# Unloading Mechanics and Energy Characteristics of Sandstone under Different Intermediate Principal Stress Conditions

Authors: Yingjie Zhang; Jiangteng Li; Gang Ma; Shuangfei Liu
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

The TRW-3000 true triaxial rock testing machine was used to conduct loading and unloading tests of sandstone under differentσ2, and the true triaxial lateral unloading mechanics and energy characteristics of sandstone under different σ2 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 σ3 path is reduced, the unloading direction has obvious expansion and deformation, and the amount of expansion increases significantly with the increase of σ2; 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 σ3 path are reduced. (3) Under the condition of unloading σ3, U,Ue, and Ud, when the specimen is broken, are all linearly positively correlated with σ2. Ud increases nonlinearly with the increase of H, and as σ2 increases, the slope of the Ud-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 σ3 path is converted into Ue, but as σ2 increases, Ud/U increases, and the energy consumed when the specimen is broken is greater.


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## Body

## 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 Figure1. 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°.Figure 1
TRW-3000 rock mechanics test system.


### 2.2. Experimental Scheme and Procedure

To explore the influence ofσ2 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 0.05MPa•s−1.Step 2: the stress was kept stable in all directions. After 30 s, whenσ2 and σ3 remained unchanged, σ1 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 0.05MPa•s−1.Step 2: the stress was kept stable in all directions. After 30 s, whenσ2 and σ3 remained unchanged, σ1 was loaded to a predetermined value (90% of the peak strength σ1max obtained by the true triaxial loading test under the same initial stress condition).Step 3:σ1 and σ2 were kept stable, and σ3 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.Figure 2
Stress path schematic diagram of the (a) true triaxial loading test and (b) true triaxial lateral unloading test.
(a)(b)Table 1
Initial stress level.
TypeInitial stress (MPa)Unloading point (MPa)Unloading rate ofσ3 (MPa·s−1)σ10σ20σ30Loading test205, 11, 17, 205––Lateral unloading test205, 11, 17, 205σ1=0.9σ1max0.05


## 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 Figure1. 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°.Figure 1
TRW-3000 rock mechanics test system.


## 2.2. Experimental Scheme and Procedure

To explore the influence ofσ2 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 0.05MPa•s−1.Step 2: the stress was kept stable in all directions. After 30 s, whenσ2 and σ3 remained unchanged, σ1 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 0.05MPa•s−1.Step 2: the stress was kept stable in all directions. After 30 s, whenσ2 and σ3 remained unchanged, σ1 was loaded to a predetermined value (90% of the peak strength σ1max obtained by the true triaxial loading test under the same initial stress condition).Step 3:σ1 and σ2 were kept stable, and σ3 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.Figure 2
Stress path schematic diagram of the (a) true triaxial loading test and (b) true triaxial lateral unloading test.
(a)(b)Table 1
Initial stress level.
TypeInitial stress (MPa)Unloading point (MPa)Unloading rate ofσ3 (MPa·s−1)σ10σ20σ30Loading test205, 11, 17, 205––Lateral unloading test205, 11, 17, 205σ1=0.9σ1max0.05


## 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 theσ1-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 σ2 when σ30 = 5 MPa and σ10 = 20 MPa. According to the loading curves, the deformation trend of the specimens under different σ2 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 σ3 curve, the bearing capacity of the specimens is reduced, and obvious lateral expansion occurs when σ1 is maintained at 0.9σ1max. Among them, the direction of σ3 is the main expansion direction, and the minimum principal strain ε3 increases. As σ2 increases, the expansion in the unloading direction increases, and the peak strength increases. Sudden brittle failure occurs at the end of unloading σ3.Figure 3
Loading and unloading stress-strain curves under differentσ2. (a) σ2 = 5 MPa. (b) σ2 = 11 MPa. (c) σ2 = 17 MPa. (d) σ2 = 20 MPa.
(a)(b)(c)(d)


### 3.2. Deformation Modulus and Unloading Ratio

To study the change law of sandstone mechanical parameters under the true triaxial unloadingσ3 condition, the concept of H is introduced to analyze the relationship between mechanical parameters and unloading degree. Considering the initial stress σ30 of the specimen and the unloading amount, H is defined as [9](1)H=σ30−σ3iσ30,where σ3i is the real-time stress value in the σ3 direction during unloading σ3 and H reflects the overall unloading degree of σ3. 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(2)E0=σ1−σ10ε1−ε10.Figure4 depicts the E0-H curves of sandstone under different σ2 during true triaxial unloading. The curves are fitted with the cubic polynomial, and the fitting coefficients R2 are all up to 0.99, a good fitting effect. The curves show that E0 gradually decreases with the increase of H during unloading σ3. 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 E0 increases. The reason is that the decline of σ3 is small, and the development of the specimen deformation is slow in the initial stage of unloading, but as H increases, the decline of σ3 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 σ2 also show certain differences: the curves are generally gentle when σ2 is small, but the curves become steeper as σ2 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.Figure 4
The deformation modulus and unloading ratio curves of the sandstone under differentσ2 during true triaxial unloading.


### 3.3. Strength Characteristics

The Mohr–Coulomb criterion is the most commonly used shear failure criterion in practice, but it only considers the influence ofσ1 and σ3 on rock failure and underestimates rock strength by ignoring the effect of σ2. 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 byσ2 and proposed the Mogi yield criterion based on the octahedral shear stress τoct and the effective intermediate principal stress σm,2 [36], which essence is still the shear failure criterion:(3)τoct=fσm,2,τoct=13σ1−σ22+σ2−σ32+σ1−σ32,σm,2=σ1+σ32.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 withσ2 [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 τoct and σm,2 was good, and then they combined it with the Mohr–Coulomb criterion and renamed it the Mogi–Coulomb criterion:(4)τoct=a+bσm,2,where a and b are fitting parameters, and the intensity parameter expression based on the Mohr–Coulomb criterion is(5)a=223c⋅cosφ,b=223sinφ.Figure5 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 R2 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 σ3 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 σ3, and the strength is lower than that under the loading path.Figure 5
Specimen strength fitting curves based on the Mogi–Coulomb criterion.Table 2
Fitting parameters of sandstone under different stress paths.
Loading methodMogi–Coulomb criterionabc (MPa)φ (°)R2True triaxial loading11.0630.79621.9057.600.996True triaxial lateral unloading10.3870.70416.5648.310.999


## 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 theσ1-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 σ2 when σ30 = 5 MPa and σ10 = 20 MPa. According to the loading curves, the deformation trend of the specimens under different σ2 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 σ3 curve, the bearing capacity of the specimens is reduced, and obvious lateral expansion occurs when σ1 is maintained at 0.9σ1max. Among them, the direction of σ3 is the main expansion direction, and the minimum principal strain ε3 increases. As σ2 increases, the expansion in the unloading direction increases, and the peak strength increases. Sudden brittle failure occurs at the end of unloading σ3.Figure 3
Loading and unloading stress-strain curves under differentσ2. (a) σ2 = 5 MPa. (b) σ2 = 11 MPa. (c) σ2 = 17 MPa. (d) σ2 = 20 MPa.
(a)(b)(c)(d)


## 3.2. Deformation Modulus and Unloading Ratio

To study the change law of sandstone mechanical parameters under the true triaxial unloadingσ3 condition, the concept of H is introduced to analyze the relationship between mechanical parameters and unloading degree. Considering the initial stress σ30 of the specimen and the unloading amount, H is defined as [9](1)H=σ30−σ3iσ30,where σ3i is the real-time stress value in the σ3 direction during unloading σ3 and H reflects the overall unloading degree of σ3. 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(2)E0=σ1−σ10ε1−ε10.Figure4 depicts the E0-H curves of sandstone under different σ2 during true triaxial unloading. The curves are fitted with the cubic polynomial, and the fitting coefficients R2 are all up to 0.99, a good fitting effect. The curves show that E0 gradually decreases with the increase of H during unloading σ3. 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 E0 increases. The reason is that the decline of σ3 is small, and the development of the specimen deformation is slow in the initial stage of unloading, but as H increases, the decline of σ3 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 σ2 also show certain differences: the curves are generally gentle when σ2 is small, but the curves become steeper as σ2 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.Figure 4
The deformation modulus and unloading ratio curves of the sandstone under differentσ2 during true triaxial unloading.


## 3.3. Strength Characteristics

The Mohr–Coulomb criterion is the most commonly used shear failure criterion in practice, but it only considers the influence ofσ1 and σ3 on rock failure and underestimates rock strength by ignoring the effect of σ2. 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 byσ2 and proposed the Mogi yield criterion based on the octahedral shear stress τoct and the effective intermediate principal stress σm,2 [36], which essence is still the shear failure criterion:(3)τoct=fσm,2,τoct=13σ1−σ22+σ2−σ32+σ1−σ32,σm,2=σ1+σ32.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 withσ2 [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 τoct and σm,2 was good, and then they combined it with the Mohr–Coulomb criterion and renamed it the Mogi–Coulomb criterion:(4)τoct=a+bσm,2,where a and b are fitting parameters, and the intensity parameter expression based on the Mohr–Coulomb criterion is(5)a=223c⋅cosφ,b=223sinφ.Figure5 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 R2 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 σ3 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 σ3, and the strength is lower than that under the loading path.Figure 5
Specimen strength fitting curves based on the Mogi–Coulomb criterion.Table 2
Fitting parameters of sandstone under different stress paths.
Loading methodMogi–Coulomb criterionabc (MPa)φ (°)R2True triaxial loading11.0630.79621.9057.600.996True triaxial lateral unloading10.3870.70416.5648.310.999


## 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,(6)U=Ud+Ue,where U is the total work done by the external force during the test, that is, the total energy input; Ud 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 Ueis the releasable elastic energy, which is used to generate elastic deformation of the specimen.This paper takes the starting point of theσ1-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), σ1, σ2, and σ3 are the maximum, middle, and minimum principal stresses, respectively, ε1,ε2,andε3 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 U0 and U0e are the total energy input and the releasable elastic energy in the initial stress loading stage.(7)U=∫0ε1σ1dε1+∫0ε2σ2dε2+∫0ε3σ3dε3−U0,Ue=12Eσ12+σ22+σ32−2μσ1σ2+σ2σ3+σ3σ1−U0e.


### 4.2. Energy-Strain Curve

Figure6 depicts the energy-strain curve of the specimen under loading and unloading σ3 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 Ue, and the part converted into Ud 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 Ue, and the other part of the energy is converted into Ud. As for the unloading curve, Ud increases significantly, while Ue 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 Ud is transformed into the kinetic energy of the falling rock.Figure 6
Energy-strain curves. (a)σ3 = 5 MPa and σ2 = 11 MPa. (b) σ3 = 5 MPa and σ2 = 17 MPa.
(a)(b)


### 4.3. Energy Analysis of Lateral Unloading

Table3 depicts the energy value of each characteristic point of the specimen under the unloading σ3 condition. As is shown in Table 3, when σ2 increases from 5 MPa to 20 MPa, all types of energy at the specimen failure point increase, among which the maximum increment is Ud: from 0.18 MJ⋅m−3 to 0.32 MJ⋅m−3, an increase of 77.8%. Meanwhile, the increment of U and Ud increases with the increase of σ2, but the increment of Ue remains unchanged. Figure 7 shows the fitted curves of energy-σ2 at the unloading failure point under the unloading σ3 condition when σ30 = 5 MPa and σ10 = 20 MPa, which depict that U,Ud,andUe are linearly related to σ2, and the linear fitting coefficients R2 are all above 0.9, a good fitting effect.Table 3
The energy of unloading and failure points under differentσ2.
Initial stress (MPa)Unloading point energy (MJ·m−3)Unloading failure point energy (MJ·m−3)Unloadingσ3 energy increase (MJ·m−3)σ10,σ20,σ30UUeUdUUeUd△U△Ue△Ud20, 5, 50.320.300.020.490.310.180.170.010.1620, 11, 50.370.330.040.550.340.210.180.010.1720, 17, 50.430.380.050.650.380.270.220.000.2220, 20, 50.490.420.070.750.430.320.260.010.25Figure 7
The fitting curves of energy-σ2 at the failure point under the unloading σ3 condition.Figure8 depicts the Ud-H curve under the unloading σ3 condition, which shows that Ud increases slowly with the increase of H in the initial unloading stage, but as H increases, Ud increases rapidly until the final specimen failure. The influence of σ2 on the change of Ud during the unloading process is mainly manifested as follows: with the increase of σ2, the slope of the curve becomes larger, and the growth rate of Ud increases; the specimen deformation consumes more energy under the same unloading amount.Figure 8
Dissipated energy-unloading ratio curves.The total energy absorbed by the specimen during the loading process is used to store asUe and convert it into Ud 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 σ3 condition when σ30 = 5 MPa and σ10 = 20 MPa, which shows that Ue/U is generally above 0.5, while Ud/U increases with the increase of σ2, indicating that most of the total energy absorbed by the specimen under the unloading σ3 path is converted into Ue and stored inside the specimen, but as σ2 increases, the proportion of Ud increases, and specimen destruction consumes relatively more energy.Figure 9
Failure point energy distribution under the unloadingσ3 condition.


## 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,(6)U=Ud+Ue,where U is the total work done by the external force during the test, that is, the total energy input; Ud 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 Ueis the releasable elastic energy, which is used to generate elastic deformation of the specimen.This paper takes the starting point of theσ1-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), σ1, σ2, and σ3 are the maximum, middle, and minimum principal stresses, respectively, ε1,ε2,andε3 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 U0 and U0e are the total energy input and the releasable elastic energy in the initial stress loading stage.(7)U=∫0ε1σ1dε1+∫0ε2σ2dε2+∫0ε3σ3dε3−U0,Ue=12Eσ12+σ22+σ32−2μσ1σ2+σ2σ3+σ3σ1−U0e.


## 4.2. Energy-Strain Curve

Figure6 depicts the energy-strain curve of the specimen under loading and unloading σ3 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 Ue, and the part converted into Ud 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 Ue, and the other part of the energy is converted into Ud. As for the unloading curve, Ud increases significantly, while Ue 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 Ud is transformed into the kinetic energy of the falling rock.Figure 6
Energy-strain curves. (a)σ3 = 5 MPa and σ2 = 11 MPa. (b) σ3 = 5 MPa and σ2 = 17 MPa.
(a)(b)


## 4.3. Energy Analysis of Lateral Unloading

Table3 depicts the energy value of each characteristic point of the specimen under the unloading σ3 condition. As is shown in Table 3, when σ2 increases from 5 MPa to 20 MPa, all types of energy at the specimen failure point increase, among which the maximum increment is Ud: from 0.18 MJ⋅m−3 to 0.32 MJ⋅m−3, an increase of 77.8%. Meanwhile, the increment of U and Ud increases with the increase of σ2, but the increment of Ue remains unchanged. Figure 7 shows the fitted curves of energy-σ2 at the unloading failure point under the unloading σ3 condition when σ30 = 5 MPa and σ10 = 20 MPa, which depict that U,Ud,andUe are linearly related to σ2, and the linear fitting coefficients R2 are all above 0.9, a good fitting effect.Table 3
The energy of unloading and failure points under differentσ2.
Initial stress (MPa)Unloading point energy (MJ·m−3)Unloading failure point energy (MJ·m−3)Unloadingσ3 energy increase (MJ·m−3)σ10,σ20,σ30UUeUdUUeUd△U△Ue△Ud20, 5, 50.320.300.020.490.310.180.170.010.1620, 11, 50.370.330.040.550.340.210.180.010.1720, 17, 50.430.380.050.650.380.270.220.000.2220, 20, 50.490.420.070.750.430.320.260.010.25Figure 7
The fitting curves of energy-σ2 at the failure point under the unloading σ3 condition.Figure8 depicts the Ud-H curve under the unloading σ3 condition, which shows that Ud increases slowly with the increase of H in the initial unloading stage, but as H increases, Ud increases rapidly until the final specimen failure. The influence of σ2 on the change of Ud during the unloading process is mainly manifested as follows: with the increase of σ2, the slope of the curve becomes larger, and the growth rate of Ud increases; the specimen deformation consumes more energy under the same unloading amount.Figure 8
Dissipated energy-unloading ratio curves.The total energy absorbed by the specimen during the loading process is used to store asUe and convert it into Ud 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 σ3 condition when σ30 = 5 MPa and σ10 = 20 MPa, which shows that Ue/U is generally above 0.5, while Ud/U increases with the increase of σ2, indicating that most of the total energy absorbed by the specimen under the unloading σ3 path is converted into Ue and stored inside the specimen, but as σ2 increases, the proportion of Ud increases, and specimen destruction consumes relatively more energy.Figure 9
Failure point energy distribution under the unloadingσ3 condition.


## 5. Conclusion

(1)
Compared with the results of the loading test, the peak strength of the sandstone under the unloadingσ3 path is reduced, and sudden brittle failure occurs at the end of unloading. During the unloading σ3 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, thec and φ values of the sandstone under the unloading σ3 path are reduced, of which the cohesive force c decreases by 24.36%, and the internal friction angle φ decreases by 16.13%.(3)
U,Ud,andUe are all positively linearly correlated with σ2 when the specimen is damaged under the unloading σ3 condition. Udincreases nonlinearly with H, and with the increase of σ2, the slope of the Ud-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 σ3 path is converted into Ue and stored inside the specimen, but as σ2 increases, the Ud/U curve rises, the proportion of Ud increases, and the energy consumed when the specimen is broken is greater.



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*Source: 5577321-2021-04-22.xml*