1 Introduction

With the gradual depletion of shallow mineral resources, deep mining has gradually become the new normal of resource development. In the future, the mineral resources development will fully enter the status quo of 1000–2000 m deep (Wu et al. 2016a; Jiang et al. 2020). The existing research shows that the ground stress will be as high as 20–50Mpa (Gong et al. 2023a; Zhou et al. 2021; Stacey and Wesseloo 2022) after the resources mining enters the deep part of 1000 m, and the rocks in underground engineering are often in the multi-directional compressive stress state (Xia et al. 2021; Ma et al. 2023). Frequent large-scale mining activities, sudden earthquakes and accidental explosions and other extreme loads will inevitably lead to different degrees of dynamic disturbance. Under the complex static-dynamic load shown in Fig. 1, the deep rock mass is easy to induce the problems of high frequency and large amount of surrounding rock catastrophic instability, which poses a great challenge to the deep rock engineering construction (Dai et al. 2016; Xia et al. 2020; He et al. 2015). Therefore, it is of great significance to study the dynamic properties and failure behavior of rocks under the coupling effect of complex in-situ stress state and impact load, for the prediction and control of deep high geostress surrounding rock disasters.

Fig. 1
figure 1

Typical dynamic and static combined load state of deep rock (Xia et al. 2021; Ma et al. 2023)

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The deformation characteristics, failure strength and fracture toughness of rock under external loads show obvious strain rate effect (Cao et al. 2023; Alneasan and Behnia 2021; Xu et al. 2023). At present, many scholars have studied the failure mechanical behavior of rocks under three loading rates of quasi-static, medium strain rate and high strain rate. In terms of quasi-static loading, the progressive failure process of rock is mainly evaluated based on acoustic emission (Ding et al. 2022; Tan et al. 2019), acoustic wave velocity (Wu et al. 2021), elastic modulus (Liu et al. 2021a) and other indicators, and the confining pressure effect and the loading rate dependence of rock brittle failure are also considered (Gong et al. 2022; Mahmoud and Mahmoud 2022). In the aspect of medium strain rate loading, full digital confining pressure servo control system (Li et al. xxxx) and drop hammer impact test system (Wang et al. 2017) are often used to study the progressive damage and fracture characteristics of brittle rock when the strain rate ranges from 10–2 to 101 s−1. In general, the dynamic response characteristics of surrounding rock mass induced by large-scale mining of deep rock mass, mechanical operation and special extreme load belong to the research category of high strain rate (101–103 s−1). In the aspect of high strain rate loading, scholars have achieved rich research results in the dynamic strength (Tao et al. 2023; Demirdag et al. 2009), fracture toughness (Chen et al. 2016; Dong et al. 2018), damage constitutive relation (Ou et al. 2019; Zhai et al. 2022), fracture fractal (Wang et al. 2022; Huang et al. 2017) and energy dissipation (Luo et al. 2020; Feng et al. 2021) of rock-like materials based on the split Hopkinson pressure bar.

The confining pressure SHPB device can be used to simulate several typical in-situ stress states of deep rocks, such as axial confining pressure, hydrostatic pressure and triaxial confining pressure. Based on this, some important research results have been made on the static and dynamic loading coupling effect of rock-like materials. In the aspect of dynamic compressive properties of rock under the combined action of axial stress (σ1 ≥ σ2 = σ3 = 0) and impact load. Li et al. (2008) found that when the axial stress was 15–60% of the quasi-static compressive strength of siltstone, it was beneficial to improve the dynamic compressive strength of rock, while when the axial confining pressure exceeded 80% of the quasi-static compressive strength, its dynamic compressive strength decreased rapidly. Zhou et al. (2020) showed that when the axial confining pressure was fixed, the dynamic stress of rock showed obvious strain rate effect, and the dynamic failure degree was positively correlated with the axial confining pressure. Zhang et al. (2023) and Liu et al. (2021b) respectively conducted experimental studies on the development process and ultimate failure mode of dynamic cracks in rocks with prefabricated parallel cracks and intact rocks under the coupling of axial confining pressure and dynamic load. In the aspect of dynamic compression properties of rock under the combined action of hydrostatic pressure (σ1 = σ2 = σ3 ≥ 0) and impact load. Relevant scholars' research showed that the dynamic compression strength and peak strain of rock under hydrostatic pressure had obvious strain rate effect, and the dynamic compression strength of rock was positively correlated with hydrostatic pressure at the same loading rate (Gong et al. 2019; Frew et al. 2010). In addition, the research of Hokka et al. (2016) showed that hydrostatic pressure had a significant influence on the dynamic compression failure mode of granite, and when hydrostatic pressure reached a certain level, the rock mainly undergo shear failure. Wu et al. (2016b) considered that the dynamic tensile strength of granite was positively correlated with the hydrostatic pressure. In the aspect of dynamic compressive properties of rock under the combined action of triaxial stress (σ1 ≥ σ2 = σ3 ≥ 0) and impact load. Ma et al. (2019) believed that when the axial stress was less than or equal to the threshold of rock elastic limit, the coupling effect of axial stress and lateral confining pressure had a significant enhancement effect on the dynamic compressive strength of rock. Li et al. (2020) found that when granite was exposed to high temperature below 600 °C, the lateral confining pressure and strain rate significantly enhanced the dynamic compressive strength of rock, while when the temperature was above 600 °C, the dynamic compressive strength decreased rapidly, and the weakening effect of thermal damage was obviously higher than the strengthening effect of confining pressure.

To sum up, when the improved confining pressure SHPB device is used to simulate the triaxial stress state of deep surrounding rock, only the conventional triaxial stress state (σ2 = σ3) can be satisfied, and the dynamic loading of rock under the constraint of true triaxial stress (σ1 > σ2 > σ3 ≠ 0) cannot be realized, nor can the coupling effect of the single lateral confining pressure and impact load be realized. Therefore, the existing laboratory experimental research can not be able to truly simulate the in-situ complex stress state of deep rock. To this end, this paper introduces a triaxial Hopkinson pressure bar test system for dynamic fracture characteristics of rock materials. Fully considering the coupling effect of the constraint states of uniaxial stress, biaxial stress and true triaxial stress and impact load, the strain rate effect and stress constraint effect of dynamic mechanical response characteristics and failure behavior of sandstone under complex static-dynamic load coupling are studied, and the progressive damage evolution law of sandstone under the coupling of true triaxial stress constraint and high strain rate is discussed.

2 Sample preparation and test system

2.1 Sample preparation

In this experiment, sandstone is taken as the research object, and the rock integrity and homogeneity are good. Figure 2a shows the scanning experiment results of sandstone slices by polarizing microscope, which are mainly composed of quartz (72%), feldspar (13%), calcite (9%), iron (3%), sericite (1%), chlorite (1%). The sample is processed into a cube of 50 × 50 × 50mm3, as shown in Fig. 2b. The non-parallelism error between the end faces of the sample is required to be less than 0.02 mm, and the maximum deviation of the perpendicularity of the adjacent faces of the cube is less than 0.25°. The longitudinal wave velocity of the air-dried sandstone is 4038–4189 m/s, and the density is 2.23–2.34 g/cm3.

Fig. 2
figure 2

Sandstone sample and its mineral components

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2.2 Test method and principle

Using the true triaxial Hopkinson bar shown in Fig. 3a can realize the multi-axial dynamic and static combined loading of concrete and rock brittle materials, and the detailed composition of the system can be found in the author's previous research (Gong et al. 2023b). In the impact direction (X-axis), there were a high-pressure air gun, a square incident bar (2.5 m), a square transmission bar (2.0 m), a hydraulic cylinder, and a wave absorber. Four square output bars with the length of 2.0 m were set along the horizontal Y-axis and vertical Z-axis. The principle of dynamic triaxial compression test for sandstone samples is shown in Fig. 3b. Before impact load is applied, static loads of any size are applied along the X-axis, Y-axis and Z-axis directions respectively, and then the bullet is given a certain speed to impact the incident bar along the X-axis direction. After the impact signal (σIn) is transmitted to the sample, the transmitted signal (σTr) and the reflected signal (σRe) will be generated respectively. At the same time, due to Poisson effect, the square bars along the Y axis and Z axis directions will also output certain dynamic signals (σy1, σy2, σz1 and σz2).

Fig. 3
figure 3

Triaxial Hopkinson pressure bar system

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The strain rate, strain and stress can be calculated according to the following formula (Gong et al. 2023c).

$$\dot{\varepsilon }_{{\text{X - dyn}}} (t) = \frac{{C_{b} }}{{L_{{{\text{X}} - {\text{S}}}} }}\left[ {\varepsilon_{{{\text{In}}}} (t) - \varepsilon_{{{\text{Re}}}} (t) - \varepsilon_{{{\text{Tr}}}} (t)} \right]$$
(1)
$$\dot{\varepsilon }_{{\text{X - dyn}}} (t) = \frac{{C_{b} }}{{L_{{{\text{X}} - {\text{S}}}} }}\int\limits_{0}^{t} {\left[ {\varepsilon_{{{\text{In}}}} (t) - \varepsilon_{{{\text{Re}}}} (t) - \varepsilon_{{{\text{Tr}}}} (t)} \right]dt}$$
(2)
$$\sigma_{{\text{X - dyn}}} (t) = \frac{{A_{0} }}{{2A_{{{\text{X}} - {\text{S}}}} }}E_{b} \left[ {\varepsilon_{{{\text{In}}}} (t) + \varepsilon_{{{\text{Re}}}} (t) + \varepsilon_{{{\text{Tr}}}} (t)} \right]$$
(3)

where A0, Eb and Cb are cross-sectional area, Young's modulus and P wave velocity of square bar respectively. Ai-S and Li–S are the cross-sectional area and length of the specimen in each direction, where i represents the X-axis, Y-axis or Z-axis respectively. εIn, εRe and εTr are incident strain, reflected strain and transmitted strain respectively.

Similarly, it is assumed that the sample is in a stress equilibrium state during the dynamic loading process along the Y axis and Z axis directions. The calculation formulas of strain rate, strain and stress are as follows:

$$\dot{\varepsilon }_{{i{\text{ - dyn}}}} (t) = \frac{{C_{b} }}{{L_{{i - {\text{S}}}} }}\left[ {\varepsilon_{i + } (t) + \varepsilon_{i - } (t)} \right]$$
(4)
$$\varepsilon_{{i{\text{ - dyn}}}} (t) = \frac{{C_{b} }}{{L_{{i - {\text{S}}}} }}\int\limits_{0}^{t} {\left[ {\varepsilon_{i + } (t) + \varepsilon_{i - } (t)} \right]dt}$$
(5)
$$\sigma_{{i{\text{ - dyn}}}} (t) = \frac{{A_{0} }}{{2A_{{i - {\text{S}}}} }}E_{b} \left[ {\varepsilon_{i + } (t) + \varepsilon_{i - } (t)} \right]$$
(6)

where εi+ and εi- are the signals received on the square bars in each direction of the Y-axis and Z-axis, respectively.

2.3 Test scheme and dynamic stress equilibrium analysis

In order to study the dynamic failure mechanical behavior and progressive damage evolution law of sandstone under complex static-dynamic load coupling, a triaxial Hopkinson pressure bar system is used to carry out the dynamic compression test of sandstone. The coupling effect of the constraint states of uniaxial stress, biaxial stress and true triaxial stress and impact load is fully considered. The specific test conditions are shown in Table 1. In order to ensure that the strain gauge in Y-axis direction can capture the lateral expansion signal of sandstone samples due to Poisson effect at σ2 = 0 MPa, the steel bar is contacted with sandstone samples during the experiment, but no stress is applied, so it is recorded as 0 + in data processing. It should be noted that in order to obtain multiple failure patterns of sandstone specimens under dynamic and static loading, the impact velocity was determined to be 22 m/s through pre-test. In addition, to reduce the dispersion of the test results, three samples were tested for each condition to corroborate the results.

Table 1 Dynamic and static combined loading test conditions
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Taking the coupling effect of true triaxial stress (σ1 = 30 MPa, σ2 = 20 MPa, σ3 = 10 MPa) constraint and impact load (22 m/s) as an example, the stress wave transmission law of sandstone samples during static and dynamic combined loading is discussed, and the dynamic stress equilibrium state during loading process is analyzed. Firstly, during the dynamic triaxial compression test of sandstone samples, the stress wave signals along the impact direction are completely recorded, as shown in Fig. 4a. The initial prestress loading level results in a tensile platform with a magnitude of approximately 31.86 MPa between the incident and reflected waves (Liu et al. 2019). Furthermore, the analysis of the three-wave method (Huang et al. 2022; Xu et al. 2020; Lv et al. 2017) shows that the incident stress wave plus the reflected stress wave is approximately equal to the transmitted stress wave (i.e. σIn + σRe ≈ σTr), as shown in Fig. 4b.

Fig. 4
figure 4

Typical stress wave transfer law of sandstone samples

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In addition, under the impact compression, the sandstone sample undergoes lateral expansion due to the Poisson effect, and the characteristic signal of the stress wave recorded thereby is shown in Fig. 4c. Furthermore, Fig. 4d shows that the dynamic stresses (σY+, σY-, σZ+ and σZ-) generated along the Y-axis and Z-axis directions are basically synchronous with time evolution. Figure 4e shows the evolution curves of total stress (σi + σi-dyn(t), i = 1, 2, 3) along the three directions of the sample during dynamic compression. The evolution of dynamic stress in the three directions has a good synchronization with time, the stress peaks are reached at about 105 μs (227.73 MPa, 53.51 MPa and 44.99 MPa, respectively). The above analysis shows that the stress wave satisfies the dynamic stress uniformity in the sandstone samples, that is, the stress balance between the incident end and the transmitted end of the samples, which verifies the validity of the test results (Zhou et al. 2012; Li et al. 2014; Wang et al. 2020).

As shown in Fig. 4f, because the sandstone sample is not completely broken under impact, the dynamic stress–strain curve has a certain degree of rebound effect (Millon et al. 2016). At the same time, the evolution law of dynamic stress–strain curves along the Y axis and Z axis directions due to Poisson effect is basically the same as that along the impact direction, but there is a big difference in peak stress.

3 Analysis of dynamic response test results of sandstone

3.1 Dynamic properties of sandstone under axial stress constraint

The loading level of axial stress σ1 is designed according to 0%, 25%, 50% and 75% of quasi-static compressive strength of sandstone (40.17 MPa), and the impact velocity is designed to be 22 m/s. It can be seen from Fig. 5 that under no axial stress constraint, the dynamic stress–strain curve shows obvious four stages of compaction, linear elastic deformation, nonlinear deformation and post-peak failure. Similar phenomena have been found in uniaxial SHPB compression tests of rock-like materials (Han et al. 2022; Li et al. 2017). However, when a certain axial stress constraint is imposed, the compaction stage disappears, because the microcracks inside sandstone samples are further compacted under initial prestress.

Fig. 5
figure 5

Dynamic stress and deformation of samples under axial stress constraints

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In addition, the axial stress constraint will also change the dynamic compressive strength and post-peak deformation characteristics of sandstone samples, and the details can be obtained from Fig. 5b. When σ1 is 0 MPa, the dynamic compressive strength of sandstone samples is 134.40 MPa, which is 3.35 times of the quasi-static compressive strength. At this time, the strain rate is 186.65 s−1, showing significant rate dependence. When σ1 increases from 0 to 30 MPa, the peak stress of sandstone samples decreases from 134.40 to 100.30 MPa (a decrease of 25.37%). This is because with the increase of axial stress, the microcracks in sandstone samples are activated. Even when the axial stress exceeds 50% of the quasi-static compressive strength, new microcracks will germinate, and a certain degree of damage has been generated before dynamic loading, thus resulting in a gradual decrease in the dynamic compressive strength of sandstone samples. Similarly, when σ1 increases from 0 to 30 MPa, the peak strain decreases from 0.0216 to 0.0181 (a decrease of 16.20%), which means that the axial stress constraint induces the rock sample to fail earlier. It is similar to the conclusion that the high ground stress rock mass has strong impact tendency in practical engineering (Ju et al. 2021).

As shown in Fig. 6, when the impact velocity is 22 m/s, the sandstone samples are completely broken into small fragments or powder. The axial stress constraint changes the strain rate. When σ1 increases from 0 to 30 MPa, the strain rate increases from 186.65 to 233.44 s−1 (an increase of 25.37%). With the increase of strain rate, fragments decrease and more powder is produced. Because the energy absorbed by the fracture sample increases with the increase of strain rate, more energy dissipates in the initiation, expansion, and penetration of new cracks, which increases the energy dissipation density of the rock sample and gradually reduces the geometric size of the fragments (Li et al. 2005; Luo et al. 2023).

Fig. 6
figure 6

Dynamic failure modes of sandstone samples under axial stress constraints

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3.2 Dynamic properties of sandstone under biaxial constraint

In the coupling loading test of biaxial stress constraint and impact load, the stress constraint conditions imposed along X-axis and Z-axis directions are kept unchanged, namely σ1 = 30 MPa and σ3 = 0 MPa respectively. The lateral (Y-axis) stress σ2 is considered to be 0 MPa, 10 MPa, 20 MPa and 30 MPa in turn. It can be seen from Fig. 7a that as σ2 increase from 0 to 30 MPa, the dynamic peak stress of sandstone samples gradually increases, and the size of σ2 directly determines the shape of the dynamic stress–strain curve at the post-peak stage.

Fig. 7
figure 7

The influence of lateral stress σ2 on the dynamic stress and deformation characteristics of sandstone samples

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The sandstone sample is compressed in the impact direction and expands laterally due to Poisson effect, as shown in Fig. 7b. It can also be seen that the lateral stress σ2 also affects the dynamic peak stress along the Y-axis direction, but does not change the shape of the dynamic stress–strain curve. More details about the influence of lateral stress σ2 on the dynamic stress and deformation characteristics of sandstone samples can be obtained from Fig. 7c, d. When σ2 increases from 0 to 30 MPa, the peak stress of sandstone samples along the impact direction increases from 137.90 to 170.08 MPa (an increase of 23.24%), the strain rate decreases from 169.34 to 136.09 s−1, and the peak stress along the Y-axis direction increases from 16.50 to 24.34 MPa (an increase of 47.52%). In addition, the lateral stress σ2 also affects the peak strain of sandstone samples. Because the sandstone samples only have small strain under dynamic impact failure, the test results have certain discreteness, but the overall evolution law is as follows. When σ2 increases from 0 to 30 MPa, the peak strain of sandstone samples along the X-axis direction decreases from 0.0175 to 0.0115 (a decrease of 34.29%), while it increases by 33.33% along the Y-axis direction. The above analysis shows that when the axial stress is fixed, the lateral stress constraint limits the crack development in the impact direction of sandstone samples, reduces the damage degree of the samples and improves the dynamic compressive strength of the samples.

Further, in order to study the strain rate effect of dynamic failure mechanical behavior of sandstone under biaxial stress constraint, the initial stress constraint state (σ1, σ2, σ3) = (30, 20,0)MPa is kept unchanged, and the impact velocity is considered as 9 m/s, 11 m/s, 13 m/s, 15 m/s, 17 m/s, 20 m/s and 22 m/s. Figure 8a, b show the stress–strain curves of sandstone samples along the X-axis and Y-axis directions respectively. The magnitude of impact velocity has obvious influence on the shape of the stress–strain curve in the X-axis direction at the post-peak stage. The initiation and propagation of microcracks can still be induced under low-velocity impact, resulting in permanent deformation occurs after unloading (Li et al. 2018; Huang et al. 2023). As the impact velocity gradually increases, the sandstone sample is irreversibly destroyed and the hysteretic loop gradually disappears.

Fig. 8
figure 8

The strain rate effect of dynamic failure mechanical behavior of sandstone under biaxial stress constraint

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More detailed data about the influence of strain rate on dynamic stress and deformation characteristics of sandstone under biaxial stress constraint can be obtained in Fig. 8c, d. With the increase of impact velocity, the strain rate shows a linear growth relationship. When the strain rate increases from 44.54 to 154.41 s−1, the dynamic compressive strength along the X-axis direction increases from 60.15 to 158.21 MPa (an increase of 163.03%). The conclusion that the increase of strain rate leads to the increase of dynamic compressive strength of rock is also consistent with the conventional SHPB test (Zhu et al. 2022; Jin et al. 2018).

Figure 9 shows the typical dynamic failure mode of sandstone samples under biaxial stress constraint. When the initial stress constraint state (σ1, σ2, σ3) = (30, 20,0)MPa remains unchanged, the magnitude of the strain rate determines the failure degree and failure mode of sandstone samples. At a low strain rate (58.10 s−1), only a slight splitting failure occurs at the stress concentration at the end of the sample. When the strain rate is between 58.10 and 126.65 s−1, the macroscopic fracture develops approximately along the direction parallel to σ1. When the strain rate reaches 147.58 s−1, the macroscopic fracture direction is approximately parallel to σ1 direction and intersects in the σ3 direction with a certain inclination angle, showing an obvious inclined shear failure mode.

Fig. 9
figure 9

Dynamic failure modes of sandstone samples under biaxial stress constraints

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3.3 Progressive damage evolution of sandstone under true triaxial stress constraint

In order to study the influence of intermediate principal stress σ2 on the dynami cfracture characteristics of sandstone samples under the true triaxial (σ1 ≥ σ2 ≥ σ3 ≠ 0) constraint, the stress constraint conditions in the X-axis and Z-axis directions are kept unchanged, namely σ1 = 30 MPa and σ3 = 10 MPa respectively, and the lateral stress in the Y axis direction is considered as 0 MPa, 10 MPa, 20 MPa and 30 MPa respectively. Figure 10a, b show the dynamic stress–strain curves of sandstone samples along the X-axis, Y-axis and Z-axis directions. With the increase of the intermediate principal stress σ2, the shape of the stress–strain curve does not change significantly, but there are only differences in dynamic peak stress and peak strain.

Fig. 10
figure 10

Dynamic peak stress and deformation characteristics of sandstone samples under the true triaxial dynamic and static loading coupling

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The detailed changes can be obtained from Fig. 10c, d. When the static load σ2 increases from 0 to 30 MPa, the peak stress of the sandstone samples along the impact direction increases from 183.32 to 204.28 MPa (an increase of 11.43%), and the growth rates of dynamic peak stress s along the Y-axis and Z-axis directions are 51.54% and 21.43% respectively. It shows that the intermediate principal stress constraint can enhance the dynamic stress of sandstone samples. In addition, because the lateral stresses (σ2 and σ3) inhibit the deformation and failure of sandstone samples along the impact direction, when the intermediate principal stress σ2 increases from 0 to 30 MPa, the peak strain along the impact direction gradually decreases (the reduction rate is 31.03%). However, the peak strains along Y-axis and Z-axis directions increase, with the growth rates of 43.68% and 18.22% respectively. It is worth noting that no matter how the intermediate principal stress σ2 changes, there are numerical differences between the dynamic strength and the peak strain generated by sandstone samples along the Y-axis and the Z-axis directions. This is due to the uneven size and distribution of microcracks or micropores in the sandstone sample, and the differences decrease with the increase of σ2.

Under the constraint of true triaxial stress (σ1 ≥ σ2 ≥ σ3 ≠ 0), it is not easy for sandstone to produce obvious macroscopic fracture under the single action of high strain rate (You et al. 2021; Liu and Zhao 2021). Therefore, the progressive damage evolution law and failure behavior of sandstone samples under the coupling effect of true triaxial stress (30, 20, 10)MPa constraint and impact load (22 m/s) are further discussed here. Figure 11a shows the dynamic stress–strain curve of sandstone samples subjected to 10 impacts. The dynamic peak stress and elastic modulus decrease with the increase of impact times. This is because the damage of sandstone samples continues to accumulate under repeated impacts and internal cracks continue to initiate and expand, resulting in a decrease in bearing capacity. Interestingly, the shape of dynamic stress–strain curve has not changed obviously under the first seven impacts. From the eighth impact, there is an obvious "plateau" phenomenon at the stress peak stage, which indicates that the failure of sandstone samples has gradually changed from brittleness to plasticity. In addition, the dynamic stress–strain curves along the Y-axis and Z-axis directions are shown in Fig. 11b. The dynamic peak stress and peak strain show a trend of first increasing and then decreasing with the increase of impact times, indicating that multiple repeated impacts affect the Poisson's ratio of sandstone samples.

Fig. 11
figure 11

The progressive damage evolution law of sandstone samples under the true triaxial dynamic and static loading coupling

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More information about the progressive evolution of dynamic peak stress and deformation characteristics of sandstone samples under repeated impacts can be obtained from Fig. 11c, d. By analyzing the dynamic stress and deformation characteristics of sandstone samples along the impact direction (X-axis), it is found that the dynamic compressive strength and elastic modulus of the samples are improved during the second impact, and the peak strain is lower than that of the first impact. It indicates that the first impact plays a compaction role, resulting in the improvement of the deformation resistance of the samples during the second impact. Subsequently, until the 8th impact, the dynamic strength decreased from 204.04 to 189.71 MPa, and the peak strain increased by 22.84%, indicating that the sandstone sample has a damage accumulative effect under repeated impacts. Under the next two impacts, the dynamic compressive strength rapidly decreases to 154.78 MPa (the reduction rate is 18.41%), and the peak strain increases by 28.92%. With the sudden increase of strain rate, the bearing capacity of sandstone samples is completely lost after the 10th impact. However, the evolution characteristics of dynamic peak stress and peak strain of sandstone samples along Y-axis and Z-axis directions with impact times are significantly different from those along X-axis direction. The main performance is that from the first impact to the eighth impact, the dynamic peak stress s along the Y-axis and Z-axis directions increase with the increase of impact times, with the growth rates of 33.83% and 34.65% respectively, and the peak strain also increases with the impact times, indicating that Poisson's ratio increases with the damage accumulation of sandstone samples. During the last two impacts, the dynamic strength and peak strain along the Y-axis and Z-axis directions decrease, indicating a decrease in Poisson's ratio of sandstone samples.

Figure 12 shows that the stress constraint state remains unchanged at (σ1, σ2, σ3) = (30, 20,10)MPa, and the impact times determine the failure degree and failure mode of sandstone samples. During the first eight impacts, only slight splitting occurs in the stress concentration area at the end of the sample. Until after the 8th impact, the significant macroscopic fracture occurs in the sample and the crack direction is approximately parallel to the σ1 direction. After the 10th impact, the sandstone sample completely lost the bearing capacity, mainly in an inclined shear failure mode.

Fig. 12
figure 12

Progressive failure behavior of sandstone under true triaxial stress constraint

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4 Discussion

4.1 Stress constraint effect of dynamic stress and deformation characteristics

The effects of the three constraint states of uniaxial stress (σ1 > σ2 = σ3 = 0), biaxial stress (σ1 ≥ σ2 > σ3 = 0) and triaxial stress (σ1 ≥ σ2 ≥ σ3 ≠ 0) on the dynamic stress and deformation characteristics of sandstone samples are compared. When the impact velocity is fixed at 22 m/s, the dynamic stress–strain curves of sandstone samples under different stress constraints are shown in Fig. 13. On the whole, the dynamic stress and deformation characteristics of sandstone samples have significant stress constraint dependence. Two interesting phenomena can be observed from Fig. 13a. First, the stress constraint state changes the shape of the dynamic stress–strain curve of the sandstone sample. Under no axial stress constraint, the dynamic stress–strain curve presents an obvious compaction stage, but when the axial stress constraint of 30 MPa is applied, the compaction stage disappears, which is due to the compaction of the microcracks inside the sandstone sample (Yang et al. 2020; Su et al. 2022).

Fig. 13
figure 13

Dynamic stress–strain curves of sandstone samples under different stress constraints

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Secondly, under the constraints of triaxial stress, biaxial stress and uniaxial stress, the enhancement effect of dynamic compressive strength and the deformation resistance of sandstone are weakened in turn. Compared with the peak stress (100.31 MPa) and peak strain (0.02467) of sandstone samples under the constraint of uniaxial stress (30,0,0)MPa, the peak stress of sandstone samples under the constraint of triaxial stress (30,30,10)MPa increases by 103.65%, and the peak strain decreases by 66.88%. It indicates that sandstone samples can produce larger fracture events when they produce smaller deformation under high ground stress and multi-axis constraints, which brings great challenges to the early warning and prevention of rock failure under high ground stress in practical engineering. In addition, as shown in Fig. 13b, under the static load constraints in three directions, a smaller dynamic peak strain is generated in the Y-axis direction, indicating that the lateral expansion of the sandstone sample under impact compression is along the direction of minimum principal stress.

4.2 Stress constraint effect of dynamic failure mode

Figure 14 shows the failure modes of sandstone samples with different stress constraints under impact load (22 m/s). When the sandstone samples are only subjected to axial stress constraint, the lateral expansion deformation of sandstone samples during dynamic compression is not limited, and crushing failure is easy to occur. However, under the condition of lateral stress constraints, sandstone samples are more likely to be broken along the free surface direction. As the lateral stress σ2 increases from 0 to 30 MPa, the damage degree of sandstone samples decreases. When the lateral stress is σ2 ≤ 20 MPa, there are obvious inclined shear cracks. The reason for this phenomenon can be explained from the perspective of energy evolution. Under normal circumstances, the stress constraint restricts the failure of sandstone, and the development of cracks in the sample requires more energy, that is, under high stress constraint, more energy needs to be input for the initiation and propagation of microcracks in sandstone samples (Chen et al. 2022; You et al. 2017).

Fig. 14
figure 14

Dynamic failure mode of sandstone samples under different stress constraints

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When the sandstone sample is in a triaxial stress constraint state, due to the absence of free surfaces, the initial triaxial constraint state improves the compactness of the sandstone sample, and the fracture cracks generated during the dynamic compression process are significantly suppressed, with almost no obvious macroscopic fracture on the surface. The above analysis shows that the stress constraint state has a significant influence on the dynamic fracture degree and failure mode of sandstone samples.

5 Conclusion

Taking full account of the coupling effect of the constraint states of uniaxial stress, biaxial stress and true triaxial stress and impact load, a large number of dynamic compression tests of sandstone samples are carried out. The stain rate effect and stress constraint effect of sandstone dynamic mechanical characteristics and failure behavior under complex static-dynamic load coupling are studied, and the cyclic impact effect of sandstone constrained by true triaxial stress is discussed. The main conclusions are as follows:

  1. (1)

    Under uniaxial impact load, the dynamic compressive strength of sandstone samples is 3.35 times higher than the quasi-static compressive strength, showing a significant rate dependence. Crushing failure occurs at high strain rates. When the axial stress σ1 is increased, the dynamic compressive strength and peak strain of sandstone samples gradually decrease, while the strain rate gradually increases, indicating that axial stress constraint is easy to induce larger failure events in sandstone samples earlier.

  2. (2)

    When the axial stress is fixed and the lateral stress σ2 is increased, the peak stress in the impact direction of the specimen is gradually increases, while the degree of damage and deformation is weakened. However, the peak strain along the Y-axis direction has an opposite effect. With the increase of strain rate, the failure mode changes from slight splitting failure to inclined shear failure.

  3. (3)

    Under the true triaxial stress constraint, the intermediate principal stress σ2 has a significant strengthening effect on the dynamic compressive strength of sandstone samples. With the increase of the intermediate principal stress σ2, the peak strain of sandstone samples along the X-axis direction decreases gradually, while the peak strains along the lateral direction increase. Under the constraints of triaxial stress, biaxial stress and uniaxial stress, the enhancement effect of dynamic compressive strength and the deformation resistance of sandstone are weakened in turn.

  4. (4)

    When the sandstone sample is under true triaxial stress constraint, it is not easy to cause obvious damage or fracture under single impact (high strain rate). After repeated impacts, sandstone samples show obvious progressive damage evolution effect, and eventually inclined shear failure occurs, resulting in complete loss of bearing capacity. Under repeated impacts, the peak strain increases gradually along the impact direction and the lateral expansion direction. When the specimen is on on the verge of crushing, the deformation and strength along the Y-axis and Z-axis directions change suddenly.