# Strain Rate Effect on Acoustic Emission Characteristics and Energy Mechanisms of Karst Limestone under Uniaxial Compression

Authors: Qingsong Wang; Jianxun Chen; Jiaqi Guo; Yanbin Luo; Yao Li; Hongyu Wang; Qin Liu
Journal: Advances in Materials Science and Engineering
 (2020)

---

## Abstract

In this paper, the strain rate effect on mechanical properties, failure modes, acoustic emission (AE) characteristics, and energy mechanism of the karst limestone was analyzed based on uniaxial compression tests with different strain rates (5 × 10−6–5 × 10−4/s). The results showed that the peak strength increased linearly and peak strain increased quadratically with the logarithm value of the strain rate. Moreover, the strain rate effect on elastic modulus was not significant. Under low strain rates, the rock was damaged seriously, AE signals appeared continuously, and the cumulative number of AE signals was high. Under high strain rates, the total quantity of the macroscopic cracks decreased, but the crack length extended with better coalescence. The AE peak significantly increased under high strain rates, while the cumulative AE activity significantly reduced. The energy evolution of the karst limestone failure process had significant stage characteristics, and the strain energy ratio presented an S-shape. The maximum value of the elastic strain energy at peak stress showed a linear relationship with the logarithm value of the strain rate.


---

## Body

## 1. Introduction

The mechanical property of the rock is the important foundation of the rock mechanics constitutive model, numerical simulation, and engineering design and construction, especially under complex loading conditions [1–4]. In engineering construction, blasting disturbance, unreasonable excavation and supporting, earth squeezing action, and earthquakes can cause different strain rate changes on the rock and can even lead to geological hazards, such as rock burst, collapse, and water inrush in tunnel construction [5–13]. Therefore, the strain rate is an important factor that affects the mechanical property of rocks. Research on the strain rate effects on rock strength and deformation evolution is helpful for construction safety and prewarning for geologic hazards [14–19]. Many scholars have conducted relevant research, mainly through lab experiments (e.g., uniaxial or triaxial compression, Brazilian splitting test, and bending test) or numerical simulation, to study the strain rate effects on mechanical properties, failure modes, acoustic emission (AE) characteristics, and energy mechanism of the rock or rock-like materials [20–30].Li and Gu [31] suggested that it belonged to static loading when the strain rate was less than 10−1 s−1 and it belonged to dynamic loading when the strain rate was larger than 10−1 s−1. Zhang et al. [32–34] investigated the strain rate effect on the mechanical properties of marble under uniaxial compression and examined its failure characteristics and energy mechanism. Qi et al. [35] demonstrated that the rock mass strength sensitivity to the strain rate may be regarded as the competition result between the coexisting thermally activated and macroviscous mechanisms. Alam et al. [36] explored the quasistatic to dynamic compression behaviors of Kota sandstone under low (0.00001–0.1/s) and medium (1/s) strain rates, and an empirical equation was developed to correlate the dynamic increase factor with strain rates. Meng et al. [37] studied the effects of the size and strain rate on red sandstone under uniaxial compression, and it turned out that the peak stress was negatively correlated with the rock size and was positively correlated with the strain rate. In addition, AE activities were enhanced and the AE quantities increased with the increase of the strain rate. Zhang et al. [38] used the bonded particle model to investigate the AE characteristics of rocks under different compressive loading rates, and the results showed that total cracks, tensile cracks, shear cracks, and AE event numbers all increased with the loading rate. Wu et al. [39] conducted the uniaxial and triaxial compression experiments on rock specimens with two preexisting flaws under different loading rates, and the results showed that the failure characteristics of the fissured rock follow the tensile shear coalescence model, crack branching occurred with the increasing loading rate, and the multisection coalescence model was verified with the increasing confining pressure. Singh et al. [40] proposed an approach to predicting the creep behavior of rock salt on the basis of the uniaxial compression test and found that viscosity was negatively correlated with the strain rate.Rock mechanical properties under static loading have constantly been a main content of rock mechanics research. In this paper, the uniaxial compression tests on the karst limestone were conducted under different strain rates (5 × 10−6–5 × 10−4 s−1). The rock was obtained from Qiyueshan Highway Tunnel, and the tunnel area belongs to the karst landscape in Western Hubei. The strain rate effect on mechanical properties, failure modes, AE characteristics, and energy mechanism of the karst limestone was analyzed and investigated. The results could provide a reference for relevant rock mechanics tests, and the deformation or failure mechanism of rock mass engineering.


## 2. Preparation of Rock Samples and Test Equipment

Karst limestone was obtained at mileage GK2730-2485 in Qiyueshan Highway Tunnel, Hubei Province. The rock density was 2.69 g/cm3, and the P-wave velocity was approximately 4900 m/s. As shown in Figure 1(a), the samples were prepared as Φ 50 mm × 100 mm cylindrical samples according to the International Society for Rock Mechanics.Figure 1
(a) Rock samples. (b) RMT-150B rock mechanics test system. (c) DS5-8B AE. (d) AE sensor arrangement.
(a)
(b)
(c)
(d)As shown in Figure1(b), the RMT-150B rock mechanics test system with a control range of strain rates of 10−2–10−6 s−1 was used. The displacement control mode was adopted in the test. The strain rates were set from 5 × 10−6 to 5 × 10−4 s−1. Four strain rates were selected, and each test was repeated 2 times to reduce test errors.As shown in Figure1(c), the DS5-8B AE detection system was used for the real-time monitoring of AE signals. The threshold was set to 50 mV, and the preamplifier gain was set to 40 dB. As shown in Figure 1(d), two AE sensors were arranged symmetrically on the sides of the sample and the contact face was coupled by using Vaseline. In the test, the AE monitoring remained synchronized with stress loading, and all the tests were conducted at room temperature.


## 3. Analysis and Discussion of Test Results

### 3.1. Stress-Strain Curve

Figure2 illustrates the stress-strain curves of the karst limestone under uniaxial compression at different strain rates.Figure 2
Stress-strain curves of the karst limestone under different strain rates.Figure2 demonstrates that the stress-strain curves of the karst limestone under uniaxial compression present an S shape, which is similar to that reported by Li et al. [41] and Guo et al. [42]. After the initial compression stage, linear elastic deformation stage, and short plastic deformation, the stress approached the peak value, and then it dropped sharply with evident cracking sound and even splashing debris. Table 1 lists the mechanical parameters of the karst limestone under different strain rates.Table 1
Mechanical parameters of the karst limestone under different strain rates.
No.
Loading rate (mm/s)
Strain rate (ε′/s)
lg(ε′)
Peak strain (10−3)
σ
peak (MPa)
E (GPa)
E
s (GPa)
E
s/E
S
11
0.0005
5 × 10−6
−5.30
2.60
91.53
54.57
29.79
0.55
S
12
0.0005
5 × 10−6
−5.30
2.08
81.54
52.52
33.62
0.64
S
21
0.002
2 × 10−5
−4.70
2.24
116.00
62.03
48.79
0.79
S
22
0.002
2 × 10−5
−4.70
2.36
116.16
66.42
41.21
0.62
S
31
0.01
1 × 10−4
−4.00
2.74
130.78
60.59
42.05
0.69
S
32
0.01
1 × 10−4
−4.00
2.81
133.16
66.25
49.72
0.75
S
41
0.05
5 × 10−4
−3.30
2.39
153.42
63.24
43.61
0.69
S
42
0.05
5 × 10−4
−3.30
3.61
149.78
64.29
32.90
0.51Table1 shows that the peak strength of the rock sample was approximately 80–150 MPa, the elastic modulus (E) was approximately 50–70 GPa, the secant modulus (Es) was approximately 30–50 GPa, and the ratio of Es/E was 0.5–0.8. Table 1 also shows that the elastic and secant moduli of the karst limestone were insensitive to the strain rate (5 × 10−6–5 × 10−4 s−1). Figure 3 exhibits the relationship between strength and strain rate and the relationship between peak strain and strain rate.Figure 3
Relationship between strength and strain rate and between peak strain and strain rate.
(a)
(b)Figure3 shows that, with the increase in the strain rate, the strength and peak strain increased rapidly and then slowly. The fitting relationship showed that strength increased linearly with the strain rate logarithm, and the peak strain presented the parabolic increasing trend with the strain rate logarithm. This phenomenon indicated that the strain rate effect on strength was partially consistent with the peak strain of the karst limestone. Notably, strength is not an intrinsic property of rock materials, which is always significantly impacted by the rock size, shape, and loading conditions.The strain rate effect on rock strength was closely related to the crack formation and propagation during the loading process. Limestone is a kind of elastic-brittle rock with a compact structure. When the strain rate was low, the mineral particles in the rock underwent large deformation and the time for the evolution of initial damage and crack growth in the rock was sufficient. Therefore, the formation and propagation of cracks would remain coordinated with the increase in loading. Under large strain rates, the crack initiation and propagation lagged behind the increase in loading, and the absorbed energy was stored in the rock material, thereby increasing strength with the strain rate [43].


### 3.2. Failure Modes

Figures4(a)–4(h) present the failure modes of the karst limestone under different strain rates. When the strain rate was low, the sample was seriously damaged with dense macrocracks. White powder was observed on the fracture surface, which indicated the strong friction effect during the loading process. Under a large strain rate, the number of macrocracks was small, but the length and coalescence were more evident.Figure 4
Failure modes of the karst limestone under different strain rates. (a) 5 × 10−6 s−1. (b) 2 × 10−5 s−1. (c) 1 × 10−4 s−1. (d) 5 × 10−4 s−1. (e) 5 × 10−6 s−1. (f) 2 × 10−5 s−1. (g) 1 × 10−4 s−1. (h) 5 × 10−4 s−1.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)The strain rate effect on rock failure modes was closely related to crack formation and propagation during loading. Under low strain rates, the rock was seriously damaged because the load-transferring effect between mineral particles was sufficient. Furthermore, crack initiation and propagation were complete, thereby increasing the number of macrocracks. Small debris was produced on fracture surfaces for the shearing slip damage. When the strain rate was large, the load-transferring effect among mineral particles was incomplete, the loading effect among particles was uneven and the dominant effect of certain primary fissures was strengthened. Therefore, the number of macrocracks decreased, whereas the length increased and crack coalescence became more apparent [43].


## 3.1. Stress-Strain Curve

Figure2 illustrates the stress-strain curves of the karst limestone under uniaxial compression at different strain rates.Figure 2
Stress-strain curves of the karst limestone under different strain rates.Figure2 demonstrates that the stress-strain curves of the karst limestone under uniaxial compression present an S shape, which is similar to that reported by Li et al. [41] and Guo et al. [42]. After the initial compression stage, linear elastic deformation stage, and short plastic deformation, the stress approached the peak value, and then it dropped sharply with evident cracking sound and even splashing debris. Table 1 lists the mechanical parameters of the karst limestone under different strain rates.Table 1
Mechanical parameters of the karst limestone under different strain rates.
No.
Loading rate (mm/s)
Strain rate (ε′/s)
lg(ε′)
Peak strain (10−3)
σ
peak (MPa)
E (GPa)
E
s (GPa)
E
s/E
S
11
0.0005
5 × 10−6
−5.30
2.60
91.53
54.57
29.79
0.55
S
12
0.0005
5 × 10−6
−5.30
2.08
81.54
52.52
33.62
0.64
S
21
0.002
2 × 10−5
−4.70
2.24
116.00
62.03
48.79
0.79
S
22
0.002
2 × 10−5
−4.70
2.36
116.16
66.42
41.21
0.62
S
31
0.01
1 × 10−4
−4.00
2.74
130.78
60.59
42.05
0.69
S
32
0.01
1 × 10−4
−4.00
2.81
133.16
66.25
49.72
0.75
S
41
0.05
5 × 10−4
−3.30
2.39
153.42
63.24
43.61
0.69
S
42
0.05
5 × 10−4
−3.30
3.61
149.78
64.29
32.90
0.51Table1 shows that the peak strength of the rock sample was approximately 80–150 MPa, the elastic modulus (E) was approximately 50–70 GPa, the secant modulus (Es) was approximately 30–50 GPa, and the ratio of Es/E was 0.5–0.8. Table 1 also shows that the elastic and secant moduli of the karst limestone were insensitive to the strain rate (5 × 10−6–5 × 10−4 s−1). Figure 3 exhibits the relationship between strength and strain rate and the relationship between peak strain and strain rate.Figure 3
Relationship between strength and strain rate and between peak strain and strain rate.
(a)
(b)Figure3 shows that, with the increase in the strain rate, the strength and peak strain increased rapidly and then slowly. The fitting relationship showed that strength increased linearly with the strain rate logarithm, and the peak strain presented the parabolic increasing trend with the strain rate logarithm. This phenomenon indicated that the strain rate effect on strength was partially consistent with the peak strain of the karst limestone. Notably, strength is not an intrinsic property of rock materials, which is always significantly impacted by the rock size, shape, and loading conditions.The strain rate effect on rock strength was closely related to the crack formation and propagation during the loading process. Limestone is a kind of elastic-brittle rock with a compact structure. When the strain rate was low, the mineral particles in the rock underwent large deformation and the time for the evolution of initial damage and crack growth in the rock was sufficient. Therefore, the formation and propagation of cracks would remain coordinated with the increase in loading. Under large strain rates, the crack initiation and propagation lagged behind the increase in loading, and the absorbed energy was stored in the rock material, thereby increasing strength with the strain rate [43].


## 3.2. Failure Modes

Figures4(a)–4(h) present the failure modes of the karst limestone under different strain rates. When the strain rate was low, the sample was seriously damaged with dense macrocracks. White powder was observed on the fracture surface, which indicated the strong friction effect during the loading process. Under a large strain rate, the number of macrocracks was small, but the length and coalescence were more evident.Figure 4
Failure modes of the karst limestone under different strain rates. (a) 5 × 10−6 s−1. (b) 2 × 10−5 s−1. (c) 1 × 10−4 s−1. (d) 5 × 10−4 s−1. (e) 5 × 10−6 s−1. (f) 2 × 10−5 s−1. (g) 1 × 10−4 s−1. (h) 5 × 10−4 s−1.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)The strain rate effect on rock failure modes was closely related to crack formation and propagation during loading. Under low strain rates, the rock was seriously damaged because the load-transferring effect between mineral particles was sufficient. Furthermore, crack initiation and propagation were complete, thereby increasing the number of macrocracks. Small debris was produced on fracture surfaces for the shearing slip damage. When the strain rate was large, the load-transferring effect among mineral particles was incomplete, the loading effect among particles was uneven and the dominant effect of certain primary fissures was strengthened. Therefore, the number of macrocracks decreased, whereas the length increased and crack coalescence became more apparent [43].


## 4. AE Characteristics

Figures5(a)–5(h) show the strain rate effect on the AE characteristics of the karst limestone under uniaxial compression. The following observations were made from Figure 5:Figure 5
AE characteristics of the karst limestone under different strain rates. (a) AE ring-down counts at 5 × 10−6 s−1. (b) AE energy rate at 5 × 10−6 s−1. (c) AE ring-down counts at 2 × 10−5 s−1. (d) AE energy rate at 2 × 10−5 s−1. (e) AE ring-down counts at 1 × 10−4 s−1. (f) AE energy rate at 1 × 10−4 s−1. (g) AE ring-down counts at 5 × 10−4 s−1. (h) AE energy rate at 5 × 10−4 s−1.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)(1)
In Figures5(a)–5(d), the AE signals continuously emerged when the strain rate was low. In the initial loading stage, a small amount of AE signals were generated because of fissure compression and friction. With the increase of stress, the sample proceeded to the elastic stage and the AE activity increased. In the stable stage of crack growth, the AE activity gradually began to be active. At the unstable stage of crack growth, macrocracks started to appear with abundant AE signals and the AE peak value became visible. Then, the rock reached the peak strength and damaged rapidly.
(2)
Figures5(e)–5(h) demonstrate that the AE signals were discontinuous when the strain rate was large (ε′ ≥ 1 × 10−4 s−1). During the initial loading and elastic stages, AE signals were less than those at the low strain rate. The AE peak was extremely higher under a large strain rate during the unstable crack growth stage. When ε′ = 5 × 10−6 s−1, the peak of AE ring-down counts was approximately 1400 and reached nearly 8000 when ε′ = 5 × 10−4 s−1, which reduced by more than 80%.
(3)
With the increase in the strain rate, the AE peak significantly increased, but the accumulated AE activity significantly reduced. Whenε′ = 5 × 10−6 s−1, the accumulative AE ring-down count was 2.14 × 106 and the accumulative AE energy was 4.16 × 106 mV·mS. When ε′ = 5 × 10−4 s−1, the accumulative AE ring-down count was 6.51 × 104 and the accumulative AE energy was 2.15 × 105 mV·mS.
(4)
The strain rate effect on the AE characteristics was closely related to crack formation and propagation. According to Gao et al. [44] and Jiang et al. [45], during the rock deformation and failure processes, the variation in AE signals caused by interior friction belonged to a continuous signal, while the AE signals related to crack development could be regarded as a mutational signal. Combined with failure modes (Figure 3) under low strain rates, the load-transferring effect among mineral particles was more sufficient with a strong friction effect, thereby causing more macrocracks. So the AE signals were continuous, and the cumulative quantity was large. Under high strain rates, the load-transferring effect among mineral particles was incomplete and the loading effect among particles was uneven. Furthermore, the dominant fracture effect of several primary fissures was strengthened. Therefore, the main failure surface became more coalescent, whereas the total number of macrocracks reduced. Therefore, the AE peak became greatly larger, and the cumulative AE activity decreased and showed mutational characteristics.Figures6(a)–6(h) show the strain rate effect on the characteristics of AE amplitude and average signal level (ASL) of the karst limestone under uniaxial compression at different strain rates.Figure 6
AE amplitude and ASL of the karst limestone under different strain rates. (a) Amplitude of AE signals at 5 × 10−6 s−1. (b) ASL of AE signals at 5 × 10−6 s−1. (c) Amplitude of AE signals at 2 × 10−5 s−1. (d) ASL of AE signals at 2 × 10−5 s−1. (e) Amplitude of AE signals at 1 × 10−4 s−1. (f) ASL of AE signals at 1 × 10−4 s−1. (g) Amplitude of AE signals at 5 × 10−4 s−1. (h) ASL of AE signals at 5 × 10−4 s−1.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)As shown in Figures6(a)–6(h), the AE amplitude of limestone under uniaxial compression was 50–200 mV and the ASL was 45–60 dB. Under low strain rates, the AE signals showed continuity and the signals with high amplitude appeared near the peak stress. With the increase in the strain rate, the AE signal changed from continuity to mutation and the amplitude of the AE signal near the peak stress increased significantly. Compared with low strain rate loading, some AE signals with high amplitude also appeared in the initial stage at high strain rates.


## 5. Energy Mechanism Analysis

### 5.1. Energy Theory

According to Xie et al. [46, 47], the energy theory of a unit volume of rock can be described as follows:(1)W=U=Ue+Ud,where W is the work done to the rock by outside, U is the total energy absorbed from outside, Ue is the elastic strain energy stored in the rock, and Ud is the dissipated energy.The strain energy relationship during the rock failure process is shown in Figure7.Figure 7
Strain energy relationship during the rock failure process [33].According to Huang et al. [33] and Zhang et al. [48], only axial stress works on the rock in the uniaxial compression test:(2)U=∫0ε1σ1dε1,Ue=12Eiσ12≈12E0σ12,in which Ei is the unloading elastic modulus, which can be replaced by the initial elastic modulus E0.


### 5.2. Energy Evolution Mechanism

Figures8(a)–8(h) display the energy evolution mechanism during the failure process of the karst limestone.Figure 8
Energy evolution mechanism of the karst limestone under different strain rates. (a) Energy curves at 5 × 10−6/s. (b) Energy ratio curves at 5 × 10−6/s. (c) Energy curves at 2 × 10−5/s. (d) Energy ratio curves at 2 × 10−5/s. (e) Energy curves at 1 × 10−4/s. (f) Energy ratio curves at 1 × 10−4/s. (g) Energy curves at 5 × 10−4/s. (h) Energy ratio curves at 5 × 10−4/s.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Figures8(a)–8(h) illustrate the energy mechanism of the karst limestone in the loading process under uniaxial compression at different strain rates. The energy ratio presents an S-shape, which is similar to a deep-buried carbonaceous slate [49]. The relationship between stress and axial strain for the samples would undergo approximately five stages:(1)
Initial closure stage: the rock absorbed mechanical energy from outside, and most energy was consumed as dissipated energy for crack closure, friction, and slipping, which constantly produced small AE signals. In this stage,Ue/U decreased to a minimum value, whereas Ud/Ue and Ud/U increased to a maximum value. Then, Ue/U increased, whereas Ud/U decreased.(2)
Elastic stage: the original cracks were completely closed, and the energy input from outside was mainly used for elastic compaction. Thus,Ue/U increased, whereas Ue remained consistent with the stress curve. Moreover, most energy was transformed into elastic strain energy stored in the rock.(3)
Stable stage of fracture growth: in the crack initiation stage, a part of the energy was consumed as surface energy and other forms of radiant energy. The AE activity constantly became active in this stage. However, considerable energy remained stored in the rock as elastic strain energy.Ue/U still increased, but the growth rate gradually reduced. The crack initiation stage still belonged to the elastic stage according to Liang et al. [50] and Jaeger and Cook [51].(4)
Unstable stage of fracture growth: macrocracks appeared and extended rapidly, which caused abundant AE activity with high energy.Ue deviated from the stress curve, and Ud gradually increased, thereby constantly corresponding to crack damage stress (σd). In this stage, Ue/U decreased, whereas Ud/U increased significantly.(5)
Failure stage: after peak stress, the stored elastic strain energy was released and reduced sharply. Dissipated energy increased rapidly, and the rock sample damaged and divided into blocks of different sizes.The maximum value of the elastic strain energy of the rock is important for engineering applications. Liu et al. [52] proposed the energy instability criterion of water-resistant strata and rock mass failure index on the basis of releasable elastic strain energy Ue. Guo et al. [53] presented a new model for predicting and classifying the rock burst considering the intrinsic relations between the elastic strain energy and the rock burst based on the criteria for strength and structural failure of rocks. Table 2 and Figure 9 display the elastic strain energy at the peak strength of rock samples.Table 2
Strain energy at peak strength under different loading rates.
Number
Strain rate (ε′/s)
lg(ε′)
U (MJ/m3)
U
e (MJ/m3)
U
d (MJ/m3)
U
e/U
U
d/U
U
d/Ue
S
11
5 × 10−6
−5.30
0.0994
0.0768
0.0226
0.7724
0.2276
0.2947
S
12
5 × 10−6
−5.30
0.0730
0.0633
0.0097
0.8674
0.1326
0.1529
S
21
2 × 10−5
−4.70
0.1239
0.1085
0.0154
0.8756
0.1244
0.1421
S
22
2 × 10−5
−4.70
0.1140
0.1016
0.0124
0.8912
0.1088
0.1221
S
31
1 × 10−4
−4.00
0.1577
0.1411
0.0165
0.8951
0.1049
0.1172
S
32
1 × 10−4
−4.00
0.1894
0.1338
0.0556
0.7065
0.2935
0.4155
S
41
5 × 10−4
−3.30
0.2492
0.1861
0.0631
0.7467
0.2533
0.3392
S
42
5 × 10−4
−3.30
0.2081
0.1745
0.0337
0.8383
0.1617
0.1929Figure 9
Relationship between strain energy and strain rate.
(a)
(b)Table2 and Figure 9 present that the total energy (U) and elastic strain energy (Ue) at peak strength increased rapidly and then slowly with the increase in the strain rate. The ration of dissipated energy (Ud/U) at peak strength also increased, indicating that rock damage before peak stress increased with the strain rate. Therefore, the peak and amplitude of AE signals increased significantly before peak stress under high strain rates. The fitting relationship showed that U and Ue at peak stress increased linearly with the strain rate logarithm:(3)Ue=0.3576+0.0054∗lgε′,R2=0.979.According to the fitting equation (3), the maximum value of elastic strain energy in the karst limestone could be obtained. This value was helpful for studying rock mechanical properties and engineering stability.


## 5.1. Energy Theory

According to Xie et al. [46, 47], the energy theory of a unit volume of rock can be described as follows:(1)W=U=Ue+Ud,where W is the work done to the rock by outside, U is the total energy absorbed from outside, Ue is the elastic strain energy stored in the rock, and Ud is the dissipated energy.The strain energy relationship during the rock failure process is shown in Figure7.Figure 7
Strain energy relationship during the rock failure process [33].According to Huang et al. [33] and Zhang et al. [48], only axial stress works on the rock in the uniaxial compression test:(2)U=∫0ε1σ1dε1,Ue=12Eiσ12≈12E0σ12,in which Ei is the unloading elastic modulus, which can be replaced by the initial elastic modulus E0.


## 5.2. Energy Evolution Mechanism

Figures8(a)–8(h) display the energy evolution mechanism during the failure process of the karst limestone.Figure 8
Energy evolution mechanism of the karst limestone under different strain rates. (a) Energy curves at 5 × 10−6/s. (b) Energy ratio curves at 5 × 10−6/s. (c) Energy curves at 2 × 10−5/s. (d) Energy ratio curves at 2 × 10−5/s. (e) Energy curves at 1 × 10−4/s. (f) Energy ratio curves at 1 × 10−4/s. (g) Energy curves at 5 × 10−4/s. (h) Energy ratio curves at 5 × 10−4/s.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Figures8(a)–8(h) illustrate the energy mechanism of the karst limestone in the loading process under uniaxial compression at different strain rates. The energy ratio presents an S-shape, which is similar to a deep-buried carbonaceous slate [49]. The relationship between stress and axial strain for the samples would undergo approximately five stages:(1)
Initial closure stage: the rock absorbed mechanical energy from outside, and most energy was consumed as dissipated energy for crack closure, friction, and slipping, which constantly produced small AE signals. In this stage,Ue/U decreased to a minimum value, whereas Ud/Ue and Ud/U increased to a maximum value. Then, Ue/U increased, whereas Ud/U decreased.(2)
Elastic stage: the original cracks were completely closed, and the energy input from outside was mainly used for elastic compaction. Thus,Ue/U increased, whereas Ue remained consistent with the stress curve. Moreover, most energy was transformed into elastic strain energy stored in the rock.(3)
Stable stage of fracture growth: in the crack initiation stage, a part of the energy was consumed as surface energy and other forms of radiant energy. The AE activity constantly became active in this stage. However, considerable energy remained stored in the rock as elastic strain energy.Ue/U still increased, but the growth rate gradually reduced. The crack initiation stage still belonged to the elastic stage according to Liang et al. [50] and Jaeger and Cook [51].(4)
Unstable stage of fracture growth: macrocracks appeared and extended rapidly, which caused abundant AE activity with high energy.Ue deviated from the stress curve, and Ud gradually increased, thereby constantly corresponding to crack damage stress (σd). In this stage, Ue/U decreased, whereas Ud/U increased significantly.(5)
Failure stage: after peak stress, the stored elastic strain energy was released and reduced sharply. Dissipated energy increased rapidly, and the rock sample damaged and divided into blocks of different sizes.The maximum value of the elastic strain energy of the rock is important for engineering applications. Liu et al. [52] proposed the energy instability criterion of water-resistant strata and rock mass failure index on the basis of releasable elastic strain energy Ue. Guo et al. [53] presented a new model for predicting and classifying the rock burst considering the intrinsic relations between the elastic strain energy and the rock burst based on the criteria for strength and structural failure of rocks. Table 2 and Figure 9 display the elastic strain energy at the peak strength of rock samples.Table 2
Strain energy at peak strength under different loading rates.
Number
Strain rate (ε′/s)
lg(ε′)
U (MJ/m3)
U
e (MJ/m3)
U
d (MJ/m3)
U
e/U
U
d/U
U
d/Ue
S
11
5 × 10−6
−5.30
0.0994
0.0768
0.0226
0.7724
0.2276
0.2947
S
12
5 × 10−6
−5.30
0.0730
0.0633
0.0097
0.8674
0.1326
0.1529
S
21
2 × 10−5
−4.70
0.1239
0.1085
0.0154
0.8756
0.1244
0.1421
S
22
2 × 10−5
−4.70
0.1140
0.1016
0.0124
0.8912
0.1088
0.1221
S
31
1 × 10−4
−4.00
0.1577
0.1411
0.0165
0.8951
0.1049
0.1172
S
32
1 × 10−4
−4.00
0.1894
0.1338
0.0556
0.7065
0.2935
0.4155
S
41
5 × 10−4
−3.30
0.2492
0.1861
0.0631
0.7467
0.2533
0.3392
S
42
5 × 10−4
−3.30
0.2081
0.1745
0.0337
0.8383
0.1617
0.1929Figure 9
Relationship between strain energy and strain rate.
(a)
(b)Table2 and Figure 9 present that the total energy (U) and elastic strain energy (Ue) at peak strength increased rapidly and then slowly with the increase in the strain rate. The ration of dissipated energy (Ud/U) at peak strength also increased, indicating that rock damage before peak stress increased with the strain rate. Therefore, the peak and amplitude of AE signals increased significantly before peak stress under high strain rates. The fitting relationship showed that U and Ue at peak stress increased linearly with the strain rate logarithm:(3)Ue=0.3576+0.0054∗lgε′,R2=0.979.According to the fitting equation (3), the maximum value of elastic strain energy in the karst limestone could be obtained. This value was helpful for studying rock mechanical properties and engineering stability.


## 6. Conclusions

The uniaxial compression test of the karst limestone was conducted under different static strain rates (5 × 10−6–5 × 10−4/s), and the strain rate effect on the mechanical properties, failure modes, AE characteristics, and energy mechanism of the karst limestone was analyzed. Several main conclusions of this research were drawn as follows:(1)
The strength of the karst limestone under uniaxial compression was 80–150 MPa. The elastic modulus (E) was approximately 50–70 GPa, and the secant modulus (Es) was approximately 30–50 GPa. Moreover, the ratio of Es to E was 0.5–0.8. The strain rate effects on elastic and secant moduli of the karst limestone were not significant. The fitting relationship showed that the strength increased linearly with the strain rate logarithm, and the peak strain presented the parabolic increasing trend with the strain rate logarithm.(2)
The strain rate significantly affected rock failure modes. Under low strain rates, the karst limestone was damaged seriously with more macrocracks and the friction effect was also strong. When the strain rate was large, the number of macrocracks reduced and the crack length was large with more significant coalescence.(3)
The AE amplitude of the karst limestone under uniaxial compression was mainly 50–200 mV, and the ASL was mainly 45–60 dB. Under low strain rates, the friction effect was strong and the AE signals showed favorable continuity. Under a large strain rate (ε′> 1 × 10−4 s−1), the fracture effect strengthened and AE signals showed significant discontinuity. With the increase in the strain rate, the AE peak increased significantly, whereas the accumulated AE activity reduced significantly.(4)
The energy evolution of the karst limestone failure process showed an evident periodic characteristic, and the energy ratio presented an S-shape. In the elastic stage, the elastic strain energy (Ue) curve remained parallel to the stress curve. In the unstable stage of crack growth, Ue deviated from the stress curve, the maximum value of the elastic strain energy ratio appeared at the crack damage stress, and the dissipated energy ratio and dissipated energy increased rapidly. The rock damage before peak stress increased with the strain rate. The fitting relationship showed that the maximum value of Ue at the peak stress showed a linear relationship with the value of the strain rate logarithm.



---

*Source: 6871902-2020-02-14.xml*