Investigation of the Energy Evolution of Tectonic Coal under Triaxial Cyclic Loading with Different Loading Rates and the Underlying Mechanism

Abstract

It is of great significance to ascertain the mechanical characteristics and deformation laws of tectonic coal that is under complex stress conditions for safe production, but the targeted research in this area is still insufficient at present. This paper performed triaxial tests under cyclic multi-level loading at different rates by using an MTS-815 Rock Mechanics Testing System. The strain characteristics, elastic modulus and energy evolution were obtained in order to explore the effects of the mechanism of loading rate on the evolution of deformation and energy parameters of tectonic coal. The results showed that the irreversible strain and plastic energy increased exponentially with the increase in the deviatoric stress, but the growth rate decreased with the increase in loading rate. Furthermore, the elastic strain increased linearly and the growth rate was essentially unaffected by the loading rate. During the compaction stage, the variation of each parameter was not sensitive to the loading rate; during the elastic and damage stage, the rate increase inhibited secondary defect propagation and improved rock strength. In addition, the stepwise and cumulative energy ratio was defined in order to describe the energy distribution during cyclic loading and unloading. It was found that the decrease in the loading rate was beneficial to the transformation of the total energy into plastic energy. The elastic modulus was the most sensitive to sample damage, but the energy density evolution was able to be used to describe the deformation damage process of tectonic coal in more detail. These findings provide important theoretical support for the tectonic coal deformation law and action mechanism in the damage process that occurs under complex stress conditions.

1. Introduction

As it is affected by tectonism such as faulting and slipping, tectonic stress destroys the original coal structure. This results in tectonic coal, which has wide distribution globally and especially in China. Because of its gas enrichment, low permeability, and strength characteristics, tectonic coal has long been known as the major cause of coal and gas outbursts [1,2,3]. Previous studies on the porosity [4,5], pore structure [6], permeability, and adsorption–desorption characteristics [1,7,8] of tectonic coal have been carried out. Based on reconstituted tectonic coal samples, the preparation information for which is shown in

Table 1. Preparation information for tectonic coal.

Particle Size (mm)	Stress	Additive	Refs.
<0.2	2.76–19.9 MPa	/	[9]
0.18–0.425	100 MPa	Water	[10]
<0.25	48 MPa	Water	[11]
0.38–0.83	100 MPa	Water	[12]
0.25–0.425	100 MPa	Water	[13]

, many scholars have performed uniaxial and triaxial tests in order to study the mechanical properties of tectonic coal [9,10,11,12,13]. However, in concrete engineering (especially in the tectonic coal mining processes), the change in stress is not monotonic [14,15,16]. The mechanical response and deformation characteristics of rocks that are under cyclic loading and unloading are completely different from those that are under monotonic loading [17,18,19]. Hence, important theoretical significance and engineering value exist for the study of the strength and deformation of rocks that are under complex stress conditions [20,21,22,23].

Currently, cyclic loading and unloading testing in the laboratory constitutes an effective method for studying the rock deformation characteristics of complex stress paths and many scholars have conducted relevant tests and theoretical studies resulting in numerous achievements [24,25,26,27,28,29]. The results of these tests have shown that the confining pressure and the amplitude, frequency, and peak value of the axial stress are important factors affecting the mechanical behaviour of rocks [14,26,30]. Wang suggested that the damage time for coal would be delayed by an increase in frequency and that the peak stress of loading had the most significant influence on coal damage [31]. Taheri performed cyclic loading experiments on lignite and observed that, with an increase in confining pressure, the specimen damage under axial loading decreased [32]. Taking the initial properties of the coal and rock and the stress environment into account, Zhang carried out cyclic loading experiments with confining pressures and found that the Poisson’s ratio and deformation of coal increased non-linearly with confining pressure [33]. Roberts, L. A. et al. carried out cyclic triaxial creep tests under various load paths and found that salt that was subjected to cyclic loading in extension was no more prone to damage than if it were subjected to compression [34]. Voznesenskii, A. S et al. established the interrelation between the acoustic emission (AE) signals and mechanical strength under cyclic loading and unloading [35]. Based on a cyclic loading experiment, Shkuratnik V. L. et al. found that stress memory in the characteristic of AE that were occurring under constant temperatures was steady [36]. Alexander Lavrov suggested that the loading rate had no significant effect on the Kaiser effect, which might be because the rate that is found in laboratory tests is higher than the in situ loading history [37]. Olovyannyy, A., and Chantsev, V. used rock salt in order to research the models of deformation and failure under various cyclic loading conditions and obtained the parameters that were needed for solving engineering problems in the context of mining [38,39].

From the perspective of thermodynamics, the essence of material damage is driven by energy, so the process of rock deformation and failure is an irreversible process of energy consumption [40]. In recent years, energy transformation and its evolution have gradually been favoured by scholars in the study of rock damage and a series of failure criteria and methods for the stability evaluation for rocks were put forward [41,42,43,44]. Zhang and Lindqvist, M. investigated the variation characteristics of rock energy dissipation in the whole process under dynamic and instantaneous dynamic effects [45,46]. Gaziev suggested that the primary element begins to fail when the strain reaches its limit value and the continuous accumulation of strain energy is a necessary condition for material failure [47]. Xie et al. and Xue et al. noted that energy dissipation leads to material deterioration and energy release results in abrupt structural collapse [48,49]. Li et al. obtained the influence behaviours of the initial confining pressure and unloading rate on the strain–energy conversion of rocks during the deformation and failure processes [50]. Peng et al. performed triaxial compression experiments on coal under different confining pressures and analysed the relationship between the failure modes and energy conversion [51]. Jiang et al. studied the AE characteristics and energy dissipation of gas-containing coal under tiered cyclic loading and unloading and established a new equation for the variable damage of coal with dissipation energy [52].

The extraction of coalbed methane from the tectonic coal reservoir is a common method that is used in order to realize the efficient utilisation and safe exploitation of resources, but borehole fractures that are caused by tectonic coal reservoir deformation are the main constraint for coal and coalbed methane production [53]. However, a majority of the research has focused on hard rocks, while research on tectonic coal is still insufficient. By utilising the MTS-815 test system, tectonic coal was subjected to a cyclic loading and unloading test under triaxial compression with different loading rates. On this basis, the damage characteristics and energy evolution of the tectonic coal during cyclic loading were explored. Moreover, the action mechanism of loading rate in the damage process was analysed. The research results provide theoretical support for methods for inducing or preventing the destruction or damage of tectonic coal.

2. Experimental Design and Schemes

2.1. Specimens

The pulverised coal that was used in this experiment came from the Huainan mining area in Anhui Province, China. The tectonic coal that was sampled underground was screened in order to obtain its particle size distribution (calculated by mass percentage), as shown in Figure 1.

Due to its low strength and weak cohesion, tectonic coal is difficult to obtain in the form of an intact core specimen. Therefore, reconstituted coal specimens have been widely used in order to investigate the mechanical and permeability properties of tectonic coal. The main process for preparing these specimens is as follows: as shown in Figure 1, the mixed particle size pulverised coal was prepared, 10% water (mass percentage) was added, and petroleum jelly was applied to the inner wall of the mould in order to reduce the friction force. Following this, the pulverised coal was put into a steel mould and pressed at 15 MPa for 2 h in order to make a cylindrical sample with a diameter of 50 mm and a height of 100 mm, as seen in Figure 2 and Figure 3. Formed pressure was provided by an electric–hydraulic serving compression machine. Subsequently, the samples were put in an oven and the weight of each sample was measured every 12 h until the change in weight was less than 0.2% of the initial weight. This process lasted approximately 96 h. Uniaxial compressive strength that was measured at 0.37 MPa, tensile strength that was measured at 0.07 MPa, and an internal friction angle of 29.2° proved that the samples had the characteristics of low strength and weak cohesion.

2.2. Test Equipment

This study used the MTS815 test system for rock mechanics from the State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou, China, as shown in Figure 4. The equipment consisted of three parts, including loading, testing, and control systems. The equipment could be used for loading and unloading testing under multiple control modes of force, stress, displacement, and strain. The rigidity of the test machine was 10.5 × 109 N/m, the maximum axial load could reach 1700 kN, the confining pressure was lower than or equal to 45 MPa, and the limiting data acquisition frequency was 2 × 104 times per second.

The CT scans were performed using the high-resolution 3D X-ray microscopy imaging system from the Advanced Analysis and Computation Center of China University of Mining and Technology, Xuzhou. The X-ray tube voltage was 30–60 KV, the X-ray power was 2–10 w, and the resolution ratio was 50 μm.

2.3. Test Procedure and Schemes

Triaxial cyclic loading and unloading procedures taking place under stress gradients were conducted in this test, with a confining pressure of 12 MPa. The axial upper limit of stress increased by 2 MPa per gradient and the lower limit of axial deviatoric stress remained constant at 1 MPa (as Figure 5). The cyclic loading and unloading was performed at different rates (0.05 MPa/s, 0.1 MPa/s, 0.15 MPa/s, 0.2 MPa/s, and 0.25 MPa/s) and the corresponding samples were named NO. i (i = 1,2,3,4,5, respectively).

2.4. Calculation of Energy Parameters

It was assumed that there was no heat exchange between the rocks and the experimental environment during the test [54]. According to Xie et al., energy input, storage, and consumption all conform to the following formulas [48]:

$$\{\begin{array}{l}U=U_e+U_d\\ U={\displaystyle \int \mathit{udV}}\\ U_e={\displaystyle \int u_e\mathit{dV}}\\ U_d={\displaystyle \int u_d\mathit{dV}}\end{array}$$

(1)

where U, U_e, and U_d are the total strain energy, elastic strain energy, and plastic strain energy, respectively; u, u_e, and u_d are the total strain energy density, elastic strain energy density, and plastic strain energy density of the rocks, respectively; and V denotes the rock volume.

As shown in Figure 6, u is equivalent to the sum of the areas that are encircled by the X-axis and loading curve; u_e is equal to the sum of the areas encircled by the X-axis and unloading curve, in numerical terms; and u_d is equivalent to the sum of the areas that are formed by the loading and unloading curves and abscissa axis, as per the following formulas [30]:

$$u={\displaystyle {\int }_{{\epsilon }_1}^{{\epsilon }_{\mathit{max}}}\sigma }d\epsilon ={\displaystyle \sum _{i=1}^N\frac{\left({\epsilon }_i-{\epsilon }_{i+1}\right)\left({\sigma }_i+{\sigma }_{i+1}\right)}{2}}$$

(2)

$$u_e={\displaystyle {\int }_{{\epsilon }_2}^{{\epsilon }_{\mathit{max}}}\sigma }d\epsilon ={\displaystyle \sum _{n=1}^N\frac{\left({\epsilon }_n-{\epsilon }_{n+1}\right)\left({\sigma }_n+{\sigma }_{n+1}\right)}{2}}$$

(3)

$$u_d=u-u_e=\frac{{\displaystyle \sum _{i=1}^N\left({\epsilon }_i-{\epsilon }_{i+1}\right)\left({\sigma }_i+{\sigma }_{i+1}\right)}-{\displaystyle \sum _{n=1}^N\left({\epsilon }_n-{\epsilon }_{n+1}\right)\left({\sigma }_n+{\sigma }_{n+1}\right)}}{2}$$

(4)

where ε₁ and ε₂ indicate the strains that were apparent when the stress σ equalled σ_min in loading and unloading processes; and ε_max denotes the strain when the stress σ equalled σ_max on the loading curve. Moreover, ε_i, ε_i+1, ε_n, and ε_n+1 indicate the strains at an integral step, while σ_i, σ_i+1, σ_n, and σ_n+1 denote the stresses at an integral step. The units of ε and σ are “ [-]” and “MPa”, respectively.

3. Results

3.1. Full Stress–Strain Curves

Figure 7 presents the curves that were measured under the cyclic loading and unloading processes. Overall, the change in behaviour was the same; as the stress gradually increased, the axial stress–strain curve experienced two stages which were described as “dense–sparse”. With the increase in deviatoric stress, the strain that was generated increased in each cycle and the growth rate accelerated gradually. With the increase in the loading rate, the total axial strains at the end of the cycle were 11.67%, 9.87%, 8.47%, 7.62%, and 6.25%.

As shown in Figure 7f, the total strain in a single cycle increased with the increased deviatoric stress, but the growth rate was inversely proportional to the loading rate. From 0.05 MPa/s to 0.25 MPa/s, the strain in a single cycle increased by an average of 0.0032, 0.0027, 0.0024, 0.0023, and 0.0018 stepwise. The total strain increased exponentially with the deviatoric stress, as shown in

Table 2. Fitting relationships of total strain with cycle numbers under different loading rates.

Loading Rate	Fitting Expression	R2
(MPa/s)
0.05	$\epsilon =0.0016\mathit{exp}0.1601{\sigma }^{\prime }$	0.99
0.1	$\epsilon =0.0017\mathit{exp}0.1505{\sigma }^{\prime }$	0.99
0.15	$\epsilon =0.0016\mathit{exp}0.1457{\sigma }^{\prime }$	0.99
0.2	$\epsilon =0.0015\mathit{exp}0.1461{\sigma }^{\prime }$	0.98
0.25	$\epsilon =0.0018\mathit{exp}0.1319{\sigma }^{\prime }$	0.98

Note: $\epsilon$ = Total strain; ${\sigma }^{\prime }$ = Deviatoric stress.

, and the correlation coefficients were all greater than 0.99. However, with the increase in loading rate, the fitting coefficients decreased from 0.16 to 0.13.

3.2. Elastic Strain and Plastic Strain

Due to the effects of tectonism, the intact coal was destroyed, causing it to be severely crushed or even pulverised [55]. Therefore, even under the lower levels of stress, the tectonic coal exhibited plastic deformation. In this experiment, the irreversible strain was equal to the strain at the end of the cycle minus the strain at the beginning of the cycle and the restorable strain was the strain which rebounded during unloading.

As described in the previous section, the characteristics of the stress–strain curves of all of the samples were consistent. The NO. 4 specimen was taken as an example to illustrate the changing laws of elastic strain and plastic strain in a single cycle, as shown in Figure 8. When the deviatoric stress was less than 5 MPa, the axial plastic strain was slightly higher than the elastic strain. This is because the original sample contained several voids, which compressed rapidly under the low axial stress that was applied during the compaction. As the voids were gradually compacted by deviatoric stress in the range of 7 MPa to 21 MPa, the elastic strain was greater than the plastic strain, therefore accounting for a higher percentage of the total strain. As the axial stress increased above 21 MPa, the growth rate of the plastic strain was much faster than that of the elastic strain. When the deviatoric stress increased from 21 MPa to 27 MPa, the plastic strain increased by 230%, while the elastic strain only increased by about 25%. Finally, the irreversible strain and elastic strain were measured at 2.22% and 0.96%. The sharp increase in the plastic strain that was observed in this stage was caused by the continuous expansion of secondary defects.

As presented in

Table 3. Fitting relationships of stepwise elastic strain and cumulative plastic strain with cycle numbers under different loading rates.

Loading Rate MPa/S	Fitting Expression
	Stepwise Elastic Strain		Cumulative Plastic Strain
	Fitting Expression	R2	Fitting Expression	R2
0.05	${\epsilon }_E=0.039{\sigma }^{\prime }-0.0376$	0.99	${\epsilon }_P=0.158\mathit{exp}0.1601{\sigma }^{\prime }$	0.99
0.10	${\epsilon }_E=0.0418{\sigma }^{\prime }-0.0529$	0.99	${\epsilon }_P=0.171\mathit{exp}0.1505{\sigma }^{\prime }$	0.99
0.15	${\epsilon }_E=0.0409{\sigma }^{\prime }-0.0604$	0.99	${\epsilon }_P=0.163\mathit{exp}0.1457{\sigma }^{\prime }$	0.99
0.20	${\epsilon }_E=0.0388{\sigma }^{\prime }-0.0627$	0.99	${\epsilon }_P=0.148\mathit{exp}0.1461{\sigma }^{\prime }$	0.98
0.25	${\epsilon }_E=0.0389{\sigma }^{\prime }-0.0666$	0.99	${\epsilon }_P=0.184\mathit{exp}0.1319{\sigma }^{\prime }$	0.98

Note: ${\epsilon }_R$—Elastic strain; ${\epsilon }_I$—Cumulative plastic strain.

and Figure 9, the plastic strain increase that occurred with the increase in the deviatoric stress can be fitted with an exponential function, the correlation coefficient of which was greater than 0.98. The fitting coefficient decreased from 0.16 to 0.13 with the increase in the loading rate, indicating that the growth rate of the cumulative plastic strain decreased with the increased loading. An increase in the loading rate inhibited the growth rate of the plastic strain. The elastic strain increased linearly and the correlation coefficients were greater than 0.99. Compared with that of the plastic strain, the fitting coefficient of the elastic strain did not change significantly with the loading rate and the effect was weaker than that of the plastic strain. However, relatively speaking, when the loading rate was less than 0.15 MPa/s, the fitting coefficient was larger, and vice versa. Therefore, for this experiment, 0.15 MPa/s can be regarded as the critical value to define low- or high-speed loading.

3.3. Evolution Trends of the Elasticity Modulus

The elastic modulus can be used to reflect the ability of a material to resist deformation under external forces [56]. In the compression process, generally, the strain state of the rock can be divided into compaction, elastic, plastic, and damage stages [57]. Commonly, the slope of the linear variation section of the stress–strain curve is taken as the elastic modulus of the rock. In this experiment, the stress–strain curve that was gathered during the loading process can be divided into an approximate elastic stage and approximate plastic stage. Therefore, the secant slope of the approximate elastic stage in the loading process was regarded as the elastic modulus.

As presented in Figure 10, the elastic modulus evolution has the following characteristics: first, the voids were compressed during early loading and the elastic modulus increased gradually. Second, as the loading rate changed from slow to fast, the corresponding elastic modulus peaks were measured at 2.18 GPa, 2.20 GPa, 2.27 GPa, 2.44 GPa, and 2.44 GPa. Furthermore, the change curves of the elastic modulus under the different loading rates were approximately parallel. In the compaction stage, there was no obvious regularity in the elastic modulus change with the loading rate. Only after the elastic modulus approached the peak did it show a positive correlation with the loading rate. In addition, the elastic modulus experienced a rising period after which it reached the maximum value, then declined immediately when the loading rate was lower than 0.15 MPa/s. However, when the loading rate was higher than 0.15 MPa/s, the change in the elastic modulus could be divided into three stages. These were the rising stage, stable stage and decay stage. At the point at which the deviatoric stress reached 9 MPa, the elastic modulus was stable at about 2.42 GPa; that was until the deviatoric stress was 17 MPa, at which point the elastic modulus began to decrease gradually.

If the stress corresponding to a significant decrease in elastic modulus was defined as the damage intensity, then 0.15 MPa/s can be considered to be the critical value. The ultimate damage strengths under low-speed loading were 19 MPa, 21 MPa, and 21 MPa; while under high-speed loading, they reached 25 MPa and 27 MPa. Under the low-speed loading conditions, the sample strength did not significantly improve with the increase in loading rate. However, for high-speed loading, the strength of the tectonic coal significantly increased; this is a finding which is consistent with the previous conclusion that high-frequency loading can enhance the strength of rock and inhibit damage propagation.

4. Discussion

4.1. Growth Laws of Strain Energy Density with Different Loading Rates

Through the use of Formulas (2)–(4), u, u_e, and u_d were obtained. Based on the analysis of the strain and elastic moduli, 0.15 MPa/s was used as the demarcation point to define the loading rate. Therefore, the samples with loading rates of 0.05 MPa/s, 0.15 MPa/s, and 0.25 MPa/s were selected to represent low-speed, medium-speed, and high-speed loading, respectively.

The strain energy density evolution curves were plotted using the deviatoric stress as the X-axis and the three strain energy densities as the Y-axis, as shown in Figure 11. Although the loading rates of the tectonic samples were different, the u, u_e, and u_d of the samples rose with the upper limit of the deviatoric stress throughout the whole cyclic loading process. By fitting the above test results through the use of previously mentioned functions, it was found that the total strain energy density and plastic strain energy density increased exponentially with the deviatoric stress, while the elastic strain energy density increased linearly, with a correlation coefficient that was greater than 0.9724, as shown in

Table 4. Variation in strain energy density with loading rate.

Specimen	Total Energy Density	R2	Plasticity Energy Density	R2	Recoverable Energy Density	R2
NO. 1	$u=0.0233\mathit{exp}0.1525{\sigma }^{\prime }$	0.99	$u_d=0.0111\mathit{exp}0.1691{\sigma }^{\prime }$	0.99	$u_E=0.0093{\sigma }^{\prime }-0.0287$	0.99
NO. 3	$u=0.0233\mathit{exp}0.1425{\sigma }^{\prime }$	0.99	$u_d=0.0106\mathit{exp}0.1563{\sigma }^{\prime }$	0.97	$u_E=0.0097{\sigma }^{\prime }-0.0333$	0.99
NO. 5	$u=0.0249\mathit{exp}0.1292{\sigma }^{\prime }$	0.99	$u_d=0.0123\mathit{exp}0.1357{\sigma }^{\prime }$	0.97	$u_E=0.0093{\sigma }^{\prime }-0.0337$	0.99

. In addition, the growth coefficients of u and u_d gradually decreased as the loading rate increased, while the growth coefficient of u_e had no obvious response to the loading rate.

Furthermore, there was evidence that both u_e, and u_d increased slowly at the initial stage and the gap between them was not obvious. However, as the stress gradually increased, the plastic energy density began to increase rapidly and inflection points appeared in the 15 MPa, 17 MPa, and 19 MPa levels of deviatoric stress.

The above analysis shows that an increase in the loading rate not only inhibits the average growth rate of plastic strain energy density, but also elevates the deviatoric stress value that is required for the accelerated growth of the plastic strain energy density. This explains why higher loading rates in cyclic loading and unloading can improve rock strength from the perspective of energy dissipation.

4.2. Laws of Energy Storage and Dissipation

Li et al. studied the energy evolution characteristics of granite under different loading and unloading paths and found that the ratio of dissipated energy could be used to describe the degree of rock deformation and failure [58]. Wang et al. [59,60] established a rockmass damage indicators based on plastic strain work and energy release, by plotted contours of energy components at different damage stages of rockmass in numerical models. To further explore the internal relationships among the three energy densities, the stepwise plastic strain energy ratio (SPER)—${\lambda }_d^i$, stepwise elastic strain energy ratio (SEER)—${\lambda }_{{}_e}^i$, cumulative elastic strain energy ratio (CEER)—${\lambda }_e^n$, and cumulative plastic strain energy ratio (CPER)—${\lambda }_d^n$ were used. These were defined according to the following formulas:

$$\{\begin{array}{l}u=u_e+u_d\\ {\lambda }_{{}_e}^i=\frac{u_{{}_e}^i}{u^i}\times 100\%\\ {\lambda }_{{}_d}^i=\frac{u_{{}_d}^i}{u^i}\times 100\%\end{array}$$

(5)

$$\{\begin{array}{l}{\lambda }_e^n=\frac{{\displaystyle \sum _{i=1}^n}{u}_e^i}{{\displaystyle \sum _{i=1}^n}{u}^i}\times 100\%\\ {\lambda }_d^n=\frac{{\displaystyle \sum _{i=1}^n}{u}_d^i}{{\displaystyle \sum _{i=1}^n}{u}^i}\times 100\%\end{array}$$

(6)

where both i and n are the number of cycles. The stepwise energy ratio reflects the percentage of elastic strain and plastic strain energy for the total strain energy in a single cycle; and the cumulative energy ratio describes the distribution of the total strain energy input from the first cycle to the end of the current cycle, i.e., it is a quantitative characterisation of energy distribution for the current state.

It can be seen from Figure 12 that ${\lambda }_d^i$ increased with the increase in deviatoric stress in the first stage and then decreased with the increase in stress in the second stage; and that the demarcation point for the deviatoric stress became 11 MPa. In the first stage, ${\lambda }_d^i$ had no obvious correlation with the loading rate, but it did have a significant positive correlation with the loading rate in the second stage. I.e., the higher the loading rate was, the higher the recoverable energy ratio. For example, for samples NO. 1, NO. 3, and NO. 5, the peak values of ${\lambda }_d^i$ were 48.40%, 56.05%, and 57.16%, respectively. At the end of the experiment these values reached low points of 13.97%, 18.30%, and 26.18% respectively. The total strain energy density was equal to the sum of the plastic and elastic strain energy densities, so the change mechanism of the plastic strain energy ratio was not found to have been repeated.

Figure 13a–c shows the relationship between ${\lambda }_{{}_e}^i$ and ${\lambda }_d^i$ of the NO. 1, NO. 3, and NO. 5 samples. It can be seen that ${\lambda }_{{}_e}^i$ had increased from 20% to 50% as the deviational stress rose to 7 and that it reached its peak value when the deviatoric stress was 11 MPa. These evolution laws seems to be independent of the loading rate. However, with the increase in the loading rate, ${\lambda }_{{}_e}^i$ occupied a higher proportion for a longer period. For the NO. 1 sample, ${\lambda }_{{}_e}^i$ decreased to 50% when the deviatoric stress was 13 MPa; while for the NO. 3 and NO. 5 samples the corresponding values of deviatoric stress were 17 MPa and 19 MPa, respectively. The evolution law of ${\lambda }_d^i$ was opposite to that of ${\lambda }_{{}_e}^i$. The measurements of ${\lambda }_d^i$ for the NO. 1, NO. 3, and NO. 5 samples were 86.03%, 80.19%, and 73.82%, respectively, when the cyclic loading ended.

Figure 13d–f shows the relationship between ${\lambda }_e^n$ and ${\lambda }_d^n$. The results indicate that, when the loading rate was 0.05 MPa/s, ${\lambda }_e^n$ was always less than 50%; however, the peak values of ${\lambda }_d^n$ for the NO. 3 and NO. 5 samples were 52.83% and 52.41%, respectively. At the end of the experiment, ${\lambda }_d^n$ was measured at 73.51%, 66.62%, and 60.550%. With the increase in loading rate, a significant downward trend was present.

4.3. Mechanism of the Loading Rate Action

Through parallel comparisons of the total plastic strain energy density, upper limit damage strength and total strain, some results can be obtained. As presented in Figure 14, the damage strength and total strain had a linear relationship with the loading rate, with a unit growth factor of 40, −26.2 and a fitting degree of 0.926, 0.985. However, the relationship between the total plastic strain energy density and loading rate can be fitted by power function and the fitting coefficient was 0.981. Therefore, even though the natural inhomogeneity of the samples was taken into account, the increase in loading rate exhibited a significant influence on the inhibition of the growth of the total plastic strain density and plastic strain energy density.

Reconstituted tectonic coal samples are traditionally composed of coal particles with different particle sizes and other fine particles, such as cement. The initial defects of the samples mainly include the inter-granular spaces and the micro-cracks in the particles, as seen in Figure 15. In the initial loading stage, these voids and cracks were compressed, so the input energy was largely converted into plastic strain properties and the sample strength increased instead of incurring damage.

The strength of a reconstituted sample is affected by its cementation strength and the confining pressure, so there are two damage thresholds. The first threshold occurs when the axial stress exceeds the cementation strength, resulting in relative displacement between the particles. Peng suggested that the increase in the loading rate essentially meant a shortened disturbance time for the external force and a greatly decreased rotation time of the crystals about the weak crystal face, hence a hardening effect was generated [22]. For the reconstituted sample, as the rate increased the rotation time of the particles was shortened, thereby inhibiting the propagation of secondary defects. However, under a lower loading rate the secondary defects were fully expanded and the damage degree increased with the plastic strain at the same stress level. At this stage, although the secondary defects began to be generated, the damage degree was always at a low level and elastic strain still occupied a high proportion of the total strain. The plastic strain and plastic strain energy began to increase, but the rate of their increase was still relatively slow.

The second threshold occurs when the axial stress exceeds the peak strength and the samples lose their bearing capacity through gradual damage accumulation. At this point, the samples broke through the confining pressure and expanded rapidly in the transverse direction, leading to macroscopic damage. Additionally, when the loading rate was low, the dislocation and climbing of the internal crystal grains in the rock were more plentiful [60]. Therefore, the plastic strain and the total plastic strain energy increased rapidly and the total strain energy was converted into plastic strain energy which drove the samples’ failures. In a finding that is consistent with the conclusion drawn in reference [59], the rapid change of elastic strain energy, plastic strain energy, and total strain energy can be used as indicators of rockburst. In this stage, the particles moved faster at a higher stress level, so the time effect that was caused by the decrease in the loading rate became more significant. The rock hardening effect was eventually replaced by a softening effect.

5. Conclusions

• As the stress gradually increased, the axial stress–strain curve exhibited two stages, which can be described as “dense–sparse”. The total strain and irreversible strain increased exponentially with the increase in the deviatoric stress; however, the elastic strain increased linearly. However, when the axial deviatoric stress was the same as the loading rate of increase, the three strains were gradually decreased.
• The elastic modulus increased with the increase in the loading rate and rapidly increased in the compressed stage. After reaching its peak, it began to decrease gradually. Higher loading rates led to a slower decrease.
• The total energy density and plastic energy density increased exponentially with the increase in deviatoric stress, but the elastic strain and elastic energy density increased linearly. At the preliminary stage, the difference between the u_e, and u_d was small. As the cyclic loading continued, u_d rose at an obviously faster rate than u_e, and the difference between them enlarged. In addition, the loading rate had a significant effect on the increase in the plastic energy density; the lower that this rate was, the higher the growth of the plastic energy density. However, there was a limited effect on the change in the recoverable energy density.
• The relative displacement of the coal particles that was driven by axial stress was the main way in which energy was consumed. A lower loading rate was beneficial to the damage and movement of particles; i.e., the higher loading rate will inhibit the conversion of input energy to plastic energy. Increasing of loading rate results in the decreases of expansion degree of secondary defects under the same stress level, which improved the energy storage capacity and strength of structural coal.