1 Introduction

Rockbursts, which are uniquely characterized by the rapid ejection of rock fragments from the surrounding rock mass, have recently become a topic of increased research in the fields of rock mechanics and rock engineering (Cook 1965b; Ortlepp and Stacey 1994; Kaiser et al. 1996; Li et al. 2007; Zhang et al. 2012b; Cai 2013; Feng et al. 2013), as the sudden and violent failure of rock masses in narrow spaces during rockbursts can kill workers and damage construction equipment. Generally, rockbursts can be classified into three major types: strainbursts, fault slip bursts and pillar bursts (Hedley 1992). Strainbursts are the main form and are caused by the release of strain energy when the tangential stress exceeds the capacity of rocks. Extensive studies have been conducted to investigate strainburst mechanisms (Cook 1965a; Vardoulakis 1984; Dyskin and Germanovich 1993; Lee and Haimson 1993; Linkov 1996; Procházka 2004; Cai 2008; He et al. 2010; Zhang et al. 2012a; Feng et al. 2013; Zhao and Cai 2014). However, precise descriptions of the failure process of strainbursts and accurate predictions of the occurrence and intensity of strainbursts remain unrealized. Therefore, more fundamental studies are necessary to further investigate the mechanisms of strainbursts.

The occurrence of strainbursts is closely related to the stress state and constraint conditions of rock masses. The excavation of underground openings exposes free surfaces and leads to stress redistribution and disturbance (i.e., radial stress release and tangential stress concentration) in the vicinity of the excavation boundary, forming a new poly-axial stress state (Fig. 1). The stress state near the excavation boundary after excavation is closely related to the far-field stress conditions, the lithology of the rocks and the shape of the excavation. It is assumed that the maximum and minimum far-field principal stresses are perpendicular to the tunnel axis and that the intermediate far-field principal stress is parallel to the tunnel axis (Fig. 1). This description is consistent with the situation where strainburst or spalling has been frequently reported in underground excavations (Lee and Haimson 1993; Ortlepp and Stacey 1994; Martin 1997; Kaiser et al. 2000; Eberhardt 2001; Feng et al. 2013). In the simplest case, linear elasticity and isotropy are assumed. Figure 2 presents the distributions of the tangential stress σ θ and the radial stress σ r around circular openings (e.g., tunnels or boreholes) along the θ = 0° direction (see Fig. 1 for more details), where σ θ peaks and the deviatoric stress is the largest, and where the rocks will most likely eventually rupture according to the Kirsch equations and experimental or field observations. Particular attention was paid to the distribution of σ r . As can be seen from Fig. 2, σ r exhibits a remarkable nonlinear increase within a certain depth from the boundary to 0.5a (a is the tunnel radius) away from the boundary, and the gradient and the peak value of σ r improve significantly as the ratio of σ 1–σ 3 increases (i.e., large stress difference), from the Kirsch equations. In addition, the smaller the a is, the higher the gradient of σ r will be.

Fig. 1
figure 1

Stress state of the surrounding rock mass and the representative rock element after excavation: σ 1, σ 2 and σ 3 are the maximum, intermediate and minimum far-field principal stresses, respectively; σ θ , σ a , σ r and τ rθ (τ θr ) are the tangential stress, axial stress, radial stress and shear stress of tunnel, respectively; θ is the angular direction measured counter-clockwise from the σ 3 direction; r is the radial distance from the axis of the hole; and a is the tunnel radius

Full size image
Fig. 2
figure 2

Stress distributions along the θ = 0° direction from the Kirsch equations (see Fig. 1 for the parameter a, r and θ)

Full size image

Extensive studies have verified the significant effect of the confining stress or the minimum principal stress on the mechanical behavior of rocks, particularly on the strength and fracture characteristics, and this effect is more marked under low confining stress conditions (Mogi 1971; Peng and Johnson 1972; Horii and Nemat-Nasser 1985; Haimson and Chang 2000; Jaeger et al. 2007; Cai 2008; Kwasniewski 2012; Hoek and Martin 2014; Tarasov 2014). Under high in situ stress conditions, especially when the difference between the major and minor far-field principal stresses is large, the radial stress shows notable gradient variations in the surrounding rock mass. Meanwhile, the asymmetric constraints, where one face is free and the other five faces are constrained, will affect the dilation and energy flow of rocks near the excavation boundary within a certain depth (Alejano and Alonso 2005; Diederichs 2007; Zhao and Cai 2010a, b), exacerbating the effect of the gradually increasing radial stress. However, few studies have addressed the effect of the radial stress gradient on strainbursts in the published literature.

Uniaxial, biaxial, conventional triaxial and true-triaxial loading (or unloading) tests on small specimens have been widely conducted to study the behavior of rocks and to investigate rockburst mechanisms near the excavation boundary (Cook 1965a; Hua and You 2001; Chen and Feng 2006; Chen et al. 2009; Yun et al. 2010; Huang and Li 2014; Jiang et al. 2015; Feng et al. 2016). The triggering conditions of strainbursts around underground openings, namely poly-axial compression with one free boundary, however, have rarely been simulated in experimental studies until an improved true-triaxial apparatus was developed by He et al. (2007). Strainburst tests were successfully reproduced under true-triaxial unloading conditions afterward (He et al. 2010, 2012a, b), and true-triaxial tests that consider a free face have become the preferred method of exploring rockbursts (Zhao and Cai 2014; Zhao et al. 2014; Gong et al. 2015). During these pioneering strainburst tests, the macroscopic failure process, microscopic fracture characteristics, and, in particular, AE frequency spectra characteristics and energy release of strainbursts were investigated in detail, and the experimental results have provided valuable insight into the characteristics of strainbursts. For example, He et al. (2007, 2010) and Zhao et al. (2014) found that under true-triaxial unloading (of σ 3) conditions, the stress level before unloading and the unloading rate significantly affect whether and when the strainburst will occur and its intensity after unloading. Unloading under a relatively high stress level or with a relatively large rate tends to initiate instantaneous, intense strainbursts. He et al. (2010, 2012a) noted that the AE activity and energy release rapidly increase from unloading to sudden rockbursts. There are much higher amplitude and lower frequency events near the rockburst failure. However, experimental studies regarding the influence of the radial stress gradient on strainbursts have not yet been conducted.

So far, most of the research that addressed the problem of rockbursts, especially experimental research, usually focused on the failure of rocks at or close to excavation boundaries. The actual failure scope of in situ rockbursts is often further away from the excavation boundary (Ortlepp 2001; Li et al. 2012; Feng et al. 2013), so we attempt to reproduce strainbursts near excavation boundaries within a certain depth and investigate the influence of the radial stress gradient. Therefore, a representative rock element (Fig. 1) is introduced in our strainburst tests (see Sect. 3.1 for more details). As presented in Fig. 1, the stress state of the representative element of the surrounding rock mass can be simplified as a special true-triaxial stress condition, i.e., one face is free and the other five faces are loaded. In addition, strainbursts induced by loading, which, in the field, occur after excavation when the tangential stress continuously increases and exceeds the capacity of rock masses, rather than those induced by unloading (He et al. 2010, 2012a, b; Zhao et al. 2014; Gong et al. 2015), which, in the field, occur soon after excavation and are caused by the release of radial stress under extremely high in situ stress conditions, are investigated in our tests (Su et al. 2016, 2017).

In this paper, strainburst tests under true-triaxial loading conditions are conducted by means of an improved true-triaxial testing system, which can realistically simulate the in situ stress state (in a non-principal stress space) and boundary conditions of the representative rock element near the excavation boundary within a certain depth. Meanwhile, different radial stress gradients are applied on granite rectangular prismatic specimens in the tests. The influence of radial stress gradients on strainbursts is investigated in terms of the ejection failure process, rock strength and deformation features, strainburst failure modes and kinetic energy of ejected fragments. The experimental results are helpful in obtaining deeper insight into the mechanisms of strainbursts.

2 True-Triaxial Rockburst Testing System

Some special functions are required for true-triaxial testing apparatuses to simulate strainbursts in underground excavations. First, unloading the minor stress and exposing one face of rectangular prismatic specimens suddenly or rapidly (six surfaces loaded, and unloading on one face) to simulate the radial unloading caused by excavation or unloading rockbursts, and continuously increasing the major stress (or simultaneously increase other stresses) with one face free (one face free, and loading on the other five faces) to simulate the gradual concentration of the tangential stress after excavation or loading rockbursts are needed, for examining the loading mode. Thus, two opposite movable pistons that are independently capable of loading/unloading are required along at least one lateral direction and are also essential to emulate the (radial) stress gradient near the excavation boundary within a certain depth. Second, a high loading capacity is required to provide sufficiently high stresses on the specimen because larger specimens are preferred to simulate strainbursts at excavation boundaries or within a certain depth. Third, high-stiffness loading frames that can help to reduce the post-peak instability of rocks resulting from the testing apparatuses (Berry 1960; Cook 1965a; Wawersik and Fairhurst 1970; Manouchehrian and Cai 2015) are required to accurately investigate the influence of in situ factors on strainbursts and quantitatively analyze the evolution of energy during the strainburst process. Moreover, these tests should be conducted in a comprehensive testing system or platform that contains various controlling and monitoring systems to record the strainburst failure process and accompanying signals.

Recently, a new true-triaxial testing system was developed by Su et al. (2016) (Fig. 3) that mainly contains a rockburst testing machine, acoustic emission (AE) monitoring system, high-speed cameras, video recorder, decibel meter, etc. The rockburst testing machine is an improved true-triaxial electrohydraulic servo pressure testing machine (Fig. 3a), which is significantly different from the true-triaxial testing apparatus, a preferred apparatus for exploring rockburst, developed by He et al. (2007). This testing machine consists of a highly rigid integral frame structure. The stiffness of the vertical (z-direction) frame is over 9000 kN/mm, and those of the horizontal (x-direction and y-direction) frames are over 5000 kN/mm. These relatively high stiffnesses can effectively reduce the influence of the deformation recovery of the loading configuration on the kinetic energy of rockbursts. Both static loading/unloading and dynamic disturbance along the x-, y- and z-directions are allowed and can be independently servo-controlled. Uniquely, two independent loading systems exist along the y-direction. Thus, this testing machine can expose one surface of the rock specimen perpendicular to the y-direction by using a rotating swing link device after the stress on this surface has been unloaded, i.e., six surfaces loaded, with abrupt unloading on one face; with the help of the shear stresses between the surfaces of the rock specimen and the rigid plates (see Sect. 3.2 for more details), this testing machine can load on only five surfaces of the rock specimen while keeping one face free, i.e., one face free, and loading on other five faces; crucially, this testing machine can perform the radial stress gradient near the excavation boundary on the rock specimen along the y-direction (Fig. 3b, c), which greatly differs from the true-triaxial apparatus developed by He et al. (2007). In addition, this testing machine has a maximum capacity of 5000 kN along the z-direction and 3000 kN along the x- and y-directions. Accordingly, the dimensions of the rock specimen in our strainburst tests (see Sect. 3.1) are allowed to be more large than those (roughly 150 × 60 × 30 mm) in previous true-triaxial strainburst tests (He et al. 2007, 2010; Gong et al. 2012; Zhao et al. 2014; Gong et al. 2015).

Fig. 3
figure 3

True-triaxial rockburst test system: a full view; b zoomed-in photograph; c schematic illustration that shows the stress state of the rock specimen-rigid plate assembly and the interior rock specimen in one-face-free true-triaxial tests

Full size image

3 Testing

3.1 Specimen Preparation

A representative rock element was introduced in our strainburst tests to reproduce strainbursts near excavation boundaries within a certain depth and demonstrate the influence of the radial stress gradient. The location and stress conditions of the representative rock element are displayed in Fig. 1. This rock element represents rocks around the excavation boundary along the θ = 0° direction (see Fig. 1) where the rocks are most prone to rockbursts.

Actually, relatively large rock specimens have been used in rock mechanical laboratory tests (Bear 1972; Cuisiat and Haimson 1992). The REV (representative elementary volume) is defined as the smallest size for which equivalence exists between a continuous material and the real rock mass. Although the representative rock specimen employed in our strainburst tests is still not large enough to consider joints, it does represent the surrounding rock mass near the excavation boundary within a certain depth, a much wider range than 100 × 100 × 200 mm, which follows the meaning of representativeness. Thus, the rock specimen in our strainburst tests is also called the representative rock element of the surrounding rock mass, or RER, distinctly different from large rock specimens in conventional tests.

Coarse gray-white granite rectangular prismatic specimens with nominal dimensions of 100 mm in length, 100 mm in thickness and 200 mm in height were prepared for the strainburst tests. A granite block was taken from a quarry in Guangdong Province, China. The intact granite was a typical hard brittle rock with major minerals of quartz, feldspar and mica. The rock was isotropic and homogenous with high strength, and its basic physical and mechanical properties are listed in Table 1.

Table 1 Basic physical and mechanical properties of the granite specimens
Full size table

3.2 Design of Loading Path

The stress redistribution and the deformation and failure of rocks interact with each other. Unloading of the radial stress and loading of the tangential stress are major triggering mechanisms of strainbursts in tunnels. Although the stress redistribution in tunnels is significantly affected by the excavation method, sequencing, footage and other factors (for example, the radial stress is unloaded instantaneously when excavating with full-face drilling and blasting, while a gradual unloading of the radial stress is used for TBM or other types of mechanical excavation), it is clear that the tangential stress concentration is a progressive process and apparently lags behind the completion of the unloading of the radial stress (Ortlepp and Stacey 1994; Xu et al. 2000; Feng et al. 2013). For example, according to in situ records during the excavation at Jinping II Hydropower Station, strainbursts usually occurred approximately several hours or several days after excavation (Feng et al. 2013). The stress path of the surrounding rock mass near the excavation boundary within a certain depth can be briefly described as follows: Before excavation, the rock masses are under true-triaxial stress conditions, i.e., far-field stress conditions (see Fig. 1); during excavation, σ r decreases rapidly (more significantly closer to the excavation boundary); after excavation, σ r exhibits gradient variations, the tunnel axial stress σ a remains essentially constant and σ θ increases continuously. The significant concentration (loading) of the tangential stress after excavation essentially dominates the occurrence of strainbursts, which is an important basis for the design of the loading path.

Given that the rapid unloading of σ r occurs when σ θ is much lower than the capacity of rocks, it has little effect on the rocks, which has been verified by a repeated loading–unloading σ 3 test. Thus, a simplified loading path was employed in our tests for higher efficiency (Fig. 4). As shown in Fig. 4, the loading path procedure can be described as follows:

Fig. 4
figure 4

Loading path

Full size image

  • Step 1 Independently load σ x and σ z at a constant rate, while σ x is loaded to the appointed value.

  • Step 2 Load σ y2 to the appointed value, keeping the opposite face free (σ y1 = 0 MPa).

  • Step 3 Keep σ x and σ y2 constant and continually load σ z until the rock specimen fails.

Moreover, strainbursts of the surrounding rock mass in our tests were reproduced in the in situ stress space, a non-principal stress space that is defined according to the spatial orientations around the excavation, rather than in the principal stress space, where most previous laboratory tests were conducted to explore the mechanical properties of rocks. Therefore, the friction or shear stress on the representative rock element from adjacent rocks does exist (Fig. 1) and should be simulated realistically (Fig. 3c). In other words, minimizing or eliminating the “end friction” is no longer necessary in this study.

3.3 True-Triaxial Strainburst Tests

The testing plan is presented in Table 2. The tunnel axial stress σ x was set to 30 MPa, and the radial stress gradient was set from 1 to 150 kPa/mm, corresponding to σ y2 from 0.1 to 15 MPa (σ y1 = 0 MPa), which is of interest for civil engineering and deep-mining applications (Lee and Haimson 1993; Martini et al. 1997; Kaiser et al. 2000; Eberhardt 2001; Feng et al. 2013).

Table 2 Testing plan
Full size table

During the tests, the modified loading path in Fig. 4 was followed. First, σ z and σ x were set to 2 MPa and 0.5 MPa, respectively, with a loading rate of 2 mm/min under displacement control. Then, σ x was increased to 30 MPa at a loading speed of 0.5 MPa/s under stress control; σ z was increased in the same fashion. Meanwhile, σ y2 was applied a short amount of time after the beginning of the normal loading (0.5 MPa/s) of σ z and σ x , with the appointed values listed in Table 2, at a loading rate of 0.25 MPa/s under stress control. Finally, σ z was continuously increased (0.5 MPa/s) until the specimen failed.

4 Results and Discussion

The failure process of the tested specimens near the free face was recorded by high-speed cameras during the tests. As shown in Fig. 5, all of the specimens presented apparent violent ejection of rock fragments. Although differences existed between the ejection failures of the tested specimens under different radial stress gradients (i.e., different σ y2), the basic process or mode was similar. During the initial loading stage, elastic deformation occurred alongside elastic strain energy accumulation, and no macro-cracks could be observed. With increasing σ z , a large number of small cracks fully developed and entered an unstable development stage, especially in the biaxial stress region (σ y  ≈ 0) near the free face. Several local failure regions randomly formed on the free face, along with the ejection of small fragments or grains. Tension micro- or meso-cracks near the free face intersected or coalesced as σ z further increased, forming large-scale fractures or macro-cracks. The rock near the free face split into slabs that were approximately parallel to the free face (i.e., the σ z –σ x plane). Afterward, the rock slabs expanded and bent outward perpendicular to the free face, and their stability gradually decreased. Before buckling, the external slabs provided a constraint for the internal slabs, so the expansion of internal slabs was restricted and shear failure may have been induced, which led to the final formation of the pot-shaped, stepwise-shaped or V-shaped rockburst pit near the free face after the ejection failure. When σ z reached the capacity of the specimen, the outermost slab bucked and broke off, which was accompanied by the subsequent bucking of the internal slabs or rupture of the internal relatively intact rocks; the elastic strain energy accumulated in the specimen rapidly released, resulting in the violent ejection of rock fragments near or further from the free face, which was accompanied by an extremely loud sound. The ejection failure process near the free face can be summarized into four stages, including grain ejection, the splitting of rocks into slabs, the bending of rock slabs and fragment ejection at the instant when the rock slabs break off.

Fig. 5
figure 5figure 5

Ejection failure near the free face under different radial stress gradients: a 1 kPa/mm (σ y2 = 0.1 MPa); b 10 kPa/mm (σ y2 = 1 MPa); c 30 kPa/mm (σ y2 = 3 MPa); d 50 kPa/mm (σ y2 = 5 MPa); e 70 kPa/mm (σ y2 = 7 MPa); f 100 kPa/mm (σ y2 = 10 MPa); g 150 kPa/mm (σ y2 = 15 MPa); the numbers in the top-left corners of the pictures indicate the time in h:m:s

Full size image

From Fig. 5, it can be observed that the intensity and scale of the ejection failure and the size of the rockburst pit markedly increased as the radial stress gradient increased. Moreover, the decibels of the sound were recorded during the strainburst tests, and the maximum decibels under different radial stress gradient conditions are shown in Fig. 6. The maximum decibels of the rockburst sound continually increased with increasing radial stress gradient, which suggested that the intensity of the rockburst increased with increasing radial stress gradient.

Fig. 6
figure 6

Maximum decibels of the rockburst sound under different radial stress gradients

Full size image

4.1 Influence of Radial Stress Gradient on Rock Strength and Deformation

Figure 7 shows the strengths of the tested specimens under different radial stress gradient conditions. The strength continuously increased, first sharply and then slightly, as the radial stress gradient increased. For instance, when the radial stress gradient was 100 kPa/mm (σ y2 = 10 MPa) or 150 kPa/mm (σ y2 = 15 MPa), the strength of the tested specimen was 290 MPa or 297 MPa, which is 21.8 or 24.8% higher than that (238 MPa) under a radial stress gradient of 1 kPa/mm (σ y2 = 0.1 MPa). This result of the one-face-free true-triaxial loading tests is consistent with numerous triaxial experimental and theoretical investigations, which demonstrated that the strength of common hard rocks significantly increases with increasing confining pressure or minimum principal stress (Peng and Johnson 1972; Horii and Nemat-Nasser 1985; Haimson and Chang 2000; Haimson 2006; Jaeger et al. 2007; Kwasniewski 2012). In addition, when the radial stress gradient increased from 1 kPa/mm (σ y2 = 0.1 MPa) to 50 kPa/mm (σ y2 = 5 MPa), the strength of the tested specimens increased from 238 MPa to 277 MPa, an increase of 39 MPa, while the strength (297 MPa) of the rock specimen under a radial stress gradient of 150 kPa/mm (σ y2 = 15 MPa) was only 7 MPa higher than that (290 MPa) under a radial stress gradient of 100 kPa/mm (σ y2 = 10 MPa). Hence, the rock strength is much more sensitive to variations in the radial stress gradient under low radial stress gradient conditions than under high radial stress gradient conditions. This is in good agreement with results in the previous studies on the effect of the confining pressure or minimum principal stress on the strength of crystalline rocks (Horii and Nemat-Nasser 1985; Haimson 2006; Diederichs 2007; Kwasniewski 2012).

Fig. 7
figure 7

Strengths of the tested specimens under different radial stress gradients

Full size image

Figure 8 shows the z-direction stress–strain curves. In the above strainburst tests, the z-direction stress and strain dominated the failure process of the tested specimens, varying over much larger ranges than those along the x-direction and y-direction. As can be seen from Fig. 8, the strength and deformation significantly varied with the radial stress gradient. In particular, the apparent yield platform, i.e., obvious plastic deformation prior to the peak stress, and its significant decrease with increasing radial stress gradient on these stress–strain curves deserve special attention. For instance, when the radial stress gradient was 1 kPa/mm (σ y2 = 0.1 MPa) or 10 kPa/mm (σ y2 = 1 MPa), a yield platform of ε z  = 7.965–10.576‰ or ε z  = 8.527–9.417‰ could be observed on the stress–strain curve, which indicates that apparent plastic deformation occurred in the tested specimen; when the radial stress gradient was 100 kPa/mm (σ y2 = 10 MPa), a yield platform of ε z  = 11.291–11.644‰ occurred, which significantly decreased but was still visible.

Fig. 8
figure 8

z-direction stress–strain curves of the tested specimens under different radial stress gradients

Full size image

Generally, granite, which is a typical hard rock, seldom exhibits apparent plastic deformation or yield platforms in the conventional triaxial or true-triaxial loading tests (Martin 1997; Haimson and Chang 2000; Kwasniewski 2012; Feng et al. 2016). Meanwhile, plastic deformation or yield platforms on granite specimens were reported in recent triaxial or true-triaxial unloading tests (Chen et al. 2009; Feng et al. 2016). It can be inferred that the relatively low confining pressure or even the free boundary after unloading dominates the occurrence of the yield platforms. The low confining stresses (or even the free boundary) tend to lead apparent yield platforms probably for the following reasons: First, at low confining pressure, immediately before and during failure, rock inelastic dilation can fully proceed along the lateral surface in triaxial compression and in particular along the free surface in true-triaxial compression. Accompanying this lateral dilation, consequent significant axial shortening likely occurs for existing angles between the cracks and the compression axis. Actually, previous experimental studies have suggested that under low confining pressure or unloading of confining pressure conditions, the strong lateral dilation can dominate the rock failure. Second, the nucleation of unstable fractures will be more obvious and gradual at low confining pressure because a large amount of local fractures are generated successively throughout the sample as failure approaches. Thus, obvious plastic deformation or large yield platforms tend to appear on the stress–strain curve.

The free boundary in our tests corresponded approximately to the flexible boundary with a hydrostatic pressure of 0 MPa (i.e., σ 3 = 0 MPa) in the confining cell of a Mogi-type true-triaxial testing apparatus (Feng et al. 2016). Consequently, an apparent yield platform was particularly likely to occur on each stress–strain curve of the tested specimens in our tests. At the same time, σ y obviously increased in the interior, further from the free face, under high radial stress gradient conditions, which created a relatively high confining pressure. Hence, the yield platform decreased with increasing radial stress gradient but could still be clearly observed because the constraint was always relatively low near the free face.

Over the last few decades, more attention has been paid to the deformation features in relation to the failure of the surrounding rock mass, specifically to post-peak deformation, such as brittle–ductile conversion, dilation (Wawersik and Fairhurst 1970; Ewy and Cook 1990; Alejano and Alonso 2005; Zhao and Cai 2010a, b). In the above strainburst tests, apparent plastic deformation or yield platforms close to the peak stress corresponded to relatively intensive progressive fractures prior to strainbursts. During this period, a large amount of meso- and macro-fractures developed and the tested specimens were notably damaged. Therefore, the apparent yield platform and its variations on the stress–strain curves of the tested specimens under different radial stress gradient conditions could significantly affect the intensity of post-peak strainbursts.

In addition, the inward radial (y-direction in tests) plastic deformation or dilation of the surrounding rock mass may have been more critical to the occurrence of strainbursts in the field compared to the tangential or circumferential (z-direction in tests) deformation. However, the y-direction deformation or dilation near the free face could not be accurately measured in our tests because of the intense impaction and destruction of the violent ejection. Hence, the influence of the radial stress gradient on the radial deformation is barely discussed here and will be systematically investigated in a future study.

4.2 Influence of Radial Stress Gradient on Strainburst Failure Mode

Figure 9 shows the failed specimens after the strainburst tests. Under different radial stress gradient conditions, not only the volume and location of the rockburst pit near the free face but also the failure that occurred both near the free face and further from the free face were significantly different. In this study, smooth cracks covered with few powders or small grains are regarded as tensile cracks, and coarse cracks mixed with a large amount of white powders or small grains are regarded as shear cracks (Bobet and Einstein 1998; Haimson 2007).

Fig. 9
figure 9

Failed specimens after the strainburst tests under different radial stress gradients: a 1 kPa/mm (σ y2 = 0.1 MPa); b 10 kPa/mm (σ y2 = 1 MPa); c 30 kPa/mm (σ y2 = 3 MPa); d 50 kPa/mm (σ y2 = 5 MPa); e 70 kPa/mm (σ y2 = 7 MPa); f 100 kPa/mm (σ y2 = 10 MPa); g 150 kPa/mm (σ y2 = 15 MPa)

Full size image

Under relatively low radial stress gradient conditions, rockburst pits usually occurred on the top or bottom of the free face, splitting fractures that were subparallel to the free face occurred on the two sides or throughout the tested specimens, slight shear or tension–shear fractures that dipped in the σ y direction occurred in the middle of the tested specimens, and the overall failure of the tested specimens was dominated by tensile splitting (Fig. 9a, b).

Under relatively high radial stress gradient conditions, rockburst pits tended to occur in the middle of the free face with a relatively large volume, splitting fractures occurred near the free face, visible shear or tension–shear ruptures occurred further from the free face, and the overall failure of the tested specimens exhibited an apparent zoning mode, that is, near the free face failure was dominated by tensile splitting, whereas further from the free face failure was dominated by shear (Fig. 9f, g).

Generally, the tested specimens exhibited two significantly different failure modes under both relatively low and relatively high radial stress gradient conditions (Fig. 10).

Fig. 10
figure 10

Two failure modes of the tested specimens: a under relatively low radial stress gradients, and b under relatively high radial stress gradients

Full size image

The failure mode of the tested specimens under relatively low radial stress gradients [e.g., 1 kPa/mm (σ y2 = 0.1 MPa) or 10 kPa/mm (σ y2 = 1 MPa)] is shown in Fig. 10a. In this case, the σ y values on the rock elements at different positions along the σ y direction were roughly equal and relatively low, so the strengths of the rock elements had no apparent differences and were not high. Tensile or tensile–shear macro-cracks continuously developed throughout the tested specimen before the overall failure, presenting an apparent yield platform on the stress–strain curve (see Fig. 8). As fractures developed or σ z further increased, rocks or rock slabs broke off and fragment ejection occurred near the free face (Fig. 5a, b); furthermore, the tested specimen totally failed, predominantly in tensile or tensile splitting on the two sides and tensile–shearing in the middle (Fig. 9a, b). The tested specimen was almost completely under biaxial stress conditions with a relatively low radial stress gradient and thus tended to fail through tensile splitting subparallel to the free face, which has been reported by theoretical, experimental and numerical research in relation to the failure of rock masses near the excavation boundary (Fairhurst and Cook 1966; Germanovich and Dyskin 2000; Diederichs 2007; Cai 2008; Yun et al. 2010; Gong et al. 2012). In addition, the slight shear failure in the middle of the tested specimen was mainly caused by constraints along the σ y direction from the rock elements near the two sides of the relatively thick (100 mm along the σ y direction) rock specimen during the dilation deformation process. Additionally, the friction between the surfaces of the rock specimen and rigid plates may have significantly changed the distribution of relatively low σ y . The above-mentioned knowledge suggests that the tensile splitting near the free face was not dense enough to form numerous apparent splitting rock slabs and further form a relatively large potential rockburst pit before the overall failure of the tested specimen with a low radial stress gradient. Actually, splitting rock slabs were seldom observed near the free face, or only extremely thick rock slabs were present, as shown on the failed specimens (Fig. 9a, b). Hence, the ejection failure was not too intensive and the rockburst pit tended to occur on the top or bottom of the free face for a relatively low radial stress gradient (Figs. 5a, b, 9a, b).

The failure mode of the tested specimens with a relatively high radial stress gradient [e.g., 100 kPa/mm (σ y2 = 10 MPa) or 150 kPa/mm (σ y2 = 15 MPa)] is presented in Fig. 10b. In this case, the σ y value or confining pressure significantly increased further from the free face along the σ y direction, increasing the strength of the rock element in the interior, farther from the free face. First, numerous apparent tensile or tensile–shear cracks progressively developed near the free face with increasing σ z , splitting the rocks into slabs because the rock elements near the free face were almost under biaxial compression and had relatively low strengths (Fairhurst and Cook 1966; Germanovich and Dyskin 2000; Diederichs 2007; Cai 2008; Yun et al. 2010; Gong et al. 2012); at the same time, the rock elements in the interior, further from the free face, were still intact. The tested specimen tended to fail in shear failure or tensile–shear failure overall and presented a relatively short yield platform on the stress–strain curve (Fig. 8). As σ z further increased, the rock slabs near the free face continuously expanded and bent, and the internal slabs that were constrained by the external rock slabs tended to fail in shear rupture, forming a relatively large potential rockburst pit. Once the outer slabs broke off, the rocks near the free face instantaneously fragmented and ejected out, creating intensive ejection failure (Fig. 5f, g); meanwhile or perhaps earlier, violent shear ruptures rapidly occurred in the interior, further from the free face, forming apparent shearing faults (Fig. 9f, g) because these rock elements were under triaxial compression and prone to shear ruptures (Mogi 1967, 1971; Horii and Nemat-Nasser 1985; Haimson and Chang 2000; Xu et al. 2000; Hoek and Martin 2014; Feng et al. 2016). In summary, except for those near the free face, the major rock elements in specimens were under triaxial compression, so the specimen exhibited a relatively high strength. As σ z continuously increased, the external slabs fully expanded and bent, while the constrained internal slabs tended to fail in shear rupture, forming a relatively large potential rockburst pit in the middle of the free face. Finally, as soon as the rock slabs near the free face broke off, the specimen failed in violent shear ruptures, creating intensive ejection failure near the free face (Figs. 5f, g, 9f, g).

Figure 11 shows the variations in the AE hits during the strainburst tests with a relatively low radial stress gradient (Fig. 11a, b) and relatively high radial stress gradient (Fig. 11c, d). Apparent differences can be observed in Fig. 11, which indicates the variations in the cracking and failure process. Before the sudden rockburst, the AE hits of the tested specimens with a relatively high radial stress gradient were generally less active than those with a relatively low radial stress gradient, which indicated that the development of the cracks was distinctly restricted (Horii and Nemat-Nasser 1985; Kawakata et al. 1999; Chang and Lee 2004). Under relatively low radial stress gradients [1 kPa/mm (σ y2 = 0.1 MPa) or 10 kPa/mm (σ y2 = 1 MPa)], in the middle and later stages of loading, the AE hits rapidly increased and remained relatively dense and stable as many tensile or tensile–shear cracks continuously developed throughout the tested specimen. This result is consistent with the above-mentioned fracture process and failure mode under relatively low radial stress gradients. Correspondingly, under relatively high radial stress gradients [100 kPa/mm (σ y2 = 10 MPa) or 150 kPa/mm (σ y2 = 15 MPa)], as σ z reached the peak stress, the AE hits continuously increased, first slightly and then sharply. This indicated that first fewer tensile or tensile–shear fractures occurred as tensile splitting took place only near the free face, and then a high degree of crack propagation and coalescence occurred as the violent shear rupture developed further away from the free face, coupled with the overall failure of the tested specimen.

Fig. 11
figure 11

Variations in AE hits during the strainburst tests under different radial stress gradients: a 1 kPa/mm (σ y2 = 0.1 MPa); b 10 kPa/mm (σ y2 = 1 MPa); c 100 kPa/mm (σ y2 = 10 MPa); d 150 kPa/mm (σ y2 = 15 MPa)

Full size image

Thus, as discussed above, under relatively low radial stress gradients, the strainburst failure of the specimens was dominated by tensile splitting, which is known as tensile strainbursts (Fairhurst and Cook 1966; Zheng et al. 1989; Dyskin and Germanovich 1993; Diederichs 2007; He et al. 2007, 2010; Gong et al. 2012); under relatively high radial stress gradients, the strainburst failure of the specimens was dominated by shear rupture, which is known as shear strainbursts (Ortlepp 2000, 2001; Xu et al. 2000; Hou et al. 2011; Feng et al. 2013). Figure 12 shows these two strainburst modes. As illustrated in Fig. 12, tensile strainbursts are characterized by the progressive bending, buckling and ejecting of tensile splitting rock slabs near the free face, and consequently, the ejection failure is not too intensive and the rockburst pit is relatively shallow (Fig. 12a); shear strainbursts are characterized by the instantaneous breaking off of multiple tensile splitting rock slabs near the free face and violent shear rupture further away from the free face, and consequently, the ejection failure is intensive or extremely intensive and the rockburst pit is relatively deep (Fig. 12b). It is, however, important to note that the shear strainbursts have seldom been simulated and reported in previous laboratory tests and that considering the radial stress gradient and further introducing the representative rock element with relatively large dimensions are key factors to successfully reproduce them in this laboratory study.

Fig. 12
figure 12

Schematic illustrations of two strainburst modes: a tensile strainbursts under relatively low radial stress gradients, and b shear strainbursts under relatively high radial stress gradients

Full size image

The failure modes of strainbursts under different radial stress gradients identified and characterized in our tests can provide insights into in situ strainbursts. Figure 13 presents the representative in situ strainbursts in the handrace or drainage tunnel at Jinping II Hydropower Station, Sichuan, China. The rockbursts in Fig. 13a, b are relatively light and accompanied by relatively shallow pot-shaped explosion pits, while those in Fig. 13c and d are relatively strong or extremely strong and accompanied by relatively deep V-shaped explosion pits. As shown in Fig. 12, these two significantly different failure phenomena of in situ strainbursts near excavation boundaries within a certain depth can be well interpreted according to the two failure modes of tensile strainbursts and shear strainbursts, respectively.

Fig. 13
figure 13

On-site photographs of strainbursts at Jinping II Hydropower Station, Sichuan, China: a the light–moderate rockburst that occurred in the drainage tunnel from chainage SK5 + 143 to SK5 + 138 (Feng et al. 2013); b the moderate rockburst that occurred in the headrace tunnel #1 at Stake K8 + 940 ~ 8 + 943 (Feng et al. 2013); c the intense rockburst that occurred in the drainage tunnel from chainage SK8 + 715 to SK8 + 718 (Liu et al. 2016); and d the “7.14” extremely intense rockburst that occurred in the headrace tunnel #4 at Stake K9 + 728 ~ 9 + 766 (Zhang et al. 2012a)

Full size image

According to the findings discussed above, it can be concluded that the relatively high radial stress gradient coupled with the sufficiently high tangential stress concentration will be likely to initiate stronger shear strainbursts in relatively deep regions around the excavation boundary in highly stressed underground openings, whereas relatively light tensile strainbursts or slabbing/spalling may occur in underground openings under relatively low in situ stress and low radial stress gradient conditions. Accordingly, different prevention and control should be advised and taken. Furthermore, it should be noted that the failure characteristics of field strainbursts are significantly affected by many factors, such as the tunnel axis stress, excavation method and rate and disturbances.

4.3 Influence of Radial Stress Gradient on Strainburst Kinetic Energy

The kinetic energy of ejected fragments is one of the most direct and effective indices to assess the intensity of rockbursts, and thus, quantitatively analyzing this energy is very important to perform laboratory or in situ investigations of the behavior of rockbursts. In this paper, the speed and kinetic energy of ejected fragments from the tested specimens were quantitatively estimated by analyzing videos of rockburst processes that were recorded by high-speed cameras.

As demonstrated in Fig. 14, two high-speed cameras were used to record the rockburst process. High-speed camera #2 was auxiliary, which helped to record the traces of some ejected fragments that were blocked by others (viewing from high-speed camera #1) and to adjust the unclear traces recorded by high-speed camera #1. The image analysis software Image-Pro Plus 7.0 was used to automatically or manually capture the trajectories of the ejected fragments, and then, the speed was calculated by dividing the distance along the trajectory by the corresponding time interval (Fig. 15).

Fig. 14
figure 14

Layout of the high-speed cameras

Full size image
Fig. 15
figure 15

Schematic of the speed measurement in Image-Pro Plus 7.0 (unit: m/s)

Full size image

The rock fragments were collected and divided into three groups according to their physical dimensions to facilitate the speed analysis: coarse fragments (maximum dimension d ≥ 9.5 mm), medium fragments (4.75 mm ≤ d < 9.5 mm) and fine fragments or grains (d < 4.75 mm), as shown in (Fig. 16). The ejection speed for the coarse fragments was the average speed of several points on the fragment. The ejection speed for the medium fragments was the average speed of several representative fragments in a certain region. The ejection speed for the fine fragments or grains was the average speed of several fragments in different locations. Then, the total kinetic energy of the ejected fragments could be calculated as follows:

$$E_{k} = \sum\limits_{i = 1}^{n} {\frac{1}{2}m_{1i} \bar{v}_{1i}^{2} + \frac{1}{2}M_{2} \bar{V}_{2}^{2} + } \frac{1}{2}M_{3} \bar{V}_{3}^{2}$$
(1)

where, the first, second and third items on the right side of the equation are the kinetic energy values of the coarse fragments, medium fragments and fine fragments, respectively; n is the number of coarse fragments; m 1i is the mass of the ith coarse fragment; M 2 and M 3 are the total masses of the medium and fine fragments, respectively; \(\bar{v}_{1i}\) is the average speed of several points on the ith coarse fragment; and \(\bar{V}_{2}\) and \(\bar{V}_{3}\) are the average speeds of the medium and fine fragments.

Fig. 16
figure 16

Ejected fragments, which are divided into three groups: coarse fragments, medium fragments and fine fragments or grains (unit: mm)

Full size image

The kinetic energy calculated here is the estimated value, as velocity and weight of each fine fragment are hard to be measured. Nevertheless, the velocity and weight of each coarse fragment can be measured. Thus, the kinetic energy of these coarse fragments, which constitutes the majority of the kinetic energy of ejected fragments, can be captured relatively accurate.

Figure 17 shows the kinetic energy of ejected fragments under different radial stress gradients. The kinetic energy of ejected fragments significantly increased with increasing radial stress gradient, roughly exhibiting a tri-linear variation. Specifically, the kinetic energy remained relatively low at approximately 2 J in Region I (radial stress gradient ≤50 kPa/mm), exhibiting a low sensitivity to variations in the radial stress gradient. In Region II (radial stress gradient ranging from 50 to 100 kPa/mm), the kinetic energy increased from 2.4 to 13.31 J, which indicates high sensitivity to variations in the radial stress gradient. In Region III (radial stress gradient >100 kPa/mm), the kinetic energy gradually increased above a relatively high level of approximately 12 J and also exhibited low sensitivity to variations in the radial stress gradient. Variations in the kinetic energy of ejected fragments under different radial stress gradients are discussed as follows.

Fig. 17
figure 17

Kinetic energy of ejected fragments under different radial stress gradients

Full size image

In Region I, the kinetic energy was relatively low. On the one hand, the tested specimens were subjected to relatively low radial stress or confining pressure, so the strength was relatively low (Fig. 7), which indicates that only a relatively small amount of elastic strain energy accumulated during the pre-peak loading period, further resulting in the relatively small post-peak kinetic energy. On the other hand, the relatively low radial stress gradients tended to initiate tensile strainbursts of the specimens. Tensile splitting uniformly occurred along the σ y direction, and macro-cracks fully developed throughout the tested specimens (Fig. 9) during the pre-peak period, consuming a large amount of elastic strain energy, which can be confirmed by the relatively long yield platform on the stress–strain curve (Fig. 8). Thus, the pre-peak elastic strain energy further decreased and the storage capacity of the elastic strain energy markedly weakened.

In Region III, the kinetic energy was relatively high. On the one hand, the tested specimens that were subjected to relatively high radial stress or confining pressure exhibited relatively high strength (Fig. 7), so much more pre-peak elastic strain energy accumulated, which can explain the occurrence of the relatively high post-peak kinetic energy. On the other hand, the relatively high radial stress gradients preferred to initiate shear strainbursts. Tensile splitting or tensile–shear fractures fully developed only near the free face (Fig. 9) before the apparent overall fracturing of the tested specimen, consuming a small amount of elastic strain energy and reducing the subsequent energy dissipation during the formation of the rockburst pit as well, which can be verified by the relatively short but visible yield platform on the stress–strain curve (Fig. 8). Thus, the pre-peak accumulated elastic strain energy was relatively high and the storage capacity of the elastic strain energy slightly decreased.

In Region II, with increasing radial stress gradient, the failure mode of the specimens changed from tensile strainbursts to shear strainbursts, coupled with the significantly increasing strength and decreasing plastic deformation, which eventually resulted in the rapid increase in the kinetic energy.

5 Conclusions

In underground excavations, the radial stress has significant effects on the strength, deformation and failure mode of surrounding rocks. However, few studies have addressed the effect of the radial stress gradient near the excavation boundary within a certain depth on strainbursts. Employing the improved rigid true-triaxial rockburst testing system and introducing the representative rock element of rocks near the excavation boundary within a certain depth, the strainburst characteristics under different radial stress gradients were investigated in this study. Rectangular prismatic granite specimens with relatively large dimensions of 100 × 100 × 200 mm, especially with a relatively large thickness of 100 mm along the radial direction, were tested based on the modified one-face-free true-triaxial loading stress path in a non-principal stress space rather than in the conventional principal stress space, which successfully simulated the structural failure process of strainbursts under different radial stress gradients ranging from 1 to 150 kPa/mm. High-speed cameras were used to capture and record the ejection failure processes during strainbursts, and the kinetic energy of ejected fragments was uniquely estimated to quantitatively assess the intensity of strainbursts.

The experimental results indicated that the radial stress gradient has an obvious influence on strainbursts. With increasing radial stress gradient, the strength of the specimens increased and the apparent yield platform prior to the peak stress that is peculiarly likely to happen in one-face-free true-triaxial loading tests decreased. Furthermore, the specimens tended to fail throughout in tensile splitting under low radial stress gradients, while they tended to fail in tensile splitting near the free face and in shear rupture further from the free face under high radial stress gradients. Thus, the strainburst mode of the specimens changed from tensile strainbursts, which are dominated by tensile splitting, to shear strainbursts, which are dominated by shear rupture. The strainburst modes under different radial stress gradients identified and characterized in our tests can explain in situ rockburst phenomena well. Consequently, the kinetic energy of ejected fragments during strainbursts significantly increased. It should be noted that a high radial stress gradient can restrain the occurrence of strainbursts but may result in much more intensive rockbursts as soon as the stress exceeds the capacity of rocks.

Although the experimental results provide insight into the influence of the radial stress gradient on the strainburst characteristics of granite, one must consider the limitations of the loading conditions when performing true-triaxial testing in the laboratory. The characteristics of in situ strainbursts under different radial stress gradients require further study. In particular, further investigations on tensile strainbursts and shear strainbursts are necessary.