Introduction

The rock in deeply buried hard rock tunnels may fail by splitting, spalling, and rock burst throughout excavation, as shown in Fig. 1. The first failure type separates the rock perpendicular to the newly created fracture surfaces and is caused by a tensile or compressive stress applied perpendicular to the fracture plane. Spalling refers to a failure mode in which the rock fractures parallel to the sidewalls of the excavation under high tangential compressive stresses and the thin rock slabs peel off and fall (Fairhurst and Cook 1966). These two failure types are static failure behaviours, and they affect the stability of deeply buried tunnels. In a rock burst, rock is ejected or remarkably displaced. Based on the triggering mechanisms, there are two types of rock bursts: strain bursts associated with overstressing and fault-slip bursts associated with the slippage of preexisting faults (Wan and Li 2022). Strain bursts are caused by the conversion of excess energy into kinetic energy as the strain energy stored in the burst rock exceeds that consumed during its fracturing. Thus, strain bursting is a dynamic failure behaviour that not only affects the stability of deeply buried tunnels but also threatens the safety of construction personnel.

Fig. 1
figure 1

Different failure forms in rock engineering: splitting (a), spalling (b), and rock bursting (c) (RE represents the rock element: σ1, σ2, and σ3 are the maximum, intermediate, and minimum principal stresses, respectively; σr and σθ represent the radial stress and tangential stress, respectively)

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The main reason for the failure of deeply buried tunnels is the formation of tensile stress and compressive stress concentration areas in the surrounding rock mass induced by excavation. Before excavation, the rock is subjected to true triaxial stress with σ1 > σ2 > σ3, where σ1, σ2, and σ3 are the maximum, intermediate, and minimum principal stresses, respectively, as shown in Fig. 1c. After excavation, compressive stress concentration areas are formed in the sidewalls perpendicular to σ1, and the stress state of an element at this position is changed to a state that is stressed on five of its six faces, with one face stress-free, as shown in rock element #1 in Fig. 1c. The radial stress (σr) on the free surface is zero, but it is slightly greater than zero on the face of this rock element opposite to the free surface. The tangential stress (σθ) gradually increases during excavation. When σθ exceeds the bearing capacity of the rock element, the rock fails.

There are two main types of laboratory tests used to study the failure behaviour of deeply buried tunnels. The first is the model test, which mainly simulates the overall failure behaviour of a deeply buried tunnel by loading a specimen with a hole. Related research mainly includes four aspects: (a) the effect of the size of the hole on breakout onset stress (Carter 1992; Cheon et al. 2011; Martin 1997); (b) the effect of the shape of the hole, such as circular, square, elliptical, and saddle, on the failure behaviour of the surrounding rock (Haimson and Lee 2004; Gong et al. 2018; Luo et al. 2019; Wang et al. 2021; Zhu et al. 2022); (c) the effect of geological conditions, such as weak interlayers, joints, bedding planes, and fractures, on the failure intensity of the surrounding rock (Sagong et al. 2011; Zhong et al. 2018; Lin et al. 2020); and (d) laboratory simulations of strain bursts and spalling (e.g., Fakhimi et al. 2002; Ohta and Aydan 2010; He et al. 2012; Hu et al. 2019; Zhang et al. 2020; Gong et al. 2020; Zhao et al. 2020; Wang et al. 2021). Among them, the fourth item is the research focus in recent years. This is due to the increase in the frequency of spalling and strain bursts at deeply buried tunnel projects. For example, Kusui et al. (2016) carried out model tests using granite and sandstone, and the specimens failed in the form of a strain burst, with a rock block ejection velocity of 3–6 m/s. Gong et al. (2018, 2019, 2020) conducted model tests with circular, arched, and rectangular hole shapes, and the model specimens exhibited both spalling failure and strain burst failure. Si et al. (2020, 2022) used model tests to study the effects of loading rate and bedding angle on rock burst intensity. The other main type of laboratory test is to carry out loading and unloading tests on intact rock specimens. Initially, this type of test was used to mainly conduct loading and unloading tests on cylindrical specimens subjected to symmetric stress states (conventional triaxial stress) or prismatic specimens subjected to asymmetric stress states (true triaxial stress) to simulate the stress adjustment process of rock elements at different positions. The corresponding experimental results were mainly used to analyse the strength, deformation, and fracture characteristics of rocks and provide data support for numerical calculations (e.g., Von Karman 1911; Cook 1963; Mogi 1967; Haimson and Chang 2000; Lu et al. 2021; Shi et al. 2022, 2023b). However, this type of test cannot simulate the dynamic failure behaviour of rock, such as strain bursting. Some novel true triaxial rock burst test devices have been developed, and some strain burst simulation methods have been proposed, such as the “single-face or multiple-faces sudden unloading test”, “single-free-face or multiple-free-faces true triaxial compression (TTC) test”, and “single-face sudden unloading coupled dynamic disturbance test” (He et al. 2012, 2021; Li et al. 2015; Su et al. 2017a; Jiang et al. 2019). In such tests, the tested specimen is prismatic in shape and considered a burst body to simulate the stress adjustment process of a rock element on the sidewall of a tunnel structure, and a high-speed camera is used to capture the development process of the strain burst. Related research involves the influence of loading path, loading rate, dynamic disturbance parameters, and size effects on rock burst intensity (e.g., Zhao and Cai 2014; He et al. 2015; Du et al. 2016; Su et al. 2017a, 2017b, 2018).

The model test and the single-free-face TTC test are both effective means to study the failure behaviour of surrounding rocks in the stress concentration area. Among them, the former considers the structural effect of deeply buried tunnels, and the latter focuses on the stress adjustment process of representative rock elements. Although the specimen specifications of the above two test methods are different, the failure of the specimen is induced by stress concentration. In previous studies, these two types of tests have been carried out independently, and no case study has provided a comprehensive consideration of both. The differences and connections between the static or dynamic failure of the same material under the two loading modes are still unclear. In addition, what is the effect of lateral stresses on the failure of the model specimens? What are the characteristics of the strain energy in the surrounding rock induced by the hole and stress concentration? Notably, some rocks in previous model tests failed by spalling, and some failed by strain bursting; thus, the triggering condition of strain bursts is also worth analysing.

In response to the above questions, model tests under both uniaxial and biaxial loading conditions were first performed on Beishan granite. During the model tests, the surface strain of the specimens and the failure process of the hole were captured by the digital image correlation (DIC) technique and a video camera, respectively. Then, a single-free-face TTC test was carried out on Beishan granite. During that test, the failure process of the specimen was monitored by a high-speed camera. The failure forms of the specimens under the three loading conditions were different, especially for the model test under biaxial compression and the single-free-face TTC test. Although the specimen failed due to compressive stress concentration under the latter two loading conditions, the failure form of the specimen was spalling in the model test and strain burst in the TTC test. The reason for this difference in failure was analysed (see the Discussion section) to better understand the triggering conditions and mechanisms of strain burst.

Specimens and experimental procedures

Specimens

The tested granite specimens in the present study were taken from the Beishan area of Gansu Province in China. This area has been preliminarily determined to be the preferred candidate site for China’s high-level radioactive nuclear waste (HLW) repository. The properties of the Beishan granite are presented in Table 1. Regarding the specimen size of the model test, this study comprehensively considered factors such as the loading capacity of the loading system, boundary effects, and size effects and referred to rock model tests carried out by other scholars (e.g., He et al. 2012; Gong et al. 2018; Hu et al. 2019). The rock model specimens were prismatic with a size of 300 mm × 300 mm × 67 mm and a central hole 57 mm in diameter, as shown in Fig. 2a. This size ensures that the lateral dimensions of the rock specimen are greater than 5 times the diameter of the hole, weakening the boundary effect. For single-free-face TTC testing, the specimen was prismatic with dimensions of 100 mm × 100 mm × 200 mm, which meets the requirements of an approximate width: length ratio of 1:2, as suggested by ISRM (Feng et al. 2019), as shown in Fig. 2b.

Table 1 Basic mechanical properties and mineral contents of Beishan granite
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Fig. 2
figure 2

Schematic diagram of the loading path and stress state of specimens in a model test (a) and a single-free-face TTC test (b)

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Experimental apparatus

The model tests were carried out on a ZJCS-5000 biaxial loading apparatus, as shown in Fig. 3a (Peng et al. 2019). The load capacity of the apparatus is 5000 kN both vertically and horizontally. A DIC system was used in the model tests to monitor the full-field strains at the front surface of the specimens. The DIC system includes two digital cameras, an LED light source, a Vic-Snap image acquisition system, and a digital processing system. Furthermore, a video camera monitored the failure process of the hole during testing.

Fig. 3
figure 3

a The biaxial loading apparatus ZJC-5000. b The novel true triaxial rock burst test apparatus (Su et al. 2017a)

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The single-free-face TTC test was conducted on a novel true triaxial rock burst test apparatus (see Fig. 3b), which has a stiffness of 9 GN/m in the vertical direction and a stiffness of 5 GN/m in the horizontal direction. The axial and lateral load capacities of the machine are 5000 kN and 3500 kN, respectively. Loads can be applied to the specimen in three orthogonal directions. In this test, the failure process of the specimen was monitored by a high-speed camera.

Experimental procedures

Regarding the model tests, the loading path and stress state of a rock specimen are shown in Fig. 2a. σ2 was 0 MPa and 5 MPa for the uniaxial and biaxial loading conditions, respectively. The loading process was divided into two stages. σ2 was first increased to a predetermined value at a loading rate of 300 N/s under load control. Then, σ2 was kept constant, and σ1 was applied at a loading rate of 0.04 mm/s under displacement control until rock failure occurred. The criteria for rock failure here were severe splitting of the model specimen or through-type failure from the hole. This is because continued loading of the specimen in these two states may result in collapse, which is not conducive to analysis of its failure mode after the test.

For the single-free-face TTC test, the loading path and stress state of a rock specimen are shown in Fig. 2b. The loading process of this test was divided into three stages. First, σ1, σ2, and σ3 were synchronously increased to 5 MPa. Then, σ1 and σ2 were simultaneously increased to 30 MPa, while σ3 remained unchanged. Finally, σ1 was increased until failure of the specimen, while both σ2 and σ3 remained unchanged. The stresses were applied at a rate of 0.5 MPa/s in all three stages.

Experimental results

Model test

Failure modes

Figure 4 shows an overview of the back side of the Beishan granite specimen after uniaxial and biaxial compression testing. The uniaxial and biaxial compression tests were terminated at σ1 = 66.8 MPa and σ1 = 120 MPa, respectively. Under uniaxial compression, the specimen formed four tensile fractures, with the longest and shortest fractures having lengths of 155 mm and 31 mm, respectively, as shown in Fig. 4a. In addition, the sidewalls of the hole under uniaxial compression did not produce breakout notches. Under biaxial compression, breakout notches were formed in the sidewalls of the hole to depths of 5.86–11.71 mm within the surrounding rock, as shown in Fig. 4b. It is obvious that the lateral stress changes the specimen failure mode from failure by tensile fracture formation to breakout notch failure of the sidewalls.

Fig. 4
figure 4

Overview of the back side of the Beishan granite specimen after testing under uniaxial (a) and biaxial compression (b)

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Failure process

Figure 5 shows the fracturing moments at a few stress levels during testing under uniaxial compression. The first fracture was initiated in the floor of the hole when σ1 = 64.2 MPa, as shown in Fig. 5a. The fracture gradually propagated as σ1 increased. Then, a macro fracture suddenly appeared on one sidewall of the hole and extended downwards when σ1 was slightly increased to 64.4 MPa (Fig. 5b). Afterwards, a further increase of 0.3 MPa in σ1 resulted in a fracture in the middle of the roof of the hole (Fig. 5c). A long fracture appeared on the other sidewall of the hole but propagated upwards when σ1 was increased to 65.2 MPa (Fig. 5d). Slight pop-up spalling occurred on one sidewall of the hole when the test was terminated at 66.8 MPa (Fig. 5e). This failure process shows that the other tensile fractures formed in a very short time after the first tensile fracture event and that the most predominant tensile fractures formed in the sidewalls of the hole rather than in the roof and floor.

Fig. 5
figure 5

Video images of the front side of the Beishan granite specimen showing the fracturing process under uniaxial compression

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Figure 6 shows the fracturing moments at a few stress levels during testing under biaxial compression. The first pop-up spalling occurred on the right sidewall of the hole when σ1 was 88.4 MPa (Fig. 6a). At σ1 = 96.7 MPa, pop-up spalling also appeared on the left sidewall, and obvious spalling appeared on the right sidewall (Fig. 6b). Afterwards, spalling extended through the entire depth of the specimen, and breakout notches gradually formed in the sidewalls as σ1 increased (Fig. 6c, d). The test was terminated at σ1 = 116.7 MPa, with some small debris having spalled from the sidewalls and accumulated on the floor of the hole. This failure process shows that the model specimen under biaxial compression failed by spalling, which was a progressive formation.

Fig. 6
figure 6

Video images of the front side of the Beishan granite specimen showing the fracturing process under biaxial compression for σ2 = 5 MPa

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Full-field strain development process

Figure 7 shows the full-field strain cloud images of the DIC in the Beishan granite specimen under uniaxial compression. The orientations of the principal strains εy and εx were vertical and lateral, respectively. Figure 7 shows that the tensile fractures in the specimen were associated with εx. The εx concentrations first appeared in the middle of both the roof and the floor in the early stage of loading, as shown in Fig. 7a. At σ1 = 64.2 MPa, that is, when the first tensile fracture occurred in the specimen (Fig. 5a), the tensile strains ranged from −0.008 to −0.012 based on the DIC image of εx, as shown in Fig. 7b. Therefore, the range of −0.008 to −0.012 may be the critical strain interval for tensile fractures in the Beishan granite. The DIC image shows that similar strains appeared in the roof, upper left shoulder, and lower right haunch of the hole (Fig. 7b). At σ1 = 64.7 MPa, all εx concentration areas were enlarged, with significant elevations in the lower right haunch, where the strains dominated in the range from −0.012 to −0.024, as shown in Fig. 7c. At this stress level, a long tensile fracture was created, and a roof fracture became visible (Fig. 5c). A fracture in the upper left shoulder suddenly became visible and extended to a length of approximately 44 mm when σ1 was slightly increased by 0.3 to 61.5 MPa, as shown in Fig. 5d. At σ1 = 66.8 MPa, εx increased from −0.024 to −0.028 in the positions of the tensile fractures (Fig. 7d), indicating further fracture propagation.

Fig. 7
figure 7

DIC images of the front side of the Beishan granite specimen showing the evolution of the principal strains εy and εx under uniaxial compression; “+” indicates compressive strain, and “−” indicates tensile strain

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Figure 8 shows the full-field strain cloud images of the DIC in the Beishan granite specimen under biaxial compression. The spalling and breakout processes in the sidewall of the hole were reflected by the changes in εy during testing. At σ1 = 88.4 MPa, the first pop-up spalling occurred on the right sidewall of the hole, and εy ranged from 0.0034 to 0.0039 in the sidewalls (Fig. 8a). At σ1 = 96.7 MPa, the spalling zones of εy = 0.0034 to 0.0039 expanded in the walls, and dark clouds of falling debris also appeared on the wall surfaces (Fig. 8b). At σ1 = 113.8 MPa, the spalling zone in the right sidewall became irregular, and the dark clouds extended deeper into the sidewalls, indicating that a good amount of debris had fallen and that breakout notches were being formed in the sidewalls, as shown in Figs. 6c and 8c. At σ1 = 116.7 MPa, the spalling zones spread from the walls to the depths, but the immediate sidewall areas had lower εy values (Fig. 8). The relief in εy was a consequence of the unloading after the crushed material in the notches fell.

Fig. 8
figure 8

DIC images of the front side of the Beishan granite specimen showing the evolution of the principal strains εy and εx under biaxial compression for σ2 = 5 MPa

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Comparing the DIC cloud images of εx in Fig. 8 with the evolution of the spalling in the sidewalls shown in Fig. 6, εx at spalling onset was in the range from −0.009 to −0.011, which is similar to the tensile fracture onset strain observed under uniaxial compression. The spalling zone defined by εx = −0.009 to −0.011 expanded with an increase in the applied stress σ1. Fig. 8 shows that relatively high εx values occurred in four zones in the shoulders and haunches of the hole in the DIC cloud images of εx, but the εx value was less than the critical strain value for spalling and splitting fracture onset. Thus, no tensile fractures were visible in the video images when the test was terminated at σ1 = 116.7 MPa.

Single-free-face TTC test

Failure mode

Figure 9a shows the failure mode of a Beishan granite specimen after testing in the single-free-face TTC test. The free surface of the specimen in the σ3 direction burst and collapsed, and fractures formed perpendicular to the σ1–σ3 plane. The fractures close to the free surface of the specimen were subparallel to the free surface and extensional, while the internal fractures were shear fractures.

Fig. 9
figure 9

Failure mode (a) (Shi et al. 2023b) and typical failure process (b) of a Beishan granite specimen in a single-free-face TTC test

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Failure process

Figure 9b shows the typical failure process on the free surface recorded by a high-speed camera in a single-free-face TTC test. Some small rock particles were ejected from the upper region of the Beishan granite after σ1 was loaded to approximately 95% of the peak stress of the tested specimen. After an approximately 0.53 s delay, a splitting fracture formed in the upper region of the specimen, and then the split rock plates were ejected away from the host rock. Afterwards, many rock fragments were ejected from the free surface and accompanied by a loud sound. Finally, a burst pit was formed on the free surface of the specimen. Obviously, the Beishan granite failed by strain bursting in the single-free-face TTC test. This failure form was significantly different from the compressive stress-induced sidewall failure behaviour of the model specimen under biaxial loading. The triggering condition of the strain bursts under the two loading modes is further analysed in the following section.

Discussion

Splitting fracture

Under uniaxial compression, the model specimen formed tensile fractures in the surrounding rock of the hole, especially at the top and bottom of the hole. These test results are helpful for interpreting fractures observed in some mountain tunnels. Figure 10a shows a short road tunnel subjected to a splitting fracture in the roof. The tunnel has a rock cover of less than 20 m and is less than 10 m from the mountain slope. The slope declines towards Trondheim Fjord, which extends 100 m below the left side of the tunnel. The lateral in situ stresses in the position of the tunnel are nearly zero because they are only a short distance from the slope. Therefore, the tangential stresses in the tunnel roof are tensile, even though their magnitudes are very low. Because of the tensile stresses, a geological discontinuity exposed on the ceiling of the southern entrance of the tunnel has opened several centimetres. Longitudinal splitting fractures appear on the roof along the entire tunnel length of 70 m. Some of the splitting fractures are marked by white lines in Fig. 10b, which shows the inside of the tunnel towards the northern exit. The rock types of the tunnel are gneiss and greenstone. The foliations in the rock mass are sub perpendicular to the tunnel length. Obviously, these splitting fractures are created by the tensile tangential stresses in the roof. Such splitting matches the tensile fracture pattern in the roof and floor of the hole in a Beishan granite specimen tested under uniaxial compression. The tensile fractures in the shoulders and floor corners of the hole in the specimen demonstrate that similar extensional fractures may exist in the rock mass surrounding tunnels close to mountain slopes where the lateral in situ stresses are approximately zero. To avoid such extensional fracturing, tunnels should be placed far from mountain slopes. A low horizontal in situ-stress could easily suppress tensile and extensional fracturing in and around the tunnel.

Fig. 10
figure 10

A road tunnel near Trondheim fjord: a splitting fracture in the roof (a) and the details of the splitting fractures (b) (marked by the white arrows) along the length of the tunnel (photograph by Charlie C. Li)

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Progressive rock failure in a model test under biaxial compression

The lateral stress changes the model specimen failure from splitting failure to breakout notch failure in the sidewalls. Under biaxial compression, Beishan granite specimen failure starts with spalling and ends with the formation of breakout notches. The breakout notches of the Beishan granite are similar to the V-shaped notches formed in the Mine-by Experiment tunnel of the AECL underground laboratory, as shown in Fig. 11 (Read 2004).

Fig. 11
figure 11

Mine-by Experiment tunnel photographs showing the final shape of V-shaped breakout notches in the crown and invert of the tunnel (a) and a close-up of notch tips at the tunnel invert (b) (Read 2004)

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The spalling onset stress and the breakout notch depth are two important parameters that describe breakout rock failure. Martin et al. (1999) proposed the following relationship between the notch depth and the maximum tangential stress (σθmax) value in the tunnel wall based on field observations:

$$\frac{R_f}{a}=0.49\left(\pm 0.1\right)+1.25\frac{\sigma_{\theta \max }}{\sigma_c}$$
(1)

where Rf is the distance from the notch tip to the tunnel centre; a is the tunnel radius; σɵmax is the maximum tangential stress, which is estimated by the Kirsch solution σɵmax = 3σ1–σ2 for circular tunnels; and σc is the uniaxial compressive strength of the rock.

Using the Kirsch solution, σɵmax of the sidewalls of the Beishan granite specimen under biaxial compression is 260 MPa. The magnitude of σɵmax/σc at the onset of spalling is 1.8, approximately 3.7 times the spalling onset stress (0.49 ± 0.1) obtained from Eq. (1). This discrepancy has also been observed in similar laboratory model tests by Gong et al. (2018) and Hu et al. (2019) (Fig. 12). In response to this problem, some scholars have shown that the curvature of the hole affects the onset of spalling and have tested specimens with different hole diameters. The magnitudes of σɵmax/σc at the onset of spalling in those tests were greater than 1 and decreased with the hole diameter, as shown in Fig. 12, confirming that the curvature of the hole significantly affects the spalling onset stress in laboratory tests. The curvature of the small holes used in laboratory tests is less than one-hundredth of the curvature of tunnels. A smaller curvature possibly results in confinement to the material around the hole such that a higher tangential stress is needed to initiate spalling, which may be the mechanical mechanism for the increase in the spalling onset stress in laboratory tests. Therefore, the empirical Eq. (1) obtained based on on-site observations is not suitable for determining the fracture initiation stress in laboratory-based small-scale model specimens.

Fig. 12
figure 12

Relationship between the maximum tangential stress at the onset of spalling and the hole diameter (Martin 1997; Gong et al. 2018; Hu et al. 2019)

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Strain energy stored in the rock surrounding a circular hole

To understand the fracture range of the model specimen, the strain energy concentration characteristics of the rock surrounding the hole are further analysed. Here, the Beishan granite is assumed to be a continuous linear elastic rock. If the specimen does not contain a circular hole, its elastic strain energy density V0 can be expressed as follows (He et al. 2016):

$${V}_0=\frac{1+v}{2E}\left[\left(1-v\right){\left({\sigma}_1+{\sigma}_2\right)}^2-2{\sigma}_1{\sigma}_2\right]$$
(2)

where E and ν are Young’s modulus and Poisson’s ratio, respectively.

Therefore, for a specimen without a circular hole, the elastic strain energy W0 stored in the annulus between b and a can be expressed as follows:

$${W}_0=\pi \left({b}^2-{a}^2\right)h{V}_0$$
(3)

where a is the radius of the hole (28.5 mm here) and b is the distance from the considered position to the centre of the specimen. h stands for the thickness of the model specimen and is 67 mm.

In fact, the model specimen contains a circular hole, so the elastic strain energy We stored in the annulus between b and a during loading can be expressed as follows (He et al. 2016):

$${W}_e=\frac{\pi h}{E}\left\{\begin{array}{c}\left(1-{v}^2\right){\int}_a^b\left[{\left({\sigma}_1+{\sigma}_2\right)}^2r+2{\left({\sigma}_1-{\sigma}_2\right)}^2\frac{a^4}{r^3}\right] dr+\\ {}2\left(1+v\right){\int}_a^b\left[\frac{{\left({\sigma}_1+{\sigma}_2\right)}^2}{4}\frac{a^4}{r^3}+\frac{{\left({\sigma}_1-{\sigma}_2\right)}^2{a}^4}{4}\left(\frac{2}{r^3}-\frac{12{a}^2}{r^5}+\frac{9{a}^4}{r^7}\right)-{\sigma}_1{\sigma}_2 rdr\right]\end{array}\right\}$$
(4)

The strain energy storage coefficient ke expresses the degree of strain energy concentration in the surrounding rocks induced by a circular hole:

$${k}_e=\frac{W_e-{W}_o}{W_o}\times 100$$
(5)

According to Eqs. (2)–(5), We, W0, and ke are calculated at different b values, and the calculation results are presented in Table 2. Notably, σ1 is the fracture onset stress of the surrounding rock of the hole. Under both the uniaxial and biaxial loading conditions, the We and Wo values of the Beishan granite increased as b increased, and ke decreased as b increased. As shown in Fig. 13, the strain energy characteristics of the Beishan granite under biaxial compression showed that ke was 67.5% at b = 1.5a, 41.6% at b = 2a, 20.1% at b = 3a, 11.9% at b = 4a, and 7.7% at b = 5a, indicating a greater strain energy concentration in the surrounding rock in the range of 1.5a. This strain energy concentration feature is also valid for the Beishan granite under uniaxial compression (ke is 64.8% at b = 1.5a), as shown in Table 2. In addition, the “Failure modes” section shows that the maximum depth of the surrounding rock notch is approximately 1.2a, which is consistent with the strain energy concentration range.

Table 2 Summary of the strain energy of the surrounding rocks at different b values in the model tests
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Fig. 13
figure 13

Strain energy characteristics of the surrounding rock at different depths for a Beishan granite specimen under biaxial compression: σ1 was 88.4 MPa

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Difference between the model test failure and single-free-face TTC test failure

The test results showed that the failure forms of the Beishan granite specimens in the model test with biaxial stresses were different from those in the single-free-face TTC test. Under a high compressive stress (i.e., in the surrounding rock elements perpendicular to σ1 in the model test), the sidewalls of the model specimen failed by spalling, that is, static failure. However, in the single-free-face TTC test, the Beishan granite acted as a sidewall element, and it failed via strain bursting, showing dynamic failure characteristics. In this section, from the perspective of the strain burst kinetic energy source, the reasons for the difference in the failure forms of the Beishan granite specimens under the above two loading methods are discussed.

Strain burst will occur when the total strain energy stored in the rock isto be ejected and the energy released from the surrounding rock is greater than the energy needed for rock fracturing. In other words, there is excess energy in the rock that is converted into kinetic energy to eject the rock. Li et al. (2022) proposed a conceptual model to analyse the source of kinetic energy in a strain burst event, as shown in Fig. 14. In Fig. 14, line OA represents the pre-peak behaviour of the strain burst body. When the burst body is loaded to the peak load point A, a small amount of energy is used to generate microcracks, that is, the dissipated energy Wd in Fig. 13. The rest is stored in the strain burst body, that is, the elastic energy Wb = (Wbf + Wbk). Among them, Wbf is used to fracture the strain burst body at the post-peak stage, and Wbk is converted into kinetic energy to eject the rock. The abovementioned Wd, Wbf, and Wbk describe the transformation of the strain energy stored by the strain burst body. In addition, in this conceptual model, the blue line AD represents the response of the surrounding rock during a strain burst, and its slope represents the overall stiffness of the surrounding rock. The strain energy released from the surrounding rock is denoted as Wm = (Wmf + Wmk), where Wmf is the portion dissipated by rock fracture and Wmk is the portion converted into kinetic energy. In summary, the kinetic energy Wk of the strain burst comes from the Wbk stored in the strain burst itself and the Wmk provided by the surrounding rock.

Fig. 14
figure 14

A conceptual model for the energy conversions during a strain burst event: the strain energy stored in the strain burst body is Wb = (Wbf + Wbk), and the strain energy released from the surrounding rock mass is Wm = (Wmf + Wmk) (Li et al. 2022)

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For the model specimen, the sidewall rock of the circular hole can be considered a potential strain burst body. The kinetic energy contributed by the potential strain burst body itself, Wbk, depends on the rock type and is a constant per unit volume for a given lithology (Tarasov and Potvin 2012; Zhang and Li 2019; Shi et al. 2023a). It can therefore be defined as the intrinsic potential energy of strain bursts and can be obtained from the uniaxial compressive stress–strain curve of the rock under servo control. The Wbk value is 1.5 kJ/m3 for the Beishan granite (Shi et al. 2023a), which is much smaller than the value for the Kuru granite of 28.3 kJ/m3 calculated by Zhang and Li (2019), indicating that the Wbk value of the Beishan granite is very small.

In addition, the “Progressive rock failure in a model test under biaxial compression” section shows that the failure onset stress of the potential strain burst body was much higher than its uniaxial compressive strength due to the influence of the curvature and structure of the circular hole. This means that the failure of the potential strain burst body in the model test required more fracture energy. Therefore, in the model test, the strain energy accumulated in the potential strain burst body of Beishan granite was mainly used for its fracturing.

For a rock with low Wbk, the key factor for triggering a strain burst in the model test is the magnitude of Wmk provided by the surrounding rock. Wmk is negatively related to the radial stiffness (Krb) of the surrounding rock mass. In other words, the smaller the Krb value of the surrounding rock mass is, the greater the Wmk provided to the strain burst body. For a circular tunnel, the Krb value of the surrounding rock mass can be expressed as follows (Zhao et al. 2021):

$${K}_{rb}=\frac{E}{1+\nu}\frac{1}{b}$$
(6)

where E = 47.2 GPa and ν = 0.28 are Young’s modulus and Poisson’s ratio of the rock, respectively, and b is the distance from the considered position to the tunnel centre.

Because the distance from the notch tip to the specimen centre was 40.21 mm, b in Eq. (6) was set to 45 mm in this study. Krb, calculated using Eq. (6), was 819 GPa/m for the Beishan granite, indicating that the Krb of the surrounding rock mass was higher. Therefore, the Beishan granite had low Wbk and high surrounding rock stiffness, and neither the potential strain burst body nor the surrounding rock mass could provide sufficient kinetic energy to induce a strain burst.

Previous studies have shown that some model specimens failed by strain bursting under multiaxial stress conditions. For example, Hu et al. (2019) carried out a similar model test on a Guangdong granite specimen with a circular hole, where the specimen was also subjected to biaxial compression with σ2 = 5 MPa. The Guangdong granite specimen experienced strain bursts during loading (see Fig. 15a, b), and breakouts formed in the sidewalls of the hole (see Fig. 15c). According to the stress–strain curve of cylindrical specimens of Guangdong granite tested under uniaxial compression (Fig. 16), its Wbk = 12 kJ/m3 is 8 times that of Beishan granite. This means that when model tests are carried out on the Guangdong granite specimen, a large portion of the accumulated strain energy in the potential strain burst body is transformed into kinetic energy to eject the rock.

Fig. 15
figure 15

Strain burst failure of a Guangdong granite specimen under biaxial compression with σ2=5 MPa: Images of strain bursts at σ1 = 95.28 MPa (a) and σ1 = 99.54 MPa (b) recorded by a high-speed camera and failure notches in one sidewall of the hole in the Guangdong granite specimen (c) (Hu et al. 2019)

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Fig. 16
figure 16

Axial stress–strain curves of cylindrical specimens of Guangdong granite under uniaxial compression

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In the single-free-face TTC test, the Beishan granite specimen simulates the stress state of a rock element on the sidewall of the tunnel, which can be called a potential strain burst body. The true triaxial test system can be regarded as the surrounding rock. The vertical and horizontal stiffnesses of the loading system used are 9 GN/m and 5 GN/m, respectively, which are greater than the 5 GN/m proposed by ISRM (Fairhurst and Hudson 1999) and correspond to a high-stiffness test apparatus. Therefore, the influence of its deformation energy on the failure of the specimen does not need to be considered. In other words, the influence of the surrounding rock can be ignored when discussing rock failure here. In this case, as long as σ1 exceeds the bearing capacity of the rock specimen and there is excess strain energy, i.e., Wbk, a strain burst can be triggered.

Engineering application

Three types of laboratory tests were conducted in the present study, and the failure forms presented by these three tests correspond to splitting, spalling, and strain burst in hard rock engineering, as shown in Table 3. Rock splitting is mainly induced by tensile stress concentration and is prone to occur in hard rock engineering without lateral stress constraints, such as in the case of the road tunnel shown in Fig. 10. To avoid such splitting failures, such a tunnel should be excavated in the rock mass with horizontal stress constraints. Rock spalling and strain burst are two very similar forms of failure, both of which are induced by compressive stress concentrations. The main difference between the two is whether the fractured rock can be ejected: if it can, a strain burst occurs; if not, spalling occurs. The analysis of the “Difference between the model test failure and single-free-face TTC test failure” section shows that the kinetic energy of rock ejection in the rock model test mainly comes from the burst rock, that is, the Wbk shown in Fig. 14. This conclusion also applies to strain burst events that occur at deep hard rock engineering sites. The typical feature of a strain burst is the small volume of the burst pit, which means that the deformation of the surrounding rock of the burst rock is relatively small. Li et al. (2022) used numerical simulation to quantitatively evaluate the effect of bust pit depth on the magnitude of kinetic energy supplied by the surrounding rock and the burst body and showed that for shallower burst pits, the kinetic energy is mainly supplied by the burst rock. Therefore, the intensity of strain bursts can be preliminarily evaluated by the intrinsic potential energy Wbk of the rock. For rocks with higher Wbk, such as Guangdong granite (Hu et al. 2019), the excavation of such rock masses is prone to triggering strain bursts, and it is suggested to consider dynamic support elements, such as yielding and energy-absorbing rock bolts, in the support system. For rocks with a low Wbk, such as Beishan granite, the excavation of such rocks is not prone to triggering spalling, and light dynamic support elements can be considered for the support system.

Table 3 Summary of test types and corresponding test results
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Conclusions

In the model test, the specimen under uniaxial loading was prone to tensile fractures, especially fractures in the shoulders and haunches of the hole, which extend remarkably and finally result in splitting of the specimen. The lateral stress transforms the tensile fractures of the specimen into breakouts in the areas where compressive stress is concentrated. The intrinsic potential energy (Wbk) of the rock determines how the breakout is formed. If the Wbk value of the rock is high, the breakout is formed by a strain burst, such as in the Guangdong granite. If the Wbk value of the rock is low, the breakout is formed by spalling, such as in the Beishan granite.

For the model specimens with a circular hole, the strain energy stored in the surrounding rock increases significantly, especially in the range of 1.5 times the radius. The strain energy storage coefficient of the surrounding rock at 1.5 times the radius is approximately 65%, indicating that the surrounding rock in this range is prone to fracturing.

In the single-free-face true triaxial compression testing, fractures perpendicular to the σ1–σ3 plane were observed in the Beishan granite specimen, slight strain bursts occurred on the free surface of the specimen, and a burst pit was formed. In this test, the triggering condition of strain bursts is that Wbk is greater than 0, and the specimens in this test were more prone to strain bursting than those in the model test.