# Experimental Investigation on the Deformation, Strength, and Acoustic Emission Characteristics of Sandstone under True Triaxial Compression

Authors: Zhaolin Li; Lianguo Wang; Yinlong Lu; Wenshuai Li; Kai Wang
Journal: Advances in Materials Science and Engineering
 (2018)

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## Abstract

The study of deformation, strength, and other mechanical characteristics of sandstone under true triaxial compression is significant for understanding failure mechanisms in rock and evaluating the stability of underground structures. Conventional and true triaxial compression tests for sandstone are conducted for different stress states in this study using the self-developed true triaxial electrohydraulic servo test system combined with acoustic emission (AE) testing. This study presents an in-depth and systematic investigation of deformation, strength, and AE characteristics. The results show significant differences in deformation, strength, and acoustic emission characteristics for the rock under conventional triaxial and true triaxial compression tests, respectively. The peak strength, axial strain, lateral strain, and incremental strain (in unstable crack growth stage) increase with increasing confining pressure under conventional triaxial compression, and the AE count gradually decreases while shear crack proportion gradually increases, indicating that increasing confining pressure gradually inhibits the shear slip effect along fractures, delays perforation of the rock shear fracture surface, and enhances the ability of the rock to withstand deformation and load. Under true triaxial compression, the peak strength increases and then decreases with increasing intermediate principal stressσ2 and the axial strain ε1 and lateral strain ε2 gradually decrease; besides, the lateral strain (expansion) of the rock is mainly in the minimum principal stress σ3 direction, and lateral expansion tends to decrease before increasing. AE events first weaken and then enhance with increasing σ2, and the proportion of shear cracks increases first and then decreases, indicating that the confining pressure gradually changes from the shear slip effect that controls crack offset to the damage effect that promotes crack tension with increasing σ2. In addition, the protective effect of confining pressure improves when σ3 increases.


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## Body

## 1. Introduction

Rock material is a kind of brittle geological material, which has a complex composition and contains numerous defects, such as joints and cracks. Rock mechanical properties change with variations in external load and environmental factors because of these inherent characteristics. Rock engineering is a fundamental part of construction and is an important support for social development and progress. These projects are all related to research on fundamental rock mechanics problems [1].The mechanical properties of the rock in Earth’s crust are closely related to the 3D stress state. The deformation, strength, and other mechanical properties can be obtained by studying the deformation and failure of various rocks. Understanding rock failure mechanisms is significant for geotechnical engineering. At present, the study of rock damage in a 3D stress state is primarily carried out using conventional triaxial tests. Domestic and foreign scholars have studied the mechanical properties of rock under conventional triaxial compression by using conventional testing machines. Karman obtained the earliest stress-strain curves for Carrara marble under different confining pressures on an ordinary testing machine [2]. Fredrich studied the mechanical properties of calcite marble under conventional triaxial compression and examined the effect of particle size on the brittle-ductile transition [3]. Yang et al. [4]. carried out conventional triaxial compression tests on sandstones by using a servo testing machine.However, the influence of the intermediate principal stress is ignored due to the setup of the conventional testing machine. A cylindrical rock sample is subjected to different axial and lateral stresses such that the rock is in an axisymmetric stress state. So, only the strength and deformation for rocks in an axisymmetric stress state are measured, and the general stress states are not represented (σ1≠σ2≠σ3). In a true triaxial compression test, each of the three principle stresses can be independently applied to the sample allowing for the failure mode to be accurately simulated and a more complex stress path to be measured. In recent years, Mogi [5, 6], Haimson et al. [7, 8], and Feng et al. [9] have carried out a large number of experimental studies on rocks using a true triaxial test machine. However, previous studies mainly focus on the effect of the intermediate principal stress on rock strength [10] and the establishment of true triaxial strength criterion to show the effect of the intermediate principal stress on rock strength [11, 12]. Furthermore, most true triaxial compression experiments examine strength and failure criteria for granite, marble, or other hard rocks, while failure and deformation characteristics for relatively soft rocks, such as sandstone and coal, remain largely unknown.Underground projects such as roadways are mostly undertaken in sedimentary structures that consist of sandstone-filled channels, and the deformation problems in the roadways are more prominent. Due to the complex microstructure of and numerous defects, its physical and mechanical properties are more sensitive to external loads. Sandstone is one of the most widely distributed rocks in East, South, and Southwest China. With the rapid development of underground construction, understanding the response of sandstone to 3D stress states is of significance for the construction of underground foundations and roadways. Examining the deformation, strength, and other mechanical properties of sandstone under true triaxial compression can provide reference for the selection of rock mechanics parameters in future geotechnical engineering projects.The rock damage induced by compression is essentially caused by the continuous development of microcracks, such that the rock fractures. Therefore, the combination of macroscopic mechanical properties such as strength, deformation of the rock, and the development of microcracks in the rock can reveal the cause of the deformation and failure. Acoustic emission (AE) is an effective method to observe the development of microcracks in rock. AE is caused by the generation and expansion of microfractures when rock materials are under an external stress, and AE is accompanied by elastic wave or stress wave release. AE information reflects the damage and failure state of the rock, which is closely related to the densification of the original internal fissures as well as the generation, expansion, and penetration of the new fissures. The influence of the intermediate principal stress on the development of internal microcracks in rocks can be revealed using AE information. Moradian et al. [13], Yang et al. [14], and Ranjith et al. [15] analyzed the evolution of internal cracks in rocks using AE. However, these studies focused on the evolution of rock fracturing under conventional triaxial compression. Therefore, it is necessary to explore the development of internal microcracks in rock based on AE to reveal the root cause of rock deformation and failure under true triaxial compression.In this study, sandstone is subjected to conventional and true triaxial tests with varying stress conditions using the self-developed true triaxial electrohydraulic servo test system. The development of internal cracks within the rock is monitored using an AE monitoring system. The deformation, strength, and AE characteristics of sandstone under three triaxial stress states are discussed. The effects of different confining pressures under conventional triaxial compression and different stresses in true triaxial compression on the variation of rock deformation, strength parameters, and internal crack development are discussed.


## 2. Experimental Methodology

### 2.1. True Triaxial Electrohydraulic Servo Test System

The tests presented in this study are carried out using the self-developed true triaxial compression test system (Figure1). The test system is primarily composed of a maximum principal stress, intermediate principal stress, minimum principal stress (σ1,σ2,σ3) three-dimensional servo control load system, true triaxial pressure cell, automatic data acquisition system, and AE monitoring system. The σ1,σ2,σ3 three-dimensional servo control load system forms an independent load frame. Piston rods are installed at one end of the three load frames, which is externally connected to the servo valve to produce independent servo loading. The three loading frames are perpendicular to each other. The positions of loading frames σ1 and σ3 are fixed, while the third loading frame (σ2) is placed on a horizontal guide rail and has the able to slide along the guide rail to facilitate the operation and placement of the sample. The other end of the two loading frames (σ2 and σ3) is installed with reaction screws, which have an adjustable length to adapt to different specimen sizes.Figure 1
True triaxial compression test system.Specimens used in this study are cuboids with a size of 50 × 50 × 100 mm. The specimen is placed in a cubic pressure chamber composed of a fully rigid structure, which consists of a box-shaped body and a loading plate. In order to avoid deformation of the rock samples leading to mutual extrusion of the loading plate during the test, a gap is left between the vertical and horizontal plates in the traditional true triaxial pressure chamber. This gap causes the uneven stress and strain in the test, resulting in a decrease in peak strength and an abnormal failure pattern in the sample [16]. In order to overcome this defect, the true triaxial pressure cell adopts a rotary interlocking platen design, in which the loading plates on the 6 faces of the sample are arranged in a misplaced method. When the specimen is compressed, the loading plate moves synchronously with the specimen to avoid interference effects in each direction (Figure 1). The relative displacement between the loading plate and the specimen produces a friction force, which negatively influences the test result. To reduce the friction between the loading plate and the sample, a 0.5 mm thick layer of lubricant is applied between the loading plate and the sample surface. The lubricant is composed of a 1 : 1 ratio of stearic acid and Vaseline [17].A spoke-type pressure sensor with a precision of 0.01 kN is installed between the piston rod and loading block. A rope-type displacement sensor with an accuracy of 0.002 mm and a measurement range of 1,000 mm is fixed to the piston rod and reaction force screw. A DS2-8B full information acoustic emission signal analyzer is used to examine the AE activity during tests. The AE system used in this study mainly consists of the AE host, preamplifier, and AE sensor. The type of the AE sensor is RS-54A, and its monitoring frequency range is 100–900 kHz. The AE sensor is installed in the loading plate of the true triaxial pressure cell (an 8 mm diameter hole is set in the side of the plate). The AE signal line is drawn out from the hole in the side of the plate, and the probe is in direct contact with the surface of the sandstone to ensure the accuracy of the AE data. The model of AE preamplifier is 20/40/60 dB which can provide three different transmission gains; furthermore, it has the advantages of low noise, wide bandwidth, impact resistance, and small volume. During the test, the threshold is set to 40 dB to avoid the possibility of electronic/environmental noise. The AE signal is amplified by the preamplifier with the gain of 40 dB. The time parameters for AE waveforms included peak definition time (PDT), hit definition time (HDT), and hit lockout (HLT), which are set to 50, 100, and 100μs, respectively.


### 2.2. Specimen Preparation

Sandstone is selected for examination in this study. Specimens are composed of feldspar and quartz, as well as a small amount of montmorillonite. The sandstone has a medium grain structure with grain size ranging from 0.1 to 0.35 mm and a relatively dense massive structure. Grains are relatively uniform with an identical macroscopic uniformity and an average density of 2380 kg/m3. Samples are 50 × 50 × 100 mm in size, and the end faces are carefully polished. Detailed sample information is given in Table 1.Table 1
Tested sandstone specimens and conditions in this research.
Loading program
Group
Specimen number
Dimensions (mm)
Quality (g)
σ
2 (MPa)
σ
3 (MPa)
CTT
C1
C1-1
100.2 × 50.0 × 49.8
622.5
10
10
C1-2
100.2 × 49.8 × 50.1
613.2
C1-3
100.3 × 50.1 × 49.9
612.9
C2
C2-1
100.0 × 49.5 × 50.0
613.0
20
20
C2-2
100.5 × 49.5 × 50.0
613.5
C2-3
99.8 × 50.3 × 50.3
605.1
C3
C3-1
100.3 × 50.0 × 50.1
636.0
30
30
C3-2
99.9 × 50.1 × 50.2
633.9
C3-3
99.6 × 50.3 × 49.6
602.3
C4
C4-1
99.9 × 50.1 × 50.2
625.3
40
40
C4-2
99.8 × 50.2 × 50.1
629.5
C4-3
100.2 × 50.0 × 49.2
639.7
TTT (σ3=10MPa)
T2
T2-1
100.1 × 50.3 × 49.8
628.1
20
10
T2-2
100.4 × 50.5 × 50.1
636.0
T2-3
100.3 × 50.6 × 50.1
646.6
T3
T3-1
100.4 × 49.9 × 50.3
630.5
30
10
T3-2
100.0 × 50.2 × 50.1
650.0
T3-3
99.8 × 50.4 × 50.5
624.7
T4
T4-1
100.4 × 49.6 × 49.6
618.8
40
10
T4-2
100.3 × 49.7 × 49.9
626.7
T4-3
100.1 × 50.4 × 50.6
607.1
T5
T5-1
100.1 × 50.1 × 49.9
615.6
50
10
T5-2
99.9 × 49.5 × 50.2
610.8
T5-3
100.2 × 50.1 × 49.9
614.5
T6
T6-1
99.7 × 49.7 × 50.3
610.4
60
10
T6-2
99.8 × 49.9 × 50.2
606.3
T6-3
100.1 × 49.8 × 50.2
595.7
TTT (σ3=20MPa)
T7
T7-1
100.3 × 49.9 × 50.2
630.5
30
20
T7-2
99.8 × 50.4 × 50.5
624.2
T7-3
99.9 × 49.5 × 50.2
612.3
T8
T8-1
100.4 × 49.8 × 50.2
599.8
40
20
T8-2
100.1 × 50.4 × 50.3
603.1
T8-3
99.7 × 50.1 × 50.3
615.4
T9
T9-1
100.3 × 49.7 × 49.9
621.7
50
20
T9-2
100.1 × 50.1 × 49.9
615.6
T9-3
100.0 × 49.7 × 50.1
630.1
T10
T10-1
99.8 × 49.9 × 50.4
626.3
60
20
T10-2
100.2 × 50.1 × 49.9
624.6
T10-3
100.3 × 49.8 × 49.9
621.7


### 2.3. Testing Procedure

In order to study the mechanical properties and AE characteristics of rock under true triaxial compression, the difference between the true triaxial test (TTT) and conventional triaxial test (CTT) is compared and the influence of different triaxial stresses on rock mechanical properties and AE characteristics is analyzed. The test setup is shown in Figure2.Figure 2
The two different loading programs: (a) CTT; (b) TTT.
(a)
(b)


#### 2.3.1. CTT

Pressure is applied to the specimen at a rate of 0.2 MPa/s up to a predetermined confining pressure (set values ofσ2 and σ3), while the specimen is maintained in a hydrostatic pressure state (oa); then, with constant confining pressure, the axial stress is applied to the specimen at a loading rate of 0.002 mm/s, until the specimen failed (ab).


#### 2.3.2. TTT

Pressure is applied to the specimen at a rate of 0.2 MPa/s up to a predetermined confining pressure (set value ofσ3), while the specimen is maintained in a hydrostatic pressure state (oa,oa′); then, with constant σ3, the values of σ1 and σ2 are continuously increased to the set value of σ2 (ab,a′b′) at the same rate. Afterwards with constant σ2 and σ3, the axial stress is applied to the specimen at a loading rate of 0.002 mm/s, until the specimen failed (bc,b′c′).Since the test discreteness is considered in this study, each test is repeated 2 times. The specific test scheme is shown in Table1.


## 2.1. True Triaxial Electrohydraulic Servo Test System

The tests presented in this study are carried out using the self-developed true triaxial compression test system (Figure1). The test system is primarily composed of a maximum principal stress, intermediate principal stress, minimum principal stress (σ1,σ2,σ3) three-dimensional servo control load system, true triaxial pressure cell, automatic data acquisition system, and AE monitoring system. The σ1,σ2,σ3 three-dimensional servo control load system forms an independent load frame. Piston rods are installed at one end of the three load frames, which is externally connected to the servo valve to produce independent servo loading. The three loading frames are perpendicular to each other. The positions of loading frames σ1 and σ3 are fixed, while the third loading frame (σ2) is placed on a horizontal guide rail and has the able to slide along the guide rail to facilitate the operation and placement of the sample. The other end of the two loading frames (σ2 and σ3) is installed with reaction screws, which have an adjustable length to adapt to different specimen sizes.Figure 1
True triaxial compression test system.Specimens used in this study are cuboids with a size of 50 × 50 × 100 mm. The specimen is placed in a cubic pressure chamber composed of a fully rigid structure, which consists of a box-shaped body and a loading plate. In order to avoid deformation of the rock samples leading to mutual extrusion of the loading plate during the test, a gap is left between the vertical and horizontal plates in the traditional true triaxial pressure chamber. This gap causes the uneven stress and strain in the test, resulting in a decrease in peak strength and an abnormal failure pattern in the sample [16]. In order to overcome this defect, the true triaxial pressure cell adopts a rotary interlocking platen design, in which the loading plates on the 6 faces of the sample are arranged in a misplaced method. When the specimen is compressed, the loading plate moves synchronously with the specimen to avoid interference effects in each direction (Figure 1). The relative displacement between the loading plate and the specimen produces a friction force, which negatively influences the test result. To reduce the friction between the loading plate and the sample, a 0.5 mm thick layer of lubricant is applied between the loading plate and the sample surface. The lubricant is composed of a 1 : 1 ratio of stearic acid and Vaseline [17].A spoke-type pressure sensor with a precision of 0.01 kN is installed between the piston rod and loading block. A rope-type displacement sensor with an accuracy of 0.002 mm and a measurement range of 1,000 mm is fixed to the piston rod and reaction force screw. A DS2-8B full information acoustic emission signal analyzer is used to examine the AE activity during tests. The AE system used in this study mainly consists of the AE host, preamplifier, and AE sensor. The type of the AE sensor is RS-54A, and its monitoring frequency range is 100–900 kHz. The AE sensor is installed in the loading plate of the true triaxial pressure cell (an 8 mm diameter hole is set in the side of the plate). The AE signal line is drawn out from the hole in the side of the plate, and the probe is in direct contact with the surface of the sandstone to ensure the accuracy of the AE data. The model of AE preamplifier is 20/40/60 dB which can provide three different transmission gains; furthermore, it has the advantages of low noise, wide bandwidth, impact resistance, and small volume. During the test, the threshold is set to 40 dB to avoid the possibility of electronic/environmental noise. The AE signal is amplified by the preamplifier with the gain of 40 dB. The time parameters for AE waveforms included peak definition time (PDT), hit definition time (HDT), and hit lockout (HLT), which are set to 50, 100, and 100μs, respectively.


## 2.2. Specimen Preparation

Sandstone is selected for examination in this study. Specimens are composed of feldspar and quartz, as well as a small amount of montmorillonite. The sandstone has a medium grain structure with grain size ranging from 0.1 to 0.35 mm and a relatively dense massive structure. Grains are relatively uniform with an identical macroscopic uniformity and an average density of 2380 kg/m3. Samples are 50 × 50 × 100 mm in size, and the end faces are carefully polished. Detailed sample information is given in Table 1.Table 1
Tested sandstone specimens and conditions in this research.
Loading program
Group
Specimen number
Dimensions (mm)
Quality (g)
σ
2 (MPa)
σ
3 (MPa)
CTT
C1
C1-1
100.2 × 50.0 × 49.8
622.5
10
10
C1-2
100.2 × 49.8 × 50.1
613.2
C1-3
100.3 × 50.1 × 49.9
612.9
C2
C2-1
100.0 × 49.5 × 50.0
613.0
20
20
C2-2
100.5 × 49.5 × 50.0
613.5
C2-3
99.8 × 50.3 × 50.3
605.1
C3
C3-1
100.3 × 50.0 × 50.1
636.0
30
30
C3-2
99.9 × 50.1 × 50.2
633.9
C3-3
99.6 × 50.3 × 49.6
602.3
C4
C4-1
99.9 × 50.1 × 50.2
625.3
40
40
C4-2
99.8 × 50.2 × 50.1
629.5
C4-3
100.2 × 50.0 × 49.2
639.7
TTT (σ3=10MPa)
T2
T2-1
100.1 × 50.3 × 49.8
628.1
20
10
T2-2
100.4 × 50.5 × 50.1
636.0
T2-3
100.3 × 50.6 × 50.1
646.6
T3
T3-1
100.4 × 49.9 × 50.3
630.5
30
10
T3-2
100.0 × 50.2 × 50.1
650.0
T3-3
99.8 × 50.4 × 50.5
624.7
T4
T4-1
100.4 × 49.6 × 49.6
618.8
40
10
T4-2
100.3 × 49.7 × 49.9
626.7
T4-3
100.1 × 50.4 × 50.6
607.1
T5
T5-1
100.1 × 50.1 × 49.9
615.6
50
10
T5-2
99.9 × 49.5 × 50.2
610.8
T5-3
100.2 × 50.1 × 49.9
614.5
T6
T6-1
99.7 × 49.7 × 50.3
610.4
60
10
T6-2
99.8 × 49.9 × 50.2
606.3
T6-3
100.1 × 49.8 × 50.2
595.7
TTT (σ3=20MPa)
T7
T7-1
100.3 × 49.9 × 50.2
630.5
30
20
T7-2
99.8 × 50.4 × 50.5
624.2
T7-3
99.9 × 49.5 × 50.2
612.3
T8
T8-1
100.4 × 49.8 × 50.2
599.8
40
20
T8-2
100.1 × 50.4 × 50.3
603.1
T8-3
99.7 × 50.1 × 50.3
615.4
T9
T9-1
100.3 × 49.7 × 49.9
621.7
50
20
T9-2
100.1 × 50.1 × 49.9
615.6
T9-3
100.0 × 49.7 × 50.1
630.1
T10
T10-1
99.8 × 49.9 × 50.4
626.3
60
20
T10-2
100.2 × 50.1 × 49.9
624.6
T10-3
100.3 × 49.8 × 49.9
621.7


## 2.3. Testing Procedure

In order to study the mechanical properties and AE characteristics of rock under true triaxial compression, the difference between the true triaxial test (TTT) and conventional triaxial test (CTT) is compared and the influence of different triaxial stresses on rock mechanical properties and AE characteristics is analyzed. The test setup is shown in Figure2.Figure 2
The two different loading programs: (a) CTT; (b) TTT.
(a)
(b)


### 2.3.1. CTT

Pressure is applied to the specimen at a rate of 0.2 MPa/s up to a predetermined confining pressure (set values ofσ2 and σ3), while the specimen is maintained in a hydrostatic pressure state (oa); then, with constant confining pressure, the axial stress is applied to the specimen at a loading rate of 0.002 mm/s, until the specimen failed (ab).


### 2.3.2. TTT

Pressure is applied to the specimen at a rate of 0.2 MPa/s up to a predetermined confining pressure (set value ofσ3), while the specimen is maintained in a hydrostatic pressure state (oa,oa′); then, with constant σ3, the values of σ1 and σ2 are continuously increased to the set value of σ2 (ab,a′b′) at the same rate. Afterwards with constant σ2 and σ3, the axial stress is applied to the specimen at a loading rate of 0.002 mm/s, until the specimen failed (bc,b′c′).Since the test discreteness is considered in this study, each test is repeated 2 times. The specific test scheme is shown in Table1.


## 2.3.1. CTT

Pressure is applied to the specimen at a rate of 0.2 MPa/s up to a predetermined confining pressure (set values ofσ2 and σ3), while the specimen is maintained in a hydrostatic pressure state (oa); then, with constant confining pressure, the axial stress is applied to the specimen at a loading rate of 0.002 mm/s, until the specimen failed (ab).


## 2.3.2. TTT

Pressure is applied to the specimen at a rate of 0.2 MPa/s up to a predetermined confining pressure (set value ofσ3), while the specimen is maintained in a hydrostatic pressure state (oa,oa′); then, with constant σ3, the values of σ1 and σ2 are continuously increased to the set value of σ2 (ab,a′b′) at the same rate. Afterwards with constant σ2 and σ3, the axial stress is applied to the specimen at a loading rate of 0.002 mm/s, until the specimen failed (bc,b′c′).Since the test discreteness is considered in this study, each test is repeated 2 times. The specific test scheme is shown in Table1.


## 3. Strength and Deformation Behavior

### 3.1. Strength and Criterion Verification

In order to analyze the strength characteristics of sandstone under conventional triaxial and true triaxial compression and to further verify the accuracy of the test results for the triaxial test system, rock strength and the strength-fitting curve under the two loading conditions are obtained (Figures3 and 4). The peak strength increases linearly with increasing confining pressure under conventional triaxial compression (Figure 3). The Mohr–Coulomb criterion is used to fit the relationship between the confining pressure and rock strength under the conventional triaxial compression. The fitting correlation coefficient is R2 = 0.994, which reflects the relationship between rock strength and varying confining pressure under conventional triaxial compression.Figure 3
Peak strengthσ1 as a function of the minimum principal stress σ3 under conventional triaxial stress and the best-fit linear function strength criterion (Mohr–Coulomb) for sandstone specimens.Figure 4
True triaxial strength for sandstone: (a) variation of peak compressive stressσ1 as a function of σ2 for the constant values of σ3=10MPa and σ3=20MPa; kpeak represents the slope of the peak strength with increasing σ2; (b) the best-fit linear function true triaxial strength criterion (Mogi–Coulomb Criterion) based on all experimental results in terms of τoct versus σm,2.
(a)
(b)Because the influence of the intermediate principal stress on the rock strength is not considered in the Mohr–Coulomb criterion, the Mohr–Coulomb criterion is only applicable to the special case in whichσ2=σ3, and it cannot accurately reflect the strength characteristics of the rock. Domestic and foreign scholars have come to realize the significance of the intermediate principal stress on ultimate compressive strength. Mogi [5, 6], Al-Ajmi and Zimmerman [18], Oku et al. [19], Lee and Haimson [20], and Chang and Haimson [21] considered that a true triaxial failure criterion can be expressed as a linear function (the Mogi–Coulomb criterion) between octahedral shear stress τoct and the average effective normal stress acting on the shear plane σm,2:(1)τoct=a+bσm,2,(2)τoct=13σ1−σ22+σ2−σ32+σ1−σ32,(3)σm,2=σ1+σ32,where τoct and σm,2 are the octahedral shear stress and the average effective normal stress on the shear plane, respectively, and a is the intercept of Mogi’s linear criterion fitting straight line and τoct axis, and b is the slope of straight line.By comparing the Mohr–Coulomb criterion and Mogi–Coulomb criterion, the relationship between them is concluded:(4)a=223ccosφ,b=223sinφ.According to the suggestions from ISRM, the Mogi–Coulomb criterion is used to fit the strength of the rock (including conventional and true triaxial strength) (Figure4(b)). According to the fitting results, the Mogi–Coulomb criterion (R2 = 0.964) reflects the strength characteristics of the rock under true triaxial loading conditions relatively well and is consistent with the test results of Mogi, Haimson, and Chang. These results are further verified by the reliability and accuracy of the experimental system used in this study.The values ofc and φ of the sandstone under two different strength criteria are shown in Table 2 and are obtained by combining (1)–(4) according to the parameters in the fitting equations in Figures 3 and 4(b). The fitting coefficient of the two strength criteria is high. Therefore, the values of c and φ of the sandstone are relatively close to each other. The Mogi–Coulomb criterion applies to both the conventional and true triaxial test and is more universal. The Mohr–Coulomb criterion is a special case of the Mogi–Coulomb criterion.Table 2
Fitting results and parameter values of two strength criteria of sandstone.
Group
Criterion
c (MPa)
φ (°)
R
2
CTT
Mohr–Coulomb
21.6
37.7
0.994
TTT
Mogi–Coulomb
18.4
40.5
0.964The strength characteristics of rock under true triaxial compression are shown in Figure4(a), and the variation is different from that of conventional triaxial condition. The peak rock strength increases and then decreases with increasing σ2 under true triaxial loading. When σ3=10MPa and 20 MPa, the peak rock strength first increases to the maximum and then gradually decreases with increasing σ2. Inflection points are located at 40 MPa and 50 MPa, respectively. The ascending segment tends to increase linearly with increasing σ2, and the slopes of the peak strength are 1.208 and 1.993, indicating that improving σ3 has a more significant protective effect on the rock.


### 3.2. Stress-Strain Curve

In this study, the stress-strain curves under conventional and true triaxial compression with different confining pressure conditions are shown in Figures5 and 6, where the yellow origin represents the peak strength of the rock. Rock stress-strain characteristics for four confining pressures under conventional triaxial compression are similar (Figure 5). The steepness of the curve in the elastic deformation stage is greatly affected by the confining pressure. The larger the confining pressure, the steeper the curve is, which indicates that the elastic modulus of the sandstone increases with increasing of confining pressure. The difference in steepness of the lateral curve is not obvious, which shows that the effect of confining pressure on the steepness of the lateral deformation of sandstone is small. The axial strain, lateral strain, and peak strength of the rock increase with increasing confining pressure, which shows that the increase in the conventional triaxial confining pressure improves the ability of the rock to withstand deformation and loading in both the axial and lateral directions.Figure 5
Stress-strain curves for sandstone from conventional triaxial compression experiments.Figure 6
Differential stress (σ1−σ3)-strain relations for sandstone under true triaxial compression conditions with a constant σ3 of 10 MPa: (a) differential stress (σ1−σ3)-axial strain (ε1); (b) differential stress (σ1−σ3)-lateral strain (ε2); (c) differential stress (σ1−σ3)-lateral strain (ε2); (d) differential stress (σ1−σ3)-volumetric strain (ε2).
(a)
(b)
(c)
(d)Figures6(a)–6(d), respectively, show the relationships between the differential stress (σ1−σ3) and principal strain (ε1, ε2, and ε3) as well as volumetric strain (εV) under true triaxial compression. Considering that different intermediate principal stresses under true triaxial compression are taken, the test process is divided into one section with a loading by force (o→aa′→bb′) and other section with a loading by displacement (bb′→cc′). To facilitate the analysis, the rock’s stress-strain characteristics are only analyzed for the loading by the displacement section.The condition of the rock in the initial compaction stage is not obvious due to the effect of loading (Figure6(a)). The rock enters the elastic stage quickly, and the axial curve is linear under different σ2 conditions. The steepness of the axial curve is effected by σ2. The steepness of the axial curve increases and gradually decreases with increasing σ2. The variation law is different from that of conventional triaxial compression. The elastic modulus gradually increases and then decreases with increasing σ2, and the variation law is consistent with peak strength. The strain during loading by displacement gradually reduces with increasing σ2, and the lateral expansion of the rock gradually plays a more dominant role.The strain curve in the direction of intermediate principal stress gradually steepens with increasingσ2 (Figure 6(b)). The increase of σ2 limits the deformation (expansion) in the σ2 direction. After the peak stress is reached, the rock experiences compression in the intermediate principal stress direction especially when σ2 equals 50 MPa or 60 MPa, indicating that brittle failure of the rock occurs under such intermediate principal stress conditions, and the rock extends rapidly along the minimum principal stress direction, which leads to the rapidly compressed in the intermediate principal stress direction after the peak.The strain curve in the direction of minimum principal stress steepens and gradually shallows with increasingσ2 (Figure 6(c)). The lateral expansion of the rock mainly in the direction of the minimum principal stress is due to the increase of σ2. Because σ2>σ3, the expansion of the rock in the direction of minimum principal stress is promoted by increasing σ2. At the same time, the lateral expansion of rock could be limited by σ2, which can be regarded as the confining pressure. Results show that when σ2 is 20 MPa∼40 MPa, the increase of σ2 plays a protective role for the rock and limits deformation in the direction of σ3 (lateral restraint effect). When σ2 is 50 MPa or 60 MPa, the increase of σ2 plays a damaging role for the rock and accelerates deformation in the direction of σ3 (damage effect).In Figure6(d), the change in rock volumetric strain (εV) is shown. εV first increases and then decreases, showing that the rock undergoes gradual compression first, and then begins to expand with increasing differential stress. The inflection point of εV is the point where rock volume stops compression and begins to expansion. The inflection point of volumetric strain is more easily reached with the increasing σ2.The stress-strain curve of the rock under atσ3=20MPa is similar to that of σ3=10MPa (Figure 7). The rock axial strain curve is steeper, which indicates that the elastic modulus of the rock increases. The peak strength, axial strain, and lateral strain of the rock increase with increasing σ3, and the strain corresponding to the inflection point of εV greatly improved. The increase of σ3 enhances the ability of the rock to bear stress and improves the carrying capacity of the rock.Figure 7
Stress-strain curve of rock at varyingσ3 (σ2=30MPa).The variation of elastic modulus under conventional triaxial and true triaxial compression is shown in Figure8. The elastic modulus increases with increasing confining pressure under conventional triaxial compression (Figure 8(a)). The variation is similar to that of strength, which shows that there are microcracks in the rock and the confining pressure closes the cracks to some extent, causing some slippage between cracks. If the confining pressure increases, cracks would close more fully, and slippage between the cracks decreases. The increase of confining pressure makes slip along the cracks reduce due to the inhibition of the frictional force. Therefore, the axial deformation of the specimen decreases, and the elastic modulus of the rock increases.Figure 8
Influence of the confining pressure on Young’s modulus of sandstone: (a) CTT; (b) TTT.
(a)
(b)The elastic modulus of the rock first increases and then decreases with increasingσ2 under true triaxial compression, in which σ3 is 10 and 20 MPa, and the values of the two peak points are both σ2=40MPa (Figure 8(b)). When σ2 is 20∼40 MPa, the number of slippage cracks in the rock is effectively reduced by improving σ2, and slip along the microcracks is inhibited, which leads to an increased elastic modulus. The effect of confining pressure is similar to that of conventional triaxial compression. When σ2 exceeds 40 MPa, the confining pressure promotes the expansion of microcracks in the rock, and the elastic modulus gradually decreases (Figure 6(c)). By comparing the variation laws of elastic modulus when σ3 is 10 MPa and 20 MPa, it is apparent that the latter elastic modulus is larger, indicating that the increase of σ3 makes the cracks close more fully and reduces slip. When σ2 is 40∼60 MPa, the decrease of the latter elastic modulus is slower than the former, which indicates that the increase of σ3, to some extent, slows down the variation trend of “first increase and then decrease” under true triaxial compression and plays a protective role to the rock.


### 3.3. Strength and Deformation Characteristic in Unstable Crack Growth Stage

The deformation properties of rock are closely related to the growth and development of internal cracks. The rock failure process is associated with the closure, fracture initiation, expansion, and interactive perforation of cracks. The failure process is divided into the following stages [22]: (1) microcrack closure compaction stage; (2) elastic deformation stage; (3) stable crack growth stage; (4) unstable crack growth stage; (5) postpeaking stage. When the axial stress gradually increases to the rock damage stress σcd, the stress-strain curve enters the unstable crack growth stage, marking the beginning of shear and expansion of the rock sample. In this stage, the cracks are unstable. Even if the rock is no longer loaded, the internal cracks in the rock specimen cannot hold stable for a significant amount of time and continue to expand over time. Therefore, the damage stress is also called the long-term strength of the rock. Examination of rock deformation, strength, and crack development in this stage is of great significance to understanding its failure mechanism. This study focuses on the analysis of rock strength, deformation, and acoustic emission characteristics in this stage.The variation rule of damage strength and peak strength under conventional triaxial and true triaxial compression is shown in Figure9. Damage strength σcd and peak strength σp of the rock under conventional triaxial compression increase with increasing confining pressure (Figure 9(a)). Because there are microcracks in the rock, development of cracks is effectively prevented, and the expansion of internal cracks of the rock is restrained laterally by improving confining pressure.Figure 9
Variation rule of rock damage strength and peak strength under conventional triaxial and true triaxial conditions.
(a)
(b)Variation of damage strengthσcd and peak strength σp for rocks under true triaxial compression (σ3=10MPa and 20 MPa) is different from rocks under conventional triaxial compression (Figure 9(b)). Damage strength σcd and peak strength σp both increase first and then decrease. When σ3 is 10 MPa, the damage stress σcd and peak strength σp show an obvious trend of “first increasing and then decreasing,” and the confining pressure constrains the expansion of internal cracks when σ2 is small. While the lateral restraint gradually transforms into a damaging force on the rock with increasing σ2, the expansion of internal cracks in the rock is promoted, leading to decreasing σcd and σp. By comparing variations in damage and peak strength under true triaxial compression, it is apparent that σ3 equal to 10 MPa and 20 MPa corresponds to σ2 equal to 40 MPa and 50 MPa, respectively. The latter values of σcd and σp are higher and vary more gradually than the former values. In particular, the values of σcd and σp decrease when the former σ2 exceeds 40 MPa, while the values of σcd and σp reduce slightly when the latter σ2 exceeds 50 MPa, indicating that the damage effect of the latter confining pressure is significantly weakened, and improving σ3 limits the development of internal cracks.The strain characteristics corresponding to damage and peak stress of the rock under conventional triaxial compression are shown in Figure10. The axial strain and lateral strain corresponding to damage and peak stress of the rock under conventional triaxial compression vary linearly with confining pressure, all of which increase with increasing confining pressure. Volumetric strain fluctuates greatly. The lateral strain of the peak point εp3 increases more quickly than the axial strain εp1 with increasing confining pressure, which indicates that the lateral strain dominates during the unstable crack growth stage and the admission value of lateral expansion increases. At the same time, the volumetric strain increment of the rock Δεv=εpv−εcdv in this stage increases with increasing confining pressure, which indicates that the ability of the rock to withstand deformation increases gradually due to increasing confining pressure under the conventional triaxial compression.Figure 10
Influence of the confining pressure on strain corresponding to each characteristic stress under conventional triaxial compression experiments: (a) axial strain; (b) lateral strain; (c) volumetric strain.
(a)
(b)
(c)Incremental strain variation in the unstable crack growth stage under the conventional triaxial compression is shown in Figure11. The axial strain increment Δε1 increases slightly with increasing confining pressure in this stage, while the lateral strain Δε3 increases quickly. The volumetric strain increment ΔεV increases quickly with increasing confining pressure due to Δεv=Δε1+2Δε3, which indicates that the lateral expansion of the rock in the unstable crack growth stage under the conventional triaxial compression is obvious, and the ability to withstand lateral deformation increases significantly due to increasing confining pressure.Figure 11
Variation in strain increment in the unstable crack growth stage under the conventional triaxial compression.The strain characteristics of rock damage point and peak point under true triaxial condition are shown in Figure12. Because the deformation characteristics of the rock in the loading segment by displacement are only analyzed, the values of σ3 and σ2 are 20 MPa under the conventional triaxial compression in this study, and the strain law is not discussed because of its abnormality. Axial strains of the damage point and the peak point under true triaxial compression decrease with increasing σ2. Variation is similar when σ3 is 10 and 20 MPa, and the latter axial strains of the damage point and peak point εcd1 and εp1 are higher than the former ones. The lateral strains εcd2 and εp2 in the direction of σ2 decrease gradually with increasing σ2. Deformation in the rock reflects expansion in the σ2 direction, and the expansion is limited by an increasing σ2. The restriction is more pronounced when σ3 equals 20 MPa (Figure 12(b)). The change laws of the two kinds of conditions where σ3 is 10 or 20 MPa are different in the direction of σ3 (Figure 12(c)). When σ3 is 10 MPa, the lateral strain in the minimum principal stress direction εp3 (expansion) of the rock peak point first increases and then decreases with increasing σ2. When σ2 is 50 MPa or 60 MPa, the lateral strain exhibits a sharp decrease, indicating that the rock reaches the peak point at a fast speed and its ability to withstand deformation is significantly weakened. When σ3 is 20 MPa, the lateral strain εp3 (expansion) of the rock peak point first increases and then decreases with increasing σ2. Here, while σ2 is 60 MPa, the lateral strain decreases slightly, indicating that the lateral strain εp3 of the peak point reduces the ability to withstand lateral strain. However, due to increasing σ3, the rock suffers only a slight decrease in lateral deformation capacity. The law of rock volumetric strain is similar to that in the minimum principal stress direction (Figure 12(d)). When σ3 is 10 MPa, the peak volumetric strain εpv first increases and then decreases with increasing σ2. When σ3 is 20 MPa, the peak volumetric strain εpv slightly decreases with increasing σ2, whose value is higher than the former one (σ3=10MPa), which indicates that the rock’s ability to support expansion and deformation could be improved by increasing σ3.Figure 12
Strain characteristics of rock damage point and peak point under true triaxial compression: (a) axial strain; (b) lateral strain (the intermediate principal stress direction); (c) lateral strain (the minimum principal stress direction); (d) volumetric strain.
(a)
(b)
(c)
(d)The variation in strain increment in the unstable stage under true triaxial compression is shown in Figure13. The axial strain increment Δε1 and lateral strain increment Δε2 in the intermediate principal stress direction in this stage are similar when σ3 is 10 and 20 MPa and decreases with increasing σ2. The difference in strain increment Δε3 in the direction of the minimum principal stress is large. Strain increments first increase and then decrease, but the former decreases significantly when σ2 is 50 MPa and 60 MPa, which indicates that the ability to bear lateral deformation significantly decreases due to damage from confining pressure. The latter decreases slightly when σ2 is 60 MPa, which indicates that the damage caused by confining pressure could be significantly weakened due to an increase in σ3. The volumetric strain increment is similar to that in the direction of σ3.Figure 13
Variation in strain increment of rock in the unstable crack growth stage under true triaxial compression.


## 3.1. Strength and Criterion Verification

In order to analyze the strength characteristics of sandstone under conventional triaxial and true triaxial compression and to further verify the accuracy of the test results for the triaxial test system, rock strength and the strength-fitting curve under the two loading conditions are obtained (Figures3 and 4). The peak strength increases linearly with increasing confining pressure under conventional triaxial compression (Figure 3). The Mohr–Coulomb criterion is used to fit the relationship between the confining pressure and rock strength under the conventional triaxial compression. The fitting correlation coefficient is R2 = 0.994, which reflects the relationship between rock strength and varying confining pressure under conventional triaxial compression.Figure 3
Peak strengthσ1 as a function of the minimum principal stress σ3 under conventional triaxial stress and the best-fit linear function strength criterion (Mohr–Coulomb) for sandstone specimens.Figure 4
True triaxial strength for sandstone: (a) variation of peak compressive stressσ1 as a function of σ2 for the constant values of σ3=10MPa and σ3=20MPa; kpeak represents the slope of the peak strength with increasing σ2; (b) the best-fit linear function true triaxial strength criterion (Mogi–Coulomb Criterion) based on all experimental results in terms of τoct versus σm,2.
(a)
(b)Because the influence of the intermediate principal stress on the rock strength is not considered in the Mohr–Coulomb criterion, the Mohr–Coulomb criterion is only applicable to the special case in whichσ2=σ3, and it cannot accurately reflect the strength characteristics of the rock. Domestic and foreign scholars have come to realize the significance of the intermediate principal stress on ultimate compressive strength. Mogi [5, 6], Al-Ajmi and Zimmerman [18], Oku et al. [19], Lee and Haimson [20], and Chang and Haimson [21] considered that a true triaxial failure criterion can be expressed as a linear function (the Mogi–Coulomb criterion) between octahedral shear stress τoct and the average effective normal stress acting on the shear plane σm,2:(1)τoct=a+bσm,2,(2)τoct=13σ1−σ22+σ2−σ32+σ1−σ32,(3)σm,2=σ1+σ32,where τoct and σm,2 are the octahedral shear stress and the average effective normal stress on the shear plane, respectively, and a is the intercept of Mogi’s linear criterion fitting straight line and τoct axis, and b is the slope of straight line.By comparing the Mohr–Coulomb criterion and Mogi–Coulomb criterion, the relationship between them is concluded:(4)a=223ccosφ,b=223sinφ.According to the suggestions from ISRM, the Mogi–Coulomb criterion is used to fit the strength of the rock (including conventional and true triaxial strength) (Figure4(b)). According to the fitting results, the Mogi–Coulomb criterion (R2 = 0.964) reflects the strength characteristics of the rock under true triaxial loading conditions relatively well and is consistent with the test results of Mogi, Haimson, and Chang. These results are further verified by the reliability and accuracy of the experimental system used in this study.The values ofc and φ of the sandstone under two different strength criteria are shown in Table 2 and are obtained by combining (1)–(4) according to the parameters in the fitting equations in Figures 3 and 4(b). The fitting coefficient of the two strength criteria is high. Therefore, the values of c and φ of the sandstone are relatively close to each other. The Mogi–Coulomb criterion applies to both the conventional and true triaxial test and is more universal. The Mohr–Coulomb criterion is a special case of the Mogi–Coulomb criterion.Table 2
Fitting results and parameter values of two strength criteria of sandstone.
Group
Criterion
c (MPa)
φ (°)
R
2
CTT
Mohr–Coulomb
21.6
37.7
0.994
TTT
Mogi–Coulomb
18.4
40.5
0.964The strength characteristics of rock under true triaxial compression are shown in Figure4(a), and the variation is different from that of conventional triaxial condition. The peak rock strength increases and then decreases with increasing σ2 under true triaxial loading. When σ3=10MPa and 20 MPa, the peak rock strength first increases to the maximum and then gradually decreases with increasing σ2. Inflection points are located at 40 MPa and 50 MPa, respectively. The ascending segment tends to increase linearly with increasing σ2, and the slopes of the peak strength are 1.208 and 1.993, indicating that improving σ3 has a more significant protective effect on the rock.


## 3.2. Stress-Strain Curve

In this study, the stress-strain curves under conventional and true triaxial compression with different confining pressure conditions are shown in Figures5 and 6, where the yellow origin represents the peak strength of the rock. Rock stress-strain characteristics for four confining pressures under conventional triaxial compression are similar (Figure 5). The steepness of the curve in the elastic deformation stage is greatly affected by the confining pressure. The larger the confining pressure, the steeper the curve is, which indicates that the elastic modulus of the sandstone increases with increasing of confining pressure. The difference in steepness of the lateral curve is not obvious, which shows that the effect of confining pressure on the steepness of the lateral deformation of sandstone is small. The axial strain, lateral strain, and peak strength of the rock increase with increasing confining pressure, which shows that the increase in the conventional triaxial confining pressure improves the ability of the rock to withstand deformation and loading in both the axial and lateral directions.Figure 5
Stress-strain curves for sandstone from conventional triaxial compression experiments.Figure 6
Differential stress (σ1−σ3)-strain relations for sandstone under true triaxial compression conditions with a constant σ3 of 10 MPa: (a) differential stress (σ1−σ3)-axial strain (ε1); (b) differential stress (σ1−σ3)-lateral strain (ε2); (c) differential stress (σ1−σ3)-lateral strain (ε2); (d) differential stress (σ1−σ3)-volumetric strain (ε2).
(a)
(b)
(c)
(d)Figures6(a)–6(d), respectively, show the relationships between the differential stress (σ1−σ3) and principal strain (ε1, ε2, and ε3) as well as volumetric strain (εV) under true triaxial compression. Considering that different intermediate principal stresses under true triaxial compression are taken, the test process is divided into one section with a loading by force (o→aa′→bb′) and other section with a loading by displacement (bb′→cc′). To facilitate the analysis, the rock’s stress-strain characteristics are only analyzed for the loading by the displacement section.The condition of the rock in the initial compaction stage is not obvious due to the effect of loading (Figure6(a)). The rock enters the elastic stage quickly, and the axial curve is linear under different σ2 conditions. The steepness of the axial curve is effected by σ2. The steepness of the axial curve increases and gradually decreases with increasing σ2. The variation law is different from that of conventional triaxial compression. The elastic modulus gradually increases and then decreases with increasing σ2, and the variation law is consistent with peak strength. The strain during loading by displacement gradually reduces with increasing σ2, and the lateral expansion of the rock gradually plays a more dominant role.The strain curve in the direction of intermediate principal stress gradually steepens with increasingσ2 (Figure 6(b)). The increase of σ2 limits the deformation (expansion) in the σ2 direction. After the peak stress is reached, the rock experiences compression in the intermediate principal stress direction especially when σ2 equals 50 MPa or 60 MPa, indicating that brittle failure of the rock occurs under such intermediate principal stress conditions, and the rock extends rapidly along the minimum principal stress direction, which leads to the rapidly compressed in the intermediate principal stress direction after the peak.The strain curve in the direction of minimum principal stress steepens and gradually shallows with increasingσ2 (Figure 6(c)). The lateral expansion of the rock mainly in the direction of the minimum principal stress is due to the increase of σ2. Because σ2>σ3, the expansion of the rock in the direction of minimum principal stress is promoted by increasing σ2. At the same time, the lateral expansion of rock could be limited by σ2, which can be regarded as the confining pressure. Results show that when σ2 is 20 MPa∼40 MPa, the increase of σ2 plays a protective role for the rock and limits deformation in the direction of σ3 (lateral restraint effect). When σ2 is 50 MPa or 60 MPa, the increase of σ2 plays a damaging role for the rock and accelerates deformation in the direction of σ3 (damage effect).In Figure6(d), the change in rock volumetric strain (εV) is shown. εV first increases and then decreases, showing that the rock undergoes gradual compression first, and then begins to expand with increasing differential stress. The inflection point of εV is the point where rock volume stops compression and begins to expansion. The inflection point of volumetric strain is more easily reached with the increasing σ2.The stress-strain curve of the rock under atσ3=20MPa is similar to that of σ3=10MPa (Figure 7). The rock axial strain curve is steeper, which indicates that the elastic modulus of the rock increases. The peak strength, axial strain, and lateral strain of the rock increase with increasing σ3, and the strain corresponding to the inflection point of εV greatly improved. The increase of σ3 enhances the ability of the rock to bear stress and improves the carrying capacity of the rock.Figure 7
Stress-strain curve of rock at varyingσ3 (σ2=30MPa).The variation of elastic modulus under conventional triaxial and true triaxial compression is shown in Figure8. The elastic modulus increases with increasing confining pressure under conventional triaxial compression (Figure 8(a)). The variation is similar to that of strength, which shows that there are microcracks in the rock and the confining pressure closes the cracks to some extent, causing some slippage between cracks. If the confining pressure increases, cracks would close more fully, and slippage between the cracks decreases. The increase of confining pressure makes slip along the cracks reduce due to the inhibition of the frictional force. Therefore, the axial deformation of the specimen decreases, and the elastic modulus of the rock increases.Figure 8
Influence of the confining pressure on Young’s modulus of sandstone: (a) CTT; (b) TTT.
(a)
(b)The elastic modulus of the rock first increases and then decreases with increasingσ2 under true triaxial compression, in which σ3 is 10 and 20 MPa, and the values of the two peak points are both σ2=40MPa (Figure 8(b)). When σ2 is 20∼40 MPa, the number of slippage cracks in the rock is effectively reduced by improving σ2, and slip along the microcracks is inhibited, which leads to an increased elastic modulus. The effect of confining pressure is similar to that of conventional triaxial compression. When σ2 exceeds 40 MPa, the confining pressure promotes the expansion of microcracks in the rock, and the elastic modulus gradually decreases (Figure 6(c)). By comparing the variation laws of elastic modulus when σ3 is 10 MPa and 20 MPa, it is apparent that the latter elastic modulus is larger, indicating that the increase of σ3 makes the cracks close more fully and reduces slip. When σ2 is 40∼60 MPa, the decrease of the latter elastic modulus is slower than the former, which indicates that the increase of σ3, to some extent, slows down the variation trend of “first increase and then decrease” under true triaxial compression and plays a protective role to the rock.


## 3.3. Strength and Deformation Characteristic in Unstable Crack Growth Stage

The deformation properties of rock are closely related to the growth and development of internal cracks. The rock failure process is associated with the closure, fracture initiation, expansion, and interactive perforation of cracks. The failure process is divided into the following stages [22]: (1) microcrack closure compaction stage; (2) elastic deformation stage; (3) stable crack growth stage; (4) unstable crack growth stage; (5) postpeaking stage. When the axial stress gradually increases to the rock damage stress σcd, the stress-strain curve enters the unstable crack growth stage, marking the beginning of shear and expansion of the rock sample. In this stage, the cracks are unstable. Even if the rock is no longer loaded, the internal cracks in the rock specimen cannot hold stable for a significant amount of time and continue to expand over time. Therefore, the damage stress is also called the long-term strength of the rock. Examination of rock deformation, strength, and crack development in this stage is of great significance to understanding its failure mechanism. This study focuses on the analysis of rock strength, deformation, and acoustic emission characteristics in this stage.The variation rule of damage strength and peak strength under conventional triaxial and true triaxial compression is shown in Figure9. Damage strength σcd and peak strength σp of the rock under conventional triaxial compression increase with increasing confining pressure (Figure 9(a)). Because there are microcracks in the rock, development of cracks is effectively prevented, and the expansion of internal cracks of the rock is restrained laterally by improving confining pressure.Figure 9
Variation rule of rock damage strength and peak strength under conventional triaxial and true triaxial conditions.
(a)
(b)Variation of damage strengthσcd and peak strength σp for rocks under true triaxial compression (σ3=10MPa and 20 MPa) is different from rocks under conventional triaxial compression (Figure 9(b)). Damage strength σcd and peak strength σp both increase first and then decrease. When σ3 is 10 MPa, the damage stress σcd and peak strength σp show an obvious trend of “first increasing and then decreasing,” and the confining pressure constrains the expansion of internal cracks when σ2 is small. While the lateral restraint gradually transforms into a damaging force on the rock with increasing σ2, the expansion of internal cracks in the rock is promoted, leading to decreasing σcd and σp. By comparing variations in damage and peak strength under true triaxial compression, it is apparent that σ3 equal to 10 MPa and 20 MPa corresponds to σ2 equal to 40 MPa and 50 MPa, respectively. The latter values of σcd and σp are higher and vary more gradually than the former values. In particular, the values of σcd and σp decrease when the former σ2 exceeds 40 MPa, while the values of σcd and σp reduce slightly when the latter σ2 exceeds 50 MPa, indicating that the damage effect of the latter confining pressure is significantly weakened, and improving σ3 limits the development of internal cracks.The strain characteristics corresponding to damage and peak stress of the rock under conventional triaxial compression are shown in Figure10. The axial strain and lateral strain corresponding to damage and peak stress of the rock under conventional triaxial compression vary linearly with confining pressure, all of which increase with increasing confining pressure. Volumetric strain fluctuates greatly. The lateral strain of the peak point εp3 increases more quickly than the axial strain εp1 with increasing confining pressure, which indicates that the lateral strain dominates during the unstable crack growth stage and the admission value of lateral expansion increases. At the same time, the volumetric strain increment of the rock Δεv=εpv−εcdv in this stage increases with increasing confining pressure, which indicates that the ability of the rock to withstand deformation increases gradually due to increasing confining pressure under the conventional triaxial compression.Figure 10
Influence of the confining pressure on strain corresponding to each characteristic stress under conventional triaxial compression experiments: (a) axial strain; (b) lateral strain; (c) volumetric strain.
(a)
(b)
(c)Incremental strain variation in the unstable crack growth stage under the conventional triaxial compression is shown in Figure11. The axial strain increment Δε1 increases slightly with increasing confining pressure in this stage, while the lateral strain Δε3 increases quickly. The volumetric strain increment ΔεV increases quickly with increasing confining pressure due to Δεv=Δε1+2Δε3, which indicates that the lateral expansion of the rock in the unstable crack growth stage under the conventional triaxial compression is obvious, and the ability to withstand lateral deformation increases significantly due to increasing confining pressure.Figure 11
Variation in strain increment in the unstable crack growth stage under the conventional triaxial compression.The strain characteristics of rock damage point and peak point under true triaxial condition are shown in Figure12. Because the deformation characteristics of the rock in the loading segment by displacement are only analyzed, the values of σ3 and σ2 are 20 MPa under the conventional triaxial compression in this study, and the strain law is not discussed because of its abnormality. Axial strains of the damage point and the peak point under true triaxial compression decrease with increasing σ2. Variation is similar when σ3 is 10 and 20 MPa, and the latter axial strains of the damage point and peak point εcd1 and εp1 are higher than the former ones. The lateral strains εcd2 and εp2 in the direction of σ2 decrease gradually with increasing σ2. Deformation in the rock reflects expansion in the σ2 direction, and the expansion is limited by an increasing σ2. The restriction is more pronounced when σ3 equals 20 MPa (Figure 12(b)). The change laws of the two kinds of conditions where σ3 is 10 or 20 MPa are different in the direction of σ3 (Figure 12(c)). When σ3 is 10 MPa, the lateral strain in the minimum principal stress direction εp3 (expansion) of the rock peak point first increases and then decreases with increasing σ2. When σ2 is 50 MPa or 60 MPa, the lateral strain exhibits a sharp decrease, indicating that the rock reaches the peak point at a fast speed and its ability to withstand deformation is significantly weakened. When σ3 is 20 MPa, the lateral strain εp3 (expansion) of the rock peak point first increases and then decreases with increasing σ2. Here, while σ2 is 60 MPa, the lateral strain decreases slightly, indicating that the lateral strain εp3 of the peak point reduces the ability to withstand lateral strain. However, due to increasing σ3, the rock suffers only a slight decrease in lateral deformation capacity. The law of rock volumetric strain is similar to that in the minimum principal stress direction (Figure 12(d)). When σ3 is 10 MPa, the peak volumetric strain εpv first increases and then decreases with increasing σ2. When σ3 is 20 MPa, the peak volumetric strain εpv slightly decreases with increasing σ2, whose value is higher than the former one (σ3=10MPa), which indicates that the rock’s ability to support expansion and deformation could be improved by increasing σ3.Figure 12
Strain characteristics of rock damage point and peak point under true triaxial compression: (a) axial strain; (b) lateral strain (the intermediate principal stress direction); (c) lateral strain (the minimum principal stress direction); (d) volumetric strain.
(a)
(b)
(c)
(d)The variation in strain increment in the unstable stage under true triaxial compression is shown in Figure13. The axial strain increment Δε1 and lateral strain increment Δε2 in the intermediate principal stress direction in this stage are similar when σ3 is 10 and 20 MPa and decreases with increasing σ2. The difference in strain increment Δε3 in the direction of the minimum principal stress is large. Strain increments first increase and then decrease, but the former decreases significantly when σ2 is 50 MPa and 60 MPa, which indicates that the ability to bear lateral deformation significantly decreases due to damage from confining pressure. The latter decreases slightly when σ2 is 60 MPa, which indicates that the damage caused by confining pressure could be significantly weakened due to an increase in σ3. The volumetric strain increment is similar to that in the direction of σ3.Figure 13
Variation in strain increment of rock in the unstable crack growth stage under true triaxial compression.


## 4. AE Behavior

AE is a phenomenon that microcracks generate during crack development, which is accompanied by the release of an elastic wave or stress wave. Deformation and failure of the rock are accompanied by the generation of the AE. AE information contains precursor information for rock failure. The AE information during compression and failure of the rock is used to predict the rock failure process, which is of great importance to monitoring deformation in mines and other geotechnical engineering [23].The typical curve of the cumulative AE events with the change of strain under different loading conditions is shown in Figure14. The characteristics of the rock in unstable crack growth stage (the same below) are only selected in this study. AE events increase sharply in this stage, and interactions among cracks increase. AE events weaken gradually with increasing confining pressure under conventional triaxial compression, and the cumulative AE counts decrease gradually when the peak value is reached. With a confining pressure of 10 MPa, the cumulative AE counts are significantly higher than the one with higher confining pressure. Since the constraint on lateral deformation with low confining pressure under conventional triaxial compression is small, it is easy to produce new microcracks and generate slip along cracks, creating a more active acoustic emission signal. High confining pressure reduces the generation of internal fractures and shear slip, producing relatively low AE counts. The higher the confining pressure, the stronger the binding effect is, which enhances the rock’s capacity to withstand deformation and loading.Figure 14
The typical curve of the cumulative number of AE events with the change in strain under different loading conditions: (a) CTT; (b) TTT (σ3=10MPa); (c) TTT (σ3=20MPa).
(a)
(b)
(c)AE count decreases first and then increases under true triaxial compression. Whenσ3 is 10 MPa, AE events increase with increasing σ2 (Figure 14(b)). When σ2 is 20∼40 MPa, the cumulative AE counts gradually reduce with increasing σ2. There is no “sudden increase” phenomenon, and the rock produces many continuous AE counts. In combination with the AE characteristics under conventional triaxial compression, numerous shear slip cracks are produced in this stage (which would be analyzed in detail below), and microcracks converge and perforate within the rock, leading to the formation of macroshear and slip surfaces. When σ2 is 50 and 60 MPa, the cumulative AE counts increase rapidly, and there is a “sudden increase” phenomenon indicating that the increase of σ2 inhibits the deformation in the σ2 direction but promotes the lateral expansion in the σ3 direction. The rock produces many tensile cracks whose convergence and perforation cause fracture surfaces to form. The confining pressure gradually evolves from a constraining force to a damaging force, greatly weakening the carrying capacity of the rock. When σ3 is 20 MPa, AE events change slightly with increasing σ2. Here, when σ2 is 30∼50 MPa, the typical AE event curves (Figure 14(c)) are similar to each other, and more shear and slip cracks are formed. The constraining effect of confining pressure to internal cracks is enhanced, and the development of cracks is suppressed by increasing σ2. When σ2 is 60 MPa, the cumulative AE counts increase rapidly, but there is no “sudden increase” phenomenon, indicating that the confining pressure damages the rock and promotes the development of rock cracks. For σ3=10MPa, the damage effect is greatly weakened by improving σ3 and the rock is protected.In order to further analyze the destruction of sandstone under different loading conditions, the value of RA (the ratio of the rise time and the amplitude) and the average frequency value AF (the ratio of the ringing count and the duration) of the AE parameters are used to describe the type of microcracks within the rock. In general, AE signals with low AF and high RA or with high AF and low RA usually represent the generation or development of shear or tensile cracks, respectively [24–26]. Energy levels are combined to analyze the distribution law for RA under three kinds of AE levels which are 0.1∼1, 1∼10, and >10 mV2·ms (ignore < 0.1) (Figure 15). Microcracks in the rock under the three loading conditions are stretched and expanded. The AE count which has an AE energy of over 10 mV2·ms, 1∼10 mV2·ms, and 0.1∼1 mV2·ms is small, large, and dense. According to the energy level, the proportion rule of the rock shear cracks under different loading conditions is shown in Figure 16.Figure 15
Variation of the RA and AF values under different loading conditions.Figure 16
Proportion characteristics of shear cracks under different loading conditions: (a) CTT; (b) TTT (σ3=10MPa); (c) TTT (σ3=20MPa).
(a)
(b)
(c)The proportion of the shear cracks under conventional triaxial compression with AE energy greater than 10 mV2·ms varied little with increasing confining pressure, which is in 27%∼32%. The proportion of shear cracks gradually increases (from 25% to 37%) with AE energy from 1 to 10 mV2·ms. The proportion of the shear cracks is relatively small and irregular with AE energy varying from 0.1 to 1 mV2·ms (only AE of more than 1 mV2·ms would be analyzed in the following paper). Although a former study believed that the rock sample reflects the shear failure characteristics under conventional triaxial compression at the macrolevel, analyses in this study show that the macroshear failure of the rock samples at the microlevel is caused by tension, expansion, or a large number of connecting internal microcracks. The presence of shear cracks is often accompanied by a large number of tensile cracks, which is consistent with the existing literature [27, 28]. The cracks need to overcome the confining pressure. The law of the proportion of shear cracks in which AE is over 1 mV2·ms increases with increasing confining pressure indicates that cracks gradually overcame the shear constraints of confining pressure. On the contrary, the shear slip is gradually inhibited due to increasing confining pressure and delaying penetration of the cracks and rupture of the rock surface, in affecting playing a protective role in rock.There are significant differences in the change laws for RA under true triaxial compression in whichσ3 is 10 and 20 MPa. When σ2 is 20∼40 MPa, the former (σ3=10MPa) proportion of the shear cracks gradually increases (from 28% to 35%) with increasing Δε3 with AE energy greater than 1Δεv. The role of confining pressure is similar to conventional triaxial compression. Slippage of rock cracks can effectively be inhibited by increasing σ2, which delays the formation of macroshear and slip surfaces and plays a protective role in the rock. When σ2 is 50 and 60 MPa, the proportion of shear cracks decreases dramatically (12% and 9% resp.) with its AE energy more than 1 mV2·ms, indicating that the rock records mainly tension and expansion, accompanied by a small amount of shear cracks. Tension cracks within the rock are constantly fused, and the macroscopic tension and failure surface instantaneously forms, which explains why the cumulative AE counts appeared as a “sudden increase” (Figure 14(b)). The effect of confining pressure on the cracks within the rock gradually evolves from a constraining force to a damage force. The latter (σ3=10MPa) has a relatively smooth change in RA. When σ2 is 20∼50 MPa, the proportion of the shear cracks gradually increases (from 26% to 37%) with increasing σ2 with AE energy of more than 1 mV2·ms. By comparing with σ3=10MPa, the proportion of shear cracks increases, indicating that the slip effect may be inhibited by increasing σ3 and the rock is restricted laterally. When σ2 is 60 MPa, the proportion of shear cracks (20%) in which AE energy is more than 1 mV2·ms decreases. It is significantly larger than the former proportion with conditions the same as σ2, indicating that the damage effect of confining pressure is weakened because of increasing σ3, although the rock is damaged by the confining pressure, to some extent, which led to the increase of the tensional cracks of the rock.


## 5. Conclusion

In this study, the conventional triaxial and true triaxial tests of sandstone under different stress states are carried out using the self-developed true triaxial electrohydraulic servo test system. Deformation, strength, AE characteristic parameters, and development of the internal cracks are studied under different confining pressures, intermediate principal stresses, and minimum principal stresses. The primary research results are summarized as follows:(1)
The strength of the sandstone increases with increasing confining pressure under conventional triaxial compression, while the strength of sandstone increases first and then decreases with increasing intermediate principal stress under true triaxial compression. The strength under the true triaxial condition is consistent with the Mogi–Coulomb criterion. The rock is protected obviously, and the peak strength of the rock is greatly improved by increasingσ3.(2)
The deformation characteristics of the rock under the two loading conditions are quite different. Under conventional triaxial compression, the axial strain and lateral strain as well as the elastic modulus increase with increasing confining pressure before failure, showing that the ability of the rock to withstand deformation capacity of the rock is gradually improved because of the increase in confining pressure under conventional triaxial compression. Under true triaxial compression (σ3 equal 10, 20 MPa), the elastic modulus of the rock first increases and then decreases, and the axial strain ε1 and lateral strain ε2 decrease gradually with increasing σ2. The lateral strain (swelling) of the rock is mainly along the σ3 direction, and the tendency of the lateral expansion is to decrease first and then to increase. In the unstable crack growth stage, the lateral strain εp3 and strain increment Δε3, ΔεV first increase and then decrease, but the former appears as “cliff” decrease when σ2 is 50 and 60 MPa, while the latter decreases slightly when σ2 is 60 MPa. The variation of strain increment Δε3, ΔεV in the stage is similar, indicating that when σ2 is large, the former ability to bear lateral deformation is weakened significantly, while the latter can greatly reduce the role of confining pressure damage at this time and provide a protective effect to the rock due to increasing σ3.(3)
In this study, the AE characteristics of rocks in unstable crack growth stage under conventional and true triaxial compression are analyzed. It is found that microcracks in the sandstone reflected mainly stretching and expansion under the two loading conditions (the characteristics of AE signals are high AF and low RA); however, the AE characteristics are significantly different. The trend of AE activity for rocks under conventional triaxial compression is “gradually reduced” with increasing confining pressure, and the shear and slip effect of the cracks is gradually inhibited. AE events first weaken and then enhance with increasingσ2 under true triaxial compression in which σ3 is 10 and 20 MPa. When the former σ2 is 20∼40 MPa, the cumulative AE counts gradually decrease with increasing σ2, and the proportion of the shear cracks increases gradually. When σ2 is 50 and 60 MPa, the cumulative AE counts increases rapidly, and there is a “sudden increase” phenomenon. The proportion of the shear cracks decreases dramatically, showing that the confining pressure gradually evolves from the shear and slip effect inhibiting shear cracks to damage the rock promoting the formation of tensional cracks with increasing σ2, resulting in an instant of macrotension rupture formation. But, the damage of confining pressure is weakened by increasing σ3 (the latter σ3=20MPa).



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*Source: 5241386-2018-04-15.xml*