with the duration of the dynamic load action, but the value is less than 0.1 J. The growth gradients of the different energies, in descending order, are: the growth gradient of incident energy, reflection energy, dissipation energy, and transmission energy. The energy inflection point appears at 60 °C. Based on the linear elastic fracture mechanics and damage mechanics theories, the expression for coal energy dissipation from the nanoscale to the microscale is established, and the relationship between energy dissipation and macroscopic mechanical behavior response of the coal samples is analyzed. The main physical components of the coal sample are calcite and kaolinite. Within the temperature range of 18–60 °C, the macroscopic failure form of the coal is horizontal tensile failure. The study results are introduced into dynamic disaster prevention and control and the surrounding rock system stability evaluation in deep mines.

1. Introduction

There are various risks in the process of coal mine production safety, such as gas explosions, coal rock dynamic disasters, etc. The scale of the enterprise, the number and quality of employees, the effective monitoring and early warning of dynamic disasters, and the effective implementation of operating procedures all significantly affect the risk level of coal mines. Cameron A. Beeche et al. developed a new calculation method for coal mine risk analysis mechanism, and the risk level of the mine has been quantitatively evaluated [1]. In the risk of mine dynamic disasters, rock burst is a common rock dynamic failure phenomenon in the fields of tunnel engineering and mining engineering [2,3], and with the deep development of coal resources, its frequency and intensity are increasing day by day, which seriously affects the safe mining of coal. From the analysis of rock mechanics, rock burst is a nonlinear dynamic process in which energy accumulates in a stable state and releases in an unstable state during the deformation and failure process of a coal and rock mass system under specific geological occurrence conditions under the influence of mining. It comprehensively reflects the external load environment, internal structure, tectonics, as well as the physical and mechanical characteristics of the systems [4]. So, it is particularly indispensable to analyze the dynamic behavior of coal under the effect of the surrounding rock temperature, static load, and dynamic load when discussing the formation process of rock burst in deep mines.

In the research of coal impulse mechanics, D.J. Frew put forward the SHPB technology method for testing brittle materials with cracks [5], and then, Chinese scholars Shan Renliang and Gao Wenjiao analyzed the stress and strain time-history characteristics of anthracite under dynamic load and found that the anthracite’s uniaxial dynamic stress was staged, and the original elastomeric modulus and ultimate strength were positively correlated with impact velocity [6,7]. Liu Xiaohui et al. examined the changes in stress, strain response, and the energy dissipation rate of coal and rock under the condition of a strain rate of 44–149 s⁻¹ using SHPB experimental installation with a diameter of 75 mm [8]. Liu Shaohong et al. analyzed the variation characteristics of coal strength under different combined static and dynamic loads. The increase in dynamic load (120–290 MPa) led to the linear enhancement of coal strength in the same static load, while the increase in static load (0, 3, 10, 18 MPa) led to an increase in coal strength first and then a decrease in the same dynamic load [9]. Yixin Zhao et al. conducted impact splitting experiments on coal samples and analyzed the correlation between coal strength and impact velocity and dynamic strain [10,11]. Based on the engineering background of the 79Z6 working face of the Qitaihe Taoshan coal mine, He Jiang et al. examined the correlation between dynamic load stress, static load stress, and thickness in regions prone to rock bursts in thin coal seams. The thickness of coal becomes thinner and the stress gradient increases sharply [12]. Li Ming et al. conducted impact experiments on coal at different impact pressures (0.30, 0.35, 0.40, 0.45, 0.50, 0.55 MPa) by using the SHPB trail system. The dynamic elastic modulus and peak stress of coal exhibited a logarithmic increase as the rate of strain increased. As the rate of strain increases, the proportion of stress–strain curve compaction stage to pre-peak curve decreases gradually, and the proportion of elastic stage to pre-peak curve increases gradually [13]. Li Chengwu et al. used LS-DYNA R8.0 software to numerically simulate and examine the dynamic damage of coal and rock. The primary damage mode of the samples is axial splitting [14]. Wang Dengke et al. analyzed the dynamic mechanical response of coal samples in the Shanxi Weiding mine when the impact pressure was 0.4–0.6 MPa, and found that the compressive strength and elastic modulus had a typical strain rate effect [15,16]. Yanbing Wang et al. discussed the influence of different load velocities and bedding angles on the dynamic mechanical parameters of semicircular bending coal samples [17]. Pan Junfeng et al. experimentally studied the dynamic failure characteristics of coal samples under different impact inclinations. The different abilities of accumulating elastic energy in coal leads to different failure processes, forms, and acoustic emission response laws [18]. Guo Deyong et al. analyzed the variation characteristics of the impact kinetic parameters of coal at different impact velocities (2.058–9.078 m/s). The coal samples at various impact loads have obvious segmented mechanical response characteristics [19]. Zhang Yuxu et al. researched the impact kinetic parameter characteristics of coal samples at various strain rates. The escalation in strain rate induces a shift in the coal’s dynamic response from hardening to softening [20]. Wang Enyuan et al. investigated the dynamic response and its evolution in triaxial coal under dynamic load. The triaxial dynamic compressive strength and strain demonstrate a strong correlation with the average strain rate [21]. Wang Haibo et al. experimentally studied the dynamic stress and strain response of coal samples under radial free and passive confining pressure when the impact air pressure is 0.15, 0.2, 0.3, 0.4, or 0.5 MPa. Under the radial free state, the hard coal’s peak stress and strain rate are positively correlated, while in the passive confining pressure state, the hard coal’s peak stress and strain rate are negatively correlated [22]. Wang Wen et al. analyzed the triaxial stress transformation characteristics of saturated coal samples by using the true triaxial SHPB experiment system, and analyzed the triaxial stress variation effect on the dynamic strength of coal samples [23]. Zhenhua Yang et al. analyzed the dynamic stress–strain changes in two types of coal with different outburst tendencies, and their stress–strain curves exhibited stage distribution characteristics [24]. Liang Weimin et al. conducted experimental analysis on the micro-pore size distribution of coal samples following impact tests at velocities ranging from 0 to 6.62 m/s. As the impact load increased, there was a decreasing trend observed in the total pore volume [25]. Xianjie Hao et al. analyzed the prestress and loading rate effect on the coal’s dynamic tensile strength. The coal’s dynamic indirect tensile strength has different trends under different prestresses [26]. J. Li et al. experimentally studied the variation characteristics and failure types of dynamic parameters of coal samples with different stress loading angles corresponding to the original bedding under the condition of biaxial stress under different impact velocities (10, 13, 17, 21 m/s). When the biaxial stress is consistent, the peak stress is positively related to the impact velocity, but the stress growth rate decreases [27]. Jiao Zhenhua et al. studied the response between coal’s dynamic stress and strain under different impact pressures (0.25, 0.30, 0.35, 0.40, 0.45, 0.50 MPa) under passive confining pressure conditions, and the peak stress and strain rate showed a linear significant correlation [28]. Fang Shuhao et al. simulated and studied the critical effective stress, strain change, and energy change in coal and rock under five different dynamic loads, and found that the change in load change rate has a significant influence on the failure type of the coal and rock [29]. Zheng Yu et al. conducted a uniaxial impact test on coal and rock at a dynamic strain rate from 20 s⁻¹ to 100 s⁻¹ and obtained a dynamic stress and strain response curve. When the strain rate is greater than 50 s⁻¹, the dynamic compressive strength increases significantly and the dynamic elastomeric modulus is between 2 and 13 GPa [30]. Zhao Hongbao et al. investigated the damage process and surface crack evolution of coal and rock under local static load and constant and increasing impulse cyclic loading using a constrained pendulum impact dynamic loading device [31]. Sun and Wu et al. researched the impact experiments of coal samples with aspect ratios of 0.5, 0.6, 0.8, and 1.0 with triaxial dynamic loading [32]. Kai Wang et al. experimentally studied the impact kinetic parameters and failure types of mortar-wrapped coal samples. Their dynamic strength has a strain rate effect, and the failure mode is characterized by axial splitting of the external mortar, and then they are completely destroyed with the increase in dynamic load [33]. Yang Ke et al. experimentally studied the mechanical parameters and failure types of coal and rock under the action of dynamic load (0.7 MPa) and a true triaxial single-faced empty state [34]. The dynamic parameter characteristics of cylindric coal specimens with a diameter of 75 mm under six impact grades and four aspect ratios were studied [35]. Xiaojun Feng et al. discussed the mechanic and acoustic launch response properties of the coal dynamic fracture process [36]. Kong Xiangguo and others studied the impact failure form and gas launch laws of gas-bearing coal under true triaxial conditions. Under the experimental conditions of low confining pressure or high impact load, gas-bearing coal showed crushing failure, while under other experimental conditions, coal samples showed an axial tensile failure mode [37]. Shen Rongxi et al. conducted research on the peak stress and strain of the dynamic parameters of coal samples under uniaxial and triaxial static load prestress and the triaxial static load prestress and strain rate [38]. Wang Lei et al. studied the dynamic stress and strain characteristics of a coal body when the gas pressure was 0, 0.5, 1.0, and 1.5 MPa. The peak stress had an exponential negative correlation with the initial gas pressure [39].

The high-stress and high-temperature environment in which the deep coal rock mass is located affects the dynamic properties of the coal rock mass, thereby significantly increasing the risk of dynamic disasters during coal mining. According to the current literature search results, scholars from various countries have conducted some experimental research on the dynamic response law of coal under temperature and dynamic and static load combinations. At present, most coal mines in China have a mining depth of over 800 m, and the ground temperature of the coal mines is close to 50 °C. As the mining depth increases, the temperature continues to increase. In this paper, an array of dynamic experiments on coal under normal temperature ~60 °C were carried out via a temperature and dynamic and static load coupling experimental system. The dynamic response and morphological change characteristics of the damage process are discussed, which provides a reference for the stability assessment of the surrounding rock system in deep mines.

2. Experimental System and Method

2.1. System Introduction

The experiment adopts the SHPB system developed by the China University of Mining and Technology, which combines dynamic load and temperature effects. The structural framework is shown in reference [40], in which the diameter of the impact bar in the system is 100 mm, the ultra-high-speed dynamic strain gauge is 8-channel LK2107B, and the high-speed camera is a Phantom Veo710 (Vision Research Inc., Wayne, NJ, USA). The installation of the experimental system is shown in Figure 1. This system can provide the real-time temperature field environment and temperature control required for coal dynamics experiments.

Aiming at the thermal effect of the interface at which the sample and the rod meet, Lindholm and Bacon et al. [41,42] studied the relationship between the temperature and wave impedance of the rod and the length of the guide rod when the temperature at the end of the rod is 950 °C. When the distance between the measuring position and the end of the rod is more than 1400 mm, the temperature and wave impedance of the elastic rod hardly change. The coal impact experiment conducted in this paper shows that the distance between the strain gauges set on the incident and transmission rods and the heating rod end was 2500 mm, which met the experimental requirements.

2.2. Coal Sample Preparation

The coal samples were taken from the Hengda coal mine, which is located in the southwest direction of Fuxin City. The main coal seam being mined is the Taixia coal seam, with an average mining depth of about 900–1000 m. The average thickness of the Taixia coal seam is 10 m, and there are relatively more and thicker interbedded layers in the coal seam, with significant changes in lithology. Generally, it is composed of siltstone, fine sandstone, and locally medium sandstone or conglomerate, with a thickness generally ranging from 0.06 to 3.5 m. The sampling location of the coal seam is the 5315 working face. According to the identification standards for coal body impact tendency [43] GB/T25217, the coal sample taken has a strong impact tendency. The coal samples were processed into a cuboid (40 × 40 × 80 mm), and the surfaces of the samples were polished to ensure that both ends were flat. Some samples are shown in Figure 2. From Figure 2, there are obvious crack defects and bedding structures on the surface of some coal samples.

2.3. Basic Properties of Coal Samples

Table 1. Physical property parameters of coal sample foundation.

SN	Experimental Type	Size (mm)	Density (kg/m3)	Wave Velocity (m/s)	Preset Axial Pressure (MPa)	Temperature (°C)	Compressive Strength (MPa)
1	Uniaxial compression	40.07 × 39.95 × 80.10	1320	1695	-	18	8.01
2		39.87 × 40.14 × 80.15	1304	1406	-	18	9.12
3		40.02 × 40.19 × 80.28	1287	1352	-	18	7.29
4		40.15 × 40.42 × 80.16	1370	1434	-	18	5.88
5		39.87 × 40.12 × 80.11	1281	1528	-	18	6.39
6	Dynamic and static load combination experiment	39.98 × 39.62 × 80.08	1372	1586	1.5	18	-
7		40.12 × 40.52 × 79.90	1308	1689	1.5	30	-
8		40.08 × 40.03 × 79.98	1271	1329	1.5	40	-
9		40.49 × 40.13 × 79.32	1424	1304	1.5	50	-
10		39.64 × 40.28 × 80.18	1334	1672	1.5	60	-

reflects the size of coal samples during static and dynamic load experiments. The density varies between 1271 and 1424 kg/m³, and the standard deviation is 45.96. The wave velocity fluctuates between 1304 and 1695 m/s, and the standard deviation is 146.38. The uniaxial compression strength under static load is between 5.88 and 9.12 MPa, with an average strength of 7.34 MPa.

3. Experimental Results of Coal Sample Dynamic Load

At the time that the coal sample is under the impact test under the combined effect of dynamic load and temperature, the preset constant temperatures are the normal temperature (18 °C), 0 °C, 40 °C, 50 °C, and 60 °C; the constant temperature action time is 4 h; the axial static load is set to 1.5 MPa; the air chamber pressure is 0.05 MPa; the impact speed is preset to 1.00 m/s; and the actual impact speed is 1.14–1.40 m/s. The experimental results are shown in Figure 3.

Figure 3 shows the dynamic stress–strain of coal at different real-time temperatures. The increase in the heating temperature of coal leads to a decrease in the first and second peak stresses of the coal, and the stress peak in the elastic section does not follow a clear temperature pattern. The coal sample at room temperature (18 °C) has an obvious compaction section under dynamic load, and its stress value is 4.09 MPa, accounting for about 20.51% of the first peak stress; then, it enters the stage of elastic–plastic deformation, and the first peak stress is increased by 170% compared with the average strength of the static load. However, under the influence of thermal stress, coal samples under other temperature conditions enter the elastic deformation stage directly under dynamic load, and as the strain increases, the stress drops rapidly after the multiple peaks appear, showing the phenomenon of multi-peaks, while reference [36] shows that the stress of coal under dynamic load at room temperature has a “double peak” phenomenon. The multi-peak phenomenon of dynamic stress of coal samples may be caused by the fact that during the impact process, the stress wave partially transmits through the coal body, and the rest of the stress wave is superimposed with the reflected wave. Additionally, the heterogeneity of the coal sample and the preheating stress lead to the rapid closure or expansion of the internal cracks. The post-peak coal sample still has the ability to resist external loads.

Table 2. Analysis results of dynamic stress section of coal sample.

SN	σC (MPa)	εC	EC (GPa)	σB (MPa)	εB	Secant Elastic Modulus EB (GPa)	σA (MPa)	εA	Secant Elastic Modulus EA (GPa)
6	13.15	0.0047	3.08	19.94	0.0090	2.22	19.84	0.0106	1.87
7	8.08	0.0011	10.45	13.86	0.0065	2.13	16.83	0.0139	1.21
8	8.81	0.0016	7.17	10.25	0.0042	2.42	14.22	0.0059	2.41
9	8.30	0.0024	4.50	-	-	-	14.16	0.0091	1.56
10	5.52	0.0014	4.39	9.99	0.0066	1.51	12.86	0.0158	0.81

analyzes the changes in stress peaks and strains of coal samples at different stages of stress variation from room temperature to 60 °C. Under the effect of temperature and dynamic load, the stress of the elastic section of the coal sample changes from 13.15 MPa to 5.52 MPa, and the elastomeric modulus changes from 3.08 GPa to 10.45 GPa. The variation range of the second peak stress is 9.99 MPa–19.94 MPa, accounting for 72.1–82.4% of the peak stress. The variation range of the first peak stress is 12.86 MPa–19.84 MPa, and the strain variation is 0.0059–0.0158. Statistical analysis of the stress and strain changes at different stages is shown in Figure 4.

Figure 4 illustrates that under various temperatures, the dynamic stress, the second peak stress, and the first peak stress of the elasticity section of the coal sample decrease, and the rate of decrease diminishes. The increase in temperature shortens the elastic stage of the dynamic load process of the coal sample, and its plastic stage is extended. The strain variation range of the first stress peak and the second stress peak increases. Compared with the relevant strain to the first peak stress, the strain corresponding to the second peak stress is between 40 and 85%. The first peak stress of the coal sample was examined. Compared with the dynamic load process at room temperature, the stress of the coal sample at 60 °C decreased by 35%, and the second peak stress decreased by 50%. With the increase in temperature, during the loading process of the coal samples the strain in the elastic section first decreases, then increases, and then decreases again. The strain change law corresponding to the second stress peak is similar, and the temperature values corresponding to the inflection point of the strain change are 30 °C and 60 °C.

Artificial neural networks [44] and other regression analysis methods can be used to analyze the predicted values of different prediction parameters. This paper uses the polynomial fitting analysis method in Origin to analyze the fitting relationship between the dynamic mechanical parameters of coal and temperature.

Table 3. Stress–temperature fitting relation of coal sample during dynamic loading process.

Factors	Fitted Equation	Correlation Coefficient	Number of Points	Standard Error
σC and Temperature	${\sigma }_c=-5.635E-4t^3+0.068t^2-2.67t+42.401$	$R=0.99$	5	0.2467
σB and Temperature	${\sigma }_B=1.185E-4t^3-0.0038t-0.5329t+30.075$	$R=0.99$	4	0.0014
σA and Temperature	${\sigma }_A=-6.792E-5t^3+0.0114t^2-0.7118t+29.439$	$R=0.9$8	5	0.3558
εC/εA and Temperature	$\frac{{\epsilon }_C}{{\epsilon }_A}=-0.0052t^3+0.6264t^2-23.5t+294.2$	$R=0.96$	5	3.6692
εB/εA and Temperature	$\frac{{\epsilon }_B}{{\epsilon }_A}=-0.0092t^3+1.0641t^2-38.047t+478.57$	$R=0.99$	4	0.0462

displays the relationship between the first peak stress, the second peak stress, strain, and temperature, which adheres to a polynomial equation with a correlation coefficient exceeding 0.9.

4. Discussion

4.1. Energy Dissipation

During the process of mining, the mechanical responses and failure forms of coal and rock under different temperatures and dynamic loads are obviously different from the static load experimental process. Xie Heping et al. [45,46] proposed that energy dissipation would lead to rock damage and strength reduction, which eventually led to rock failure. As a special rock with double pore characteristics, coal body can be calculated by using the research results of Lundberg B. [47]. The vicissitudes of incident energy (W_i), reflection energy (W_r), and transmission energy (W_t) of the coal body in the process of dynamic impact are shown in Figure 5. Based on the principle of energy balance, the dissipation energy (W_s) of the coal samples conforms to Formula (1), and the unit volume dissipation energy is introduced to reflect the energy dissipation capacity of the coal body (Formula (2)),

$$w_s=w_i-w_r-w_t$$

(1)

$$E_s=w_s/V$$

(2)

where V is the volume of the coal sample.

In Figure 5, when the coal sample experiences dynamic loading, the incident energy, reflection energy, and dissipation energy gradually increase and stabilize at a set value, while the transmission energy shows a rising trend throughout the duration of the dynamic loading. Among the four kinds of energy, the growth gradient of incident energy > the growth gradient of reflection energy > the growth gradient of dissipation energy > the growth gradient of transmission energy, which is similar to the energy dissipation process of fine sandstone, and the transmission energy is very small, less than 0.1 J. The correlation between incident energy, reflection energy, dissipation energy, and transmission energy at different temperatures is analyzed (Figure 5f). When the temperature of the coal sample is room temperature, the incident energy is 52.31 J, the reflection energy is 42.36 J, the dissipated energy is 9.85 J, accounting for 18.83%, and the dissipated energy rate is 0.0777 J/dm³. When the coal sample temperature is 30 °C, the incident energy is 35.96 J, the reflected energy is 27.94 J, the dissipated energy is 7.94 J, accounting for 22.08%, and the dissipated energy rate is 0.0611 J/dm³. When the coal sample temperature is 40 °C, the incident energy is 13.69 J, the reflected energy is 7.14 J, the dissipated energy is 6.51 J, accounting for 47.55%, and the dissipated energy rate is 0.0507 J/dm³. When the coal sample temperature is 50 °C, the incident energy is 54.20 J, the reflected energy is 51.26 J, the dissipated energy is 6.51 J, accounting for 47.55%, and the dissipated energy rate is 0.0507 J/dm³. When the coal sample temperature is 60 °C, the incident energy is 35.88 J, the reflected energy is 21.53 J, the dissipated energy is 14.11 J, accounting for 39.33%, and the dissipated energy rate is 0.1103 J/dm³. From the perspective of energy dissipation per unit volume of coal, as the temperature increases, the energy dissipation of the coal first decreases and then increases, and the energy inflection point occurs at a temperature of 60 °C. The abnormal change in energy reflects that thermal damage causes the disorder of pore and crack propagation in the coal body, which further leads to the increase in internal anisotropy and relatively weakens the ability of the coal body to resist external loads.

4.2. Energy Dissipation Model

From the perspective of energy dissipation and crack propagation, the macroscopic failure of coal is essentially a process in which small-scale cracks continue to expand and aggregate to form large-scale cracks under the action of energy. At the nanoscale, the internal crack propagation of coal can be considered as a fracture phenomenon caused by the atomic bond oscillating and jumping to a far position (exceeding the critical position) at the equilibrium position under the action of a dynamic load and thermal stress. At the nanoscale, from the perspective of atomic lattice dynamics, when coal atoms absorb energy, energy can promote atomic transitions across a certain height of energy barrier, corresponding to the crack propagation phenomenon at the nanoscale. The rate process theory is introduced to characterize the internal crack propagation of coal at the nanoscale. The rate process theory suggests that under external temperature and dynamic load excitation, a few atoms located at the bottom of the potential well can absorb sufficient energy, produce transition phenomena, and trigger crack propagation. Based on linear elastic fracture mechanics, the energy dissipation ∆E required for isotropic propagation of plane cracks at the nanoscale can be characterized by Formula (3), and the crack propagation rate [48] can be characterized by Formula (4):

$$∆E=L_a\eta {\mathcal{G}}_a$$

(3)

$$v=L_{a}f\approx \frac{∆E{L}_a}{h}{e}^{-\frac{E_0}{kT}}$$

(4)

In the formula, ${\mathcal{G}}_a$ is the energy release rate of the nanoscale, η is the crack geometry constant, $L_a$ is the crack propagation distance of the nanoscale, $f$ is the net barrier crossing frequency, $k$ is the Boltzmann constant, h is the Planck constant, T is the temperature, and E₀ is the energy barrier height.

Based on damage mechanics, when the crack propagation distance is $L_a$, the corresponding damage increment is $D_b$ and the energy dissipation is denoted by $∆E_b$. According to the research results of Ding Z.D. [49] and others, the dissipation energy described by the two processes is equal, that is,

$$∆E=∆E_b=V_b{\int }_L^{L+L_a}\dot{E}\mathrm{d}b\approx D_b{V}_b\dot{E}$$

(5)

where $V_b$ is the damage volume and $\dot{E}$ is the damage energy release rate.

Combining Formulas (3)–(5), the energy dissipation rate of single crack propagation can be obtained:

$$\dot{E_{f,1}}={\mathcal{G}}_{a}v=\frac{{∆E}^2}{\eta h}{e}^{-\frac{E_0}{kT}}$$

(6)

The general number of cracks contained in the microscopic unit of the coal body is described by the crack hierarchy model [47]. Assuming that there are s levels from the nanoscale to the microscale, and each level contains n_i cracks, the total number of cracks N [50] can be expressed as

$$N=n_1{n}_2\dots n_s$$

(7)

where n_i is the number of cracks in each level, N is the total number of cracks.

Then, the total cumulative dissipation energy can be expressed by the following formula:

$$E_{f,\tau }={\int }_0^{\tau }\dot{E_{f,s}}dt={\int }_0^{\tau }\frac{N\overline{\dot{E_{f,1}}}D}{{\overline{V}}_b}$$

(8)

where $\tau$ is time, $\overline{\dot{E_{f,1}}}$ is the average of the crack energy expansion rate after statistics, and D is the damage factor.

Combining Formulas (6)–(8), we can obtain:

$$E_{f,\tau }={\int }_0^{\tau }\frac{N{D}_b^2{\overline{V}}_b}{\eta h}{e}^{-\frac{E_0}{kT}}Ddt$$

(9)

When the coal body is under dynamic load, the dissipation energy generated by the coal body during the duration of the stress wave is:

$$w_s=A_{0}CE\left({\int }_0^{\tau }{\epsilon }_i^2\left(t\right)dt-{\int }_0^{\tau }{\epsilon }_r^2\left(t\right)dt-{\int }_0^{\tau }{\epsilon }_t^2\left(t\right)dt\right)$$

(10)

where C is the wave velocity of the stress wave in the wave guide rod; $A_0$ is the cross-sectional area of the sample; E is the elastic modulus of the wave guide rod; and ${\epsilon }_i$, ${\epsilon }_r$, ${\epsilon }_t$ correspond to the incident strain, reflection strain, and transmission strain, respectively.

Therefore, whether the coal body will produce macroscopic failure behavior can be judged by Formula (11):

$$\left\{\begin{array}{c}{w}_{s,t}>E_{f,t} \\ w_{s,t}\le E_{f,t}\end{array}\right. 0<t\le \tau$$

(11)

4.3. Development Morphology of Surface Cracks in Coal Samples at Different Temperatures

According to Figure 5, when the coal sample is subjected to dynamic load, the action time of stress waves in the coal body is within 500 μs. High-speed photography was used to obtain the dynamic load failure process of the coal samples, and typical damage and failure forms of the coal samples under dynamic load were selected, as shown in Figure 6. The failure of the coal samples under different temperatures and dynamic loads is mainly in the form of horizontal tensile failure. The direction of the main crack is consistent with the direction of stress wave propagation, and the main crack is basically horizontally developed and runs through the sample. Some of the crack initiation positions are located near the end face of the incident rod (Figure 6. 18 °C, 30 °C, 50 °C), while others are located in the middle of the sample (Figure 6. 40 °C, 60 °C), which is mainly related to the distribution of surface defects in the coal sample. The physical composition is related to the development of internal horizontal bedding. As the temperature increases, the number of secondary cracks near the main crack on the surface of the sample increases, and the effect of temperature degradation on the coal body is significant, weakening the macroscopic mechanical behavior of the coal body.

4.4. Physical Composition of Coal Samples

Figure 7 shows the X-ray spectra of coal samples under dynamic loading at different temperatures. The mineral components in this group of coal samples include calcite (CaCO₃), quartz (SiO₂), and kaolinite (Al₂Si₂O₅ (OH)₄), with the main components being calcite and kaolinite, with a mass fraction of over 90%. Based on the information of the X-ray diffraction patterns, within the temperature range of 18 °C~60 °C, the mineral composition inside the coal sample is stable, mainly undergoing physical changes manifested by the precipitation of free water in mineral crystals. At the same time, calcite often contains elements such as Mg, Fe, Mn, etc. In the coal sample, it coexists with the coal matrix in a parallel or approximate form in the form of flakes (plates) or fibers, forming layered weak planes and joint planes. This is one of the reasons why the macroscopic cracks and fractures on the surface of coal samples under temperature and dynamic and static loads are characterized as river-like fracture morphologies.

5. Conclusions

In this paper, the variation characteristics of dynamic stress and strain of coal samples under temperature and dynamic and static loads are studied experimentally. The correlation between the first peak stress, the second peak stress, and the temperature during the loading path of coal samples is established. The energy dissipation expression of the coal samples at the nanoscale is constructed, and the conditions for the macroscopic failure behavior of coal are determined by comparing with the energy dissipation under the effect of the stress wave.

• When the coal body at different temperatures is under dynamic load, the stress curve characterizes the phenomenon of multiple stress peaks. The reason may be that during the impact process, the stress wave partially transmits in the coal body, and the remaining stress wave is superimposed with the reflected wave. Coupled with the heterogeneity of the coal sample and the preheating stress, the internal crack is caused by the extremely fast closure or expansion.
• Under nearly identical bullet incident velocities, the elastomeric stage of the coal sample decreases while the plastic deformation stage increases with rising temperature. Additionally, both the first peak stress and the second peak stress decrease to varying extents.
• When the coal test specimen is under a dynamic load, the abnormal changes in incident energy, reflected energy, and dissipated energy are at 60 °C, and the thermal damage causes the disorder of the internal pores and crack propagation of the coal, which weakens the ability of the coal to resist external loads.
• The macroscopic cracks on the surface of coal samples during impact are characterized by horizontal development, which is consistent with the direction of stress wave propagation. The reasons for crack propagation are analyzed from the aspects of coal physical composition and external joints.

With the increase in coal mine ground temperature and surrounding rock stress field, the risk of coal rock dynamic disasters significantly increases. The next step is to conduct dynamic stress and surrounding rock temperature testing and analysis during the mining process at the Hengda coal mine and analyze their correlation with the manifestation of onsite dynamic disasters. At the same time, in situ core technology should be used to truly study the dynamic experiments of coal under in situ stress and temperature field conditions and to provide basic data for the prevention and control of coal rock dynamic disasters.