#### Abstract

The TRW-3000 true triaxial rock testing machine was used to conduct loading and unloading tests of sandstone under different , and the true triaxial lateral unloading mechanics and energy characteristics of sandstone under different were studied. The experimental results show the following: (1) compared with the results of the loading test, the peak strength of the sandstone under the unloading path is reduced, the unloading direction has obvious expansion and deformation, and the amount of expansion increases significantly with the increase of ; sudden brittle failure occurs at the end of unloading. *E* gradually decreases with the increase of *H*, and it performs well to use the cubic polynomial to fit the curve of *E-H*. (2) The Mogi–Coulomb strength criterion can accurately describe the true triaxial strength characteristics of sandstone under loading and unloading conditions. Compared with the results of the loading test, the values of *c* and *φ* obtained based on this criterion under the unloading path are reduced. (3) Under the condition of unloading , *U,**,* and , when the specimen is broken, are all linearly positively correlated with . increases nonlinearly with the increase of *H*, and as increases, the slope of the *-H* curve becomes larger, and the specimen consumes more energy under the same unloading amount. Most of the energy absorbed by the specimen under the unloading path is converted into , but as increases, increases, and the energy consumed when the specimen is broken is greater.

#### 1. Introduction

The stress redistribution induced by the excavation of underground engineering and slope engineering results in obvious unloading deformation of the excavation face [1], which will cause serious engineering problems such as landslides, mine roof fall, slabs, and goaf collapses. If the surrounding rock is in a high-stress environment, the sudden release of the internally accumulated energy in the excavation process may also cause a large-scale rock burst [2]. The mechanical behaviors of rocks under unloading are different from those of rocks under loading [3]. Therefore, it is important to study the mechanical and energy properties of rocks under the unloading condition. The rock mass in actual excavation projects is mostly anisotropic and presents three-dimensional unequal characteristics [4]. Scientists and engineers assumed that the intermediate principal stress has an important effect on the mechanics and failure characteristics of rock mass under true triaxial loading [5–7]. Therefore, it is important to investigate the lateral unloading mechanics and energy characteristics of rocks subjected to different intermediate principal stresses, which is helpful to understand the unloading failure mechanism of rock masses under real geostress states.

In recent years, a substantial number of laboratory tests were conducted to reveal the macromechanical and failure behaviors of rock masses under different loads, e g., uniaxial compression, conventional triaxial compression [8], and true triaxial compression [9, 10]. Mogi [11] first employed precise testing methods to study the influence of the combined stress system, particularly of the intermediate principal stress, on rock failure. Ma et al. [12] studied the unloading failure and deformation feature of siltstone under different intermediate principal stress coefficients *b* and discussed the strength feature of the rock mass based on the Mogi–Coulomb strength criterion. At the same time, many scholars have investigated rock failure from the perspective of energy [13]. Xie et al. [14–16] proposed the overall rock failure criterion based on energy dissipation according to the characteristics of energy dissipation and release in the process of rock deformation and failure. Actually, rock masses are discontinuous media containing initial defects, such as grain boundaries, microcracks, and pores; initiation and propagation of microcracks significantly affect the failure of rock materials [17]. In order to investigate the influence of initiation and propagation of microcracks on the failure of rock materials, micromechanical methods [18–20] were employed to understand the failure of rock materials and establish the corresponding strength criteria and constitutive relations. Zhou et al. [21] proposed a nonlinear three-dimensional strength criterion for rock-like materials based on the micromechanical methods, in which the effects of the intermediate principal stress on the failure of rock-like materials are taken into account. Zhou et al. [22] used the real-time CT technique to obtain the computerized tomography (CT) images and CT values for the process of unloading, microcracking, and dilation up to the failure of limestone specimens under different unloading levels and established a damage model based on the CT value and a new failure criterion. Besides, due to the dispersion of rock materials and the expensive and time-consuming laboratory equipment, the rapid development of computers has given scholars new methods to study the properties of rock materials [23]. Different numerical simulation methods such as the finite element method (FEM) [24, 25], discrete element method (DEM) [26–29], smoothed-particle hydrodynamics (SPH) [30], and peridynamics (PD) [31, 32] were proposed to investigate the mechanical behaviors and failure characteristics of rock specimens under different compressive loading conditions. Zhang et al. [33] employed a 3D particle-based discrete element methodology to investigate the mechanical and failure characteristics of fissured marble specimens in true triaxial compression with a new loading stress path.

In this paper, the true triaxial loading and unloading test was carried out with sandstone as the object, in which the effects of the intermediate principal stress on unloading failure and energy dissipation of the rock material are considered. It is of great value for theoretical research and engineering practice.

#### 2. Materials and Methods

##### 2.1. Materials and Equipment Used in the Tests

The experiment process used the TRW-3000 rock mechanics test system of Central South University, as is shown in Figure 1. To ensure the homogeneity of the specimens, the specimens were taken from the same sandstone rock block with good homogeneity. The specimen size is 100 mm × 100 mm × 100 mm, the unevenness is less than 0.05 mm, and the nonperpendicularity is less than 0.25°.

##### 2.2. Experimental Scheme and Procedure

To explore the influence of on the true triaxial unloading mechanics and energy characteristics of sandstone, a true triaxial loading and unloading test was carried out. The test was divided into two parts as follows: Part I: true triaxial loading: Step 1: all directional stresses were loaded to the initial stress level at a rate of 0.05 . Step 2: the stress was kept stable in all directions. After 30 s, when and remained unchanged, was loaded until specimens were destroyed. Part II: true triaxial lateral unloading: Step 1: all directional stresses were loaded to the initial stress level at a rate of 0.05 . Step 2: the stress was kept stable in all directions. After 30 s, when and remained unchanged, was loaded to a predetermined value (90% of the peak strength obtained by the true triaxial loading test under the same initial stress condition). Step 3: and were kept stable, and was unloaded until specimens were destroyed. The initial stress level is shown in Table 1. The test stress path diagram is shown in Figure 2.

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#### 3. Results and Discussion of the Mechanical Characteristics

##### 3.1. Stress-Strain Curve

The rock material is typically inhomogeneous, containing initial defects, such as grain boundaries, microcracks, and pores. Due to the dependence of the load path, the mechanical behaviors of rocks under unloading are different from those under loading. The rock material contains a number of randomly oriented preexisting microcracks. The preexisting microcracks will be closed under loading, but they will be open under unloading, so the deformation of the rock material under unloading is more than under loading. The frictional sliding on preexisting cracks will occur under loading, but the frictionless sliding on preexisting cracks may occur under unloading. The strength of the rock material under unloading is less than under loading. In the micromechanical approach, the nucleation, growth, and coalescence of microcracks dominate the failure and macroscopic properties of the rock under loading and unloading paths [3].

In this paper, the starting point of the -loading stage is used as the reference point for data processing. Figure 3 shows the loading and unloading stress-strain curves of sandstone under different when = 5 MPa and = 20 MPa. According to the loading curves, the deformation trend of the specimens under different is the same. The early stage is approximately elastic deformation, and the later stage shows obvious plastic deformation until the final failure. As for the unloading curve, the bearing capacity of the specimens is reduced, and obvious lateral expansion occurs when is maintained at . Among them, the direction of is the main expansion direction, and the minimum principal strain increases. As increases, the expansion in the unloading direction increases, and the peak strength increases. Sudden brittle failure occurs at the end of unloading .

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##### 3.2. Deformation Modulus and Unloading Ratio

To study the change law of sandstone mechanical parameters under the true triaxial unloading condition, the concept of *H* is introduced to analyze the relationship between mechanical parameters and unloading degree. Considering the initial stress of the specimen and the unloading amount, *H* is defined as [9]where is the real-time stress value in the direction during unloading and *H* reflects the overall unloading degree of . The initial confining pressure and the magnitude of the unloaded stress in the unloading direction have an obvious impact on the rock deformation, failure, and strength parameters’ degradation during the unloading process [34]. Quantitative research on the change law of mechanical parameters during rock unloading by *H* has strong theoretical and practical significance [35].

Define the true triaxial unloading deformation modulus as

Figure 4 depicts the - curves of sandstone under different during true triaxial unloading. The curves are fitted with the cubic polynomial, and the fitting coefficients are all up to 0.99, a good fitting effect. The curves show that gradually decreases with the increase of *H* during unloading . At the initial stage of unloading, the curves are relatively gentle, as *H* increases, the curves gradually become steeper, and the rate of decrease of increases. The reason is that the decline of is small, and the development of the specimen deformation is slow in the initial stage of unloading, but as *H* increases, the decline of increases, the restraint of the specimen becomes weaker, the bearing capacity of the specimen decreases, and the deformation accelerates. At the same time, the curves under different also show certain differences: the curves are generally gentle when is small, but the curves become steeper as increases, and when the initial stress level is high, a small unloading ratio can cause a significant deformation of the specimen, indicating that when the initial stress level is higher, more energy is stored in the specimen. Under the unloading condition, the energy stored in the specimen is released suddenly, and the failure deformation of the specimen is more significant.

##### 3.3. Strength Characteristics

The Mohr–Coulomb criterion is the most commonly used shear failure criterion in practice, but it only considers the influence of and on rock failure and underestimates rock strength by ignoring the effect of . To solve this defect, scholars have proposed many rock strength criteria based on the three-dimensional force [36–38].

Among them, through the analysis of numerous true triaxial test data, Mogi found that the yield or failure of the rock is affected by and proposed the Mogi yield criterion based on the octahedral shear stress and the effective intermediate principal stress [36], which essence is still the shear failure criterion:

As the distortional strain energy is proportional to the octahedral shear stress, this criterion is equivalent to asserting that failure will occur when the distortional strain energy reaches some critical value that increases monotonically with [39]. The function in the Mogi yield criterion is often regarded as a nonlinear power function. The parameters obtained based on formula (3) cannot well connect with the strength parameters *c* and *φ* of the Mohr–Coulomb criterion. To solve this problem, Al-Ajmi and Zimmerman [39, 40] put forward the view that the *f* function is linear and used numerous test data to verify it. They found that the linear fitting effect of and was good, and then they combined it with the Mohr–Coulomb criterion and renamed it the Mogi–Coulomb criterion:where *a* and *b* are fitting parameters, and the intensity parameter expression based on the Mohr–Coulomb criterion is

Figure 5 is the fitting curve of sandstone strength based on the Mogi–Coulomb criterion under true triaxial loading and lateral unloading conditions, which is in good agreement with the experimental data, and the fitting coefficients are all higher than 0.99, a good fitting effect, indicating that the criterion can well describe the strength relationship of sandstone. The strength parameters *a* and *b* can be approximated from a set of triaxial loading and unloading tests, and then the strength parameters *c* and *φ* of sandstone can be calculated based on this strength criterion. As is shown in Table 2, compared with the values under the loading path, the *c* and *φ* values of sandstone under the unloading path are all reduced. The cohesive force *c* decreases by 24.36%, and the internal friction angle *φ* decreases by 16.13%, that is, the specimen is softened during unloading , and the strength is lower than that under the loading path.

#### 4. Results and Discussion of the Energy Characteristics

##### 4.1. Energy Calculation Principle

The rock produces deformation during loading, and the heat exchange between the rock and outside during this process is ignored; that is to say, the specimen can be considered as a closed system. According to the first law of thermodynamics,where *U* is the total work done by the external force during the test, that is, the total energy input; is the dissipated energy, which is used to form internal damage and plastic deformation of the specimen, and its change satisfies the second law of thermodynamics, that is, the internal state change conforms to the increasing trend of entropy; and is the releasable elastic energy, which is used to generate elastic deformation of the specimen.

This paper takes the starting point of the -loading stage as the base point of data processing and regards the energy input and dissipation in the initial stress loading stage as changes in the internal energy of the specimen, so it is ignored. The energy of each part of the specimen under the complex stress states can be expressed as equation (7) [14–16].

In equation (7), , , and are the maximum, middle, and minimum principal stresses, respectively, are the strains in the directions of the principal stresses, *μ* is Poisson’s ratio, *E* is the unloading elastic modulus, which is replaced by the elastic modulus in the calculation, and and are the total energy input and the releasable elastic energy in the initial stress loading stage.

##### 4.2. Energy-Strain Curve

Figure 6 depicts the energy-strain curve of the specimen under loading and unloading conditions, which shows that the initial forces of the two stress paths are the same, so the early changes of the energy-strain curve under the two paths are similar. That is to say, the energy is mainly stored as , and the part converted into is very small. However, because the forces of the two stress paths change in the later stage, there are obvious differences in the later stage of the energy-strain curves: according to the loading curve, one part of the energy is stored as , and the other part of the energy is converted into . As for the unloading curve, increases significantly, while remains stable with minor changes, indicating that the energy input in the later stage is used for internal damage and plastic deformation of the specimen, and the excess is transformed into the kinetic energy of the falling rock.

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##### 4.3. Energy Analysis of Lateral Unloading

Table 3 depicts the energy value of each characteristic point of the specimen under the unloading condition. As is shown in Table 3, when increases from 5 MPa to 20 MPa, all types of energy at the specimen failure point increase, among which the maximum increment is : from 0.18 to 0.32 , an increase of 77.8%. Meanwhile, the increment of *U* and increases with the increase of , but the increment of remains unchanged. Figure 7 shows the fitted curves of energy- at the unloading failure point under the unloading condition when = 5 MPa and = 20 MPa, which depict that are linearly related to , and the linear fitting coefficients are all above 0.9, a good fitting effect.

Figure 8 depicts the *-H* curve under the unloading condition, which shows that increases slowly with the increase of *H* in the initial unloading stage, but as *H* increases, increases rapidly until the final specimen failure. The influence of on the change of during the unloading process is mainly manifested as follows: with the increase of , the slope of the curve becomes larger, and the growth rate of increases; the specimen deformation consumes more energy under the same unloading amount.

The total energy absorbed by the specimen during the loading process is used to store as and convert it into for the initiation and propagation of cracks inside the specimen. Therefore, even if the energy input during the loading process is the same, different energy distributions will cause the specimen failure mode to change. Figure 9 is the energy distribution diagram of the specimen under the unloading condition when = 5 MPa and = 20 MPa, which shows that is generally above 0.5, while increases with the increase of , indicating that most of the total energy absorbed by the specimen under the unloading path is converted into and stored inside the specimen, but as increases, the proportion of increases, and specimen destruction consumes relatively more energy.

#### 5. Conclusion

(1)Compared with the results of the loading test, the peak strength of the sandstone under the unloading path is reduced, and sudden brittle failure occurs at the end of unloading. During the unloading process, the unloading direction of the specimen expands significantly, *E* gradually decreases with the increase of *H*, and the relationship between *E* and *H* can be fitted with a cubic polynomial, a good fitting effect.(2)It is accurate to describe the true triaxial strength characteristics of specimens under different stress paths by using Mogi-Coulomb criterion. Compared with the values under the loading path, the *c* and *φ* values of the sandstone under the unloading path are reduced, of which the cohesive force *c* decreases by 24.36%, and the internal friction angle *φ* decreases by 16.13%.(3) are all positively linearly correlated with when the specimen is damaged under the unloading condition. increases nonlinearly with *H*, and with the increase of , the slope of the *-H* curve becomes larger, and the specimen consumes more energy under the same unloading amount. Most of the total energy absorbed by the specimen under the unloading path is converted into and stored inside the specimen, but as increases, the curve rises, the proportion of increases, and the energy consumed when the specimen is broken is greater.

#### Abbreviations

: | Maximum principal stress |

C: | Cohesive force |

: | Intermediate principal stress |

Φ: | Internal friction angle |

: | Minimum principal stress |

U: | Total energy |

H: | Unloading ratio |

: | Elastic energy |

E: | Deformation modulus |

: | Dissipation energy. |

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This paper obtained funding from projects (no. 51979293) supported by the National Natural Science Foundation of China. The authors wish to acknowledge the support.