1 Introduction

Ductile–brittle behaviours are the key basic mechanical behaviours of rock, especially for deep engineering rock (Hoek and Bieniawski 1965; Bürgmann and Pollard 1992; Kaiser et al. 2010). Brittle rock behaviour is a necessary condition for engineering disasters such as spalling and rockburst in deep surrounding rock, and the stronger the brittleness is, the greater the tendency of the above disasters to occur (Cai 2013; Eberhardt et al. 2020; Li et al. 2022; Pan et al. 2022; Zhao et al. 2022a, b, 2023). Many deep projects, such as the Canadian Atomic Energy of Canada Limited’s underground research laboratory (Martin et al. 1997), the world's deepest China Jinping Underground Laboratory Project II (CJPL-II) (Feng et al. 2018) and the deeply buried Tibet railway tunnel project in China, have suffered severe spalling and rockburst disasters (Feng et al. 2018). These deep projects are located in regions with the high three-dimensional (3D) geostress conditions. Compared with 1D or 2D stress conditions, the mechanical properties of rock under 3D stress conditions are very different, especially the ductile–brittle behaviours, which are significantly different. However, to date, there is a serious lack of ductile–brittle evaluation research suitable for rock under 3D stress conditions. Reasonable quantitative evaluation of the effect of 3D stress on the ductile–brittle behaviours of deeply buried rocks is an urgent research task that has distinct engineering significance and important research value.

Many researchers have pointed out that rock ductile–brittle behaviours are related characteristics that describe their post-peak behaviours (Kaiser et al. 2010; Tarasov and Potvin 2013; Nicolas et al. 2016; Meng et al. 2021). In addition to the influence of rock intrinsic characteristics on brittleness, the stress state plays a key role in rock brittleness (Fredrich et al. 1989; Rybacki et al. 2016; Wang et al. 2020a, b). Obtaining the complete stress–strain curve, including the post-peak section, is crucial for the study of rock brittleness. Previously, there have been abundant research results on rock brittleness based on the results of uniaxial and conventional triaxial tests (Dueck and Borgesson 2015; Akinbinu 2016, 2017; Wang et al. 2020a, b; Tao et al. 2020; Liu et al. 2021a, b; Abdallah et al. 2021). The influence of confining pressure on rock brittleness has been investigated (Tarasov and Potvin 2013; Peng et al. 2017; Zhang et al. 2021a, b; Liu et al. 2022). Research has shown that with the increase in confining pressure, the brittleness of rock decreases and the ductility increases, which is manifested by the increase in nonlinear plastic deformation before the peak strength and the decrease in the stress drop rate in the post-peak stage. In other words, rock brittleness is dependent on confining pressure (Byerlee 1968; Liu et al. 2021a, b; Zhao et al. 2022a, b). Other scholars established brittleness indexes based on rock mechanical properties under uniaxial and conventional triaxial compression, such as brittleness indexes based on strength characteristics (Hucka and Das 1974; Munoz et al. 2016), brittleness indexes based on pre-peak deformation (Gong and Wang 2022), and brittleness indexes based on post-peak stress drop (Meng et al. 2021; Liu et al. 2021a, b). However, the above understanding of rock brittleness and evaluation methods based on uniaxial and conventional triaxial compression (CTC, σ1 > σ2 = σ3) test results are difficult to directly use to understand and evaluate the brittleness of deep engineering rock under true 3D stress states (σ1 > σ2 > σ3), due to neglecting the influence of the intermediate principal stress, the brittleness of deep engineering rock is not reasonably assessed, further increasing the difficulty of risk assessment of engineering disasters induced by brittle failure.

At present, true triaxial compression (TTC) tests mainly have been performed to study rock strength and failure angle (Mogi 1967; Haimson and Rudnicki 2010; Ingraham et al. 2013; Zheng et al. 2019; Wang et al. 2020a, b; Duan et al. 2021; Zheng et al. 2023a, b, c), and some scholars have carried out true triaxial tests for rockburst (He et al. 2010, 2021; Su et al. 2017; Li et al. 2018) and multifield coupling tests for permeability (Frash et al. 2014; Zheng et al. 2023d; Li et al. 2016). The test results of complete stress–strain curves of rock, including the post-peak stage under true triaxial stress, are very rare, and the quantitative evaluation method for the influence of true 3D stress states on rock brittleness based on post-peak characteristics are scarce and need to be studied urgently. In addition, there are often significant differences in lithology between different projects or different project zones (Rojat et al. 2009). It is also necessary to analyse the brittleness response of different rocks to true 3D stresses and to explore lithology and stress, which control brittleness.

A series of TTC tests were carried out on marble and four kinds of granite with a high-pressure true triaxial test system. The complete stress–strain curves and the post-peak energy conversion characteristics were obtained and analysed. A new normalized index for rock evaluation was proposed, and the influence of the true 3D stress states on ductile–brittle behaviours of rock was quantitatively studied. The differences in rock ductile–brittle behaviours under true triaxial and conventional triaxial stress states were compared and analysed. The ductile–brittle behaviours of four different granites under true triaxial stresses and the control factor of brittleness were investigated. The ductile–brittle evolution of the surrounding rock in deep engineering projects with 3D high-stress states was further studied. This research provides a basis for the evaluation of the ductile–brittle failure tendency of surrounding rock in deep underground engineering.

2 Experimental preparation

In this experiment, Jinping marble (samples from CJPL-II) was the main research object to study the influence of 3D stress states on ductile–brittle behaviours, and four kinds of granites were selected to consider the influence of lithological factors. These five kinds of rock samples were fabricated from thick blocks, with good homogeneity and no visible cracks or bedding defects. The prismatic specimens were processed into 50 mm × 50 mm × 100 mm blocks according to the International Society for Rock Mechanics and Rock Engineering (ISRM) suggested method (Fairhurst and Hudson 1999; Feng et al. 2019). The specimen size deviation was ± 0.02 mm, the perpendicularity deviation of two adjacent faces was ± 0.25°, and the parallelism deviation of the two end was ± 0.02 mm. The typical specimens of five kinds of rocks and their main mineral components and microstructures are shown in Fig. 1 and Table 1.

Fig. 1
figure 1

Typical rock specimens and their main mineral components and microstructure (polarized light photograph). a Jinping marble, b granite A, c granite B, d granite C, e granite D

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Table 1 Basic properties of five kinds of rocks
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A high-pressure true triaxial test system was used for the experimental tests (Feng et al. 2019; Zheng et al. 2019, 2022, 2023b). This system can realize the independent control of σ1, σ2, and σ3, and information such as force and displacement during testing is automatically recorded at high frequencies. The high-speed and high-frequency closed-loop microservo control system was operated based on these recorded force and deformation data. In addition, the stiffness of the frame loaded in the σ1 direction was estimated to be 6 MN/mm, which is higher than the ISRM suggested value of 5 MN/mm (Fairhurst and Hudson 1999). Therefore, the true triaxial test system can realize good control of strong brittle rock failure and obtain complete stress–strain curves, and the deformation curves and failure characteristics have good repeatability (Zheng et al. 2019).

Stress control and deformation control models are the important control methods for rock mechanics test in the laboratory. Stress control can easily realize the control of three principle stress paths and levels. The advantage of deformation control is that it can obtain the post-peak curve of rock, especially the ε3-controlled model. Since the ε3 direction is the most sensitive to deformation in rock failure, the stress–strain curve in the post-peak stage can be well obtained by this loading control model. Therefore, the two control methods are combined in this test, and ε3-controlled model is used near the peak. Before that, stress control is used to realize the combined loading of three principle stress settings. The true triaxial testing stress paths are shown in Fig. 2, which can be divided into four typical stages: before point A, keeping σ1 = σ2 = σ3, all three principal stresses are loaded at a rate of 0.5 MPa/s; Stage A-B, keeping σ3 constant, loading of σ1 and σ2 at the previous rate; Stage B-C, keeping σ3 and σ2 constant, loading of only σ1; after point C (corresponding to approximately 75% of the peak strength σp, peak strength is first obtained by stress-controlled compression test), the σ1-controlled model is converted to the ε3-controlled model (the corresponding rate is 0.015 mm/min), and σ1 loading continues until the residual stage. The true triaxial tests were carried out on the marble and granite with the stress paths in Fig. 2, and the specific test stress levels are shown in Table 2 (Zheng et al. 2019). The values of σ3 and σ2 in Table 2 are the maximum values maintained after point A and point B, respectively, as shown in Fig. 2.

Fig. 2
figure 2

The stress paths under TTC; stress-controlled and ε3-controlled modes are adopted before and after point C (corresponding to approximately 75% of the peak strength σp), respectively

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Table 2 Testing stress levels of five kinds of rocks under TTC
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3 Peak-peak deformation characteristics of ductile–brittle failure under true triaxial stresses

Figure 3 describes the complete stress–strain curves of marble corresponding to ductile–brittle failure under different true triaxial stress states. As shown in Fig. 3a–d, σ3 = 5 MPa and σ2 = 5, 30, 65, and 135 MPa, with increasing σ2, the brittleness of marble increases gradually, and the stress–strain relations in three directions (differential stress σ1-σ3 vs ε3, ε2, and ε1, i.e., σ1-σ3 vs ε3, σ1-σ3 vs ε2, and σ1-σ3 vs ε1) show distinct characteristics. For the σ1-σ3 vs ε3 curves, due to the σ1-compressed lateral expansion of rock, the strain is always negative and decreases monotonically. With increasing σ2, the post-peak deformation curve changes from a smooth state to a gradually enhanced state with multiple small stress drops. After multiple stress drops, the strain in the ε3 direction increases significantly (note that the multiple obvious small stress drops are not sudden uncontrolled stress drops (Hudson et al. 1972; Fairhurst and Hudson 1999) but match the progressive brittle cracking of rock after the peak). The σ1–σ3 versus ε2 curve characteristics are obviously different from the σ1–σ3 versus ε3 curve characteristics. When σ2 = σ3 = 5 MPa (Fig. 3a), the situation corresponds to the conventional triaxial stress state. ε2 and ε3 completely coincide in the pre-peak stage, a difference appears when the post-peak cracking reaches a certain degree, and the overall trend is still similar. When σ2 = 30 MPa (Fig. 3b), ε2 first compresses due to the increase in σ2. When σ2 reaches the predetermined value, ε2 gradually decreases due to σ1-compressed lateral expansion, changes from a positive value to a negative value, and ultimately rebounds, gradually increasing as σ1 decreases after the peak. When σ2 = 65 MPa and 135 MPa (Fig. 3c, d), the overall trend of ε2 is similar to that when σ2 = 30 MPa (Fig. 3b). The difference appears because the larger σ2 is, the larger the forward compression in this direction, and ε2 is always positive. For the σ1-σ3vs ε1 curves, due to the continued compression under σ1, the strain values are always positive. With increasing σ2, the nonlinear irreversible deformation in the pre-peak stage decreases gradually, and the shapes of the curves in the post-peak stage change greatly. The change trend of ε1 in the post-peak stage increases from a continuous increase (σ2 = 5 MPa) to an insignificant change (σ2 = 30 MPa) and then to an overall decrease (σ2 = 135 MPa); that is, the shape of the post-peak curve changes from Class I (ε1 increases monotonically in the whole failure process) to Class II (ε1 increases first and then reverses in the post-peak stage). The above analysis shows that with increasing σ2, the overall variation characteristics of strain in the ε3 direction are not obvious in the ε2 and ε1 directions. ε2 is complex due to the sign changes before the peak and the change trend of the curve shape after the peak. The deformation in the ε1 direction can meet and is relatively easy to describe the characterization of the change trend of ductile–brittle failure. The previous studies based on the post-peak curve shapes under conventional triaxial stresses also use ε1 for description (Wawersik and Brace 1971; Okubo et al. 1990; He et al. 1990;  Tarasov and Potvin 2013; Liu et al. 2022). Therefore, for the convenience of description and comparison, the subsequent analysis is based on the deformation curve results in the direction of ε1.

Fig. 3
figure 3

a–c The typical stress–strain curves and e corresponding failure characteristics of marble in different stress states under TTC (modified from Zheng et al. 2020). E is the elastic modulus; H is the post-peak modulus, which is the whole slope of post-peak curve shape

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4 New ductile–brittle evaluation method based on post-peak energy conversion under true triaxial stresses

The deformation and failure of rock are driven by its stored internal energy. Meanwhile, internal stored energy, external energy intake and energy consumption occur in a dynamic balance, which drives the evolution of rock ductile–brittle fractures. Figure 4a, b show two kinds of typical complete stress–strain curves of rock under true triaxial stress: Class I and Class II. As the difference in the ductile–brittle fracturing of rock is obvious in the post-peak stage, the energy balance and dynamic conversion of rock in the post-peak stage are shown in the two figures. The energy form in the rock in the post-peak stage can be divided into three types: the elastic energy, the fracture energy, and the energy absorbed from the outside (Class I) or the energy released to the outside (Class II). The elastic energy is the energy stored in the rock by the external work. Fracture energy is defined as the energy consumed in the post-peak fracture evolution.

Fig. 4
figure 4

Conceptual graph of energy conversion in rock fracture process. a Class I curve corresponding to low brittle failure, b Class II curve corresponding to high brittle failure

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For the Class I case, as shown in Fig. 4a, the elastic energy at peak \({{dU}}_{{{\text{ec}}}}\), which is the maximum value, corresponds to the light blue region SABC, and the elastic energy at the residual state corresponds to the light green region SDEF. The difference between \(dU_{{{\text{ec}}}}\) and \(dU_{{{\text{er}}}}\) is the released elastic energy in the post-peak stage \(dU_{{{\text{ep}}}}\), which is equal to the area SABC–SDEF. \(dU_{{{\text{ap}}}}\) = SCBDF is the energy absorbed from the outside in the post-peak stage. The fracture energy \(dU_{{{\text{fp}}}}\) include the elastic energy at the peak and the energy absorbed from the outside in the post-peak stage, which is equal to the area SACFD = SABC + SCBDF. Fracturing is maintained by absorbing energy from the outside. It is a non-self-sustaining fracture type and reflects a weak brittleness.

For the Class II case, as shown in Fig. 4b, the elastic energy released in the post-peak stage \(dU_{{{\text{ep}}}}\) (the difference between peak elastic energy \(dU_{{{\text{ec}}}}\) and residual elastic energy \(dU_{{{\text{er}}}}\)) is similar to the Class I case, which is equal to the area SABC − SDEF. The difference is that the energy released to the outside in the post-peak stage \(dU_{{{\text{rp}}}}\) corresponding to the area of the pink zone, SCBDF. Timely, the fracture energy \(dU_{{{\text{fp}}}}\) is only the remaining part of the elastic energy released to the outside, SAEFC = SABC – SDEF – SBCFD. The post-peak fracture evolution is completely driven by the energy inside the rock, which is a self-sustaining failure state and reflects a strong brittleness.

Therefore, for the Class I and Class II cases, the released elastic energy in the post-peak stage \(dU_{{{\text{ep}}}}\) and the area SABC–SDEF in Fig. 4a, b can be calculated as follows:

$$dU_{{{\text{ep}}}} =dU_{{{\text{ec}}}} - dU_{{{\text{er}}}} =\frac{{{\sigma }_{\text{p}}^{2} - \sigma_{\text{r}0}^{2} }}{{2{{E}}}}$$
(1)

where \(dU_{{{\text{ec}}}}\) and \(dU_{{{\text{er}}}}\) are the elastic energy at the peak and residual stages, respectively, E is the elastic modulus, σp is the peak strength, and σr0 is the initial residual stress.

For the Class I curve, the absorbed energy in the post-peak stage \(dU_{{{\text{ap}}}}\), the area SCBDF in Fig. 4a, can be calculated in Eq. 2:

$${{dU}}_{\text{ap}} =\frac{{{\sigma }_{\text{p}}^{2} - \sigma_{\text{r}0}^{2} }}{{2{{H}}}}$$
(2)

where H is the post-peak modulus as shown in Figs. 3 and 4, which is the whole slope of post-peak curve shape.

For the Class II curve, the released energy in the post-peak stage \(dU_{{{\text{rp}}}}\), the area SCBDF in Fig. 4b, can be calculated as follows:

$$dU_{\text{rp}} =\frac{{{\sigma }_{\text{p}}^{2} - \sigma_{\text{r}0}^{2} }}{{2{{H}}}}$$
(3)

For the Class I curve, the fracture energy in the post-peak stage \(dU_{{{\text{fp}}}}\), the area SADFC in Fig. 4a, can be calculated as follows:

$${{dU}}_{\text{fp}} = {{ dU}}_{{\text{ep}}} - {{ dU}}_{{\text{ap}}} =\frac{{{{(\sigma }}_{\text{p}}^{2} - \sigma_{\text{r}0}^{2} )({{H}} - {{E}})}}{{2{{EH}}}}$$
(4)

For the Class II curve, the fracture energy in the post-peak stage \(dU_{{{\text{fp}}}}\), the area SAEFC = SABC – SDEF – SBCFD in Fig. 4b, can be calculated as follows:

$${{dU}}_{{\text{fp}}} = {{dU}}_{{\text{ep}}} - {{dU}}_{{\text{rp}}} =\frac{{{{(\sigma }}_{\text{p}}^{2} - \sigma_{\text{r}0}^{2} )({{H}} - {{E)}}}}{{2{{EH}}}}$$
(5)

Rock post-peak failure evolution is the dramatic transformation process from the high-energy state at the peak stress to the low-energy state at the residual stage, accompanied by the above energy forms, such as the released elastic energy, the fracture energy, and the absorbed energy (Class I) or released energy (Class II) in the post-peak stage. The fracture energy is greater than the released elastic energy in the Class I case and less than the released elastic energy in the Class II case. Their relationship can clearly express the post-peak energy conversion characteristics. Therefore, the establishment of the ductile–brittle index based on the relationship between fracture energy and released elastic energy can reflect the essence of the ductile–brittle failure of rock. The new ductile–brittle index IBD is proposed according to the exponential function of the released elastic energy \(dU_{{{\text{ep}}}}\) and the fracture energy in the peak-peak stage \(dU_{{{\text{fp}}}}\) as follows:

$$I_{{{\text{BD}}}} =f( dU_{{{\text{fp}}}} , \, dU_{{{\text{ep}}}} ) = A^{{ - d U_{\text{fp}} /d U_{{\text{ep}}} }}$$
(6)

Substituting Eqs. 1–5 into Eq. 6, a simplified form for the ductile–brittle index can be expressed in Eq. 7:

$$I_{{{\text{BD}}}} =A^{{\left( {E - H} \right)/H}}$$
(7)

where A is a constant parameter.

As shown in Fig. 5, for different post-peak curve shapes, H has different values, and the value of (E–H)/H ranges from − ∞ to 0, so the value of ductile–brittle index IBD ranges from 0 to 1. The larger the value of IBD is, the stronger the brittleness. IBD is a normalized index for easy application and comparison. For A, theoretically, it can take any value. In fact, it affects the decay rate of IBD with (E–H)/H. Through the analysis of different values of A, combined with the test results, the corresponding IBD is calculated, and a value of 2 for A is a relatively suitable value. Notably, according to the typical relationship between E and H, IBD has several special boundary values. As shown in Fig. 5, for H = E, (E–H)/H = 0, IBD = 20 = 1, corresponding to the strongest brittleness; for H = 2E, (E–H)/H = −0.5, IBD = 2–0.5 ≈ 0.707; for H =  ± ∞, (E–H)/H = − 1, IBD = 2–1 = 0.5; for H = − 2E, (E–H)/H = −1.5, IBD = 2–1.5 ≈ 0.354; for H = 0, (E–H)/H = − ∞, IBD = 2−∞ = 0, corresponding to the strongest ductility.

Fig. 5
figure 5

Ductile–brittle index IBD based on complete stress–strain curve characteristics

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Therefore, the rock brittleness can be specifically determined by Eq. 7 based on the stress–strain curves (Fig. 4 has shown the determination of E and H in different actual curve types) for given stresses and divided into four types, namely, ductile-brittleness, transitional, brittleness and super-brittleness, which shown in Fig. 5 and expressed in Eq. 8 as follows:

$$\left\{ {\begin{array}{ll} 0 \le I_{{{\text{BD}}}} \le 0.354&\quad {\text{Ductile-brittleness }} \\ 0.354 < I_{{{\text{BD}}}} \le 0.5&\quad {\text{Transitional }} \\ 0.5 < I_{{{\text{BD}}}} \le 0.707&\quad {\text{Brittleness }} \\ 0.707 < I_{{{\text{BD}}}} \le 1&\quad {\text{Super-brittleness }} \\ \end{array} } \right.$$
(8)

where IBD = 1, corresponding to the strongest brittleness, is defined as absolute brittleness; IBD = 0, lacking brittleness and representing the ideal elastic‒plastic state, is defined as absolute ductility.

According to the proposed calculation method and classification method, the brittle‒ductile evolution characteristics of rock under different true triaxial stress levels (σ2, σ3) were quantitatively and qualitatively investigated.

5 Quantitative analysis of the influence of true triaxial stresses on the ductile–brittle behaviours of rock

5.1 Influence of σ 3 on the ductile–brittle behaviours of rock

Figure 6 shows the variations in the ductile–brittle behaviours of marble with σ3 under true triaxial stresses. For a constant σ2 (= 65 MPa), when σ3 = 0 and 2 MPa, the marble undergoes super-brittle failure; when σ3 = 5 MPa, the marble experiences brittle failure; and when σ3 = 10 MPa, the marble exhibits a transition from brittle failure to ductile–brittle failure. When σ3 = 20 MPa, marble is close to ductile–brittle failure; when σ3 = 30 and 40 MPa, marble experiences ductile–brittle failure. That is, when σ2 is constant, IBD first decreases rapidly with increasing σ3, the rate of decrease keeps decreasing gradually, and there is a trend of gradual stabilization after reaching a certain value. As σ3 increases from a low value to a high value, the marble changes from super-brittleness to ductile-brittleness.

Fig. 6
figure 6

Influence of σ3 on marble ductile–brittle behaviours

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5.2 Influence of σ 2 on the ductile–brittle behaviours of rock

Figure 7 shows the variations in the ductile–brittle behaviours of marble with σ2 under true triaxial stresses. For constant σ3 (= 5 MPa), when σ2 = 5 MPa, the marble undergoes super-brittle failure; when σ2 = 30 MPa, the marble is close to the transitional stage from brittle to ductile–brittle failure; when σ2 = 65 MPa, the marble exhibits brittle failure. When σ2 = 100 and 135 MPa, the marble reaches super-brittle failure; That is, when σ3 is constant, IBD first increases rapidly with increasing σ2, the growth rate keeps decreasing continuously, and there is a trend of gradual stabilization after reaching a certain value. As σ2 increases from a low value to a high value, marble changes from undergoing ductile–brittle failure to super-brittle failure.

Fig. 7
figure 7

Influence of σ2 on marble ductile–brittle behaviours

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5.3 Control effect of high 3D stress conditions on ductile–brittle behaviours of different rocks

Figure 8 shows the ductile–brittle characteristics of the four granites under different true triaxial stress levels. At the same σ3 (= 5 MPa), the variation in ductile–brittle behaviours of the four granites with increasing σ2 is consistent: with increasing σ2 (or differential stress σ2–σ3), the brittleness increases rapidly at first, and the rate of increase decreases gradually and finally tends to be super-brittleness. The difference is that the initial points of IBD of different granites are different, and the evolution rates from low brittleness to super-brittleness are also different. When the differential stress σ2–σ3 is 0 MPa or the ratio σ2/σ3 is 1, that is, under the conventional triaxial stress state, granite A and granite B show ductile-brittleness, and the other two granites show brittleness. When the differential stress σ2–σ3 is approximately 20 MPa or the ratio σ2/σ3 is 5, the four granites are at least brittle; when the differential stress σ2–σ3 reaches 40 MPa or the σ2/σ3 ratio is 9, the four kinds of granites are super-brittle. That is, when the difference between σ2 and σ3 is small, the effect of lithology on ductile–brittle behaviour is obvious; when the difference between σ2 and σ3 is large, the stress plays a dominant role in brittleness. In summary, the change in stress controls the evolution of the ductile–brittle behaviour of rock. When the difference between σ2 and σ3 is large, all kinds of rocks can achieve super-brittleness, and a high stress controls rock brittleness.

Fig. 8
figure 8

Comparison of ductile–brittle behaviours of four granites under different true triaxial stresses

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6 Discussion

6.1 Difference comparison of ductile–brittle behaviours under true triaxial and conventional triaxial stresses

To compare the difference in rock ductile–brittle behaviours under true triaxial and conventional triaxial stress states, Fig. 9 shows the ductile–brittle evolution of marble under the two stress conditions. The brittleness corresponding to true triaxial stress states is significantly higher than that corresponding to conventional triaxial stress states at the same σ3, and the decay rate of IBD is fast with increasing σ3 under conventional triaxial stress states. When σ3 increases to 10 MPa, IBD decreases to near the lowest value of 0. Under true triaxial stress states, when σ3 increases to 20 MPa, IBD = 0.367, and the rock is still brittle. By comparing the two stress states, it can be concluded that σ2 induces an increase in rock brittleness and causes the decay rate of brittleness to decrease with increasing σ3 and that σ2 increases the upper limit of σ3 for rock brittle failure, which means that the depth of brittle failure of the deep surrounding rock under the high 3D stress state is larger. The enhancement effect of σ2 on rock brittleness must be considered when evaluating the brittle failure of the deep surrounding rock under high-stress conditions.

Fig. 9
figure 9

Comparison of ductile–brittle behaviours under true triaxial and conventional triaxial stresses

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6.2 Inspiration for ductile–brittle evolution of the surrounding rock in deep engineering under conditions of high 3D stress

The CJPL-II (burial depth of approximately 2400 m) is one of the deepest underground research laboratories. The three principal stress components σ3, σ2, and σ1 of high geostress are approximately 25, 67, and 69 MPa, respectively (Feng et al. 2018). Figure 10a shows the step-by-step excavation scheme of the laboratory experiments: Step I is the excavation of the top-heading, Step II is excavation of the sidewall in the upper layer, and Step III is excavation of the middle groove and sidewall in the bottom bench. After the excavation of Step I, the high-stress in the surrounding rock underwent complex adjustments, such as stress unloading and stress concentration. Severe brittle failure, such as deep spalling, occurred at a certain depth range of the surrounding rock (the maximum depth was close to 3 m), as shown in Fig. 10b. Due to the complexity of engineering geological conditions and high-stress adjustment processes, it was very difficult to obtain an accurate stress field of the surrounding rock. In this project, as the axial direction of the excavation tunnel was approximately parallel to the σ2 component of the geostress, according to the classical mechanical theory, it was assumed that σ2 changes little and that σ3 is unloaded. From the deep undisturbed area of the surrounding rock to the surface layer of the tunnel sidewall, σ3 decreased from the initial geostress level to a small value or even zero. Therefore, the test results of marble with σ2 = 65 MPa and σ3 = 0.5, 5, 10, 15, and 20 MPa and the new proposed ductile–brittle index IBD can be used to evaluate the ductile–brittle characteristics of the surrounding rock from the surface to the interior. Meanwhile, to compare with the traditional understanding based on the conventional triaxial test results, the ductile–brittle behaviour of the surrounding rock under conventional triaxial stress states under the same σ3 was also calculated, as shown in Fig. 10c. Based on the true triaxial test results, the surface layer (σ3 = 0.5 MPa) and shallow layer (σ3 = 2 MPa) of the surrounding rock is super-brittle. With the increase in the depth of the surrounding rock, the brittleness of the rock mass gradually decreases, and the weakening rate is slow. The deeper surrounding rock (σ3 = 5 and 10 MPa) still has high brittleness, which provides a necessary condition for the brittle failure of the surrounding rock and can better explain the strong brittle spalling failure phenomenon in a certain depth range of the surrounding rock. The conventional triaxial test results show that the brittleness of the surface and shallow layers of the surrounding rock is low, which is inconsistent with the fact that the surrounding rock displays a deep brittle spalling failure phenomenon. The evaluation of the brittleness of the surrounding rock based on the results of the conventional triaxial test may underestimate the brittleness of the rock mass and the risk of brittle or even catastrophic failure of the surrounding rock in deep underground engineering.

Fig. 10
figure 10

a The excavation sequence and b the in situ deep spalling failure of the CJPL-II (modified from Feng et al. 2018), c ductile–brittle evolution behaviours of the surrounding rock of the CJPL-II under true triaxial stresses or conventional triaxial stresses

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7 Conclusion

In this study, true triaxial compression tests of marble and granite were carried out to obtain the stress–strain curves, including the processes of pre- and post-peak failure, and the new ductile–brittle evaluation method was proposed. The influence of true triaxial stress states on rock ductile–brittle behaviour and the control effect of high-stress on rock brittleness were analysed. The ductile–brittle evolution of the surrounding rock in deep underground engineering was explored. The main conclusions are as follows:

  1. 1.

    A new normalized ductile–brittle evaluation index was proposed based on the post-peak energy conversion characteristics of rock under true triaxial stresses, which can better quantitatively evaluate the influence of the true triaxial stress state (σ2, σ3) on the ductile–brittle behaviours of rock.

  2. 2.

    As σ2 increases or σ3 decreases (that is, the differential stress σ2–σ3 increases), the brittleness of rock increases, and its rate of increase gradually decreases and tends to be stable, transforming from ductile-brittleness to transitional, brittleness and ultimately super-brittleness. As a result, super-brittleness is easily induced by low-σ3 and high-σ2 conditions.

  3. 3.

    When the differential stress σ2–σ3 is small, the intrinsic characteristics of the rock itself have an obvious influence on ductile–brittle behaviours. When the differential stress σ2–σ3 is large, stress plays a dominant role in rock brittleness. The change in stress controls the evolution of the rock ductile–brittle behaviours, and high-stress controls rock brittleness.

  4. 4.

    The rock brittleness under true triaxial stress is significantly higher than that under conventional triaxial stress at the same σ3. σ2 induces an increase in rock brittleness and causes the decay rate of brittleness to decrease with increasing σ3, and σ2 increases the upper limit of σ3 for rock brittle failure. The enhancement effect of σ2 on rock brittleness must be considered when evaluating the brittle failure of deep surrounding rock under high stress states; otherwise, the risk of brittle failure may be underestimated.

The proposed normalized ductile–brittle evaluation index was established based on the true triaxial post-peak energy conversion characteristics of rock, which reflects the intrinsic driving essence of rock brittle failure from the perspective of energy, and is suitable for the brittleness evaluation of surrounding rock in 3D high stress deep engineering which respectcan; moreover it ranges from 0 to 1, which is beneficial to the comparison and application of different lithology and stress state. However, it is also necessary to point out that the post-peak test results of rock under true triaxial are not abundant, and the experimental basis for the application of this index still needs to be strengthened.