# Energy Evolution Law and Fractal Characteristics of Different Rock Specimen Sizes on Dynamic Compression

Authors: Jun Zhou; Wensong Xu; Guangming Zhao; Xiangrui Meng; Yingming Li; Xukun Wu; Yuguang Li; Xiang Cheng
Journal: Geofluids
 (2022)

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## Abstract

To explore the broken energy change and the specimen fragment influence of granite where the length-to-diameter ratio is 0.5–2, the SHPB device was used to perform dynamic loading on the granite specimens. The rock energy evolution law was analyzed by the energy time history curve, and according to rock fragment characteristics, the rock fractal dimension was calculated. The experimental results show that the rock energy-time history curve can be divided into four stages. The incident energy is independent of the length-to-diameter ratio of the rock specimens. When the length-to-diameter ratio of the rock specimens is 0.5–0.9, the difference of incident energy, transmitted energy, and reflection energy of rock specimens is small. With the length-to-diameter ratio increasing, the rock fragment size became larger. These rock fragments have good self-similarity. The rock specimen fractal dimension is at least 1.94, and the maximumD is 2.536. And with the length-to-diameter ratios increasing, the fractal dimension of rock specimens decreases. With the specimen fractal dimension increasing, the energy dissipation density of rock specimens also increases. The higher the energy dissipation density of a rock specimen, the more uniform rock fragments are.


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## Body

## 1. Introduction

Rock fragmentation law under impact load is the basis for the study of mining, geotechnical blasting, and so on [1–3]. The process of rock fracture under impact load is also the process of energy-driven transformation from a stable state to an unstable state. Rock damage degree is closely related to energy dissipation. Rock size is one of the factors affecting the mechanical properties of rock, and the size affects the degree of rock failure. It is of great significance to understand the dissipation characteristics and broken characteristics in rock, as they are helpful for ore yield and energy utilization rate.Rock size effects have been studied in many ways at home and abroad. Tanusree [4] used SHPB to impact different rocks to study the size effect and combined it with static experiments to analyse the size effect. Jin [5] established a constitutive model and explained the concrete size effect on tensile strength. Chen [6] studied the concrete size effect under axial compression and proposed a related stress-strain constitutive model. Li et al. [7] studied the rock fragment characteristics at different strain rates and established the fragmentation model to predict the rock fragment size.It is great significance to study the energy evolution law, fragmentation form, and fragment distribution characteristics of rock fragmentation under impact load [8–11]. Zhang et al. [12] studied the fracture and energy dissipation characteristics of rocks at different loading rates and analyzed the relationship between dissipation energy and absorption energy. Grady [13] used the principle of constant energy, based on the assumption that all the dissipated energy of rocks is used for rock fractures, the rock block and energy formula are established, which can well describe the relationship between ductility and brittle material. Lundberg [14] studied the rock distribution and energy absorption characteristics through the SHPB device. According to the characteristics of rock fragmentation that established two models and analyzed that the greater the incident energy, the more thoroughly the rock fragmentation. Hong et al. [15] used a 75 mm SHPB device to impact rock and analyzed the energy absorption rate and fragmentation form of rocks at different sizes, and it was concluded that the rock damage is closely related to rock size. After Mandelbrot [16] founded the fractal theory, Tyler et al. [17, 18] successively established and developed a mass fractal model for the particle size distribution of rock fragments and was widely used in rocks [19, 20].Granite is a common rock in engineering. In this study, many granites were impacted by SHPB, and the influence of length-to-diameter ratio on incident energy, reflection energy, and transmission energy was analyzed. The distribution characteristics of its fragments are screened and the fractal characteristics of rock specimens with different length-to-diameter ratios are discussed, which has certain engineering significance for revealing the characteristics of rock fragmentation under impact loads.


## 2. Test Scheme Design and Testing Apparatus

### 2.1. Processing and Preparation of Granite Specimens

This test takes granite as the research object. The unified diameter is 50 mm, and the length is 25 mm, 30 mm, 35 mm, 40 mm, 45 mm, 50 mm, 60 mm, 75 mm, and 100 mm, respectively. These length-to-diameter ratios (LDRs) of specimens are 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.2, 1.5, and 2.0, respectively. In the compression test, the mechanical characteristics and fragments of granite with different high diameter ratios are studied. The rock specimens are processed in accordance with the corresponding specifications. Through a series of procedures such as cored extraction, cutting, and polishing, the rock specimens that meet the requirements are obtained. The plane at both ends of the rock sample is nonparallel ≤±0.3 mm, and the axis deviation of the vertical specimens at the end surface is ≤±0.25°. Test specimens were performed under dynamic compression testing under 0.75 MPa impact loading.


### 2.2. Testing Apparatus and Principle

The test equipment adopts the SHPB device of the dynamic and static combination loading system, which is shown in Figure1. The input bar, transmission bar, and absorption bar are made of 40Cr alloy steel, with a density of 7820 kg/m3, Young's modulus of 210 GPa, the diameter of the bar is 50 mm, and elastic limit of the bar is 800 MPa. In order to reduce the equipment waveform dispersion, a 1 mm rubber pad is used as a pulse shaping device at the end of the incident bar. Test specimens are sandwiched between the incident bar and the transmission bar, in order to reduce the friction effect between the pressure bar and the contact surface of the specimen. A Vaseline lubricant is evenly applied to the contact surface of the specimens and the bar.Figure 1
SPHB experimental device.A high-speed bullet was fired by the SHPB cavity and formed a stress pulse in the incident bar end. The stress pulse forward transmits to the incident bar and specimen interface, due to the difference in the wave impedance of the incident bar and rock specimens. A reflection wave and transmission wave are generated, respectively, at the contact surface. Specimens are much shorter than the bar. The time of the impact pulse transmitted repeatedly is 2L/C. After a few times, the stress and strain of the specimens’ surfaces are basically the same. Through recording the waveform data, we can get the stress-strain relationship of specimens.The input strain pulse signal, reflected strain pulse signal, and transmitted strain pulse signal are denoted and collected. The three-wave formula to calculate the stress, strain, and average strain rate of the specimens in the impact process is used. The calculation principle as follows [10]:(1)σs=AE2ASεtt+εrt+εitε.t=CLεtt+εrt−εitεtt=CL∫0tεtt+εrt−εitdt,where σs is the stress of the specimens, ε is the strain of the specimens, and ε. is the strain rate of the specimens. A, E, and C are the area, the elastic modulus, and the wave velocity of the bar, respectively. As and L are the area and length of specimens, respectively. εit,εrt,andεtt are the incident strain pulse, the reflected strain pulse, and the transmitted strain pulse at time t, respectively.


### 2.3. Typical Waveform

According to the strain gauge on the incident bar and transmission bar, the incident signal is collected first, and then the transmitted and reflected wave signals are collected. When the LDR of rock specimens is less than 1.2, it can better meet the stress wave balance, and the stress wave equilibrium diagram is shown in Figure2.Figure 2
Stress wave equilibrium.


## 2.1. Processing and Preparation of Granite Specimens

This test takes granite as the research object. The unified diameter is 50 mm, and the length is 25 mm, 30 mm, 35 mm, 40 mm, 45 mm, 50 mm, 60 mm, 75 mm, and 100 mm, respectively. These length-to-diameter ratios (LDRs) of specimens are 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.2, 1.5, and 2.0, respectively. In the compression test, the mechanical characteristics and fragments of granite with different high diameter ratios are studied. The rock specimens are processed in accordance with the corresponding specifications. Through a series of procedures such as cored extraction, cutting, and polishing, the rock specimens that meet the requirements are obtained. The plane at both ends of the rock sample is nonparallel ≤±0.3 mm, and the axis deviation of the vertical specimens at the end surface is ≤±0.25°. Test specimens were performed under dynamic compression testing under 0.75 MPa impact loading.


## 2.2. Testing Apparatus and Principle

The test equipment adopts the SHPB device of the dynamic and static combination loading system, which is shown in Figure1. The input bar, transmission bar, and absorption bar are made of 40Cr alloy steel, with a density of 7820 kg/m3, Young's modulus of 210 GPa, the diameter of the bar is 50 mm, and elastic limit of the bar is 800 MPa. In order to reduce the equipment waveform dispersion, a 1 mm rubber pad is used as a pulse shaping device at the end of the incident bar. Test specimens are sandwiched between the incident bar and the transmission bar, in order to reduce the friction effect between the pressure bar and the contact surface of the specimen. A Vaseline lubricant is evenly applied to the contact surface of the specimens and the bar.Figure 1
SPHB experimental device.A high-speed bullet was fired by the SHPB cavity and formed a stress pulse in the incident bar end. The stress pulse forward transmits to the incident bar and specimen interface, due to the difference in the wave impedance of the incident bar and rock specimens. A reflection wave and transmission wave are generated, respectively, at the contact surface. Specimens are much shorter than the bar. The time of the impact pulse transmitted repeatedly is 2L/C. After a few times, the stress and strain of the specimens’ surfaces are basically the same. Through recording the waveform data, we can get the stress-strain relationship of specimens.The input strain pulse signal, reflected strain pulse signal, and transmitted strain pulse signal are denoted and collected. The three-wave formula to calculate the stress, strain, and average strain rate of the specimens in the impact process is used. The calculation principle as follows [10]:(1)σs=AE2ASεtt+εrt+εitε.t=CLεtt+εrt−εitεtt=CL∫0tεtt+εrt−εitdt,where σs is the stress of the specimens, ε is the strain of the specimens, and ε. is the strain rate of the specimens. A, E, and C are the area, the elastic modulus, and the wave velocity of the bar, respectively. As and L are the area and length of specimens, respectively. εit,εrt,andεtt are the incident strain pulse, the reflected strain pulse, and the transmitted strain pulse at time t, respectively.


## 2.3. Typical Waveform

According to the strain gauge on the incident bar and transmission bar, the incident signal is collected first, and then the transmitted and reflected wave signals are collected. When the LDR of rock specimens is less than 1.2, it can better meet the stress wave balance, and the stress wave equilibrium diagram is shown in Figure2.Figure 2
Stress wave equilibrium.


## 3. Energy Analysis of Granite with Different Length-to-Diameter Ratios

### 3.1. Dynamic Stress-Strain Curves of Granite with Different Length-to-Diameter Ratios

The voltage signals were collected, the data were processed with the three-wave method, and the stress-strain curve of the specimen was obtained, as shown in Figure3.Figure 3
Stress-strain curves of granite.Stress-strain curves show four characteristic stages. The first stage curves are called the microcrack closure stage. Specimens remain intact at this stage. In the second stage, the curves slope are nearly linear, which is called elastic deformation stage. Specimens appear small cracks in this stage. In the third stage of the curves which occurs in the upper sections, the peak stress is over 70% and the curves slope gradually decreases to 0. This is called the plastic deformation stage. Specimens appear to have a lot of cracks in this stage. In the fourth stage of curves, the curves slope are negative, which is called the failure stage. Specimens are damaged in this stage.The stress peak values are close in different sized specimens. The prepeak curves of different LDR specimens show good consistency, and the postpeak curves are more obvious in larger LDR specimens. The postpeak curves of smaller size specimens show more obvious plastic properties with secondary peaks; the specimen strain also increases.


### 3.2. Energy Calculation

Cai et al. [21] studied that the lateral inertia effect has few influence on test results when the strain rate is lower than some certain value. The test strain rates are all less than 200 s−1, so the consumed energy of transverse movement can be ignored. Combined the one-dimensional elastic stress wave theory and the stress uniform assumption, the energy is performed by the following equation [21]:(2)Wi=AρC∫0tσi2tdtWr=AρC∫0tσr2tdtWt=AρC∫0tσt2tdt,Wd=Wi−Wr+Wt,where Wi, Wr, Wt, and Wd are incident energy, reflection energy, transmission energy, and absorbed energy, respectively.The statistical distribution of specimens’ energy is shown in Table1. Under the 0.75 MPa impact pressure, the absorbed energy has no relation with the length-to-diameter ratio and remains stable with the increase of the length-to-diameter ratio. Because specimen volume is different, the energy dissipation of rock broken per unit volume is also different. Therefore, in order to reflect the energy absorption of the specimen [15], the energy dissipation density (wWd) is as follows:(3)wwd=WdV.Table 1
Statistical distribution of specimens’ energy.
Impact air (MPa)LDRσ(MPa)Absorbed energyWd(J)Reflected energyWi(J)Incident energyWr(J)Transmitted energyWt(J)Energy dissipation densityWWd(J·cm−3)0.750.5334.25246.84229.157549.46773.475.0280.6346.678327.11105.87596.616163.6365.5530.7348.25323.7193.082576.332159.544.710.8349.65298.3776.562599.985225.0533.7990.9358.403318.8998.889575.203157.4243.6091.0356.125304.19245.137594.57545.2483.0981.2344.749291.18138.642549.197119.3752.4711.5351.225330.34155.83578.95492.7842.2432343.696307.04209.654579.81863.1241.563


### 3.3. Energy Time History Curves and Rock Damage

Incident energy time history curves of different volume specimens overlap. Transmission energy and reflection energy time history curves of different LDR specimens are different. Transmitted energy and reflected energy time history curves are near when the length-to-diameter ratios are 0.5–0.9 of specimens. A specimen’s transmitted energy is higher than its reflected energy. And with the length-to-diameter ratios of specimens increasing, the transmitted energy decreases. The specimens’ reflection energy is higher than the transmitted energy when specimens’ length-to-diameter ratios are 1–2. In the process of specimens being broken, the stress state changes constantly. Different volume specimens are in different states at the same time. If the specimen is too short, it is easy to cause a friction effect in the experiment. And if the specimen is too long, it is easy to cause an inertia effect. In the experiment, when specimen length-to-diameter ratios are in 0.5–0.9, the time history curves of reflection energy and transmission energy are almost the same. So, when the granite specimens have a diameter of 50 mm and the length-to-diameter ratios are 0.5–0.9, the specimens are appropriate. Through a lot of experiments, Davies and Hunter [22] studied that the most appropriate length-to-diameter ratio is L/R=3/2, and the LDR value is 0.5–0.9.Compared with Figure3 stress-strain curves, Figure 4 energy-time history curves can also be divided into four stages. Such as incident energy, stage 1, when time is in the range of 0–75 μs, the incident energy rises rapidly, and the absorbed energy as a kind of elastic energy is stored in the specimen. Stage 2, when time is in the range of 75–150 μs, the incident energy increases linearly. In the meantime, specimen internal microcracks propagate rapidly and a large number of new microcracks are generated. Stage 3, when time is in the range of 150–175 μs, the incident energy tends to a stable value. In the meantime, granite specimens crack and undergo axial tensile split, releasing elastic strain energy. Stage 4, after the incident energy is 175 μs, specimens are completely broken, and incident energy are stable.Figure 4
Different energy time history curves of different length-to-diameter ratios of granite specimens.


## 3.1. Dynamic Stress-Strain Curves of Granite with Different Length-to-Diameter Ratios

The voltage signals were collected, the data were processed with the three-wave method, and the stress-strain curve of the specimen was obtained, as shown in Figure3.Figure 3
Stress-strain curves of granite.Stress-strain curves show four characteristic stages. The first stage curves are called the microcrack closure stage. Specimens remain intact at this stage. In the second stage, the curves slope are nearly linear, which is called elastic deformation stage. Specimens appear small cracks in this stage. In the third stage of the curves which occurs in the upper sections, the peak stress is over 70% and the curves slope gradually decreases to 0. This is called the plastic deformation stage. Specimens appear to have a lot of cracks in this stage. In the fourth stage of curves, the curves slope are negative, which is called the failure stage. Specimens are damaged in this stage.The stress peak values are close in different sized specimens. The prepeak curves of different LDR specimens show good consistency, and the postpeak curves are more obvious in larger LDR specimens. The postpeak curves of smaller size specimens show more obvious plastic properties with secondary peaks; the specimen strain also increases.


## 3.2. Energy Calculation

Cai et al. [21] studied that the lateral inertia effect has few influence on test results when the strain rate is lower than some certain value. The test strain rates are all less than 200 s−1, so the consumed energy of transverse movement can be ignored. Combined the one-dimensional elastic stress wave theory and the stress uniform assumption, the energy is performed by the following equation [21]:(2)Wi=AρC∫0tσi2tdtWr=AρC∫0tσr2tdtWt=AρC∫0tσt2tdt,Wd=Wi−Wr+Wt,where Wi, Wr, Wt, and Wd are incident energy, reflection energy, transmission energy, and absorbed energy, respectively.The statistical distribution of specimens’ energy is shown in Table1. Under the 0.75 MPa impact pressure, the absorbed energy has no relation with the length-to-diameter ratio and remains stable with the increase of the length-to-diameter ratio. Because specimen volume is different, the energy dissipation of rock broken per unit volume is also different. Therefore, in order to reflect the energy absorption of the specimen [15], the energy dissipation density (wWd) is as follows:(3)wwd=WdV.Table 1
Statistical distribution of specimens’ energy.
Impact air (MPa)LDRσ(MPa)Absorbed energyWd(J)Reflected energyWi(J)Incident energyWr(J)Transmitted energyWt(J)Energy dissipation densityWWd(J·cm−3)0.750.5334.25246.84229.157549.46773.475.0280.6346.678327.11105.87596.616163.6365.5530.7348.25323.7193.082576.332159.544.710.8349.65298.3776.562599.985225.0533.7990.9358.403318.8998.889575.203157.4243.6091.0356.125304.19245.137594.57545.2483.0981.2344.749291.18138.642549.197119.3752.4711.5351.225330.34155.83578.95492.7842.2432343.696307.04209.654579.81863.1241.563


## 3.3. Energy Time History Curves and Rock Damage

Incident energy time history curves of different volume specimens overlap. Transmission energy and reflection energy time history curves of different LDR specimens are different. Transmitted energy and reflected energy time history curves are near when the length-to-diameter ratios are 0.5–0.9 of specimens. A specimen’s transmitted energy is higher than its reflected energy. And with the length-to-diameter ratios of specimens increasing, the transmitted energy decreases. The specimens’ reflection energy is higher than the transmitted energy when specimens’ length-to-diameter ratios are 1–2. In the process of specimens being broken, the stress state changes constantly. Different volume specimens are in different states at the same time. If the specimen is too short, it is easy to cause a friction effect in the experiment. And if the specimen is too long, it is easy to cause an inertia effect. In the experiment, when specimen length-to-diameter ratios are in 0.5–0.9, the time history curves of reflection energy and transmission energy are almost the same. So, when the granite specimens have a diameter of 50 mm and the length-to-diameter ratios are 0.5–0.9, the specimens are appropriate. Through a lot of experiments, Davies and Hunter [22] studied that the most appropriate length-to-diameter ratio is L/R=3/2, and the LDR value is 0.5–0.9.Compared with Figure3 stress-strain curves, Figure 4 energy-time history curves can also be divided into four stages. Such as incident energy, stage 1, when time is in the range of 0–75 μs, the incident energy rises rapidly, and the absorbed energy as a kind of elastic energy is stored in the specimen. Stage 2, when time is in the range of 75–150 μs, the incident energy increases linearly. In the meantime, specimen internal microcracks propagate rapidly and a large number of new microcracks are generated. Stage 3, when time is in the range of 150–175 μs, the incident energy tends to a stable value. In the meantime, granite specimens crack and undergo axial tensile split, releasing elastic strain energy. Stage 4, after the incident energy is 175 μs, specimens are completely broken, and incident energy are stable.Figure 4
Different energy time history curves of different length-to-diameter ratios of granite specimens.


## 4. Study on Fractal Fracture of Different Length-to-Diameter Ratios in Granite

### 4.1. Specimens Fragments Sieving

Sieving mesh with sizes of 2.5 mm, 5 mm, 10 mm, 20 mm, and 50 mm was selected for screening specimen fragments, and the mass of fragments on the sieving mesh was weighed by precision electronic balance. The sieving test results of specimen fragments are shown in Table2.Table 2
The results of sieving test of granite specimens fragments.
LDRAccumulated mass percentage on each sieving mesh (%)Fractal dimension DFragment average particle size(mm)0–2.5 mm2.5∼5 mm5–10 mm10–20 mm20–50 mm0.515.79115.78931.57836.84202.3578.6380.619.17215.34935.32130.15802.427.9860.727.7612.21420.02340.00302.5368.2460.815.38611.53826.92338.4617.6922.33611.0520.913.5158.10816.21643.24318.9182.28514.772123.53111.76417.64729.41117.6472.49812.5661.29.0926.81811.36315.90956.8182.21223.4631.59.6794.8389.67727.41948.3872.19422.06124.1262.0614.12313.40276.2881.9429.135


### 4.2. Specimen Length-to-Diameter Ratio and Dynamic Broken Characteristics

Specimen fragments are divided into five grades that are 0–2.5 mm, 2.5∼5 mm, 5–10 mm, 10–20 mm, and 20–50 mm, respectively. The distribution curves of granite specimen fragments in different length-to-diameter ratios are shown in Figure5.Figure 5
Fragment distribution of different length-to-diameter ratios and cumulative mass percentage on the sieving mesh.In order to quantify specimen fragment size, the fragment average particle sizedS is used to characterize the specimen fragment degree [23, 24], which is as follows:(4)ds=∑ηidi∑ηi,where di is the fragment average size in different sizes sieving mesh; ηi is the fragment mass percentage of di.The results of the average fragment size are shown in Table2. With length-to-diameter ratios increasing, the ds increases gradually, indicating that specimen damage gradually decreases. The ds can calculate specimen fragment degree directly, but it cannot reflect the fragment distribution characteristics intuitively. In other words, the same ds does not mean that the fragments’ weight is the same in different sieving mesh. So, it is not possible to quantify the fragment distribution characteristics truly. Many scholars’ results show that rock fragment fractal characteristics are in many kinds of rock [25]. According to the mass-fragment correlation, the rock fragment distribution equation is as follows:(5)Y=mxmt=xxm3−D,where parameters x and xm are the fragment particle size and the maximum particle size, respectively;mx is the cumulative fragment mass that is smaller than x. mt is the total specimen fragment mass. D is the fragment distribution fractal dimension. Take the logarithm for (5), the following equation is obtained:(6)lgY=lgmxmt=3−Dlgxxm,where 3−D is the fitting line slope in Figure 6. The result of fractal dimension are shown in Table 2.Figure 6
lgmx/mt−lgx/xm curve of fragmentation distribution of different length-to-diameter ratios.As shown in Figure6, the fitting line shows that the obvious fractal distribution characteristic is in granite fragments. Gao et al. [26] deemed rock macroscopic breakages to consist of many groups of small breakages. Small breakages are consist of lots of microcracks. There is a self-similarity characteristic in the specimen crack evolution process, and specimen broken fragments also have the same characteristic. So, the fragments’ fractal characteristic can be described as a rock broken process. Under impact loading conditions, the rock fragment fractal dimension is in some certain ranges, and the most granite fragment fractal dimension is in the 2–3 range. Fractal dimension is not only related to the experiment condition but also related to rock characteristic. And fractal dimension change is closely related to the change of rock characteristic [27].As shown in Figure7, with the increase of specimens’ length-to-diameter ratios, specimens' average fragment increases gradually. When specimen length-to-diameter ratios are to grow from 0.5 to 2, the specimen average fragments are to grow from 8.638 to 29.135. With the specimen length-to-diameter ratios decreasing, the specimen’s fragments are more homogeneous. When the length-to-diameter ratios are 0.5, 0.6, and 0.7, the specimens broken fragment sizes are almost the same. It is shown that under the same impact loading condition, within a certain length-to-diameter ratio range, the higher the specimen energy utilization rate, the rock specimen broken fragments are more homogeneous. Outside the range, specimen fragments will become less homogeneous.Figure 7
Correlation between specimen length-to-diameter ratios and average specimen fragmentation.


### 4.3. Fractal Dimension and Dynamic Loading Parameters

The specimen evolution process was to happen from microscopic damage to macroscopic fracture. Both the geometric feature and the digital feature of mechanical properties show good self-similarity [25]. After specimen fracture, the specimen broken fragments distribution is self-similar in Figure 4. As Figure 8 shows, the fractal dimension (D) decreases as the specimen length-to-diameter ratio (LDR) increases. Both D and LDR show a certain linear correlation. In the experiment, the length-to-diameter ratio was increased from 0.5 to 2, and the fractal dimension was decreased from 2.357 to 1.94.Figure 8
Correlation between specimen fractal dimension and specimen length-to-diameter ratios.With the specimens’ LDR increasing, the energy dissipation density decreases. When LDR is 0.5, the specimen energy dissipation density is 5.028 J/cm−3. When LDR is 2, the specimen energy dissipation density is 1.563 J/cm−3. With specimen energy dissipation density increasing, the specimen damage is more thorough. It can explain the result in Figure 4. When the specimen size is small, the specimen is broken more thoroughly than large-sized specimen fragments. It can be seen from the fitting curve that there is a critical LDR value; that is, the relationship between energy dissipation density and LDR is relevant in a certain LDR range.In the experiment, energy dissipation density can reflect the rock’s toughness characteristic, and energy dissipation density means the dissipated energy per unit volume of rock. The energy dissipation density parameter takes many influence factors into account in the test, and it is close to the actual toughness value. As shown in Figure9, with specimen fragment fractal dimension also increases, specimen energy dissipation density increasing. In other words, as the fractal dimension increases, the specimens are broken more thoroughly. Lots of cracks developed and expanded which caused rock damage. The energy dissipation density is increasing gradually, the crack extension velocity is increasing more rapidly, while the fractal dimension is also increasing gradually. Fractal geometry can well describe the regularity of irregular things in nature. Rock fragment fractal characteristics can reflect rock broken mode, and rock fragment fractal dimension shows some physical mechanism in the rock broken evolution process [28–30]. Under a 0.75 MPa impact loading condition, rock will suffer damage and failure. The rock broking process is also called the fractal developed process. Besides, the fragment fractal dimension is an ideal statistical parameter of rock fragments.Figure 9
Correlation between specimen fractal dimension and specimen broken energy density.With the fractal dimension increasing, the energy dissipation density is also increasing. It is shown that when the specimens are broken more thoroughly, the energy dissipated is higher. There is a positive correlation between fractal dimension and energy dissipation density.


## 4.1. Specimens Fragments Sieving

Sieving mesh with sizes of 2.5 mm, 5 mm, 10 mm, 20 mm, and 50 mm was selected for screening specimen fragments, and the mass of fragments on the sieving mesh was weighed by precision electronic balance. The sieving test results of specimen fragments are shown in Table2.Table 2
The results of sieving test of granite specimens fragments.
LDRAccumulated mass percentage on each sieving mesh (%)Fractal dimension DFragment average particle size(mm)0–2.5 mm2.5∼5 mm5–10 mm10–20 mm20–50 mm0.515.79115.78931.57836.84202.3578.6380.619.17215.34935.32130.15802.427.9860.727.7612.21420.02340.00302.5368.2460.815.38611.53826.92338.4617.6922.33611.0520.913.5158.10816.21643.24318.9182.28514.772123.53111.76417.64729.41117.6472.49812.5661.29.0926.81811.36315.90956.8182.21223.4631.59.6794.8389.67727.41948.3872.19422.06124.1262.0614.12313.40276.2881.9429.135


## 4.2. Specimen Length-to-Diameter Ratio and Dynamic Broken Characteristics

Specimen fragments are divided into five grades that are 0–2.5 mm, 2.5∼5 mm, 5–10 mm, 10–20 mm, and 20–50 mm, respectively. The distribution curves of granite specimen fragments in different length-to-diameter ratios are shown in Figure5.Figure 5
Fragment distribution of different length-to-diameter ratios and cumulative mass percentage on the sieving mesh.In order to quantify specimen fragment size, the fragment average particle sizedS is used to characterize the specimen fragment degree [23, 24], which is as follows:(4)ds=∑ηidi∑ηi,where di is the fragment average size in different sizes sieving mesh; ηi is the fragment mass percentage of di.The results of the average fragment size are shown in Table2. With length-to-diameter ratios increasing, the ds increases gradually, indicating that specimen damage gradually decreases. The ds can calculate specimen fragment degree directly, but it cannot reflect the fragment distribution characteristics intuitively. In other words, the same ds does not mean that the fragments’ weight is the same in different sieving mesh. So, it is not possible to quantify the fragment distribution characteristics truly. Many scholars’ results show that rock fragment fractal characteristics are in many kinds of rock [25]. According to the mass-fragment correlation, the rock fragment distribution equation is as follows:(5)Y=mxmt=xxm3−D,where parameters x and xm are the fragment particle size and the maximum particle size, respectively;mx is the cumulative fragment mass that is smaller than x. mt is the total specimen fragment mass. D is the fragment distribution fractal dimension. Take the logarithm for (5), the following equation is obtained:(6)lgY=lgmxmt=3−Dlgxxm,where 3−D is the fitting line slope in Figure 6. The result of fractal dimension are shown in Table 2.Figure 6
lgmx/mt−lgx/xm curve of fragmentation distribution of different length-to-diameter ratios.As shown in Figure6, the fitting line shows that the obvious fractal distribution characteristic is in granite fragments. Gao et al. [26] deemed rock macroscopic breakages to consist of many groups of small breakages. Small breakages are consist of lots of microcracks. There is a self-similarity characteristic in the specimen crack evolution process, and specimen broken fragments also have the same characteristic. So, the fragments’ fractal characteristic can be described as a rock broken process. Under impact loading conditions, the rock fragment fractal dimension is in some certain ranges, and the most granite fragment fractal dimension is in the 2–3 range. Fractal dimension is not only related to the experiment condition but also related to rock characteristic. And fractal dimension change is closely related to the change of rock characteristic [27].As shown in Figure7, with the increase of specimens’ length-to-diameter ratios, specimens' average fragment increases gradually. When specimen length-to-diameter ratios are to grow from 0.5 to 2, the specimen average fragments are to grow from 8.638 to 29.135. With the specimen length-to-diameter ratios decreasing, the specimen’s fragments are more homogeneous. When the length-to-diameter ratios are 0.5, 0.6, and 0.7, the specimens broken fragment sizes are almost the same. It is shown that under the same impact loading condition, within a certain length-to-diameter ratio range, the higher the specimen energy utilization rate, the rock specimen broken fragments are more homogeneous. Outside the range, specimen fragments will become less homogeneous.Figure 7
Correlation between specimen length-to-diameter ratios and average specimen fragmentation.


## 4.3. Fractal Dimension and Dynamic Loading Parameters

The specimen evolution process was to happen from microscopic damage to macroscopic fracture. Both the geometric feature and the digital feature of mechanical properties show good self-similarity [25]. After specimen fracture, the specimen broken fragments distribution is self-similar in Figure 4. As Figure 8 shows, the fractal dimension (D) decreases as the specimen length-to-diameter ratio (LDR) increases. Both D and LDR show a certain linear correlation. In the experiment, the length-to-diameter ratio was increased from 0.5 to 2, and the fractal dimension was decreased from 2.357 to 1.94.Figure 8
Correlation between specimen fractal dimension and specimen length-to-diameter ratios.With the specimens’ LDR increasing, the energy dissipation density decreases. When LDR is 0.5, the specimen energy dissipation density is 5.028 J/cm−3. When LDR is 2, the specimen energy dissipation density is 1.563 J/cm−3. With specimen energy dissipation density increasing, the specimen damage is more thorough. It can explain the result in Figure 4. When the specimen size is small, the specimen is broken more thoroughly than large-sized specimen fragments. It can be seen from the fitting curve that there is a critical LDR value; that is, the relationship between energy dissipation density and LDR is relevant in a certain LDR range.In the experiment, energy dissipation density can reflect the rock’s toughness characteristic, and energy dissipation density means the dissipated energy per unit volume of rock. The energy dissipation density parameter takes many influence factors into account in the test, and it is close to the actual toughness value. As shown in Figure9, with specimen fragment fractal dimension also increases, specimen energy dissipation density increasing. In other words, as the fractal dimension increases, the specimens are broken more thoroughly. Lots of cracks developed and expanded which caused rock damage. The energy dissipation density is increasing gradually, the crack extension velocity is increasing more rapidly, while the fractal dimension is also increasing gradually. Fractal geometry can well describe the regularity of irregular things in nature. Rock fragment fractal characteristics can reflect rock broken mode, and rock fragment fractal dimension shows some physical mechanism in the rock broken evolution process [28–30]. Under a 0.75 MPa impact loading condition, rock will suffer damage and failure. The rock broking process is also called the fractal developed process. Besides, the fragment fractal dimension is an ideal statistical parameter of rock fragments.Figure 9
Correlation between specimen fractal dimension and specimen broken energy density.With the fractal dimension increasing, the energy dissipation density is also increasing. It is shown that when the specimens are broken more thoroughly, the energy dissipated is higher. There is a positive correlation between fractal dimension and energy dissipation density.


## 5. Conclusions

This study used SHPB tests and studied dynamic granite fragmentation characteristics of different length-to-diameter ratios. The stress-strain curves and rock fragmentation characteristics were obtained. The sieving mesh is used to screen the rock fragments, and the rock fragmentation distribution curve and the fractal dimension (D) were obtained by the statistical date. Finally, the research analyzed the correlation between rock dynamic parameters and fractal dimension. The central findings are as follows:(1)
The granite energy-time history curve and the stress-strain curve show similar regular before the peak. The difference is that the stress-strain curve shows a downward trend in the postpeak stage, while the energy time history curve remains stable in the postpeak stage.(2)
The specimen fragment degree is negatively correlated with the specimen length-to-diameter ratio. The specimen energy dissipation density and the specimen fragment degree increase as the specimens’ length-to-diameter ratio increases. If the length-to-diameter ratio of specimens is 0.5–0.9, the time history curve is closer than others. The length-to-diameter ratios of rock specimens of 0.5–0.9 are more suitable for SPHB testing.(3)
With the specimen length-to-diameter ratios increasing, the rock specimen fractal dimension decreases. The granite specimens’ fractal dimensions are in the range of 1.94–2.536. The result shows that granite fragment distribution similarity is good.(4)
The specimen fractal dimension is positively related to the specimen’s energy dissipation density. With the fractal dimension increasing, the specimen energy dissipation density increases. In other words, the more energy per unit volume the specimen absorbs, the more thoroughly the specimen is broken.



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