1 Introduction

Rock pillars under high stress are common in deep mines, wherein they are used to maintain the stability of underground engineering excavations (Cording et al. 2015; Renani and Martin 2018; Wang and Cai 2021). Constructions in deep mines inevitably involve excavation-unloading processes, which frequently induce rock brittle failure, such as spalling and rockburst, and lead to major safety hazards on-site (Ortlepp and Stacey 1994; Martin 1997; Martin et al. 2003; Zhang et al. 2012). Moreover, estimating rock pillar strength under high stress is very important for deep mines construction; however, evaluating pillar strength is highly challenging owing to several factors, including the in situ stress, excavation-unloading method, and rock mass quality (Sengani 2018; Martin and Maybee 2000; Esterguizen et al. 2011). Figure 1a–c shows the characteristics of rock pillar failure in deep mines on-site (Sengani 2018). Understandably, a pillar failure disaster makes stability in deep mines highly challenging. In addition, rock pillar failure and strength characteristics are related to the changes in the stress state before and after the excavation process (Martin and Maybee 2000). However, literature on rock pillar failure and strength analysis from the perspective of a high-stress unloading path is scarce. Such a path refers to a deep rock in a three-dimensional (3D) initial high stress before the excavation-unloading process. Under this condition, the formation of a rock pillar is similar to two-dimensional (2D) stress unloading, and the rock pillar is subject to a stress adjustment caused by the stress redistribution, which induces pillar failure. Figure 1d shows a schematic of an entire stress path during the unloading process of rock pillar excavation, and the high-stress unloading path is summarized as “3D initial high stress + 2D stress unloading + stress adjustment”. Figure 1e–g shows schematics of the high-stress rock pillar in a 1D stress state after excavation unloading, respectively (Sinha and Walton 2021a; Sankhanee and Walton 2018; Li et al. 2019). However, recently, few related studies have been conducted; thus, the innovation of this study is investigating the failure characteristics and strength reduction of the high-stress rock pillar caused by a high-stress unloading path: 3D initial high stress + 2D stress unloading + 1D stress adjustment.

Fig. 1
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Rock pillar failure: a–c rock pillar failure on-site (Sengani 2018); d schematic of entire stress path of rock pillar; e–g schematics of rock pillar bearing stress after excavation and unloading (Sinha and Walton 2021a; Sankhanee and Walton 2018; Li et al. 2019)

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Until now, research on deep mining engineering has typically been prominent, and abundant results have been obtained. Research on rock failure characteristics under excavation unloading focuses primarily on true triaxial tests (He et al. 2021; Jiang et al. 2021; Si and Gong 2020; Li et al. 2017; Liu et al. 2021; Zhao and Cai 2015). Si and Gong (2020) conducted a true triaxial unloading test on a cubic granite specimen. After confining pressure unloading, the specimen suffered severe rockburst failure under stress adjustment and the failure mode of the specimen changed. He et al. (2021) used self-developed true triaxial equipment to conduct an experimental study on the strain rockburst characteristics of sandstone under true triaxial loading and double-face unloading in one direction. In addition, considerable research has been performed to consider the effect of stress adjustment on rockburst (Cai 2008a, b; He et al. 2014; Hu et al. 2019; Gong et al. 2018, 2023; Luo et al. 2019; Wu et al. 2020). He et al. (2014) conducted true triaxial compression tests on prefabricated hole specimens by simulating the sidewall rockburst failure process under vertical stress adjustment after excavation. Accordingly, they conducted numerous true triaxial compression tests on prefabricated specimens and systematically explored the sidewall failure characteristics under stress adjustment after excavation unloading. However, the abovementioned tests considered only the effect of triaxial unloading or stress adjustment on rock failure under a single factor, and most of them simulated sidewall failure, whereas there are very few studies on rock pillars.

At present, although studies on pillar failure and strength considering triaxial unloading and stress environment are relatively few, relevant research has not stagnated. Regarding pillar failure, studies primarily consider pillar failure characteristics under loading conditions (Li et al. 2007; Gong et al. 2010; Xu et al. 2012; Deng et al. 2019; Deng 2021). Li et al. (2007) examined the mechanical response mechanism of the rockburst failure under the dynamic disturbance of a high pre-static pillar after excavation by performing numerical simulations. Similarly, Gong et al. (2010) conducted 1D coupled static–dynamic tests to analyze the energy evolution characteristics of pillar rockburst instability and founded the “Energy release” indoor rockburst phenomenon during sandstone specimen failure. In addition, Huang et al. (2017) theoretically derived the basic mechanical properties required for pillar rockburst. Furthermore, pillar strength is critical to the long-term stability of underground engineering (Esterhuizen 2006; Chen et al. 2009; Kaiser et al. 2011; Sankhanee and Walton 2018; Li et al. 2019; Sinha and Walton 2021b). Wang and Cai (2021) established a numerical model of the strength and deformation of rock pillars subjected to time-dependent and analyzed the long-term stability of a rock pillar. Sankhaneel and Wallton (2018) focused on studying the progressive S-shaped strength criterion and demonstrated its ability to reproduce observed pillar behavior. Li et al. (2019) used finite discrete element models to study rock pillar failure behavior in laboratory and in situ scales. They found that the post-yield behavior obtained from rock pillar models using laboratory parameters was more brittle than expected in practice. In summary, the abovementioned research on pillar failure and strength characteristics is limited to the loading condition, and the influence of an entire stress path (Fig. 1d) on the stability of the rock pillar is basically not considered.

In this study, rock pillar failure and strength characteristics were investigated under triaxial unloading paths of 3D initial high stress followed by 2D stress unloading and 1D stress adjustment. Two different 2D stress unloading paths were used: horizontal bidirectional sequential unloading path (YU test), and horizontal bidirectional simultaneous unloading path (TU test). The 3D initial high stresses (σ1 = 60 MPa, σ2 = σ3 = 10, 20, 30, 40, and 50 MPa) were applied to cubic granite specimens (50 × 50 × 50 mm). Additionally, uniaxial compression (UC) tests were performed for comparison. The peak strength, rockburst failure process, and failure model were obtained and compared. Finally, the mechanical behavior of the deep rock pillar under the triaxial unloading paths was examined based on the test results.

2 Experimental Procedures

2.1 Mechanical Properties and Preparation of Zhumadian Granite

Rock pillar failure tests under different unloading paths of high stresses were performed on a fine-grained homogeneous Zhumadian granite (Henan Province, China). Optical analyses of thin sections of the granite using the polarizing microscope to determine the mineral composition and structure, and the results are shown in Fig. 2a and b. It can be observed that the granite has a blocky structure without remarkable defects, and that the average mineral particle size is approximately 0.2–0.3 mm. The petrographic studies according to Fig. 2c reveals that the Zhumadian granite is composed primarily of quartz (33%), K-feldspar (32%), plagioclase (30%), biotite (3%), and other minerals (2%).

Fig. 2
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Thin-section observation of granite material (Wu et al. 2022): a single and b orthogonally polarized light; c mineral composition

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All standard cubic specimens of size 50 × 50 × 50 mm used in the triaxial unloading path tests were cored from the same Zhumadian granite block in the same direction to reduce the influence of anisotropy, as illustrated in Fig. 3a. The blocks were carefully polished to reduce the influence of dimensional errors, where the dimensional tolerance of all end faces was controlled within 50 ± 0.0025 mm and their perpendicularity tolerance was 0.025 mm (Feng et al. 2019). The stress–strain curves and typical failure modes of the cubic specimens under UC tests are shown in Fig. 3b and c. In the process of uniaxial compression failure of the cube specimens, they emitted loud sound accompanied by violent rockburst characteristics, such as rock fragments ejecting, which was similar to the failure characteristics of cylinder specimen under uniaxial compression (Wu et al. 2022). The average UC strength (UCS) of each cubic specimen was 140.00 ± 0.45 MPa, and the UC failure mode was tension splitting. Additionally, each cubic specimen had a density of 2600.52 ± 2.79 kg/cm3, P-wave velocity of 3879.18 ± 74.64 m/s, Young’s modulus of 40.48 ± 1.35 GPa, and indirect tensile strength of 7.50 ± 0.04 MPa.

Fig. 3
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Basic information of cubic sample: a specimen geometry; b and c typical static axial stress–strain curves and corresponding rockburst failure characteristics

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2.2 Testing Apparatus

As shown in Fig. 4, the rock pillar failure tests under different triaxial unloading paths were performed on a true triaxial testing system, which comprised a data acquisition system and independent loading and unloading units in the X-, Y-, and Z-directions. The true triaxial equipment has high stiffness, in which the horizontal and vertical loading frame stiffness is 400 kN/mm and 1000 kN/mm, respectively, belonging to the rigid testing machine. The maximum load capacity in X- and Y-directions is 2000 kN, whereas that in the Z-direction is 3000 kN, and the loading rate of force is 10 N/s–10 kN/s. In the displacement aspect, the measurement range is 0–200 mm in the X-direction, and 0–100 mm in the Y- and Z-directions. The measurement accuracy in all three directions was <  ± 0.5% and the resolution was 0.001 mm. The deformation measurement range was 0–10 mm, with a measuring accuracy of <  ± 0.5% and a resolution of 0.0005 mm. The micro-camera captures and records video at a sampling rate of 30 frames per second, and the image resolution is set to 160 × 120 pixels. More details about the test system can be found in the references Gong et al. (2018).

Fig. 4
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Test equipment: a triaxial system; b corresponding 3D stress loading chamber

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2.3 2D Unloading Path Design

It is well known that excavation unloading changes the stress state of the rock and induces stress redistribution, which is also affected by the stress unloading path. Therefore, the stress change process of the rock pillar in deep mining engineering occurs in a 3D initial high stress state followed by excavation unloading and 1D stress adjustment. This entire stress change process can be summarized as 3D initial high stress + 2D stress unloading + 1D stress adjustment. A 2D stress unloading process of a rock pillar (Fig. 1d) can be divided into sequential and simultaneous unloading paths in the horizontal bidirectional. Thus, rock pillar failure processes were conducted under two triaxial unloading paths on the triaxial testing system. As illustrated in Figs. 5 and 6, these are horizontal bidirectional sequential and simultaneous unloading tests (YU and TU tests, respectively). Table 1 lists the cubic granite specimen labels for the two triaxial unloading paths. In order to eliminate the end effect as much as possible, we applied a lot of lubricant between the rock specimen and the loading platens, but it may not be possible to completely eliminate the end effect. The detailed test process was as follows.

Fig. 5
figure 5

Schematic of 3D stress state unloading to 1D stress state under two different modes: (1) → (2) → (3) → (5) represents YU test and (1) → (4) → (5) depicts TU test

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Fig. 6
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Schematics of high-stress unloading path: a YU and b TU tests (σ1, σ2, and σ3 represent X-, Y- and Z-direction stresses, respectively)

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Table 1 Different types of tests
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2.3.1 (1) YU Test

In Fig. 5, steps (1) → (2) → (3) → (5) schematically show the triaxial unloading path by which the YU tests were conducted on the triaxial system, and the corresponding stress paths are shown in Fig. 6a. In the 3D initial stress loading stage, the three directions (X, Y, Z) were loaded with the preset values at a loading rate of 0.4 MPa/s and subsequently maintained for 5 min (step (1) in Fig. 5). Note that the X- and Y-directions were subjected to equal confining pressure loading, and the preset confining pressures were 10, 20, 30, 40, and 50 MPa, respectively, whereas the preset threshold in the Z-direction was 60 MPa. Subsequently, the loading in Y- and Z-directions was kept unchanged and the X-direction was unloaded to 0 MPa at an unloading rate of 1.0 MPa/s, which was maintained for 2 min (step (2) in Fig. 5). Following this, the same unloading process was performed in the Y-direction, wherein the specimen was subjected to stress only in the Z-direction, and the specimen was in a 1D stress state (step (3) in Fig. 5). Finally, the Z-direction stress was activated again and increased at a loading rate of 0.4 MPa/s until the specimen failed (step (5) in Fig. 5).

2.3.2 (2) Tu Test

In Fig. 5, steps (1) → (4) → (5) schematically show the entire process by which the TU tests are conducted on the triaxial system, and the corresponding stress paths are shown in Fig. 6b. Noticeably, although steps (1) and (5) are the same as in the YU tests, there are significant differences in the 2D stress unloading process. Specifically, the horizontal X- and Y-directions were simultaneously unloaded to 0 MPa at a rate of 1.0 MPa/s in the TU tests (step (4) in Fig. 5).

3 Results and Analysis

A series of triaxial unloading path tests of “3D initial high stress + 2D stress unloading + 1D stress adjustment” was performed to reproduce the rock pillar failure. This section focuses on the stress characteristics, pillar rockburst failure process, and pillar strength-reduction effect (also known as strength-weakening effect) under the triaxial unloading path.

3.1 Stress Characteristics of Rock Pillar Under Triaxial Unloading Path

The stress–time and stress–strain curves can reflect the actual stress–loading paths and deformation characteristics of the cubic granite specimens in the triaxial unloading tests, respectively, as shown in Fig. 7. Figure 7a–d depict the typical stress–time and stress–strain in the three directions of cubic granite specimens YU-10 and TU-30. Figure 7e and f show the Z-direction stress–strain characteristics of all cubic granite specimens in the YU and TU tests. Figure 7a and c show that two triaxial unloading tests are conducted in strict compliance with the test scheme in Fig. 6. Noticeably, the Z-direction stress–strain curves of all cubic granite specimens can be divided into compaction, elastic, inelastic, and failure segments (seen in Fig. 7e and f). In comparison, those of the X- and Y-directions show compression, elastic, and unloading segments (seen in Fig. 7b and d). Additionally, stresses were not constrained in the X- and Y-directions after the 2D stress unloading, whereas the Z-direction stress remained unchanged; therefore, a notable stress step was observed in the Z-direction owing to the Poisson effect. Moreover, remarkable stress hysteresis loops were formed in X- and Y-directions, and the corresponding strains did not return to the original values, which suggests that the 2D stress unloading induced notable local damage in a rock pillar specimen. Therefore, compared to the stress–strain curves obtained under UC tests (Fig. 3b), those from the YU and TU tests present a distinct yield segment near the peak strength owing to the unloading effect (Fig. 7e and f). This suggests that the mechanical properties of a rock pillar specimen are affected by the unloading effect, such as plastic enhancement and brittleness reduction. It can be concluded that the mechanical behavior of the cubic granite specimens under “3D initial stress + 2D stress unloading + 1D stress adjustment” becomes more complex. In summary, the mechanical characteristics of a rock pillar specimen under the considered triaxial unloading paths are essentially different from those under uniaxial loading, i.e., the physical characteristics of rock pillar are changed under triaxial unloading.

Fig. 7
figure 7

Stress characteristics of samples under different high-stress unloading paths: a and b stress–time and stress–strain curves of specimen YU-10; c and d stress–time and stress–strain curves of specimen TU-30; e and f Z-direction stress–strain curves from YU and TU tests

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3.2 Rockburst Failure Process of Rock Pillar Under Triaxial Unloading Path

During the 2D stress unloading process, no remarkable macroscopic damage was caused to the cubic granite specimen surface, and it remained intact, due to the axial load being below the UCS. However, the axial strain will increase and the stress step appears in Fig. 7, which indicates that the triaxial unloading causes a certain degree of damage to the specimen. This indicates that the triaxial unloading did not immediately induce rock pillar macroscopic failure until the external stress was adjusted to the maximum bearing capacity of the cubic granite specimen. Figures 8 and 9 illustration the severe dynamic rockburst failure abruptly occurred as the Z-direction stress was adjusted to the maximum bearing capacity of the cubic granite specimen. The triaxial unloading paths did not seem to significantly affect the rockburst failure characteristics of the cubic granite specimens, which reflects that the rockburst failure processes in the YU and TU tests are very similar. Due to the narrow inside of the true triaxial apparatus, it is impossible to obtain the range and distance of rock bursting, which leads to the inability to reflect the intensity of rockburst. In the YU and TU tests, with increase in the Z-direction stress, remarkable macroscopic cracks were first observed on the surfaces of the cubic granite specimens (Figs. 8a1–e1 and 9a1–e1). After 0.04 s, numerous fine and dust particles were ejected on the surface and accompanied by a high noise; after 0.08 s, the damage phenomenon intensified, and increasing number of fine particles and rock flakes were ejected. After 0.12 s, the surface ejection phenomenon became slightly gradual, but new fine particles and rock flakes were still observed to fly out. The duration of the dynamic rockburst failure progress of all specimens was very short. The yield stages of the stress–strain curves under the unloading confining pressure of 50 MPa were very significant, and the strain increased rapidly under the extremely small stress change (Fig. 7). This reflects an increase in the plasticity of the rock specimen and a decrease in the brittleness. This demonstrates that the confining pressure unloading may affect the rockburst intensity, particularly the high-confining pressure unloading. It also suggests that the triaxial unloading affects its mechanical characteristics of deep rock pillar.

Fig. 8
figure 8

Rockburst failure process in YU test

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Fig. 9
figure 9

Rockburst failure process in TU test

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3.3 Post-peak Failure Mode of Rock Pillar

Understanding the post-peak failure mode of the rock is helpful in analyzing the failure mechanism (Kate and Reches 2004). Figure 10 shows the post-peak failure characteristics of the cubic granite specimens obtained in the UC, YU, and TU tests, respectively. Figure 10a presents the post-peak failure characteristics of the cubic granite specimens in the UC tests, primarily in the tension-splitting failure mode, and it shows that all specimens remained basically intact. In the YU tests, the shear failure mode was the main mode under low-confining pressure unloading (Fig. 10b1 and b2), and the shear–tension failure mode dominated as the confining pressure increased (Fig. 10b3 and b4), which is similar to the failure mode of surrounding rock after deep mining unloading (Li 2021). Similarly, the shear–tension failure mode was dominant in the TU tests (Fig. 10c). Furthermore, Wang and Cai (2021) reproduced the shear failure characteristics of a rock pillar by numerical simulations in combination with field engineering cases, and results are shown in Fig. 11a. Sankhanee and Walton (2018) also concluded that the failure mode of a rock pillar is mainly shear failure, which is shown in Fig. 11b. The failure mode of a rock pillar under the high-stress triaxial unloading path simulated in this study is in agreement with these conclusions, which shows that the simulation tests are reliable. Although two triaxial unloading stress paths do not significantly affect the failure modes of cubic granite specimens, the strength-reduction effect directly indicates that the unloading path has a very significant influence on the strength characteristics of the rock specimens (Fig. 7). Compared with the UC test, triaxial unloading changes the failure mode of the cubic granite specimens; additionally, the unloading path and confining pressure affect the failure mode.

Fig. 10
figure 10

Post-peak failure characteristics of specimens under a UC tests; b YU tests; c TU tests

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Fig. 11
figure 11

Typical shear failure mode of rock pillar (Wang and Cai 2021; Sankhanee and Walton 2018)

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3.4 Strength-Reduction Effect of Rock Pillar Under Triaxial Unloading Path

Table 2 lists the peak strength values of the cubic granite specimens obtained from two triaxial high-stress unloading path tests. Compared to the UCS, the peak strengths of the cubic granite specimens under the “3D initial high stress + 2D stress unloading + 1D stress adjustment” stress path was significantly reduced, and they were affected by the triaxial unloading path. The peak strengths of all cubic granite specimens were lower than those of the UCS, and the peak strengths significantly decreased with increase in the confining pressure, regardless of the unloading path, thereby exhibiting a significant strength-reduction effect. This is observed from Fig. 12a. In the YU tests, with increase in the confining pressure, the peak strengths of cubic granite specimens ranged between 138.07 and 102.22 MPa, and compared to the uniaxial compressive strength (i.e., UCS = 140.00 MPa), the peak strengths decreased in the range of 1.93–37.78 MPa. Similarly, the peak strengths of the cubic granite specimens fluctuated from 137.48 to 87.60 MPa with increasing confining pressure in the TU tests, and the peak strengths decreased from 2.52 to 52.40 MPa. In addition, under the same confining pressure, the peak strength of the granite in the TU test was lower than that in the YU test, and the stress difference between them increased with increase in the confining pressure. These trends suggest that the triaxial unloading path has a significant effect on the strength-reduction degree of a rock pillar.

Table 2 Test results
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Fig. 12
figure 12

Strength variation characteristics: a relationship between peak strength and confining pressure; b peak strength difference of granite specimens in TU and YU tests at same confining pressure

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To quantitatively express the strength-reduction effect of a rock pillar, the strength-weakening rate (SWR) is proposed to characterize the strength-reduction degree as (Si and Gong 2020; Wu et al. 2022):

$$SWR = \frac{{UCS - \sigma_{i}^{{}} }}{UCS} \times 100\% ,$$
(1)

where UCS is the uniaxial compressive strength and σi is the peak strength in the YU or TU test under a particular confining pressure, i = YU, TU test.

The SWR values under two triaxial unloading paths are listed in Table 2. Figure 12a reveals the relationship between the SWR and the confining pressure, i.e., the strength-reduction degree of the cubic granite specimen increases with increasing unloading confining pressure. The SWR values from the TU tests were significantly higher than those from the YU tests, which indicates that the reduction degree of the peak strength by the simultaneous unloading stress path is larger than that by the sequential unloading stress path. Moreover, the peak strength difference between two triaxial unloading paths increased with increasing unloading confining pressure, as shown in Fig. 12b.

These trends suggest that high-stress unloading induces rock pillar strength reduction, and the triaxial unloading path affects its weakening intensity. Therefore, the strength-reduction effect of a rock pillar can be summarized as follows. First, the excavation-unloading process changes the stress state of the surrounding rock from a 3D stress state to a 1D stress state, which causes the rock pillar strength to sharply decrease. This is the dominant factor and inevitable. The strength-reduction effect induced by bidirectional simultaneous unloading is more significant than that by sequential unloading. When the bidirectional simultaneous excavation-unloading method is adopted for the construction of underground engineering, the stress adjustment rate of the surrounding rock is higher than when the bidirectional sequential unloading method is used. Therefore, the stress concentration is more significant in the former case, which makes the surrounding rock more prone to induce damage. As the excavation-unloading path is controllable during the excavation process, the strength-reduction degree can be adjusted in this part.

Compared with the stress–strain curves and tension-splitting failure modes of the cubic granite specimens in pure UC tests, those in triaxial unloading path show a remarkable yield stage, and shear failure is the main failure mode (Figs. 7 and 10). In addition, the peak strengths of the cubic granite specimens under triaxial unloading paths show a notable strength-reduction effect. These results indicate that triaxial unloading affects the rock strength parameters. Generally, shear failure characteristics of rock materials can be described well by the Mohr–Coulomb strength criterion. This strength criterion is composed primarily of cohesion \(c\) and internal friction angle \(\varphi\) variation with the normal stress, i.e., \(t = c + \tan \varphi\). The triaxial compress strength of a rock (σTUS) is typically higher than pure UC (σUCS). Accordingly, Mohr–Coulomb stress circles under four different stress conditions are shown in Fig. 13, wherein circles 1–4 represent the triaxial compression, UC, YU, and TU tests, respectively. In this study, changes in strength parameters (c and \(\varphi\)) with the Mohr–Coulomb stress circles were examined to analyze the influence of triaxial unloading path, as follows. (1) It is assumed that \(\varphi_{1}\) of a rock under loading and unloading conditions remains unchanged. The variation characteristics of \(c\) after 2D stress unloading are: \(c_{3} > c_{4} > c_{1}\), i.e., the cohesion of a rock increases under confining pressure unloading. (2) It is assumed that \(c_{1}\) of a rock under loading and unloading conditions remains unchanged. The variation characteristics of \(\varphi_{1}\) after confining pressure unloading are as follows: \(\varphi_{3} > \varphi_{4} > \varphi_{1}\), i.e., the internal friction angle of a rock increases under confining pressure unloading. (3) Compared with a UC test, after confining pressure unloading, c or \(\varphi\) decreases. This suggests that confining pressure unloading affects the strength parameters of rock. However, we did not examine the specific influence of unloading on the strength parameters (c and \(\varphi\)) and the constitutive relationship, and this will be the focus of our future research.

Fig. 13
figure 13

Mohr–Coulomb stress circles under four different stress conditions

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4 Discussion

4.1 Pillar Strength-Reduction Mechanism Under Triaxial High-Stress Unloading Path

In this study, the strength-reduction effect of a rock pillar was reproduced from the perspective of triaxial unloading path (i.e., 3D initial high stress + 2D stress unloading + 1D stress adjustment). Moreover, the strength-reduction effect of a rock pillar was attributed to the 3D initial stress and the 2D stress unloading path. Compared with a pure UC test, under a high-stress path, the cubic granite specimens experienced a 3D initial high stress and 2D stress unloading process. Triaxial unloading under different confining pressures induces rock local damage, thereby reducing its bearing capacity (Huang and Li 2014; Xu et al. 2019; Cai 2008a, b; Li et al. 2014; Martino and Chandler 2004). Moreover, it can be seen from Fig. 7 that the stress step will appear in the axial direction after unloading, indicating that the axial strain is increasing, which further indicates that the triaxial unloading causes damage to the rock, so the rock is more prone to failure. Therefore, the peak strengths in the triaxial unloading path tests were lower than those in the pure UC tests. When the confining pressure reached 50 MPa, the yield stage under high-confining pressure unloading gradually were very significant (Fig. 7). This is because after triaxial unloading under a high confining pressure, a wide damage zone appears in the surrounding rock, which results in numerous microcracks (Yan et al. 2011, 2015). Compared with the splitting failure mode of the cubic granite specimens under pure UC (Fig. 10a), those under triaxial unloading path are shear and shear–tension failure modes (Fig. 10b and c). Furthermore, the peak strengths of the cubic granite specimens under high-confining pressure unloading were significantly lower than those under low-confining pressure unloading, regardless of the triaxial unloading stress path (Fig. 12). These results show that high-confining pressure unloading may change the rock properties (such as reduce the plastic-enhanced brittleness, as shown in Figs. 7 and 13), thus changing the failure mode of the rock and reducing both the peak strength and rockburst intensity. This also reflects that the 3D initial stress determines the strength-reduction level of a rock pillar. Second, the peak strengths of the cubic granite specimens in the TU tests were significantly lower than those in the YU tests, which indicates that the 2D stress unloading path affects the weakening intensity (Fig. 12b). In the TU tests, the X- and Y-directions were unloaded to 0 MPa simultaneously; therefore, the stress release was faster and the rock local damage was more severe. In the YU tests, the X- and Y-directions were unloaded to 0 MPa successively, i.e., the Y-direction stress remained unchanged, whereas the X-direction stress was reduced to 0 MPa. In these tests, the lateral stress suppresses the rock local damage. Thus, the peak strength of a rock pillar under simultaneous unloading is lower than that under sequential unloading, which indicates that the triaxial unloading path affects its weakening intensity.

In conclusion, the strength-reduction mechanism of a rock pillar under a high-stress path can be summarized as follows. The 3D initial stress determines the strength reduction level of the pillar, whereas the triaxial high-stress unloading induces a strength reduction effect and the unloading path affects its weakening intensity.

4.2 Engineering Significance of Considering the Rock Pillar Strength Reduction

Pillars are used in deep and high-stress mines to maintain the stability and integrity of openings (Cording et al. 2015; Renani and Martin 2018; Wang and Cai 2021). The design and construction of rock pillar in traditional deep mining engineering are basically based on the mechanical properties of rock under pure uniaxial compression. However, one aspect rarely considered in the analysis of rock pillar strength, is the influence of the stress path imposed on the pillar due to the excavation-unloading sequence of the pillars. This article presents a study of pillar strength from the perspective of 3D initial high stress + 2D stress unloading + 1D stress adjustment by simulating laboratory-scale rock specimens with a pillar structure. Our test results reveal that the initial high stresses unloading reduce the pillar strength and induces the strength reduction effect (Fig. 12). Further, the pillar strength under the simultaneous unloading path is lower than that under the sequential unloading path (Fig. 12b). In this case, it seems hazardous to adopt the pillar strength under pure static loading as the sole reference for deep mine engineering design. Therefore, for the safe construction and operation of deep mining engineering, it is necessary to establish a new design and construction standard considering the excavation-unloading effect, wherein necessary to measure the initial high stresses, determine the unloading path, and even the unloading rate adopted, and the possible reduction degree of the pillar strength induced by unloading should be pre-estimated.

Size effects also have a significant influence on the results of the pillar models. In this study, we used the original rock for the test and the failure and strength characteristics of rock pillars were studied under triaxial unloading paths of 3D initial high stress followed by 2D stress unloading and 1D stress adjustment. Although the results of laboratory-scale rock pillar model test simulation are similar to those of on-site deep mining engineering, can provide some insights for pillar design and hazard prevention in deep mines, it is hard to up-scale the laboratory-scale pillar model size to a full-size pillar model on-site. Thus, this study only conducts qualitative analysis. Future research is needed to address these challenging issues.

5 Conclusions

In this study, a series of simulated pillar rockburst and strength characteristics tests under a “ 3D initial high stress + 2D stress unloading + 1D stress adjustment” stress path were considered from the perspective of an entire stress path. Their results were compared to those obtained in pure UC tests. The effects of triaxial high-stress unloading paths on the mechanical and failure characteristics of cubic granite specimens were examined in detail, and the main conclusions were as follows.

  1. (1)

    Strength-reduction effect of rock pillar under triaxial high-stress unloading path. The compressive strength under triaxial unloading was significantly lower than the UCS, which indicates a notable strength-reduction effect. SWR was proposed to represent the characteristics of the strength reduction. It was found that the SWR increased significantly with increase in the confining pressure, thus indicating that a high-confining pressure during triaxial unloading implies a significant strength-reduction degree.

  2. (2)

    The triaxial unloading path has a significant influence on the rock pillar strength-reduction effect. The peak strength in the horizontal bidirectional sequential unloading was higher than that in the horizontal bidirectional simultaneous unloading. The peak strength difference increased gradually with increasing confining pressure, and the maximum difference observed was 14.62 MPa. In addition, the SWR values were approximately 26.99% and 37.43% under the horizontal bidirectional sequential and simultaneous unloading, respectively. These results indicate that the unloading path affects the weakening intensity.

  3. (3)

    Violent rockburst occurred in all specimens, and the deep rock pillar strength-reduction mechanism is summarized as follows. The 3D initial stress determines the strength-reduction level of the rock pillar, whereas the triaxial high-stress unloading induces a strength-reduction effect, and the specific unloading path affects its weakening intensity.