# Size Effect of Concrete Specimens on the Acoustic Emission Characteristics under Uniaxial Compression Conditions

Authors: Jianbo Wu; Enyuan Wang; Xuekun Ren; Mingwei Zhang
Journal: Advances in Materials Science and Engineering
 (2017)

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## Abstract

Acoustic emission (AE) experiments under uniaxial compression and cyclic loading-unloading compression conditions were performed using different sizes of cubic concrete specimens. The influences of the loading methods and the concrete sizes on the mechanical parameters and the concrete AE activities were analyzed. The loading method was found to have great impact on the deformation, failure, and energy dissipation of concrete materials. With the increase of the material size, the uniaxial compressive strength of the concrete specimens gradually decreased, while the corresponding strain of peak strength increased first and then decreased. The elasticity modulus fluctuated irregularly. Under the uniaxial compression conditions, five AE patterns corresponding to the deformation and failure of the concrete materials were observed. A significant nonlinear relationship was found between the AE and the stress level. The cumulative AE rings at the peak stress showed nonlinear growth with the increase of the concrete size. Based on an established relationship between the cumulative AE rings and the stress level, the necessary conditions for the existence of the quiet AE period were given. Under the uniaxial cyclic loading-unloading compression conditions, the Felicity ratio decreased first and then increased as the stress increased. The research results have some guiding significance to AE-based monitoring of internal stress evolution of coal, rock, and concrete materials and thereby enable assessment of their stability.


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## Body

## 1. Introduction

During the deformation and failure evolution of coal and rock materials, internal strain energy will be transmitted and dissipated continuously. Acoustic emission (AE) is not only one of physical responses to the deformation and failure behaviors under loads but also one of the energy dissipation pathways during the deformation and failure evolution of coal and rock materials. AE event monitoring has become an important means for real-time monitoring of the failure evolution of materials [1]. AE event is a high-frequency and low-energy body wave [2]. Grain-scale motions, such as dislocations, twin crystal, grain rearrangements, and friction, as well as macroscale motions, such as quantity growth and size expansion of microcracks, could lead to AE events [3]. Deformation and failure behaviors of coal, rock, and concrete materials are complicated processes. It is necessary to explore the fracture process from the microscopic perspective to discuss the rupture failure mechanism. The AE phenomenon is one means to study rupture failure on the microscale or grain-scale [2, 3]. Based on the AE activities, precursor information and laws of buckling failure of coal and rock mass have been found [4–7].Many research studies on the AE characteristics of coal, rock, and concrete materials during the deformation and failure process have been reported since the Kaiser effect was discovered [8]. Moura et al. [9] viewed rock rupture as a second-order phase transition and proposed a new method to forecast sudden rock rupture based on the AE phenomenon. Baud et al. [10] studied the relationships between the localization of sandstone stress deformation and the AE evolution and believed that the localization pattern could be discovered through AE imaging. Based on the concept of the AE trigger threshold of rock mass, Cai et al. [11] proposed a retrieval method of rock strength based on the AE parameters. Zong et al. [12] analyzed crack expansion in the damage evolution process of sandstone materials based on the AE activities and established the corresponding damage evolution model. Wasantha et al. [13] studied the energy dissipation characteristics of layered sandstones with different dips under the uniaxial compression condition and found that released AE energy decreased with the increase of the bedding dip because small-scale rupture on the microlevel is similar to the large-scale rupture on the macrolevel [14–16]. Therefore, the large-scale rupture momentum of dynamic disasters, such as rock burst, could be simulated by the small-scale rupture process of rock specimens [17–19]. Study of the AE characteristics from small-scale rock failure to large-scale rock dynamic disasters as well as precursor laws of microseism involves the size effect of coal and rock mass. Moreover, failure and the AE response laws of coal and rock masses with different sizes are of important significance for monitoring and forecasting dynamic disasters in deep underground engineering.At present, scholars have performed many studies on electromagnetic radiation, AE, surface potential change law, and other phenomena in the rupture failure process of coal and rock masses; these studies have achieved tremendous research results that have been applied in engineering practice for the monitoring and forecasting of dynamic disasters. Nevertheless, few research studies have discussed the impact of the size of coal and rock masses on the electromagnetic radiation, AE, and other characteristics; this lack of study is a problem that requires urgent solution.Research studies on the size effect of coal and rock masses under the uniaxial compression condition are being performed worldwide [20]. However, only slow progress has been made, and some uncertain research results have been reported. The first reason for this lack of progress is that coal and rock materials are a heterogeneous media with uneven distribution of internal defects and have great individual differences. In particular, coal and rock specimens have distinct features. Even coal and rock specimens with the same size exhibited great differences in the experimental results. The second reason is that size effect studies require a substantial size range, including many size categories, which are often difficult to implement in practice.Therefore, to eliminate the individual effect of coal and rock specimens and provide a more direct, specific, and objective reflection of influences of pure size effect on electromagnetic radiation and AE characteristics of specimens, this study used concrete blocks that were fabricated using the same formula under equal conditions in the simulation study of similar materials. As an artificial prefabricated rock material, concrete can reduce the physical differences of sample individuals to the maximum extent. On this basis, AE responding laws of cubic concrete specimens with different sizes in the uniaxial loading failure process and the influences of size on the basic mechanical parameters were explored.


## 2. Experimental Design and Scheme

The concrete specimens used in this experiment were prepared mainly by cement, sand, and water according to the mass fraction of 1 : 1.60 : 0.58. The cement was P.O52.5R Portland cement, and the sand was screened by a fine sieve, with a maximum particle size of 0.42 mm; clean tap water was used. These raw materials were evenly mixed at predetermined proportions. Next, the mixture was poured into a cast iron mold and then cultured in the dark for 28 days. The concrete specimens were divided into four groups according to size, namely, 50 mm, 70 mm, 100 mm, and 150 mm cubes. There were 4 concrete specimens in each group. Images of some specimens are shown in Figure1.Figure 1
Different concrete specimens used in the experiment.The uniaxial loading experiment of the concrete specimens was carried out on a SANS microcomputer controlled electrohydraulic servo press. This press system is mainly composed of the PowerTestV3.3 control program, hydraulic oil pump, and DCS controller. The maximum load that the press can apply is 3,000 KN, and the press can apply loads linearly to concrete specimens at a uniform velocity. In addition, the CTA-1 AE data acquisition system manufactured by the Physical Acoustics Company (USA) was used in the experiment to achieve real-time monitoring of AE activities of the concrete specimens at a sampling frequency of 500 kHz. This system is equipped to perform a variety of functions, such as parameter setting, data acquisition, and A/C conversion. Structures of the experimental system are shown in Figure2. The layout of the AE sensors is shown in Figure 3.Figure 2
Loading system and AE monitoring system applied in the experiment.Figure 3
Arrangement of the AE sensors on the specimen.The entire experiment under uniaxial compression conditions adopted the displacement control mode. To ensure the same strain rate, the loading rate was proportional to the specimen size. The specific experimental scheme is presented in Table1.Table 1
Experimental scheme in detail.
Number
Loading condition
Loading rate
50 × 50 × 50
A1
Uniaxial loading with displacement control
0.6 mm/min
A2
A4
0.4 mm/min
A3
Cyclic loading
—
70 × 70 × 70
B1
Uniaxial loading with displacement control
0.4 mm/min
B2
B4
0.4 mm/min
B3
Cyclic loading
—
100 × 100 × 100
C1
Uniaxial loading with displacement control
0.28 mm/min
C2
C4
0.4 mm/min
C3
Cyclic loading
—
150 × 150 × 150
D1
Uniaxial loading with displacement control
0.2 mm/min
D2
D4
0.4 mm/min
D3
Cyclic loading
—


## 3. Results and Discussion on the AE Activities of Concrete Specimens

Stress-strain curves of concrete specimens under the uniaxial compression and uniaxial cyclic loading-unloading compression conditions are shown in Figures4 and 5, respectively. Figure 4 shows that, under the uniaxial compression condition, the stress deformation curves of concrete specimens could be divided into five stages: consolidation stage, linear elastic stage, yield softening stage before the failure, critical buckling stage, and residual strength stage. As shown in Figure 5, the displacement control mode (A3 and B3) was adopted in the uniaxial cyclic loading-unloading compression conditions. The loading and unloading curves basically coincided, and the plastic deformation was weak, indicating the small amount of energy dissipation, while under the force control conditions, the area formed by loading and unloading curves and the horizontal axis was large. The large amount of plastic deformation implied that the loading control method can significantly influence the damage deformation and energy dissipation of concrete materials.Figure 4
Stress-strain curves of concrete specimens under the uniaxial compression conditions.Figure 5
Stress-strain curves of concrete specimens under the uniaxial cyclic loading-unloading compression conditions.
(a)
A3
(b)
B3
(c)
C3
(d)
D3The size effects on the mechanical properties of concrete materials are shown in Figures6 and 7. It can be known that, under the conditions of uniaxial direct and cyclic compression, the variations of the uniaxial compressive strength and the cubic specimen size were basically consistent. Furthermore, the uniaxial compressive strength of C3 significantly deviated from those of C1 and C2. However, specimens of the same size exhibited a small difference between σc and σcc. Hence, the loading method (uniaxial loading or cyclic loading) is not thought to obviously influence the uniaxial compressive strength of concrete specimens. When the size effect on the compressive strength was taken into account, the loading method could be neglected. However, the loading method significantly influenced the corresponding strain (εc).Figure 6
Influence of the size scale on the uniaxial compressive strength, the corresponding strain, and the elastic modulus of concrete specimens.Figure 7
Influence of the size scale on the compressive strength and the corresponding strain of concrete specimens under uniaxial cyclic loading-unloading compression.σcc represents the uniaxial compressive strength, and εcc denotes the corresponding strain.From Figure6, under the conditions of direct uniaxial compression, with the increase of size scale, the uniaxial compressive strength of concrete specimens gradually decreased while the corresponding strain of peak strength increased first and then decreased. The elasticity modulus fluctuated to some extent as the size increased. Also seen from Figure 7, the compressive strength (σcc) and the corresponding strain (εcc) of the peak strength of concrete specimens gradually decreased.


### 3.1. AE Response Models of Concrete Specimens under Uniaxial Compression

The AE response of concrete specimens under uniaxial compression condition is shown in Figure8. It can be found that AE events occurred in the uniaxial compressive failure process of concrete specimens and the AE activities showed different characteristics in different deformation stages. Through the experimental analysis, in view of the relationship between the time sequences of AE events and the main fractures, five AE patterns of concrete specimens under the uniaxial compression conditions were observed, as shown in Figure 9.Figure 8
AE activities of concrete specimens under the uniaxial compression conditions.
(a)
Stress, strain, AE energy, and AE cumulative energy of A1
(b)
Stress, strain, AE count, and AE cumulative counts of B1
(c)
Stress, strain, AE count, and AE cumulative counts of B2
(d)
Stress, strain, AE count, and AE cumulative counts of C1
(e)
Stress, strain, AE count, and AE cumulative counts of D1Figure 9
The AE mode of concrete specimens under uniaxial compression.
(a)
(b)
(c)
(d)
(e)Pattern A accounts for a high proportion of the behavior under uniaxial loading, reaching as high as 37.5% (Figure8(c)). At the beginning of loading, some AE events occur, possibly because there are many small holes in the concrete mass caused by bubbles during the specimen preparation and these holes cause some microcracks in the compression stage. After such small holes are compressed, the AE activities weakened. With the increase of stress level, the microcracks tended to be active and the AE activities strengthened and quickly reached a counting peak. Later, the AE activities weakened, possibly because the internal microcrack density of the concrete materials reached a limit and the occurrence frequency of microcracks decreased. At this moment, the accumulated energy in the material was still inadequate to connect and gather microcracks into macrocracks, manifested by the weakening AE activities. After the AE activities remained stable for a short period, they gradually became active again with the increase of the loads. Accordingly, the AE energy or counts grew very quickly and the accumulated energy reached the energy limitation that the material could tolerate; as a result, the microcracks quickly connected together into macrocracks, leading to primary fracture. Subsequently, the severe friction between weakened fracturing surfaces of materials kept the AE events strong. This phenomenon explains why the main fracture occurred before the AE counts reached the peak value.Pattern B accounts for 31.25% of the AE rings in the uniaxial compression of concrete specimens, which is a common pattern (Figure8(e)). In the very short duration at early loading, strong AE events occurred. Subsequently, the AE events gradually enhanced with the load increase, and the maximum AE activities were achieved at the fracture. Under some circumstances, the peak of AE activities occurred first, and then the AE strength weakened. Concrete failure occurred in the weakening process of the AE activities, that is, after the maximum AE rings.Pattern C accounts for 18.75% of the AE rings in the uniaxial compression of concrete specimens; AE rings often occur under a high loading rate (Figure8(a)). At the beginning of the loading, few AE events were observed. In addition, the AE activities violently fluctuated as the loading process continued. In some periods, AE events were strong, whereas they were weak in other periods. Such states were maintained before 70% of the peak stress. Beyond the 70% of peak stress, the AE strength continued to grow with the increase in the stress level until the main fracture is developed. At this moment, the AE strength reached the maximum, possibly due to the fact that prior to 70% of peak stress, the internal damage of materials had been undergoing complicated evolutions. The occurrence and expansion of internal microcracks in materials are not continuous, but intermittent.Pattern D accounts for 6.25% of the AE rings in the uniaxial compression of concrete specimens (Figure8(b)). The AE activities enhanced with the increase of stress and showed a sudden weakening (or quiet period) at approximately 90% of the stress peak before the fracture. After this quiet period, the AE activities became even more active than those before the quiet period, and the specimens quickly broke. The occurrence of the quiet AE period implies that the adjustment of internal damage evolution of the materials is related to the reduction of strain energies released from concrete damage [21], which has important significance for forecasting dynamic disasters of coal and rock masses.Pattern E accounts for 6.25% of AE rings in the uniaxial compression of concrete specimens (Figure8(d)), which is relatively similar to pattern B. However, the main fracture of pattern E develops in the weakening process of the AE activities after the energy peak. Moreover, there is another AE peak after the peak count of AE activities.


### 3.2. Relationships between the AE Characteristics of Concrete Specimens and the Loads under Uniaxial Compression Conditions

The AE phenomenon in the deformation and failure processes of concrete specimens is caused by the development, expansion, and connection of microcracks. Whether microcracks will expand is determined by whether the stress intensity factor at the tip of the cracks has exceeded its fracture toughness. The stress intensity factor of cracks is a function of the crack size, shape, and far-field stress. In other words, whether cracks will expand is determined to a large extent by the stress that the material has suffered. Therefore, the AE activities are closely related to the stress level.Define the probability density of time sequence of AE counts asfφ in the process when stress level increases from φ to φ+dφ [22]:(1)fφdφ=dNN.N is the accumulated AE rings from the beginning of loading to the stress level φ, and dN denotes the AE rings produced when the stress level increased from φ to φ+dφ. Thus, the probability density function of AE fφ could be expressed as [23](2)fφ=aφ+b,where a and b are experimental constants. The former constant is related to the initial damage and cracks in the specimens, and the latter is related to the size scale and the material type. According to (1) and (2), the accumulated AE rings could be obtained:(3)N=cφaexp⁡bφ.Based on the differentiation of (3), it can be determined that(4)dN=cφa-1exp⁡bφada+φ2db+a+bφdφ,where c is an integration constant and a, b, and c can be determined from the curve fitting.Figure10 shows the relationship between the accumulated AE rings and the stress level φ under -0.3≤a≤1, c=100000, and b=-0.5, 0.5, and 2. When a>0, N is positively related to a, whereas it is negatively related to a when a<0. As b increases, the accumulated AE rings increase. When the stress level φ increases, the variation trend of the accumulated AE rings is related to a and b and exhibits nonlinear changes.Figure 10
Relationships between the cumulative AE counts, the stress levels and parametersa and b.Figure11 shows the fitting relationship between the stress levels and the accumulated AE rings; a good fitting effect is observed, indicating that there is a significant nonlinear relationship between the AE activities and the stress levels. On this basis, it is feasible to retrieve the stress level that concrete materials can bear according to monitored AE counts.Figure 11
Fitting curves of the cumulative AE counts and the stress levels.
(a)
A1-Ch2
(b)
B1-Ch1
(c)
C1-Ch1
(d)
D1-Ch2For the B1-Ch1 sample,a=-0.15, whereas a of A1-Ch2, C1-Ch1, and D1-Ch2 are all larger than 0. It can be found that the AE level of B1-Ch1 during the initial loading period is relatively lower, whereas the AE levels of A1-Ch2, C1-Ch1, and D1-Ch2 during the initial loading period are relatively higher. This indicates that positive and negative values of a could represent the intensity of the AE level in the initial loading period.The theoretical value in Figure12 is the accumulated AE events at the peak stress determined according to (4), and the actual value is the accumulated AE rings at the peak stress measured in the experiment. Except for A1-Ch2 (which has a great gap between the theoretical and actual values), the specimens or channels exhibit minor differences between the theoretical values and the actual values. On this basis, it is feasible to estimate the accumulated AE rings at peak stress according to (3) in most cases. In addition, a nonlinear relationship between the accumulated AE rings and the size scale of the concrete specimens is found. From the beginning of loading to the peak stress, the accumulated AE rings generated from the weak deformation to the entire failure of the concrete specimens increase nonlinearly with the size expansion.Figure 12
The theoretical value and actual value of the cumulative AE counts and its relationship with size at the stress peak.Before the main fracture of the specimens, the AE activities may suddenly weaken (pattern D in Figure9), and the variation of the accumulated AE events tends to be stable. Therefore, quantitative analysis and description of the AE variation of concrete specimens with stress as well as the judgment of the quiet AE period could be carried out according to (3).Calculating the second derivative of Equation (3), we have(5)∂2N∂φ2=cφa-2exp⁡bφaa-1+2abφ+b2φ2.The zero point is(6)φ1=-a+ab,φ2=-a-ab.Because0<φ<1, the value ranges of φ1, a, and b are b>-a+a (0<a<1) or b<-a+a (a>1). The effective regions are A and AB. For φ2, b<-a-a (a>0) and the effective regions are B and AB.Therefore, one necessary condition for the existence of the quiet AE period could be gained: whena and b meet the appropriate conditions in the region AB in Figure 13, the quiet AE period will begin at the stress level φ1 and end at the stress level φ2.Figure 13
Quiet AE period and the correspondinga   and  b.


### 3.3. AE Response Characteristics of Concrete Specimens under Uniaxial Cyclic Loading-Unloading Compression Conditions

As shown in Figure14, the peak loading stress of specimens A3 and C3 gradually increases from the first cycle to the fifth cycle. In each cycle, the AE events are relatively stronger during the loading period, whereas they are relatively weaker in the unloading period. In loading periods of the fifth cycles, the peak loading stress of specimen D3 increases first and then remains at the postpeak stage, which exceeds the uniaxial compressive strength of concrete specimens. The Felicity ratio of the material AE characteristics is the ratio between the stress level corresponding to strong AE events and the maximum stress level at the previous cycle. The Felicity ratio could be used to measure the Kaiser effect and the Felicity effect of materials. Generally, the Kaiser effect is effective when the Felicity ratio is greater than or equal to 1.0 [24].Figure 14
AE response of concrete specimens under the uniaxial cyclic loading-unloading compression conditions.
(a)
Stress and AE energy for A3-Ch2
(b)
Stress and AE count for C3-Ch2
(c)
Stress and AE energy for D3-Ch2As seen from Figure15, in the entire cyclic loading-unloading compression process, the Felicity ratio decreases first and then increases with the increase of the stress level. Before 43.1% of the peak stress, the Felicity ratio of specimen A3 is negatively correlated with the stress level, which indicates that, in two adjacent cycles, the stress level of the second cycle significantly decreases and the occurrence of AE events is relatively ahead of schedule. The Felicity ratio is low (0.827) at approximately 43.1% of the peak stress when the Kaiser effect becomes invalid. Subsequently, the Felicity ratio increases as the stress level increases. This phenomenon may occur because, within a certain stress range, the amount of damage that is recoverable in the unloading period increases in each cycle as the stress level increases, thus resulting in stress reduction for producing strong AE events in the loading period of the next cycle, that is, reduction of the Felicity ratio. After the stress level exceeds a certain value, the amount of unrecoverable damage caused by the plastic deformation increases, which further leads to the relative hysteresis of loading stress for producing strong AE events in the loading stage of the next cycle. This is manifested by the increase of the Felicity ratio. In the postpeak stage, the Felicity ratio continuously increases with the increase of the number of cycles.Figure 15
Variation of the Felicity ratio with the increase of the stress level.


## 3.1. AE Response Models of Concrete Specimens under Uniaxial Compression

The AE response of concrete specimens under uniaxial compression condition is shown in Figure8. It can be found that AE events occurred in the uniaxial compressive failure process of concrete specimens and the AE activities showed different characteristics in different deformation stages. Through the experimental analysis, in view of the relationship between the time sequences of AE events and the main fractures, five AE patterns of concrete specimens under the uniaxial compression conditions were observed, as shown in Figure 9.Figure 8
AE activities of concrete specimens under the uniaxial compression conditions.
(a)
Stress, strain, AE energy, and AE cumulative energy of A1
(b)
Stress, strain, AE count, and AE cumulative counts of B1
(c)
Stress, strain, AE count, and AE cumulative counts of B2
(d)
Stress, strain, AE count, and AE cumulative counts of C1
(e)
Stress, strain, AE count, and AE cumulative counts of D1Figure 9
The AE mode of concrete specimens under uniaxial compression.
(a)
(b)
(c)
(d)
(e)Pattern A accounts for a high proportion of the behavior under uniaxial loading, reaching as high as 37.5% (Figure8(c)). At the beginning of loading, some AE events occur, possibly because there are many small holes in the concrete mass caused by bubbles during the specimen preparation and these holes cause some microcracks in the compression stage. After such small holes are compressed, the AE activities weakened. With the increase of stress level, the microcracks tended to be active and the AE activities strengthened and quickly reached a counting peak. Later, the AE activities weakened, possibly because the internal microcrack density of the concrete materials reached a limit and the occurrence frequency of microcracks decreased. At this moment, the accumulated energy in the material was still inadequate to connect and gather microcracks into macrocracks, manifested by the weakening AE activities. After the AE activities remained stable for a short period, they gradually became active again with the increase of the loads. Accordingly, the AE energy or counts grew very quickly and the accumulated energy reached the energy limitation that the material could tolerate; as a result, the microcracks quickly connected together into macrocracks, leading to primary fracture. Subsequently, the severe friction between weakened fracturing surfaces of materials kept the AE events strong. This phenomenon explains why the main fracture occurred before the AE counts reached the peak value.Pattern B accounts for 31.25% of the AE rings in the uniaxial compression of concrete specimens, which is a common pattern (Figure8(e)). In the very short duration at early loading, strong AE events occurred. Subsequently, the AE events gradually enhanced with the load increase, and the maximum AE activities were achieved at the fracture. Under some circumstances, the peak of AE activities occurred first, and then the AE strength weakened. Concrete failure occurred in the weakening process of the AE activities, that is, after the maximum AE rings.Pattern C accounts for 18.75% of the AE rings in the uniaxial compression of concrete specimens; AE rings often occur under a high loading rate (Figure8(a)). At the beginning of the loading, few AE events were observed. In addition, the AE activities violently fluctuated as the loading process continued. In some periods, AE events were strong, whereas they were weak in other periods. Such states were maintained before 70% of the peak stress. Beyond the 70% of peak stress, the AE strength continued to grow with the increase in the stress level until the main fracture is developed. At this moment, the AE strength reached the maximum, possibly due to the fact that prior to 70% of peak stress, the internal damage of materials had been undergoing complicated evolutions. The occurrence and expansion of internal microcracks in materials are not continuous, but intermittent.Pattern D accounts for 6.25% of the AE rings in the uniaxial compression of concrete specimens (Figure8(b)). The AE activities enhanced with the increase of stress and showed a sudden weakening (or quiet period) at approximately 90% of the stress peak before the fracture. After this quiet period, the AE activities became even more active than those before the quiet period, and the specimens quickly broke. The occurrence of the quiet AE period implies that the adjustment of internal damage evolution of the materials is related to the reduction of strain energies released from concrete damage [21], which has important significance for forecasting dynamic disasters of coal and rock masses.Pattern E accounts for 6.25% of AE rings in the uniaxial compression of concrete specimens (Figure8(d)), which is relatively similar to pattern B. However, the main fracture of pattern E develops in the weakening process of the AE activities after the energy peak. Moreover, there is another AE peak after the peak count of AE activities.


## 3.2. Relationships between the AE Characteristics of Concrete Specimens and the Loads under Uniaxial Compression Conditions

The AE phenomenon in the deformation and failure processes of concrete specimens is caused by the development, expansion, and connection of microcracks. Whether microcracks will expand is determined by whether the stress intensity factor at the tip of the cracks has exceeded its fracture toughness. The stress intensity factor of cracks is a function of the crack size, shape, and far-field stress. In other words, whether cracks will expand is determined to a large extent by the stress that the material has suffered. Therefore, the AE activities are closely related to the stress level.Define the probability density of time sequence of AE counts asfφ in the process when stress level increases from φ to φ+dφ [22]:(1)fφdφ=dNN.N is the accumulated AE rings from the beginning of loading to the stress level φ, and dN denotes the AE rings produced when the stress level increased from φ to φ+dφ. Thus, the probability density function of AE fφ could be expressed as [23](2)fφ=aφ+b,where a and b are experimental constants. The former constant is related to the initial damage and cracks in the specimens, and the latter is related to the size scale and the material type. According to (1) and (2), the accumulated AE rings could be obtained:(3)N=cφaexp⁡bφ.Based on the differentiation of (3), it can be determined that(4)dN=cφa-1exp⁡bφada+φ2db+a+bφdφ,where c is an integration constant and a, b, and c can be determined from the curve fitting.Figure10 shows the relationship between the accumulated AE rings and the stress level φ under -0.3≤a≤1, c=100000, and b=-0.5, 0.5, and 2. When a>0, N is positively related to a, whereas it is negatively related to a when a<0. As b increases, the accumulated AE rings increase. When the stress level φ increases, the variation trend of the accumulated AE rings is related to a and b and exhibits nonlinear changes.Figure 10
Relationships between the cumulative AE counts, the stress levels and parametersa and b.Figure11 shows the fitting relationship between the stress levels and the accumulated AE rings; a good fitting effect is observed, indicating that there is a significant nonlinear relationship between the AE activities and the stress levels. On this basis, it is feasible to retrieve the stress level that concrete materials can bear according to monitored AE counts.Figure 11
Fitting curves of the cumulative AE counts and the stress levels.
(a)
A1-Ch2
(b)
B1-Ch1
(c)
C1-Ch1
(d)
D1-Ch2For the B1-Ch1 sample,a=-0.15, whereas a of A1-Ch2, C1-Ch1, and D1-Ch2 are all larger than 0. It can be found that the AE level of B1-Ch1 during the initial loading period is relatively lower, whereas the AE levels of A1-Ch2, C1-Ch1, and D1-Ch2 during the initial loading period are relatively higher. This indicates that positive and negative values of a could represent the intensity of the AE level in the initial loading period.The theoretical value in Figure12 is the accumulated AE events at the peak stress determined according to (4), and the actual value is the accumulated AE rings at the peak stress measured in the experiment. Except for A1-Ch2 (which has a great gap between the theoretical and actual values), the specimens or channels exhibit minor differences between the theoretical values and the actual values. On this basis, it is feasible to estimate the accumulated AE rings at peak stress according to (3) in most cases. In addition, a nonlinear relationship between the accumulated AE rings and the size scale of the concrete specimens is found. From the beginning of loading to the peak stress, the accumulated AE rings generated from the weak deformation to the entire failure of the concrete specimens increase nonlinearly with the size expansion.Figure 12
The theoretical value and actual value of the cumulative AE counts and its relationship with size at the stress peak.Before the main fracture of the specimens, the AE activities may suddenly weaken (pattern D in Figure9), and the variation of the accumulated AE events tends to be stable. Therefore, quantitative analysis and description of the AE variation of concrete specimens with stress as well as the judgment of the quiet AE period could be carried out according to (3).Calculating the second derivative of Equation (3), we have(5)∂2N∂φ2=cφa-2exp⁡bφaa-1+2abφ+b2φ2.The zero point is(6)φ1=-a+ab,φ2=-a-ab.Because0<φ<1, the value ranges of φ1, a, and b are b>-a+a (0<a<1) or b<-a+a (a>1). The effective regions are A and AB. For φ2, b<-a-a (a>0) and the effective regions are B and AB.Therefore, one necessary condition for the existence of the quiet AE period could be gained: whena and b meet the appropriate conditions in the region AB in Figure 13, the quiet AE period will begin at the stress level φ1 and end at the stress level φ2.Figure 13
Quiet AE period and the correspondinga   and  b.


## 3.3. AE Response Characteristics of Concrete Specimens under Uniaxial Cyclic Loading-Unloading Compression Conditions

As shown in Figure14, the peak loading stress of specimens A3 and C3 gradually increases from the first cycle to the fifth cycle. In each cycle, the AE events are relatively stronger during the loading period, whereas they are relatively weaker in the unloading period. In loading periods of the fifth cycles, the peak loading stress of specimen D3 increases first and then remains at the postpeak stage, which exceeds the uniaxial compressive strength of concrete specimens. The Felicity ratio of the material AE characteristics is the ratio between the stress level corresponding to strong AE events and the maximum stress level at the previous cycle. The Felicity ratio could be used to measure the Kaiser effect and the Felicity effect of materials. Generally, the Kaiser effect is effective when the Felicity ratio is greater than or equal to 1.0 [24].Figure 14
AE response of concrete specimens under the uniaxial cyclic loading-unloading compression conditions.
(a)
Stress and AE energy for A3-Ch2
(b)
Stress and AE count for C3-Ch2
(c)
Stress and AE energy for D3-Ch2As seen from Figure15, in the entire cyclic loading-unloading compression process, the Felicity ratio decreases first and then increases with the increase of the stress level. Before 43.1% of the peak stress, the Felicity ratio of specimen A3 is negatively correlated with the stress level, which indicates that, in two adjacent cycles, the stress level of the second cycle significantly decreases and the occurrence of AE events is relatively ahead of schedule. The Felicity ratio is low (0.827) at approximately 43.1% of the peak stress when the Kaiser effect becomes invalid. Subsequently, the Felicity ratio increases as the stress level increases. This phenomenon may occur because, within a certain stress range, the amount of damage that is recoverable in the unloading period increases in each cycle as the stress level increases, thus resulting in stress reduction for producing strong AE events in the loading period of the next cycle, that is, reduction of the Felicity ratio. After the stress level exceeds a certain value, the amount of unrecoverable damage caused by the plastic deformation increases, which further leads to the relative hysteresis of loading stress for producing strong AE events in the loading stage of the next cycle. This is manifested by the increase of the Felicity ratio. In the postpeak stage, the Felicity ratio continuously increases with the increase of the number of cycles.Figure 15
Variation of the Felicity ratio with the increase of the stress level.


## 4. Conclusions

The AE characteristics and evolution rules of concrete materials under the conditions of uniaxial compression and uniaxial cyclic loading-unloading compression conditions were analyzed and discussed. The findings of this study are as follows.(1) Concrete materials with different sizes have similar deformation and failure process under uniaxial loads. The loading control method has a great impact on the deformation of concrete materials and the concomitant energy dissipation. With the increase of the specimen size, the uniaxial compressive strengthσc of the concrete specimens gradually decreases while the corresponding strain εc at the peak strength increases first and then decreases. The elasticity modulus E fluctuates to some extent as the size increases.(2) Under uniaxial compression conditions, five AE patterns in the failure process of concrete materials were observed. In addition, the AE activity exhibited different characteristics at different deformation stages. There is a significant nonlinear relationship between the AE rings and the stress level. The quantitative relationship between the accumulated AE rings and the stress level was established asN=cφaexp⁡bφ. On this basis, a necessary condition for the existence of quiet AE period was given: a>1, b<-a-a. The accumulated AE rings caused by concrete failures grow nonlinearly with the increase of size.(3) Under the uniaxial cyclic loading-unloading conditions, the AE activities in the loading period of each cycle are relatively stronger but are relatively weaker in the unloading period of each cycle. Felicity ratio changes continuously with the increase of the stress level, which generally decreases first and then increases. In the postpeak period, the Felicity ratio significantly increases with the increase of cycles.(4) The research results not only provide significant guidance to stress level monitoring and stability assessment of coal, rock, and concrete materials based on AE characteristics but also have positive significance for discovery of the precursor information for coal, rock, and concrete materials and thereby enable prediction of their macroscale failures.



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*Source: 7652313-2017-10-02.xml*