1 Introduction

In many underground rock engineering applications, such as traffic tunnel engineering, hydraulic engineering and mining, the rock masses are subjected to initial stress before construction. When the stressed rock masses are excavated with the drilling and blasting method, the rock masses are subjected to combined static–dynamic loading during the rock excavation. Taking traffic tunneling engineering as an example, cyclical footage is usually set to 0.5–3 m, and then the dynamic loading of blasting operations can be regarded as a cyclic process, as shown in Fig. 1. The rock is subjected to continuous dynamic disturbance. The energy generated by the blasting impact is input to the internal rock, leading to energy dissipation that causes the development of internal microcracks and irreversible deformation. Additionally, the continuous accumulation of internal damage leads to a significant degradation in its mechanical properties. Meanwhile, the rock is less able to withstand external impact loads, weakening the surrounding rock’s load-bearing capacity and stability. Moreover, the dynamic properties of the rock, which have a critical impact on the safety assessment of many underground rock projects, are severely affected once it is subjected to static stress. Dynamic properties of the rock are closely related to the evolution of energy, and it is necessary to investigate how the dissipated energy affects the dynamic properties of rocks under the action of the confining pressure with energy analysis methods. Hence, it is crucial to investigate the dynamic behaviors and energy evolution of rock under coupled static–cyclic impact loading to better understand the mechanical characteristics of the rock and provide basic data for the safety evaluation of tunnel engineering.

Fig. 1
figure 1

Diagram of rock subjected to coupled static–cyclic impact loading in tunnel engineering

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The Split-Hopkinson Pressure Bar (SHPB) is an important experimental means for studying the dynamic performance of rock (Frew et al. 2001; Hokka et al. 2016; Li et al. 2005; Tong et al. 2019). At present, the dynamic properties of rock specimens under cyclic impact are widely studied by researchers (Li et al. 2018a, b; (Wang et al. 2019a, b, 2022a, b; Shu et al. 2019; Dai et al. 2022). However, engineered rock masses are in a state of compressive stress before they are subjected to cyclic impact loading (Zhou et al. 2022; Xie et al. 2020), which will significantly affect the dynamic response of the rock (Xie et al. 2020; Gong et al. 2019). Several researchers have investigated the effect of the confining pressure on the dynamic properties of the rock. Wang et al. (2019a, b, 2021a, b) conducted cyclic impact tests on sandstone under confining pressures, and the results showed that the dynamic strength of the rock increased while the peak strain decreased with the increasing confining pressure. At the same confining pressure, the dynamic strength decreases while the peak strain increases with increasing impacts. The intact sandstone underwent compression-shear failure under confining pressure after cyclic impact. Tian et al. (2020) found that under the same cyclic impact load, with the increase of the confining pressure, the lower the strain rate of the rock, the more total impact cycles it can withstand. Moreover, the confining pressure inhibits the development of internal cracks, resulting in fewer specimen fragments in case of shear failure (Tian et al. 2020; Yu et al. 2020a, b). Meanwhile, a triaxial Hopkinson bar system (Liu et al. 2019; Xie et al. 2021; Liu and Zhao 2021; Luo et al. 2022) was used to conduct dynamic triaxial compression tests of rocks subjected to cyclic impact loading recently. Liu et al. (2021) carried out dynamic triaxial compression tests to study the degradation process and dynamic characteristics of the rock under triaxial cyclic impact loading. Luo et al. (2022) performed a true triaxial SHPB test on the granite subjected to repetitive impact loading to investigate the cumulative damage characteristics of the rock.

The energy evolution of rock under impact loading has also received considerable attention from researchers. There are many studies on the energy evolution of rock under single uniaxial impact, such as the effect of incident energy on the dissipation energy of rock (Liu et al. 2012; Lundberg 1976; Wang et al. 2022a, b; Gong et al. 2020) and the degree of damage (Whittles et al. 2006). However, most studies are based on the energy evolution of rock under static-single impact loading tests (Yin et al. 2012; You et al. 2022; Ma et al. 2021; Li et al. 2019a, b; Du et al. 2020) and only few studies are based on coupled static–cyclic impact loading tests, as summarized in Table 1. Wang et al. (2021a, b) conducted triaxial cyclic impact tests on sandstone, and the results show that the dissipated energy increase under the confining pressure with the increase of impact times, and the cumulative absorption energy increases with the increase of confining pressure (Wang et al. 2019a, b; Yu et al. 2020a, b) discovered that the propagation of original cracks and the initiation of new cracks are the internal reasons for the decrease in transmission energy and the increase in reflection energy. In their study, Liu and Zhao (2021) showed that the dissipated energy of rock increases nonlinearly with the increase in crack surface area and volume. Luo et al. (2022) found that the increase in the rock’s dissipated energy density is a significant contributor to the increase in its progressive damage. Research by Gong et al. (2023) found that axial prestressing results in a higher ratio of transmitted energy and a lower amount of energy absorbed in the rock. Consequently, it is crucial to study the dynamic mechanical properties, failure characteristics and energy evolution of rock subjected to coupled static–cyclic impact loading.

In this paper, dynamic tests were carried out on granite specimens using a modified split Hopkinson pressure bar (SHPB) to study the dynamic characteristics and energy evolution of the granite rock under a static axial pre-stress of 4 MPa, a confining pressure of 2 MPa, 4 MPa and 6 MPa, respectively, and at a repeated impact velocity of 19 m/s. The effect of confining pressure and impact times on mechanical property degradation, failure model and energy evolution characteristics of the rock was investigated.

Table 1 Previous studies on energy evolution of rock under impact loading
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2 Experiments

2.1 Sample preparation

Granite specimens with intact texture and good uniformity were used in the experiments. All the rock specimens were cored from one homogeneous granite block to avoid possible scattering of experimental results. The granite block is taken from the Qinling tunnel, Shaanxi Province, China. According to the rock mechanical performance test requirements, the specimens were processed and polished into cylindrical ones of φ 100 mm×100 mm, with both ends facing the ground and polished to ensure a non-parallelism and non-perpendicularity of less than 0.02 mm. Figure 2 shows the processed granite specimens.

Meanwhile, the mineral composition of the granite was analyzed using the rotating target X-ray diffractometer. Figure 3 shows the XRD patterns of the mineralogical features of the granite in this test.

Fig. 2
figure 2

Prepared granite specimens

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After the samples were processed, their mass, density and longitudinal wave velocity were checked to select granite samples with similar values for the repeated impact test (Table 2). In this study, three granite samples were prepared for an impact test under each confining pressure to ensure the reliability of the test results. Notably, the average quasi-static uniaxial compressive strength of the rock is 187.33 MPa.

Fig. 3
figure 3

Mineralogical features of granite

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Table 2 Granite samples and test conditions
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2.2 Test device

The SHPB system used in this experiment is shown in Fig. 4. The SHPB test system mainly includes a power system, bars, an axial pressure device, an active confining pressure device and a data acquisition system. The power system consists of high-pressure gas cylinders, an air gun and a striker bar. The striker bar and the bars have a diameter of 100 mm, and are both made of 40 Cr alloy steel, with a density of 7850 kg/m3, elastic modulus of 210 GPa, and longitudinal wave velocity of 5172 m/s. While the striker bar is only 800 mm long, the other bars, which include the incident bar, transmission bar and energy-absorbing bar, have a length of 5000 mm, 3000 mm and 2000 mm, respectively. A limiting device is installed along the incident bar close to the striker bar. While the incident bar and the limiting device move in the direction of the dynamic impact pulse, the displacement limiting device fixed to the ground prevents the incident bar from moving in the opposite direction. When the SHPB system is used for coupled static–cyclic impact loading test, axial stress is applied by hydraulic device at the end of the energy-absorbing bar, which generates axial stress in the specimen, incident bar, transmission bar and energy-absorbing bar. In the process of static load loading, axial pressure is applied before confining pressure. In the test, the impact velocity of the striker bar was measured by a laser velocimeter, and the NUXI-1008 ultra-dynamic strain gauge was used to collect the strain gauge signals.

Fig. 4
figure 4

Schematic diagram of SHPB system

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2.3 Test method

The SHPB coupled static–dynamic loading system is used in this test to conduct cyclic impact compression tests on granite specimens under different confining pressures. In the cyclic impact test, the granite samples were tested under a confining pressure of 2 MPa, 4 MPa and 6 MPa, and these confining pressures were selected to represent depths of approximately 80 m, 160 and 240 m, respectively. The value of axial pressure, 4 MPa, which falls between the maximum and the minimum confining pressures, was chosen for the test to better study the effect of confining pressure on the dynamic properties of the rock under combined dynamic-static load. During the test, the impact speed of the striker bar was controlled by adjusting the impact air pressure. The impact pressure of the air gun was a constant 1 MPa, and the impact speed was 19.0 m/s, as the speedometer displayed.

Based on the assumption of one-dimensional stress wave and stress uniformity (Kolsly 1949; Li et al. 2008), as well as the incident signal εI(t), reflected signal εR(t) and transmitted signal εT(t) collected by the strain gauge, the stress, strain and strain rate formulas of the specimens as a function of time could be obtained as follows:

$$\sigma (t)=\frac{{{E_b}{A_b}}}{{2{A_S}}}[{\varepsilon _{\text{I}}}(t)+{\varepsilon _{\text{R}}}(t)+{\varepsilon _{\text{T}}}(t)]$$
(1)
$$\varepsilon (t)=\frac{{{C_b}}}{{{L_S}}}\int_{0}^{t} {[{\varepsilon _{\text{I}}}(t) - {\varepsilon _{\text{R}}}(t) - {\varepsilon _{\text{T}}}(t)]\text{d}t}$$
(2)
$$\dot {\varepsilon }(t)=\frac{{{C_b}}}{{{L_{\text{S}}}}}[{\varepsilon _{\text{I}}}(t) - {\varepsilon _{\text{R}}}(t) - {\varepsilon _{\text{T}}}(t)]$$
(3)

where Ab is the cross-sectional area of the bars, Eb is the elastic modulus of the bars, Cb is the longitudinal wave velocity of the bars, Ls is the length of the specimen, and As is the cross-sectional area of the specimen.

In the SHPB test, the state of stress balance inside the specimen is a prerequisite for the validity of the test results (Han et al. 2022). Before the test, lubricant was evenly applied on both ends of the specimen to ensure good contact between the specimen and the bar’s end faces. A circular copper sheet was placed at the center of the impact end face of the incident bar as a wave shaper to improve the oscillation and dispersion effect of the rectangular incident wave. The pulse shapers in the SHPB tests are 4 cm in diameter and 0.1 cm in thickness. Figure 5 shows the typical stress balance in the test. In order to illustrate the stress balance problem in Fig. 5, we introduced the stress balance factor η (Li et al. 2014). As shown in Fig. 5, at 0µs, the stress wave propagates to the incident end of the sample, but it does not reach the transmission end of the sample, and the stress balance factor is 0.19. As time passes, the stress balance factor starts to decrease and fluctuates around zero within a small range. The result confirms that stress balance has been achieved during the loading process, thereby ensuring the accuracy of the test results.

Fig. 5
figure 5

Typical dynamic stress balance

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3 Test results

3.1 Dynamic stress–strain curve

The schematic diagram of the differential stress is shown in Fig. 6. The striker bar hits the incident bar to generate the incident stress σI, which is transmitted through the rock to the transmission bar to generate the dynamic total axial stress σ1. The SHPB test investigates the one-dimensional stress wave propagation in the rock. σ1 is the axial coupled stress in the rock under coupled load, which includes the axial pressure σ3 and the dynamic stress. To study the dynamic mechanical properties of the rock under coupled loading, the calculated differential stress (σ1–σ3) (Gong et al. 2019) is taken as the dynamic stress of the rock.

Fig. 6
figure 6

Schematic diagram of differential stress

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Figure 7 shows the stress–strain curves of the granite specimens under different confining pressures, and the impacts and their corresponding strain rate are given in the upper right corner of the figures. The dynamic peak stress is higher than the static stress. The resistance of the specimen against external impact load increases with the increase of the confining pressure. It can be seen from Fig. 7 that under the same cyclic impact load, when the confining pressure is 2 MPa, 4 MPa and 6 MPa, respectively, the number of impacts received by the specimens is 7, 10 and 15, respectively. The stress–strain curves of the specimens under the three confining pressures have the same variation trend, i.e., a “rebound” during the cycle impact. The stress–strain curves move to the lower right as a whole, and the dynamic peak stress decreases with the increase of impact times while the corresponding strain increases accordingly.

Fig. 7
figure 7

Cyclic impact dynamic stress–strain curves for specimens with the confining pressure of a 2 MPa; b 4 MPa; c 6 MPa

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3.2 Peak stress

Figure 8 shows the change of dynamic peak stress of the specimens under different confining pressures with the number (n) of cyclic impacts. It can be seen from the figure that the dynamic peak stress of the specimens under different confining pressures decreases gradually as the number of impacts grows. The fitting equation of the relationship between the dynamic peak stress and the number of impacts in the figure shows the strongest linear relationship between them when the confining pressure is 6 MPa, and in the meantime, the absolute value of the slope of the equation is the minimum. That is to say, the dynamic peak stress of the specimen drops at the lowest rate when the confining pressure is 6 MPa. Accordingly, when the confining pressure is 2 MPa and 4 MPa, the dynamic peak stress decreases faster than under 6 MPa.

Fig. 8
figure 8

Relationship between dynamic peak stress and the number of impacts

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Evidently from the above analysis, the existence of confining pressures has a positive effect on the increase of dynamic peak stress of the specimens. In order to explore the influence of confining pressure during cyclic impacts, the dynamic increase factor (DIF) ((Li et al. 2018a, b) is introduced to reflect the positive response of the dynamic mechanical properties of rock materials, which is defined as follows:

$$DIF=\frac{{{f_\text{d}}}}{{{f_\text{s}}}}$$
(4)

where fd and fs represent the dynamic peak stress and quasi-static compressive strength of the granite specimens, respectively.

Figure 9 shows the variation of DIF with the number of cyclic impacts on the granite specimens under different confining pressures. It can be seen that the DIF of the specimens during cyclic impact ranges from 1.2 to 1.7, indicating that the peak stress of the specimens during cyclic impact is higher than their quasi-static compressive strength of them. Under the same confining pressure, the DIF shows a decreasing trend with the increase of repeated impact numbers. As the confining pressure grows, the DIF increases, and the specimens’ dynamic peak stress increases more greatly.

Fig. 9
figure 9

Relationship between DIF and the number of impacts

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3.3 Peak strain

Figure 10 shows the relationship between the peak strain of the granite specimens and the number of cyclic impacts. It can be seen that the peak strain increases linearly with the increase in the number of impacts under different confining pressures. With the increase of confining pressure, the slope of the fitting function of the peak strain and the number of impacts decreases, and the increased peak strain rate gradually slows down.

Fig. 10
figure 10

Relationship between peak strain and the number of impacts

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3.4 Dynamic elastic modulus

The dynamic elastic modulus of rock is one of the most critical dynamic parameters (Carbajo et al. 2022). Since the stress–strain curve of rock changes nonlinearly during the cyclic impact, the dynamic elastic modulus of rock is not constant. During the dynamic impact process, there is no prominent compaction section in the stress–strain curve of the specimen, indicating good elastic deformation of it in the pre-peak stage. In this paper, 30% and 70% of the peak stress of the specimens were selected to calculate the slope of the line connecting the two points for the dynamic elastic modulus. As shown in Fig. 11, the dynamic elastic modulus is defined as follows:

$${E_d}=\frac{{{\sigma _{70}} - {\sigma _{30}}}}{{{\varepsilon _{70}} - {\varepsilon _{30}}}}$$
(5)

where σ70 and σ30 are 70% and 30% of the dynamic peak stress, respectively; and ε70 and ε30 are the strain values corresponding to σ70 and σ30.

Fig. 11
figure 11

Schematic diagram of dynamic elastic modulus

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Figure 12 shows the variation of the dynamic elastic modulus with the number of impacts. It can be seen that the dynamic elastic modulus varies from 30 to 160 GPa under different confining pressures. The dynamic elastic modulus reflects the mechanical properties of rock under impact loads. In the cyclic impact process, the rock’s dynamic elastic modulus decreases nonlinearly with the increase in impact times under the same confining pressure, and the mechanical properties of the specimens continuously deteriorate. The variation of the dynamic elastic modulus of the first three impacts has no dependence on the confining pressure. After the fourth impact, when the confining pressure is more significant, the value of the dynamic elastic modulus is smaller. However, when the specimens are finally destroyed, little difference is observed between their dynamic elastic modulus under different confining pressures. According to Fig. 12, the elastic modulus of the specimens under the confining pressure of 2 MPa, 4 MPa and 6 MPa are 41.03 GPa, 39.47 GPa and 34.45 GPa, respectively, at the last impact.

Fig. 12
figure 12

Dynamic elastic modulus with the number of impacts under different confining pressures

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3.5 Damage evolution

The rock’s damage evolution in the cyclic impact loading process is analyzed from the perspective of energy dissipation. The dissipated energy in the SHPB system can be calculated according to the “Energy evolution characteristics” section. The damage variable Di is defined as the total energy dissipated after each cyclic impact divided by the cumulative energy dissipated after rock impact failure (Wang et a1. 2021a, b):

$${D_i}=\frac{{\sum\nolimits_{{i=1}}^{n} {{W_\text{Si}}} }}{\varsigma }$$
(6)

where WSi and ζ can be obtained from Sect. 5.

Figure 13 shows the relationship between rock damage and the number of impacts defined based on energy dissipation. Under different confining pressures, this relationship can be fitted with a power function, indicating the nonlinear increase of the internal damage of rock with the increase in the number of impacts. The short dotted lines in Fig. 13 represent the change of the slope of the fitting equations, which shows the variation of the damage increase rate more intuitively. As the confining pressure grows, the slope of the fitting equations decreases, i.e., the damage increase gets slower, indicating that the confining pressure limits the expansion of internal cracks of the rock.

Fig. 13
figure 13

Variation of damage with the number of impacts

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4 Failure morphology of the specimens

The failure mode of the granite specimens under coupled static–cyclic impact loading is obviously different from that under one-dimensional impact loading, and the existence of static load changes the failure mode of the specimens. The presence of axial static pressure and circumferential confining pressure limits the deformation of the specimens and affects the crack development inside them. The failure forms of the granite specimens under the coupled static–cyclic impact loading are shown in Fig. 14. The confining pressure changes the crack propagation path of the specimens, and the failure mode is mainly a tensile failure, with no obvious axial splitting. This is because when the incident stress wave is transmitted to the rock-bar contact surface, part of the compressive stress wave is reflected to form a tensile stress wave. Under the action of the tensile stress wave, the granite will eventually undergo axial tensile failure. When the confining pressure is 2 MPa and 4 MPa, the specimens break into two relatively complete parts due to tensile failure, and the broken pieces can still be put together into a complete shape of the initial specimen without any obvious crack in the axial direction and with several fine cracks on the end faces. Meanwhile, all specimens have a wide radial crack that almost runs through their side face. However, when the confining pressure is 6 MPa, fragments are fall off from the tensile failure position of the specimens, obvious longitudinal cracks are observed in the axial direction, and the end face failure is mainly concentrated at the edges.

Fig. 14
figure 14

Failure modes of granite specimens

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5 Energy evolution characteristics

In the SHPB test, lubricant was applied to the contact surface between both ends of the specimens and bars. Therefore, the energy consumed by friction could be ignored. According to the SHPB loading mechanism and the law of energy conservation, the energy of the specimens can be calculated according to the following equations (Peng et al. 2020):

$${W_{\text{I}}}=\frac{{{A_{\text{b}}}{C_{\text{b}}}}}{{{E_{\text{b}}}}}\int {\sigma _{{\text{I}}}^{{\text{2}}}\text{d}t}= {A_{\text{b}}}{C_{\text{b}}}{E_{\text{b}}}\int {\varepsilon _{{\text{I}}}^{{\text{2}}}\text{d}t}$$
(7)
$${W_{\text{R}}}=\frac{{{A_{\text{b}}}{C_{\text{b}}}}}{{{E_{\text{b}}}}}\int {\sigma _{{\text{R}}}^{{\text{2}}}dt=} {A_{\text{b}}}{C_{\text{b}}}{E_{\text{b}}}\int {\varepsilon _{{\text{R}}}^{{\text{2}}}dt}$$
(8)
$${W_{\text{T}}}=\frac{{{A_{\text{b}}}{C_{\text{b}}}}}{{{E_{\text{b}}}}}\int {\sigma _{{\text{T}}}^{{\text{2}}}\text{d}t}= {A_{\text{b}}}{C_{\text{b}}}{E_{\text{b}}}\int {\varepsilon _{{\text{T}}}^{{\text{2}}}\text{d}t}$$
(9)
$${W_{\text{S}}}={W_{\text{I}}} - {W_{\text{R}}} - {W_{\text{T}}}$$
(10)

where WI, WR and WT are the incident energy, reflection energy, and transmission energy, respectively, and WS is the dissipation energy, it is dissipated by the rock during every cyclic impact loading, causing the development of more micro-cracks in the rock (Zhou et al. 2022), i.e., increased internal damage of the rock (Xie et al. 2009).

The data obtained from the test were analyzed and processed, and the energy change results of the granite specimens during cyclic impact were obtained, as shown in Table 3.

Table 3 Energy of specimens under coupled static–cyclic impact load
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5.1 Time history curve of energy

In order to further understand the dynamic mechanical properties of the granite specimens in the process of cyclic impact loading, the energy evolution of the specimens was analyzed from the perspective of energy dissipation. Due to the limited length of the paper, only the energy evolution of the granite specimens with time at failure is calculated, as shown in Fig. 15. Under different confining pressures, the energy evolution of the specimens during dynamic impact compression follows roughly the same trend, which can be divided into three stages:

In the first stage, the loading stress inside the specimens is about 20% of the dynamic peak stress in the initial loading stage. The energy values of each part are small and change very slightly.

In the second stage, the stress wave is gradually loaded until the peak incident stress (i.e., the dynamic peak stress corresponding to the stress–time curve of the specimens) is reached, and all energies increase significantly, of which the incident energy has the largest increase, followed by the transmitted energy. When the specimen is loaded with the peak stress of the incident stress wave, the reflection energy reaches about 70% of the maximum value. At this point, the wave impedance of the specimen and the bar increases compared with that before loading. According to the one-dimensional wave propagation theory, the reflected wave mainly depends on the wave impedance ratio between the bars and the specimen. The bars always remain elastic, and their generalized wave impedance is constant, indicating that the generalized wave impedance of the specimen decreases gradually. According to the definition of generalized wave impedance, the longitudinal wave velocity of the specimen decreases gradually during this process. Meanwhile, the dissipated energy of the specimen in this stage also increases.

In the third stage, with the gradual decrease of the incident stress wave, the stress of the specimens also decreases, and the incident energy, reflected energy and transmitted energy grow at a lower rate, while the dissipated energy grows faster than that in the previous stage.

Figure 15 shows that when the confining pressure is 2 MPa, 4 MPa and 6 MPa, the total reflected energy of the rock as a percentage of its total incident energy after the last impact is 8.48%, 8.34% and 7.75%, respectively. With the increase of the confining pressure, the proportion of reflected energy decreases and the reduction of generalized wave impedance decreases gradually. When the specimen fails, the greater the confining pressure, the smaller the reduction of the longitudinal wave velocity inside the rock, indicating that the confining pressure limits the internal crack propagation.

Fig. 15
figure 15

Evolution of energy over time: a 2 MPa; b 4 MPa; c 6 MPa

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5.2 Law of energy dissipation

The rock requires a certain amount of energy input to fail (Du et al. 2022; Yin et al. 2012). In the early stages of cyclic dynamic loading, each impact cycle’s incident energy is insufficient to cause rock rupture. The rock dissipates part of the incident energy, reducing the number of expanded internal microcracks. In other words, the increase of the internal damage of the rock is inadequate to cause rock rupture, but is enough to reduce the dynamic mechanical properties of the rock. As the number of cyclic impact loading increases, the incident energy continues to input into the damaged rock, and the damaged rock continues to dissipate energy, causing further expansion of microcracks. Eventually, the accumulated internal damage increases to the extent that the rock ruptures. With the increase in the number of impacts, the dissipated energy WS of the specimens is continuously accumulated, known as the cumulative dissipated energy ζ, which is defined as follows:

$$\zeta =\sum\limits_{{i=1}}^{n} {{W_{\text{Si}}}}$$
(11)

where WSi is the dissipation energy at the ith impact cycle, n is the number of cyclic impacts.

Figure 16 shows the variation of the dissipated energy and cumulative dissipated energy of the granite specimens with the number of impacts during cyclic impact loading under different confining pressures. The cumulative dissipated energy of the specimens increases with the number of impacts. The larger the confining pressure, the smaller the slope of the fitting line of the cumulative dissipated energy with the number of impacts and the slower the growth of the cumulative dissipated energy. However, the higher the confining pressure, the more impact cycles needed for the failure of the specimens and the greater the cumulative dissipated energy during failure. It can be seen from Fig. 16 that when the confining pressure is 2 MPa, 4 MPa and 6 MPa, the cumulative dissipated energy values at failure are 7.21 kJ, 9.88 kJ and 11.22 kJ, respectively.

Fig. 16
figure 16

Change of energy dissipation with the number of impacts: a 2 MPa; b 4 MPa; c 6 MPa

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5.3 Characteristics of the energy distribution ratio

In the SHPB test, the total incident energy depends on the velocity of the striker bar hitting the incident bar, while the reflected energy, transmitted energy and dissipated energy reflect the internal properties of the rock. Therefore, the reflected energy ratio δR, transmitted energy ratio δT and dissipated energy ratio δS are calculated to reflect the energy variation pattern of the granite specimens. The formula to calculate the energy ratios is defined as follows:

$${\delta _{\text{R}}}=\frac{{{W_{\text{R}}}}}{{{W_{\text{I}}}}}$$
(12)
$${\delta _{\text{T}}}=\frac{{{W_{\text{T}}}}}{{{W_{\text{I}}}}}$$
(13)
$${\delta _{\text{S}}}=\frac{{{W_{\text{S}}}}}{{{W_{\text{I}}}}}$$
(14)

Rock failure in its nature is the result of energy evolution, which involves the accumulation, dissipation and release of energy, the reasons for the degradation of rock’s mechanical properties and strength (Tong et al. 2019; Xie et al. 2005). Under cyclic impact load, the variation of the reflected energy ratio, transmitted energy ratio and rock’s dissipated energy ratio are shown in Fig. 17. According to Table 2, values of the incident energy under different impact cycle numbers are relatively stable for the same confining pressure. The variance coefficient of the incident energy with the confining pressures of 2 MPa, 4 MPa, and 6 MPa are 0.0933%, 0.1209%, and 0.0821%, respectively. As observed in Fig. 17, the values of δT decrease with the increase of the number of impacts, while the values of δR and δS present an increasing trend under the same confining pressure, indicating that the internal damage of the specimens continues to accumulate during the cyclic impact. With the same confining pressure, the values of δR, δT and δS are different at the same number of impacts: δT > δS > δR. The main reason is that the incident energy during the cyclic impact is insufficient to destroy the specimen. Specifically, most of the incident energy transfers to the transmission bar through the specimen, while the rest is reflected into the incident bar. At the same number of impacts, the value of δT increases with the increasing confining pressure, indicating the remarkable effect of the confining pressure in enhancing the dynamic mechanical property of the rock.

Fig. 17
figure 17

Variation of energy distribution with the number of impacts: a 2 MPa; b 4 MPa; c 6 MPa

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5.4 Dissipated energy density and cumulative specific energy dissipation

During the impact failure process, the dissipated energy is strongly related to the damage and failure of the specimen. Throughout the cyclic impact loading process, the rock is inside the confining pressure device and is subjected to the combined effect of the confining pressure and axial pressure. According to the obtained dynamic stress–strain curves of the rock specimens obtained, the maximum axial strain of the rock at each impact is in the range of about 0.004–0.012, which means that the axial compression length of the rock accounts for 0.4–1.2% of its total axial length (100 mm). Such a small proportion of axial deformation has little effect on the volume change of the rock. During the last impact, after the input of incident energy into the rock, the rock dissipates energy first and then breaks. Therefore, assuming that the volume of the specimen is constant during cyclic impact (Li et al. 2021; Lundberg 1976; Yu et al. 2020a, b), the dissipated energy density is introduced to represent the energy absorbed per unit volume of the specimen under impact load.

$${\omega _{\text{s}}}=\frac{{{W_{\text{s}}}}}{{{V_{\text{s}}}}}$$
(15)

where ωs is the dissipated energy density and Vs is the volume of the specimen.

The relationship between the dissipated energy density of the granite specimens and the number of impacts, as shown in Fig. 18, can be expressed as a power function. As indicated in the figure, the dissipated energy density of the specimens under different confining pressures tends to increase nonlinearly with the number of impacts. The slope variation curves of the power function for the dissipated energy density and the number of impacts under the three confining pressures are added as short-dotted lines in Fig. 18. The slope of the power function is used to indicate its variation trend with the increase of the number of impacts. The gradual decrease in the slope of the power function indicates a slower increase in the dissipated energy density. During repeated impact, the dissipated energy density decreases with the increase in the confining pressure. Hence, the higher the confining pressure, the stronger the restriction on deformation and mechanical property degradation of the rock.

Fig. 18
figure 18

Relationship between dissipated energy density and the number of impacts

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In order to better characterize the energy evolution of granite during cyclic impact, the cumulative specific energy dissipation (Li et al. 2019a, b) is defined as:

$$\xi =\sum_{{i=1}}^{n} {{\omega _{\text{si}}}}$$
(16)

where ξ is the cumulative specific energy dissipation, n is the number of cyclic impacts, and ωsi is the dissipated energy density in each cyclic impact.

Figure 19 shows the variation of cumulative specific energy dissipation with the increase in the number of cyclic impacts under different confining pressures. With the same confining pressures, the cumulative specific energy dissipation increases nonlinearly with the number of impacts. With the increase of confining pressure, cumulative specific energy dissipation growth gradually slows down. With the same number of impacts, the higher the confining pressure, the less the cumulative specific energy dissipation. The existence of a higher confining pressure increases the cumulative specific energy dissipation at the time of rock failure. For example, when the confining pressure changes from 2 to 6 MPa, the cumulative specific energy dissipation value grows from 9.19 to 14.3 J cm−3.

Fig. 19
figure 19

Variation of cumulative specific energy dissipation with the number of impacts

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6 Discussions

In this paper, the effect of confining pressure on the dynamic characteristics and energy evolution of rock is studied through cyclic impact loading tests conducted using the coupled static–dynamic SHPB equipment. When the rock is under circumferential confining pressure and axial pressure, more impact cycles to the rock lead to a decrease in its peak stress and an increase in its peak strain. As the confining pressure increases, the rock has a higher peak stress and can withstand more cycles of impact. Under confining pressure, the rock is broken into two large blocks perpendicular to the loading direction, and each block remains relatively intact with few fragments. By analyzing the energy in the coupled static–dynamic SHPB system, it can be concluded that when the rock receives approximately the same amount of incident energy, (a) the more impact cycles to it, the more energy reflected and dissipated in the system, and thus the less energy transmitted; and (b) the higher the confining pressure, the more energy transmitted and the less energy reflected and dissipated.

To analyze the development of internal damage and cracks of the rock under coupled static–dynamic cyclic impact, the static load on the rock should be removed by the end of each impact cycle before taking the rock out of the confining pressure device. During the cyclic impact, the rock should be put back into the device to apply static load. Both loading and unloading will change the stress state inside the rock (Chen et al. 2020; Li et al. 2017). Therefore, it is difficult to record the dynamic fracturing process inside the rock or capture high-speed fragmentation of it during the cyclic impact process using techniques such as X-ray CT and high-speed cameras. The damage evolution of rock is closely related to the evolution of energy. Under cyclic impact, the incident energy at each cycle cannot cause rock breaking, but it will cause energy dissipation, stimulating further development of internal damage. On this basis, the internal damage of rock is defined. With the increase in the number of impacts, the internal damage of the rock increases nonlinearly under different confining pressures. When the confining pressure is increased, the internal damage of the rock develops at a lower rate, and more impact cycles are required before the internal damage develops to the same level.

In order to better evaluate the safety of tunnel projects, and analyze when rock fracture occurs, non-contact techniques such as acoustic emission or numerical simulation methods can be used to further investigate the dynamic fracturing inside the rock under coupled static–dynamic cyclic impact loading.

7 Conclusions

  1. 1.

    The stress–strain curves of the specimens under the three confining pressures have the same variation trend, i.e., a “rebound” during the cycle impact, and the stress–strain curves move to the lower right as a whole.

  2. 2.

    In the combined dynamic–static cyclic impact test, the dynamic strength of granite increases significantly with the increase of confining pressure compared with the static load strength. When the confining pressure is 2–6 MPa, the dynamic increase factor (DIF) of granite is 1.2–1.7. Under the same confining pressure, the DIF decreases while the peak strain increases linearly with the number of cyclic impacts. However, confining pressure has no significant effect on dynamic elastic modulus.

  3. 3.

    The effect of the confining pressure on affecting the crack development of the rock is apparent, the increased rate of internal damage slows down with the increase in confining pressure. The confining pressure could change the crack propagation path of the samples, and the failure mode is mainly a tensile failure, with no obvious axial splitting.

  4. 4.

    The energy evolution of granite under the combined dynamic–static cyclic impact loading can be divided into three stages. When the dynamic stress of granite is 20% of its dynamic peak stress, the changes in dissipative energy, reflective energy and transmitted energy are negligible. When the dynamic stress reaches its peak value, all the energies increase significantly. As the dynamic stress decreases, the reflected energy and transmitted energy grow at a lower rate while the growth of dissipative energy accelerates.

  5. 5.

    Under the same confining pressure, the transmitted energy ratio decreases linearly. In contrast, the cumulative dissipated energy, the reflected energy ratio, the dissipated energy ratio, the dissipated energy density and the cumulative specific energy dissipation tend to increase with the increasing number of repeated impacts. By increasing the confining pressure, the energy dissipation capacity of the granite rock is significantly improved, and the failure process of the rock could be slowed down.