Experimental Research on Anisotropy Characteristics of Shale under Triaxial Incremental Cyclic Loading and Unloading

Abstract

Shale is a common rock type that is associated with underground engineering projects, and several important factors, such as bedding structure, confining pressure, and the loading and unloading path, significantly influence the anisotropy of shale. Triaxial monotonic loading tests and triaxial incremental cyclic loading and unloading tests of shale under three kinds of confining pressures and five types of bedding inclination angles (θ) were thus performed to investigate the anisotropy of shale in terms of mechanical behavior, acoustic emission (AE), and energy evolution, and reveal the mechanism by which shale anisotropy is weakened. The results show that (1) the compressive strength and elastic modulus of shale decrease and then increase as the θ increases, and that both σ₃ and incremental cyclic loading and unloading reduce the anisotropy in terms of the compressive strength and elastic modulus of shale, with the ratio of plastic strain to total strain reaching its maximum at a θ of 60° during each loading and unloading cycle. (2) The failure modes of shale with θ of 0°, 30°, and 90° under triaxial monotonic loading are similar to the counterparts under triaxial incremental cyclic loading and unloading, while the failure modes of shale with θ of 45° and 60° differ significantly under the two loading conditions, and interestingly, the degree to which the bedding plane participates in shale crack evolution under incremental cyclic loading and unloading is considerably lower than that under triaxial monotonic loading. (3) The cumulative AE count and AE b-value of shale first decrease and then increase as the θ increases, while the Felicity ratio decreases as the number of cycles increases. (4) As the θ increases, the total energy density U₀ and the parameter m, which reflects the accumulation rate of elastic energy, first decrease and then increase, with both reaching a minimum at a θ of 60°. (5) The mode by which cyclic loading and unloading leads to failure in shale with a θ of 60° is similar to that at a θ of 0° and is the main mechanism by which shale anisotropy weakening occurs as a result of cyclic loading and unloading. The results provide experimental support and a theoretical basis for safer and more efficient underground engineering projects that involve shale.

1. Introduction

Shale has a unique mineral arrangement and bedding structure owing to diagenetic processes such as clay compaction, dehydration, and recrystallization, resulting in significant anisotropy [1,2,3]. Its wide distribution in southwestern China means that it commonly surrounds underground engineering projects such as mines and tunnels [4,5]. Mining processes, such as ore blasting, pillar mining, and backfilling, as well as repeated hydraulic fracturing, reservoir impounding, and dewatering in shale gas mining, all subject the shale to incremental cyclic loading and unloading stages [6,7,8]. Therefore, understanding shale anisotropy under triaxial incremental cyclic loading and unloading conditions has important theoretical and engineering implications.

Many scholars have conducted in-depth research into the mechanical properties or anisotropy of shale under uniaxial, triaxial monotonic loading, and uniaxial cyclic loading and unloading conditions, with significant progress made. Wang et al. [9] examined the Longmaxi Formation shale, Zheng et al. [10] studied organic rich shales, Jia et al. [11] investigated the Hunan shale, Niandou et al. [12] studied the Tournemire shale, and Cho et al. [13] looked into the Boryeong shale to obtain a better understanding of the effects of uniaxial and triaxial monotonic loading, particularly with regard to the mechanical anisotropic properties of shale. With the increase in bedding inclination angles, the compressive strength tends to decrease first and then increase again. Results indicated that the lowest compressive strength is at a bedding angle of around 30° and that the peripheral pressure has a significant effect on the anisotropic characteristics of shale, revealing its damage morphology and mechanism. Chen et al. [14] studied the acoustic emission (AE) activity characteristics of shale under uniaxial compression and found that as the bedding inclination angle increased from 0° to 90°, the proportion of shear events in the shale decreased, while the proportion of tensile events increased. Hu et al. [15] studied the anisotropy of energy evolution in shale that was subjected to triaxial compression and found that the total energy, elastic energy, and dissipated energy were all minimized when the angle between the bedding plane and the maximum principal stress surface was 60°. In studying uniaxial cyclic loading and unloading, Kong et al. [16] found that the maximum irreversible deformation of shale occurred when the angle between the bedding plane and the maximum principal stress surface was 45°, and that the compressive strength of shales with different bedding angles decreased to varying degrees under cyclic loading and unloading conditions. Xie et al. [17] found that the bedding inclination angle significantly affected the mechanism and energy release rate of fractures in shale under uniaxial cyclic loading and unloading. Wei et al. [18] found that the failure mode of cracked shale was generally mixed under triaxial cyclic loading, and that the compressive strength of the shale increased after multiple cycles. Yin et al. [19] studied the effects of triaxial cyclic loading and unloading on the compressive strength of shale. They found that when the bedding plane dominates the shale failure, the effect of cyclic loading and unloading on the failure mode is weakened. Jiang et al. [20] studied the fracture characteristics of shale under pore water pressure and triaxial cyclic loading and unloading and found that shear failure was the main failure mode in shale, with axial cyclic loading and unloading leading to the development of complex secondary fractures. Jiang et al. [21] studied the deformation characteristics of shale under triaxial constant-amplitude cyclic loading and found that the deformation modulus and Poisson’s ratio fluctuated violently near failure. Li et al. [22] conducted triaxial cyclic loading and unloading tests on shale under different confining pressures and found that the confining pressure has little effect on the distribution ratio of pre-peak energy parameters but has a certain enhancing effect on the post-peak energy accumulation ability of shale.

In summary, there is currently no consensus on the effects that cyclic loading and unloading have on the compressive strength, elastic modulus, and failure mode of shale at different bedding angles. The anisotropic characteristics and weakening mechanisms of shale under triaxial incremental cyclic loading and unloading remain unclear, and in-depth research is still lacking. Therefore, in this study, triaxial monotonic loading and triaxial incremental tests were performed under three confining pressures (σ₃) and bedding inclination angles (θ) to analyze the anisotropic characteristics of the mechanical behavior, AE activity, and energy evolution in shale, allowing the mechanism by which shale anisotropy is weakened to be revealed. The results provide experimental support and a theoretical basis for safer and more efficient construction in shale underground engineering.

2. Experimental Materials and Methods

2.1. Sample Preparation

Shale was obtained from the roof of a deep Mn orebody in the Datangpo Formation, which lies in the Lower Nanhua Formation, eastern Guizhou. Samples were collected at a depth of approximately 1200 m, where the bedding is well developed. The results of nuclear magnetic resonance and X-ray diffraction tests showed an average porosity of 0.38%, density of 2.57 g/cm³, and a mineral composition comprising quartz (37.03%), mica (34.16%), albite (21.54%), chlorite (4.98%), and pyrite (2.29%). There is a natural water content of 0.46 ± 0.02%. Due to the minimal variation in water content of each sample, the impact of differences in water content on the test results can be ignored.

Take large rock samples on site and drill shale specimens with different θ, as shown in Figure 1a,b. The angle between the bedding plane and the maximum principal stress surface (horizontal plane) is denoted as θ, as shown in Figure 1c. Tests were therefore conducted on shales with θ of 0°, 30°, 45°, 60°, and 90°. For testing, rock cores were processed into standard cylinders that were 100 mm in height and 50 mm in diameter. The parallelism of the upper and lower surfaces was within 0.05 mm and the surface flatness within 0.02 mm, thus meeting the testing standards of the International Society for Rock Mechanics (ISRM). Uniaxial compression testing showed average uniaxial compressive strengths of 141.64, 98.24, 90.99, 85.24, and 125 MPa at 0°, 30°, 45°, 60°, and 90°, respectively.

2.2. Experimental Procedure

Mechanical tests were performed using a TAJW-2000 rock triaxial testing system form Changchun, China, with a maximum axial force of 2000 kN and maximum confining pressure of 80 MPa. Axial and radial displacement were measured using displacement sensors. The real-time development of internal cracks was monitored using a DS5 AE system from Beijing Softland Times company, Beijing, China, a sampling frequency of 1.5 MHz, a maximum signal amplitude of 100 dB, and a dynamic range > 40 dB. Figure 2 is a schematic of the experimental setup used.

The successful implementation of triaxial incremental cyclic loading and unloading tests is dependent on the compressive strength (σ_p) of the shale under triaxial monotonic loading. To determine this, a triaxial monotonic loading test was conducted using five θ and three σ₃ (10, 20, 30 MPa), with four pre-peak unloading points at 0.2σ_p. 0.4σ_p. 0.6σ_P, and 0.8σ_P. Triaxial incremental cyclic loading and unloading tests were then performed under the same conditions. When the specimen begins to be loaded formally, AE monitoring is performed simultaneously. The main steps used were as follows:

(1) Shale samples were sealed and fixed in place within a triaxial pressure chamber under an axial load of 2 kN. Good contact between the sample and the pressure-measuring element was ensured.

(2) Confining pressure was applied to the designed value at a rate of 0.05 MPa/s using the force control mode. The obtained confining pressure was then maintained for the entire test period.

(3) The axial pressure was increased to 0.2σ_p using the displacement control mode (with a loading rate of 0.05 mm/min). The axial pressure was then unloaded to 5 MPa at the same rate to complete the first loading and unloading cycle, followed by sequential unloading points to 0.4σ_p, 0.6σ_P, and 0.8σ_P for the second, third, and fourth cycles.

(4) The axial pressure continued until the specimen failed.

Because of the high constant confining pressure that was applied during the experiment, rock samples were subjected to relatively small circumferential displacement, resulting in lower circumferential strain energy. However, as stated in [23], changes in the circumferential energy of a rock sample can be disregarded. Figure 3 displays the strain and energy indicators for shale during a single loading and unloading cycle, including the total strain (ԑ^tot), elastic strain (ԑ^e), and plastic strain (ԑ^irr). The total energy density absorbed by the shale (external input energy and work performed by the testing machine) is denoted as U₀, the elastic energy density accumulated inside the shale is denoted as U_e, and the dissipated energy density due to the compaction of pores or microcracks and the connection and expansion of microcracks is denoted as U_d. U₀ and U_e represent the areas under the loading and unloading curves for each loading and unloading cycle, respectively, and U_d is the difference between U₀ and U_e.

3. Basic Mechanical Behaviors of Anisotropic Shale under Incremental Triaxial Cyclic Loading and Unloading

3.1. Cyclic Stress–Strain Curve

The triaxial incremental cyclic loading and unloading stress–strain curves of shale under the three σ₃ and five θ conditions are shown in Figure 4. The typical characteristics of the curve are shown in Figure 4a. The stress–strain curve during the first loading is in the form of a non-linear upward concave line, whereas the subsequent loading and unloading stages are approximately linear. The hysteresis loop area generated by loading and unloading gradually increases as the number of cycles increases, and its arrangement transitions from dense to sparse, indicating that the internal damage of shale is gradually increasing. The rapid decrease in stress observed when the compressive strength is reached is a typical brittle failure characteristic. The results also show anisotropy in the hysteresis loop widths for shales with different θ that are subjected to the same axial stress, with the widths of the hysteresis loops at bedding orientations of 0° and 90° being relatively small compared to those obtained under bedding orientations of 30°, 45°, and 60°.

3.2. Anisotropy of Compressive Strength

The variation curves obtained for shale compressive strength at different θ under triaxial monotonic loading (ML) and triaxial incremental cyclic loading and unloading (CL) are shown in Figure 5. As θ increases, the compressive strength under two loading conditions tends to decrease and then increase again, with the maximum and minimum compressive strengths under the two loading conditions observed at 0° and 60°, respectively. Compared to triaxial monotonic loading, the average compressive strength of shale with θ of 0° and 90° decreased by 8.81% and 3.11%, while that with θ of 45° and 60° increased by 7.36% and 11.84%, respectively, under triaxial incremental cyclic loading and unloading. A slight increase of 0.5% was observed in the average compressive strength for shale with a θ of 30°. The degree of anisotropy in the compressive strength R_c [24] is defined using:

$$R_C=\frac{{\sigma }_{c(0)}}{{\sigma }_{c(\min )}},$$

(1)

where σ_c(0) and σ_c(min) represent the compressive strength of shale with a 0° bedding inclination angle and the minimum compressive strength of shale with different bedding inclination angles under the same test conditions, respectively.

The anisotropy R_c in the shale compressive strength under the three types of confining pressures is shown in Figure 6. The results show, under triaxial monotonic loading, the R_c decreases from 1.62 to 1.33 as the confining pressure increases from 10 to 30 MPa, which is a decrease of 17.9%. However, the R_c initially increases followed by a decrease as the confining pressure increases from 10 to 30 MPa under triaxial incremental cyclic loading and unloading, with a difference of 8.5% between the maximum and minimum values. Compared with monotonic loading, the R_c decreased by 29.69%, 11.13%, and 11.59% at confining pressures of 10, 20, and 30 MPa, respectively, under triaxial incremental cyclic loading and unloading, suggesting weakened anisotropy in the shale compressive strength under triaxial incremental cyclic loading and unloading conditions.

3.3. Anisotropy in the Elastic Modulus

The linear segment of the stress–strain curve in each triaxial incremental loading and unloading test was used to calculate the elastic modulus during each loading and unloading cycle. The formula used is as follows:

$$E=\frac{\Delta \sigma }{\Delta \epsilon },$$

(2)

where E is the elastic modulus, Δσ is the stress difference of the linear segment, and Δε represents the difference in strain that corresponds to Δσ.

The elastic modulus of shale with the same θ under the three different pressures followed a similar trend. Taking σ₃ = 10 MPa as an example, the change curves describing the elastic modulus of shale under cyclic loading and unloading with different θ are shown in Figure 7. The results indicate that during loading and unloading, the elastic modulus of shale reaches a maximum value when the θ is 0° and a minimum value when the θ is 60°. In addition, the unloading modulus of elasticity exceeded the loading modulus of elasticity in each cycle. A gradual increase was observed in the loading elastic modulus over each successive cycle, leading to a decrease in the difference between loaded and unloaded elastic moduli; however, the rate of increase for the loading elastic modulus deceased over the same period.

The degree of anisotropy, R_E, in the elastic modulus during loading and unloading is defined using the following:

$$R_E=\frac{E_{(0)}}{E_{(\min )}},$$

(3)

where E₍₀₎ represents the elastic modulus of shale with a 0° bedding inclination and E_(min) denotes the minimum elastic modulus of shale with different bedding inclinations at the same stage under the same testing conditions.

The variation curves obtained for R_E during cyclic loading and unloading under the three tested confining pressures are shown in Figure 8. The results show that the difference in R_E was more significant during the first loading stage than the other loading stages. Additionally, the R_E decreased as the confining pressure increased, with reductions of 38.79, 33.96, and 29.05% for confining pressures of 10, 20, and 30 MPa, respectively, observed during the fifth loading stage as compared to the first. The results thus indicate that the elastic modulus anisotropy of shale is weakened as the number of cycles of triaxial incremental cyclic loading and unloading increases; however, weakening as a result of confining pressure was observed only during the first loading and unloading stage.

3.4. Anisotropy of Deformation

The change rule of plastic strain per cycle for shale with the same laminar inclination under different confining pressures was similar. Taking σ₃ = 10 MPa as an example, the ratio of irrecoverable strain ԑ^irr to total strain ԑ^tot under four loading and unloading cycles is shown in Figure 9. The ԑ^irr/ԑ^tot increases and then decreases as the θ increases, with a significant increase at 45° to 60° and the maximum at 60°. The ratio of ԑ^irr/ԑ^tot decreases as cycling continues, with gradual slowing observed in the rate of decrease. The shale with a θ of 60° exhibited the highest value in each unloading cycle, with ԑ^irr/ԑ^tot decreasing and a gradual decrease in the magnitude observed as loading and loading continued. The shale with a θ of 60° exhibited an average ԑ^irr/ԑ^tot value of 62.28% after each loading and unloading cycle, while the other shales showed average ԑ^irr/ԑ^tot values of 53.07 to 56.16%. This result suggests that shale with a laminated dip angle of 60° is more prone to plastic deformation.

3.5. Anisotropy in the Failure Pattern

Shale failure is closely related to its matrix and bedding structures, confining pressure level, and loading and unloading stress paths. The failure patterns of shale under triaxial monotonic loading and triaxial incremental cyclic loading and unloading are shown in Figure 10 and Figure 11, respectively.

Few fracture surfaces penetrated the shale under triaxial monotonic loading, and all show significant anisotropy. For 0° shale, failure occurs via tensile failure through the bedding planes, mixed tensile–shear failure through the bedding planes, and shear failure through the bedding planes resulting from two intersecting shear rupture surfaces at confining pressures of 10, 20, and 30 MPa, respectively. At 30°, shale failure occurs via tensile failure, mixed tensile shear failure, and shear failure through the bedding planes at confining pressures of 10, 20, and 30 MPa, respectively. At 45°, shale failure occurs via shear slip failure along the bedding planes, shear protrusion failure along the bedding planes, and shear failure through the bedding planes at confining pressures of 10, 20, and 30 MPa are, respectively. At 60°, shale failure occurs via shear slip failure along the bedding plane, with a shear crack along the bedding plane and a secondary crack extending from the main crack at all confining pressures. At 90°, failure occurs via tensile failure along the bedding plane, with the axial stress loading direction parallel to the direction of the bedding planes at 10 MPa and 20 MPa. In these cases, the increasing axial stress leads to an increase in the circumferential displacement, resulting in lateral expansion stress and the formation of tensile cracks along the bedding plane. However, at 30 MPa, failure occurs via mixed tensile–shear failure that is composed of tensile failure along the laminar surface and shear failure through the bedding plane.

The results differed considerably under triaxial incremental cyclic loading and unloading, with large numbers of fracture surfaces penetrating the shale and strong anisotropy in the failure patterns. Failure was similar to that observed under triaxial monotonic loading at θ = 0° and 90° for all confining pressures. However, these results differed for different θ, with 30° leading to tensile–shear mixed failure that penetrated the bedding plane at all confining pressures; 45° resulting in Y-shaped shear failure through the bedding planes, two shear cracks through the bedding planes, and tensile–shear mixed failure composed of multiple tensile and shear cracks through the bedding planes at 10, 20, and 30 MPa, respectively; and 60° leading to shear failure through the bedding planes at all confining pressures, with the number of shear cracks increasing as the confining pressure increased.

Comparing the two loading conditions, the failure patterns of shale are similar when θ is 0°, 30°, and 90°, while a significant difference is observed in the failure patterns of shale when θ is 45° and 60°, manifested by a lower degree of involvement of the bedding planes in shale crack evolution under incremental cyclic loading and unloading conditions. In addition, the number of macroscopic fracture surfaces and secondary cracks that were easily and partially connected along the bedding planes at the upper end of the shale increased significantly, and the degree of fragmentation was more severe under triaxial incremental cyclic loading and unloading conditions than it was under triaxial monotonic loading.

It is worth noting that the degree to which the bedding planes are involved in crack evolution is highest during shale failure at a θ of 60°, resulting in lower compressive strength and elastic modulus. This indicates that the bedding planes of shale are weak mechanical surfaces and are the root of the anisotropy observed in the shale mechanical behavior.

4. Evolution of the Shale Characteristics of AE

In the process of loading and unloading, the cracks inside the rock continuously form, develop, expand, and penetrate, finally leading to damage, which is accompanied by the release of AE signals [25,26]. Similar AE characteristics were observed in the confining pressures under the same θ, and in this section, we analyze the evolution of the shale characteristics of AE under the five different θ, using σ₃ = 10 MPa as an example.

4.1. Evolution Characteristics and Anisotropy of AE Ring Counts

The changes in the AE ring count, cumulative ring count, dynamic b-value, and differential stress over time during triaxial incremental cyclic loading and unloading are shown in Figure 12. Overall, similar evolution characteristics were observed for the AE ring count and cumulative ring counts during triaxial incremental cyclic loading and unloading under five θ values.

The results show that the AE signal was already active during the first loading and unloading cycle, and the cumulative ringing count rapidly increases. The AE signal was also active near the unloading points during the second, third, and fourth cycles, with the cumulative ring count increasing steadily. The fifth loading cycle, which is associated with an increase in axial stress, also shows an active AE signal, with the cumulative ring count increasing steeply near the point of damage.

The total sum of ring count of AE varies with θ, as shown in Figure 13, decreasing and then increasing as θ increases and maximum and minimum values at 0° and 60°, respectively; however, the total sum of ring count for θ of 0° and 30° is much higher than that observed under the other θ conditions. Based on the shale failure mode in Section 3.5, it is apparent that the shear failure mechanism is stronger when θ is 60° as compared to the other θ conditions. The AE signal released during shear failure is less than that released during tensile failure, which may be the reason for the minimum c total sum of ring count in shale with θ = 60°.

4.2. Felicity Effect

The phenomenon in which a large number of AE signals are generated before the previous maximum stress is reached is known as the Felicity effect [27]. The Felicity ratio R_F, which occurs during repeated loading and unloading, is defined using the following:

$$R_F(\mathrm{i})={\sigma }_{AE}-\sigma (\mathrm{i}-1),$$

(4)

where R_F(i) is the Felicity ratio of the ith cycle, σ_AE is the stress level when the AE activity starts in the ith loading process, and σ(i − 1) is the highest stress level suffered during the previous period. R_F can be used to reflect the damage state within a rock, with smaller R_F values indicating more serious damage.

The variation in the shale R_F under different numbers of load and unloading cycles and five θ is shown in Figure 14. The significant differences observed for different θ indicates that the damage evolution is anisotropic. During the second loading cycle, a large number of AE signals were generated only after the previous unloading stress level was exceeded, indicating a significant Kaiser effect. However, a large number of AE signals were generated before the previous maximum stress was reached in the third and fourth loading cycles, indicating the Felicity effect. The gradual decrease in the total R_F indicates that the degree of damage inside the shale gradually increases with the number of loading–unloading cycles and axial stress. The R_F of shales with different θ varied significantly during the second and third loading cycles and was closer during the fourth and fifth loading cycles, indicating that the number of cyclic loading and unloading cycles weakens the anisotropy of shale.

4.3. Characteristics of AE b-Values

The b-value that is associated with AE originates from seismology studies. Gutenberg and Richter [28] first proposed the well-known G–R relationship between earthquake frequency N (AE event count was identified by the system according to specific acoustic emission characteristics) and magnitude M, with the similarity between the distribution characteristics of AE events during rock compression and the mechanisms of earthquake evolution resulting in the following equation:

$$\left.\begin{array}{l}\lg N=a-bM\\ M=A_{dB}/20\end{array}\right\},$$

(5)

where a and b are constants and A_dB is the maximum amplitude of the AE event. The AE b-value is not only a statistically analyzed parameter but is also closely related to the damage pattern of a rock. For shale, the further a primary crack penetrates, the less fracture energy is released per unit crack area, resulting in a larger AE b-value; however, if shear dominates, more energy is dissipated by shear friction, resulting in lower fracture energy release and a larger AE b-value. Similarly, the more secondary cracks, the more energy is dissipated into the shale, resulting in larger AE b-values.

To ensure statistical accuracy, an amplitude interval ΔM of 5 dB was used, and the b-value was determined by fitting all AE events during the shale failure process, as shown in Figure 15a. The variation curve for the b-value with the θ is shown in Figure 15b; the results indicate that with the increase in θ, the b-value of shale decreases from 0° to 45° and increases from 45° to 90°. According to the failure mode, a θ of 0° results in two main cracks running through the shale, resulting in a larger b-value, while all other θ values result in only one main crack. Although a tensile crack is observed in shale with a θ of 90°, the associated b-value is higher than it is for the other θ conditions. This may be due to the fact that considerably less fracture energy is released when failure occurs along rather than through the bedding plane. In addition, the range of b-values for the shale specimens with different θ values is 0.6~0.8, which indicates a higher proportion of small amplitude AE events during the failure of shale samples, which mainly consist of small-scale cracks.

To further analyze the crack evolution characteristics of shale during incremental cyclic loading and unloading, the dynamic b-value variation curve over time is shown in Figure 12. A window length of 10% and step size of 1% were used to calculate the total number of AE events. The dynamic b-value of shale corresponds to the ring count response; thus, the b-value is observed to fluctuate. The significant decrease followed by an increase near the first four unloading points remains stable during unloading and reloading, indicating that the large cracks are mostly generated around the unloading points of each cycle. Finally, a through-crack is generated at the point of destruction, with the corresponding dynamic b-value decreasing sharply to its lowest value. Compared with the cases in which the θ is 0° and 90°, the time window during which the dynamic b-values remain stable before failure is shorter when θ = 30°, 45°, or 60°. These results indicate that the distribution of large- and small-scale microfractures within the sample is more balanced at a θ of 0° and 90°.

5. Anisotropy of Shale Energy Evolution

Energy, which is the core driving force associated with rock failure [29], is a universal and reliable means of evaluating rock deformation and failure processes.

5.1. Anisotropy in the Total Energy Density U₀

A similar variation pattern was observed for U₀ with axial strain in shale with the same θ under the three different confining pressures. Taking σ₃ = 10 MPa as an example, the relationship curve between U₀ and axial strain under each loading and unloading cycle of shale with five different θ is shown in Figure 16a. Overall, U₀ shows non-linear growth with axial strain, and the growth rate during the third and fourth loading and unloading cycles, as well as the fifth reloading cycle, is significantly higher than it is during the first two cycles, indicating that the ability of the shale to resist axial deformation increases as the axial strain increases.

The variation curve of U₀ with θ during cyclic loading and unloading is shown in Figure 16b. The U₀ of shale with the same θ increases with the confining pressure, indicating that increasing the confining pressure improves the energy storage limit of the sample. At the same confining pressure, U₀ shows an overall trend of first decreasing and then increasing as the θ increases, with the minimum value at 60° indicating that the least input energy is required for failure at a θ of 60°.

5.2. Anisotropy of the Energy Conversion

The energy absorbed by a loaded rock specimen in a closed experimental system is mainly converted into elastic and dissipated energy, with the energy conversion directly affecting the damage to the rock specimen. The ratio of U_d to U₀ is defined as the dissipated energy ratio and the ratio of U_e to U₀ as the elastic energy ratio. Taking σ₃ = 10 MPa as an example, the curves for the elastic energy and dissipated energy ratios of shale with five different θ values as a function of axial strain are shown in Figure 17. The evolution curves of the elastic energy ratio and dissipated energy ratio can be divided into four stages: stage I, in which the elastic energy ratio is smaller than the dissipation energy ratio; stage II, during which the elastic energy ratio increases rapidly and the dissipation energy ratio decreases rapidly; stage III, in which the rates at which the elastic energy increases and the dissipation energy decreases slows down; and stage IV, when the elastic energy ratio starts to decrease and the dissipation energy ratio to increase. These four stages correspond to the micropore compaction, elastic and stable crack development, and accelerated crack propagation stages.

When θ = 0°, 30°, and 90°, the compressive strength is lower than that under triaxial monotonic loading conditions. When θ = 45° and 60°, the compressive strength is higher than that under triaxial monotonic loading conditions, possibly because the shale is still undergoing stable crack propagation following the fourth cycle.

5.3. Evolution of Damage Factors

Rock damage is accompanied by energy dissipation, which can be described using the energy release rate Y [30] and the corresponding damage factor D as follows:

$$Y=\frac{{\displaystyle {\sum }_l^i{U}_{\mathrm{e}}(i)}}{1-D},$$

(6)

$$D=\frac{{\displaystyle {\sum }_l^i{U}_{\mathrm{d}}(i)}}{U_D},$$

(7)

where U_e(i) (i = 1, 2, 3, 4) is the elastic energy density in each loading and unloading cycle, U_d(i) (i = 1, 2, 3, 4) is the dissipated energy density in each loading and unloading cycle, and U_D is the dissipated energy density over the entire shale damage process.

The evolution law for the relationship between shale Y and D with the same θ under three confining pressures is similar. Taking σ₃ = 10 MPa as an example, the relationship curves between Y and D for shale with five different θ values are shown in Figure 18. As Y increases, D first increases sharply, with the rate of increase for shales with different θ basically the same; this is followed by a gradual decrease in the rate at which D increases, and variation in the rate of increase in D for shales with different θ. The fastest D increase at θ = 45° and 60° is followed by gradual stabilization.

Taking σ₃ = 10 MPa as an example, the damage factor of shale with θ =0° is taken as a benchmark. The ratio of the damage factor of shale with other θ values to this damage factor is shown in Figure 19. Compared with other θ conditions, the variation pattern of D and Y for shale with a θ of 90° is closest to that of shale with θ = 0°, which may be related to their close compressive strength. When Y is small, and the same Y conditions predominate, significant differences in shale D with different θ indicate significant anisotropy. However, the anisotropy gradually weakens with increasing Y, which is consistent with the Felicity ratio. This indicates that the higher the degree of damage, the weaker the anisotropy of the shale.

5.4. Elastic Energy Density Evolution Model

The energy accumulation effect of the rock is triggered only when the energy absorbed by the rock reaches the minimum activation energy U_d0, with U_d0 and the total energy U₀ satisfying the following criteria [31]:

$$U_0-U_{\mathrm{d}0}>0.$$

(8)

The more the external work applied, the more energy is absorbed, which is conducive to energy storage. Therefore, the cumulative rate of change in the elastic energy density dU_e/U_e is related to the total energy U₀, and the elastic energy density U_e is the axial strain increment dԑ₁, i.e.,

$$\frac{1}{U_{\mathrm{e}}}\frac{d{U}_{\mathrm{e}}}{d{\epsilon }_1}\propto \left(U_0-U_{\mathrm{d}0}\right).$$

(9)

Accumulated elastic energy has an inhibitory effect on the subsequent energy storage. This inhibitory effect increases sharply as the amount of accumulated elastic energy increases [32] in a manner that is non-linear and can be described using the following:

$$\frac{1}{U_{\mathrm{e}}}\frac{d{U}_{\mathrm{e}}}{d{\epsilon }_1}\propto -U_{\mathrm{e}}.$$

(10)

Finally, the following equation is used to establish the evolution of shale energy density:

$$\frac{1}{U_{\mathrm{e}}}\frac{\mathrm{d}{U}_{\mathrm{e}}}{\mathrm{d}{\epsilon }_1}=r(U_0-U_{\mathrm{d}0})-s{U}_{\mathrm{e}},$$

(11)

where r and s are the coefficients representing the degree of acceleration and limitation of the internal energy conversion process, respectively. Different values were obtained for different rock samples and energy conversion processes.

Integrating Equation (11) yields

$$U_{\mathrm{e}}=\frac{k}{1+e^{-m{\epsilon }_1-c}}.$$

(12)

Equation (12) is the energy evolution equation for shale failure. The elastic strain energy exhibits an exponential relationship with axial strain, where c is a constant and $m=r(U_0-U_{\mathrm{d}0})$, $k=\frac{r(U_0-U\mathrm{d}0)}{s}=\frac{m}{s}$.

Regression analysis of the energy evolution curve of the shale loading process was performed to obtain the fitting curve between elastic energy density and the axial strain during the three-axis incremental cyclic loading and unloading processes, as shown in Figure 20. The ranges of the fitted parameters are 0.18 ≤ m ≤ 0.49, 758.8 ≤ k ≤ 1211.7, and −6.71 ≤ c ≤ −5.93. The elastic energy density calculation results for the shale-loading process with three confining pressures and five θ values are in good agreement with the experimental data, with all correlation coefficients being above 0.97, indicating that the theoretical model can accurately describe the variation in shale elastic energy with axial strain under different confining pressures and θ values.

In the shale elastic energy density evolution model, m represents the rate at which elastic energy accumulates, with a larger m indicating faster rates of accumulation. The variation curve of m with θ under three confining pressures is shown in Figure 21. When the confining pressure is the same, m shows a U-shaped variation as θ increases, with the minimum value observed at 60°, indicating that the energy accumulation rate has clear dependence on θ. When the θ is the same, the value of m increases with the confining pressure, which is because the pressure compacts the initial pores, suppresses the circumferential deformation, reduces crack development, and improves the accumulation rate of elastic energy density.

6. Weakening Mechanism of Shale Anisotropy

The experimental data show that triaxial incremental cyclic loading and unloading weakens the anisotropy in shale, with the considerable differences in the mechanical behavior, AE characteristics, and energy evolution characteristics observed at θ values of 0° and 60°, indicating the use of these θ as typical for measuring shale anisotropy. The anisotropy of shale is closely related to its failure mode and the key control factors. Based on Figure 10 and 11, the shale failure description method in Figure 22 was used to analyze the failure mode and bedding plane relationship of shale with 0° and 60° θ under three confining pressures and two loading conditions, as shown in

Table 1. Shale failure modes with bedding inclination angles of 0° and 60° under two loading conditions.

Confining Pressure/MPa	Shale Failure Mode
	θ = 0°		θ = 60°
	Monotonic Loading	Cyclic Loading and Unloading	Monotonic Loading	Cyclic Loading and Unloading
10
20
30

. The failure modes and main control factors are listed in

Table 2. Failure modes and main control factors of shale with bedding inclination angles of 0° and 60° under two loading conditions.

Bedding Dip Angle θ/°	Confining Pressure/MPa	Triaxial Monotonic Loading		Triaxial Incremental Cycle Loading and Unloading
		Destruction Mode	Main Control Factor	Destruction Mode	Main Control Factor
0	10	Tensioning failure through bedding planes	Matrix body	Tensioning failure through bedding planes	Matrix body
	20	Mixed failure of tension and shear across bedding planes	Matrix body	Shear failure	Matrix body
	30	Conjugate shear failure	Matrix body	Conjugate shear failure	Matrix body
60	10	Shear slip failure along bedding planes	Bedding plane	Shear failure through bedding planes	Matrix and bedding planes
	20	Shear failure along bedding planes and through bedding planes	Matrix and bedding planes	Shear failure through bedding planes	Matrix and bedding planes
	30	Shear slip failure along bedding planes	Bedding plane	Shear failure through bedding planes	Matrix and bedding planes

. It can be seen from the above that the failure modes of shale under triaxial monotonic loading and triaxial incremental cyclic loading and unloading are similar when the θ is 0°, with the same main control factor and matrix. The damage caused by repeated disturbance under cyclic loading and unloading resulted in a lower compressive strength than that observed under monotonic loading. When the θ is 60°, the shale failure mode is shearing slip failure along the bedding plane under monotonic loading conditions, with the main control factor being the bedding plane, while failure is in the form of shear failure through the bedding plane under cyclic loading and unloading, and the main control factors are the matrix and the bedding plane. The higher shear strength of the matrix as compared to the bedding plane means that the compressive strength of shale is higher under cyclic loading and unloading than it is under monotonic loading.

In summary, the mechanism by which cyclic loading and unloading weakens shale anisotropy mainly works by changing the failure mode and main control factor of shale with a θ of 60°, rendering it closer to the failure mode and main control factor of shale with a θ of 0°.

The changes in the failure mode and main control factor of shale that is subjected to incremental cyclic loading and unloading, which lead to weakening of the anisotropy, occur for two main reasons: (1) during triaxial incremental cyclic loading and unloading, the microstructure between the shale layers and unoriented mineral particles is subjected to periodic stretching and compression, which gradually destroys the bedding structure, rendering the internal structure of the shale more uniform and the location of cracks more random and weakening the anisotropy of the shale; (2) after unloading, the stress distribution in the shale specimens is redistributed, thereby relieving the local stress concentration that results from the loading. After multiple loading and unloading cycles, the stress distribution inside the specimen becomes more uniform, weakening the anisotropy. In addition, each loading and unloading cycle produces an irreversible strain, which is also a mechanism by which the anisotropy can be weakened.

7. Conclusions

In this paper, triaxial monotonic loading and triaxial incremental cyclic loading and unloading tests were conducted on shale with three confining pressures and five θ. Analysis of the anisotropic characteristics of the shale mechanical behavior, AE activity, and energy evolution were analyzed, revealing the mechanism of shale anisotropy weakening. The main conclusions are as follows:

(1) As the θ increases, the compressive strength and elastic modulus of shale first decrease and then increase. The confining pressure and incremental cyclic loading and unloading reduce the anisotropy in the compressive strength and elastic modulus. Compared with triaxial monotonic loading, the compressive strength of shale with θ of 0° and 90° decreases under triaxial incremental cyclic loading and unloading, while the compressive strength increases at θ of 45° and 60°. The compressive strength of shale with a θ of 30° is essentially unchanged. The average ratio of plastic strain to total strain for shale with a θ of 60° is higher after each loading and unloading cycle than it is for shale with other θ, indicating that irreversible deformation is more likely to occur in shale with a θ of 60°.

(2) Under two loading conditions, the failure modes of shales with θ of 0°, 30°, and 90° are similar, while the failure modes of shales with θ of 45° and 60° differ significantly, as evidenced by the lower degree of involvement that the lower bedding planes have in shale crack evolution under incremental cyclic loading and unloading conditions. In particular, shale bedding planes with a θ of 60° are more involved in crack evolution than any of the other θ conditions.

(3) The cumulative ringing number and AE b-value of the shale show a trend of first decreasing and then increasing as the θ increases. As the number of cycles increased, the Felicity ratio gradually decreased, and shale damage accumulated, mainly as the result of small-scale cracks.

(4) The total energy density U₀ of shale with the same θ increases with the confining pressure. For the same confining pressure, U₀ generally shows a tendency to first decrease and then increase as the θ increases. The minimum value is observed at a θ of 60°. Based on the evolution model of shale elastic energy density, it was found that under constant confining pressure, an increase in the θ will lead to the parameter m, which reflects the accumulation rate of elastic energy density, showing first decrease and then increase. When the θ is 60°, the minimum value is reached. When the θ is the same, the value of m increases with the confining pressure.

(5) The anisotropy of shale is closely related to its failure mode and is the main control factor. The main mechanism by which cyclic loading and unloading weakens the anisotropy of shale is to change the failure mode and main control factor of shale with a θ of 60°, rendering it closer to the failure mode and main control factor of shale with a θ of 0°.