Mechanical properties and fracture mode transformation of rocks subjected to asymmetric radial stresses

Abstract

Based on the analysis of the radial stress distribution in surrounding rocks of tunnel, different asymmetric radial stresses were loaded, which can more truly reflect the stress state of surrounding rocks of tunnel. The stress state and boundary conditions of representative surrounding rock element were simulated by sandstone in this study. The mechanical response and fracture mode transformation were studied. The experimental results showed that as the depth of surrounding rocks from the free surface of tunnel increased, the properties of surrounding rocks showed the characteristics of transition from brittleness to ductility. The strengths of sandstone specimens gradually increased with the increase of the depth, and a power function can well characterize the relationship between the strength and the depth. The fracture modes of sandstone specimens transformed from tensile splitting fracture to tensile-shear complex fracture, and finally to shear fracture with the increase of the depth, which was mainly caused by the asymmetric radial stresses. According to the characteristics of fracture modes under different asymmetric radial stress conditions, four types of failure zones in the radial direction of surrounding rocks can be divided, i.e., tensile splitting failure zone, tensile-shear complex failure zone, shear failure zone, and slight shear failure zone. The shear fracture in deep surrounding rocks and the accompanying large deformation of surrounding rocks should be paid more attention in the support of tunnel. In addition, the accuracy of time-failure model was verified, and it can accurately predict the time needed for rock failure.

Introduction

With the gradual depletion of shallow resources and the development of underground constructions, the excavation of the tunnel gradually develops to the deep. More and more attentions have been paid to the stability of deep tunnel surrounding rocks. As we know that after tunnel excavation, a free surface is formed, and the radial stress in surrounding rocks at the boundary of tunnel is approximately zero. With the increase of the distance from the excavation-free surface, the radial stress gradually increases and finally tends to far-field stress. The characteristics of radial stress distribution lead to that surrounding rocks in different depths from the excavation free surface are subjected to different radial stresses. The stress state of surrounding rocks changes from biaxial stress at the boundary of tunnel (Su et al. 2017b) to poly-axial stress state with one face free and other five faces constrained near the excavation boundary and finally to the true triaxial stress state at a certain depth. Figure 1 shows the radial stress distribution curve of surrounding rocks along the radial direction of a circular tunnel. Based on the Kirsch equations and filed observations, the value of radial stress at the excavation free surface is zero, and it increases nonlinearly with the increase of the distance from the free surface. The distribution of radial stress shows notable gradient variations, and the radial stress gradient is more obvious under high in situ stress conditions, especially when the difference between the far-filed principal stresses is large.

Fig. 1

Radial stress distribution along the θ = 0° direction (θ is the angular direction that rotates counterclockwise from the x-coordinate)

Extensive studies have been conducted on the mechanical behavior of surrounding rocks, and a large number of loading or unloading tests under uniaxial, biaxial, conventional triaxial, and true triaxial stress conditions have been carried out (Chen et al. 2009; Feng et al. 2016; Hua and You 2001; Huang and Li 2014; Jingtao and Xiating 2006; Yun et al. 2010a). In these studies, true triaxial tests considering the free surface have become the preferred method for investigating the stability of tunnel, especially the problems of rockburst (He et al. 2007, 2010, 2012a, b; Su et al. 2017a; Zhao and Cai 2014). The results of true triaxial tests considering the free surface show that this poly-axial stress state near the excavation boundary significantly affects the mechanical behavior, deformation, and energy characteristics of surrounding rocks (Alejano and Alonso 2005; Diederichs 2007; Zhao and Cai 2010a, b). He et al. (2010) unloaded the minimum principal stress and simulated the process of rockburst under different true triaxial stresses. The characteristics of rockburst and acoustic emission (AE) signals were studied. The experimental results showed that the stress level before unloading and unloading rates has a significant effect on the occurrence time of rockburst and the compressive strength of rocks. A high stress level or an unloading rate can trigger transient and intense strainburst. AE signals showed that the AE activity and energy release rapidly increased from unloading to the occurrence of rockburst. True triaxial stress states of surrounding rocks with different radial stress gradients after tunnel excavation were simulated by Su et al. (2017b). The influences of radial stress gradients on the strainburst were studied from the aspects of failure process, compressive strength, failure mode, and the kinetic energy of fragments. The experimental results were helpful to better understand the mechanism of strainburst. Su et al. (2018) conducted true triaxial rockburst tests using an improved rockburst test system and analyzed the evolution characteristics of AE and sound signals during the rockburst process. They found that on the eve of rockburst, the main frequency of AE and sound signals showed a downward trend. The research results provided powerful information for the prediction of rockburst. The true triaxial tests under different initial stresses and unloading rates were carried out to investigate the mechanisms of the deformation and failure of marble (Xu et al. 2019). The test results implied that the damage degree of marble gradually increased with the increase of unloading strain rate, and the failure mode of marble changed from mixed shear and tensile failure to tensile failure. In addition, more transgranular cracks were observed through microscopic images. Due to excavation, the unloading effect was simulated through the true triaxial experiments under different confining pressures and unloading rates (Si and Gong 2020), and it was found that under the same unloading rate, the greater the initial confining pressure, the greater the compressive strength and stress drop after failure, and the more intense the rockburst.

So far, the stability of surrounding rocks has been studied by many researchers. However, most of these studies focused on the failure of surrounding rocks, which are at the free surface of tunnel or near it. In the actual situation, the damage induced by the structural instability, rockburst, and other dynamic disasters after the tunnel excavation is not only confined to the vicinity of the free surface, but also spreads to deep surrounding rocks (Feng et al. 2013; Li et al. 2012; Ortlepp 2001). The free surface will be formed after tunnel excavation, and the stress in surrounding rocks will be redistributed. Then, the radial stress σ_r gradually increases with the increase of the distance between surrounding rocks and free surface. In addition, due to the constraint of adjacent rock elements, the shear stress exists on the surface of representative rock element. When the tangential stress σ_θ exceeds the capacity of surrounding rocks, the rocks at free surface or even a certain depth may be damaged or even cause the rockburst. The above stress conditions considering the variation of radial stress appears in the non-principal stress space (Su et al. 2017b), and it allows the existence of shear stress on the surface of the representative rock element. Thus, as the distance from the free surface increases, the radial stress increases nonlinearly, which leads to that surrounding rocks are subjected to asymmetric radial stresses with unequal stresses at the two faces in the radial stress direction, as shown in Fig. 1. In Fig. 1, the radial stress σ_ri is less than σ_rj, which causes the existence of the stress gradient in the rock.

However, there are few reports on the effect of radial asymmetric constraints on the mechanical properties and fracture modes of surrounding rocks. Generally, the rock mechanics tests show that the stresses at the two parallel faces are equal, e.g., σ_ri = σ_rj. It cannot truly reflect the stress state of surrounding rocks, which results in that the damage degree of the deep surrounding rocks is unknown when the spalling, splitting failure, or even rockburst occurs. Moreover, the support depth of tunnel is closely related to the damage degree of deep surrounding rocks. Therefore, it is necessary to carry out the experimental research to truly simulate the stress state of surrounding rocks in the radial stress direction and boundary conditions, so as to obtain the influences of radial asymmetric constrains on the mechanical properties, and fracture modes of surrounding rocks, and deepen the understanding of rockburst and the damage of deep surrounding rocks.

Experimental preparation

Specimen preparations

Sandstone specimens were used in this study to simulate surrounding rocks. The sandstone specimens are gray-white without obvious cracks, defects, or beddings on the surface. The homogeneity of them is good, and it can effectively reduce the test dispersion. Before taking the tests, the sandstone specimens are ground to cubes with 100 mm × 100 mm × 100 mm in length, width, and height, respectively. The squareness of each cube specimen surfaces is within 0.02 mm. The cube sandstone specimens are presented in Fig. 2. Before the test, the uniaxial compression and Brazilian splitting tests were carried out to obtain the basic physical parameters of sandstone, as shown in Table 1.

Fig. 2

Cube sandstone specimens

Table 1 Basic physical parameters of sandstone

True triaxial equipment

A newly developed true triaxial equipment (TTG) by the Chongqing University in China was used in this study (Li et al. 2016). The 3D schematic diagram and physical diagram of TTG are shown in Fig. 3. In addition, the relationship between the loading heads in true triaxial cell and the rectangular coordinate system is shown in Fig. 3. Each stress loading head in true triaxial cell can independently load the specimen under the displacement or force control mode, which provides operability for the application of asymmetric true triaxial stress state in this study. Among the six stress loading heads, four stress loading heads in the X and Y directions can apply a maximum pressure of 6000 kN, and two stress loading heads in the Z directions can apply a maximum pressure of 4000 kN (Zhao et al. 2020). This system can truly simulate the stress path under different engineering backgrounds and fully meet the technical requirements of the tests in this study.

Fig. 3

True triaxial equipment (TTG)

Experimental scheme

The deep tunnel with a radius of a and a buried depth of 1000 m is taken as an example. According to the existing research results (Gong et al. 2017, 2018; Luo et al. 2019, 2020), the vertical and horizontal principal stresses can be obtained through the calculation formula of in situ stress by Stephansson et al. (1986), and the expressions are as follows:

$$\begin{gathered} \sigma_{v} = \gamma H \hfill \\ \sigma_{h}^{\max } = 6.7 + 0.0444H \hfill \\ \sigma_{h}^{{{\text{m}} ix}} = 0.8 + 0.0329H \hfill \\ \end{gathered}$$

(1)

where γ is the average bulk density of rock, and taking as 27kN/m³, H is the depth of tunnel, σ_v is the vertical principal stress (MPa), σ^max_h is the maximum horizontal principal stress (MPa), and σ^mix_h is the minimum horizontal principal stress (MPa). According to Eq. (1), we can obtain that σ_v = 27 MPa, σ^max_h = 51.1 MPa, and σ^mix_h = 33.7 Mpa.

Generally, three ways of tunnel layout are applied in practical engineering, i.e., the axis of tunnel along the direction of σ^max_h, σ^mix_h, or along a angle with the direction of σ^max_h. In this study, the axis of tunnel along the σ^max_h direction is taken as an example (Gong et al. 2018). Based on the Kirsch equations, the distribution of radial stress σ_r around the tunnel excavation boundary can be calculated, and the curve of the distribution of radial stress σ_r is shown in Fig. 4. According to the values of radial stresses σ_r, eight different ranges of surrounding rocks are divided along the direction of radial stress σ_r with the increment of 5 MPa. In Fig. 4, the radial stress σ_r expresses an obvious nonlinear increase. The value of radial stress σ_r on the free surface of tunnel is zero. When the distance L = 0.115a, radial stress σ_r is about 5 MPa; the distance L = 0.257a, radial stress σ_r is about 10 MPa; the distance L = 0.447a, radial stress σ_r is about 15 MPa; the distance L = 0.733a, radial stress σ_r is about 20 MPa; the distance L = 1.243a, radial stress σ_r is about 25 MPa; and the distance L = 2.717a, radial stress σ_r is about 30 MPa. When the distance L is greater than 2.717a, the radial stress σ_r gradually approaches to the far field stress 33 MPa. Due to the range of surrounding rocks where the radial stress σ_r from 30 to 33 MPa is large, the radial stress σ_r is set as 30 MPa in this range. It should be noted that the plane strain problem is considered by the Kirsch equations and the axis stress σ_a along the axis of tunnel basically remains unchanged in the actual excavation process. Thus, the variation of axial stress σ_a before and after excavation is not considered in this study.

Fig. 4

Radial stress distribution along the θ = 0° direction of the circular tunnel with buried depth of 1000 m

According to the above analyses, it can be seen that two faces of the rock element in the direction of radial stress are subjected to different radial stresses, and this special boundary condition is called asymmetric stress condition in our study in the non-principal stress space. Generally, the unloading in the process of tunnel excavation can easily cause the instability of surrounding rocks or even the rockburst. However, the formation of free surface and the increase of tangential stress in surrounding rocks are also the main factors that lead to the failure of surrounding rocks and rockburst. A large number of engineering practices show that the failure of surrounding rocks and rockburst induced by the increase of tangential stress often occur within a few hours or 1–3 days after tunnel excavation (Feng et al. 2013). Thus, the stress load path in this study is designed as follows:

• Step 1: Load σ_θ and σ_a to preset value with force control of 1 kN/s, while σ_r is loaded to σ_ri with force control of o.5 kN/s.

• Step 2: Continually load σ_θ and σ_a. Keep σ_ri on one face of specimen unchanged and increase the stress on the opposite face to σ_ji.

• Step 3: Keep σ_a, σ_ri, and σ_ji constant and continually increase σ_θ with displacement control of 0.001 mm/s until specimen failure.

For convenience, in the following analysis, σ_z, σ_y, and σ_x are used to represent σ_θ, σ_a, and σ_r, respectively. The schematic diagram of experimental scheme and stress loading curve under the stress condition of σ_xi = 5 MPa and σ_xj = 10 MPa are shown in Fig. 5. The specific experimental scheme is shown in Table 2.

Fig. 5

Stress loading curves: a schematic diagram of experimental scheme; b stress loading curve under the stress condition of σ_xi = 5 MPa and σ_xj = 10 MPa

Table 2 Experiment plan

Experimental results and analysis

Stress–strain curves of sandstone specimen under different asymmetric radial stress conditions

Figure 6 shows the stress–strain curves under different asymmetric radial stress conditions. As shown in Fig. 6, since the cracks in the sandstone specimens have been compacted during the stage of initial stress loading, the stage of crack compaction in the stress–strain curve under different asymmetric radial stress conditions is not obvious. Table 3 lists the drop amplitudes and the rates of stress decrease under different asymmetric radial stress conditions. It can be seen that when σ_xi was loaded to 0 ~ 15 MPa and σ_xj was loaded to 0 ~ 20 MPa, the stress decline was obvious after the stress reached the peak and the drop amplitude was great. It indicated that a intense brittle failure of sandstone specimen occurred. When σ_xi was loaded to 20 ~ 30 MPa and σ_xj was loaded to 25 ~ 30 MPa, the rate of stress drop was low, and the drop amplitude was small. This phenomenon was the most obvious when σ_xi = 25 MPa, σ_xj = 30 MPa and σ_xi = 30 MPa, σ_xj = 30 MPa. The sandstone specimen expressed obvious ductility characteristics. To quantitatively described the brittleness or ductility of sandstone specimen, the brittleness index was proposed by Bishop (1967) and denoted by K. The expression is as follows:

$$K = \frac{{\sigma_{zp} - \sigma_{zr} }}{{\sigma_{zp} }} \times 100\%$$

(2)

where σ_zp is the peak stress (MPa) and σ_zr is the residual strength (MPa).

Fig. 6

Stress–strain curves under different asymmetric radial stress conditions: a stress–strain curve in the tangential direction (σ_z); b stress–strain curve in the axial direction (σ_y); and c stress–strain curve in radial direction (σ_x)

Table 3 Drop amplitudes and rates of stress decrease under different radial stress conditions

According to Eq. (2), we obtained the brittleness indexes K of surrounding rocks at different depths from the free surface, and the columnar diagram of brittleness indexes K was shown in Fig. 7. a represents the radius of tunnel. From Fig. 7, it can be found that the brittleness of surrounding rocks near the free surface was relatively large, which led to a intense failure of surrounding rocks. With the increase of L, the brittleness indexes K gradually decreased, the ductility of surrounding rocks performed more obviously. To sum up, with the increase of L, the properties of surrounding rocks showed a characteristics of transformation from brittleness to ductility.

Fig. 7

Columnar diagram of brittleness indexes K

From Fig. 6b, the strain of sandstone specimen near the free surface (i.e., σ_xi = σ_xj = 0 MPa and σ_xi = 0, σ_xj = 5 MPa) was positive in the σ_y direction which implied that the compression deformation dominated the deformation of surrounding rocks in the σ_y direction. When σ_xi ≥ 5 MPa and σ_xj ≥ 10 MPa, the strain of sandstone specimen in the σ_y direction was negative. It indicated that the deformation of surrounding rocks which was far away from the free surface mainly exhibited tensile expansion in the σ_y direction. Compared with Fig. 6b and c, we can find that the surrounding rock which was far away from the free surface exhibited tensile expansion in the horizontal direction, and under the same stress condition, the tensile expansion of sandstone specimen in the σ_x direction was greater than that in the σ_y direction. Figure 8 shows the absolute values of the strain difference △ε between ε_x and ε_y under different stress conditions. It can be observed that △ε gradually decreased with the increase of σ_xi and σ_xj, which indicated that the tensile expansion in the σ_y direction was gradually close to that in the σ_x direction. That was to say, with the increase of the distance L from the free surface, the expansion deformation of surrounding rocks in the horizontal direction changed from mainly concentrated in the σ_x direction to the uniform distribution in the σ_x and σ_y direction.

Fig. 8

Absolute values of the strain difference △ε between ε_x and ε_y under different stress conditions

Characteristics stresses of sandstone under different asymmetric radial stress conditions

The characteristic stresses of surrounding rocks, including the peak strength of surrounding rocks σ_zp, the crack initiation stress σ_ci, and the damage stress σ_cd, are helpful to understand the damage degree of surrounding rocks along the radial direction. In this study, the strengths of sandstone specimens were the overall response of rock under different asymmetric radial stress conditions, which can truly reflect the bearing capacity of surrounding rocks within a certain depth from the free surface. Table 3 lists the sandstone strengths under different asymmetric radial stress conditions. In Table 3, when σ_xi = σ_xj = 0 MPa, i.e., the surrounding rock at the free surface of tunnel, the strength of sandstone specimen was the lowest (96.75 MPa). With the increase of σ_xi and σ_xj, the strength gradually increased. When σ_xi and σ_xj increased to 30 MPa, i.e., the surrounding rock was in the original rock stress area, the strength was the largest (221.35 MPa). This was consistent with the previous results that the rock strength increased with the increase of confining pressure or minimum horizontal principal stress (Horii and Nemat-Nasser 1985). However, the stress constraint conditions in our study were obviously different from those applied in the previous conventional triaxial or true triaxial mechanical tests, which more truly reflected the strength characteristics of surrounding rock within a certain depth from the free surface.

In addition, in Table 3, we can see that the strength of S8 was greater than that of S7. The sandstone specimen (S7) was considered to consist of four thin rock plates (1–4 in Fig. 9). Due to the fact that there was a stress gradient in the specimen, the stress acting on the rock plate in the σ_x direction gradually increased, resulting in a gradient change of its strength. The strength of rightmost rock plate (4) was approximately equal to that of S8, and the strengths of other three rock plates were less that that of S8. Thus, when the four thin rock plates were combined into a whole rock (S7), the strength was less than that of S8.

Fig. 9

Schematic diagram of rock strength change under asymmetric radial stress conditions

The in situ brittle failure of tunnel surrounding rocks usually occurs near the excavation boundary and gradually develops along the radial direction to deep surrounding rocks. Previous studies have shown that the surrounding rock failure or rockburst caused by the tangential stress concentration not only occurred near the excavation free surface, but also caused different degrees of damage in deep surrounding rocks. Generally, the failure process of rock materials is usually accompanied by crack closure and propagation. Base on the characteristics of crack deformation, the mode of crack strain was proposed by Martin (1997), and it is widely used to identify the crack initiation stress σ_ci and crack damage stress σ_cd at present (Cai et al. 2004; Wang et al. 2012). According to the mode of crack strain, the crack strain is equal to the measured strain minus the elastic strain, as shown in Eq. (3).

$$\begin{gathered} \varepsilon_{z}^{c} { = }\varepsilon_{z} { - }\frac{1}{E}\left[ {\sigma_{z} - v\left( {\sigma_{y} + \sigma_{x} } \right)} \right] \hfill \\ \varepsilon_{y}^{c} { = }\varepsilon_{y} { - }\frac{1}{E}\left[ {\sigma_{y} - v\left( {\sigma_{z} + \sigma_{x} } \right)} \right] \hfill \\ \varepsilon_{x}^{c} { = }\varepsilon_{x} { - }\frac{1}{E}\left[ {\sigma_{x} - v\left( {\sigma_{z} + \sigma_{y} } \right)} \right] \hfill \\ \varepsilon_{v}^{c} { = }\varepsilon_{z}^{c} { + }\varepsilon_{y}^{c} { + }\varepsilon_{x}^{c} { = }\varepsilon_{v} { - }\frac{1 - 2v}{E}\left( {\sigma_{z} + \sigma_{y} + \sigma_{x} } \right) \hfill \\ \end{gathered}$$

(3)

where ε_z, ε_y, and ε_x are the measured strains in the directions of σ_z, σ_y, and σ_x, respectively. ε^c_z, ε^c_y, and ε^c_x are the crack strains in the directions of σ_z, σ_y, and σ_x, respectively. ε_v is the measured volumetric strain, and ε^c_v is the volumetric crack strain.

Figure 10 shows the measured volumetric strain and crack volumetric strain of sandstone during the compression failure under asymmetric radial stress conditions. According to Fig. 10, the process of rock failure can be divided into four stages: stage I—the cracks gradually close. Stage II—in this stage, only elastic compression deformation occurs, which is shown as a horizontal straight line on the crack volumetric strain, i.e., line AB in Fig. 10. When the existing cracks in rock propagate and the new cracks appear, the measured volumetric strain includes elastic volumetric strain and crack volumetric strain, which makes the curve of crack volumetric strain begin to deviate from the straight line, i.e., point B in Fig. 10. The stress corresponding to the deviation point is the crack initiation stress σ_ci. The stage III is the steady growth stage of cracks. In this stage, the cracks in the rock gradually increase, and the crack volumetric strain shows the effect of propagation and increase. At the end of this stage (point C), the prototype of shear plane begins to form in the rock, and the cracks begin to unsteadily propagate. The expansion deformation in the horizontal direction is greater than the compression deformation in the vertical direction, which leads to the appearance of the inflection point (point D) of the measured volumetric strain curve. The stress corresponding to this inflection point is the crack damage stress σ_cd.

Fig. 10

Curves of the measured volumetric strain ε_v and crack volumetric strain ε^c_v

According to the above method, σ_xi and σ_xj can be obtained, as shown in Table 4. Due to the fact that the strain in the σ_x direction cannot be measured, σ_ci and σ_cd under the stress condition of σ_xi = 0 MPa and σ_xj = 5 MPa were not shown in Table 4.

Table 4 Crack initiation stress σ_ci and damage stress σ_cd under different radial stress conditions

The mean values of the ranges of surrounding rocks corresponding to different σ_xi and σ_xj were caculated. For example, the surrounding rock under the stress condition of σ_xi = 5 MPa and σ_xj = 10 MPa was located in the range of 0.115a ~ 0.257a from the free surface, then the mean value of this range was taken, i.e., the distance L of the surrounding rock from the free surface is 0.186a, as shown in Table 4. Thus, the relationship between the distance L and the strength of surrounding rock, the measured volumetric strain, and crack volumetric strain of sandstone were obtained, as shown in Fig. 11. It can be seen that the characteristic stresses gradually increased with the increase of L, which indicates that the restraining effect of the depth of surrounding rocks on crack initiation and propagation is gradually strenthened. When L increased from 0.186a to 0.988a, the values of σ_zp, σ_ci, and σ_cd increased by 37.15 MPa, 32.04 MPa, and 44.66 MPa, respectively. When L increased from 0.988a to 2.717a, the values of σ_ci and σ_cd increased by 35.47 MPa, 28.38 MPa, and 40.04 MPa, respectively. This phenomenon indicates that the rate of increase of the characteristic stresses in shallow surrounding rocks is higher that in deep surrounding rocks. This result can be explained by the rough contact model of primary cracks in surrounding rocks, as shown in Fig. 12a.

Fig. 11

Variation curves of characteristic stresses of surrounding rocks at different depth from excavation free surface

Fig. 12

Schematic diagram of crack initiation and propagation model of fractured rock under different constraint effects

It can be assumed that the primary crack surfaces are saw-tooth shaped. When the distance between the surrounding rock and free surface is small, the primary cracks are affected by low confinement (low σ_r). Rock elements will slide along the slope of serrated boss, resulting in dilatancy, which leads to crack initiation and propagation. In this case, there is no cohesive force on the surface of crack, and the resistance ${\tau }_{R}$ against the relative slide between crack surfaces can be expressed as

$$\tau_{R} { = }\sigma_{n} \tan (\phi_{b} + \alpha )$$

(4)

where σ_n is the normal stress, Φ_b is the basic friction angle of the crack surface, and a denotes the ramp angle of boss. When the constraint effect reaches the critical value, the serrated boss will be sheared, that is to say, the intact rock will be sheared, and the dilatancy will disappear. In this case, ${\tau }_{R}$ can be expressed as

$$\tau_{R} = c + \sigma_{n} \tan \phi_{r} { = }\sigma_{n}^{^{\prime}} \tan (\phi_{b} + \alpha ) + (\sigma_{n} - \sigma_{n}^{^{\prime}} )\tan \phi_{r}$$

(5)

where c is the cohesive force of intact rock, Φ_r is the residual friction angle of intact rock, and ${\sigma }_{n}^{^{\prime}}$ is the normal stress that corresponds to the critical confinement. According to Eqs. (4) and (5), when σ_n is limited at a low level, the increase of ${\tau }_{R}$ is mainly dependent on Φ_b + a, while when σ_n is at a high level, the increase of ${\tau }_{R}$ is dependent on Φ_r. In this section, the axial stress is fixed, and the radial stress σ_xi and σ_xj dominate the change of constraint effect, and the constraint effect on the primary crack increases with the increase of σ_xi and σ_xj. Since Φ_b + a is significantly greater than Φ_r (Fig. 12b), when σ_xi and σ_xj increase at a low stress level, the resistance to crack initiation and propagation will increase at a high rate. Thus, when σ_xi and σ_xj increase at a low stress level, the rates of the increase of the characteristic stresses are greater than those at a high stress level.

According to the results of the above analyses, the characteristic stresses of surrounding rocks along the radial direction can be summarized as follows: with the increase of the depth from the free surface, the characteristic stresses of surrounding rocks gradually increase, and the surrounding rock at or near the free surface is more susceptible to the radial stress than that far away from the free surface. According to the magnitude of the radial stress, the tunnel can be roughly divided into two regions along the radial direction: region 1 (σ_xi and σ_xj are relatively small) and region 2 (σ_xi and σ_xj is relatively large). Compared with region 2, the increase rates of characteristic stresses in region 1 are higher with the increase of radial stress. It should be noted that the division of region 1 and region 2 needs to determine the characteristic stresses of surrounding rocks according to the specific stress environment and lithology. For example, in this chapter, according to Fig. 11, the critical depth of region 1 is about 0.590a ~ 0.988a.

Fracture mode of sandstone under different asymmetric radial stress conditions

Figure 13 shows the physical images and CT scanning diagrams of fracture modes under different asymmetric radial stress conditions after sandstone specimen failure. It needs to be noted that since the sandstone specimen under the stress condition of σ_xi and σ_xj = 0 MPa is seriously damaged, the fracture mode can be clearly observed by using binary image instead of CT scanning diagram. As it can be seen from Fig. 13, the fracture modes of sandstone specimens showed obvious differences under different σ_xi and σ_xj. In this test, the distributions of σ_x simulated the stress conditions of surrounding rocks in the radial direction, which were located in an area from the free surface to the original rock stress area. The two opposite faces of sandstone specimen were subjected to different radial stresses σ_x, i.e., σ_xi ≤ σ_xj, which led to the different fracture modes of surrounding rocks located near and far away from the free surface. In our study, the tensile fractures are characterized by smooth surface of fracture and few powders or small grains distribution, and the coarse fractures with a large amount of powders or small grains are regarded as shear fractures (Haimson 2007).

Fig. 13

Failed specimens under different asymmetric radial stress conditions: (a) σ_xi = 0 MPa, σ_xj = 0 MPa; (b) σ_xi = 0 MPa, σ_xj = 5 MPa; (c) σ_xi = 5 MPa, σ_xj = 10 MPa; (d) σ_xi = 10 MPa, σ_xj = 15 MPa; (e) σ_xi = 15 MPa, σ_xj = 20 MPa; (f) σ_xi = 20 MPa, σ_xj = 25 MPa; (g) σ_xi = 25 MPa, σ_xj = 30 MPa; (h) σ_xi = 30 MPa, σ_xj = 30Mpa

The results in Fig. 13 showed that under biaxial compression, the fracture modes of sandstone specimens exhibited tensile splitting failure. Several splitting fractures appeared in sandstone specimen along the σ_x direction, and rock plates which approximately parallel to the σ_z and σ_y direction or the free surface were formed. This was consistent with the previous experimental results that focused on the failure of surrounding rocks near the excavation boundary (Cai 2008; Diederichs 2007; Gong et al. 2012; Read and Martin 1996; Yun et al. 2010b).

The fracture mode of sandstone specimen under the stress condition of σ_xi = 0 MPa and σ_xj = 5 MPa is presented in Fig. 13b. In this case, near the free surface, several splitting fractures through the specimen appeared, while the shear fractures were formed far away from the free surface. Since the stress gradient existed in the sandstone specimen in the σ_x direction, we considered the specimen as a geometry consisting of four thinner rock plates, as shown in Fig. 14a. The first rock plate, which was near the free surface, was subjected to small radial stress and was almost in a state of approximate biaxial compression stress. Therefore, the strength of the first rock plate was low and easy to form the tensile cracks. When the tensile cracks spread through the sandstone specimen, the split rock plates were formed as shown in Fig. 14a. As the distance of the rock plates from free surface increased, rock plates were subjected to a large stress. The stress state can be regarded as true triaxial stress state. Thus, the rock strength was relatively large, and the formation of tensile cracks was inhibited, and shear fractures were more likely to generate (Feng et al. 2016; Haimson and Chang 2000; Hoek and Martin 2014), as shown in Figs. 13b and 14b. Therefore, the fracture mode of sandstone specimen under the stress condition of σ_xi = 0 MPa and σ_xj = 5 MPa exhibited tensile-shear complex failure.

Fig. 14

Schematic diagram of fracture mode transformation

When σ_xi was in the range of 5 ~ 20 MPa and σ_xj was in the range of 10 ~ 25 MPa, it can be seen in Fig. 13c ~ f that the ultimate failure of sandstone specimens was mainly dominated by the shear fractures. The main reason for this phenomenon might be that when σ_xi and σ_xj are relatively large, the four rock plates are all in a true triaxial stress condition (Fig. 14b). The initiation and propagation of tensile cracks in the σ_x direction are effectively restrained, so that all rock plates exhibit shear fracture (Feng et al. 2020), which results in shear failure of sandstone specimen as a whole.

When σ_xi and σ_xj increased close to the far-filed stress, i.e., σ_xi = 25 MPa, σ_xj = 30 MPa and σ_xi = 30 MPa, σ_xj = 30 MPa, the number of fractures on the surface of sandstone specimens was relatively less, and the length of them was short. Through CT scanning and three-dimensional reconstruction of sandstone specimens under the above stress conditions (Fig. 15), we found that in the σ_x direction, the shear fractures appeared on both front and back surfaces of sandstone specimen. But these fractures did not fully coalesce to form the macroscopic fracture surface, and the degree of sandstone failure was relatively low. This might be due to that the micro tensile cracks in the rock have almost the same opportunity to expand along the σ_xi and σ_xj directions. However, due to the great constraint, the crack propagation time is short. After the cracks coalesce, the slight shear fractures are formed along the σ_xi and σ_xj directions.

Fig. 15

Three-dimensional reconstruction of failed specimens

According to the above analyses, it can be seen that with the increase of the distance L, the fracture modes of sandstone specimens varied significantly and gradually changed from tensile splitting failure at the boundary of free surface to tensile-shear complex failure near the free surface, and then to shear failure at a certain depth from the free surface, and finally to slight shear failure in the area which was close to the original rack stress area. This kind of variation of fracture modes had a good consistency with the failure mode of surrounding rock after tunnel instability or rockburst in the field (Jiang 2017). Based on the transformation of fracture modes under different asymmetric radial stress conditions, the four types of failure zones of surrounding rocks in the radial direction were divided, i.e., tensile splitting failure zone, tensile-shear complex failure zone, shear failure zone, and slight shear failure zone, as shown in Fig. 16.

Fig. 16

Schematic diagram of failure zone of surrounding rock in the radial direction

It should be noted that we need to pay more attention to the shear fracture in our tests. When the tangential stress is large enough, the surrounding rocks at and near the free surface will be destroyed first, forming the split rock plates. As the tangential stress continues to increase, the size of rock plates reaches its bucking size, and then it is broken, and the rock fragments are ejected into the tunnel space, resulting in a strong dynamic failure phenomenon, i.e., rockburst. Finally, the rockburst pits are formed on the surface of tunnel, as shown in Fig. 17a. At this time, the shear fractures induced by the relatively large tangential stress may be formed in deep surrounding rock, and the residual elastic energy in the surrounding rock is great. Thus, damage range of surrounding rocks caused by rockburst is not only limited to the depth of rockburst pits, but also extends to deep surrounding rocks, as shown in Fig. 17b. The tunnel with a radius of a is taken as an example, and we assume that the formation of rockburst pits is caused by the fracture of rock plates. According to Fig. 13b, the range of tensile splitting is about one-third of the total width of sandstone specimen. Thus, it can be calculated that the range of tensile splitting near the free surface is about 0 ~ 0.0383a, i.e., the depth of rockburst spits is 0.0383a. The depth of surrounding rocks with shear fracture is greater than 0.0383a. Therefore, in the support of tunnel surrounding rocks, it is not only necessary to strengthen surrounding rocks near the free surface to avoid the tensile splitting fracture caused by high tangential stress in the tunnel support, but also to pay more attention to the shear fracture in deep surrounding rocks and the accompanying large deformation of surrounding rocks.

Fig. 17

Schematic diagram of fracture mode of surrounding rock with different depths in the radial direction: a tensile splitting failure near the free surface; b shear fracture in deep surrounding rock

Discussion

Analysis of progressive damage process of surrounding rock under asymmetric radial stress conditions

During tunnel excavation, the surrounding rocks are damaged by the drilling and blasting excavation (D&B) or mechanical excavation (TBM) methods, and the physical and mechanical properties of surrounding rocks are changed. Thus, the excavation damaged zone (EDZ) is formed in surrounding rocks. After tunnel excavation, the stress in surrounding rocks is redistributed. The tangential stress continuously increases and leads to the development of EDZ from free surface to the deep. It often causes the spalling failure, large volume collapse, or even the rockburst, which seriously threatens the safety of life and property. Therefore, a thorough understanding of EDZ and its damage degree to surrounding rocks are an important guarantee to maintain the stability of tunnel.

In the following analysis, a tunnel with radius of a was taken as an example to study the progressive failure process of surrounding rocks. Before analyzing the progressive failure process of surrounding rocks, since the sandstone specimen is firstly destroyed under the stress condition of σ_xi = 0 MPa and σ_xj = 0 MPa, the strength under this stress condition is taken as a reference in the following progressive damage analysis. In addition, for convenience, the surrounding rock located at the surface of tunnel is referred to as No. 1 surrounding rock, and the surrounding rock within the range of σ_xi = 0 MPa and σ_xj = 5 MPa is referred to as No. 2 surrounding rock, and so on, as shown in Fig. 18.

Fig. 18

Marks of surrounding rock at different distances from free surface

In this study, the stress of crack initiation and propagation is taken as the starting point to calculate the EDZ of surrounding rock. When the tangential stress increases to 96.75 MPa, the surrounding rock at the free surface of tunnel are firstly destroyed by tensile splitting, and the damage has already been produced in No. 2 surrounding rock. According to the gradient distribution of radial stress, it can be inferred that the damage is mainly concentrated in the surrounding rock which is near the free surface. The radial stress in this part of the surrounding rock is small, and the rock is almost subjected to biaxial compression state. Thus, it is reasonable to infer that the failure mode of this part of the surrounding rock is tensile splitting failure. However, the tangential stress does not reach the bearing capacity of the surrounding rock which is far away from the free surface, and the overall failure of No. 2 surrounding rock does not occur, but the cracks in the rock are expanding unsteadily. At this moment, the depth of EDZ is 0.0383a. When the tangential stress increases to 108.45 MPa, No. 2 surrounding rock becomes unstable as a whole, and the failure mode is tensile-shear complex failure, while No. 3 surrounding rock and deeper surrounding rock are still in the elastic stage, and almost no damage is produced. The depth of EDZ is 0.115a. As the tangential stress continues to increase to 115.51 MPa, No. 3 surrounding rock enters the stage of unstable crack propagation, and the damage rapidly accumulates until shear failure occurs, and the depth of EDZ is 0.257a. With the increase of tangential stress to 127.74 MPa, No. 4 surrounding rock begins to fail at an accelerating rate, and No. 5 surrounding rock is still in the elastic stage. The depth of EDZ is 0.447a. Finally, when the tangential stress increases to 196.03 MPa, No. 1 ~ 6 surrounding rocks have failed. No. 7 surrounding rock is in the stage of crack stable expansion. The EDZ is close to the original rock stress area and equals to 2.717a. Since No. 8 surrounding rock is located in the original rock stress area, the range of this stress state is large. Thus, the damage range in this position is not considered in our study.

Through the above analysis, the damage of surrounding rocks in the radial direction is a progressive process in time and space. With the increase of the depth from free surface, the time needed for the damage of surrounding rocks gradually increases. The progressive process in space shows that the damage of surrounding rocks gradually develops from the free surface to the deep. The ratio of maximum stress σ_max acting on the rock to the uniaxial compressive strength σ_c can be used to characterize the difficulty of rock failure. According to the experimental results, we can get the relationship between the EDZ of surrounding rocks and the tangential stress acting on sandstone specimens, as shown in Fig. 19 and Eq. (6). Equation (6) is similar to the empirical formula proposed by Kaiser and Martin based on field tests. It can be seen that the area of EDZ of surrounding rocks is closely related to the mechanical properties of rock mass, tunnel size, and original rock stress. However, there are some gaps in the correlation coefficient, which may be caused by the homogeneity of the rock mass tested in the laboratory and the existence of end effect, resulting in the difference in the characteristic stresses between them and the in situ heterogeneous rock mass.

$$\frac{D}{a}{ = 2}{\text{.04}}\frac{{\sigma_{\max } }}{{\sigma_{c} }} - 2.73$$

(6)

$$\frac{D}{a}{ = 1}{\text{.34}}\frac{{\sigma_{\max } }}{{\sigma_{c} }} - 0.57( \pm 0.05)$$

(7)

$$\frac{D}{a}{ = 1}{\text{.25}}\frac{{\sigma_{\max } }}{{\sigma_{c} }} - 0.51( \pm 0.1)$$

(8)

where D is the depth of EDZ, a is the equivalent radius of tunnel, σ_max is the maximum tangential stress at the boundary of tunnel, and σ_c is the uniaxial compressive strength of rock. Equations (7) and (8) are the empirical formula proposed by Kaiser and Martin, respectively.

Fig. 19

Relationship between tangential stress and depth of EDZ

Time prediction of surrounding rock fracture under asymmetric radial stress conditions

In the aspect of preventing the rockburst and failure of tunnel surrounding rocks, it does not only need to monitor the stress distribution and deformation of surrounding rocks, but also need to predict the failure time according to the stress and the change law of deformation, so as to strengthen the support timely and effectively before the failure of surrounding rocks occurs. Due to the fact that the failure of surrounding rocks induced by the increase of tangential stress after tunnel excavation was simulated in this study. The rock failure is essentially caused by the accumulation of damage under the effect of deviator stress. Thus, the time-failure model proposed by Voight can be used to predict rock failure (Voight 1988; Ying et al. 2009):

$$\frac{{d^{2} \Omega }}{{dt^{2} }} = A\left( {\frac{d\Omega }{{dt}}} \right)^{\alpha }$$

(9)

where t is the real-time moment of the experiment recorded by the experimental apparatus in s, A is the constant, α is an index to measure the degree of nonlinear, usually 1 ~ 2, and Ω is related to precursory strain. Based on the previous study, the minimum principal strain ε_x is relatively sensitive to crack propagation (Kong et al. 2018) and is closely related to the failure of surrounding rock. Thus, ε_x was used to reply Ω in our study, and Eq. (9) can be written as follows:

$$\frac{{d^{2} \varepsilon_{x} }}{{dt^{2} }} = A\left( {\frac{{d\varepsilon_{x} }}{dt}} \right)^{\alpha }$$

(10)

When the internal damage of rock accumulates to a certain extent, the cracks begin to rapidly expand out of control, and the rock begins to enter the stage of nonlinear expansion until rock failure, which exhibits a rapid increase of ε_x. At this moment, the value of α is equal to 2. Substitute it into Eq. (10):

$$\left( {\frac{{d\varepsilon_{{\text{x}}} }}{dt}} \right)^{ - 1} = \left( {\frac{{d\varepsilon_{x} }}{dt}} \right)_{0}^{ - 1} - A\left( {t - t_{0} } \right)$$

(11)

When the left side of Eq. (11) is equal to zero ((dε_x/dt)⁻¹ = 0), the line represented by Eq. (11) intersects with the abscissa, as shown in Fig. 20. It indicates that the cracks in the rock rapidly expand, coalesce with each other, and tend to failure. Take the test data into Eq. (11), and the predicted time of failure can be obtained. Table 5 lists the actual time of failure t_act and predicted time of failure t_pre under different asymmetric radial stress conditions. Note that ε_x in the σ_x cannot be measured under the stress condition of σ_xi = σ_xj = 0 MPa and σ_xi = 0 MPa, σ_xj = 5 MPa; thus, the predicted time of failure is not shown in Table 5. It can be seen from the data in Table 5 that the predicted time t_pre of failure is close to the actual time t_act of failure, which implies that the time-failure model can be used to effectively predict the failure time of surrounding rocks. In practical engineering, the deformation and stress in the radial direction can be obtained by technical means. For example, by embedding optical fiber in surrounding rocks of tunnel, the distributed optical fiber monitoring technology is used to measure the stress and deformation (Chai et al. 2016; Yuan et al. 2014), so as to predict the failure time of surrounding rocks and achieve the purpose of stress release or surrounding rock support in advance.

Fig. 20

Prediction fracture time curves of sandstone specimen under different asymmetric radial stress conditions: a σ_xi = 5 MPa, σ_xj = 10 MPa; b σ_xi = 10 MPa, σ_xj = 15 MPa; c σ_xi = 15 MPa, σ_xj = 20 MPa; d σ_xi = 20 MPa, σ_xj = 25 MPa; e σ_xi = 25 MPa, σ_xj = 30 MPa; f σ_xi = 30 MPa, σ_xj = 30Mpa

Table 5 Actual and prediction fracture times of sandstone specimen under different asymmetric radial stress conditions

Preliminary study on the influence of discontinuities on failure characteristics of surrounding rocks

Through the above test results and analyses, the tension cracks approximately parallel to the excavation surface are formed in the surrounding rock near the tunnel boundary. These tension cracks further expand, penetrate, and have tension displacement, cutting the surrounding rock into plate-shaped rock slab. After the tangential concentrated stress further acts on the rock plate and exceeds its ultimate strength, the rock plate will be unstable and damaged, forming rock mass ejection, while the failure of the surrounding rock far from the tunnel boundary are dominated by fracture. It should be noted that the test results in this paper are based on the intact rocks. Figures 16 and 17 show the failure mode of homogeneous and intact surrounding rock under the action of tangential concentrated stress. However, in the actual stratum, the rock mass often contains structural planes with different occurrences, location, and scale, showing significant deformation anisotropy. During the development of the failure from the rock mass near the tunnel boundary to the deep surrounding rock, the normal force of the structural plane gradually decreases, making the closed state of the structural plane change to active, and the structural plane begins to expand under the action of tangential concentrated stress. The failure of surrounding rocks is significantly influenced by the structural plane in this case. The influence of structural plane is illustrated below.

Case 1: When the structural plane is approximately parallel to the tunnel axis or distributed at a small included angle, as shown in Fig. 21a, the expansion of the fracture to the deep part of surrounding rocks is blocked and restricted by the structural plane, and the expansion of the structural plane further reduces the stability of the structure of surrounding rocks. In this case, the failure of surrounding rocks is mainly caused by the energy accumulation in the rock plate or external disturbance, and the structural plane controls the depth and shape of the failure of surrounding rocks.

Fig. 21

Sketch of mechanisms of structural plane in the formation of failure

Case 2: When the included angle between the structural plane and the tunnel axis is relatively large, as shown in Fig. 21b, the concentrated shear stress in the vertical direction can be decomposed into shear stress along the structural plane and normal stress perpendicular to the structural plane. When the shear stress applied to the structural plane is greater than its shear strength, the shear dislocation of the structural plane occurs, and the violent energy is released (Zhou et al. 2015a), which induces the overall instability and failure of surrounding rocks. Thus, the shear dislocation of structural plane becomes the inducing factor of the failure.

Case 3: For the unexposed structural plane, as shown in Fig. 21c, the rock mass between the structural plane and the tunnel boundary is still intact. After the excavation, the high shear stress is induced by the stress redistribution, resulting in shear dislocation along the structural plane. The intact rock mass between the structural plane and the tunnel boundary is subjected to shear failure, and the structural plane is connected with the tunnel boundary. During this failure process, the rigid filler or convex plate between the contact surfaces is sheared off. According to the research of (Ryder 1988; Zhou et al. 2015b), the stress decline caused by shear failure in weak planes such as structural plane is about 5 ~ 10 MPa, while the shear failure in intact rock mass is about 20 MPa, Therefore, it is reasonable to infer that the energy released in this failure process is greater than that in case 2 and less than the energy released by the failure of intact rock.

The above analyses are a preliminary understanding of the effect of structural plane on the failure of surrounding rocks based on the orientation and location of structural plane. However, in the actual stratum, the distribution of structural plane is random, and it has more complex morphology and different mechanical properties. Thus, it needs to be deeply analyzed and verified through laboratory tests, numerical simulation, and physical model tests.

Conclusions

After tunnel excavation, the free surface was formed, and the radial stress in the surrounding rocks increased gradiently. In the non-principal stress space, the gradient characteristics of radial stress are simulated by applying different stresses on the front and back faces of cubic sandstone. The strength response and mechanism of fracture mode transformation were investigated. The main results can be concluded as follows:

• 1. Under different asymmetric radial stress conditions, with the increase in σ_xi and σ_xj, the ductility of sandstone specimens performed more obviously, and the peak strengths of sandstone specimens gradually increased. That is to say, the properties of surrounding rocks show characteristics of transformation from brittleness to ductility with the increase of depth from the free surface. The relationship between the strength and the depth of surrounding rocks can be well described by a power function.

• 2. The fracture modes of sandstone specimens were significantly influenced by the asymmetric radial stress conditions, showing a transformation from tensile splitting fracture to tensile-shear complex fracture and finally to shear fracture with the increase of σ_xi and σ_xj. Thus, four types of failure zones in the radial direction can be divided, i.e., tensile splitting failure zone, tensile-shear complex failure zone, shear failure zone, and slight shear failure zone.

• 3. With the progressive failure of surrounding rocks, the excavation damage zone was gradually deepened and had a linear increasing relationship with the tangential stress. In addition, the time required for failure of sandstone specimens can be predicted by a time-failure model.

It should be pointed out that although the test results based on the intact rocks are different from the failure of surrounding rocks containing structural planes in the actual stratum, the failure modes in the tests are basically consistent with the existing research results. In addition, structural planes do not exist everywhere in surrounding rocks of the whole tunnel. Therefore, the research results of this paper are still helpful to understand the evolution process of the progressive damage failure and the mechanism of fracture mode transformation of surrounding rocks along the radial direction and can provide guidances for the stability control of surrounding rocks.