Experimental study on the effect of strain rate on rock acoustic emission characteristics

Experimental study on the effect of strain rate on rock acoustic emission characteristics

International Journal of Rock Mechanics & Mining Sciences 133 (2020) 104420 Contents lists available at ScienceDirect International Journal of Rock ...

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International Journal of Rock Mechanics & Mining Sciences 133 (2020) 104420

Contents lists available at ScienceDirect

International Journal of Rock Mechanics and Mining Sciences journal homepage: http://www.elsevier.com/locate/ijrmms

Experimental study on the effect of strain rate on rock acoustic emission characteristics Xiling Liu a, Zhou Liu a, Xibing Li a, Fengqiang Gong b, Kun Du a, * a b

School of Resources and Safety Engineering, Central South University, Changsha, 410083, China School of Civil Engineering, Southeast University, Nanjing, 211189, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Rock acoustic emission Strain rate Uniaxial compression test SHPB Characteristic parameter analysis

To explore the variation in rock acoustic emission (AE) characteristics with strain rate, uniaxial compression tests at different loading rates and impact loading tests were conducted on granite using a MTS322 rock mechanical test system and split Hopkinson pressure bar (SHPB) system, respectively. The effect of the strain rate on the AE characteristic parameters, rock fracture properties, and destruction evolution were systematically analyzed. The results demonstrated that with increasing strain rate, the cumulative AE count decreases as a power function, and the variation between the cumulative AE count and strain rate can be fitted log-linearly with a slope of 0.48. The peak frequencies of the AE signals are mostly distributed in the zones of 0–100 kHz, 175–250 kHz, and 400–550 kHz. The signal proportion in the 0–100 kHz zone gradually increases with strain rate, while the signal pro­ portions in the 175–250 kHz and 400–550 kHz zone exhibit decreasing trends. A transition of a sudden increase in the RA-value and decrease in the AF-value occurs when the stress reaches a certain level, and the stress level corresponding to this transition will increase with strain rate. Meanwhile, the RA–AF distribution is mostly concentrated on the abscissa in the low strain rate tests, but gradually concentrates on the longitudinal axis as the strain rate increases. This indicates that tensile cracking becomes the dominant fracture mode with increasing strain rate. The b-value decreases with increasing strain rate in the uniaxial compression tests; however, the bvalue in the impact loading tests is higher than that in the uniaxial compression tests. Furthermore, to distinguish the signals generated by stress wave propagation from the signals generated by rock fracturing in the impact loading tests, a four-parameter k-means algorithm is used to conduct a clustering analysis. The results indicate that the signals can be classified into four clusters: tensile fracturing signals (cluster A), mixed stress wave and shear fracture signals (cluster B), mixed stress wave and tensile fracture signals (cluster C), and stress wave signals (cluster D).

1. Introduction It has been established that the failure mechanism of rock under different loading rates is not consistent with variations in the fracture strength, brittle-ductility, elastic modulus, degree of fragmentation, and fracture pattern. The loading rate reflects the speed of the force applied from a loading device, while the strain rate is generally used to describe the mechanical response characteristics of rock at various loading rates. Generally, strain rates below 10-4 s-1 are considered low strain rates; 104 –102 s-1 are considered intermediate strain rates, of which the range from 10-4–10-2 s-1 is referred to as “quasi-static” and 10-2–102 s-1 is referred to as “quasi-dynamic”; strain rates greater than 102 s-1 are considered high strain rates.1–4 The mechanical response of rock to the

strain rate is also a topic of much interest in rock mechanics.5,6 Accordingly, a substantial amount of effort has been expended toward investigating the strain rate dependence of the mechanical behavior of rock materials. As summarized by Paterson and Masuda et al.7,8 the rock fracture strength has been observed to increase with the strain rate. Subse­ quently, many studies have been performed on different loading con­ ditions, including dynamic loading tests,2,3,9 triaxial compression tests,5,10 uniaxial compression tests,11 and tensile tests.12,13 These studies have investigated the effect of the strain rate on the rock strength, toughness, and deformation characteristics over a wide range of strain rates, and the results demonstrate that the strength and toughness of rock decrease as the strain rate decreases. Liang et al.4

* Corresponding author. E-mail address: [email protected] (K. Du). https://doi.org/10.1016/j.ijrmms.2020.104420 Received 27 August 2019; Received in revised form 27 May 2020; Accepted 20 June 2020 Available online 7 July 2020 1365-1609/© 2020 Elsevier Ltd. All rights reserved.

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considered that differences in the energy absorption cause the difference in mesoscopic crack propagation modes, ultimately resulting in the rate-dependence of the uniaxial compressive strength. In addition, the increase in tensile strength with the strain rate is generated by the in­ homogeneity of rock and crack arrests originating from the generation of a large number of micro-cracks. Notably, besides the above strain rate dependence of the rock strength and toughness, the degree of rock rupture,2,4,14 delay fracture time,15 peak strain, and modulus of elas­ ticity10 all have a better correlation with the strain rate. As a type of polycrystalline material, the mechanical behavior of rock during deformation is closely related to its internal structure; fractures can arise from intergranular and trans-granular cracks on the microscopic scale and the separation of mineral grains and joints on the macroscopic scale. The fracturing of rock during deformation is actually a process of the generation, nucleation, and propagation of cracks, and this process will emit energy outward as elastic waves, which are referred to as acoustic emission (AE) activity. The collected waveforms of the AE signals carry information about the fractures in the rock, and the characteristic pa­ rameters such as the amplitude, frequency, counts, and energy derived from these waveforms are widely used to reveal the rock failure mech­ anism.16–22 Importantly, because variation in the strain rate will lead to a change in the fracture mechanism, the AE signal produced by frac­ turing rock will differ under various strain rates. Modern AE technology originated from the observations of Kaiser,23 while the systematic evaluation of the fracture process in rocks using AE technology arose in the 1960s.16,17 To date, numerous studies have been conducted on AE characteristics in rock.18,22,24–27 Among them, a few have focused on the strain rate dependence of rock AE signal charac­ teristics. Lavorov28 studied the Kaiser effect of limestone at three loading rates—0.05, 1, and 5 mm min-1—and found that the number of AE hits increased with increasing strain rate. However, Filimonov et al.29 studied the rate-dependence of rock salt at loading rates in the range of 0.05–1 MPa s-1, and demonstrated that the AE hit rate was considerably higher at higher loading rates, but the cumulative number of AE hits was smaller. This result was confirmed by the results obtained by Jiang et al. at strain rates ranging from 2 � 10-5–2 � 10-3 s-1.30 Moreover, a few studies have found that the decrease in AE hits and count numbers with increasing loading rate follow a power func­ tion.31,32 This phenomenon can be attributed to the limitation of the development of AE activities by a high loading rate according to Zhang et al.,33 who investigated the effect of the loading rate on the AE counts of three rock types: limestone, sandstone, and rock salt. In addition, Backer et al.12 noted that the amplitude–frequency distribution of AE events remained similar for all low loading rates from 5�10-7 to 5�10-3 mm s-1, whereas the distribution would be shifted to higher amplitudes at higher loading rates. Similarly, Zhang et al.34 suggested that the strain rate has no significant influence on the amplitude–frequency distribu­ tion of AE events in the low strain rate range. In addition, Liu et al.35,36 conducted AE tests on four rock types using a split Hopkinson pressure bar (SHPB) system, and concluded that high-amplitude signals (greater than 70 dB) appeared in the early stage of fracturing under high strain rate loads, and the amplitude–frequency distribution was much different than that in low strain rate loading tests. Based on the above research, it is generally acknowledged that the AE hit rate increases with increasing strain rate, while the total number of AE hits decreases. However, most of the previous research has focused on the low strain rate loading range. Few studies have investigated the whole strain rate range from static to impact loading to elucidate the AE rate-dependent characteristics of rock. This study aims to explore the rock AE rate-dependence at different strain rates; low strain rate AE tests are conducted on a MTS-322 loading device, and high strain rate AE tests are conducted with a SHPB loading system. The variation in characteristic parameters such as the AE counts, energy, RA and AF values, frequency, and b-values of AE signals with the strain rate is analyzed, and the relationship between these AE parame­ ters and the fracture mechanisms at different strain rates are discussed.

2. Experimental setup A coarse granite containing mainly quartz, feldspar, and a small amount of biotite was used in these tests. The rock samples were pro­ cessed into standard specimens according to the method recommended by the International Society for Rock Mechanics (ISRM).37 The sample specifications were Ø50 mm � H100 mm for the uniaxial compression tests, and Ø50 mm � H25 mm for the impact loading tests. High strain rate loading tests were performed using a 50-mm-diam­ eter SHPB system (as shown in Fig. 1(a)) with a spindle-shaped striker that could produce a stable half-sine-wave, which would reduce oscil­ lation.38,39 A PCI-2 system and one ultra-mini PICO type sensor with a resonant frequency of 550 kHz were used to collect the AE signals in SHPB tests. The AE signals detected by the sensor were pre-amplified by 40 dB, and the detection threshold and sampling rate were set at 45 dB and 40 Msps, respectively. A MTS322 rock mechanical test system was used to carry out the uniaxial compression tests, as shown in Fig. 1(b). For the uniaxial compression tests, the displacement control mode was used with loading rates of 2.5�10-3, 2.5�10-2, 0.25, 2.5, and 25 mm s-1. Four PICO sensors were arranged on the cylindrical surface of the rock sample, the AE signals were pre-amplified by 40 dB, and the detection threshold and sampling rate were set at 45 dB and 10 Msps, respectively. In addition, the value of PDT, HDT and HLT was set at 200 μs, 800 μs and1000 μs respectively in impact loading tests, and was set at 50 μs, 200 μs and 300 μs respectively in uniaxial compression loading tests. Notably, the average strain rates in the SHPB tests were indirectly calculated using the stress wave data measured by strain gauges attached to the incident and transmission bars. The equations for calculating the basic dynamic mechanical parameters of the rock sample are as follows38,39:

σ ðtÞ ¼

Ae Ee ½εI ðtÞ þ εR ðtÞ þ εT ðtÞ� 2As

��

ε t ¼ �

εðtÞ ¼

Ce Ls

Z

t

½εI ðtÞ

εR ðtÞ

εT ðtÞ�Þdt;

0

Ce ½εI ðtÞ Ls

εR ðtÞ

εT ðtÞ�;

(1) (2) (3)

where t is a time point in the dynamic loading process; σ(t), ε(t), and ἐ(t) are the stress, strain, and strain rate of rock, respectively, at a certain moment t; εI (t), εR (t), and εT (t) are the incident, reflected, and trans­ mitted strains, respectively, at a certain moment t; Ce, Ee, and Ae are the wave velocity, elastic modulus, and cross-sectional area of the elastic bar, respectively; and As and Ls are the cross-sectional area and length of the rock sample, respectively. 3. Results and discussion 3.1. Mechanical behavior of granite at different strain rates The stress–strain curves and peak stress variation of granite at various strain rates are plotted in Fig. 2. It can be clearly seen that the elastic slope of the curve and the peak stress increase with increasing strain rate because there is insufficient time for the rock samples to release the accumulated energy under higher strain rate loads. These results reflect the commonly observed rate dependence characteristics of rock, which have previously been reported in the literature.2,9,10 In addition, it can be observed that the curve of the pre-peak stress versus strain shifts from an upward concave shape to an upward convex shape (Fig. 2(a)). This suggests that the state of the rock sample is converted from ductile to elastic in the uniaxial compression loading process, while the opposite conversion occurs in the impact loading process (Fig. 2(b)). This demonstrates that the granite sample exhibits brittleness at low strain rates, while it exhibits ductility at high strain rates. This behavior of the rock samples due to varying strain rates is considered to be the 2

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International Journal of Rock Mechanics and Mining Sciences 133 (2020) 104420

Fig. 1. Schematics of the impact and uniaxial compression loading systems with the AE testing device.

cause of stress concentration and redistribution during rock fracturing.34

X

3.2. Variation in AE counts, amplitude, and energy with the strain rate

NðtÞ ¼ α � tβ ; R2 ¼ γ; ðβ > 1Þ;

(4)

where α and β are constants, γ is the goodness of fit of the curve, and the power exponent, β, can represent the cumulative count rate. However, the above common trend is not obvious as the strain rate increases to 10-2 s-1 or higher, as shown in Fig. 3(d), (e), and (f). These P results clearly show that the curve of N(t) changes to a convex shape, P leading to the insignificant characteristic of a slow increase in N P during the initial loading stage. Moreover, the curves of N(t) in Fig. 3 (d), (e), and (f) can be also fitted with a power function as follows: X NðtÞ ¼ α � ðt t0 Þβ ; R2 ¼ γ; ðt > t’; 0 < β < 1Þ; (5)

The variation in AE parameters such as the energy, counts, and amplitude, as well as their relationship with the cracking process, are evaluated. The AE counts are represented herein as N and can be used to reflect information regarding micro-ruptures and the damage process of rock samples. As observed in Fig. 3, at lower strain rates, the initial AE signals mainly have low amplitude and energy, and the recorded crack generation and growth processes are obvious. In contrast, in the prox­ imity of the ultimate failure load, the AE energy increases noticeably P with a marked rise in the slope of the cumulative counts ( N) due to the occurrence of large-scale ruptures with high amplitude (Fig. 3(a), (b)). However, as the strain rate increases continuously, the AE signals generated by crack growth and propagation are minimal, and the recorded signals are mainly those produced by macroscopic fractures on a larger scale. This indicates that the micro-cracks within the rock sample are unable to develop sufficiently as a result of the high strain P rate. Meanwhile, the shape of the cumulative counts ( N) versus time curve also changes as the strain rate increases. In the low strain rate tests P (3 � 10-5–2.5�10-3 s-1), the curve of N versus time can be fitted by a power law function (red dashed line in Fig. 3) with the following form:

where t’ is the time corresponding to points D, E, and F in Fig. 3(d), (e), P and (f), respectively; these are the inflection points of the N(t) curve. The constants for the fitting curves described by Equations (4) and (5) at the six different strain rates in Fig. 3 are listed in Table 1. At lower strain rates, the power exponent, β, is greater than 1, while β is less than P 1 at higher strain rates where the shape of the curve of N(t) changes from concave to convex; β gradually decreases with increasing strain P rate. In addition, the variation of N with the stain rate can be fitted log-linearly with a slope of 0.57 in the uniaxial compression tests and a slope of 0.48 over the whole strain rate range, as shown in Fig. 4. It can P be seen in Fig. 4 that N is sensitive to lower strain rates, but less 3

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International Journal of Rock Mechanics and Mining Sciences 133 (2020) 104420

Fig. 2. Curves describing the mechanical characteristics of granite samples in uniaxial compression tests and impact loading tests (Images a’ and b’ represent the relationship between the peak stress and strain rate under uniaxial compression and impact loading, respectively).

sensitive to high strain rates. This is because there is no generation, propagation, and transfixion process for micro-cracks in the higher strain rate loading tests, which causes an obvious decrease in the AE signal. This is why a few AE events of large-scale ruptures will be acti­ vated during a slower deformation process, but not activated during a faster process. Hence, the AE signals generated by crack generation, propagation, and transfixion account for the majority of the total number in the whole process of low strain rate loading. However, the crack generation, propagation, and transfixion become less significant with increasing strain rates owing to the shortened time before the peak P stress is reached, which leads to a decrease in N. Notably, when the strain rate reaches 10-2 s-1, the higher probability of instantaneous collision and extrusion between the rock sample and steel plate rather than the cracking process itself leads to the earlier appearance of a large number of low-energy AE signals (see the portions of Fig. 3(d) and (e) framed by the red line); these signals are abnormal, and their characteristics need to be further analyzed. Given this, an analysis of the correlation between the AE counts and the amplitude is used because it has been developed for fatigue crack and propagation monitoring in a variety of materials.40–42 As observed in Fig. 5, the AE amplitude versus count curve also follows a power law relationship. The count and amplitude have a better correspondence in a logarithmic coordinate system when the strain rate is 3�10-5, 3�10-4, and 2.5�10-3 s-1. However, at higher strain rates, some of the AE signals, shown as the red dots in Fig. 5(d) and (e), are not subject to this relationship and are distributed on a path other than that for rock fracturing; these signals are those framed by the red line in Fig. 3(d) and (e) that appear in the initial loading stage. These signals are obviously different from the others, and their peak frequencies are all 68 kHz. Therefore, it is suggested that they are generated by the instantaneous collision and extrusion between the rock sample and steel plate, and should thus be expurgated in the sub­ sequent analysis. Furthermore, this also demonstrates that the correla­ tion between the count and amplitude can be used to identify abnormal signals in rock AE tests. In addition, in the impact loading tests, the AE sensor will collect signals generated by rock fracturing as well as those generated by stress

wave propagation. A correlation analysis of the AE count versus amplitude is also used to analyze the differences in two typical signals, as shown in Fig. 5(f). The stress wave signals collected by the sensor attached to the incident steel bar are distinct from the signals collected on the rock sample. Therefore, the AE parameter characteristics can be used to distinguish the stress wave signals from the rock fracturing signals in impact loading tests; this will be further analyzed in the following section. 3.3. RA–AF distribution and its relationship to the fracture mode The reciprocal of the gradient of the AE signal waveform is expressed by the RA-value, as shown in Fig. 6. RA is a useful value for AE source mechanism characterization,26 and a combined analysis of RA and AF (average frequency) is commonly used in the fracture mode classifica­ tion of rock and concrete materials.43–45 A low RA-value associated with a high AF-value results from the propagation of cracks in the tensile mode, which involve energy release mainly in the form of fast longitu­ dinal waves. Inversely, a higher RA-value associated with a lower AF-value is caused by the propagation of cracks in the shear mode, which comprises an energy release mainly in the form of slower shear waves.46 In particular, the variation in the AF value from high to low may indicate a change in the fracture mode from tensile to shear and could also be associated with the opening of large cracks.47,48 Fig. 7 shows the typical variation in the RA and AF values with time in tests at different strain rates. At lower strain rates, the increase in the RA-value associated with the decrease in the AF-value can be attributed to the occurrence of the shear mode. More importantly, as shown in Fig. 7(a), a transition to a sudden increase in the RA-value and decrease in the AF-value occurs after reaching a stress level of 60%; this indicates a transition from tensile to shear fracture. In uniaxial compression tests, tensile fractures are dominant and are randomly distributed in the rock sample in the initial loading stage. As the stress increases to a certain level, the cracks will concentrate on a certain shear fracturing plane, which will become the final fault plane; this certain stress level is considered to be the demarcation between pre-nucleation and post4

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Fig. 3. Curves for the AE counts, amplitude, and energy with time at different strain rates under uniaxial compression and impact loading: (a)–(e) show the results of uniaxial compression tests at five strain rates; (f) shows the results of the impact loading tests. The letters A, B, C, D, E, and F on each figure represent the points at which the amplitude increases sharply for various strain rates. 5

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dominant failure mode of rock changes from shear failure to tensile failure, in agreement with the results reported in reference.31.

Table 1 Exponents of fitting equations for the AE cumulative count curves at different strain rates. Strain rate -5 -1

3�10 s 3�10-4 s-1 2.5�10-3 s-1 1�10-2 s-1 0.9�10-1 s-1 97.6 s-1

Equation type

α

β

t’

γ

Eq. (4)

235 9691 120681 178540 107263 4593

1.5601 1.6167 1.1738 0.661 0.576 0.01

– – – 1.35 1.92 2

0.9202 0.9404 0.8878 0.8719 0.9674 0.9599

Eq. (5)

3.4. Peak frequency characteristics Frequency is a key parameter for characterizing rupture source properties, and it can be used to infer changes in the internal state of the rock and reveal the rock fracture mechanism.16,17 For rock AE, different types of fractures produce different scales of sources, which release AE signals with different frequencies. The signals generated by large-scale cracks contain significant low-frequency components, while the sig­ nals generated by small-scale cracks contain significant high-frequency components.46–51 In spectral analysis, the peak frequency, which is the frequency with the maximum energy, can be approximately regarded as the main frequency of the signal, and this peak frequency has commonly been used to characterize the source type.52 As shown in Fig. 10, the distribution of the peak frequency of AE signals exhibits obvious zonal features, and these distinct peak frequencies fall mostly in ranges of 0–100 kHz, 175–250 kHz, and 400–550 kHz. To analyze the variation in the AE signal peak frequency with the strain rate more specifically, the distribution percentages of the four peak frequency bands (I: 0–100 kHz, II: 100–150 kHz, III: 175–250 kHz, IV: 400–550 kHz) at different strain rates are plotted in Fig. 11. As the strain rate increases, the signals characterized by low peak frequencies (0–100 kHz) gradually increase, but the proportions of signals with frequencies in the range from 175–250 kHz and 400–550 kHz exhibit a decreasing trend. On one hand, as mentioned above, increasing the strain rate will shorten the time for crack initiation and propagation in the early stage of loading, resulting in fewer of the small-scale fracturing signals marked by high frequencies. On the other hand, the rapid macrofailure of the rock sample under high strain rate loads will cause high AE rates, which will also result in many small-scale fractures being masked by large-scale fractures; thus, the individual events cannot be fully recorded at such high AE rates. It should be noted that the large amount of low-frequency signals in the range of 0–100 kHz may also derive from the propagation of stress waves in the impact loading tests.

Fig. 4. Variation in the cumulative counts versus the strain rate (the black dotted line is the log-linear fitting curve for the uniaxial compression tests, and the blue dotted line is the log-linear fitting curve for the uniaxial compression and impact loading tests). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

nucleation.18 Hence, the stress level corresponding to the sudden in­ crease in the RA-value and decrease in the AF-value can be regarded as the transition point from the pre-nucleation phase to post-nucleation phase during the cracking process. The higher the strain rate, the higher the stress level at this transition point will be, as shown in Fig. 7 (a), (b), and (c), which indicates that the proportion of tensile fractures increases with increasing strain rate. However, this sudden variation in RA and AF is not obvious in the loading tests at higher strain rates, as shown in Fig. 7(d), (e), and (f). An additional useful method for the qualitative classification of cracking behavior is to employ the RA–AF distribution,43–45 in which tensile cracks tend to generate AE signals with low RA values and high AF values, whereas shear cracks exhibit the opposite tendency. As shown in Fig. 8, the RA–AF values are mostly distributed along the abscissa at low strain rates, which indicates that shear cracking is the dominant fracture mode. However, with increasing strain rates, the RA–AF values tend to distribute along the longitudinal axis, which means that tensile cracking gradually becomes the dominant fracture mode over shear cracking; the recorded AE events are mostly due to tensile fracturing along the direction of the axial stress as opposed to events of a stick-slip nature that would be generated by sliding along the shear failure sur­ face. This characterization of the cracking behavior with the strain rate is confirmed by the failure morphology of granite at different strain rates, as shown in Fig. 9. At low strain rates, as micro-cracks grow and coalesce, the main crack tends to propagate along weak planes within the rock, eventually forming shear bands and leading to failure. How­ ever, with increasing strain rate, these shear planes are more difficult to characterize, particularly in the impact loading tests, where the cracks basically propagate along the radial direction of the rock sample, resulting in split fragments. Thus, with increasing strain rate, the

3.5. Variation in the amplitude–frequency distribution The amplitude–frequency distribution scaled by the b-value is commonly used to measure the relative numbers of small and large amplitude signals.17,18,22 The amplitude–frequency distribution and the calculated b1-values for various loading tests are presented in Fig. 12 and Table 2, respectively. It is clear that the b1-values decrease with increasing strain rate, which indicates an increase in large-scale frac­ tures. However, the goodness-of-fit, R21, also decreases, and particularly in the impact loading tests, the value of R21 drops to an unacceptable value. In b-value estimation, one of the important factors affecting the fitting result is incomplete data acquisition, which is mainly caused by the masking of small-scale fractures by larger ones, particularly during the final loading stage when avalanche destruction occurs. With increasing strain rate, the AE rate also increases, and this masking effect will be more significant in higher strain rate loading tests. It can be seen from Fig. 12(c)–(f) that the signals above 80 dB exhibit an increasing trend, which deviates from the Gutenberg-Richter relationship in the amplitude–frequency distribution; this will lead to a poor fitting result. Thus, a b2-value is also calculated using the data excluding signals with amplitudes above 80 dB. As listed in Table 2, the b2-values still decrease with increasing strain rate in the uniaxial compression tests, and the corresponding goodness-of-fit, R22, is significantly improved. This in­ dicates that the proportion of larger-scale AE events is higher in the high strain rate loading tests; this phenomenon has also been observed in three-point bending tests.12 The reason for this phenomenon is the same as discussed above: increasing the strain rate will shorten the time for crack initiation and propagation in the early stage of loading, thus reducing the occurrence of small-scale fracturing signals; in addition, 6

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International Journal of Rock Mechanics and Mining Sciences 133 (2020) 104420

Fig. 5. Log-log distribution of the count vs. amplitude at different strain rates. (a)–(e): uniaxial compression tests at five strain rates; (f): impact loading tests. The red points in (d) and (e) are the continuous low-amplitude signals with weak energy that appear in large numbers at the initial loading stage, whereas the red points in (f) are the signals collected on the incident steel bar in the impact loading tests. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 6. AE waveform and characteristic parameter extraction diagrams.

the rapid macroscopic failure of the rock sample under high strain rate loads will cause the masking of many small-scale fractures by larger ones. Furthermore, the temporal variation in the b-value in Fig. 13 in­ dicates that the b-value is relatively stable before the stress reaches a certain level, and then decreases dramatically in the loading tests at strain rates of 3�10-5 and 3�10-4 s-1. This decrease in the b-value has traditionally been regarded as the precursor to macroscopic fail­ ure.17,18,22 However, the variation in the b-value in higher strain rate loading tests does not exhibit the same trend. This confirms that small-scale cracks in the early stage of loading cannot be sufficiently formatted, and incomplete data acquisition is significant in high strain

rate loading tests. In addition, compared with the variation trend in the b1-value, the b2-value in the impact loading test is greater than that in the uniaxial compression tests, which indicates that the amplitudes in the range of 80–100 dB have a significant effect on the b-value in the impact loading test. It is worth noting that in the impact loading test, stress wave propagation signals will also be collected by the AE sensor. Thus, if the signals with amplitudes above 80 dB are generated by stress wave propagation, the larger b2-value would indicate that the fractures in the impact loading test are mostly small-scale fractures. Therefore, the AE signals in the impact loading test need to be analyzed further. 7

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Fig. 7. Trends in the mean RA and AF parameters during uniaxial compression and impact loading tests in granite. (a)–(e): uniaxial compression tests at five strain rates; (f): impact loading tests.

8

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Fig. 8. Distribution of RA–AF values in granite at different strain rates under uniaxial compression and impact loading. (a)–(e): uniaxial compression tests at five strain rates; (f): impact loading test.

Fig. 9. Photographs of granite fragments in uniaxial compression and impact loading tests at different strain rates. (a)–(e) represent strain rates of 3�10-5, 3�10-4, 2.5�10-3, 10-2, and 0.9�10-1 s-1 in the uniaxial compression tests, respectively; (f)–(j) represent strain rates of 79.8, 91.8, 97.6, 103.4, and 111.6 s-1 in the impact loading tests, respectively.

3.6. Classification of AE signals in the impact loading test using the kmeans algorithm

types of signals in the impact loading test. Here, an extensively used clustering method called the k-means algorithm is applied to classify the signals collected in the impact loading test. Four major characteristic parameters of the AE signal—the RA value, AF value, amplitude, and peak frequency—are used for the clustering analysis. These parameters are normalized to have a zero mean value and unit standard deviation before analysis to eliminate the effect of the physical dimension on the classification. The task of the k-means algorithm is to partition a set of n data points in m-dimensional space into k clusters with k clustering centers, C. The clustering process is conducted using the Euclidean distance, d, between the data and the centers. This distance, d, can be described as

As mentioned above, stress wave propagation signals were collected in the impact loading test. The stress wave propagation signal and rock fracturing signal were discussed using waveform correlation and spec­ trum characteristic analysis in references,35,36 which found that the signal with the highest energy and amplitude is the stress wave signal. However, the signal with the highest energy and amplitude is not the only stress wave propagation signal collected by the AE sensor in the impact loading tests, and thus, neither waveform correlation analysis nor individual AE parameter analysis can fully classify the different 9

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International Journal of Rock Mechanics and Mining Sciences 133 (2020) 104420

Fig. 10. Distribution of the peak frequency at a strain rate of 3�10-5 s-1; similar distributions are also observed for the other strain rate tests. (a): peak frequency distribution versus time; (b): distribution percentage of four distinct peak frequency bands; (c): corresponding density cloud map.

Fig. 11. Variation in the four frequency bands at different strain rates (I: 0–100 kHz, II: 100–150 kHz, III: 175–250 kHz, IV: 400–550 kHz; the letters A–E refer to uniaxial compression tests at strain rates of 3�10-5, 3�10-4, 2.5�10-3, 10-2, and 0.9�10-1 s-1, respectively, while F refers to the impact loading test).

follows53,54: m � X �xik dij2 ð2Þ ¼

�2 xjk � ði 6¼ jÞ;

determined by means of a validity criterion such as the maximum of the silhouette-value, S, combined with the sum of squared error (SSE) index. These criteria can be expressed as: � n ai min bij 1X � �; � (8) S¼ n 1 max min bij ; ai

(6)

k¼1

d2 ð2Þ ¼

k X X

kxn

Cm k2 →min;

(7)

m¼1 xn 2Cm

SSE ¼

where x is the vector composed of the original and standardized pa­ rameters of the AE signals. Based on the nearest mean in Equation (7), new centroids are calculated, and the k-means algorithm is repeated until there are no further changes in these center locations. Because the number of clus­ ters, k, from 2 to 10 is previously unknown, the optimal value of k is

k X X ðx

Ai Þ2 ;

(9)

i¼1 x2Ci

where ai is defined as the cohesion, which is the average distance from sample xi to other sample points in the same cluster, k; bij is the degree of separation, which is the distance from sample xi to all points in the other cluster, k; Ci represents the set of the ith cluster; and Ai is the average 10

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International Journal of Rock Mechanics and Mining Sciences 133 (2020) 104420

Fig. 12. Amplitude–frequency distributions in the uniaxial compression tests (a)–(e) and impact loading test (f).

distance in ith cluster. An elbow point in SSE versus k indicates a good clustering result. Fig. 14(a) shows the silhouette and SSE values versus the classifi­ cation number, k. When the AE signals are classified into four types, the silhouette value has maximum of 0.7523, corresponding to the elbow point in the curve of SSE versus k, which confirms the efficiency of the clustering. Hence, four classification types for AE signals in the impact loading tests is reasonable. Fig. 14(b), (c) shows the distribution of the peak frequency versus the amplitude and the RA-value versus AF-value for these four clusters (labeled A, B, C, and D); the corresponding range of values for each parameter is listed in Table 3. It is clear that clusters A, B, C, and D can easily be classified through the peak frequency­ –amplitude distribution and RA–AF distribution. The signals in cluster A have amplitudes ranging from 45–60 dB, while their peak frequencies range from 400–650 kHz, which are higher than the signals in the other clusters. Similarity, in the RA–AF distribution, the signals of cluster A have the lowest RA-values and highest AF-values compared with the other clusters. This indicates that cluster A can be regarded as signals generated by tensile fracture. Another distinct type of signals with moderate RA-values and low AF-values, and peak frequencies is classi­ fied as cluster D, which can be considered as stress wave signals. As Liu35 previously noted that the signal with highest amplitude and energy in SHPB rock AE tests is the stress wave signal, this highest amplitude and energy signal in the impact loading tests in this study is one of the signals in cluster D, and the signals in cluster D typically appear in the early stage of the loading process. This confirms that the signals in cluster D are those generated by the propagation of stress waves, and the signals with amplitudes above 80 dB that were excluded in b2-value estimation in the impact loading tests are in cluster D. This means that small-scale cracks are less likely to grow and coalesce into larger cracks, but simultaneously appear in the whole rock sample in the impact loading tests. However, the stress wave signals will be collected in the whole loading process, as discussed above, and the masking of various signals will be significant in such high strain rate loading tests. Thus, the two types of signals in clusters B and C may be the mixed signals. The signals in cluster B are more likely stress waves mixed with shear fracture sig­ nals, while the signals in cluster C are more likely stress waves mixed with tensile fracture signals. The differences in the characteristic pa­ rameters of the signals in the four clusters can also be found in Fig. 15,

Table 2 b-values at different strain rates. Loading conditions Uniaxial compression

Impact loading

Strain rate -5 -1

3�10 s 3�10-4 s-1 2.5�10-3 s-1 10-2 s-1 0.9�10-1 s-1 91.8 s-1 103.4 s-1 111.6 s-1

b1-value

b2-value

R21

R22

0.9762 0.7882 0.6781 0.6276 0.5543 0.5601 0.5883 0.6094

1.0945 0.9163 0.8919 0.7398 0.4973 1.2446 1.3309 1.1541

0.9754 0.9325 0.8026 0.7565 0.8659 0.6723 0.6777 0.7488

0.9990 0.9911 0.9666 0.9758 0.9176 0.9149 0.9213 0.9345

Remarks: b1 and b2 refer to b-values calculated in ranges of 40–100 dB and 40–80 dB in uniaxial compression loading tests respectively, and calculated in ranges of 45–100 dB and 45–80 dB in impact loading tests respectively.

Fig. 13. Temporal changes in the dynamic b-value in uniaxial compres­ sion tests. 11

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International Journal of Rock Mechanics and Mining Sciences 133 (2020) 104420

Table 3 Dominant ranges of AE parameters in each cluster. cluster

fracture mechanism

Amplitude (dB)

Peak frequency (kHz)

RAvalue (ms⋅v-1)

AF-value (kHz)

A

Tensile fracture signal Stress wave mixed shear fracture signal Stress wave mixed tensile fracture signal Stress wave signal

45–60

400–500

0–0.2

200–500

60–75

0–50

5–35

0–50

45–70

0–200

0–7

0–300

80–99

0–50

0–7.5

0–100

B C D

interaction between the rock sample and steel platen in the high strain rate loading tests; these noise signals can be better elimi­ nated through a correlation analysis between the counts and amplitude. (2) With increasing strain rate, tensile cracking gradually becomes the dominant fracture mode over with shear cracking, and the recorded AE events are mostly due to tensile fracturing along the direction of the axial stress as opposed to events with a stick-slip nature that are generated by sliding along the shear failure sur­ face. In addition, at lower strain rates, a sudden increase in the RA-value associated with a sharp decrease in the AF-value occur simultaneously at a certain stress level corresponding to the transition of the cracking process from the pre-nucleation phase to post-nucleation phase; this transition stress level increases with increasing strain rate. However, this sudden variation in RA and AF is not obvious when the strain rate exceeds 10-3 s-1. (3) The experimental results in this study show that the AE events generated during the process of crack generation, propagation, and transfixion before the occurrence of macroscopic failure become less significant with increasing strain rate. In addition, the high AE rate in higher strain rate loading tests will also cause small-scale fractures to be masked by larger ones, leading to markedly incomplete AE data acquisition. Incomplete data acquisition severely limits the application of AE technology for rock fracture analysis. Thus, if the whole process of rock behavior from crack germination to macroscopic failure is to be completely analyzed, or the AE characteristics of rock under different loading conditions are to be compared, the strain rate in the rock AE tests should be less than 10-5 s-1. (4) The Gutenberg-Richter relationship describing the rock AE amplitude–frequency distribution is an intrinsic characteristic of AE events indicating that the number of small-scale ruptures is much greater than the number of large-scale ruptures. However, a deviation of the rock AE amplitude–frequency distribution from the Gutenberg-Richter relationship will appear because of incomplete AE data acquisition, which is mainly related to the masking of small-scale ruptures by larger ones. This deviation will be more significant during the final loading stage when avalanche destruction occurs and in high strain rate loading tests. This intrinsic characteristic of the AE events is determined by the internal structure and loading condition of the rock, determining a reliable value to represent it is key. The estimation of b2-values in this paper is a meaningful attempt that aims to diminish the effect of incomplete data acquisition by excluding the signals with amplitudes above 80 dB. The results of this study show that this attempt greatly improves the goodness-of-fit of the b2-values and makes the b2-values more reliable for representing the intrinsic characteristic of AE events. Meanwhile, the variation in the b2-value also means that this intrinsic characteristic of AE events is highly dependent on the strain rate.

Fig. 14. Distribution of four clusters of AE signals under impact loading (labeled A, B, C, and D). (a): Silhouette and SSE values as a function of the number of clusters, k; (b): peak frequency–amplitude distribution; and (c): RA–AF distribution.

which shows the typical signal waveform for each cluster. 4. Conclusions (1) The cumulative AE counts gradually decrease with increasing strain rate, and the variation between the cumulative AE counts and strain rates can be fitted log-linearly with a slope of 0.48. Additionally, noise signals are inevitably generated from the 12

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International Journal of Rock Mechanics and Mining Sciences 133 (2020) 104420

Fig. 15. Typical signal waveforms of the four clusters. (a): typical burst signal in cluster A, which has the shortest rise time and duration, highest frequency, and fast attenuation, exhibiting obvious features of signals generated by tensile fracture; (b): typical continuous signal in cluster B, which has a longer rise time and duration and lower frequency; (c): typical signal in cluster C, which has a shorter rise time, longer duration, and higher frequency; (d): typical stress wave propagation signal in cluster D, which has a longer rise time and duration, and the highest amplitude and energy.

References

(5) It is a difficult task to carry out rock AE experiments with the SHPB system. Not only is it difficult to protect the AE sensor on the rock sample, but the AE acquisition system and SHPB loading system can also not be synchronized; this is why the stress versus time curves in Fig. 3(f) and Fig. 7(f) are not integrated with other AE parameters. At the same time, the stress wave propagation signals will be collected by the AE sensors during the whole impact loading process, and the AE data analysis will inevitably involve the stress wave propagation signals. It is certain that the stress wave propagation signals will influence the analysis of the rock dynamic fracture mechanism with the AE technique; therefore, distinguishing between the signals generated by stress wave propagation and rock fracturing is necessary. An obvious characteristic of stress wave propagation signals is their low frequency. In the k-means clustering analysis in this paper, three of the selected AE parameters (peak frequency, AF-value, and RAvalue) are related to the signal frequency, and the signals are also properly classified. However, the thorough discrimination of these two types of signals in rock impact loading tests seems to be extremely difficult. However, if the signals generated by stress wave propagation are regarded as an inherent property of the rock sample under impact loads, they should be treated as a distinct feature of rock AE during impact loading tests.

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Declaration of competing interest The authors declare that they have no conflicts of interest to this work. The authors also declare that they do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 41630642, 51774326). The authors are grateful for the financial support provided by these funds. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.ijrmms.2020.104420.

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