#### 1. Introduction

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°.

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

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

Define the true triaxial unloading deformation modulus as

##### 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 [3638].

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

#### 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) [1416].

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.

##### 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.

#### 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.