#### Abstract

Deep mining involves complex geological environments. Moreover, along with strong disturbance, rockbursts and other severe dynamic hazards can occur frequently. Energy theory is widely regarded as the most appropriate method for understanding the mechanism of deep dynamic problems. When modeling dynamic disasters, energy theory includes the energy storage, energy accumulation, and energy transfer. To study the energy transfer characteristics in rock, a series of split-Hopkinson pressure bar (SHPB) impact tests were conducted with long granite specimens (400 mm in length and 50 mm in diameter) and modified incidence bars (having the same cross-sectional area but different shapes). The test results indicate that the impact energy decays exponentially with an energy attenuation coefficient of −0.42. For the scattering characteristics of energy in the rock, the scattering distance is found to be approximately three times the specimen diameter, which is very similar to Saint-Venant’s principle in elastic mechanics.

#### 1. Introduction

Deep mining is gradually becoming more prevalent owing to the lack of shallow mineral resources. In the deep underground, the geological environment becomes complex, particularly the high geostress. Under these circumstances, rockbursts and other dynamic disasters are quite severe and are frequently induced by strong disturbances. Recently, rockbursts have become the biggest threat in deep mining and have attracted wide concern. Shan and Yan investigated the locations and modes of rockbursts and evaluated methods to reduce their effect during the excavation of deep tunnels for the Jinping II Hydropower Station [1]. He and Dou established a gradient principle for rockbursts induced by horizontal stress in coal mining and proposed a criterion for layer dislocation rockbursts induced by horizontal stress [2]. Zhou et al. analyzed the mechanism of rockbursts induced by structural planes in deep tunnels and classified the rockbursts into three types [3]. Dou et al. studied the monitoring, forecasting, and prevention of rockbursts in underground coal mining in China [4]. Chen et al. investigated the effect of temperature on rockbursts in the hard rock of deep-buried tunnels and determined that the likelihood of a rockburst increased as the temperature increased [5]. Cai et al. conducted a quantitative analysis of seismic velocity tomography for rockburst hazard assessment [6].

A large number of studies have verified that energy theory is the most appropriate approach to describe the mechanism of dynamic disasters [7]. Thus, an increasing number of researchers have started to investigate rockbursts from an energy perspective. Jiang et al. employed a new energy index to investigate rockburst characteristics and their numerical simulation [8]. Kornowski and Kurzeja predicted rockburst probability using the given seismic energy and other factors defined by the expert method of hazard evaluation [9]. Sirait et al. used the energy balance and induced stress method to predict the rockburst of a cut and fill mine [10]. Feng et al. analyzed the fractal behavior of microseismic energy associated with immediate rockbursts in deep hard rock tunnels [11]. Wang et al. studied the characteristic energy factor of a deep rock mass under weak disturbance [12].

For considering dynamic disasters, energy theory includes the energy storage, energy accumulation, and energy transfer. Many studies have focused on stored energy in the rock. Hamiel et al. investigated the nonlinear behavior of damaged materials and conducted a large number of calculations and simulations of their elastic strain energy [13]. He et al. divided rockbursts into impact-induced bursts and strain bursts according to their stress path, and uniaxial compressive tests were performed to analyze the stored strain energy in strain bursts [14]. Li et al. carried out uniaxial compression tests at nine strain rates (ranging from 10^{−5} to 10^{−1} s^{−1}) and determined the increase of strain energy with the strain rate. The test results also showed that the elastic strain energy stored before reaching the peak led to brittle failure of the specimens [15]. Weng et al. used the energy index and strain energy density to analyze the energy accumulation and dissipation characteristics during the failure process of rock [16]. Zhou and Yang employed the strain energy method to investigate the dynamic damage localization features in a crack-weakened rock mass [17]. As discussed above, many studies have focused on strain energy storage, energy accumulation, and dissipation characteristics, but few have investigated energy transfer and propagation.

At present, Li et al. [18], Chen et al. [19, 20], Chai et al. [21], and Shao and Pyrak-Nolte [22] have studied stress wave propagation characteristics in rock. By employing a testing method for stress wave propagation, the energy propagation and scattering characteristics of rock specimens can be studied through laboratory tests. In this study, a series of split-Hopkinson pressure bar (SHPB) impact tests were conducted with long granite specimens (400 mm in length and 50 mm in diameter) and modified incidence bars (having the same cross-sectional area but different shapes). The results show that the impact energy exhibits an exponential decay, and the energy attenuation coefficient is −0.42. For the energy-scattering characteristics of rock, the scattering distance is approximately three times the rock diameter, which is very similar to Saint-Venant’s principle in elastic mechanics. Using numerical simulations, the test results are displayed and verified. These experimental results and numerical analyses can provide scientific guidance for in situ engineering projects.

#### 2. Laboratory Test of Energy Propagation

##### 2.1. Experimental Setup and Specimens

One-dimensional dynamic impact tests were conducted using an SHPB system (Figure 1) to determine the propagation and attenuation characteristics of the impact energy, as this is regarded as the most efficient approach for studying the dynamic characteristics of rocks.

To analyze the propagation and attenuation characteristics accurately, the rock specimens used must be sufficiently long. Here, the rock specimens were obtained from the Sanshandao gold mine in China. To reduce the influence of specimen heterogeneity on the results, all the cylindrical specimens were cut from a single block of deep granite rock without any visible cracks. The dimensions of the specimens are 400 mm in length and 50 mm in diameter. The rock block and rock specimens are shown in Figure 2.

For unconventional SHPB tests, due to the stress disequilibrium, traditional principles are not applicable. Scholars have recognized this problem and proposed some estimating methods [23] and monitoring approaches [24]. In tests, strain gauges were pasted on both specimens and bars. Based on the diameter of the rock specimen (*D*), strain gauges were pasted on the specimens at distances of 50 mm, 100 mm, 150 mm, 250 mm, and 350 mm (1*D*, 2*D*, 3*D*, 5*D*, and 7*D*) from the impacted end, as shown in Figure 3. An ultradynamic acquisition instrument was used to directly monitor the dynamic strains of specimens during the tests.

The shape and duration of the incident wave is one of the determinants of energy attenuation and scattering characteristics. To obtain stable sinusoidal dynamic strain curves and eliminate the dispersion effect, a fusiform bullet was used in the tests, as shown in Figure 4. A typical incident wave is shown in Figure 5.

As shown in the conceptual sketch in Figure 3, the transmitted bar and damper are indispensable components to prevent the reflected wave from affecting the test results. In this study, the length of the transmitted bar is 2 m. According to the actual measurements, for a wave velocity *c *= 5189 m/s, the time of the transmitted wave to propagate to the damper and then reflect back to the specimen is approximately 0.00077 s. Therefore, if the duration of the incident wave is less than 0.00077 s, there will be no reflection from the transmitted bar affecting the test results. The typical incident wave is shown in Figure 5, and the duration of the incident wave is approximately 0.00044 s.

##### 2.2. Analysis of Energy Propagation along the Axial Direction

For brittle failure of rock materials such as granite, there must be a damage threshold under impact loading. Li investigated the relationship between the minimum strain rate and bar diameter using the Holmquist–Johnson–Cook (HJC) model with the LS-DYNA software [25]. The fitted curves describing the relationship between the minimum strain rate and bar diameter are shown in Figure 6.

In this study, the diameter of the bar is 50 mm, and thus, the strain rate at the damage threshold is between 100 s^{−1} and 150 s^{−1} for the rock specimens. To eliminate plastic energy loss, the granite specimen should be kept in the linearly elastic state; hence, the impact velocity in the experiments is controlled at 4–6 m/s. After 12 sets of impact energy tests, the attenuation characteristics of the dynamic strain (maximum strain value) under different impact velocities were obtained, as shown in Figure 7.

In the elastic state, the energy in the specimen can be calculated using the following equation:where is the dynamic elastic energy, is the dynamic elastic modulus of the specimen, and is the dynamic strain.

After performing this calculation, the propagation and attenuation characteristics of the impact energy along the axial direction in the cylindrical specimens are fitted, as shown in Figure 8.

It can be seen that the impact energy decreases sharply at 1*D*, 2*D*, and 3*D*. When the distance exceeds 3*D*, the impact energy levels off gradually. In other words, the impact energy initially attenuates dramatically, and the attenuation value can be as high as 40%. However, as the distance continues to increase, the energy variation becomes more gradual. To describe this special characteristic of the elastic energy attenuation, the attenuation law was fitted.

After fitting, the energy attenuation law could be represented by a power function. The uniform expression is as follows:where *A* is the fitting coefficient, *N* is the monitoring position, and the exponent equal to −0.42 represents the average value of 12 fitted power functions.

The values of the fitting coefficient, *A*, are summarized in Table 1.

As Table 1 indicates, the fitting coefficient is approximately equal to the elastic energy at the 1*D* position. Hence, to ensure that each parameter in the above expression has an explicit physical meaning and to consider the acceptable error, Equation (2) is rewritten as follows:

Thus, the energy attenuation law for the rock is a power function based on the initial elastic energy and an exponent of −0.42.

#### 3. Laboratory Tests of Energy Scattering

##### 3.1. Laboratory Testing Method for Energy Scattering

Research has indicated that energy scattering is related to the impact mode and propagation distance. To determine the energy-scattering characteristics, a series of modified SHPB tests were carried out using incident bars with ends having the same cross-sectional area but different shapes, as shown in Figure 9. The outer diameters of the modified ends are 25 mm, 30 mm, 35 mm, and 40 mm (designated , , , and ), corresponding to 5/10, 6/10, 7/10, and 8/10 the diameter of the original incident bar () in the SHPB system, respectively. The dimensions of the modified ends are listed in Table 2.

##### 3.2. Analysis of Energy Scattering along the Radial Direction

By changing the position of action of the impact energy, the energy-scattering characteristics can be confirmed. For each modified end, tests were carried out four times under an impact velocity of 4–6 m/s. The dynamic strains at the 1*D*, 2*D*, 3*D*, 5*D*, and 7*D* locations were monitored. The calculated energy for the different ends is shown in Figure 10.

It can be observed that there are obvious differences in the trends from 1*D* to 3*D* with the different incident bars. That is to say, the impact energy scatters significantly from 1*D* to 3*D* (particularly at 1*D* to 2*D*), and the scattering characteristics are closely related to the impact modes. However, as the distance continues to increase, the energy variations are basically the same, which indicates that the different-shaped ends have little influence on the energy scattering at longer distances. This energy-scattering characteristic is very similar to Saint-Venant’s principle in elastic mechanics. Therefore, it can also be concluded that the impact energy with different action positions can only produce different effects before the distance of 3*D*. Once the distance exceeds 3*D*, the action modes have little effect on the energy.

#### 4. Numerical Simulations of SHPB Tests

With the development of numerical simulation techniques, some experimental results can be demonstrated visually. Here, using the numerical simulation software LS-DYNA, the energy propagation and scattering process was simulated. The parameters and dimensions used in the numerical simulation are the same as those of the rock specimens used in the lab tests. Figure 11 shows the energy propagation process along the axial direction at different times. It can be clearly seen that the impact energy propagates and attenuates with time.

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The energy-scattering characteristics can also be represented by the numerical simulations, as shown in Figure 12. It can be observed that, under different impact modes, the energy-scattering characteristics differ near the impact end; however, farther from the impact end (i.e., distances greater than 3*D*), the energy-scattering characteristics exhibit little difference. The numerical analysis results are thus consistent with the results of the tests.

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#### 5. Conclusions

In this study, a series of SHPB impact tests were conducted using long granite specimens and modified incidence bars to investigate the energy propagation and scattering characteristics in long cylindrical rock specimens. The main conclusions of this study are as follows.

Under impact, energy propagates and attenuates in the specimen along the axial direction. The experimental results suggest that the impact energy can be represented by a power function decay using the initial value and an energy attenuation coefficient of −0.42.

The impact energy scatters at distances of 1*D*–3*D*, and the scattering characteristics are closely related to the impact modes. However, at greater distances, the energy-scattering characteristics exhibit little difference. This energy-scattering characteristic is similar to Saint-Venant’s principle in elastic mechanics.

The energy propagation and scattering characteristics revealed by this study can be applied to the analysis of rockbursts using energy theory and are particularly relevant to the energy superposition problem caused by blasting or other excavations in deep mining, which can provide scientific guidance for in situ engineering projects.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by the State Key Research Development Program of China (No. 2016YFC0600703) and the National Natural Science Foundation of China (Nos. 51704014 and 51674013).