Post-peak deformation and failure behaviour of Jinping marble under true triaxial stresses

Engineering Geology 265 (2020) 105444

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Post-peak deformation and failure behaviour of Jinping marble under true triaxial stresses

T

⁎

Zhi Zhenga,c, Xia-Ting Fenga,b, , Cheng-Xiang Yangb, Xi-Wei Zhangb, Shao-Jun Lia, Shi-Li Qiua a

State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China Key Laboratory of Ministry of Education on Safe Mining of Deep Metal Mines, Northeastern University, Shenyang 110819, China c University of Chinese Academy of Sciences, Beijing 100049, China b

A R T I C LE I N FO

A B S T R A C T

Keywords: True triaxial compression Post-peak behaviour σ2 effect Meso- and micro-failure Acoustic emission

The post-peak behaviour of rocks plays an important role in studying the stability of surrounding rocks. A series of true triaxial compression tests in high-low stress states were carried out on 58 Jinping marble specimens. The characteristics of complete stress–strain curves, including the post-peak stage, the corresponding meso- and micro-failure characteristics with scanning electron microscopy (SEM), and the whole progressive failure processes with acoustic emission (AE), were analysed. With either decreasing σ3 or increasing σ2, the shapes of the post-peak curves change from Class I to Transition and then to Class II. Additionally, the changes have obvious stress boundaries. As σ2 increases at low-constant σ3 or σ3 decreases at constant σ2, the failure mode changes from tensile–shear mixed failure or shear failure to tensile failure, and these changes correspond with the changes in the post-peak curves. Micro-failure of marble changes from intergranular failure at Class I to transgranular failure at Class II. The AE characteristics of the three typical post-peak progressive failure processes are significantly different, matching the brittle failure behaviour from weak to strong. This study enhances the knowledge of in situ failure mechanisms and the stability of deep-buried tunnels and caverns.

1. Introduction Deformation and failure are important in rock mechanics and engineering, where they control stability, safety and economy. After excavation of tunnels and caverns, high stresses can induce some rocks to reach the post-peak stage, endangering engineering safety (Kaiser et al., 2010; Hoek et al., 1995). In recent decades, the considerable progress made in the research of rock post-peak behaviour has had a profound influence on the stability and support mechanisms of surrounding rocks (Hajiabdolmajid et al., 2002; Cai et al., 2007; Peng et al., 2017; Masoudi and Sharifzadeh, 2018). The China Jinping Underground Laboratory Phase II (CJPL-II, burial depth of approximately 2400 m) was built in a hard brittle marble zone under three-dimensional (3D) highgeostress condition. After excavation, the surrounding rocks have faced engineering problems such as a deep excavation damage zone (EDZ), spalling, and rockbursts (Feng et al., 2018a, 2018b). The post-peak behaviour under 3D stress states plays an important role in the study of the above problems and still involves complexity (such as the intermediate principal stress (σ2) effect). This topic thus deserves to be explored further in depth.

The study of the post-peak behaviour of rocks requires complete stress–strain curves. These curves can be obtained by large-scale in situ field tests (Bieniawski, 1968); however, these tests can only be carried out near the excavation boundary of surrounding rocks, where σ3 is approximately zero, and the post-peak curves of the higher σ3 zone within the surrounding rocks cannot be determined. In addition, in situ testing is difficult to install, expensive, and time consuming compared with laboratory testing and, as such, has not been widely used. With the development of stiffness servo-controlled systems in the 1960s, complete stress–strain curves have been obtained under laboratory uniaxial or conventional triaxial compression (CTC, σ1 > σ2 = σ3) (Hudson et al., 1972; Fairhurst and Hudson, 1999; Hajiabdolmajid et al., 2002; Akinbinu, 2017; Peng et al., 2017; Liu et al., 2019). Some scholars carried out further research based on the aforementioned curves. Tarasov and Stacey (2017) established a brittleness index based on post-deformation to evaluate the brittleness of rock at failure. Feng et al. (2018a) established a failure approach index (FAI) to calculate the post-peak failure degree of surrounding rocks and optimise support work. The numerical constitutive model also requires reduced parameters in the post-peak stage (Hajiabdolmajid et al.,

⁎ Corresponding author at: State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China. E-mail address: [email protected] (X.-T. Feng).

https://doi.org/10.1016/j.enggeo.2019.105444 Received 16 October 2018; Received in revised form 17 September 2019; Accepted 29 November 2019 Available online 30 November 2019 0013-7952/ © 2019 Published by Elsevier B.V.

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2002), etc. In addition, some scholars have classified post-peak curves (Hoek et al., 1995; Sharifzadeh et al., 2017), one of which is typically classified as Class I and Class II according to the monotonicity of strain ε1 (Wawersik, 1968; Okubo and Nishimatsu, 1985; He et al., 1990; Tarasov and Stacey, 2017). For Class I, ε1 increases monotonically, and the failure process needs to absorb external energy to continue, showing stable failure characteristics. For Class II, ε1 reverses in the post-peak stage, and the failure process needs to release excess energy to avoid uncontrolled catastrophic failure, showing self-sustaining failure characteristics. The effect of confining pressure on the shapes of post-peak curves has been studied, and reducing confining pressure resulted in changes from Class I to Class II (He et al., 1990; Tarasov and Stacey, 2017). Jiang et al. (2016) summarised that failure modes of Jinping marble transformed from shear failure to splitting tensile failure with decreasing confining pressure. Akinbinu (2016, 2017) found that Class II rocks were more fractured and finer than Class I rocks under uniaxial compression tests, and the relationship between fragmentation and brittleness for Class II rocks was stronger than that for Class I rocks. Li et al. (1998) found that stress–strain behaviour was related to fracture under compression. These studies imply that there are certain relationships between the confining pressure and post-peak curve and failure characteristics. These results are based on uniaxial or conventional triaxial compression, which cannot simulate the complex asynchronous 3D stress redistribution processes after deep underground cavern excavation (σ1 > σ2 > σ3). True triaxial tests can be used to study this phenomenon and have achieved significant results. Most studies involved true triaxial compression (TTC) tests to study the effects of σ2 on strength characteristics and failure angle (Mogi, 1967; Handin et al., 1967; Takahashi and Koide, 1989; Haimson and Rudnicki, 2010; Zheng et al., 2019). Some studies involved true triaxial unloading failure to study spalling and rockburst (He et al., 2010; Zhao et al., 2014; Li et al., 2018). Some studies involved multi-physical field coupling such as hydraulic fracturing (Frash et al., 2014), and permeability measurement (Goodfellow et al., 2015; Shi et al., 2017). Few studies have provided post-peak behaviour results of rocks. Complete post-peak curves such as Class I under TTC remain sparse (Mogi, 2007; Goodfellow et al., 2015). Complete post-peak curves such as Class II under TTC, especially at very low σ3 and high σ2, are rarely reported. Due to insufficient experimental data, the post-peak behaviour of rock under TTC has not been systematically studied, and the effect of σ2 on this behaviour is still unclear. Therefore, it is novel to study the postpeak deformation and corresponding failure characteristics under true triaxial stresses, especially the effect of σ2 at low σ3. The purpose of this research is to deeply study the post-peak behaviour of rocks under various true triaxial stresses and to clarify the influence of σ2 on post-peak deformation and failure, especially in the case of low σ3 and high σ2. Therefore, TTC tests are carried out under various triaxial stresses to obtain complete stress–strain curves, especially for the post-peak stage, and the corresponding failure characteristics with SEM and AE.

Table 1 Properties of the CJPL marble. ρ (g/cm3)

2.78

UCS (MPa)

192

E (GPa)

53

υ

0.30

Major mineral components Dolomite

Calcite

88%

12%

ρ, density; UCS, Uniaxial compressive strength; E, Elastic modulus; υ, Poisson's ratio.

Northeast University, China (Feng et al., 2016). This system consists of an independent triaxial loading system, a computer servo-controlled system, a data acquisition system, and an AE monitoring system. The triaxial loading system can be controlled independently in three directions. The maximum loading capacities in the horizontal (σ1) and vertical (σ2) directions are 3000 kN and 6000 kN, respectively. The σ3 direction can be hydraulically loaded up to 100 MPa, and the accuracy is ± 0.1 MPa. Information such as force and displacement is automatically recorded during the test, and the data sampling rate is 5 kHz. The maximum principal strain ε1 and the intermediate principal strain ε2 are measured by LVDTs, and the minimum principal strain ε1 is measured by a strain gauge. The resolution is 0.001 mm. The highspeed and high-frequency closed-loop servo-controlled system is operated based on the above data. In addition, the stiffness of the σ1 loading frame is estimated to be 6 MN/mm (higher than the ISRM suggested value of 5 MN/mm (Fairhurst and Hudson, 1999)). Therefore, this system can achieve good control in post-failure stage to acquire complete stress–strain curves, especially for high-brittle failure such as that exhibited in Class II curves. The curves and failure have good repeatability (Feng et al., 2016). In addition, the AE monitoring system (American Acoustic Physics Company PCI-2) can automatically record AE events at failure. The preamplifier gain is 40 dB, the amplitude threshold is 45 dB, the frequency range of the collected signal is 20 kHz~2 MHz, and the sampling frequency is 10 MHz. 2.2. Test scheme and loading stress paths The CJPL-II was built in a 3D high-geostress zone (σ3, σ2, and σ1 of approximately 25, 67, and 69 MPa, respectively). To cover the geostress level and stress redistribution range after excavation as well as fully understand post-peak behaviour of rocks, it is necessary to carry out TTC tests under various 3D stress states (especially low σ3 and high σ2). Moreover, σ2 and σ3 range from 0 to 380 MPa and 0 to 40 MPa, respectively. The testing plan is shown in Table 2. The loading stress paths under TTC are divided into four stages (Fig. 1): stage I is the hydrostatic loading stage (σ1 = σ2 = σ3); stage II is the biaxial loading stage (σ1 = σ2 loading, keeping σ3 constant); stage III is the σ1 stress-controlled true triaxial loading stage (σ1 loading, keeping σ2 and σ3 constant); and stage IV is the ε3 strain-controlled true triaxial loading phase (σ1 loading, keeping σ2 and σ3 constant). Point A, at approximately 75% of the peak strength σc, demarcates the change from stress-controlled to ε3 strain-controlled testing (the direction of ε3 is the most sensitive direction for fracturing) (Hudson et al., 1972). During the test, the stress was applied at 0.5 MPa/s, and the ε3 straincontrolled rate was 0.015 mm/min.

2. Test setup 2.1. Specimens and true triaxial testing system

3. Test results and analysis

The marble samples taken from the CJPL-II are grey-white, have a fine crystal structure, without cracks and bedding, and are homogeneous. The samples were cut to cubic specimens measuring 50 mm × 50 mm × 100 mm according to the International Society for Rock Mechanics and Rock Engineering (ISRM) suggested method (Fairhurst and Hudson, 1999). The basic physical behaviour and mechanical parameters of the specimens are shown in Table 1. The test platform is a true triaxial test system for measuring the complete stress–strain curves of high-pressure hard rock developed by

The stress–strain curves obtained by the testing system have good reproducibility in the post-peak stage (Feng et al., 2016). The results of high-brittle repeatability tests on Jinping marble are shown in Fig. 2. Fig. 2(a) shows that the complete stress–strain curves (σ1-σ3versus ε1, ε2, and ε3) coincide in the pre-peak stage and are similar in the post-peak stage at σ3 = 5 MPa and σ2 = 30. Fig. 2(b) shows that the failure surface morphologies and failure modes match well. Fractures are 2

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reversed. When σ1 reaches a certain value, the strain in the three directions becomes non-linear. In the post-peak stage, the curves in the three directions are relatively smooth at σ2 = σ3. ε2 and ε3 are monotonic and negative; thus, these two directions continue to expand. Both are consistent in the early post-peak stage and then differ in the latter stage due to some differences in meso-cracking in both directions. The characteristics of the stress–strain curves at σ2 > σ3 are quite different from those at σ2 = σ3. On the one hand, the post-peak curves are not as smooth as before, and there are multiple obvious stress drops. Of course, these drops are not sudden uncontrolled stress drops (Hudson et al., 1972) but instead match the gradual brittle cracking failure of marble. Although stress drops under TTC have been shown (Goodfellow et al., 2015), no detailed description has been provided. This phenomenon is closely related to the stress state (σ2, σ3) and is elaborated in Sections 4.1 and 5.3. On the other hand, there is an inflection point in the σ2 deformation. Before the inflection point, the increment is negative owing to expansion. After the inflection point, the increment is positive due to the rebound mainly related to the decrease in σ1. Moreover, as σ2 increases, the value of σ1 corresponding to the inflection point of ε2 increases until reaching σc. The change in ε1 is classified below and is related to the stress states, as explained in Section 4.1. According to the characteristics of post-peak curves, combined with early research on ε1, post-peak curves can be divided into three types under TTC (Figs. 3 and 4): Class I, Class II and Transition. Class I and Class II were conceived by the pioneers in this field (Wawersik, 1968; Okubo and Nishimatsu, 1985; He et al., 1990). The Transition is that strain ε1 is stable in the post-peak stage, not monotonically increasing (Class I) or reversed (Class II). The failure state does not need to absorb or release energy and is transitional from stable failure to self-sustaining failure. The stress range under TTC is much wider than that under CTC and needs to be analysed as an independent behaviour (Fig. 7). The three curve types evolve with the stress states (σ2, σ3) (Sections 4.1 and 4.2). Here, we mainly study the stress conditions and the corresponding failure characteristics of three typical curves.

Table 2 True triaxial compression testing plan. Number

σ3(MPa)

σ2(MPa)

Number

σ3(MPa)

σ2(MPa)

No.1 No.2 No.3 No.4 No.5 No.6 No.7 No.8 No.9 No.10 No.11 No.12 No.13 No.14 No.15 No.16 No.17 No.18 No.19 No.20 No.21 No.22 No.23 No.24 No.25 No.26 No.27 No.28 No.29

0 0 0 0 0 0 1 2 2 2 2 2 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 15 15

0 15 30 50 60 75 30 2 15 30 65 100 5 7.5 15 30 50 65 100 135 10 20 30 65 100 135 200 15 22.5

No.30 No.31 No.32 No.33 No.34 No.35 No.36 No.37 No.38 No.39 No.40 No.41 No.42 No.43 No.44 No.45 No.46 No.47 No.48 No.49 No.50 No.51 No.52 No.53 No.54 No.55 No.56 No.57 No.58

15 15 15 15 15 20 20 20 20 20 20 20 20 30 30 30 30 30 30 30 30 30 30 30 40 40 40 40 40

30 65 100 135 200 20 30 50 65 100 135 200 300 30 40 50 65 80 100 120 150 200 300 380 40 50 65 100 200

4. Post-peak deformation characteristics of marble under TTC 4.1. Relationship between post-peak curve shapes and 3D stress states Some scholars have studied the effect of confining pressure on the change from Class I to Class II under uniaxial or conventional triaxial compression (He et al., 1990; Okubo et al., 1990; Tarasov and Stacey, 2017); however, after the excavation of deep underground caverns, σ3 redistributes from the initial geostress at the original rock zone to approximately zero at the free surface, and the change in σ2 is not synchronous with that in σ3. To express the change in the post-peak curves, according to previous results (Tutluoglu et al., 2015; Sharifzadeh et al., 2017), the typical stress–strain curve of marble can be divided into the pre-peak stage, peak ductile section, post-peak strain softening section, brittle failure section and residual stage (as shown in Fig. 5). Fig. 6 shows the variation in the post-peak curves with different σ2 for a constant σ3. As σ2 increases, the post-peak curves change from smooth to exhibiting multiple stress drops, and the overall trend changes from Class I to Class II. There are also some differences between high-and low-σ3 conditions. For high-constant σ3 (=20 MPa) and low σ2 shown in Fig. 6(b), when σ1 reaches σc, σ1 does not decrease immediately but has a long peak ductile section and post-peak strain softening section with a relatively smooth brittle decline. With increasing σ2, the peak ductile section and post-peak strain softening section shorten, and the brittle failure section shows multiple low-amplitude stress drops. These stress states mainly correspond to ductile failure, and the post-peak curves are of Class I. As σ2 continues to increase, the peak ductile section and strain softening section further decrease or even disappear, and the brittle failure section exhibits multiple high-amplitude stress drops.

Fig. 1. The loading stress path sketch under TTC. σ1 stress-controlled and ε3 strain-controlled are applied before and after point A (0.75σc), respectively. I, II, III and IV represent the hydrostatic loading stage, the biaxial loading stage, and the σ1 stress-controlled and ε3 strain-controlled true triaxial loading phases, respectively.

parallel to σ2 in σ1-σ3 space, consistent with previous studies (Mogi, 2007; Haimson and Rudnicki, 2010). Figs. 3(a)~(c) show typical complete curves for different σ2 values at σ3 = 5 MPa. These curves exhibit the deformation characteristics in the whole progressive failure processes, especially for the post-peak stage. In the pre-peak stage, the curves show the relationships between the differential stress and strain in the three directions. The deformation characteristics at σ2 = σ3 are the same as those observed in conventional triaxial tests. For σ2 > σ3, in loading stage II, ε1 and ε2 are consistent and positive due to compression in both directions, and ε3 is negative due to expansion in that direction. In stage III, the trends of ε1 and ε3 are the same as those in stage II. As σ1 increases, with constant σ2, there is a tendency to expand in the direction of σ2, and ε2 is 3

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Fig. 2. The repeatability tests on Jinping marble at σ3 = 5 MPa and σ2 = 30 MPa. (a) The complete stress–strain curves; (b) The failed specimens: fractures are parallel to σ2 in σ1-σ3 space in specimen A (left) and B (right).

These stress states mainly correspond to strong brittle failure. ε1 is stable, and the post-peak curves are of the Transition type. When σ2 increases to a certain extent, the peak ductile section and the strain softening section have completely disappeared, and the low-amplitude stress drops appear first and then increase in amplitude or this process directly enters a high-amplitude stage after the peak. These stress states correspond to extremely strong brittle failure. ε1 is reversed, and the post-peak curves are of Class II. For low-constant σ3 (=5 MPa) shown in Fig. 6(a), the curve has no ductile and exhibits brittle characteristics. As σ2 increases, the brittleness increases, and the post-peak curves change from Class I to Transition and then to Class II. Fig. 7 shows the variation in the post-peak curves with σ2 at constant σ3. Figs. 7(a) and (b) show that independent σ3 affects the shapes of the post-peak curves, yet the effect is opposite to that of σ2. When σ2 is low (=30 MPa), the peak ductile section and the post-peak strain softening section decrease with decreasing σ3, and the brittle failure section changes from smooth to multiple stress drops and from ductile failure to extremely strong brittle failure. The overall trend of the postpeak curves changes from Class I to Transition and then to Class II. When σ2 is high (=100 MPa), it is characterised by brittle failure. As σ3 decreases, the post-peak curves also change from low- to high-amplitude multiple stress drops, and the overall trend changes from Class I to Class II.

Fig. 4. The three typical post-peak deformation sketch: Class I (stable failure), Transition, and Class II (self-sustaining failure). (revised after Wawersik (1968) and Fairhurst and Hudson (1999)).

stress states can induce changes in the post-peak curves. Fig. 8 shows that there are obvious stress zoning behaviours for the three typical curves and clear boundaries from Class I to Transition and then to Class II. The stress boundary from Class I to Transition can be expressed approximately by a straight line. This linear Eq. (1) is as follows:

4.2. Stress boundaries of the three typical post-peak curve shapes The post-peak curves can be classified into three categories, and

Fig. 3. The three typical complete stress–strain curves in different stress states under TTC at different σ2 and constant σ3. 4

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Fig. 5. Several sections of the typical stress–strain curve of marble.

Fig. 7. The variation in post-peak deformation behaviour with σ3 for each constant σ2 under TTC. (a) differential stress versus axial strain under σ2 = 30 MPa and σ3 = 1, 2, 5, 5, 10, 15, 20, and 30 MPa; (b) differential stress versus axial strain under σ2 = 100 MPa and σ3 = 5, 10, 20, 30, and 40 MPa.

σ2 = a σ3 − b

(1)

where a and b are constants. For approximately zero values or very low of σ3 (σ3c0 of approximately 1 MPa), the post-peak curves exhibit Class II characteristics. For relatively high σ3, the boundary from Transition to Class II can also be represented by a straight line. This boundary Eq. (2) is as follows:

⎧ σ2 = c σ3 − d ⎨ ⎩ σ3 = σ3c0

σ3 > σ3c0 σ3 ≤ σ3c0

(2)

where c and d are constants. Therefore, the post-peak curve types can be judged on the basis of the two boundaries for the given stress states. Fig. 6. The variation in post-peak deformation behaviour with different σ2 for each constant σ3 under TTC. (a) differential stress versus axial strain under σ3 = 5 MPa and σ2 = 5, 7.5, 50, 100, and 135 MPa; (b) differential stress versus axial strain under σ3 = 20 MPa and σ2 = 20, 30, 50, 65, 100, 135, 200, and 300 MPa.

5. Failure characteristics of marble under TTC 5.1. Mesoscopic failure The stress states (σ2, σ3) can induce the conversion from Class I to Transition and then to Class II or from stable failure to Transition and 5

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Fig. 10 shows that the micro-failure characteristics of marble are related to the 3D stress states as well as the post-peak curves. For highconstant σ3 (=20 MPa, Fig. 10(a)), σ2 is 30, 65, and 300 MPa, respectively, corresponding to Class I, Transition, and Class II curves. When σ2 = 30 MPa, the crystalline nature is clearly identifiable; microcracks appear at the crystal boundaries and rarely inside the crystals (Fig. 11(a)). The crystal surface is relatively flat, showing obvious intergranular failure characteristics. When σ2 = 65 MPa, the crystal boundaries are relatively fuzzy; some micro-cracks are intergranular, while the others are transgranular, and the latter are more obvious. When σ2 = 300 MPa, the crystal boundaries are completely indistinguishable, and the cleavage patterns on the crystal surface are ubiquitous, showing typical transgranular failure characteristics. For lowconstant σ3 (=5 MPa, Fig. 10(b)), σ2 is 5, 30, and 100 MPa, respectively, corresponding to Class I, Transition, and Class II curves. When σ2 = 5 MPa, there are many transgranular cracks, and intergranular cracks are also more numerous. When σ2 = 30 MPa, the number of intergranular cracks decreases, and the transgranular characteristics are obvious. When σ2 = 100 MPa, the crystals are indistinguishable, and step-wise cleavage patterns are very obvious (Figs. 11(b) and (c)), showing typical transgranular failure characteristics. Fig. 10(c) shows the variation in micro-failure characteristics with σ3 at a constant σ2. When σ2 = 30 MPa, σ3 is 20, 5, and 1 MPa, respectively, corresponding to Class I, Transition, and Class II curves. When σ3 = 20 MPa, there are mainly intergranular cracks and fewer transgranular phenomena. When σ3 = 5 MPa, the number of intergranular cracks decreases, and the number of transgranular cracks increases. When σ3 = 1 MPa, the crystal surface exhibits transgranular fracture, and step-wise cleavage patterns (Fig. 11(d)) are easily observed, showing typical transgranular failure characteristics.

Fig. 8. The relationship between post-peak curve types and stress states (σ2, σ3) under TTC. Black solid and dash line represent the stress boundaries from Class I to Transition and Transition to Class II, respectively.

then to self-sustaining failure. Fig. 9 shows that this conversion is accompanied by variations in the mesoscopic failure surfaces. Previous studies of failure surfaces mainly relied on the failure angle; however, here, the failure morphology at low σ3 indicates multiple failure surfaces (shown in Figs. 9(a) and (b)), and the failure angle is difficult to define. In addition, a main failure surface forms at high σ3 (Fig. 9(c)), consistent with previous results (Mogi, 2007; Haimson and Rudnicki, 2010). Therefore, the following description of the failure characteristics mainly relies on the mesoscopic morphology of the failure surfaces, such as the number and general tendency of fractures, and failure modes. Figs. 9(a)~(c) show the variation in failure surfaces with different σ2 values at high- or low-constant σ3. For low-constant σ3 (=2 and 5 MPa), with increasing σ2, the number of through-fractures ranges from 0–1 in Class I to 2–3 in Transition and then to 3–4 in Class II. The overall trend of the fractures evolves from inclined to substantially parallel to σ1. The number of non-through fractures decreases, the length increases, and there is a tendency for the fractures to evolve into through-fractures. The failure modes change from tensile-shear mixed failure to mainly tensile failure. For high-constant σ3 (=20 MPa), there is usually one main through-fracture surface. With increasing σ3, the number of non-through fractures decreases, and the main failure mode remains shear failure. Fig. 9(d) shows the variation in failure surfaces with different σ3 values at a constant σ2. For a constant σ2 (=65 MPa), as σ3 gradually decreases from 40 MPa to 2 MPa, the post-peak curves transit from Class I to Class II, and the meso-failure morphology also changes significantly. The number of main through-fractures has increased, and the overall trend of the through-fractures evolves from inclination to parallel to σ1. The number and length of non-through fractures have also increased significantly. The failure mode evolves from shear failure to tensile failure.

5.3. AE characteristics of progressive failure processes During deformation and failure of rocks, dislocation sliding, fracturing of crystals, initiation, propagation, and coalescence of microcracks and micro-pores inside rocks can all trigger AE signals (Ingraham et al., 2013; Goodfellow et al., 2015). There are a few studies of AE characteristics in the whole failure processes (including the post-peak stage) under 3D stress states. The following is an analysis of the AE characteristics of the typical progressive failure processes under TTC corresponding to Class I (stable failure), Transition, and Class II (selfsustaining failure). Fig. 12 shows the relationships between σ1 and AE signals with time for the three typical curves. AE signals can be characterised by using AE count rate, cumulative count, and cumulative energy. The whole failure process can be investigated by dividing it into pre-peak and post-peak stages (Tutluoglu et al., 2015). The progressive pre-peak failure process of rocks can be divided into four stages: an initial compaction stage, linear elastic stage, and stable crack-extension and unstable crack-extension stages. The AE characteristics in the pre-peak stage are similar for the three typical curves. In the initial compaction stage (σ1 < σcc), a few AE signals appear mainly because of cracking and damage induced by closure effects in internal defects. In the elastic stage (σ1 > σcc), there are few AE signals owing to elastic deformation of the rock skeleton. In the stable extension stage (σ1 > σci), micro-cracks and micro-pores extend and result in the activation of AE signals. In the unstable extension stage (σ1 > σcd), AE signals are not recorded frequently because the non-linear deformation of marble in this stage should be mainly caused by the plastic deformation of crystals. A few sparse AE signals appear when σ1 ≈ σc, and this phenomenon is probably because dislocation sliding of crystals and further microcracking occur when the plastic deformation of crystals reaches a certain level. The AE characteristics in the post-peak stage for the three typical curves (Class I, Transition and Class II) are distinctly different. As shown in Fig. 12(a), the post-peak curve in Class I contains a stable

5.2. Microscopic failure Many studies have researched the micro-failure of rocks by SEM of fracture surfaces (Nicolas et al., 2016). Micro-cracks inside marble can be divided into intergranular and transgranular cracks according to their locations at the boundaries or interior of crystals. The micro-crack types can be used for rock brittle-ductility failure assessment (Nicolas et al., 2016), while brittle-ductility has good correspondence with the post-peak curves (Meng et al., 2015; Tarasov and Stacey, 2017). Therefore, we link the post-peak curves and micro-failure characteristics under 3D stress states. 6

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Fig. 9. The relationship between meso-failure morphologies (σ1–σ3 surfaces), failure modes, and post-peak curve types under TTC. (a)~(c) meso-failure surfaces for different σ2 and σ3 = 2, 5, and 20 MPa, respectively; (d) meso-failure surfaces for different σ3 and σ2 = 65 MPa. The black dash-dot line and the blue dashed line are the through-fractures and the non-through fractures, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 7

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Fig. 10. Micro-failure characteristics under typical true triaxial stress states corresponding to stable failure, transitional and self-sustaining failure (Class I, transition, and Class II, respectively) (magnification 1000×). (a) and (b) variation in micro-failure with σ2 at σ3 = 20 MPa and 5 MPa, respectively; (c) the relationship between micro-failure and σ3 at σ2 = 30 MPa.

count rate is intermittent with a very low amplitude after the peak. Subsequently, rock failure enters the multiple stress drop process with a low amplitude at first and then with a high amplitude. The AE count rate becomes dense and increases abruptly at stress drops, especially with high amplitude. After several high-amplitude stress drops, local meso-cracks develop into failure surfaces. Fig. 12(c) shows that the post-peak curve in Class II shows a brittle failure section (multiple high-amplitude stress drops (up and down)). When σ1 = σc, the AE count rate abruptly increases, and the cumulative count and cumulative energy increase step-wise. Here, the mutual coalescence of micro-cracks inside the marble should form local cracks or even fractures. Then, the bearing capacity begins to quickly decrease, and the AE count rate shows a mutation. Subsequently, σ1 enters its increasing section. When the local peak is reached, the bearing capacity suddenly decreases, accompanied by a sudden increase in the AE count rate and cumulative energy. Similar multiple stress drops and AE characteristics are observed subsequently.

strain softening section and brittle failure section (multiple low-amplitude slow stress drops) at σ3 = σ2 = 5 MPa. This curve is smooth in the stable strain softening section: the AE count rate varies, increasing intermittently with moderate amplitude, and the AE cumulative count and cumulative energy increase approximately linearly. Additionally, many micro-cracks initiate, propagate, and coalesce, and even local cracks gradually form. Then, the failure process enters a low-amplitude slow stress drop stage. At the stress drops, the AE count rate is dense and increase, and the AE cumulative count and cumulative energy increase accordingly. Timely continuous propagation and coalescence of micro-cracks result in local mesoscopic cracking. After several slow stress drops, propagation and coalescence of local cracks form throughfractures. Fig. 12(b) shows that the post-peak curve in the Transition type includes a relatively stable strain softening section and brittle failure section (multiple low and high stress drops (step-shaped)) at σ3 = 2 MPa and σ2 = 15 MPa. This curve oscillates slightly, and the AE 8

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Fig. 11. Typical micro-failure characteristics under TTC (enlarged view). (a) Intergranular cracks at σ3 = 20 MPa and σ2 = 30 MPa (magnification 2000×); (b)~(d) Step-wise cleavage patterns and transgranular cracks (magnification 5000×), (b) and (c) at σ3 = 5 MPa and σ2 = 100 MPa, (d) at σ3 = 1 MPa and σ2 = 30 MPa.

6. Discussion

schemes based on in situ monitoring results and intelligent back analysis methods (including sensitivity analysis of parameters, project topic uncertainty and probability analysis (Vu-Bac et al., 2016; Hamdia et al., 2017; Juang et al., 2019)).

Many scholars have studied the in situ failure characteristics of underground tunnels and caverns subjected to high in situ stresses, in which fractures were parallel to the excavation face (spalling failure) (Hajiabdolmajid et al., 2002; Martino and Chandler, 2004; Kaiser et al., 2010). These scholars thought that σ3 near the excavation face was very low or even zero, and tensile cracks played a key role. Cai (2008) analysed the effects of σ2 on cracking using a numerical method. The simulation results showed that the generation of micro-cracks and fractures parallel to the cavern excavation surface was attributed to the condition of relatively high σ2 and zero to low σ3. In the CJPL-II, spalling failure was also observed near the excavation surface after excavation (Fig. 13). Here, TTC tests were conducted covering stress states such as σ3 = 2 MPa (σ3 was very low near the excavation face) and σ2 = 65 MPa (approximately the σ2 component value of the geostress), and multi-parallel fractures appeared (Figs. 9 and 13). The test results are in agreement with Cai's (2008) numerical results. In addition, the micro-failure (step-wise cleavage transgranular failure) within the parallel fractures of the specimen is consistent with that within in situ spalling fractures. In the future, a new numerical mechanical model that can accurately model stress–strain curves and fractures of specimens will be established based on this study. Then, this model will be extended to simulate in situ failures in the CJPL-II and optimise excavation support

7. Conclusions This paper focuses on the in-depth study of the post-peak behaviour of Jinping marble under various true triaxial stresses. Large amounts of post-peak behaviour results are very valuable for rock mechanics and geotechnical engineering. The influence of σ2 on post-peak deformation and failure was innovatively investigated, especially at very low σ3. The detailed conclusions are as follows: (1) With decreasing σ3 at constant σ2 or increasing σ2 at constant σ3, the post-peak curves change significantly, the peak ductile section and the post-peak strain softening section decrease, and the brittle failure section changes from smooth to exhibiting multiple stress drops. The post-peak curve type changes from Class I to Transition and then to Class II. Moreover, the changes have obvious stress boundaries. (2) With increasing σ2 at low-constant σ3 or decreasing σ3 at constant σ2, the lengths of non-through fractures increase, evolving to through-fractures. The number of through-fractures increases, gradually evolving into fractures parallel to σ1. The failure mode 9

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Fig. 12. AE characteristics of the three typical whole progressive failure processes under TTC. (a) AE events during stable failure (corresponding to Class I) at σ3 = σ2 = 5 MPa; (b) AE events during the transitional from stable failure to self-sustaining failure (corresponding to Transition) at σ3 = 2 MPa and σ2 = 15 MPa; (c) AE events during self-sustaining failure (corresponding to Class II) at σ3 = 5 MPa and σ2 = 135 MPa.

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Fig. 13. Comparison of the in situ macroscopic spalling failure and corresponding micro-failure (Feng et al., 2018b) with the lab mesoscopic multi-parallel fractures and corresponding micro-failure.

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This study deepens the understanding of the post-peak behaviour of rocks under true triaxial stresses. This work is conducive to research on the in situ failure mechanism, the stability of deep underground caverns and the optimisation of excavation support schemes.

Declaration of Competing Interest The authors confirm that there are no conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

Acknowledgments The authors sincerely acknowledge the financial support received from the National Natural Science Foundation of China under Grant Nos. 51621006 and 51579043 and the Chinese Academy of Sciences Key Research Program of Frontier Sciences under Grant No. QYZDJSSW-DQC016.

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