Evaluation of the Energy Consumption and Fractal Characteristics of Different Length-Diameter Ratios of Coal under Dynamic Impact

Abstract

Coal samples having the same diameter (50 mm) and different length-diameter ratios (l/d), i.e., 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 were tested under dynamic uniaxial impact compression using the Split Hopkinson Pressure Bar (SHPB) experimental system. This study evaluates: (a) The effects of l/d on the energy consumption law and fractal characteristics of coal crushing; (b) The effects of l/d and stress balance on energy dissipation; (c) The effects of l/d and energy consumption density on the fractal characteristics of coal crushing. The findings under different l/d are as follows: (1) The coal samples show similar stress–strain curve shapes in stages including elastic, plastic, and failure stage, which is an “open” shape, the proportion of plastic stage increases, and strain-softening occurs; (2) The dynamic compression dissipation energy and energy consumption ratio of coal shows the same trend, showing two stages with the increase of length-diameter ratio, which increases linearly in the first stage and overall decreases step-by-step; (3) The average particle size increases while fractal dimension of fragmentation decreases linearly, which endorses the decreasing trend of fragmentation degree; (4) It is determined that there is a power relationship between fractal dimension and energy dissipation density; (5) A new index Crushing Density Energy Efficiency (CDEE) is proposed, which can be used to characterize the rock-breaking efficiency of crushing energy consumption under different conditions. This index is inversely proportional to l/d. The research results can provide a basis for the design of top coal caving mining, and the determination of blasting parameters.

1. Introduction

There are various types of impact dynamics problems in mining and tunnel engineering, i.e., blasting excavation, rock burst, and roof blasting [1,2]. Yin and Nikolenko studied the occurrence mechanism of dynamic disasters in coal mines, and achieved some research results [3,4]. Explosive blasting, fault slip, or roof fracture will produce impact loads [5,6]. During impact load, the internal micro-cracks of coal rock continue to develop, expand, penetrate, and destroy. The entire deformation and failure process contains energy transfer and dissipation [7,8]. Weng et al. studied siltstone saturated with water and found the relationship between energy dissipation below 0 °C and dynamic fractal dimension. The energy dissipation characteristics and fragment distribution characteristics of rock dynamic crushing are analyzed [9]. Khan et al. discovered the relationship between energy and infrared radiation in the process of rock fracture [10]. Furthermore, Ma et al. used an intelligent decision-making method to study the prediction dilatancy point of the infrared radiation characteristics of rocks [11,12]. The intensity of dynamic impact effects by coal pillar width, working face length, and roof (or top coal) blasting parameters [13,14]. Therefore, combined fractal theory, the energy evolution law, and crushing fractal characteristics of coal with different length-diameter ratios (l/d) under impact load are important to study. The relationship between coal crushing shape and fractal dimension is investigated, and the coupling effect of size parameters and energy absorption is explored.

The Split Hopkinson Pressure Bar (SHPB) system was invented by Kolsky in 1949 [15]. SHPB experimental technology has been widely applied to brittle materials such as rock and concrete [16,17]. Li, Yu, Liu, Weng, and Zhou respectively studied the dynamic characteristics of granite, sandstone, coal, siltstones, and other materials [18,19,20,21,22]. The findings show the effect of strain rate on material strength, which are directly proportional to each other. However, limited literature is available on the energy evolution under dynamic load and the size effect of fractal characteristics of crushing. Li et al. carry out SHPB tests on limestone specimens with the same diameter and different l/d [23]. Their findings revealed that rock crushing strength is inversely proportional to the l/d of the specimen. Krauthammer, Elfahal, and other scholars, have jointly studied concrete specimens with the same l/d and different diameters [24,25]. The results of the drop weight test and explicit ABAQUS simulation show that the apparent strength of concrete decreases with the increase of specimen size, while the dynamic and static impact is different in the same specimens size. Hong carried out SHPB tests on rocks with the same ratio and different diameters under different strain rates, the results revealed that dynamic strength increases approximately by a power function with strain rate, also, the sensitivity of dynamic strength to strain rate is more significant in larger specimen sizes [26]. Energy dissipation is playing a significant role in material destabilizing [27]. Coal produces irreversible energy dissipation in the process of shock wave action. Liu et al. carried out SHPB tests on coal and rock at different strain rates and analyzed the fractal characteristics of impact fragmentation of coal and rock [28]. They found that the relationship between dissipated energy and strain rate was a weak power function or linear distribution, and that the fractal dimension of fragmentation of coal and rock was linearly and positively correlated with strain rate. Ma et al. conducted an experimental study on fragment distribution, fractal dimension, and energy dissipation of stable soil under impact loading, and analyzed the characteristics of crushing energy dissipation and the law of fragmentation distribution [29]. The fractal dimension showed a logarithmic relationship between energy dissipation density.

The fractal geometry established by French mathematician Mandelbrot provides a systematic, quantitative method to study superficially complex and disordered similar systems, i.e., crack distribution and fracture distribution in the rock mass [30]. Tyler et al. established and developed a quality fractal model of soil particle size distribution [31,32]. Xie et al. established a fractal model of the relationship between dynamic mechanical behavior of crack propagation and mesostructures of rock based on fractal geometry and studied the fragmentation distribution of rock mass under static load using fractal theory [33,34]. Zhang et al. studied the dynamic failure characteristics of deep sandstone pretreated at high and low temperatures using the SHPB device and analyzed the effect of temperature on the failure degree and energy dissipation of sandstone [35]. Zhang et al. analyzed the energy dissipation and fragmentation fractal dimension of coal specimens impacted by different strain rates [36]. It was found that the coal crushing dissipation energy increases exponentially with the strain rate, and the fractal dimension increases logarithmically with the strain rate and the dissipative energy density. Liu et al. used fractal dimension to evaluate the change in dynamic compressive strength of rock specimens during impact crushing, and also investigated the effect of impact velocity on the fractal dimension of crushing [37].

Specimen failure is an energy-driven process. Cao and Xie studied the law of energy dissipation and the characteristics of fragment distribution in the failure process of the specimen, and they found that it can macroscopically explain the failure mechanism of the specimen and evaluate the crushing efficiency of coal and rock [38,39]. Zuo and Zhao, respectively, studied the energy evolution characteristics of surrounding rock failure processes of a deep coal mine roadway and the damage evolution law of coal measures sandstone under cyclic loading [40,41]. Therefore, taking coal with different l/d as the research object, this paper studies the coupling relationship between energy evolution, fractal dimension characteristics, and the effect of l/d under the same impact load. The research results can provide a basis for mining design and determination of blasting parameters in the process of fully mechanized top coal caving mining.

2. Materials and Methods

2.1. Sample Preparation

In this experiment, the coal blocks were collected from the Shenmu mining area, Shenmu City, Shaanxi Province. The depth of sample collection was −500 m. We tried to avoid the area affected by faults and cracks when sampling. Cylindrical coal samples with a diameter of 50 mm were extracted from coal boulders in a vertical direction. The samples are divided into eight groups on the basis of l/d, i.e., 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0. The samples were prepared according to the recommended standard of the International Society of Rock Mechanics (ISRM) [42]. To keep the integrity of the sample high, non-parallelism was kept within 0.02 mm and unevenness to less than 0.05 mm by carefully polishing the faces of the specimens. Each group comprised three samples, as shown in Figure 1. To check the coal samples’ damage, P-wave velocity was measured using a U510 non-metal ultrasonic detector, as shown in Figure 2. The samples with obvious damage and wave speed deviations of more than 10% were discarded and not used any further in the experimental process. Each group of samples were numbered, for example, 0.5–2 would represent the second specimen with an l/d of 0.5.

2.2. Loading Plan

A dynamic compression test was carried out by the SHPB test system, as shown in Figure 3. The SHPB test system uses a cylindrical bullet of 300 mm in length, a thickness of 3 mm, and a side length of 10 mm; rubber square sheet shaper; an incident rod; and transmission rod of 50 mm in diameter made of alloy steel and having a density and a longitudinal wave velocity of 7810 kg/m³ and 5190 m/s, respectively. Before the test, it was ensured that the incident and transmission rod were horizontal and concentric. To ensure there were no friction, petroleum jelly was used as a lubricant in the empty rod. In this study, we used 0.35 MPa impact pressure and a laser velocimeter in front of the incident rod to measure the impact velocity of the bullet.

2.3. Processing Experimental Data

The square rubber sheet was used as the pulse shaper, to get the dynamic characteristics of the coal specimen precisely. To prevent transverse strains, petroleum jelly was applied to both ends of coal specimens. During the impact process, the SDY2107A ultra-dynamic strain gauge was used to measure the strain changes of the incident and transmission rod. Figure 4 shows that the stress wave is smooth, and there is no apparent dispersion effect, which contests the theory of stress wave propagation, which assumes that a stress wave is one-dimensional.

During testing, the dynamic parameters of the SHPB test system were set as the following: Firstly, according to the principle of the strain electric measurement method, the voltage signal of the oscilloscope was converted into a strain value. Secondly, based on the SHPB experimental theory, the stress σ(t) and strain ε(t) of the sample were calculated using Equation (1), and the stress–time history curve and strain–time history curve were obtained. Finally, the dynamic strength of the sample was obtained.

$$\{\begin{array}{l}\sigma (t)=\frac{EA}{A_s}{\epsilon }_{\mathrm{t}}(t)\\ \epsilon (t)=-\frac{2c}{l_s}{\displaystyle {\int }_0^t{\epsilon }_{\mathrm{r}}(t)}\mathrm{d}t\end{array}$$

(1)

where ε_r(t) and ε_t(t) are respectively the strain in the bar corresponding to the independent propagation of reflected and transmitted waves at t time, A is the cross-sectional area of the compression bar, E is the elastic modulus, c is the longitudinal wave velocity of the elastic compression bar, A_s is the cross-sectional area of samples, and l_s is the original length of the specimen.

During the loading process of the SHPB test system, the energy carried by incident wave, reflected wave and transmitted wave were W_I, W_R, and W_T, respectively. These energies can be calculated using Equation (2).

$$\{\begin{array}{l}{W}_{\mathrm{I}}(t)=AEC{\displaystyle {\int }_0^t{\epsilon }_{\mathrm{I}}^2}(t)\mathrm{d}t\hfill \\ W_{\mathrm{R}}(t)=AEC{\displaystyle {\int }_0^t{\epsilon }_{{}_{\mathrm{R}}}^2}(t)\mathrm{d}t\hfill \\ W_{\mathrm{T}}(t)=AEC{\displaystyle {\int }_0^t{\epsilon }_{{}_{{}_{\mathrm{T}}}}^2}(t)\mathrm{d}t\hfill \end{array}$$

(2)

Part of the energy absorbed by the specimen is called specimen absorption energy (dissipated energy), W_S. According to the law of energy conservation, the W_S can be calculated using Equation (3):

$$W_{\mathrm{S}}=W_{\mathrm{I}}-(W_{\mathrm{R}}+W_{\mathrm{T}})$$

(3)

The specimen absorption energy, W_S, is divided into three parts: crushing energy dissipation W_FD, crushing kinetic energy, and other forms of dissipative energy, i.e., sound and radiant energy. The other forms of dissipative energy are usually negligible and crushing kinetic energy only accounts for about 5% of the total absorbed energy [23]. Equation (3) can be modified to:

$$W_{\mathrm{S}}=W_{\mathrm{FD}}$$

(4)

Crushing energy dissipation is correlated to volume, and energy dissipation per unit volume may better represent the specimen’s crushing energy absorption. As a result, the crushing energy consumption proportion P, and crushing energy dissipation density e_d, have been defined and can be calculated using Equations (5) and (6), respectively [23].

$$P=\frac{W_{\mathrm{FD}}}{W_{\mathrm{I}}}$$

(5)

$$e_d=\frac{W_{\mathrm{FD}}}{V}$$

(6)

where V is the volume of the specimen.

The process of coal crushing includes a relationship of functional transformation, and there is an essential relationship between the evolution characteristics of energy and the crushing effect [43]. Macroscopic fractures are composed of micro-fractures, which form smaller defects and fissures [44,45]. This self-similarity behavior inevitably leads to the self-similar characteristics of fragment size after crushing, and the fractal characteristics of fragmentation distribution are closely related to rock mesostructures, loading mode, sample shape, and size. The broken fractal dimension D can quantitatively reflect the degree of fragmentation. The more broken the specimen is, the larger the fractal dimension is, and the smaller the particle size. Fractal dimension cannot only characterize the degree of fragmentation, but also comprehensively reflect the material structure, loading mode, size, shape, and other factors, including rich physical connotations. The distribution equation of fragments produced by coal under impact load is as follows:

$$M(x)/M_{\mathrm{T}}={\left(x/x_m\right)}^{3-D}$$

(7)

where D is the fractal dimension of fragments, x and x_m represent the minimum and maximum particle size respectively, and M_T and M(x) represent the total mass and cumulative mass under pore diameter, respectively.

The logarithm of both sides of Equation (7) was taken at the same time, and in the double logarithmic coordinate system of lg[M(x)/M_T]-lg(x/x_m), the slope of the fitting line is (3-D), thus the fractal dimension D of rock fragmentation distribution is obtained.

At the same time, the average particle size d_m, which is used to describe the degree of fragmentation, is compared with the fractal dimension, D. the average particle size d_m can be calculated using Equation (8):

$$d_{\mathrm{m}}=\frac{{\displaystyle \sum (r_i{d}_i)}}{{\displaystyle \sum r_i}}$$

(8)

where d_i is the average size of fragments in different grades of standard sieves; r_i is the corresponding percentage of the mass of fragments.

3. Results

3.1. Stress–Strain Curve

The stress–strain curve of coal at different stages, i.e., elastic, plastic, and failure, shows similarities with various l/d, but the curve characteristics are clearly distinct [46]. Figure 5 shows the features of the various l/d curves: The stress–strain curves of various l/d specimens have an “open” form. The proportion of the curve in the plastic phase rises with l/d and strain-softening occurs. At the same time, the dynamic strength progressively declines, the post-peak portion of the curve gradually reduces, and the residual strength gradually rises. With a small l/d, the elastic deformation of the sample is more adequate, and its capacity to resist deformation is greater. There is a significant displacement among the particles after the specimen has been plastically deformed. The load-bearing capacity of long specimens declines fast and strain rises rapidly, meanwhile, at this stage, the deformation space of the long specimen is greater than that of the short specimen.

3.2. Energy Dissipation

According to Equations (1)–(6), the electrical signal of the ultra-dynamic strain gauge is converted into the stress, strain, and energy parameters of the specimen. The dynamic strength and energy parameters of samples with different l/d ratios are described in

Table 1. Dynamic mechanical properties and energy indexes of samples with different l/d.

Serial Number	L (mm)	σ (MPa)	$\overline{\mathit{\sigma }}$(MPa)\ Std. Dev.	WI (J)	${\overline{\mathit{W}}}_{\mathbf{I}}\left(\mathbf{J}\right)$ \ Std. Dev.	WFD (J)	${\overline{\mathit{W}}}_{\mathbf{FD}}$(J)\ Std. Dev.	ed (J·cm−3)	${\overline{\mathit{e}}}_{\mathit{d}}$(J·cm−3)\ Std. Dev.	P	$\overline{\mathit{P}}$ \Std. Dev.
0.3–1	14.53	41.70	38.54\2.57	82.26	88.55\2.89	28.68	31.74\3.23	1.01	1.07\0.078	0.35	0.37\0.026
0.3–2	15.71	35.40		89.31		36.20		1.18		0.41
0.3–3	15.13	38.52		85.09		30.34		1.02		0.36
0.4–1	19.94	35.46	39.92\3.86	82.65	81.71\2.21	33.04	30.93\1.63	0.85	0.79\0.042	0.40	0.38\0.022
0.4–2	20.16	39.41		78.66		30.67		0.78		0.39
0.4–3	19.89	44.89		83.83		29.08		0.75		0.35
0.5–1	25.31	35.71	36.57\2.11	81.07	83.31\1.86	32.22	35.19\2.40	0.65	0.71\0.049	0.40	0.42\0.016
0.5–2	25.21	39.48		83.22		35.24		0.71		0.42
0.5–3	25.18	34.51		85.64		38.10		0.77		0.44
0.6–1	30.44	32.08	32.13\4.97	80.29	83.86\2.54	26.08	23.43\1.90	0.44	0.39\0.034	0.32	0.28\0.031
0.6–2	30.37	26.35		85.20		21.67		0.36		0.25
0.6–3	30.28	38.52		86.08		22.53		0.38		0.26
0.7–1	35.34	30.31	31.75\1.65	88.84	85.22\2.73	27.49	26.53\2.37	0.40	0.38\0.034	0.31	0.31\0.024
0.7–2	35.16	34.06		82.22		23.27		0.34		0.28
0.7–3	35.24	30.88		84.59		28.84		0.42		0.34
0.8–1	40.24	29.85	32.3\1.83	72.73	83.26\7.79	22.63	24.77\2.54	0.29	0.31\0.031	0.31	0.30\0.019
0.8–2	40.18	34.25		85.71		23.34		0.30		0.27
0.8–3	40.14	32.79		91.35		28.35		0.36		0.31
0.9–1	45.65	30.94	31.92\0.72	88.15	86.20\1.41	28.69	27.94\0.61	0.32	0.31\0.008	0.33	0.32\0.005
0.9–2	45.63	32.66		84.85		27.19		0.30		0.32
0.9–3	45.76	32.15		85.59		27.93		0.31		0.33
1.0–1	50.50	25.53	27.69\5.26	82.26	82.60\1.17	25.42	25.09\2.07	0.26	0.25\0.021	0.31	0.30\0.021
1.0–2	50.56	34.95		81.37		22.39		0.23		0.28
1.0–3	50.49	22.60		84.18		27.45		0.28		0.33

.

3.3. Crushing Morphology

Sample fragments were screened and divided into eight groups: <0.125 mm, 0.125~0.25 mm, 0.25~0.5 mm, 0.5~1 mm, 1~2 mm, 2~3 mm, 3~6 mm, and >6 mm groups. Figure 6 shows the dynamic failure patterns and debris screening results of some coal samples with different l/d. In terms of crushing shape, an increase in l/d, presents various forms of crushing failure, block fragmentation, block splitting, and so on. In terms of fragment size, with increasing l/d, the crushing degree of coal decreases, and the size of fragmentation increases gradually. The proportion of fine particles decreases gradually.

3.4. Characteristics of Fragmentation Size Distribution

The average particle size and fractal dimension of fragments were calculated respectively, as shown in

Table 2. Characteristic table of particle size distribution and fractal dimension of specimens with different l/d.

Specimen Number	Quality of Fragments between Sieve Holes (g)								dm (mm)	${\overline{\mathit{d}}}_{\mathit{m}}$ \Std Dev	D	$\overline{\mathit{D}}$ \Std Dev	R2
	<0.125 mm	0.125~0.25 mm	0.25~0.5 mm	0.5~1 mm	1~2 mm	2~3 mm	3~6 mm	>6 mm
0.3–1	0.21	0.49	1.53	3.73	6.93	3.40	13.00	4.51	3.95	4.06\ 0.084	1.86	1.83\ 0.045	0.982
0.3–2	0.20	0.49	1.28	2.99	5.00	2.52	14.16	8.56	4.15		1.87		0.991
0.3–3	0.12	0.47	1.29	3.12	6.03	2.71	14.21	6.34	4.09		1.77		0.983
0.4–1	0.17	0.57	1.63	3.33	6.42	3.51	16.35	15.22	4.23	4.21\ 0.078	1.80	1.81\ 0.058	0.988
0.4–2	0.14	0.48	1.28	3.10	6.25	3.20	16.74	15.79	4.30		1.75		0.991
0.4–3	0.30	0.68	1.65	3.95	7.34	4.11	18.07	10.78	4.11		1.89		0.993
0.5–1	0.44	0.53	1.37	3.25	6.19	3.58	20.10	24.40	4.40	4.17\ 0.165	1.92	1.79\ 0.090	0.992
0.5–2	0.18	0.75	2.43	6.18	12.02	5.62	23.30	9.99	4.02		1.73		0.976
0.5–3	0.16	0.88	2.29	5.69	10.43	5.40	23.83	12.27	4.09		1.73		0.976
0.6–1	0.21	0.57	1.67	4.20	7.50	4.38	23.21	30.58	4.43	4.38\ 0.187	1.75	1.70\ 0.096	0.993
0.6–2	0.07	0.49	1.66	3.94	6.71	3.92	20.22	36.88	4.58		1.57		0.978
0.6–3	0.26	1.07	2.71	6.43	11.95	5.27	27.25	19.92	4.13		1.79		0.983
0.7–1	0.40	0.58	1.33	3.09	5.72	2.95	19.30	51.53	4.70	4.61\ 0.069	1.89	1.69\ 0.146	0.981
0.7–2	0.08	0.41	1.38	3.83	7.41	4.07	25.45	40.39	4.62		1.54		0.988
0.7–3	0.15	0.58	1.57	4.10	8.21	5.19	30.36	33.66	4.53		1.65		0.994
0.8–1	0.15	0.52	1.27	3.56	6.04	3.04	19.90	62.81	4.80	4.75\ 0.056	1.68	1.70\ 0.016	0.986
0.8–2	0.18	0.65	1.68	3.91	7.73	3.12	26.40	54.00	4.67		1.70		0.989
0.8–3	0.17	0.56	1.58	3.40	6.22	2.94	17.62	63.92	4.77		1.72		0.983
0.9–1	0.11	0.40	1.01	2.23	4.45	2.55	15.00	87.64	5.12	4.98\ 0.104	1.63	1.64\ 0.019	0.974
0.9–2	0.16	0.53	1.53	3.36	5.92	3.10	24.99	71.71	4.87		1.67		0.984
0.9–3	0.11	0.54	1.30	3.03	5.42	2.73	20.00	78.50	4.95		1.63		0.980
1.0–1	0.24	0.47	1.38	3.37	7.17	3.56	27.00	79.10	4.89	4.88\ 0.008	1.70	1.62\ 0.054	0.984
1.0–2	0.12	0.59	1.66	3.92	7.36	3.69	30.24	79.89	4.87		1.59		0.985
1.0–3	0.13	0.46	1.38	3.50	7.34	4.22	30.14	74.62	4.88		1.58		0.990

.

4. Discussion

4.1. Proportion of Crushing Energy Consumption and Energy Consumption

Figure 7 shows the distribution of incident energy and crushing energy consumption under dynamic compression of coal with different length to diameter ratios. The incident energy of samples with different ratios is roughly the same under the same impact pressure, ranging from 81.71 J to 86.20 J, with a maximum difference of 5.5%, and the range is 4.49 J, which is due to the influence of the change of launcher pressure and the friction between bullet and barrel.

The crushing energy consumption variation can be divided into two stages: l/d 0.3–0.5 and l/d 0.6–1. The crushing energy consumption is positively proportional to the l/d in both stages, with ranges of 30.93–35.19 J and 23.43–27.94 J, respectively. When the l/d is 0.5 to 0.6, the crushing energy consumption decreases from 35.19 J to 23.43 J, a 33.42% decrease.

To explore the energy absorption capacity of coal samples with different l/d, the variation characteristics of the energy consumption ratio of coal samples with different l/d were analyzed. Figure 8 shows the variation law of the dynamic compression energy consumption ratio of coal with different l/d. The trend of energy consumption ratio with l/d is similar to that of crushing energy consumption with l/d. The energy consumption ratio trend with different length-diameter is divided into two stages: l/d 0.3–0.5 and l/d 0.6–1. The internal dissipation ratio and the l/d increase linearly in each stage, and their value ranges are 0.37–0.42 and 0.28–0.32 in 0.3–0.5 and 0.6–1, respectively. However, the growth rate of the l/d of 0.6–1 was slightly lower than that of 0.3–0.5.

The experimental results show that an increase of l/d in a certain range (within two stages) is beneficial to enhancing the energy absorption capacity of the specimen. However, when the l/d is between 0.5 and 0.6 (where the two stages connect), the proportion of energy consumption decreases from 0.42 to 0.28, a decrease of 33.33%. It shows a stepped decline. Moreover, it also shows that the energy consumption ratio in the overall length-diameter gives a declining relationship. The crushing energy consumption and the ratio of energy consumption increase linearly at each stage. But why is there a steep decline on the whole? And is there a relationship between the l/d?

It is worth noting that the l/d of the specimen is closely related to the degree of stress balance [47]. The dispersion effect caused by transverse inertia and the end friction effect of the specimen affects the stress balance degree of the specimen. From the author’s previous research results [48], it is revealed that the stress balance coefficient ξ can characterize the degree of stress balance at both ends of the specimen in the whole process of dynamic compression.

Table 3. Relationship of average stress balance coefficient of specimens and different l/d.

Parameter Name	Numerical Value
l/d	0.3	0.4	0.5	0.6	0.7~1.0
Average stress balance coefficient ξ	87.3%	64.6%	28.7%	16.4%	0

shows that ξ decreases gradually with l/d. By adding the average stress balance coefficient ξ of the specimens with different l/d to Figure 7, it can be found that when l/d increases from 0.3 to 0.6, the stress balance coefficient decreases from 87.3% to 16.4%, and when the l/d is 0.7–1.0, the stress balance coefficient is 0. Hence, it is concluded that with an increase in the l/d of the specimen, the degree of stress balance gradually decreases.

There are two reasons why the characteristics of energy dissipation change with the l/d: (1) As the l/d of the specimen rises, the degree of fragmentation of the loaded specimen increases, fragmentation (or particle size) decreases, and friction between the particles increases, increasing friction energy consumption and the proportion of energy consumed. This is why, as each step progresses in each stage, the amount of crushing energy consumption and energy consumption rises linearly. (2) The specimen l/d and end friction effect are inversely related. Hence, the proportion of energy consumption decreases and the more significant dispersion effect means an increase in the proportion of energy consumption. However, as a whole, the stress balance coefficient ξ decreases, and it is more difficult for the specimen to reach an equilibrium state. The specimen has been destroyed when it does not reach stress equilibrium, and the incomplete failure leads to the abnormal decrease in the proportion of energy consumption in the decrease of stress balance coefficient ξ process. Therefore, it is inferred that during the decrease in stress balance coefficient ξ of a coal specimen, there is a certain threshold (between 28.7% and 16.4%), so that when the stress balance coefficient of the specimen is lower than this value, the proportion of energy consumption decreases step-by-step.

4.2. Energy Dissipation Density

The volume of specimens with different l/d is different, and the energy absorption capacity of a specimen is closely related to volume. Figure 9 shows the relationship between dynamic compression energy dissipation density of coal with different l/d. This revealed that the dissipation density is inversely proportional to the l/d. The fitting relationship is expressed in Equation (9):

$$e_d=\frac{0.363}{l/d}-0.122\hspace{1em}{R}^2=0.964$$

(9)

The findings show that when the ratio increased from 0.3 to 1.0, the e_d decreased from 1.08 J·cm⁻³ to 0.22 J·cm⁻³, a 79.6% decrease. The capacity of the specimen to absorb energy to perform work is progressively decreased per unit volume, signifying a weakening of the specimen’s dynamic stability. This variation is mostly due to changes in energy distribution, such as crushing energy consumption and friction energy dissipation, induced by the bearing specimen’s variable crack growth and fragment fracture degree. As the l/d increases, the crushing energy dissipation density drops, and the rate of decline slows down. It is hypothesized that in a limited length, the reduction in crushing energy consumption density will decrease with increasing l/d, exactly as the inverse proportional function tends to a certain value.

Figure 10 illustrates the relationship between the average particle size of each group when l/d is modified. The average particle size, d_m, grows in proportion to the ratio. Moreover, the average particle size, d_m, and ratio show a linear relationship with a correlation value of 0.936. This indicates that the average size of the fragment increases with the ratio, while the crushing degree decreases.

Although the fact that the average particle size quantitatively reflects the total degree of fragmentation, it overlooks the general features of the fragmentation size distribution. As a result, the fractal theory was utilized to analyze the fragmentation of the specimen to explain the distribution features of the fragmentation. The logarithmic curve of particle mass distribution of various ratio fragmentation fractals is shown in Figure 11. According to the sample’s fragmentation distribution law, the correlation coefficient of the coal specimen’s fragmentation distribution is more than 0.97, indicating a high degree of self-similarity.

Figure 12 shows the average fractal dimension and l/d of each group. This revealed a negative linear relationship between the average fractal dimension and ratio with a correlation coefficient of 0.932. Furthermore, it verified that when the l/d rises, the average particle size increases, and the degree of fragmentation reduces.

4.3. Crushing Density Energy Efficiency

Coal fragmentation is the result of the continuous initiation, development, expansion, and propagation of cracks, which is a process of energy dissipation. Therefore, the distribution characteristics of specimen fragments and their energy dissipation are associated inherently and inevitably. According to Hong’s study [23], the crushing energy consumption of rock fragmentation is mostly dependent on incident energy and increases linearly. The amount of energy used is proportional to the material characteristics and microstructures. It has nothing to do with the loading rate or sample size when the ratio is the same. In terms of fractal dimension features, the crushing energy consumption per unit volume of the specimen is more directly linked to its crushing degree. Figure 13 shows the fractal dimension–energy consumption density variation curve for specimens of various ratios.

In general, with increasing energy dissipation density, the fractal dimension D progressively increases. The fractal dimension D increases from 1.62 to 1.83 when the energy dissipation density e_d is increased from 0.25 J·cm⁻³ to 1.07 J·cm⁻³, a 12.96% increase. A fitting curve was generated which reflects the relationship between fractal dimension and energy dissipation density, which is expressed in Equation (10):

$$D=1.828e_d^{0.078}$$

$$R^2=0.942$$

(10)

The fitting curve shows a power function relation between e_d and D having a coefficient of determination of 0.942. This shows that increasing the energy dissipation density can improve the crushing degree of the sample. However, under the higher energy dissipation density, the increasing trend of the crushing degree gradually decreases. Furthermore, rock failure is caused by the development of various fracture defects. The crack process of crack initiation, development, expansion, and penetration is also dependent upon the development of initial meso-damage to macro-fracture in a rock mass. The more energy the rock absorbs, the more seriously crack propagation is affected by the absorbed energy. Therefore, the fragmentation produced by rock fragmentation increases with the increase of crushing energy absorption, and the degree of fragmentation also increases, which finally leads to a greater value of fractal dimension. There is a close relationship between degree of rock fragmentation and energy dissipation, but there is a lack of a unified relationship between the two and the quantitative analysis of loading parameters such as strain rate and the l/d of specimens, as well as material properties, strength, and other material characteristics.

There is a negative linear correlation between the fractal dimension D and the l/d of the specimen, but a positive correlation between D and the crushing degree. Hence, the fractal dimension D is selected as the physical quantity to characterize the crushing degree of the coal specimens. To unify the crushing degree and energy dissipation characteristics of materials and quantitatively analyze the evolution law of crushing degree of specimens with different l/d, a new index, Crushing Density Energy Efficiency (CDEE), is proposed, which can be calculated using Equation (11):

$$CDEE=\frac{D}{V\cdot W_{FD}}$$

(11)

CDEE can represent the crushing degree of specimens with the same crushing energy consumption per unit volume and measure the efficiency of rock breaking with the crushing energy consumption under certain working conditions. Figure 14 shows the CDEE variation of specimens with different l/d. This reveals that CDEE decreases with an increase in the ratio, which means the CDEE is inversely proportional to ratios. The fitting curve is expressed as:

$$CDEE=\frac{0.521}{l/d}+0.174$$

$$R^2=0.949$$

(12)

Figure 14 shows that, as the ratios are increased from 0.3 to 1.0, the CDEE is reduced from 1.94 kJ⁻¹ cm⁻³ to 0.68 kJ⁻¹ cm⁻³, a decrease of 64.95%. The rate of CDEE decreases with increased ratios. Furthermore, it also shows that the absorbed energy for coal samples crush decreases with the ratio. This ultimately gradually decreases the weakening effect. The larger the l/d of the specimen, the smaller the corresponding CDEE. In other words, as the l/d of the specimen rises, the efficiency of crushing energy consumption for coal breaking decreases gradually under the condition of constant energy consumption per unit volume.

Physically, CDEE is defined as the size of fragmentation of a unit volume of a specimen to absorb the same amount of energy. It streamlines the energy absorption process and allows users to directly assess the coupling impact between crushing energy consumption and fragment fractal dimension, as well as paying closer attention to the relationship between crushing degree and total energy. At the same time, CDEE can eliminate the effect of volume on fragmentation size distribution while preserving the general characteristics of the sample’s crushing degree. From energy absorption to damage, damage crack propagation, and complete failure, the process of coal crushing under impact load is a fractal development process. Energy dissipation is inextricably linked to the degree of crushing. CDEE may combine the features of coal crushing and energy dissipation, making it a suitable statistic for calculating sample crushing’s energy consumption and rock-breaking efficiency.

4.4. Principles for Determining Pre-Splitting Blasting Parameters of Top Coal

Top coal caving, under the conditions of a hard and thick coal seam, is recognized as a technical problem in coal production [48]. The factors affecting the mining efficiency of top coal in fully mechanized caving mainly include coal seam occurrence (coal seam thickness and burial depth, etc.) and coal body strength (fracture density and distribution in coal body and gangue distribution in top coal). When the top coal strength is too high (f > 3) or the crack is not formed, deep hole pre-cracking blasting is often used to enhance the crack and lose coal strength, thus improving top coal recovery in fully mechanized caving mining. The stress wave generated by explosive blasting damages top coal and causes many fractures. Furthermore, the fragmentation degree of top coal is increased under the effect of mining stress generated by working face mining, which ultimately causes the block of top coal to reach an acceptable range and achieves the benefit of improving the recovery rate of coal resources. Equation (10) establishes the corresponding relationship between the fragmentation fractal dimension D and the energy dissipation density e_d, and the energy dissipation density e_d has a corresponding relationship with the explosive quantity, e_d = (detonating energy E_D + explosive energy E_E)/rock breaking volume V_S. That is, when the volume of the top coal to be blasted is determined, the larger the total explosive energy, the smaller the fragmentation. As shown in Figure 15, the pre-splitting blasting method for a fully mechanized caving face is often used in actual projects. It divides the coal body into coal blocks with a fixed l/d with rows of blast holes on both sides of a special blasting roadway set in the top coal. The energy produced by an explosive explosion propagates in the coal body in the form of a stress wave. Then the absorption of energy by coal directly leads to plastic failure, crack initiation, crack propagation, and penetration of micro-defects in coal.

The fractal theory analysis result of the SHPB energy dissipation Equation (12) establishes the relationship between CDEE and l/d, and clearly shows that CDEE is inversely proportional to l/d. It is concluded that when the l/d of the specimen increases, the efficiency of crushing energy dissipation for rock breaking diminishes. The CDEE of the specimen with an l/d of 0.3 is up to 1.94 kJ⁻¹ cm⁻³ within the range of the testing l/d, suggesting that the stress wave has a greater influence on specimens with a small ratio. Taking into account the economic costs of drilling and explosives, the length and width dimensions of the divided coal can be changed by rationally arranging the row spacing between blast holes to adjust the ratio of the rock mass to be blasted, thereby increasing the degree of coal fragmentation and improving economic benefits under the condition of a limited number of blast holes. Adjusting the number of explosives and the distance between blast holes using CDEE, fractal dimension, and crushing energy density, can control the fragmentation of top coal, consequently optimizing the crushing efficiency of deep hole pre-splitting blasting.

5. Conclusions

In this research, the SHPB experimental system is used to study the energy evolution and fractal characteristics of 8 groups of coals of different l/d i.e., 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0. The following main conclusions are drawn:

(1)

The stress–strain curve of coal in different stages i.e., elastic, plastic, and failure, show similar shape (open) with various l/d. The plastic stage of the curve increases as the l/d ratio increases, and the phenomenon of strain-softening appears.

(2)

Dynamic compression of coal with different l/d shows similar changes in the dynamic compression energy dissipation and energy consumption ratio. There are two stages as the l/d increases, with a linear increase within the stages, and a step-by-step decline as a whole. It is the dispersion effect of the lateral inertia and the internal and external friction effects that cause the change in the energy absorption distribution of the specimen. The energy consumption density of the specimen is inversely proportional to the l/d.

(3)

The impact crushing characteristics of coals with different l/d were described using the average particle size of the fragments and the fractal dimension of broken fragmentation, which mutually confirmed that the degree of crushing gradually decreases as the l/d increases. The fractal dimension and the energy consumption density of coal specimens with varying l/d are proven to have a power relationship.

(4)

A new index, CDEE, is proposed to measure the efficiency of crushing energy utilized in rock breaking. The l/d is found to be inversely proportional to CDEE, revealing the weakening effect of longer coal samples on the efficiency of absorbing energy for crushing, but the effect decreases as the l/d increases.

(5)

To enhance the crushing efficiency of deep hole pre-splitting blasting, a technique for estimating the parameters of top coal pre-splitting blasting based on CDEE is proposed to tackle the issues of undeveloped top coal cracks, completeness, high hardness, and poor caving in top coal mining.