1 Introduction

After excavation of underground caverns and tunnels, surrounding rocks suffer the development of the excavation damaged zone (EDZ), which still exhibits a certain bearing capacity (Hudson 1989; Hoek et al. 1995; Kaiser 2016). The studies of post-peak characteristics and residual strength play a crucial role in evaluating the bearing capacity (Cai et al. 2007a, b; Gao and Kang 2017). In recent decades, research on residual strength has made great progress and this has exerted a profound influence on the optimal design of supporting structures (Peng et al. 2017). However, the China Jinping Underground Laboratory Phase II (CJPL-II, with a burial depth of about 2400 m) is constructed under a state of three-dimensional (3D) high-geostress (Feng et al. 2018). The residual strength characteristics in this 3D stress state are not yet clear [e.g., the influence of intermediate principal stress (σ2)], which deserved to be explored further.

There has been considerable research on the complete stress–strain curve of rock obtained by uniaxial or conventional triaxial laboratory tests (σ2 = σ3) (Wawersik and Brace 1971; Hudson et al. 1972; Liao and Hsieh 1999; Labuz and Dai 2000; Hashiba et al. 2006; Barla et al. 2010; Peng et al. 2017), which provide a basis for investigating residual strength. Based on these data, the residual strength characteristics of rock have been studied and described by some strength criteria and models. Some scholars have researched the relationship between residual strength and confining pressure using existing 2D strength criteria. The linear relationship was studied by Mohr–Coulomb criterion (Cai et al. 2007a, b; Yang et al. 2012) and the non-linear relationship was explored by Hoek–Brown (Cai et al. 2007a; Gao and Kang 2017) and a second-order polynomial function (Joseph 2000). Others (Fang and Harrison 2001; Zhang et al. 2010) proposed a reduction index to convert peak strength to residual strength, which was related to confining pressure. By taking peak strength, confining pressure, and plastic deformation as variables, others (Kaiser 2016; Peng et al. 2017) have established a mobilised post-peak strength model, which could predict the residual strength of rock at sufficient strain: however, the aforementioned two models describe peak strength by using 2D criteria, thus still investigating the relationship between residual strength and confining pressure.

With redistribution of 3D stresses after underground cavern excavation, σ3 and σ2 change non-synchronously. The aforementioned residual strength data under uniaxial and conventional triaxial compression (σ2 = σ3) fail to evaluate contributions of independent σ3, let alone the influence of σ2. A true triaxial system can be used to investigate the aforementioned topic theoretically. However, true triaxial tests are mainly carried out to study peak strength characteristics, strength criteria (Mogi 1971; Al-Ajmi and Zimmerman 2005), failure-plane angle (Mogi 1973; Ma and Haimson 2016), and other aspects such as unloading-induced spalling and rockburst (He et al. 2015, Su et al. 2017; Li et al. 2018), etc. The literature on post-peak behaviour of rock remains sparse with regard to provision of residual strength data in 3D stress states.

Based on a true triaxial testing platform for complete stress–strain process of hard rock (Feng et al. 2016), a series of the true triaxial compression tests under different stresses were carried out on a total of 41 marble specimens from the CJPL-II to investigate their strength and deformation characteristics, with particular interest in the residual strength.

2 Testing

2.1 Specimens and Test System

The sample is a thick-layer, fine-grain white marble with single lithology, taken from the CJPL-II. The average density is 2.82 g/cm3 and major mineral components contain dolomite and calcite. According to the ISRM suggested method (Fairhurst and Hudson 1999), specimens were made into cuboids measuring 50 mm × 50 mm × 100 mm, with a tolerance of ± 0.02 mm and a maximum vertical deviation of ± 0.01 mm.

The true triaxial system was established by Northeastern University, China (Feng et al. 2016; Kong et al. 2018). The stiffness of loading framework of the system along the σ1 direction is 6 MN/mm (higher than the suggested value of 5 MN/mm (Fairhurst and Hudson 1999)). By applying a closed-loop servo-controlled system with extremely rapid and flexible servo valve, the post-peak deformation and failure of rock can be controlled to allow acquisition of the complete stress–strain curve and residual strength. Additionally, the system was equipped with the PCI-2 acoustic emission (AE) monitoring device from the US Physical Acoustics Corporation.

2.2 Test Scheme and Loading Stress Paths

The CJPL-II was constructed in a deep zone with high geostress (σ3, σ2, and σ1 of about 25, 67, and 69 MPa, respectively). The surrounding rocks experience complex redistribution of 3D stress after excavation. Therefore, it is necessary to conduct true triaxial compression under 3D stress state (σ3, σ2): σ3 and σ2 are between 0 and 50 MPa, and 0 to 200 MPa, respectively. The test scheme and stress levels are summarised in Table 1.

Table 1 Experimental results of the CJPL-II marble under true triaxial compression
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The stress paths during true triaxial compression are illustrated in Fig. 1 and the main loading steps are shown as follows:

Fig. 1
figure 1

True triaxial loading paths: σ1 stress control and ε3 strain control are applied before and after point A (about 0.75σc), respectively

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  1. 1)

    The hydrostatic pressure (σ1 = σ2 = σ3) was applied at a constant rate of 0.5 MPa/s to the set value;

  2. 2)

    Keeping σ3 unchanged, σ1 and σ2 were synchronously increased at the same rate of 0.5 MPa/s to the set values;

  3. 3)

    Keeping σ3 and σ2 constant, σ1 was further increased while the strain rate in the ε3-direction was measured. When the rate reached 0.5 × 10− 5 (point A, about 75% of peak strength σc), the control mode for σ1 was changed to strain control along the ε3 direction. Moreover, σ1 was further applied at the constant ε3 strain rate until the residual strength of rock was mobilised.

3 Test Results and Analysis

3.1 Characteristics of the Complete Stress–Strain Curve Under True Triaxial Compression

Figure 2 shows several typical complete stress–strain curves (deviatoric stress versus strain in three directions) of marble under true triaxial compression. When σ3 = σ2, these are relatively smooth post-peak curves. With increasing σ2 or decreasing σ3, multiple stress drops appear in the post-peak stage, as found elsewhere (Mogi 1973) that brittle failure was accompanied by a marked stress drop. Here, the stress drops do not refer to sudden uncontrollable stress drops (Wawersik and Brace 1971; Hudson et al. 1972), but the servo-controlled stress drops match crack-induced brittle failure in rock. In the failure state, the peak strength and residual strength can thus be investigated.

Fig. 2
figure 2

Typical complete stress–strain curves of CJPL-II marble under true triaxial compression: a curves in conventional triaxial stress states (σ2 = σ3); b and c curves in true triaxial stress states (σ2 > σ3)

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The test results of all specimens are listed in Table 1. Mechanical properties of rock are then derived from the stress–strain curve, as shown in Fig. 3. σc refers to the peak strength, εc is the peak strain corresponding to σc, σr0 is defined as the initial residual strength when the rock just enters its residual stage, and εr0 is the strain corresponding to σr0. Subsequent deformation along the ε1-direction is defined as residual slip deformation X. As the residual strength significantly decreases with X, Δσr is the reduction of residual strength relative to σr0. Determination of σr0 has definite physical meaning: Fig. 3 shows that the point at σ1 = σr0 is the sharp turning point in the complete stress–strain curve from the post-peak brittle failure stage to the residual stage. AE activity is significantly different before and after this point. Furthermore, the failure mechanisms of the two stages are also obviously different. The former mainly involves the propagation and coalescence of microcracks and local cracks inside rock, resulting in a through-fractured surface, while the latter is mainly the local rupture of asperities on the failure surfaces. Therefore, σr0 is considered to be the strength of the failed rock that has just formed a through-fractured surface under true triaxial compression.

Fig. 3
figure 3

Determination of peak strength σc and corresponding to strain εc, initial residual strength σr0 and corresponding to strain εr0, and residual slip deformation X, reduction of residual strength Δσr from the complete stress–strain curve

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3.2 Peak Strength Characteristics Under True Triaxial Compression

Figure 4 shows the peak strength characteristics of marble in different stress states under true triaxial compression. It can be seen that σ2 has a significant effect on peak strength (σc). For each tested σ3, σc increases with σ2 at a gradually decreasing rate, until a plateau is reached at some σ2 value, which conforms to previous conclusions (Mogi 1971; Ma and Haimson 2016; Zhao et al. 2018). In the test, σ2 is not large enough to result in a decline in σc, but it satisfies engineering requirements.

Fig. 4
figure 4

Peak strength characteristics of the CJPL-II marble under true triaxial compression. The solid lines show the trend line of peak strength and the best-fit polynomial

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4 Residual Strength of Marble Under True Triaxial Compression

4.1 Deformation-Dependence on Residual Strength and Its AE Characteristics

By carrying out conventional triaxial compression, some scholars (Wawersik and Brace 1971; Cai et al. 2007a; Peng et al. 2017) suggested that residual strength of rock shows deformation-dependence, gradually declining with increasing deformation in residual stage. However, there are few test results showing the complete stress–strain curves under true triaxial compression at present and therefore investigating the deformation-dependent residual strength of marble under 3D stress state and its AE characteristics are novel.

Figure 2 also demonstrates the relationships of σ1–σ3 with ε1, ε2, and ε3 in the residual stage. The deformation along the ε1 and ε3-directions constantly increases whether the residual strength decreases or not. Under true triaxial stresses (σ2 > σ3), the deformation along the ε2-direction does not increase when the residual strength decreases, which is different from that in conventional triaxial stresses (σ2 = σ3). Because when σ2 > σ3, the failure surfaces which have been formed before the residual stage are parallel to σ2 in the σ1–σ3 plane (Fig. 7). This is in accordance with previous conclusions (Mogi 1973; Ma and Haimson 2016). Therefore, during slip along the failure surface, deformation along the directions of σ1 and σ3 significantly increases while that along the direction of σ2 only evinces the elastic rebound caused by reduction in σ1.

To show deformation-dependence of residual strength, several typical curves of the variations of residual strength with X are displayed in Fig. 5a. It can be seen that the residual strength of marble is subjected to multiple reductions at small amplitude with increasing X. A quantitative description of this reduction is given in Fig. 5b, which reveals the relationship between Δσr/σr0 and ε1/εc. As the deformation increases, the residual strength significantly decreases compared with σr0. For example, the value of Δσr/σr0 corresponding to ε1/εc = 5.2 at σ3 = 2 MPa and σ2 = 15 MPa reaches 45.2%, which is still not the final value. Therefore, the residual strength shows significant deformation-dependence in 3D stress states.

Fig. 5
figure 5

Deformation-dependent residual strength of marble in several typical 3D stress states: a variation of residual strength with residual slip displacement X; b the relationship between Δσr /σr0 and ε1/εc

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Figure 6 reveals AE characteristics in the residual stage at σ3 = 2 MPa and σ2 = 15 MPa. The AE count rate is low and when failure reaches the stress reductions, AE activity increases and the AE count rate matches the rate of σ1 decrease. From Fig. 7a, it can be seen that the failure surface is not a simple plane but one with significant asperities. After undergoing significant slip deformation, the failure plane with obvious asperities may show local crushing and even contain secondary fractured surfaces probably due to frictional effects (Fig. 7b), resulting in generation of strengthened AE signals as well as corresponding reductions in residual strength.

Fig. 6
figure 6

AE characteristics in the residual stage at σ3 = 2 MPa and σ2 = 15 MPa. 1, 2, 3, and 4 represent the first, second, third, and fourth stress reductions, respectively

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Fig. 7
figure 7

Macroscopic failure morphology under true triaxial compression at σ3 = 2 MPa and σ2 = 15 MPa: a the failed marble specimen photograph; b optical microscopy observation (magnified ×30)

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4.2 The Influence of σ 3 on Residual Strength

As an absolutely stable residual strength cannot be truly acquired by laboratory testing due to deformation-dependence, there is no universal definition of residual strength at present (Cai et al. 2007a; Gao and Kang 2017; Peng et al. 2017). Therefore, σr0 owing to the obvious physical meaning (as explained in Sect. 3.2), can be investigated under true triaxial compression. The residual strength mentioned below also refers to this value.

There has been much research on residual strength under conventional triaxial stress conditions, which was consistent with the Mohr–Coulomb or Hoek–Brown criteria. In Fig. 8a, the residual strength data at σ3 = σ2 were fitted by the linear Mohr–Coulomb strength criterion:

Fig. 8
figure 8

The relationship between the residual strength σr0 and σ3: a test values of residual strength at σ2 = σ3 and predicted values based on the M–C strength criterion; b variation of σr0 with σ3 at σ2 = 15, 30, 65, and 100 MPa

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$${\sigma _{{\text{r0}}}}=a{\text{ }}{\sigma _3}+b,$$
(1)

where a and b are constants, which are related to the cohesion c and internal friction angle \(\varphi\), and can be expressed as follows:

$$a{\kern 1pt} ={\text{ }}(1+\sin \varphi ){\kern 1pt} {\kern 1pt} /{\kern 1pt} (1 - \sin \varphi ),{\text{ }}b{\text{ }}={\text{ }}(2c\;\cos \varphi )/(1 - \sin \varphi ).$$
(2)

Table 2 displays the fitting result. The linear correlation coefficient R2 between the residual strength and confining pressures at σ3 = σ2 is 99.8%, thus evincing high conformance with the Mohr–Coulomb criterion, which is consistent with previous scholars’ conclusions (Cai et al. 2007a, b; Yang et al. 2012). However, this is the residual strength characteristics at σ3 = σ2. After excavation of deep underground caverns, σ3 gradually decreases from geostress level in primitive rock zone to near zero at the free face, while the variation of σ2 is not synchronous with that of σ3. Therefore, it is necessary to evaluate the contribution of independent σ3 and σ2 to residual strength, respectively.

Table 2 Linear fitting result between σr0 and σ3
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Figure 8b reveals the variation of σr0 with σ3 for each constant value of σ2. The residual strength obtained for different σ3 at a given σ2 was also fitted by linear Mohr–Coulomb criterion. The fitting results show that residual strength exhibits a favourable linear relationship as all the R2 exceed 98.1%. When σ2 is 15 MPa, 30 MPa, 65 MPa, and 100 MPa, the corresponding values of a are 6.713, 6.660, 6.139, and 4.571, respectively, decreasing with increasing σ2, which decreases the effect of σ3 on increasing residual strength.

The effect of σ3 on residual strength parameters (cohesion c and internal friction angle φ) is evaluated for each constant σ2, by examining the corresponding values at that σ2 with its magnitude when σ2 = σ3 (Table 2). When σ2 ≤ 65 MPa, the internal friction angle for a given σ3 is slightly larger than that when σ3 = σ2 while the cohesion exhibits the opposite trend. When σ2 > 65 MPa, the internal friction angle is slightly smaller than that when σ3 = σ2 while the opposite applies to the cohesion.

4.3 The Influence of σ 2 on Residual Strength

As shown in Table 1 and Fig. 9a, σ2 exhibits a certain influence on residual strength in true triaxial stresses. Although the influence is inferior to the effect of σ3, it shows certain regularity. To study the σ2 effect and conveniently conduct a comparison with the result under conventional triaxial stresses, a new coefficient ui for residual strength is defined:

$$u{{\kern 1pt} _i}=({\sigma _{{\text{r}}ij}} - {\sigma _{{\text{r}}ic}})/{\sigma _{{\text{r}}ic}},$$
(3)

where σric and σrij represent the residual strength in conventional and true triaxial stresses, respectively. When σ2 = σ3, ui = 0.

Fig. 9
figure 9

The σ2 effect on residual strength. a The relationship between residual strength σr0 and σ2; b variation of ui with σ2 for each constant σ3 value

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It can be seen from Fig. 9b that ui varies with σ2. For a high σ3 (> 10 MPa), coefficient ui significantly decreases from zero to a low value at some σ2 value with increasing σ2, then returns to near-zero values. For a low σ3 (≤ 10 MPa), ui decreases with increasing σ2. The lowest values of ui at σ3 = 2, 5, 10, 15, 20, 30, and 40 MPa are − 0.178, − 0.163, − 0.147, − 0.174, − 0.136, and − 0.157, respectively.

In other words, the residual strength of marble exerts a certain σ2 effect. It decreases at first and then increases with σ2 for a high σ3 while declining with σ2 for a low σ3. It is worth noting that the residual strength in true triaxial stress states is generally lower (by 13.6–17.8% at most) than that in conventional triaxial stresses at the same σ3. The reasons for the σ2 effect on residual strength of marble will be analysed in Sect. 5.1.

5 Discussion

5.1 Causal Analysis of σ 2 Effect on Residual Strength

When rock enters its residual stage, the macroscopic through-fractured surfaces have formed and the frictional resistance related to the characteristics of failure surfaces greatly determines residual strength (Cai et al. 2007a; Liang et al. 2017). Figure 10 shows that the failure morphology varies with σ2 at the same σ3, which is not a simple plane but one with obvious asperities and multiple fractures. The σ2 effect on residual strength may relate to the variation of the morphology of fractures with σ2. Taking stress states such as different σ2 at σ3 = 30 and 40 MPa as typical examples of high σ3 although they cannot represent all conditions. The morphology can be approximately divided into two types: one is the composite of a primary fracture and V-shaped secondary fractures (Fig. 10a, c, d, f), the other is a single primary fracture (Fig. 10d) or single primary fracture trend (Fig. 10b). For low σ2 at σ3 = 30 and 40 MPa, the failure morphology is of the composite type. With increasing σ2, the morphology evolves into the single type at some σ2 value, and then returns to the composite type. In terms of the composite type, the residual strength is high, not only including frictional resistance along the primary fracture but also that along V-shaped secondary fractures and corresponding mechanical resistance to embedment. For the single type, the residual strength only includes frictional resistance along the single primary fracture, resulting in a lower residual strength.

Fig. 10
figure 10

Macroscopic failure morphology of marble under true triaxial compression with different σ2 at σ3 = 30 MPa (a–c) and 40 MPa (d–f). The black solid lines show the primary fractures, and the blue dashed lines show the secondary fractures. a, c, d, f are the composite of a primary fracture and V-shaped secondary fractures; b, e represent the single primary fracture or trend

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5.2 The Influence of Residual Strength Under 3D Stress States on the Stability of Deep Caverns

Cai et al. (2007a) and Yu et al. (2013) suggested that the weakening of residual strength can enlarge the EDZ after tunnel excavation by using numerical methods. Tiwari and Latha (2017) found that the influence of uncertainty of residual strength parameters on engineering safety sensitivity is greater than that of peak strength parameters. The value of σ3 gradually decreases from the primary rock zone to free faces after deep underground cavern excavation. Timeous strengthening support installation is conducive to increasing σ3, especially for free faces nearby, which can rapidly elevate the residual strength to thus strengthen the bearing capacity of surrounding rocks and reduce the range of the EDZ. However, the residual strength shows an obvious σ2 effect, resulting in a reduction thereof compared with that in the conventional triaxial stresses in general, and a significant deformation-dependence. The malign influence of the two effects on residual strength, not considered in aforementioned numerical analysis of tunnels excavation, should be taken into account in analysis of the EDZ of deep caverns. The research further improves recognition of strength of the failed rock, which is significant when controlling damage to surrounding rocks and designing rock-support schemes.

6 Concluding Remarks

A series of true triaxial compression tests were carried out on the CJPL-II marble, aimed at obtaining the complete stress–strain curves (including post-peak phase) and then investigating the peak strength, especially residual strength characteristics. The residual strength under 3D stress states was provided. The deformation-dependence and influence of dependent σ3 on residual strength were explored. Moreover, its σ2 effect was innovatively revealed. The main conclusions are as follows:

  1. 1.

    The residual strength in true triaxial stress states exhibits significant deformation-dependence and multiple small-amplitude reductions as the residual deformation increases. During these reductions, AE count rates increase, which may be related to the local rupture of asperities on the failure surfaces.

  2. 2.

    The residual strength of marble shows a significant σ3 effect, increasing linearly with σ3 at the same value of σ2. The residual strength also displays a certain σ2 effect. It decreases at first and then increases with increasing σ2 for a high given σ3 while gradually declining with σ2 for a low given σ3. Therefore, the residual strength in true triaxial stresses is generally lower (by 13.6–17.8% at most) than that in conventional triaxial stresses at the same σ3. The σ2 effect may be related to the variations of morphology of failure surfaces with σ2.

  3. 3.

    The deformation-dependence and σ2 effect on the residual strength of rock should be taken into account in engineering analysis and design. Otherwise, the residual strength may be over-estimated, resulting in increasing risk to safe engineering operations.