Influence of initial stresses and stress paths on the deformation and failure mechanism of sandy slate

Abstract

Rocks exhibit various mechanical properties under different stress conditions, with changes during unloading having significant implications for geological engineering safety. This study carried out triaxial loading and unloading mechanical tests on sandy slate to investigate its mechanical properties, deformation characteristics, and failure mechanisms at different initial stress levels and stress paths. The results showed that during the unloading process, the deformation modulus (E) of the sandy slate decreased, and the Poisson's ratio (μ) gradually increased. This indicates that significant volume expansion of the rock is the dominant factor in its deformation and failure. The exponential function can be used to describe the evolution of E and μ with confining pressure during unloading. The damage stress of the rock under unloading conditions was lower than that under loading conditions, suggesting that unloading led to an earlier onset of volumetric expansion in the sandy slate. Under conditions where the initial axial stress level approached the damage stress, the mechanical properties were most significantly affected by unloading. Compared to loading conditions, when the initial axial stress level was at 70% of the peak strength, the cohesion (c) decreased by 15.77 to 29.37%, while the internal friction angle (φ) increased by 1.88 to 5.14%. The rock's failure process can be divided into four stages based on the development of microcracks. Unloading during the stages of microcrack initiation and propagation can lead to tensile cracks of varying degrees, resulting in different mechanical properties and failure characteristics under loading and unloading conditions.

Introduction

Studying the mechanical properties of rock mass not only forms the core of rock mechanics theory but is also an essential aspect of implementing rock engineering projects. Traditionally, research in rock mechanics focused on conditions under loading, where related theories have become relatively mature. However, with the increase in engineering projects and the widespread occurrence of rock mass unloading phenomena, the research focus has gradually shifted towards unloading rock mechanics^1–3. During the excavation and unloading of large underground caverns, the re-distribution of stress in the rock surrounding the cavern often leads to collapse, spalling, and slabbing in the vault, arch shoulders, and high sidewall areas^4,5. Due to different stress responses in the crown, sidewall, and floor sections, underground cavern rock displays complex deformation and failure characteristics during excavation. In order to determine the stability of the surrounding rock mass in underground caverns, it is imperative to study the deformation and failure behavior of the rock mass under different levels of initial stress and stress paths.

Numerous scholars have embarked on experimental studies focusing on the impact of various unloading stress paths (constant axial stress and unloading lateral stress, axial stress loading and lateral stress unloading, axial and lateral stress unloading) and unloading rates on rock mechanical properties. Researchers including Li et al.⁶, Peng et al.⁷, Xu et al.⁸, and Zhao et al.⁹ have each conducted triaxial unloading tests on rock samples. By comparing mechanical and deformation characteristics across various unloading stress paths, these studies have explored the effects on rock strength and deformation parameters, as well as the mechanisms of unloading-induced fracture. Concurrently, experiments by Huang et al.^10,11, Cong et al.¹², and He and Zhao² have demonstrated that rock samples exhibit an increase in bearing strength capacity with rising unloading confining pressure rates. These findings consistently highlight that different loading and unloading paths significantly influence rock mechanical properties and rupture mechanisms.

In the study of deformation parameters and strength during rock unloading, researchers such as Chen et al.¹³, Zhang et al.¹⁴, Zhao et al.¹⁵, and Gao et al.¹⁶ have conducted triaxial unloading tests on materials like marble, coal, and sandstone. Studies have shown that rock deformation parameters deteriorate upon unloading, with cohesion values typically lower than those measured during loading. In contrast, the angle of internal friction has been observed to increase. Hou et al.¹⁷ noted that in sandstone unloading tests under varying confining pressures, Poisson's ratio and Young's modulus exhibit negative and positive correlations with the initial confining pressure, respectively. Moreover, Poisson's ratio gradually decreases during the unloading of confining pressure. Chen et al.¹⁸ demonstrated that the initial confining pressure and initial axial pressure affect the volumetric dilatation characteristics of rocks under triaxial unloading conditions. Specifically, the dilatancy angle is inversely related to the initial confining pressure and directly related to the initial axial pressure. Lv et al.¹⁹ explored the degradation effects of mechanical strength parameters of granite during unloading, indicating that Poisson's ratio exhibits an exponential increase during the unloading of surrounding rock, while the modulus of deformation abruptly drops near the yield failure point of unloading.

The rock failure modes under unloading paths are complex. Numerous triaxial unloading tests have shown that unloading can lead to more severe brittle failure in rocks compared to when loading, coupled with a notable phenomenon of volumetric dilatation^20–22. Gong et al.²³ found that granite primarily fails in shear under both triaxial loading and unloading conditions, with coplanar cracks under unloading exhibiting a mixed mode of tension and shear. Studies by Huang et al.²⁴, and Zhu and Huang²⁵ have revealed that confining pressure and unloading rate significantly influence the failure mode of rocks, categorizing unloading failure into three types: shear, conjugate shear, and splitting accompanied by shear. In order to elucidate the mechanisms of rock failure under unloading, scholars have investigated from the perspective of microcrack evolution under different stress paths. For instance, Zhao et al.²⁶ analyzed crack formation in Beishan granite at varying unloading rates using acoustic emission and high-speed camera techniques. Liu et al.²⁷ used fracture volume strain to categorize the crack evolution process. Huang et al.²⁸ employed three-dimensional scanning technology to analyze the loading and unloading failure mechanisms of mudstone sandstone.

In summary, extensive research has been conducted on the mechanical behavior, deformation properties, strength parameters, and failure modes and mechanisms of rock masses under unloading conditions. Due to the influence of tectonic stress or structural plane, the stress field of rock mass is more complex. However, most studies set the initial axial stress as the only value when conducting unloading mechanical properties tests, ignoring the influence of the initial axial stress on the unloading mechanical properties of rock. In this paper, the sandy slate of the underground main powerhouse of Kala hydropower station is the research object. Three stress paths of axial stress loading under constant lateral stress, axial stress loading and lateral stress unloading, axial and lateral stress unloading are used to simulate the stress changes of surrounding rock in different parts of the underground cavern during excavation. The mechanical properties of sandy slate under different initial stress levels and different stress paths are tested. Based on the test results, the variation characteristics of axial strain, circumferential strain and volume strain are analyzed, and the basic mechanical parameters of rock under different stress paths are determined. The evolution law of deformation parameters under unloading paths is discussed, and the failure mechanism under different initial stress levels and different stress paths is explored in combination with the failure characteristics. The results can provide a theoretical basis for the stability analysis of underground engineering under complex stress conditions during construction.

Test overview and stress path analysis

Sample preparation

Kala hydropower station is located in Muli Tibetan Autonomous County, Liangshan Yi Autonomous Prefecture, Sichuan Province, China. The rock samples for the study were sandy slate from the surrounding rocks of the exploratory adits of the underground powerhouse of the Kala Hydropower Station. Standard cylindrical samples with a diameter of 50 mm and a height of 100 mm were prepared by onsite sampling, laboratory drilling, cutting, and polishing. Sample dimensions were processed following International Society for Rock Mechanics (ISRM) standards, allowing for a ± 0.3 mm diameter tolerance and ± 0.5 mm flatness for the upper and lower end faces. To reduce the dispersion of the test results due to the rock's own defects and heterogeneity, samples with obvious defects, color differences, and impurities were first screened out. Then, using an electronic balance and a vernier caliper, the test mass and dimensions were measured to calculate density, and an RSM-SY5(T) non-metallic ultrasonic testing instrument was used to measure the sample's wave velocity. Based on the density and wave velocity results of the sandy slate (shown in Fig. 1), samples from the concentrated area with similar wave velocity and density were selected for testing. The selected samples had a wave velocity of 2.9 ~ 3.5 km/s and a density of 2.655 ~ 2.755 g/cm³.

Fig. 1.

Sandy slate samples and their density and wave velocity distribution: (a) standard samples, (b) density distribution, (c) wave velocity distribution.

Testing equipment

The experimental apparatus utilized in this study was the ROCK600-50 adaptive multi-field coupled triaxial testing system, as depicted in Fig. 2. This apparatus is capable of conducting multi-field coupled tests on rock and concrete, involving combinations of thermal, fluid, mechanical, and chemical interactions. The system primarily consists of a loading system, temperature control system, measurement system, and control system. The loading system is equipped with independent axial, confining, and pore pressure systems, capable of applying a maximum axial stress up to 500 MPa and maximal confining and pore pressure up to 60 MPa. This allows for simultaneous control of these stresses across different testing scenarios. The measurement system, composed of axial and confining pressure sensors, pore pressure sensors, as well as axial and radial displacement, and temperature sensors, enables real-time monitoring and recording of fundamental data such as the stress and strain of the samples. This device allows for the simulation of complex stress paths encountered in engineering, and determines the deformation stress–strain curves and mechanical parameters under different stress paths for sandy slate.

Fig. 2.

ROCK600-50 adaptive multifield coupled triaxial testing system.

Determination of the stress path and test scheme

Analysis of the relationship between stress path and rock mass stability

In rock masses under initial stress states, instability may occur with changes in axial and confining pressures. Based on the stability of the rock mass under different stress paths, the area around the initial stress value of the rock mass can be divided into seven distinct zones. These zones are further classified into three categories according to their stability: extremely unstable zone, unstable zone, and stable zone, as shown in Fig. 3. Zones a and b are extremely unstable, zones c and d are unstable, and zones e, f, and g are stable. Each of the seven zones is characterized by the following features: (1) Zones a and b are areas of axial loading and confining pressure unloading. The rock mass is more prone to instability and failure when the axial loading rate exceeds the confining pressure unloading rate, meaning the stress path in zone a leads to greater instability compared to zone b. (2) Zones c and e are areas where both axial pressure and confining pressure are being loading. When the axial loading rate is higher than that of the confining pressure, the increase in deviatoric stress may cause rock failure; conversely, the rock remains stable if the confining pressure loading rate exceeds the axial loading rate. (3) Zones d and f are regions of axial unloading and confining pressure unloading. For rock failure to occur, not only must the axial unloading rate be less than the confining pressure unloading rate, but also the initial axial pressure must be high enough to induce failure upon complete unloading of the confining pressure. The rock mass remains stable if the axial unloading rate is greater than the confining pressure unloading rate. (4) Zone g is the area of axial unloading and confining pressure loading, where the rock mass remains stable as long as the principal stress does not reverse.

Fig. 3.

Classification of rock mass stability under different stress paths.

Test scheme

Studies^29,30 using model experiments and numerical simulations to study stress variations in the rock surrounding underground caverns, suggests that that following stress adjustment, the rock in the vault area typically experiences an increase in tangential stress coupled with a decrease in radial stress. In comparison, the sidewalls and floor experience a decrease in both tangential and radial stresses. With this understanding, two stress unloading paths were identified: axial stress loading and lateral stress unloading (LAUL), and axial and lateral stress unloading (UAUL). Additionally, the conventional triaxial compression test serves as the base case for comparison. Based on this foundation, triaxial loading and unloading tests on sandy slate under different stress paths were conducted.

As illustrated in Fig. 3, the proximity between the initial stress state and the strength envelope dictates the rate of rock failure under various stress paths. Consequently, it is essential to consider the different initial stress states along with the stress paths. Rock undergoes various stages prior to failure, including crack closure stage, elastic stage, stable crack expansion stage and crack accelerated expansion stage³¹. The initiation points for stable and unstable crack propagation are the crack initiation stress (σ_ci) and the crack damage stress (σ_cd), respectively, as shown in Fig. 4. Through the triaxial compression test, scholars³² have determined that for granodiorite, the σ_ci is approximately 0.35 to 0.45 times the peak stress (σ_f), and the σ_cd is about 0.68 to 0.75 times the peak stress (σ_f). If the rock stress is below the crack initiation stress, the stress path for simultaneous axial and lateral stress unloading is not feasible. To explore the effect of different initial stress levels on the unloading mechanical properties of the rock, the unloading point is set above the crack initiation stress. Consequently, the initial axial stress was established as 50%, 70%, and 90% of the peak stress of the rock under various confining pressures for the unloading tests. According to the in-situ stress results of the underground powerhouse at the Kala Hydropower Station, the powerhouse area is considered to be within a medium to low stress region, with the first principal stress (σ₁) ranging from 5.68 to 11.30 MPa and the third principal stress (σ₃) from 4.24 to 5.50 MPa³³. Thus, the confining pressures for triaxial loading and unloading tests are set at 3 MPa, 6 MPa, 9 MPa, and 12 MPa. The experimental schemes (shown in Fig. 4) are described below:

Fig. 4.

Triaxial loading and unloading path analysis under different initial stress levels.

Scheme I Conventional triaxial compression test. As the base test for comparison with unloading tests, the test consists of two stages: ① Hydrostatic loading stage, using force-confining pressure control mode, applying axial and confining pressures simultaneously at a rate of 0.2 MPa/s up to the designed test value (3 MPa, 6 MPa, 9 MPa, and 12 MPa). ② Axial loading stage, maintaining the confining pressure constant, employing a stress control mode to apply axial stress at a rate of 0.2 MPa/s until the rock sample fails completely.

Scheme II Axial stress loading and lateral stress unloading (LAUL) triaxial test. Using 50%, 70%, and 90% of the peak stress (σf) of sandy slate under different confining pressures from the scheme I as the initial axial stress, the test is conducted in three stages: ① Hydrostatic pressure loading stage, with a stress path consistent with the scheme I. ② Axial loading stage, maintaining the confining pressure unchanged, applying axial stress at a rate of 0.2 MPa/s up to the initial axial stress. ③ Axial loading and confining pressure unloading stage, applying axial load and unloading confining pressure at a rate of 0.2 MPa/s simultaneously until the rock sample fails completely. If the rock sample does not fail after unloading the confining pressure to 0 MPa, then continue applying axial load until failure.

Scheme III Axial and lateral stress unloading (UAUL) triaxial test. The test comprises three stages: ① and ② are consistent with Scheme II. ③ Axial and confining pressure unloading stage, unloading axial pressure at a rate of 0.2 MPa/s and unloading confining pressure at a rate of 0.4 MPa/s. If the rock sample does not fail after the confining pressure is unloaded to 0 MPa, then an axial load is applied at a rate of 0.2 MPa/s until the rock sample fails.

Analysis of the deformation characteristics of sandy slate

Stress–strain curve analysis of sandy slate

The stress–strain curve of the rock can intuitively reflect its mechanical and deformation characteristics, providing a basis for rock mechanics and deformation parameters calculation. Figures 5, 6, and 7 present stress–strain curves for three testing schemes. The volumetric strain (ε_v) is calculated from the axial strain (ε₁) and the circumferential strain (ε₃). It is positive during compression and negative during dilation. The horizontal dashed line in the figures represents the unloading starting point. Figures 5, 6, and 7 show:

1. During loading, the stress–strain curves can be divided into 4 stages: ① Compaction stage, characterized by an upward concave curve of axial stress–strain. ② Linear elastic deformation stage, where axial, circumferential, and volumetric strains nearly linearly increase with stress. ③ Plastic deformation stage, where rates of axial and circumferential strains gradually accelerate, and the curve shows a downward concave shape, with a turning point in the volumetric strain curve marking the transition from compression to expansion. ④ Failure stage, where the stress–strain curve slope turns negative, the rock's load-bearing capacity drops rapidly, and deformation increases swiftly.

2. Under different unloading paths and initial axial stress levels, the characteristics of the stress–strain curves for sandy slate differ. When the initial axial stress is 50%σ_f, except for the LAUL test with an initial confining pressure of 12 MPa, none of the rock samples failed during unloading of confining pressure. Compared to loading, since unloading of the confining pressure in the two stress paths with an initial axial stress of 50%σ_f occurs during the linear elastic deformation stage, the features of the stress–strain curves are essentially consistent in stages ①③④ and before unloading of confining pressure in stage ②. After unloading, there are mainly three characteristic changes in the stress–strain curve: ②₁ unloading linear elastic deformation stage, during which axial strain changes in a pattern consistent with the linear elastic deformation stage of loading, while the circumferential strain rate suddenly increases and then stabilizes. ②₂ unloading plastic deformation stage, during which slopes of axial and circumferential strains gradually increase, and the curve shows a upward concave feature. ②₃ post-unloading linear elastic deformation stage, where upon completion of confining pressure unloading and subsequent increase in axial pressure, rates of axial and circumferential strains decrease relative to phase ②₁ and then stabilize, with phase ②₂ not occurring before this stage. When the initial axial stress is 50%σ_f and no failure occurs during unloading, the stress–strain curves of the rock samples under the initial confining pressures of 3 MPa, 6 MPa, and 9 MPa underwent stages ① → ② → ②₁ → ②₃ → ③ → ④, while the curves in the axial pressure confining pressure unloading test with an initial confining pressure of 12 MPa went through stages ① → ② → ②₁ → ②₂ → ③ → ④.

3. When the initial axial stress is 90%σ_f, the rock samples failed under both unloading paths, since the rocks were already in the plastic deformation stage during unloading under confining pressure. Thus, the features of the stress–strain curves under unloading are largely consistent with those during loading in stages ①②④ and before unloading in stage ③. After unloading, there is a significant increase in the rate of axial and circumferential strains relative to loading, leading more quickly to failure in sequence ① → ② → ③ → ③₁ → ④. When the initial axial stress is 70%σ_f, the characteristics of the stress–strain curves of the rocks that did not fail during unloading of confining pressure were largely in line with when the initial axial stress was 50%σ_f, whereas the curves of the failed samples underwent stages ① → ② → ②₁ → ②₂ → ④.

4. Different stress paths and initial stress levels have a significant impact on the axial, circumferential, and volumetric strains of the rock: ① In terms of axial strain ε1, in the loading test, the failure axial strain increases with confining pressure, exhibiting more ductility, while in the two unloading tests, the failure axial strain value decreases under the same confining pressure, showing more brittleness. The axial strains under UAUL are less than those under LAUL, with lower initial axial stress levels and smaller confining pressures at failure indicating stronger brittleness. ② For the circumferential strain ε₃, during unloading, ε₃ expands outward, and this deformation becomes more pronounced with higher initial axial stress levels. At initial stresses of 50%σ_f and 70%σ_f, the failure circumferential strains under both unloading paths are smaller than those under corresponding loading conditions. While at an initial stress of 90%σ_f, the failure circumferential strain increases, with the strain under axial pressure loading with confining pressure unloading generally greater than that under axial pressure confining pressure unloading. ③ Regarding volumetric strain ε_v, in unfailed samples with initial stresses of 50%σ_f and 70%σ_f, pre-peak dilation is not apparent. However, with increased initial confining pressure and initial axial stress level, samples that failed during unloading of confining pressure under both stress paths showed a marked increase in dilation.

Fig. 5.

Conventional triaxial compression test stress–strain curve.

Fig. 6.

Stress–strain curve of LAUL triaxial test: (a) initial confining pressure 3 MPa, (b) initial confining pressure 6 MPa, (c) initial confining pressure 9 MPa, (d) initial confining pressure 12 MPa.

Fig. 7.

Stress–strain curve of UAUL triaxial test: (a) initial confining pressure 3 MPa, (b) initial confining pressure 6 MPa, (c) initial confining pressure 9 MPa, (d) initial confining pressure 12 MPa.

Analysis of unloading strain variation rules for sandy slate

The deformation of sandy slate fundamentally changes during the process of confining pressure unloading, with different initial stress levels having a significant impact on the sample's unloading deformation. Figure 8 presents the strain variation curves of the samples during the unloading process under different initial stress levels, where the solid and dashed lines represent the strain curves during the unloading process under different initial confining pressures σ₃ for LAUL and UAUL tests, respectively. Figure 8 reveals:

1. Under both unloading paths, axial and circumferential strains increase upon unloading of confining pressure. Under the same initial confining pressure conditions, the increase rate of axial and circumferential strains after unloading at an initial axial stress level of 90%σ_f is significantly faster than that at 50%σ_f and 70%σ_f. This suggests that a high axial stress value at the start of unloading accelerates rock failure. Increasing the initial confining pressure at a constant initial axial stress level significantly accelerates the rate of strain change in the rock sample. This suggests that the sample's deformation is acutely sensitive to unloading effects when subjected to high stress levels.

2. When both the initial confining pressure and the initial axial stress values are high, the strain rate of the rock sample increases gradually during unloading, and the rock sample is prone to failure. The volume strain of the rock samples undergoing unloading failure shifts towards negative values, with the rate of increase in circumferential strain significantly higher than that of the axial strain, indicating that circumferential strain plays a dominant role during the unloading process.

3. Comparing the trends of stress variations under two different unloading paths, it is observed that the strain increment rate and magnitude in LAUL are greater than those in UAUL. Since the circumferential constraint is consistent during confining pressure unloading, the deviatoric stress values formed by axial loading are greater than those formed by axial unloading, resulting in a more intense response to both axial and circumferential deformations. This phenomenon is particularly evident at high initial confining pressures with initial axial stress levels at 50%σ_f and 70%σ_f. The primary reason is that in the initial stages of unloading, the sample is in the elastic deformation stage. As unloading of the high confining pressure proceeds, the rock transitions towards a plastic state. In this process, LAUL can prompt the rock to enter into a plastic state more rapidly, resulting in more significant plastic deformation. When the initial axial stress level is 90%σ_f, the rock is already in a state of plastic deformation, and axial unloading, as opposed to axial loading, slows down the trend of plastic deformation in the sample.

Fig. 8.

Strain variation curves of rock samples during unloading under different initial axial stress levels: (a) Initial axial stress 50%σ_f, (b) initial axial stress 70%σ_f, (c) initial axial stress 90%σ_f.

In order to analyze the influence of confining pressure unloading on deformation under different stress paths and initial axial stress levels, an evaluation is conducted using the strain confining pressure increment ratio. The strain confining pressure increment ratio is the ratio of unloading amount Δσ₃ to strain increment Δε₃ (i = 1, 3, v) in the confining pressure process³⁴. The calculation formula is as follows:

$$\mathrm{\Delta }{\dot{\epsilon }}_i=\frac{\mathrm{\Delta }{\epsilon }_i}{\mathrm{\Delta }{\sigma }_3}\left(i=1,3,v\right)$$ 1

where $\mathrm{\Delta }{\dot{\epsilon }}_1$, $\mathrm{\Delta }{\dot{\epsilon }}_3$ and $\mathrm{\Delta }{\dot{\epsilon }}_v$ represent the strain confining pressure increment ratios of axial, circumferential and volumetric, respectively.

The greater the absolute value of the strain confining pressure increment ratio, the more sensitive is the strain to unloading effects. Table 1 presents the strain confining pressure increment ratios for rock samples under different unloading paths and initial axial stress conditions. It is evident from Table 1 that the order of the strain confining pressure increment ratio under different unloading paths is $\mathrm{\Delta }{\dot{\epsilon }}_1$ < $\mathrm{\Delta }{\dot{\epsilon }}_3$ < $\mathrm{\Delta }{\dot{\epsilon }}_v$, indicating that during the unloading process, the sensitivity of the rock sample's circumferential strain changes is greater than that of the axial strain, with a more pronounced volumetric dilation phenomenon. Under the same unloading stress path, the strain confining pressure increment ratio generally shows an increasing trend with the increase in the initial axial stress level. The radial confining pressure increment ratio of axial compression and confining pressure unloading is greater than that of axial compression unloading, suggesting that this unloading path has the greatest impact on circumferential deformation, which is the most sensitive. The variation law of the strain confining pressure increment ratio of sandy slate under different unloading paths is basically consistent with the unloading strain change law. The primary cause of sample failure under the unloading path is the accelerated circumferential deformation rate induced by unloading, promoting the volumetric dilation of the rock sample. Moreover, the stress path of LAUL, higher initial axial stress levels, and higher initial confining pressures increase the rate of change in circumferential deformation, thereby accelerating the volumetric dilation failure process of the rock sample.

Table 1.

Results of strain confining pressure increment ratio under different unloading paths.

Test scheme	σ3 /MPa	σ1 /MPa	Δσ3 /MPa	Δε1 /10–3	Δε3 /10–3	Δεv /10–3	$\mathrm{\Delta }{\dot{\epsilon }}_1$ /10–3·MPa-1	$\mathrm{\Delta }{\dot{\epsilon }}_3$ /10–3·MPa-1	$\mathrm{\Delta }{\dot{\epsilon }}_v$ /10–3·MPa-1
II	3	50%σf	2.60	0.33	0.27	0.21	0.13	0.10	0.08
		70%σf	2.66	0.45	1.34	2.23	0.17	0.51	0.84
		90%σf	1.66	0.46	2.99	5.52	0.27	1.80	3.32
	6	50%σf	5.68	0.39	0.56	0.73	0.07	0.10	0.13
		70%σf	2.16	0.15	1.30	2.45	0.07	0.60	1.13
		90%σf	1.21	0.28	2.32	4.36	0.23	1.92	3.61
	9	50%σf	8.62	0.69	2.05	3.40	0.08	0.24	0.39
		70%σf	3.73	1.20	2.66	4.11	0.32	0.71	1.10
		90%σf	1.92	0.52	2.73	4.94	0.27	1.42	2.58
	12	50%σf	8.58	0.29	3.04	5.79	0.03	0.35	0.67
		70%σf	4.96	0.60	3.70	6.81	0.12	0.75	1.37
		90%σf	2.46	0.47	1.97	3.46	0.19	0.80	1.40
III	3	50%σf	2.68	0.04	0.38	0.61	0.02	0.12	0.23
		70%σf	2.70	0.13	0.75	1.37	0.05	0.28	0.51
		90%σf	2.05	0.16	1.62	3.07	0.08	0.79	1.50
	6	50%σf	5.68	0.12	0.36	0.61	0.02	0.06	0.11
		70%σf	3.54	0.86	1.33	1.81	0.24	0.38	0.51
		90%σf	1.81	0.28	2.06	3.84	0.15	1.14	2.12
	9	50%σf	8.54	0.21	0.49	0.78	0.02	0.06	0.09
		70%σf	5.20	0.73	2.02	3.31	0.14	0.39	0.64
		90%σf	2.24	0.34	2.18	4.02	0.15	0.97	1.80
	12	50%σf	11.55	0.57	2.23	3.88	0.05	0.19	0.34
		70%σf	6.53	0.31	2.99	5.52	0.05	0.46	0.85
		90%σf	3.10	0.14	2.77	5.40	0.04	0.89	1.75

Deformation parameters change characteristics of sandy slate

The deformation parameters of rock directly reflect its deformation characteristics under different stress states. The commonly used deformation parameters include the modulus of deformation (E) and Poisson's ratio (μ). These parameters are generally obtained through uniaxial compression tests. Poisson's ratio is the ratio of the circumferential strain to the axial strain under stress, denoted as μ = ε₃/ε₁. The modulus of deformation is the ratio of stress to strain in rock, expressed as E = σ₁/ε₁, which, in the elastic stage of the rock, is also known as the elastic modulus. Under triaxial stress conditions, especially during triaxial unloading, there is a pronounced circumferential deformation and bulk expansion. Therefore, calculating the E and μ solely based on uniaxial compression tests is not reasonable. To obtain deformation parameters at each stress state during the loading and unloading process, it is assumed that the rock samples still obey the generalized Hooke's law throughout the triaxial loading and unloading process³⁵.

$$\left\{\begin{array}{c}{\epsilon }_1=\left[{\sigma }_1-\mu \left({\sigma }_2+{\sigma }_3\right)\right]\left./\phantom{{\epsilon }_1=\left[{\sigma }_1-\mu \left({\sigma }_2+{\sigma }_3\right)\right]E}\right)E\\ {\epsilon }_2=\left[{\sigma }_2-\mu \left({\sigma }_1+{\sigma }_3\right)\right]\left./\phantom{{\epsilon }_2=\left[{\sigma }_2-\mu \left({\sigma }_1+{\sigma }_3\right)\right]E}\right)E\\ {\epsilon }_3=\left[{\sigma }_3-\mu \left({\sigma }_1+{\sigma }_2\right)\right]\left./\phantom{{\epsilon }_3=\left[{\sigma }_3-\mu \left({\sigma }_1+{\sigma }_2\right)\right]E}\right)E\\ \end{array}\right)$$ 2

In the triaxial test where σ₂ = σ₃, transformation via Eq. (2) yields the following result:

$$\left\{\begin{array}{c}E=\left({\sigma }_1-2\mu {\sigma }_3\right)\left./\phantom{\left({\sigma }_1-2\mu {\sigma }_3\right){\epsilon }_1}\right){\epsilon }_1\\ \mu =\left(B{\sigma }_1-{\sigma }_3\right)\left./\phantom{\mu =\left(B{\sigma }_1-{\sigma }_3\right)\left[{\sigma }_3\left(2B-1\right)-{\sigma }_1\right]}\right)\left[{\sigma }_3\left(2B-1\right)-{\sigma }_1\right]\\ B={\epsilon }_3\left./\phantom{{\epsilon }_3{\epsilon }_1}\right){\epsilon }_1\\ \end{array}\right)$$ 3

Figure 9 presents the relationship curves between the deformation modulus and deviatoric stress of rock samples under various unloading paths. As depicted in Fig. 9, the following observations can be made:

1. Before unloading the confining pressure, the E of sandy slate decreases sharply with the increase of axial stress and then gradually diminishes to a stable value. The E under different confining pressures remains essentially stable before the axial stress level reaches 50%σ_f. The stable section of the modulus before the unloading of surrounding rock is selected as the elastic modulus of rock under different confining pressures. The elastic modulus, derived from the average values of multiple rock samples at identical confining pressures, increases with the rise in confining pressure. Specifically, the elastic moduli are as follows: 26.67 GPa at 3 MPa, 27.43 GPa at 6 MPa, 27.45 GPa at 9 MPa, and 28.24 GPa at 12 MPa.

2. Throughout the process of unloading the confining pressure along both unloading paths, the E decreases and the magnitude of the decrease becomes significantly greater with the increase in initial confining pressure and initial axial stress level. When unloading at low initial confining pressure and axial stress levels and no rock failure occurs, the E slightly recovers after unloading the confining pressure to 0 MPa. When rock samples fail during unloading at higher initial confining pressures and axial stress levels, the E initially exhibits a linear trend during the early stages of confining pressure unloading. However, as the unloading of the confining pressure continues, it gradually transitions to a nonlinear behavior, with the nonlinearity becoming more pronounced at higher initial axial stress levels.

3. The E for both unloading paths is similar under low initial confining pressure, with initial axial stress levels at 50%σ_f and 90%σ_f. This similarity arises because the unloading point at 50%σ_f is within the elastic range, where the rock sample predominantly undergoes elastic deformation during unloading. In contrast, the unloading point at 90%σ_f is within the plastic range, where the rock sample primarily exhibits plastic deformation during unloading. Consequently, the influence of the two unloading paths on the deformation modulus is relatively minor. When the unloading point is at 70%σ_f, it is positioned within the transitional zone from elasticity to plasticity. LAUL causes the rock sample to enter plastic deformation prematurely, which is manifested by the deformation modulus entering the non-linear range earlier and exhibiting a greater reduction. Based on the above analysis, it can be observed that the impact of the initial confining pressure on the deformation modulus intensifies with increasing pressure. The influence of the unloading path on the E follows the order: LAUL > UAUL. Under different unloading paths, the influence of the initial axial stress level on the E is sequenced as: 70%σ_f > 90%σ_f > 50%σ_f.

Fig. 9.

Variation curves of the deformation modulus of sandy slate under different stress paths: (a) Initial confining pressure 3 MPa, (b) Initial confining pressure 6 MPa, (c) Initial confining pressure 9 MPa, (d) Initial confining pressure 12 MPa.

Figure 10 presents the Poisson’s ratio versus deviatoric stress curves for rock samples under different unloading paths. From Fig. 10 it can be inferred that:

1. The μ generally shows a trend of initially decreasing and then stabilizing with the increase of deviatoric stress. It reaches a stable state before the axial stress level attains 50%σ_f, as indicated by the calculated μ values at confining pressures of 3 MPa, 6 MPa, 9 MPa, and 12 MPa being 0.27, 0.26, 0.27, and 0.26, respectively. This suggests that confining pressure has minimal influence on sandy slate Poisson's ratio.

2. During the unloading process, both unloading paths cause an increase in the Poisson's ratio. The increasing trend and pattern of the μ resemble the decreasing trend of the deformation modulus. The increase in initial confining pressure and axial stress level accentuates the non-linear characteristics of the μ during unloading. In conditions of low initial confining pressure and axial stress levels, the rate of change of μ for undamaged rock samples recovers after the completion of confining pressure unloading. This recovery continues until the rock sample enters the yielding phase, at which point the rate of change of Poisson's ratio rapidly increases. In LAUL test, the rate of change and the overall change in Poisson's ratio during the unloading process are greater than those in UAUL test, suggesting that this stress path has a more pronounced effect on the rock's Poisson's ratio. After the rock enters plasticity during unloading, the μ for some portions of the rock exceeds 0.5 (the limit of deformability for an elastic–plastic material) and continues to rise. This indicates that, in addition to elastic recovery deformation in the direction of unloading, fracture deformation also occurs. At this point, the μ is no longer a characteristic of the material in the conventional sense.

Fig. 10.

Poisson's ratio variation curve of sandy slate under different stress paths: (a) Initial confining pressure 3 MPa, (b) Initial confining pressure 6 MPa, (c) Initial confining pressure 9 MPa, (d) Initial confining pressure 12 MPa.

Evolution equation of deformation parameters

The Poisson’s ratio and deformation modulus are normalized by the corresponding values (μ₀ and E₀) at the beginning of unloading, and the relationship curves between deformation parameters and confining pressure under different unloading stress paths are obtained, as shown in Fig. 11.

Fig. 11.

Deformation parameters variation curve of rock sample during unloading process: (a) Poisson’s ratio (LAUL), (b) Poisson’s ratio (UAUL), (c) deformation modulus (LAUL), (d) deformation modulus (UAUL).

Under the same stress path, the increase rate of Poisson's ratio and the decrease rate of deformation modulus during unloading process increase with the increase of initial axial stress and confining pressure. The unloading causes the Poisson's ratio to increase by up to about 4 times of the initial Poisson’s ratio, and the deformation modulus to decrease by up to about 0.7 times of the initial deformation modulus. The relationship between Poisson’s ratio and deformation modulus with confining pressure during confining pressure unloading can be expressed by exponential function:

$$\frac{\mu }{{\mu }_0}=1+A_{1}exp\left(\frac{-{\sigma }_3}{A_2}\right)$$ 4

$$\frac{E}{E_0}=1-B_{1}exp\left(\frac{-{\sigma }_3}{B_2}\right)$$ 5

In the formula, A₁, A₂, A₃, B₁, B₂ and B₃ are the fitting parameters and have clear physical meaning. 1 + A₁ and 1-B₁ are the Poisson’s ratio and deformation modulus change times after the confining pressure is unloaded to 0 MPa, respectively. A₂ and B₂ control the growth and decay rates of Poisson’s ratio and deformation modulus, respectively.

The relationship between Poisson's ratio and deformation modulus of rock samples during confining pressure unloading is fitted by Eqs. (4) and (5). As shown in Fig. 11, the fitting effect is good, and the fitting parameters are shown in Table 2.

Table 2.

μ and E evolution equation parameter values of sandy slate under different stress paths.

Test scheme	σ3/MPa	σ1/MPa	1 + A1	A2	1-B1	B2
II	3	50%σf	1.20	1.21	0.90	1.80
		70%σf	2.29	0.93	0.93	1.45
		90%σf	12.82	0.61	0.75	0.94
	6	50%σf	1.17	1.60	0.93	2.26
		70%σf	22.01	0.87	-0.19	1.34
		90%σf	740.10	0.66	-32.69	0.74
	9	50%σf	2.99	2.34	0.73	3.80
		70%σf	50.38	1.39	-2.90	1.96
		90%σf	1093.45	1.00	-155.45	0.85
	12	50%σf	10.90	3.02	0.36	5.08
		70%σf	107.62	2.11	-7.03	2.93
		90%σf	91,789.37	1.81	-198.74	1.24
III	3	50%σf	1.49	2.83	0.89	1.71
		70%σf	2.08	1.05	0.91	1.37
		90%σf	3.11	0.74	0.77	1.17
	6	50%σf	1.44	2.67	0.91	2.29
		70%σf	2.13	2.42	-0.24	1.31
		90%σf	47.34	1.02	-5.28	0.99
	9	50%σf	1.62	2.95	0.86	4.51
		70%σf	5.22	2.19	-1.09	2.17
		90%σf	114.11	0.63	-31.58	1.19
	12	50%σf	2.43	1.12	0.72	5.54
		70%σf	36.37	1.91	-3.92	3.37
		90%σf	241.47	0.68	-58.18	1.92

It can be seen from Table 2 that after the confining pressure is unloaded to 0 MPa, 1 + A₁ increases with the increase of initial confining pressure and initial axial stress, and 1 − B₁ decreases with the increase of initial confining pressure and initial axial stress. The absolute values of 1 + A₁ and 1 − B₁ are larger when the initial confining pressure and initial axial stress are unloaded, mainly because the rock is destroyed during the unloading process, and the rate of increase and attenuation before failure is significantly increased. The changes of A₂ and B₂ indicate that the parameters growth and attenuation rate increase with the increase of initial confining pressure and initial axial stress. The increase of Poisson’s ratio, the decrease of deformation modulus, the increase rate of Poisson’s ratio and the decrease rate of deformation modulus in the process of LAUL are larger than those in UAUL. This suggests that LAUL exerts a more significant impact on the strength and deformation characteristics of rock, particularly under conditions of high initial stress.

Mechanical characteristic analysis of sandy slate

Characteristic stress analysis of sandy slate

The deformation and failure process of rock can be divided into five stages³⁶: initial crack closure (I), elastic deformation (II), stable crack expansion (III), crack accelerated expansion (IV), failure and post-peak behavior (V). The beginning points of III, IV, and V correspond to three distinct characteristic stresses: the crack initiation stress (σ_ci), the damage stress (σ_cd), and the peak stress (σ_f), as shown in Fig. 12(a). The σ_ci indicates the beginning of new microfracture appearance within the rock, i.e., the commencement of crack volume increase, corresponding to the inflection point on the crack volume strain curve. The σ_cd represents the initiation stress for sliding-type crack expansion, at which point the volumetric strain shifts from reduction to increase, signifying the beginning of dilatancy in the rock sample, corresponding to the inflection point on the volumetric strain curve. The σ_f represents the axial stress at the point of rock failure.

Fig. 12.

Characteristic strength, volumetric strain and crack strain curves of sandy slate: (a) Scheme I, σ₃ = 9 MPa, (b) Scheme III, σ₃ = 3 MPa, σ₁ = 50%σ_f, (c) Scheme II, σ₃ = 9 MPa, σ₁ = 70%σ_f, (d) Scheme II, σ₃ = 12 MPa, σ₁ = 90%σ_f.

The methods mainly used to determine characteristic stresses include strain measurement, acoustic emission testing, and the crack volume strain method^37–39. Among these, the crack volume strain method can directly determine the stresses based on the volume change characteristics of the rock during the test, which is easier to implement compared to the acoustic emission method and clearer in concept than the strain measurement method. Therefore, it is the chosen method for determining characteristic stresses. The ε_v of a rock is primarily comprised of elastic volumetric strain (${\epsilon }_v^e$) and crack volume strain (${\epsilon }_v^c$). The ${\epsilon }_v^c$ is derived after subtracting the elastic part from the total volumetric strain and is caused by the closure, generation, and expansion of rock pores and fractures under stress⁴⁰.

$${\epsilon }_v={\epsilon }_v^e+{\epsilon }_v^c$$ 6

Based on the continuum mechanics theory of elasticity, elastic strain in different directions can be determined as follows:

$$\begin{array}{c}{\epsilon }_1^e=\frac{1}{E}\left[{\sigma }_1-\mu \left({\sigma }_2+{\sigma }_3\right)\right]\\ {\epsilon }_2^e=\frac{1}{E}\left[{\sigma }_2-\mu \left({\sigma }_1+{\sigma }_3\right)\right]\\ {\epsilon }_3^e=\frac{1}{E}\left[{\sigma }_3-\mu \left({\sigma }_1+{\sigma }_2\right)\right]\\ \end{array}$$ 7

where ${\epsilon }_i^e$ (i = 1, 2, 3) represent the elastic strains in three orthogonal directions of the rock, E is the elastic modulus of the elastic stage, μ is the Poisson’s ratio.

From Eqs. (6) and (7), the expressions of ${\epsilon }_v^e$ and ${\epsilon }_v^c$ of rock under triaxial loading can be determined:

$${\epsilon }_v^e=\frac{1-2\mu }{E}\left[{\sigma }_1+2{\sigma }_3\right]$$ 8

$${\epsilon }_v^c={\epsilon }_v-\frac{1-2\mu }{E}\left[{\sigma }_1+2{\sigma }_3\right]$$ 9

Figure 12 shows the characteristic strength, volume strain and crack strain curves of sandy slate under different stress paths. Table 3 shows the characteristic stress results of sandy slate under different stress paths. From Fig. 12 and Table 3, it can be seen that:

1. The crack initiation stress σ_ci is significantly less than the lowest initial axial stress level of 50%σ_f. Thus, the three testing paths do not affect the σ_ci. The experimental results indicate that the σ_ci of sandy slate increases with the increase in confining pressure. Based on the peak stress σ_f from conventional triaxial compression tests under corresponding confining pressure, the range of σ_ci for sandy slate is determined to be 28.33%σ_f to 38.90%σ_f.

2. As shown in Fig. 12(b), during the unloading process of samples that did not fail, the crack strain increases significantly, and the volume shows clear expansion. After unloading is completed, the crack strain gradually stabilizes, and the volumetric strain re-enters a compression state. This indicates that only fissure initiation and development occur during unloading without interconnection. Hence, the turning point of volumetric strain after unloading should be considered as the criterion for determining damage stress σ_cd. A comparison of the σ_cd results from different stress paths reveals that non-failed samples under both unloading paths have lower σ_cd values. This is attributed to the additional cracks generated during unloading compared to loading, and the release of confining pressure promotes crack development, thus accelerating sample damage and leading to a reduction in the damage stress σ_cd.

3. When the initial confining axial stress is 50%σ_f and 70%σ_f, damaged samples during unloading exhibit conditions as shown in Fig. 12(c), with the crack strain sharply increasing and the volume expanding until failure occurs. In such cases, the σ_cd cannot be accurately determined between the unloading and failure points. However, when the initial confining axial stress is 90%σ_f, as shown in Fig. 12(d), the sample is already in the volumetric expansion phase, so unloading does not affect the σ_cd. Based on the determinable σ_cd results, the range of loading σ_cd is calculated to be 77.51%σ_f to 82.93%σ_f. The σ_cd of non-failed samples during unloading under both paths is 56.72%σ_f to 78.92%σ_f from loading tests at corresponding confining pressure, showing that the σ_cd decreases after unloading.

Table 3.

Statistical results of characteristic stress of sandy slate.

Test scheme	Initial confining pressure σ3/MPa	Initial axial stress σ1/MPa		Crack initiation stress σci/MPa	Damage stress σcd/MPa	Peak stress σf/MPa
I	3	/	33.27	85.16	111.56
	6	/	42.58	120.32	145.58
	9	/	57.48	135.68	159.70
	12	/	65.21	152.42	182.17
II	3	50%σf	55.78	36.41	82.08	93.16
		70%σf	78.09	36.02	78.09	92.62
		90%σf	100.40	33.11	90.93	101.99
	6	50%σf	72.79	45.40	82.57	89.67
		70%σf	101.91	47.79	\	104.03
		90%σf	131.02	44.43	114.84	131.96
	9	50%σf	79.85	55.43	\	91.71
		70%σf	111.79	55.65	\	115.94
		90%σf	143.73	51.09	127.87	145.88
	12	50%σf	91.09	52.05	\	98.09
		70%σf	127.52	67.19	\	132.07
		90%σf	163.96	60.90	148.30	165.51
III	3	50%σf	55.78	35.06	81.71	92.98
		70%σf	78.09	36.79	88.05	92.70
		90%σf	100.40	31.61	87.22	99.15
	6	50%σf	72.79	46.13	82.75	91.66
		70%σf	101.91	47.75	\	99.78
		90%σf	131.02	46.80	112.84	129.68
	9	50%σf	79.85	54.33	\	91.45
		70%σf	111.79	62.13	\	108.46
		90%σf	143.73	59.28	130.82	142.48
	12	50%σf	91.09	68.39	\	90.63
		70%σf	127.52	69.44	\	124.28
		90%σf	163.96	70.68	151.07	161.87

Analysis of strength parameters of sandy slate

The Mohr–Coulomb strength criterion posits that rock's shear strength is the sum of its cohesion and the friction generated by the normal stress on the shear plane. The cohesion (c) and internal friction angle (φ) of rocks can be determined through the stress state at the onset of shear failure, making it the most widely applied strength criterion in rock mechanics. The Mohr–Coulomb criterion, expressed in terms of principal stresses, is given by the following equation.

The Mohr–Coulomb strength criterion posits that rock's shear strength is the sum of its cohesion and friction generated by normal stress on the shear plane, which is the most widely used strength criterion in rock mechanics. The c and φ of the rock can be obtained from the stress state of the rock when shear failure. The Mohr–Coulomb strength criterion, expressed in terms of principal stresses, is expressed as follows⁴¹:

$${\sigma }_1=A{\sigma }_3+B$$ 10

where σ₁ and σ₃ represent the maximum and minimum principal stresses at the failure of the rock mass, respectively, $A=\frac{1+sin\phi }{1-sin\phi }$, $B=\frac{2ccos\phi }{1-sin\phi }$.

Utilizing Eq. (10) for regression analysis of Table 3 data, parameters A and B were obtained. The rock shear strength parameters c and φ can be determined by the following expression:

$$\phi =arcsin\frac{A-1}{A+1}$$ 11

$$c=\frac{B\left(1-sin\phi \right)}{2cos\phi }$$ 12

Based on the triaxial loading and unloading test data of sandy slate presented in Table 3, regression analyses were conducted using Eq. (10), as illustrated in Fig. 13. Subsequently, the shear strength parameters of sandy slate under various stress paths and initial axial stress levels were determined using the calculation methods for shear strength parameters. To study the impact of different initial axial stress levels on the shear strength parameters of rock samples along the same unloading path, experimental data with initial axial stress levels at 70%σ_f and 90%σ_f for both unloading paths were processed as described, resulting in shear strength parameters under different test conditions as shown in Table 4. For the test with an initial axial stress level of 50%σf, the confining pressure was virtually 0 MPa at rock failure, hence the shear strength parameters could not be obtained.

Fig. 13.

Regression analysis of the Mohr–Coulomb strength criterion.

Table 4.

Shear strength parameters of sandy slate.

Shear strength parameters	Scheme I	Scheme II			Scheme III
	Total	Total	70%σf	90%σf	Total	70%σf	90%σf
c/MPa	16.99	15.96	12.00	16.75	15.83	14.31	16.99
φ/°	49.96	49.65	52.53	50.34	50.63	50.90	50.19

As seen from Table 4, compared to the loading condition, the c of sandy slate decreases while the φ increases under unloading conditions, more notably during the LAUL test. This observation is closely related to rocks' deformation and failure characteristics during the loading and unloading processes. Rocks primarily fail in compression-shear during loading and tension-shear during unloading. Local tensile fractures that occur before rocks reach their peak strength can cause the destruction of rock particles and cementation, resulting in a reduction in cohesion on the shear plane. Tension-shear fracture surfaces are generally rougher than compression-shear ones, which is why the internal friction angle tends to increase during unloading.

Compared to loading tests, the decrease in c and the increase in φ are more significant at an initial axial stress level of 70%σ_f under both unloading paths, while at an initial axial stress level of 90%σ_f, the changes in shear strength parameters are minimal. This suggests that unloading the surcharge at 70%σ_f leads to the extensive development of tensile fractures within the rock, significantly altering the c and φ values. However, when unloading at 90%σ_f, the rock is primarily dominated by fully formed shear planes, and the additional tensile fractures from unloading the confining pressure have scarcely any effect on the failure of the rock along these shear planes. When the initial axial stress level is at 70%σ_f, compared with the triaxial compression test, LAUL and UAUL test lead to a reduction in c by 29.37% and 15.77% respectively, and an increase in φ by 5.14% and 1.88% respectively, indicating that the unloading approach of LAUL test has a more substantial influence on the shear strength parameters of the rock sample.

Study on failure characteristics and mechanism of sandy slate

Analysis of failure characteristics of sandy slate

Figure 14 presents sandy slate failure characteristics under various stress paths and initial conditions. Rock samples exhibit the following failure modes under three different stress paths: ① Shear failure. The samples have one or two distinct oblique shear planes, which are narrow and exhibit good matching. ② Tensile splitting failure. Multiple vertical tensile cracks are present in rock samples, without conspicuous shear surfaces. ③ Combined shear-tension failure. Distinct shear planes are identifiable in the samples, with numerous irregularly sized tensile cracks developed along and adjacent to these shear planes, while the shear surfaces are undulating. ④ Combined tension-shear failure. With the sample containing numerous lengthy or through-going tensile cracks and pronounced shear planes, which are either tensile or only visible at the ends of the sample. Comparative analysis of rock samples' failure characteristics under different stress paths and initial conditions reveals that:

1. In the loading tests, the rock samples failed with narrow fractures that exhibited good conformity. These fractures were predominantly governed by the development and coalescence of compressive shear cracks, leading to a shear-dominated failure mode. At lower confining pressures, minor tensile fractures were present at the ends of the samples. With the increase in confining pressure, these tensile fractures gradually disappeared, and the angle between the shear plane and the direction of stress application increased progressively.

2. Tensile fractures of varying sizes were observed in rock samples at the time of failure under two different unloading paths, with the number and length of these fractures being closely associated with the initial stress conditions. When the initial axial stress level is at 50%σ_f, the rock samples primarily exhibit tensile splitting failure due to the confining pressure being nearly 0 MPa at the moment of failure. At higher confining pressures, a combined tensile-shear failure characteristic appears. At an initial axial stress level of 70%σ_f, the rock samples exhibited both tensile cracks and shear planes. The shear failure feature became more apparent with increased initial confining pressure, indicating a gradual transition from tensile-shear to shear-tensile combined failure. At an initial axial stress level of 90%σ_f, the predominant failure mode of the rock samples was shear-tensile combined failure with significant shear planes visible, akin to conventional triaxial compression tests. The angle of fracture increased with the confining pressure, but under the unloading path, there were notably more tensile fractures.

3. The rock samples that failed during the unloading process exhibited a combination of tensile fracture surfaces, shear fracture surfaces, and micro-tensile cracks of varying sizes. The tensile nature of shear fracture surfaces progressively diminished with increased confining pressure. Compared to conventional triaxial compression tests, the failure modes and characteristics of the rock samples under the unloading path were more complex, mainly reflected in two aspects: (i) the pronounced development of tension cracks of different scales and mechanical mechanisms, and (ii) the presence of shear fracture surfaces cutting through rock bridges along tension cracks, as well as continuous development of tensile fracture surfaces along shear fracture surfaces.

Fig. 14.

Failure characteristics of sandy slate.

Failure process and mechanism analysis of sandy slate

The stress–strain curves, deformation characteristics, shear strength parameters, and failure features of sandy slate under various initial stresses and stress paths are investigated to generalize the fracturing process and mechanism, as illustrated in Fig. 15. The stress–strain curve is divided into four stages based on three characteristic stress points (σ_ci, σ_cd, σ_f), which mark the development of microcracks from different perspectives: the closed microcrack stage (A), the microcrack initiation and development stage (B), the microcrack coalescence stage (C), and the macro-fracture formation stage (D). The analysis of the crack development process and mechanism under different stress paths for each stage is as follows:

Fig. 15.

Schematic representation of the failure process and mechanism of sandy slate.

Closed microcrack stage (A) Axial stress does not exceed rock crack initiation strength σ_ci. During this phase, the inherent microcracks within the rock are compressed and closed, with no new micro-fracturing occurring in the rock particles or cementing agents.

Microcrack initiation and development stage (B) Axial stress exceeds the σ_ci, causing the formation of microcracks within the rock. With increasing axial stress, these microcracks gradually develop. When constant confining pressure is applied and axial stress reaches 50%σ_f, the confining pressure's lateral constraint causes the development of a few shear cracks within the rock. If the confining pressure is maintained and axial stress continues to be applied, these shear cracks will gradually develop. Upon the commencement of confining pressure unloading, the rate of shear crack development slows while tensile cracks begin to form. As the confining pressure is progressively reduced to 0 MPa, shear cracks stop developing, and vertical tensile cracks expand. Eventually, with increasing axial stress, the tensile cracks coalesce, leading to a clear spalling failure of the rock sample. When constant confining pressure is applied and axial stress reaches 70%σ_f, the shear cracks within the rock have developed to a certain extent but are not yet interconnected. During the initial stages of unloading, the internal shear cracks continue to develop, while tensile cracks also start to form at the ends of the shear cracks and in other areas. As the confining pressure is gradually reduced, two scenarios may occur: one is where tensile cracks that develop at the ends of shear cracks progressively interconnect with other tensile cracks, leading to a tensile-shear composite failure of the rock sample. The other is where rock bridges between tensile cracks are sheared off, forming a shear-tensile interconnected fracture zone resulting in shear-tensile composite damage. Thus, compared to conventional triaxial compression tests, rocks' failure surfaces under unloading conditions are more complex, with greater roughness, undulation, and waviness, and the rock exhibits lower cohesion and higher friction angles.

Microcrack coalescence stage (C) Axial stress surpasses the damage stress σ_cd, internal shear cracks within the rock become apparent and progressively interconnect. If unloading occurs at this point, tensile cracks will form within the rock. However, the rock still primarily manifests shear crack coalescence during the initial stages of unloading. Therefore, the rock's failure mode is characterized by clear shear failure, alongside the formation of certain tensile cracks near the shear fracture surfaces. Consequently, during unloading tests where the initial axial stress level is 90%σ_f, the rock's mechanical properties, deformation characteristics, strength parameters, deformation parameters, and failure modes are similar to those obtained from conventional triaxial compression tests.

Macro-fracture formation stage (D) When loading exceeds the failure strength, the rock's load-bearing capacity starts to decline, and microcracks evolve into macro-cracks, gradually forming a macro-fracture surface. At this stage, rock particles that obstruct the sliding of the shear plane on the failure surface are sheared off or worn away, resulting in a reduction in both cohesion and friction strength.

Conclusion

To investigate the mechanical behavior of rock surrounding underground caverns during excavation unloading process, a systematic analysis was conducted on the relationship between stress path and rock stability. Mechanical characteristic experiments on sandy slate were carried out, with a focus on the influence of different initial stress levels and stress paths on the mechanical and deformation characteristics, strength and deformation parameters, and failure features of sandy slate. The main conclusions are as follows:

1. The stress path and the initial stress state of unloading both have significant impacts on rock deformation. Unloading confining pressure leads to an increased rate of axial and circumferential strain, causing noticeable volumetric dilation and making the rock sample more prone to deformation and failure. During unloading, the deformation modulus (E) and Poisson's ratio (μ) exhibit a marked diminishing and increasing trend, respectively. The influence order of unloading path on deformation parameters is LAUL > UAUL, with the initial axial stress influence order being 70%σ_f  > 90%σ_f  > 50%σ_f. This indicates that unloading near the plastic yield point exerts a greater influence on the deformation parameters of the rock sample.

2. The crack initiation stress (σ_ci) of sandy slate ranges from 28.33 to 38.90%σ_f, with damage stress (σ_cd) under loading being 77.51 to 82.93%σ_f, and σ_cd under unloading falling between 56.72%σ_f and 78.92 %σ_f. This suggests unloading induces earlier volumetric expansion in sandy slate, consequently diminishing its mechanical performance. The shear strength parameters c and φ of sandy slate under different stress paths are closely related to the initial axial stress level. When the initial axial stress is at 70%σ_f, it significantly influences the shear strength parameters, while at 90%σ_f this influence is less pronounced. Compared to the loading test, when the initial axial stress is at 70%σ_f and subjected to LAUL triaxial test, the value of c decreases by 29.37%, and φ increases by 5.14%. In the case of UAUL test, the c value decreases by 15.77%, and φ increases by 1.88%. These changes indicate that unloading has a greater impact on the shear strength parameters, especially under the stress path of LAUL.

3. In the loading tests, the rock samples exhibit distinct shear failure surfaces post-failure. Under the unloading path, the sample failure surfaces tend to show heterogeneously sized tensile cracks with surfaces characterized by considerable roughness, variability, and undulation. The mode of failure under the unloading path is closely related to the initial axial stress level. At the 50%σ_f, the samples mainly demonstrate tensile splitting failure. At the 70%σ_f, the primary failure mode is a combination of tensile and shear. At the 90%σ_f, the failure predominantly involves a composite of shear and tensile mechanisms. The failure mode correlates with the initiation, expansion, and coalescence of internal fractures under different stress paths.

Author contributions

Tianzhu Huang: carried out experiments, collated and analyzed data, and wrote papers; Xiaoliang Xu: supervised experiments, paper writing guidance, revised and polished manuscripts; Lehua Wang and Jianlin Li: put forward ideas, theoretical analysis, financial support; Jianwen Xu: Participated in the experiment and assisted in sorting out data and pictures. All authors agreed to the format and presentation of the final manuscript.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.