Energy characteristics of saturated Jurassic sandstone in western China under different stress paths-深地科学

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Energy characteristics of saturated Jurassic sandstone in western China under different stress paths

Abstract

To study the energy evolution and failure characteristics of saturated sandstone under unloading conditions, rock unloading tests under different stress paths were conducted. The energy evolution mechanism of the unloading failure of saturated sandstone was systematically explored from the perspectives of the stress path, the initial confining pressure, and the energy conversion rate. The results show that (1) before the peak stress, the elastic energy increases with an increase in deviatoric stress, while the dissipated energy slowly increases first. After the peak stress, the elastic energy decreases with the decrease of deviatoric stress, and the dissipated energy suddenly increases. The energy release intensity during rock failure is positively correlated with the axial stress. (2) When the initial confining pressure is below a certain threshold, the stress path is the main factor influencing the total energy difference. When the axial stress remains constant and the confining pressure is unloading, the total energy is more sensitive to changes in the confining pressure. When the axial stress remains constant, the compressive deformation ability of the rock cannot be significantly improved by the increase in the initial confining pressure. The initial confining pressure is positively correlated with the rock's energy storage limit. (3) The initial confining pressure increases the energy conversion rate of the rock; the initial confining pressure is positively correlated with the energy conversion rate; and the energy conversion rate has a high confining pressure effect. The increase in the axial stress has a much greater impact on the elastic energy than the confining pressure. (4) When the deviatoric stress is small, the confining pressure mainly plays a protective role. Compared with the case of triaxial compression paths, the rock damage is more severe under unloading paths, and compared with the case of constant axial stress, the rock damage is more severe under increasing axial stress.

Highlights

The effects of confining pressure and stress paths on rock failure characteristics and energy evolution were studied.
The initial confining pressure is positively correlated with the rock energy storage limit and energy conversion rate.
The increase in axial pressure has a much greater impact on the elastic energy of rocks than the confining pressure.
Compared with the case of conventional triaxial compression paths, the damage to rocks under unloading paths is more severe.

1 INTRODUCTION

In recent years, with an increasing demand for coal resources, coal mining in China has gradually shifted to the western region. Most of the coal mines in the western region pass through the Jurassic and Cretaceous strata, which are typically featured with weak cementation and poor rock structure stability. With the continuous increase in mining depth, it is imperative to study the safety and stability of underground engineering such as mines and tunnels (Bagde et al., 2005; Cheng et al., 2019; Deng et al., 2022; Feng et al., 2020; Labiouse et al., 2014; Lin, 2022; Munoz et al., 2017). In practical on-site engineering, due to the complexity of the rock stress and high difficulty of detection, most academic efforts have focused on the stress–strain curves, mechanical properties, and failure characteristics of the rocks under different conditions through indoor experiments (Duan et al., 2021; Feng et al., 2019; Gu et al., 2023; Munoz et al., 2016; Taheri et al., 2017; Zhou et al., 2019). However, the stress–strain curve of the rock cannot fully describe the different energy evolution laws and failure forms during the rock failure process. Moreover, the deformation and failure of rocks can be more clearly described from the perspective of energy (Geranmayeh Vaneghi et al., 2020; Peng et al., 2015; Wasantha et al., 2014).

Some scholars have studied the energy evolution characteristics of rocks through triaxial tests. The process of rock failure from an energy perspective has been analyzed, and various energy forms during the rock failure process have been listed and classified (Arzúa & Alejano, 2013; Arzúa et al., 2014; Liu et al., 2016; Mahanta et al., 2017; Zhang et al., 2019). With more in-depth research, scholars have gradually shifted from the conventional triaxial compression tests to triaxial unloading tests. Some scholars have conducted different studies on triaxial unloading tests on rocks and achieved a series of results (Jiang et al., 2018; Qin et al., 2020; Yin et al., 2019), while others have conducted triaxial compression unloading tests under different conditions. The mechanical behavior and energy changes of rocks during unloading have been analyzed, with the relationship between energy conversion and rock mass failure determined and that between the dissipated energy and damage variables established. The strength and failure characteristics of rocks vary significantly under different loading rates, and some scholars have conducted uniaxial loading and unloading tests under different loading rates. The energy evolution mechanism during the rock failure process under different loading rates has been analyzed, and the deformation and failure characteristics of rocks have been explored (Mahanta et al., 2018; Meng et al., 2016; Si & Gong, 2020; Walton, Gaines, 2023; Zhang et al., 2023; Zhao et al., 2020). The unloading rate is also one of the factors affecting the energy evolution of rocks. Different unloading rates may render different failure characteristics of rocks, leading to different energy evolution trends (Yang et al., 2017). Many scholars have also conducted a series of studies on marble (Feng et al., 2020; Li et al., 2014; Qiu et al., 2014). They analyzed the change in energy with strain during the failure process, explored the influence of the strain rate on the characteristic stress, deformation characteristics, and strain energy conversion of marble, and determined the relationship between the volume expansion and elasticity parameters with unloading damage. Under saturated conditions, the mechanical properties of rocks undergo significant changes in terms of mechanical properties, energy density parameters, and energy evolution mechanisms. Some scholars have conducted research on this issue and explored the energy evolution mechanism of saturated rocks (Fu et al., 2023; J. Li et al., 2020; T. Li et al., 2015; Luo et al., 2023; Yılmaz, 2006; Zhao et al., 2023).

In summary, efforts have been made to explore the energy evolution mechanism of sandstone failure from perspectives such as the stress path, unloading rate, and unloading level. However, further research is needed to establish the relationship between energy evolution, deviatoric stress, and crack development during the unloading failure of sandstone.

In view of this, in this study, triaxial unloading tests were conducted on saturated sandstone under different stress paths. The energy evolution mechanism of sandstone under different stress paths was analyzed using the energy method. Specifically, the energy evolution mechanism of the unloading failure of saturated sandstone under different stress paths was studied in terms of the stress path, initial confining pressure, energy conversion rate, and rock damage characteristics to reveal the unloading failure mechanism of western saturated sandstone from the perspective of energy evolution. The results provide a theoretical reference for the construction of vertical shafts and tunnels in coal mines of western China.

2 MATERIALS AND METHODS

2.1 Sample preparation

The Jurassic sandstone used in the tests of this study was collected from the Balasu Coal Mine in Yulin City, Shanxi Province, China. The sampling depth was 443.6–461.4 m. According to a geological report from the mine shaft inspection, the strata are mainly composed of mudstone, sandy mudstone, and sandstone, which are typical Jurassic strata. The sandstone core, sealed with cling film, was immediately transported to the laboratory to be processed into standard samples.

All the rock samples were processed into standard cylinders with a diameter of 50 mm and a length of 100 mm, according to the requirements of the Standard for Tests Methods of Engineering Rock Masses (GB/T50266-2013). To prevent the rock sample from being subjected to eccentric compression during the testing process, both ends of the sample were polished after processing to ensure that the surface flatness was within ±0.05 mm.

The samples with obvious cracks on the surface were not used, and longitudinal wave velocity testing was conducted on the processed standard samples. The longitudinal wave velocity range of the sandstone was between 2250 and 2800 m/s. The samples were numbered based on the longitudinal wave velocity test results (Figure 1).

Details are in the caption following the image — Figure 1
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Test material and the tested longitudinal wave velocity. (a) Processed samples. (b) Longitudinal wave velocity value of the samples.

2.2 Experiment scheme

A ZTCR-2000 rock triaxial testing system was used in the experiments. This instrument can be used to conduct uniaxial and triaxial compression, triaxial unloading, and other tests on rocks at low and normal temperatures. The maximum axial pressure is 2000 kN, and the maximum confining pressure is 40 MPa. Figure 2 presents a schematic diagram of the experimental instrument.

Before the excavation of the roadway section, the rock mass is in the original stress state. As the excavation depth increases, the maximum principal stress $𝜎_{1}$ near the tunnel surface often increases, and the confining pressure at greater depths remains basically unchanged. Due to different stress field environments, there may be one unloading or simultaneous unloading of the intermediate and minimum principal stresses. According to the analysis, the following three stress paths were selected: conventional triaxial compression (Scheme I), constant axial pressure and unloading of the confining pressure (Scheme II), and unloading of the confining pressure and axial loading (Scheme III).

To simulate the stress state at different depths, five different confining pressures were selected: 0, 5, 10, 15, and 20 MPa. To ensure the accuracy of the test data, two rock samples were tested at each stress level. Specifically, (1) first, the hydrostatic pressure was loaded to the predetermined value at a rate of 0.05 MPa/s. (2) Using the load control method, the confining pressure was held constant and the axial pressure was increased to the unloading critical point (This value is 80% of the peak strength of the rock under triaxial loading under the same confining pressure.) at a rate of 0.05 MPa/s. (3) The test was continued at a rate of 0.05 MPa/s according to the different test schemes until the rock sample was destroyed (Figure 3).

3 TEST RESULTS AND ANALYSIS

In the triaxial unloading test, the rock is still in a triaxial stress state. The difference from the triaxial compression test is the unloading of the confining pressure, so the method of calculating the strain energy is the same as in the triaxial compression test. For the triaxial test in this study, the axial stress and confining pressure were loaded simultaneously during the hydrostatic loading process. The test instrument performed positive work on the rock. After the hydrostatic loading was completed, the axial stress was continuously increased to perform positive work on the rock; the hoop strain was continuously expanded; and the confining pressure performed negative work on the rock. According to previous research on energy, the total energy

𝑈

of rocks during failure consists of two parts, elastic energy

𝑈_{e}

and dissipative energy

𝑈_{d}

, which can be expressed as follows (Zhang et al., 2019):

𝑈 = \int 𝜎_{1} d 𝜀_{1} + 2 \int 𝜎_{3} d 𝜀_{3} = 𝑈_{e} + 𝑈_{d} = 𝑈_{1} + 𝑈_{3},

(1)

where

𝜎_{𝑖}

denotes the principal stress;

𝜀_{𝑖}

is the principal strain corresponding to the principal stress;

𝑈_{1}

represents the energy absorbed by the rock during compression deformation; and

𝑈_{3}

is the energy consumed during the process of hoop strain expansion.

Equation ( 1) can be used to divide the region into countless small strip regions based on the definite integral limit mechanism (Figure 4) (Yang et al., 2017). Accordingly, Equation ( 1) is transformed as follows:

𝑈_{1} = \sum_{𝑖 = 1}^{𝑛} \frac{1}{2} (𝜎_{1}^{𝑖} + 𝜎_{1}^{𝑖 + 1}) (𝜀_{1}^{𝑖 + 1} - 𝜀_{1}^{𝑖}),

(2)

𝑈_{3} = \sum_{𝑖 = 1}^{𝑛} (𝜎_{3}^{𝑖} + 𝜎_{3}^{𝑖 + 1}) (𝜀_{3}^{𝑖 + 1} - 𝜀_{3}^{𝑖}) .

(3)

The elastic energy

𝑈_{e}

per unit volume can be calculated using the following equation:

𝑈_{e} = \frac{1}{2 𝐸} [𝜎_{1}^{2} + 2 𝜎_{3}^{2} - 2 𝜇 𝜎_{3} (2 𝜎_{1} + 𝜎_{3})],

(4)

where

𝜎_{1}^{𝑖}

𝜎_{3}^{𝑖}

𝜀_{1}^{𝑖}

, and

𝜀_{3}^{𝑖}

denote the axial stress, confining pressure, axial strain, and hoop strain of the rock at each point during the deformation process, respectively, and

𝐸

and

𝜇

represent the deformation modulus and Poisson's ratio of the rock, respectively.

3.1 Energy evolution characteristics of rock under different stress paths

According to the test results, the energy change curve of the western sandstone under the unloading stress path was obtained. As can be seen, the energy evolution curve trend of the rock is basically the same under different unloading paths (Figure 5). The energy values are presented in Table 1.

Table 1. Unloading energy values.

Scheme	Initial confining pressure (MPa)	Initial value (kJ/m3)			Destruction point energy value (kJ/m3)			Residual value (kJ/m3)
Scheme	Initial confining pressure (MPa)	U	Ue	Ud	U	Ue	Ud	U	Ue	Ud
I	5	7.36	4.60	2.76	100.74	65.61	35.13	185.77	19.04	166.32
	10	13.84	8.21	5.63	136.50	90.49	46.01	306.10	16.99	289.11
	15	15.28	10.56	4.72	225.64	87.60	140.04	435.35	42.27	393.08
	20	17.67	13.12	4.55	262.33	120.97	141.36	616.23	29.96	586.27
II	5	8.19	5.35	2.84	35.86	22.89	12.97	57.49	6.94	50.55
	10	11.66	7.33	4.33	42.07	28.20	13.87	112.04	11.09	100.95
	15	16.07	11.14	4.93	48.60	31.80	16.80	102.67	5.13	97.54
	20	18.07	13.29	4.78	121.80	35.20	92.96	245.90	9.40	236.50
III	5	8.01	5.34	2.67	45.06	35.63	9.43	72.65	8.80	63.85
	10	11.98	7.38	4.60	68.90	55.89	13.01	121.23	13.07	108.16
	15	13.98	10.77	3.21	108.90	61.40	47.50	172.90	8.40	164.50
	20	17.62	13.33	4.29	137.90	84.54	53.36	380.80	16.84	363.96

In this test scheme, the rock suddenly fails under the controlled load. To prevent the displacement sensor from being crushed, thus damaging the test data, the test was ended at the stage where the stress rapidly dropped after the rock failed. Therefore, there is no continuous residual strength stage of the rock in the stress–strain curve obtained from the test.

As can be seen in Figure 5, the elastic energy initially increases and then decreases, which is consistent with the trend of the stress–strain curve. The dissipated energy gradually increases with increasing strain, and the total energy gradually increases. To analyze the energy evolution during deformation and failure, it is divided into three stages:

The first stage is the process of axial stress being loaded to the unloading critical point. In this stage, the stress state of the rock is the same under different stress paths: the total energy $𝑈$ and elastic energy $𝑈_{e}$ of the rock increase with increasing axial strain, and the dissipated energy $𝑈_{d}$ remains basically unchanged. The elastic energy accounts for the main proportion of the total energy, while the dissipated energy accounts for a small proportion, indicating that the rock mainly undergoes elastic deformation at this stage, because as the stress increases, the rock mainly produces axial compression, and the amount of hoop strain expansion is very small. There is basically no crack propagation inside the rock, resulting in most of the energy absorbed by the rock being stored in the form of elastic energy. The energy gradually accumulates and dissipates slightly.

The second stage is the process from unloading to failure of the rock. The stress states of the rock along the different stress paths during this stage are different. As the axial strain gradually increases, the elastic energy initially increases linearly at a faster rate, and the growth rate decreases when the rock is about to yield and fail. During this stage, the elastic deformation of the rock is still dominant, and most of the energy is still stored as elastic energy. Although the dissipated energy increases slightly, the increase extent is small. Because the deviatoric stress on the rock continues to increase during this process, the rock sample continues to be compressed, but the confining pressure still has a binding influence on it, and the volume strain increases slowly.

The third stage is the process of rock failure, in which the dissipated energy of the rock increases suddenly. This is because when the rock is about to fail, the internal structure of the rock changes, and numerous microcracks rapidly expand and aggregate. At the same time, pore water pressure enters the interior along the cracks in the form of tension, acting on both sides and at the tips of the cracks, promoting further development of the cracks and increasing the surface energy. This leads to the formation of penetrating cracks along a certain direction and the sudden release of a large amount of the elastic energy absorbed within the rock. There is a sudden increase in the dissipative energy and a sudden decrease in the elastic energy.

The intensity of the energy release during rock failure varies under different stress paths. The energy release is the most intense under path $d 𝜎_{3} < 0, d 𝜎_{1} > 0$ , slightly gentler under path $d 𝜎_{3} = 0, d 𝜎_{1} > 0$ , and the gentlest under path $d 𝜎_{3} < 0, d 𝜎_{1} = 0$ . The comparison shows that the intensity of the rock energy release is closely related to axial stress, and there is a positive correlation between the axial stress and the intensity of the energy release. The reason for this phenomenon is that the application of axial stress exacerbates the generation of internal cracks in the rock. Before the rock fails, penetrating cracks will not form through crack aggregation. However, once the rock fails, microcracks quickly aggregate to form through cracks, and the internal energy is instantly released. The reduction of confining pressure will exacerbate this process. This indicates that increasing the axial stress under unloading conditions and reducing the confining pressure under conventional triaxial conditions will increase the degree of rock failure and energy release. In engineering construction, such situations should be avoided so as to prevent the occurrence of engineering accidents.

3.2 Influence of initial confining pressure on the energy evolution of rocks

Figure 6 presents the relationship between energy at the failure point and the initial confining pressure under different paths. Overall, as the initial confining pressure increases, the total, elastic, and dissipated energies of the rock will increase. Regarding the energy values of the three paths, the energy value of path $d 𝜎_{3} = 0, d 𝜎_{1} > 0$ is the largest, followed by paths $d 𝜎_{3} < 0, d 𝜎_{1} > 0$ , and $d 𝜎_{3} < 0, d 𝜎_{1} = 0$ .

Regarding the total energy, the difference between the total energy at the rock failure point in the conventional path and the unloading path remains basically unchanged when the initial confining pressure is low. As the initial confining pressure increases, the difference gradually increases. This indicates that when the initial confining pressure is below a certain threshold, the difference between the total energy of the rock in the conventional triaxial path and the unloading path is less affected by the confining pressure, but it is greatly affected by the stress path. Notably, the difference in the total energy of the rock under the two unloading confining pressure paths initially increases and then decreases. This indicates that when the initial confining pressure is below a certain threshold, the difference in the total energy of the rock under the unloading path is gradually increased by the confining pressure. Once the confining pressure exceeds the threshold, its influence gradually decreases. In addition, the sensitivity of the total energy to the confining pressure varies under different stress paths. The total energy increases by 160% from 5 to 20 MPa under path $d 𝜎_{3} = 0, d 𝜎_{1} > 0$ , by 239.7% under path $d 𝜎_{3} < 0, d 𝜎_{1} = 0$ , and by 206.1% under path $d 𝜎_{3} < 0, d 𝜎_{1} > 0$ . The total energy of the rock is more sensitive to the increase in the confining pressure under path $d 𝜎_{3} < 0, d 𝜎_{1} = 0$ .

As shown in Figure 6, the dissipative energy $𝑈_{d}$ at the rock failure point increases with increasing initial confining pressure. When the initial confining pressure is low, its influence on the increase in the rock dissipation energy is not significant. When the confining pressure increases to a certain value, the dissipation energy suddenly increases. This indicates that there is a sudden change point in the impact of the initial confining pressure on the rock dissipation energy. When the confining pressure does not reach this sudden change point, the dissipation energy released during rock failure is less, and the degree of rock failure is also relatively small. Once the confining pressure reaches the sudden change point, more dissipated energy is released during the rock failure process, and the degree of rock failure is also more severe. A possible reason for this phenomenon is that as the initial confining pressure continues to increase, the strength of the rock also increases, and the number of internal cracks generated during rock failure increases, leading to a sharp increase in the dissipated energy.

The elastic energy of rock $𝑈_{e}$ increases with increasing initial confining pressure. However, under paths $d 𝜎_{3} = 0, d 𝜎_{1} > 0$ , and $d 𝜎_{3} < 0, d 𝜎_{1} > 0$ , the elastic energy of the rock increases significantly with increasing confining pressure, while under path $d 𝜎_{3} < 0, d 𝜎_{1} = 0$ , the elastic energy does not increase significantly with increasing confining pressure. This indicates that increasing the confining pressure under constant axial stress does not significantly improve the ability of the rock to deform and store elastic energy. In addition, under paths $d 𝜎_{3} = 0, d 𝜎_{1} > 0$ , and $d 𝜎_{3} < 0, d 𝜎_{1} > 0$ , elastic energy shows synchronous increasing trends, and the energy difference remains basically unchanged. This indicates that the decrease in confining pressure under the condition of increasing axial stress does not change the trend of the elastic energy of the rock. The impact of the confining pressure on the elastic deformation of the rock is consistent under both paths.

From the initial energy accumulation to the later energy release, the elastic strain energy has a maximum value, that is, the energy accumulated at the peak point, known as the energy storage limit $𝑈_{\max}^{e}$ (Zhao et al., 2020). Figure 7 shows the relationship between the energy storage limit of the rock and the initial confining pressure. As the confining pressure increases, the energy storage limit of the rock increases as well, indicating that the increase in the confining pressure improves the internal energy storage capacity of the rock. The initial confining pressure is positively correlated with the energy storage limit of the rocks.

3.3 Conversion rate of rock strain energy under different stress paths

The conversion rate of the strain energy is related to the degree of rock damage to a certain extent. Based on the energy curve, the strain energy conversion rate

𝑣

at different stages can be obtained. The strain energy conversion rate is the ratio of the strain energy increment

𝛥 𝑈

at this stage to the corresponding time

𝛥 𝑡

(Dai et al., 2016)

𝑣 = \frac{𝛥 𝑈}{𝛥 𝑡} .

(5)

The stress state of the rock in the first stage is the same under the different stress paths, so Figures 8 and 9 only show the variation curves of the strain energy conversion rate with the initial confining pressure in the second and third stages.

As shown in Figure 8, in stage II, the conversion rate of the total energy $𝑈$ , elastic energy $𝑈_{e}$ , and dissipated energy $𝑈_{d}$ of the rock increases with increasing initial confining pressure. The initial confining pressure increases the conversion rate of the internal energy of the rock, and the initial confining pressure is positively correlated with the energy conversion rate. When the confining pressure remains at a low level, the difference in the energy conversion rate for paths $d 𝜎_{3} < 0, d 𝜎_{1} = 0$ , and $d 𝜎_{3} < 0, d 𝜎_{1} > 0$ is not significant, and the increase in the energy conversion rate is relatively small compared with that of path $d 𝜎_{3} = 0, d 𝜎_{1} > 0$ . This indicates that in the prepeak stage, a low confining pressure has a lower impact on the energy conversion rate of the rock along the unloading path and a greater impact along the conventional path. This is because in the unloading test, the confining pressure on the rock gradually decreases, and the current initial confining pressure is quickly removed. However, the confining pressure of the rock remains unchanged along the conventional path, and the initial confining pressure has a greater impact on it.

Regarding the conversion rate of $𝑈_{e}$ , path $d 𝜎_{3} < 0, d 𝜎_{1} = 0$ is different from the other two paths. As the initial confining pressure increases, the increase in the energy conversion rate is slow. Based on the previous analysis, the total increase in $𝑈_{e}$ is also low. This indicates that for path $d 𝜎_{3} < 0, d 𝜎_{1} = 0$ , the initial confining pressure has little effect on the elastic energy increase. Comparing the other two stress paths, it can be inferred that the impact of the axial stress on the elastic energy is much greater than that of the confining pressure.

Under a high confining pressure, the energy conversion rate of the rock shows a sudden increase, and the confining pressure at the sudden increase point varies among the different stress paths. Overall, the sudden increase only occurs under a high confining pressure. This indicates that the increase in the rock energy conversion rate has a high confining effect, and it is sensitive to high confining pressures. Compared with path $d 𝜎_{3} < 0, d 𝜎_{1} = 0$ , under the other two paths, the confining pressure is slightly lower when the conversion rate of the rock $𝑈_{d}$ suddenly increases. This indicates that under the same confining pressure, the increase in axial stress accelerates the conversion rate of the rock $𝑈_{d}$ . This may be because the increase in the axial stress leads to an increase in the rate of crack generation and propagation inside the rock, the process of converting the elastic energy into dissipative energy is accelerated, and the conversion rate of the elastic energy into dissipated energy is accelerated by the increase in axial stress.

As can be seen from Figure 9, the trend of the energy conversion rate with the initial confining pressure in stage III is basically the same as that in stage II, but there is a significant difference in the magnitude of the values. The energy conversion rate increases significantly after failure. In comparison, it can be seen that regarding the magnitude of the postpeak energy conversion rate, that of path $d 𝜎_{3} = 0, d 𝜎_{1} > 0$ is the largest, while that of path $d 𝜎_{3} < 0, d 𝜎_{1} = 0$ is the smallest. For path $d 𝜎_{3} = 0, d 𝜎_{1} > 0$ , the energy conversion rate after failure is very high, but the protection of the confining pressure under this path reduces the danger of rapid energy release after failure. For path $d 𝜎_{3} < 0, d 𝜎_{1} > 0$ , the confining pressure has been removed during failure, and the rapid energy release without the protection of confining pressure can lead to rock burst. Although the conversion rate of $𝑈_{d}$ under path $d 𝜎_{3} < 0, d 𝜎_{1} = 0$ increases slightly, its harm is much lower than that under path $d 𝜎_{3} < 0, d 𝜎_{1} > 0$ . Therefore, the degree of rock failure is greatly improved by increasing the axial stress during the unloading of confining pressure. This situation should be avoided during engineering construction.

4 DISCUSSION

Along the three stress paths, rocks generally show shear failure characteristics, but their shear crack morphology and fracture angle are related to the experimental conditions. For $d 𝜎_{3} = 0, d 𝜎_{1} > 0$ , rock failure is normal shear failure; for $d 𝜎_{3} < 0, d 𝜎_{1} = 0$ , although some tensile cracks are formed along the direction of the maximum principal stress during the unloading of the rock, shear failure still dominates from the fracture point; and for $d 𝜎_{3} < 0, d 𝜎_{1} > 0$ , after the rock is damaged, some fragments will fall off the surface, and there are abundant rock powder and fragmented rock blocks between the cracks.

Next, rock failure is analyzed from the perspective of the damage variable. When the material is not damaged, the dissipative energy is 0. When the material is severely damaged, the dissipative energy is infinite. No matter how much energy is consumed, the stress will not increase. At this point, the value of the damage variable is 1. Therefore, the damage variable can be defined as follows based on the relationship between the stress change process and the dissipative energy (Dai et al., 2016):

𝐷 = \frac{2}{π} arctan \frac{𝛥 𝑈_{d}}{𝛥 (𝜎_{1} - 𝜎_{3})},

(6)

where

𝛥 𝑈_{d}

denotes the increment of the dissipative energy and

𝛥 (𝜎_{1} - 𝜎_{3})

is the increment of the deviatoric stress.

Figure 10 presents the damage evolution curves of the rock under different stress paths. Based on these curves, the overall trends of the damage evolution of the rock under the three different stress paths can be described. In the initial stage of stress loading, the slope of the rock damage variable curve is relatively flat, and that of $d 𝜎_{3} = 0, d 𝜎_{1} > 0$ is the gentlest. This indicates that when the deviatoric stress is low, the increase in the rock damage is not significant. This is because the confining pressure mainly serves a protective function when the deviatoric stress is low, thereby reducing the increase in rock damage. As the deviatoric stress increases, the damage evolution of the rock gradually intensifies. When the deviatoric stress increases to the point where the rock is about to yield, the rock damage increases linearly and its bearing capacity sharply decreases, which is followed by rock failure.

Under the same deviatoric stress, the damage value of the rock is greater under the unloading path than the conventional triaxial path. The damage value of the rock is greater under increasing axial stress during the unloading process than under constant axial stress. This indicates that compared with conventional loading paths, the damage to rocks is more severe under unloading paths, and compared with constant axial stress, the damage to rocks is more severe under increasing axial stress conditions. This is because the decrease in the confining pressure intensifies the lateral volume expansion of the rock, and the cracks inside the rock begin to increase. On this basis, further increasing the axial stress leads to faster crack expansion and greater damage. This is precisely consistent with the failure patterns of the rock under different stress paths.

5 CONCLUSIONS

In this study, the unloading failure and the energy evolution mechanism of the saturated sandstone under different stress paths were analyzed. Based on the experimental and analytical results, the main conclusions are as follows.

1.
The energy evolution trends of the rock are basically the same under different unloading paths. The elastic energy initially increases and then decreases, and the dissipative energy initially increases slowly and then suddenly. In stage I, the rock mainly undergoes elastic deformation, and most of the energy absorbed by the rock is stored in the form of elastic energy.
2.
The intensity of the energy release during rock failure varies under the different stress paths. That of path $d 𝜎_{3} < 0, d 𝜎_{1} > 0$ is the most intense, and that of path $d 𝜎_{3} < 0, d 𝜎_{1} = 0$ is the smoothest. The intensity of the energy release from the rock is positively correlated with the axial stress.
3.
As the initial confining pressure increases, the total, elastic, and dissipative energies of the rock increase. When the initial confining pressure is below a certain threshold, the stress path is the main factor affecting the difference in the total rock energy. The sensitivity of the total energy of the rock to the confining pressure varies under the different stress paths. The total energy of the rock under path $d 𝜎_{3} < 0, d 𝜎_{1} = 0$ is more sensitive to the increase in the confining pressure.
4.
Increasing the initial confining pressure under a constant axial stress does not significantly improve the storage of elastic energy and the compressive deformation capacity of the rocks. Under increasing axial stress, reducing the confining pressure has little effect on the change trend of the elastic energy of the rock, and the initial confining pressure is positively correlated with the energy storage limit.
5.
There is a positive correlation between the initial confining pressure and the energy conversion rate. The increase in the energy conversion rate of the rock has a high confining pressure effect. The increase in the axial stress has a much greater impact on the elastic energy of the rock than the confining pressure, accelerating the conversion rate of the elastic energy into dissipative energy.
6.
When the deviatoric stress is small, the confining pressure mainly plays a protective role, reducing the increase in the rock damage. Compared with conventional loading paths, the rock damage is more severe under unloading paths, and compared with constant axial stress, the rock damage is more severe under increasing axial stress conditions.

ACKNOWLEDGMENTS

This project was supported by the National Natural Science Foundations of China (Nos. 52304073 and 52004003), the Opening Foundation of Anhui Province Key Laboratory of Building Structure and Underground Engineering (KLBSUE-2022-04), and the Anhui Natural Science Foundation Youth Program (2208085QE142).

CONFLICT OF INTEREST STATEMENT

The authors declare no conflict of interest.

Biographies

Dr. Jian Lin is a lecturer and master's supervisor at the School of Civil Engineering at Anhui Jianzhu University. He mainly engages in research and teaching on the mechanical properties of soft rock, the design theory and construction technology of coal mine shaft lining, and other aspects. He has presided over one National Natural Science Foundation project and one Anhui Provincial Natural Science Foundation project, published more than 20 academic papers, and authorized more than 20 patents.
Dr. Yun Wu is a lecturer at China University of Mining and Technology. He graduated in geological engineering with a PhD degree from Nanjing University. His main research interests cover nuclear waste disposal, underground energy storage and deep geothermal exploitation. Until now, He hosted National Natural Science Foundation of China and related research projects, has published more than 30 peer-reviewed papers.