Mechanical and Acoustic Emission (AE) Characteristics of Rocks under Biaxial Confinements

Abstract

The surrounding rocks of underground engineering are generally subjected to a biaxial compressive stress condition. The failure properties of rocks under such a stress condition are worthy of being studied to ensure the stability of surrounding rock. This study aims to investigate the mechanical characteristics and acoustic emission (AE) properties of granite, marble, and sandstone in biaxial compression tests. Under biaxial confinements, it is evident that the elastic moduli of the three types of rocks decrease, and the plasticity increases monotonously with the increase of the intermediate principal stress σ₂. As σ₂ increases, the biaxial compressive strength σ_bcs of rock increases initially and subsequently decreases. The lateral strain ε₂ of rock under biaxial confinement is controlled by both σ₁ and σ₂, and the restrain degree in the development of microcracks and the constrain extent in the expansion along the direction of σ₂ are both enhanced gradually with increase in σ₂. The sharp increase points of AE hit and AE count indicate that the failure will occur soon. The AF-RA distribution of AE signals shows that the increase of σ₂ causes more tensile cracks in rock. According to the dip failure angle of macro-cracks in rock under biaxial confinement, the failure modes of granite and marble are slabbing, while failure mode of sandstone is shear. In addition, the σ₂ has a positive effect on the mass ratio of large size fragments after rock failure. An exponent relationship between the σ_bcs and σ₂ was found, and the inner apices–inscribed Drucker–Prager criterion can be used to predict the σ_bcs of rock.

1. Introduction

The underground rocks in the vicinity of underground engineering boundaries are subjected to a static biaxial compressive geo-stress state (the minimum principal stress σ₃ unloading to 0, and the maximum principal stress σ₁ and the intermediate principal stress σ₂ reaching a new balanced state) after the excavation operations [1,2,3,4,5,6], as shown in Figure 1a. Thus, the studies of the load-carrying ability and failure behaviors of rocks under biaxial confinements are fundamental for a better understanding of the rock disasters and for effective design and construction of underground engineering [7,8,9].

Many researchers have made great efforts in this field and have achieved many results. The biaxial compression tests were initially carried out using a hollow cylinder and cubes coal specimens by Hobbs [10], but not long after that, the cubic rock specimens became more widely used, and a growing number of research findings in biaxial compression tests have been reported [11]. The cubic sandstone specimens with a side length of 75 mm have been used in biaxial compression tests to determine the biaxial compressive characteristics of rocks. It was found that the biaxial compressive strength σ_bcs is significantly increased with the increase of lateral confinement. Besides, the failure is generally initiated by rock spalling at the free faces at the early stage of loading, and the ultimate failure occurs in a form of an out-of-plane shear failure [12]. Under a high σ₂, the microcracks and fractures in the rocks can only develop in the plane of σ₁-σ₂, which leads to the anisotropic failure behaviors of rocks [13]. Furthermore, the σ₂ has a limited influence on the peak strength of the rock near the excavation boundary [14]. The σ₂ in a plastic potential function has a significant impact on the evolution of volumetric plastic strain, thereby exerting an influence on the overall stability of rock mass structures [15]. The specimen’s failure and crack coalescence are at the same time under compression, while crack coalescence occurs first, and then failure occurs in the state of biaxial confinements [16]. More energy would be required for rock failure under equal biaxial compression than under unequal biaxial compression [17]. Under biaxial confinements, structural failure occurred mainly in the specimens when the number of natural cracks in the rock is small, and the specimen appears local failure with the increase of the number of natural cracks, and under the state of low initial stress can accelerate the propagation of a large number of cracks [18,19].

The σ_bcs-σ₂ envelopes of rocks in biaxial compression tests (σ₁ ≥ σ₂ ≥ 0, σ₃ = 0) from other literature [19,20,21,22] are shown in Figure 1b, and the results are concluded as follows: (1) The uniaxial compressive strengths σ_ucs of limestone and granite are approximately equal to the biaxial compressive strengths σ_bcs under the biaxial compessions with σ₁ = σ₂, while the σ_bcs of red sandstone, sandstone, and salt are much bigger than σ_ucs; (2) The σ_bcs,peak (the maximum value of σ_bcs) of limestone, red sandstone, granite, sandstone, and salt is 21%, 53%, 63%, 85%, and 87% higher than their σ_ucs, respectively; (3) When the σ₂ is about 0.5σ_ucs, the σ_bcs of limestone and granite reaches the σ_bcs,peak, and the σ_bcs,peak of red sandstone, sandstone, and salt can be obtained under the condition of biaxial compression with σ₂ = 0.8σ_ucs, σ₂ = σ_ucs, and σ₂ = σ_ucs, respectively. Because the testing conditions are different in kinds of literature, so the change regular of the σ_bcs is not clear for different rocks. It is necessary to carry out in-depth studies on the σ_bcs of rocks.

The end friction between the loading platen and rock specimen has a great effect on the failure properties of rocks in biaxial compression tests [23]. It is necessary to take anti-friction measures when the biaxial compression tests are conducted. Figure 1c shows two typical groups of biaxial compressive strength of rocks, and there are clear differences between the strength results without anti-friction measures and these with anti-friction measures. In general, brush platen and solid steel platen pasted with molybdenum disulfide (MoS₂) powder were the most used measures to reduce end friction in biaxial compression tests and it was observed that the end friction could increase the σ_bcs, and the use of molybdenum disulfide to eliminate the end friction is more effective than brush platen [24]. It was also found that the 1:1 mixture of stearic acid and vaseline produces the lowest coefficient of friction compared to Teflon, graphite, and MoS₂ [22]. In the biaxial compression tests, the failure surfaces of specimens without anti-friction measures are coarse and fine, and most of them are curved, indicating that the friction effect at the end limits the free propagation of cracks, while the failure of most specimens is spalling along the direction of σ₁, and the failure surface is large and straight when “1 mm thick PTFE gasket pasted with vaseline” is used to anti-friction [25]. Additionally, the end friction effects markedly weakened with increasing confining pressure under the same anti-friction measures [8,26,27,28].

Acoustic emission (AE) is a nondestructive inspection technique utilizing the transient elastic energy which results mainly from crack propagation events inside a material [29,30,31]. AE has been broadly applied to rock to obtain the information about crack initiation and propagation, failure mode, abrupt failure of the structure, and correspondingly some of the phenomena in time sequence for a detailed and systematic analysis [32,33,34,35]. Aggelis proposed that the distinct signature of the cracking modes is reflected on AE waveform parameters like the amplitude, RA-value (where RA is the ratio of rising time to the amplitude of AE wave), and average frequency, and achieved the conclusion that AE waveform signals generated in shear tests have a greater RA-value and a smaller average frequency than that of AE waveform signals generated in tensile tests [36,37]. Cai investigated the generation time of the initial crack in terms of the information of the AE events generated in the failure process of rock specimens and AE characteristics of rocks in lab experiments and accordingly established the generalized AE initiation and damage thresholds [38]. Lei et al. studied AE activity during a catastrophic fracture of fault in rock and obtained the detailed time-space distribution of AE events from lab tests [39]. Tham et al. discovered that the granite and marble specimens in tensile tests have different fracture surfaces and AE characteristics, and rock lithology played a vital role [40]. AE was applied to investigate the fracture and damage mechanisms of rock induced by the propagation of microcracks under triaxial compression test, and it was found that shear failure was a major microscopic failure mechanism of rock under triaxial compression [41]. Tsuyoshi et al. demonstrated that the precision and sensitivity of AE technology are high for detecting crack propagation in the rock under a direct shear test [42,43]. In engineering fields, AE technique is regarded as a promising and flexible tool, which has been widely used in monitoring and even quantifying dynamic processes within loaded structural elements [44,45].

The aforementioned efforts had provided constructive reference values and information for a better understanding of the relationship between rock fracture processes and AE characteristics. However, most of the previous studies mentioned above concentrated on the AE characteristics of single lithology rock under a certain test condition, and the ranges of intermediate principal stress were limited in most biaxial compression tests. Thus, more investigations of the influences of rock lithologies and the full range of intermediate principal stress on AE characteristics are needed. Therefore, in this study, three kinds of cubic rock specimens, i.e., granite, marble, and sandstone, were prepared, and the value of σ₂ was preset from 0 (uniaxial compression) to σ₁ = σ₂ to analyze the detailed mechanical characteristics and AE properties of rocks under biaxial confinements. The results of this study will improve the theoretical system of damage mechanisms and AE characteristics of rocks in underground engineering, and provide a reference for engineering damage prediction and monitoring.

In this paper, the mechanical characteristics of rocks under biaxial compression, such as stress–strain curves (lateral strain and axial strain) and the variation of biaxial compression strength with σ₂, were first analyzed, and then the AE characteristics monitored by the two AE sensors installed on the free-surface of rock specimen, such as AE hit rate, AE count rate, b-value, RA-AF, and peak frequency, were studied. In addition, the failure modes of rocks were investigated, and finally, the D-P strength criterion was also verified.

2. Experimental Methodology

2.1. Rock Specimens

The intact marble, sandstone, and granite were selected as research materials and processed into cubic specimens with a side length of 50 mm for considerations of the end effect and the limited loading capacity of the apparatus in this study, as shown in Figure 2a.

Before the biaxial compression testing, the mineral compositions of the three rocks were achieved by petrographic thin-section tests. The test results are shown in Figure 2b–d. The analysis results showed that the main mineral components of granite are feldspar (50–60%) and quartz (20–30%) (Figure 2b), the main mineral components of marble are magnesite (35~45%), calcite (40~50%), and interstitial material (5~25%) (Figure 2c), whereas the main mineral components of sandstone are quartz (50–70%) and feldspar (10–30%), as well as a little of biotite and muscovite (Figure 2d).

2.2. Stress Paths

Two kinds of stress loading paths in biaxial compression tests were applied in this study. The detailed loading steps of Path I were as follows: Firstly, the maximum principal stress σ₁ and the intermediate principal stress σ₂ are loaded synchronously to the preset value of σ₂ at the same loading rate (0.2 MPa/s). Then, the σ₂ remained at the predefined levels, and the σ₁ continued to increase until the final failure of the rock specimen (Figure 3a). Path II was applied to check the bearing capacity of rocks under σ₁ = σ₂. The σ₁ and σ₂ are loaded synchronously at the same loading speeds (0.2 MPa/s) until rock failure (Figure 3b). The peak values of the σ₁ were defined as the biaxial compressive strength σ_bcs of rocks. The σ_bcs of rocks under σ₂ = 0 was also called the uniaxial compressive strength σ_ucs.

The preset values of σ₂ varied for different rocks in Path I, which were 0, 5, 10, 20, 40, and 60 MPa for granite, 0, 5, 10, 20, and 40 MPa for marble, and 0, 5, 10, and 20 MPa for sandstone, as shown in

Table 1. Testing parameters in Path I.

Rock Types	${\mathit{\sigma }}_2$ (MPa)	${\mathit{\sigma }}_3$ (MPa)	Loading Rate (MPa/s)
Granite	0, 5, 10, 20, 40, 60	0	0.2
Marble	0, 5, 10, 20, 40	0	0.2
Sandstone	0, 5, 10, 20	0	0.2

.

2.3. Anti-Friction Measures

Before testing, some strict anti-friction measures, containing two layers of polythene sheet and two layers of vaseline, were applied in this study to reduce the end friction effect on the test results. The detailed steps were as follows: (1) The vaseline and polythene sheets with a cross-section of 50 × 50 mm were prepared; (2) Cleaning the four loading surfaces of specimens, and then evenly applying the first layer of vaseline, and sticking the first layer of polythene sheet on the first layer of vaseline. (3) Applying the second layer of vaseline evenly on the first layer of polythene sheet, and then sticking the second layer of polythene sheet on the second layer of vaseline (Figure 4).

2.4. Testing Apparatus

All tests were conducted using the TRW-3000 true-triaxial testing apparatus at Central South University, China, which can load in three mutually perpendicular directions (σ₁, σ₂, and σ₃), respectively. The loading capacities along the three directions are over 2000 kN, and the loading accuracy is ±1%, as shown in Figure 5a. In this study, the cubic rock specimens were placed in the σ₁-σ₂ plane, as shown in Figure 5c,d. Before testing, each specimen was stuck with two GTMicro-300 type AE sensors numbered as “S1” and ”S2” on the same free surface to obtain the AE signals. Generally, the AE data from sensor S1 was selected for analysis, and the sensor S2 was just installed for backup and comparative verification, as shown in Figure 5b,e. Besides, two extensometers were applied to test the deformation of the specimens along the direction of σ₁ and σ₂ (Figure 5f). The AE system and loading apparatus started to record data synchronously until rock failure.

3. Results and Discussion

3.1. Basic Mechanical Parameters of Rocks

The uniaxial compression tests using cubic specimens were carried out first to determine the basic mechanical parameters of granite, marble, and sandstone, as shown in

Table 2. Basic mechanical parameters of granite, marble, and sandstone.

Rock Types	${\mathit{\sigma }}_{\mathbf{ucs}}$ (MPa)	${\mathit{E}}_{\mathbf{y}}$ (GPa)	$\mathit{\mu }$	${\mathit{V}}_{\mathit{L}}$$(\mathbf{km}·{\mathbf{s}}^{-1})$
Granite	89.86	52.25	0.25	4.26
Marble	45.39	27.67	0.27	3.58
Sandstone	27.71	12.07	0.31	2.34

. Granite has the highest uniaxial compressive strength $\sigma$_ucs and elastic modulus $E_y$, while sandstone has the lowest $\sigma$_ucs and $E_y$. The Poisson’s ratio μ of granite, marble, and sandstone are 0.25, 0.27, and 0.31, respectively. The P-wave velocities $V_L$ of granite, marble, and sandstone measured by an ultrasonic vibration are 4.26, 3.58, and 2.34 km/s, respectively.

3.2. Stress–Strain Properties

3.2.1. Axial Strain

The σ₁-ε₁ (ε₁ is the strain along the σ₁ direction) curves of granite, marble, and sandstone under biaxial compression are shown in Figure 6a–c, and it indicates that the slopes of σ₁-ε₁ carve, i.e., the elastic modulus, decrease with the increase of σ₂. The plastic deformations of rocks increase with the increase of σ₂. Moreover, the decrease of elastic modulus of granite and marble is more obvious than that of sandstone, as shown in Figure 6d. It can be concluded that the σ₂ plays a positive role in large deformation failures of rocks under biaxial compressions, which shows that the deformation and failure modes of rocks gradually change from the elastic-brittle to the elastic-plastic as the σ₂ increases.

3.2.2. Lateral Strain

The σ₁-ε₂ (ε₂ is the strain along the σ₂ direction) curves of granite, marble, and sandstone are shown in Figure 7. The ε₂ is an important index to study the mechanical properties of rocks under biaxial compression, which can well reflect the changing trend of crack propagation under biaxial confinements. The main conclusions of the ε₂ are as follows: (1) During the simultaneously loading process of the σ₁ and σ₂, the ε₁ and ε₂ both increase in the positive direction, which indicates that the internal natural microcracks are compressed in both lateral and axial directions; (2) In the process of “keeping σ₂ and increasing σ₁”, the ε₂ and σ₁ initially show a positive correlation when σ₂ is at a low level (≤10 MPa), whereas ε₂ is almost constant with the increase of σ₁ when σ₂ comes to a high level (>10 MPa), which proves that the higher σ₂ can restrict the development of lateral strain; (3) Figure 7a,d,g shows that the ε₂ is negatively correlated with σ₁ when σ₁ is increasing continuously under σ₂ of 5 MPa, which proves that a smaller σ₂ has a lower influence on lateral constraints, and the specimens expand laterally until it failed. With the increase of σ₂, the larger expansion constraint on the lateral direction of rock induces the less obvious increases of ε₂ in the negative direction (Figure 7b,c,e,f,h,i).

3.3. Biaxial Compressive Strength

The biaxial compressive strength σ_bcs of granite, marble, and sandstone specimens are listed in

Table 3. Biaxial compressive strengths of granite, marble, and sandstone.

Rock Type	Specimen NO.	${\mathit{\sigma }}_3$ (MPa)	${\mathit{\sigma }}_2$ (MPa)	${\mathit{\sigma }}_1$ (MPa)	${\mathit{\sigma }}_{\mathbf{bcs}}$ (MPa)
Granite	g-0-1	0	0	86.12	86.12
	g-0-2	0	0	93.68	93.68
	g-5-1	0	5	103.00	103.00
	g-5-2	0	5	99.95	99.95
	g-10-1	0	10	134.73	134.73
	g-10-2	0	10	138.98	138.98
	g-10-3	0	10	138.00	138.00
	g-20-1	0	20	145.50	145.50
	g-40-1	0	40	130.80	130.80
	g-40-2	0	40	140.40	140.40
	g-60-1	0	60	127.40	127.40
	g-T-1	0	${\sigma }_1$	98.10	98.10
	g-T-2	0	${\sigma }_1$	99.80	99.80
Marble	m-0-1	0	0	49.80	49.80
	m-0-2	0	0	40.98	40.98
	m-5-1	0	5	68.85	68.85
	m-5-2	0	5	57.30	57.30
	m-10-1	0	10	76.00	76.00
	m-10-2	0	10	65.50	65.50
	m-10-3	0	10	75.02	75.02
	m-20-1	0	20	69.90	69.90
	m-20-2	0	20	67.56	67.56
	m-40-1	0	40	72.30	72.30
	m-T-1	0	${\sigma }_1$	66.11	66.11
	m-T-2	0	${\sigma }_1$	62.40	62.40
Sandstone	s-0-1	0	0	28.80	28.80
	s-0-2	0	0	24.20	24.20
	s-0-3	0	0	30.14	30.14
	s-5-1	0	5	37.06	37.06
	s-5-2	0	5	39.30	39.30
	s-5-3	0	5	38.30	38.30
	s-10-1	0	10	47.20	47.20
	s-10-2	0	10	43.60	43.60
	s-10-3	0	10	48.24	48.24
	s-20-1	0	20	39.49	39.49
	s-20-2	0	20	43.40	43.40
	s-T-1	0	${\sigma }_1$	38.40	38.40

and shown in Figure 8a. The σ_bcs of the three rocks increase initially and then decrease with the increase of σ₂. There exists a maximum value of the σ_bcs, named as σ_bcs,peak, which is the peak value of σ_bcs in the variation range of σ₂. The ratios of the σ_bcs,peak to the uniaxial compressive strengths of granite, marble, and sandstone are approximately the same as 1.5–1.6. The peak points of σ_bcs corresponding to σ₂ for granite, marble, and sandstone are about 20, 20, and 10 MPa, respectively, which is much smaller than that in the literature [18,19,20,21,22]. The first derivative curves of the fitting function of σ_bcs of three rocks are drawn as Figure 8b, which show clearly that the increasing rate of σ_bcs decreases with the increase of σ₂. Furthermore, the increasing rate of σ_bcs for granite is always greater than that of marble and sandstone, and the increasing rate for sandstone is the smallest. The maximum increasing rate σ_bcs for granite, marble, and sandstone are 4, 2.5, and 1.5, respectively, which indicates that the σ₂ has a lower effect on the σ_bcs of rocks with stronger plasticity.

For all rocks, the increasing rate of σ_bcs is 0 at the σ₂ corresponding to the turning point of σ_bcs. There is an optimal solution of σ_2,peak corresponding to the maximum of σ_bcs, and the σ_2,peak of granite, marble, and sandstone are 25.2, 24.6, and 12.3 MPa, respectively. The σ_2,peak is of great significance for disaster prevention and control in the stress adjustment area after underground engineering excavation. At the stage of the increasing rate of σ_bcs is negative, the curve of each of the rock shows a similar trend with decreasing first and then increasing slowly, and asymptotically approaches to zero, which demonstrates that the σ_bcs decrease slowly when it increases to a certain extent. The minimum increasing rate σ_bcs for granite, marble, and sandstone are −1, −0.5, and −0.5, respectively, which indicates that the σ₂ has less influence on σ_bcs for the rock with lower strength under higher σ₂.

Based on the results of the biaxial compressive strengths σ_bcs of the three rocks, a relationship between the σ_bcs and σ₂, i.e., an exponent strength criterion, was proposed as shown in Equation (1), which has a high degree of adaptability to the strength data of the three rocks obtained from the tests, and has a good reference value when evaluating the σ_bcs of the rocks:

$${\sigma }_{\mathrm{bcs}}={\sigma }_{\mathrm{ucs}}+\mathrm{A}{\sigma }_{\mathrm{ucs}}{\sigma }_2\exp \left(1-\mathrm{B}{\sigma }_2\right),$$

(1)

where both A and B are material constants.

3.4. AE Characteristics

3.4.1. AE Hit Rate, AE Count Rate, and Dynamic b-Value

AE hit rate and AE count rate refers to the cumulative AE hit and AE count of the specimen in one second during the process of rock failure, and they reflect the damage degree of the specimen under loading. The dynamic b-value refers to the ratio of AE hit with low amplitude signal to AE hit with high amplitude signal during loading, and the higher dynamic b-value reflects the greater proportion of AE hit with low amplitude signal [28,46,47,48]. The AE data of the two sensors have a good consistency for most specimens in the biaxial compressive tests. Therefore, the data from “S1” was presented in this paper. According to Gutenberg–Richter (G-R) distribution [49,50], the relation between the number of earthquakes above a given magnitude, N(M), and the magnitude, M, can be drawn as:

$$\lg N\left(M\right)=a-b\left(M-M_1\right);M\ge M_1,$$

(2)

where $M_1$ is a lower threshold, below which magnitudes are not considered; a is a constant and b is referred to as the b-value.

When calculating the AE b-value of rock damage, AE amplitude is generally used to reflect AE signal, and the equivalent earthquake magnitude is obtained by dividing AE amplitude by 20. So, Equation (2) can be modified into Equation (3):

$$\lg \left[N\left(A/20\right)\right]=a-b\left(A/20\right),$$

(3)

where A is the amplitude of the AE hit; and N is the cumulative number of AE hits with an amplitude of A.

The AE amplitude data set of each rock specimen was evenly divided into 13 parts (A1, A2, A3,…, A13) according to the overall loading time, and the dynamic b-value of two adjacent parts was calculated successively from A1 to A13 at a pace of 1.

The changes of the AE hit rate, AE count rate, and dynamic b-value with time are shown in Figure 9, and the stress–strain curves of rock specimens are also drawn. In the initial stage of loading, the natural cracks in the rock are mainly closed without significant damage, so there are few AE signals at this stage. With the continuous increase of σ₁, the internal cracks in the rocks begin to develop and propagate, and AE signals at this stage gradually increase. Based on the analysis of data of AE hit rate, AE count rate, and dynamic b-value, the main conclusions are summarized as follows:

(1) In the same rock, the AE hit rate and AE count rate have roughly the same change law during loading, while the value of AE count rate is 10 times as much as AE hit rate. Under uniaxial compression, many peaks appear in the AE hit rate and AE count rate curves (Figure 9a,d,g), which indicates that the internal cracks are well developed, and more fracture surfaces are produced, which makes the rock failure more thorough.

(2) The AE hit rate and AE count rate increase rapidly when σ₁ approaches σ_bcs until the failure occurring, which indicates that the failure of the rock specimens is caused by the increase of its internal cumulative damage to a specific threshold value. With the increase of σ₂, the peak values of AE hit rate and AE count rate appear later, indicating that σ₂ can slow down the growth rate of the damage degree of rock under the increasing load (Figure 9b,c,e,f,h,i).

(3) The dynamic b-value of the three types of rocks are all distributed in the range of 1–3 (mainly in 1–2), which is related to the AE hit rate and AE count rate. The dramatic increase in the AE hit rates is always accompanied by a significant decrease in the dynamic b-value, which is a phenomenon that also occurs before the rock failure where the peaks of AE hit rate and AE count rate appear. This behavior has been observed in brittle materials [51,52,53]. During the whole process of loading, the overall trend of the dynamic b-value is declining, which can be used to predict the failure of rocks (Figure 9).

3.4.2. AF-RA Distribution

The AF-RA (where AF is the average frequency of AE waveform) density relationships of granite, marble, and sandstone are shown in Figure 10, in which the red area has higher data density, and the blue or purple area is with lower data density. The data distribution changes from sparse to dense with the color changing from blue-purple to red. It was proved that tension cracks generated during loading with a long side along with AF and a short side along with RA, while shear cracks with a short side along with AF and a long side along with RA [37].

It can be known from the figures that the high-density area gradually converges to the AF axis having low RA value and a large range of AF value with the increase of σ₂, which indicates that the greater the σ₂ is, the more the tensile cracks generate in rock specimens during biaxial loading condition. The range of AF value in the red area of granite is almost 50 and 100 kHz more than that of marble and sandstone, respectively, which indicates that the tensile failure degree of granite is the highest while that of sandstone is the lowest.

The behavior of two typical AE parameters with time for the different types of rock is depicted in Figure 11. The RA values of granite, marble, and sandstone specimens mainly maintain at a low level of 0–100 ms/V during loading and show a strong increase before specimen failure, which indicates that few shear cracks in the rock are formed before the failure stage, but the number of shear cracks increases obviously during the failure stage. Moreover, the AF is distributed in the whole range of 0–450 KHz and the density gets higher as the loading going on, which demonstrates that the tension cracks in the rock are generated during the entire loading process, but the rate is increasing with the increase of σ_1.

3.4.3. Peak Frequency

The distributions of peak frequencies of granite, marble, and sandstone with time during the axial loading at the σ₂ of 10 MPa are shown in Figure 12a–c, respectively. The peak frequencies of these three types of rocks are mostly distributed between 100–300 kHz, and the signals increase obviously with the process of loading. Furthermore, the range of peak frequency signal is extended from 100–300 kHz to 50–350 kHz with the increase of σ₁. The reason for the above result is that the cracks and joint surfaces inside the rock specimens are compressed and closed at the beginning stage of loading, resulting in some weak AE signals. Subsequently, AE signals increase and enhance gradually with the abnormal signals caused by the defects inside the rocks to generate new cracks. The macroscopic crack is formed to cause the damaged rock to slide relatively along the failure surface, which is the main reason for more abnormal peak frequencies of AE signals occurring during this period.

The distributions of peak frequencies of AE signals for granite, marble, and sandstone are shown in Figure 13a–c, respectively. The peak frequencies of the three types of rocks are mainly distributed in the range of 200–300 kHz, followed by 0–100 kHz, which has the tendency of increasing and then decreasing with the increase of σ₂. In the range of 200–300 kHz of peak frequencies, the peak frequencies of granite, marble, and sandstone have their maximum values at confining stresses of 20, 10, and 10 MPa, respectively. The increase of σ₂ leads to more cracks propagating along the direction of σ₁ when confining stress is small, while the increasing σ₂ will inhibit the propagations of cracks and reduce the peak frequency.

3.5. Failure Mode

The process from the initiation and propagation of micro-cracks in rocks to the formation of macro-cracks is the essence of rock failure and plays an important role in strain energy consumption [18,54]. In the biaxial compression tests, the propagation direction of macro-cracks in rocks with high dip fracture angles are generally perpendicular to the free surface, the main macro-crack surfaces are located at the σ_1-σ₂ plane, and the axial load can aggravate the propagation of macro cracks [24].

The dip fracture angle and the mass and size of rock fragments after failure under different σ₂ are statistically analyzed in this part. The fragments can be divided into 5 groups according to their size: >20 mm (big pieces), 12–20 mm (small pieces), 3–12 mm (coarse grain), 1–3 mm (fine grain), and <1 mm (powder). The fragments and the dip fracture angles of the three types of rocks after failure are shown in Figure 14. The weight contents of the fragments with different size ranges are shown in Figure 15.

Because of σ₃ = 0, the propagation directions of the fracture planes of macro-cracks are generally perpendicular to the direction of σ₃, as shown in Figure 14. The main micro-cracks of granite and marble obliquely intersect with the direction of σ₁ at a small angle, and the angle tends to decrease with the increase of σ₂, which indicates that the larger σ₂ leads to a greater lateral constraint and the larger influence on the generation and propagation of vertical micro-cracks. The failures of granite and marble are more caused by tension cracks. The main micro-cracks of sandstone are in a shape of “Λ”, and the dip failure angles are smaller than that of granite and marble, which indicates that the failure of sandstone is more caused by shear cracks. With the increase of σ₂, the weight of big pieces increases at first and then decreases after the failure of granite, marble, and sandstone, indicating that the σ₂ at a higher value can promote the generation and propagation of cracks in rocks. After the failure of marble specimens, the weight proportion of powder and small particles is obviously higher than that of granite and sandstone, indicating that the rock lithology has a great influence on the failure of rocks.

4. Strength Criterion

Drucker–Prager (D-P) criterion [55] is an extension of the Mohr–Coulomb criterion [56], which accounts for the effects of all principal stresses and overcomes the main defect of underestimating the intermediate principal stress of the Mohr–Coulomb criterion. The generalized form is written as:

$$\alpha I_1+J_2^{1/2}=k,$$

(4)

where $I_1$ and $J_2$ are the first and second stress invariants, and $\alpha$ and k are material constants of the rock, respectively.

$$I_1={\sigma }_1+{\sigma }_2+{\sigma }_3,$$

(5)

$$J_2=\frac{1}{6}\left[{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2\right].$$

(6)

For the outer apices–circumscribed D–P criterion (ODP):

$$\alpha =\frac{2\sin \phi }{\sqrt{3}\left(3-\sin \phi \right)},$$

(7)

$$k=\frac{6c\cos \phi }{\sqrt{3}\left(3-\sin \phi \right)}.$$

(8)

For the inner apices–inscribed D–P criterion (IDP):

$$\alpha =\frac{2\sin \phi }{\sqrt{3}\left(3+\sin \phi \right)},$$

(9)

$$k=\frac{6c\cos \phi }{\sqrt{3}\left(3+\sin \phi \right)}.$$

(10)

Drucker–Prager failure criterion for biaxial confinements condition is achieved by setting σ₃ = 0, so the Equation (4) can be written as:

$$\alpha \left({\sigma }_1+{\sigma }_2\right)+\sqrt{\frac{{\sigma }_1^2-{\sigma }_1{\sigma }_2+{\sigma }_2^2}{3}}=k.$$

(11)

The cohesions c for granite, marble and sandstone were calculated according to triaxial compression data with σ₂ = σ₃ [57] to be 50.32, 24.1, and 20.54 MPa, respectively. The internal frictional angles $\phi$ were 37°, 32°, and 29°, respectively. In Figure 16, failure stress data is plotted against two theoretical solutions (ODP and IDP), indicating that the outer apices–circumscribed Drucker–Prager criterion overestimates the σ_bcs of the three types of rocks. The inner apices–inscribed Drucker–Prager criterion can accurately predict the σ_bcs of granite and marble when σ₂ is small $(\le$20 MPa for granite and $\le$10 MPa for marble), whereas it loses accuracy under high σ₂ value. As for sandstone, OPD and IPD both overestimate the σ_bcs, and the gap between the lab data and the theoretical data is larger than that of granite and marble.

5. Conclusions

In this study, the mechanical and AE characteristics of cubic granite, marble, and sandstone specimens were investigated in the biaxial compression tests. The following conclusions can be drawn:

(1) In the biaxial compression tests, the elastic moduli of granite, marble, and sandstone show a decreasing trend with the increase of σ₂. The intermediate principal stress σ₂ has a significant effect on lateral strain of rock under biaxial confinement. For the three types of rocks in this study, the σ₂ over 10 MPa can restrain the development of micro-cracks along the direction of σ₂.

(2) The biaxial compressive strengths (σ_bcs) of granite, marble, and sandstone increased initially and then subsequently decreased with the increase of σ₂. The turning-points σ_2-t of σ₂ corresponding to peak biaxial compressive strengths σ_bcs,peak for different rocks are different, and the harder rock has a greater σ_2-t.

(3) The AE hit rate and AE count rate can reflect the damage degree of rock under biaxial compression. The sharp increases of the AE hit rate and AE count rate indicate the rapid accumulation of internal damage, which is always accompanied by a decrease in b-value. In the loading process, the dynamic b-value fluctuates, and it presents a downward trend on the whole.

(4) From the distribution characteristics of AF-RA, it can be found that few shear cracks in the rock are formed before the failure stage, but the number of shear cracks increases obviously during the failure stage. Tensile cracks in the rock are generated during the entire loading process, but the generation rate of cracks is increasing with the increase of σ_1.

(5) More tensile cracks are produced in rock during the loading process with the increase of σ₂. The failures of granite and marble are mainly caused by tension cracks, while the failure of sandstone is mainly caused by shear cracks.