Energy dissipation rate: An indicator of coal deformation and failure under static and dynamic compressive loads

Energy dissipation rate: An indicator of coal deformation and failure under static and dynamic compressive loads

International Journal of Mining Science and Technology xxx (2017) xxx–xxx Contents lists available at ScienceDirect International Journal of Mining ...

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International Journal of Mining Science and Technology xxx (2017) xxx–xxx

Contents lists available at ScienceDirect

International Journal of Mining Science and Technology journal homepage: www.elsevier.com/locate/ijmst

Energy dissipation rate: An indicator of coal deformation and failure under static and dynamic compressive loads Feng Junjun a,⇑, Wang Enyuan b, Chen Xia c, Ding Houcheng a a

Civil Engineering and Architecture, Anhui University of Technology, Maanshan 243002, China Key Laboratory of Gas and Fire Control for Coal Mines, China University of Mining and Technology, Xuzhou 221116, China c School of Mathematics and Physics, Anhui University of Technology, Maanshan 243002, China b

a r t i c l e

i n f o

Article history: Received 4 May 2017 Received in revised form 22 August 2017 Accepted 10 November 2017 Available online xxxx Keywords: Energy dissipation Stress drop Split Hopkinson pressure bar (SHPB) Stress-strain Uniaxial compressive strength

a b s t r a c t Dynamic disasters in Chinese coal mines pose a significant threat to coal productivity. Thus, a thorough understanding of the deformation and failure processes of coal is necessary. In this study, the energy dissipation rate is proposed as a novel indicator of coal deformation and failure under static and dynamic compressive loads. The relationship between stress-strain, uniaxial compressive strength, displacement rate, loading rate, fractal dimension, and energy dissipation rate was investigated through experiments conducted using the MTS C60 tests (static loads) and split Hopkinson pressure bar system (dynamic loads). The results show that the energy dissipation rate peaks are associated with stress drop during coal deformation, and also positively related to the uniaxial compressive strength. A higher displacement rate of quasi-static loads leads to an initial increase and then a decrease in energy dissipation rate, whereas a higher loading rate of dynamic loads results in larger energy dissipation rate. Theoretical analysis indicates that a sudden increase in energy dissipation rate suggests partial fracture occurring within coal under both quasi-static and dynamic loads. Hence, the energy dissipation rate is an essential indicator of partial fracture and final failure within coal, as well as a prospective precursor for catastrophic failure in coal mine. Ó 2017 Published by Elsevier B.V. on behalf of China University of Mining & Technology. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction China, the world’s largest producer and consumer of coal, produced about 3.41 billion tons of coal and consumed nearly 2.70 billion tons coal in 2016. However, dynamic disasters, such as rockburst and coal and gas outburst, present a serious threat to coal productivity in Chinese coal mines. With the increase in mining depth in recent years, the number of accidents in Chinese coal mines has been on the rise and this has drawn extensive attention from both government and research institutes. Consequently, substantial efforts have been carried out to investigate the mechanism underlying these accidents to implement effective measures for their prevention and control [1,2]. In general, dynamic disasters involve an abrupt failure of coal under external loads generated by various mining-induced activities, such as roof fall, fault slip, and blasting. It is therefore of utmost importance to understand the deformation and failure process of coal under different types of external loads.

⇑ Corresponding author. E-mail address: [email protected] (J. Feng).

Theoretical and experimental studies have already indicated that the energy concepts play a significant role in describing the deformation and failure process of rock materials. For example, the energy concept ‘‘specific energy”, defined as the energy required to excavate a unit volume of rock, was considered as a useful index to identify the critical failure mode transition depth in rock cutting [3]. The energy concept ‘‘dissipated energy” was closely related to the fatigue deformation of rock under cyclic loading [4]. In addition, other types of energy involved in rock engineering have also been extensively investigated. The seismic energy released during brittle rock failure and rock blasting results in strong tremor, eventually causing serious destruction to the underground tunnel [5]. The fragmentation energy is intrinsically associated with the breakage and the creation of new surfaces of rock blocks [6]. Researches in recent years have demonstrated that the deformation and failure of coal are irreversible processes involving energy dissipation, which eventually cause continuous damage and material deterioration during the loading process [7]. The mechanism of these processes has already been proven in many experimental studies on energy dissipation of coal under different loads. Quasi-static experimental studies of coal under uniaxial

https://doi.org/10.1016/j.ijmst.2017.11.006 2095-2686/Ó 2017 Published by Elsevier B.V. on behalf of China University of Mining & Technology. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article in press as: Feng J et al. Energy dissipation rate: An indicator of coal deformation and failure under static and dynamic compressive loads. Int J Min Sci Technol (2017), https://doi.org/10.1016/j.ijmst.2017.11.006

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compression conditions evaluate the positive relationship between dissipated energy and electromagnetic/acoustic energy emitted during coal fracture and failure [8]. Additionally, the results of these studies also confirm the link between energy dissipation and coal deformation and failure under conventional triaxial compression conditions [9]. The deformation and failure of coal under dynamic loads are usually conducted using the split Hopkinson pressure bar (SHPB) system, which is an innovative technique for achieving highstrain-rate loads in dynamic compressive tests [10]. Significant progress has been made in rock dynamic tests over the past few years, a series of key loading techniques has been reported in the literature, including the pulse shaping, momentum-trap [11], strain controlling, coupled static-dynamic loading, and axialradial confining loading techniques [12]. Various measurement techniques have also been introduced to modify the SHPB system, such as the X-ray micro- computed tomography [13], laser gap gauge, and digital image correlation [14]. Moreover, many dynamic properties of rock materials have been studied using the modified SHPB system, including the dynamic deformation and energy dissipation, the dynamic fracture toughness [15], and the dynamic crack propagation [16]. The experimental studies of coal dynamic properties demonstrate that the energy dissipation is closely related to coal fragmentation caused by the dynamic loads [17]. However, the aforementioned energy dissipation refers to the total energy dissipated during the whole loading process, rather than the instantaneous energy consumed at any time during the loading. Thus, the traditional energy concepts are not appropriate for quantitatively describing the instantaneous energy dissipation of coal. Besides, the energy dissipation of coal in quasi-static tests and in dynamic tests has not yet been compared in previous studies. In this study, experiments of coal under quasi-static loads and dynamic loads were conducted using the MTS C60 tests and SHPB systems, respectively, and the energy dissipation rate was

proposed to quantitatively describe the energy dissipation process and the deformation process of coal. The relationship between stress-strain, uniaxial compressive strength, displacement rate, loading rate, fractal dimension, and energy dissipation rate under both quasi-static and dynamic loads were experimentally investigated and theoretically discussed. 2. Experimental method 2.1. Specimen preparation Raw coal materials were obtained directly from the same working face at a depth of over 550 m in Sanhejian coal mine located in Jiangsu Province, and then processed uniformly by subjecting them to drilling, slicing, and polishing to obtain cylindrical specimens. The dimensions of the specimens are Ø50  50 mm for dynamic tests and Ø50  100 mm for quasi-static tests according to the ISRM suggested methods [18]. Both ends of the specimens were lubricated with vacuum grease to eliminate the inertial effects (i.e., the axial inertial effect and the radial inertial effect) and the interfacial friction effect, which affect the homogeneity of the sample deformation [18]. In addition, physical properties (true density and apparent density), thermal property (volatile matter), proximate analysis result (moisture and ash content), and wave velocities (dilatational wave and shear wave) of the coal specimens are obtained through laboratory tests, and the results are listed in Table 1. 2.2. Experimental system The quasi-static tests were conducted using an MTS C60 tests system (Fig. 1). The entire load and displacement histories in the quasi-static tests are measured using linear variable displacement transducers, and the following displacement rates were chosen for

Table 1 Physical properties, thermal property, proximate analysis result, and wave velocities of coal specimens. True density (kg/m3)

Apparent density (kg/m3)

Moisture (%)

Ash (%)

Volatile matter (%)

Calorific value (MJ/kg)

Dilatational wave speed (m/s)

Shear wave speed (m/s)

1446

1541

2.10

26.27

37.53

23.15

2126.89

1074.24

Fig. 1. Photographs of MTS C60 tests system. (a) Operation platform and load frame; (b) Compression space in load frame. ① Upper crosshead, ② Lower crosshead, ③ Test table, ④ Actuator.

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Fig. 2. Schematic diagram of the split Hopkinson pressure bar system. In. bar denotes incident bar; Tr. bar denotes transmitted bar.

the quasi-static tests: 0.3 mm/min, 0.5 mm/min, 0.7 mm/min, and 0.9 mm/min. A camera (model: NV-GX7) is placed near the specimen to record the deformation of coal under the compressive load. The dynamic tests were conducted using a split Hopkinson pressure bar system (Fig. 2), which consists of a striker bar (Ø50  600 mm), an incident bar (Ø50  2400 mm), and a transmitted bar (Ø50  1200 mm), all made of LC4 aluminum, with a Young’s modulus of 200 GPa and an elastic wave velocity of 5060 m/s. The coal specimen is sandwiched between the incident and transmitted bars, and two strain gauges are mounted on the incident and transmitted bars. During the tests, the striker bar driven by a high-pressure gas (nitrogen) hits the free end of incident bar with a compressive force, which induces the propagation of P waves (longitudinal compressive wave) in both directions. The leftpropagating P wave reflects at the free end of the striker bar and separates the striker bar and incident bar after reaching the striker-incident bar interface. Part of the right-propagating P wave in the incident bar reflects at the bar-specimen interface, and the remaining wave passes through the specimen to the transmitted bar. The impact velocity of the striker bar is measured by an optical method and the strain signals recorded by the strain gauges glued on the incident and transmitted bars are stored by the dynamic strain-acquisition instrument (sampling rate, 10 Msps). The deformation of coal during the loading process was recorded by a highspeed camera with a high-intensity flash light installed near the specimen. The setting of the high-speed camera was adjusted for the dynamic tests as 192  192 pixels and 100,000 frames per second (10 ls interframe time). The strain data acquisition was preset to be triggered by the initial part of the incident pulse generated by strain gauge glued on the incident bar; additionally, the high-speed camera was also triggered at the time of data acquisition with a delay of not more than 1 ls.

2.3. Data acquisition and processing The main purpose of data acquisition and processing in this study was to obtain the strain energy and dissipated energy of coal during the loading process. The method for calculating the strain energy and dissipated energy of the rock materials was reported in previous studies and adopted extensively by several researchers [9,19]. During the quasi-static tests, the stress in specimen, rs, is calculated by dividing the compressive load on the specimen by the initial cross-sectional area. The axial strain, es, is calculated

by dividing the change in measured axial length by the original measured axial length according to the ISRM-suggested methods [20]. The total work of the external load is calculated by the integration of the stress-strain curve, and can be divided into the following two parts: the releasable elastic strain energy, Ue s, and the dissipated energy, Ud s. The dissipated energy at any time t during the quasi-static loading can be obtained as follows [7]:

Z U ds ðtÞ ¼ W s  U es ¼

es ðtÞ 0

1 2

rs ðtÞdes  rs ðtÞes ðtÞ

ð1Þ

where Ws is total work of the external load; rs(t) is stress of the specimen; and es(t) is strain of the specimen at any time t during the quasi-static loading. During the dynamic tests, the strain signals were first filtered using an infinite impulse response (IIR) with a two-order Butterworth low-pass filter and a 5 kHz cutoff frequency, because the raw signals collected in the experiment contain a lot of background noise. The incident wave ei(t), reflected wave er(t), and transmitted wave et(t) were subsequently separated from the filtered strain signal, and the histories of strain rate, strain, and stress within the coal specimens were calculated using the following equations [18]:

e_ d ðtÞ ¼

C ½ei ðtÞ  er ðtÞ  et ðtÞ L0

ed ðtÞ ¼ 

rd ðtÞ ¼

C L0

Z 0

ð2Þ

t

½er ðtÞ þ et ðtÞ  ei ðtÞdt

1 A E½ei ðtÞ þ er ðtÞ þ et ðtÞ 2 A0

ð3Þ

ð4Þ

where έd(t) is the strain rate history; ed(t) is the strain history; rd(t) is the stress history of specimen; C is the elastic wave velocity of bars; E is the Young’s modulus; L0 is the sample thickness; A is the cross-sectional area of bar; A0 is the cross-sectional area of specimen; e denotes strain; and the subscripts i, r, and t refer to the incident, reflected, and transmitted waves, respectively. The strain rate of specimen is determined by the average value obtained from Eq. (2), and the dissipated energy of coal specimen under dynamic loads can be calculated using the following equation [21]:

W c ¼ W i  ðW r þ W t Þ

ð5Þ

where Wc is the dissipated energy of coal specimen; Wi, Wr, and Wt are energy carried by the incident, reflected, and transmitted wave, respectively, which can be calculated by Eqs. (6)--(8).

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Wi ¼

Ae

Z s

qe ce

Wr ¼

Wt ¼

Ae qe ce Ae qe ce

0

Z s 0

Z s 0

r2i ðtÞdt ¼ Ae Ece r2r ðtÞdt ¼ Ae Ece r2t ðtÞdt ¼ Ae Ece

Z s 0

Z s 0

Z s 0

e2i ðtÞdt

ð6Þ

e2r ðtÞdt

ð7Þ

e2t ðtÞdt

ð8Þ

0.5 mm/min, 0.7 mm/min, and 0.9 mm/min, respectively. The energy dissipation rate of coal under quasi-static loads was obtained using Eq. (9) and the typical test results are shown in Fig. 3. The results obtained indicate that the energy dissipation rate initially remains steady before the failure point (i.e., the peak stress of coal) and then reaches its maximum near the failure point; it is to be noted that the maximum of energy dissipation rate varies with the displacement rate. Additionally, every peak in the energy dissipation rate curve is associated with a sudden drop in the stress-strain curve, from which we interpret that the peaks of energy dissipation rate are related in some manner to the stress drop within coal under quasi-static loads. The dynamic tests were performed using the SHPB system (Fig. 2). The strain, stress, and energy dissipation rate of coal under dynamic loads were obtained using Eqs. (3), (4), and (10), respectively. The results obtained show that the energy dissipation rate reaches its maximum before the peak stress with the loading rate ranging from 56.96 GPa/s to 115.45 GPa/s, and the stress drop occurs after the energy dissipation rate peak, which agrees well with the results obtained in quasi-static tests (Fig. 4). Additionally, the strength dose not increase significantly from loading rate 93.32 GPa/s to 115.45 GPa/s in Fig. 4c and d, and this is due to the inevitable variation in dynamic compressive tests, which causes the dynamic strength being quite scattered and has been reported in various experimental results [22].

where s is the duration of stress wave; qe is the density of bars; ce is the elastic wave velocity of the bars; Ae is the cross-sectional area of the bars; r(t) denotes stress in bar at time t; and the subscripts i, r, and t refer to the incident, reflected, and transmitted wave, respectively. The dissipated energy calculated from Eqs. (1) and (5) is a traditional indicator of energy dissipation, or more specifically, the result of the total energy consumed under external loads. To quantitatively describe the energy dissipation process, innovative indicators of the energy dissipation under quasi-static and dynamic loads, the energy dissipation rate dUd s and dUd d, have been proposed in this study, which can be calculated by Eqs. (9) and (10), respectively.

dU ds ¼

dU dd

¼

@U ds 1 1 @ rs ðtÞ ¼ rs ðtÞ  es ðtÞ 2 @ es ðtÞ @ es ðtÞ 2 @U dd @ ed ðtÞ

¼ Ae Ece

@

Z s 0

ð9Þ

ðe2i ðtÞ  e2r ðtÞ  e2t ðtÞÞdt

ð10Þ

@ ed ðtÞ

3.2. Energy dissipation rate with uniaxial compressive strength and loading rate

3. Results The peak stress, usually known as ‘‘strength,” of coal under different types of loads is an important mechanical property. The results presented in Fig. 5 shows that the peak stress of coal under quasi-static loads ranges from 1.12 MPa to 6.01 MPa, whereas the peak stress of coal under dynamic loads ranges from 4.89 MPa to

3.1. Energy dissipation rate variation with stress and strain The quasi-static tests were conducted using the MTS C60 tests system (Fig. 1) with the displacement rate set at 0.3 mm/min, 150 Crack Displacement rate 0.3mm/min

Displacement rate 0.5mm/min

2 50

1

6

0.003

0.006

0.009

0.012

Stress 50 2

0

0 0.000

0.015

0.003

Strain 400

Displacement rate 0.9mm/min 300

Stress (MPa)

8 200 Crack

Stress

4 100 2

Energy dissipation rate 0

0 0.000

0.003

0.006 Strain

0.009

0.012

8

Stress (MPa)

Failure

Crack

6

400 10

Displacement rate 0.7mm/min 10

0.009

(d)

12 Energy dissipation rate (MJ/m3)

(c)

0.006 Strain

Failure 300

Crack

Crack

6

200

Stress 4

100 2

Energy dissipation rate

0

Energy dissipation rate (MJ/m3)

0.000

Crack

4

Energy dissipation rate

0

0

Failure 100

Stress (MPa)

Stress (MPa)

100 Stress

Energy dissipation rate (MJ/m3)

Failure

3

Energy dissipation rate

150

(b)

Energy dissipation rate (MJ/m3)

(a) 4

0 0.000

0.003

0.006

0.009

0.012

0.015

Strain

Fig. 3. Stress-strain curves and energy dissipation rate of coal specimens under quasi-static loads with different displacement rates. (a) 0.3 mm/min; (b) 0.5 mm/min; (c) 0.7 mm/min; (d) 0.9 mm/min.

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4 2 2 0 Stress Energy dissipation rate loading rate 56.96 GPa/s

0

0.000

0.004

0.008

-2

8

6 4 4 2

2 Stress Energy dissipation rate loading rate 62.05 GPa/s

0

-4 0.016

0.012

0.000

0.004

Strain

0.008

0

0.012

-2 0.016

Strain

(c) 12

(d) 12

Stress (MPa)

8

4

4

0

Stress Energy dissipation rate loading rate 93.32 GPa/s

0

0.000

0.004

0.008

8 8 Stress (MPa)

8

Energy dissipation rate (MJ/m3)

12

4

4

0

Stress Energy dissipation rate loading rate 115.45 GPa/s

0

-4

Energy dissipation rate (MJ/m3)

Stress (MPa)

4

Stress (MPa)

6

6

Energy dissipation rate (MJ/m3)

(b)

6

Energy dissipation rate (MJ/m3)

(a)

-4 0.012

0.016

0.000

0.004

0.008

0.012

-8 0.016

Strain

Strain

Fig. 4. Stress-strain curves and energy dissipation rate curves of coal specimens under dynamic loads with different loading rates. (a) 56.96 GPa/s; (b) 62.05 GPa/s; (c) 93.32 GPa/s; (d) 115.45 GPa/s.

The displacement rate listed in Fig. 5 ranges from 0.3 mm/min to 0.9 mm/min in quasi-static tests and loading rate from 42.30 GPa/s to 142.47 GPa/s in dynamic tests; the energy dissipation rate varies with the loading rate in both quasi-static tests and dynamic tests. The results in Fig. 6 shows that a higher displacement rate in quasi-static tests leads to an initial increase and a later decrease in peak stress and energy dissipation rate, whereas higher loading rate in dynamic tests results in larger peak stress and energy dissipation rate. Thus, it can be concluded that the loading rate is a significant external factor influencing the energy dissipation process, as well as the strength of coal.

14 y = 0.59x+2.78 R2 = 0.62

12

Peak stress (MPa)

10 8 y = 0.52x+0.25 R2 = 0.87

6 4

Dynamic loads Linear fit of Dynamic loads Static loads Linear fit of Static loads

2 0

2

4

6

8

10

12

14

3.3. Energy dissipation rate with fragmentation 16

Energy dissipation rate (MJ/m3) Fig. 5. Energy dissipation rate and peak stress under quasi-static loads and dynamic loads.

11.54 MPa; the energy dissipation rate at failure point ranges from 2.02 MJ/m3 to 10.75 MJ/m3 under quasi-static loads and 3.91 MJ/ m3 to 14.89 MJ/m3 under dynamic loads. The strength of coal under dynamic loads is generally higher than that under quasistatic loads. The relationship between peak stress and the energy dissipation rate at failure point was analyzed under both quasistatic loads and dynamic loads conditions. The results shown in Fig. 5 demonstrate that the peak stress (i.e., the uniaxial compressive strength) is positively associated with the corresponding energy dissipation rate, which indicates that energy dissipation rate is an alternative indicator of mechanical property.

The failure behavior is mainly reflected by fragmentation caused by different types of external loads. The fragments of coal under quasi-static loads and dynamic loads were collected and sieved after the tests, and the following sieve sizes were chosen: 30 mm, 20 mm, 10 mm, 3 mm, 0.5 mm. Based on sieve sizes, the size of fragments, r, were divided into the following six ranges: I, r > 30 mm; II, 30 > r > 20 mm; III, 20 > r > 10 mm; IV, 10 > r > 3 mm; V, 3 > r > 0.5 mm; VI, 0.5 > r. The typical sieved fragments of coal under different loading rates are shown in Figs. 7 and 8. The cumulative mass fraction of fragments passing through a certain sieve size was determined by weighing the fragments and the results are shown in Fig. 9. The proportion of small fragments (r < 20 mm) produced by dynamic loads is larger than that produced by quasi-static loads (Fig. 9), and the size of fragments produced by quasi-static loads is mostly in the range of r > 30 mm, whereas the size of fragments produced by dynamic loads is mostly in the range of r < 20 mm.

Please cite this article in press as: Feng J et al. Energy dissipation rate: An indicator of coal deformation and failure under static and dynamic compressive loads. Int J Min Sci Technol (2017), https://doi.org/10.1016/j.ijmst.2017.11.006

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6

16 2 12 0 8 -2 4

Static loads Energy dissipation rate

-4 0.3

0.4

0.5

0.6

0.7

0.8

Peak stress (MPa)

Peak stress (MPa)

4

Energy dissipation rate (MJ/m3)

20

12

20

8

16

4

12

0

8

Dynamic loads Energy dissipation rate

-4 0.9

40

60

Displacement rate (mm/min)

80

100

120

140

4

Energy dissipation rate (MJ/m3)

(b)

(a)

160

Loading rate (GPa/s)

Fig. 6. Peak stress and energy dissipation rate at failure point with different displacement rates and loading rates. (a) Quasi-static tests result; (b) Dynamic tests result.

Fig. 7. Photograph of sieved fragments of coal under quasi-static loads.

The distribution of rock fragmentation has been studied extensively and it is well-known that fragmentation usually results in a fractal distribution, which is generally characterized by a fractal dimension, D, defined as follows [23]:

D¼3a

ð11Þ

where a is a Schumann distribution constant, which could be obtained from the cumulative frequency size-distribution curves shown in Fig. 9. The fractal dimension is an indicator of fragmentation level, with larger fractal dimension indicating high levels of fragmentation. The results in Fig. 10 shows that the fractal dimension of coal fragments under quasi-static loads ranges from 1.9 to 2.1, whereas the fractal dimension of coal fragments under dynamic loads ranges from 2.1 to 2.5, which indicates that the coal specimens are much more fragmented under dynamic loads than under quasi-static loads. Fig. 10 also indicates that the relationship between fractal dimension and energy dissipation rate in quasi-static tests is different from that in dynamic tests. A comparison between distribution of fractal dimension in quasi-static tests and dynamic tests shows

that the fractal dimension in dynamic tests is mostly proportional to the energy dissipation rate ranging from 3.91 MJ/m3 to 14.89 MJ/m3, whereas the fractal dimension in quasi-static tests is discretely distributed with the energy dissipation rate ranging from 2.02 MJ/m3 to 10.75 MJ/m3. 4. Discussion 4.1. Relationship between energy dissipation rate and stress drop Energy dissipation is associated with partial damage and irreversible strength deterioration within rock materials, and thus plays an important role in rock deformation and failure [7]. In this study, the energy dissipation rate was proposed to characterize the energy dissipation process quantitatively, and the relationship between energy dissipation rate and stress-strain, strength, and fragmentation distribution was investigated through quasi-static tests and dynamic tests using the MTS C60 tests and SHPB systems, respectively. Experimental results indicate that the energy dissipation rate peaks are related to stress drops within coal, and

Please cite this article in press as: Feng J et al. Energy dissipation rate: An indicator of coal deformation and failure under static and dynamic compressive loads. Int J Min Sci Technol (2017), https://doi.org/10.1016/j.ijmst.2017.11.006

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Fig. 8. Photograph of sieved fragments of coal under dynamic loads.

(a)

(b)

80 20

50

10

25

0

0 0

D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12

75

M%(
90

M%(
100

S1 S2 S3 S4 S5 S6 S7 S8

100

10

20

30

40

0

Sieve size(mm)

10

20

30

40

Sieve size(mm)

Fig. 9. Cumulative frequency size distribution curves. M% (
the maximum energy dissipation rate is noted near the catastrophic failure of coal (Figs. 3 and 4). To investigate the relationship between energy dissipation rate peak and stress drop within coal, the deformation of the coal specimen under both quasi-static and dynamic loads are recorded by a high-speed camera. A comparison of different frames recorded in the quasi-static tests indicates that the stress drops are mainly related to shear crack propagation (Fig. 11a). The crack propagation is generally triggered by a tensile crack opening at the crack tip, followed by shearing immediately along the crack surfaces, which simultaneously results in a stress drop (Fig. 11c). Therefore, the energy dissipation rate peak in quasi-static tests is naturally associated with crack propagation, especially the shear slip along the crack surfaces (Fig. 11b). The experimental results shown in Fig. 12a demonstrate that the tensile crack opens along the loading direction, and the crack opening displacement increases continuously from t = 90 ls to 210 ls. At the beginning of the tensile crack opening, the energy dissipation rate peak arises and then the stress drop happens continuously from phase 2 to 4 (Fig. 12b). Therefore, the energy dissipation rate peak in dynamic tests is also intrinsically associated

with crack propagation, which is the same as that in quasi-static test. For crack propagation under quasi-static loads, the stress drop is related to the static stress intensity factors by:

2

3 2 3 Dr I DK I c I 6 7 6 7 4 DrII 5 ¼ 4 DK II cII 5 DrIII DK III cIII

ð12Þ

where Dr is the corresponding stress drop during crack propagation; the subscripts I, II, III refer to three basic modes of cracks within coal; KI, KII, and KIII are static stress intensity factors of three basic modes of crack; cI, cII, and cIII are constants associated with the geometry of cracks. For crack propagation under dynamic loads, the three components of the stress drop are related to the dynamic stress intensity factors by:

2

3 2 3 2 3 Dr I DK Id cI DK I kI ðv ÞcI 6 7 6 7 6 7 4 DrII 5 ¼ 4 DK IId cII 5 ¼ 4 DK II kII ðv ÞcII 5 DrIII DK IIId cIII DK III kIII ðv ÞcIII

ð13Þ

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J. Feng et al. / International Journal of Mining Science and Technology xxx (2017) xxx–xxx

related to the relative slip of crack surfaces based on the rock fracture mechanics [24].

2.6

Dynamic loads Static loads

2.5

3 3 2 DuI Dr I 1=2 7 2ð1  tÞ 2 6 7 6 DrII ða  x2 Þ 5 4 DuII 5 ¼ 4 l DuIII DrIII =ð1  tÞ 2

Fractal dimension

2.4 2.3 2.2

Dynamic

2.1

Static

2.0 1.9 1.8 0

2

4

6

8

10

12

14

16

Energy dissipation rate (MJ/m3) Fig. 10. Energy dissipation rate at failure and fractal dimension of fragmentations of coal under quasi-static loads and dynamic loads.

where KId, KIId, and KIIId are dynamic stress intensity factors of three basic modes of crack; cI, cII, and cIII are constants associated with the geometry of cracks; kI(v), kII(v), and kIII(v) are universal functions associated with the crack speed and defined pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi as kI ðv Þ  ð1  v =C R Þ= 1  v =C d , kII ðv Þ  ð1  v =C R Þ= 1  v =C s , pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kIII ðv Þ ¼ 1  v =C s ; v is the crack speed; Cd, Cs, and CR are velocities of the dilatational, shear, and Rayleigh wave, respectively. It can be noted from Eqs. (12) and (13) that the stress drops in both quasi-static and dynamic tests are determined by the stress intensity factors and the constants associated with the crack geometry and speed. Furthermore, the stress drop in rock fracture is

ð14Þ

where Du is defined as the difference between particular displacement at adjacent points across the crack (also known as ‘‘crack slip”); Dr is the stress drop in both quasi-static and dynamic test; the subscripts I, II, III refer to three basic modes of cracks within coal; t is Poisson’s ratio; l is the shear modulus; a is the half width of crack; and x is the distance between crack center and the origin of coordinate. The energy dissipation during the crack propagation is essentially strain energy release, and the strain energy release within coal for each crack slip can be calculated by [25]:

1 DU ¼ Dr 2

Z

a

a

Dudx

ð15Þ

where DU is the strain energy release; Dr is the stress drop; Du is the crack slip. Eq. (15) suggests that the strain energy release depends on the stress drop and slip of crack surfaces. Assuming that the crack slip Du is distributed uniformly on crack surfaces, the relationship between strain energy release and stress drop is obtained by substituting Eq. (14) into Eq. (15), that is, 1=2

DU ¼ 2½ð1  tÞ=l½Dr2I þ Dr2II þ Dr2III =ð1  tÞ2 ða2  x2 Þ

ð16Þ

The expression is applicable for both quasi-static and dynamic tests. The difference is that the stress drop in quasi-static and dynamic tests should be determined by Eq. (12) and Eq. (13), respectively. It can be concluded from Eq. (16) that the strain energy release, as well as the dissipated energy, is positively related to the stress drop. Because it is well-known that the final

Fig. 11. (a) Photograph of crack propagation under quasi-static loads; (b) Energy dissipation rate and stress of coal under quasi-static loads; (c) Schematic diagram of crack propagation. A thin layer of white paint was applied to the coal samples for clarity.

Please cite this article in press as: Feng J et al. Energy dissipation rate: An indicator of coal deformation and failure under static and dynamic compressive loads. Int J Min Sci Technol (2017), https://doi.org/10.1016/j.ijmst.2017.11.006

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Fig. 12. (a) Photograph of crack propagation under dynamic loads; (b) Energy dissipation rate and stress of coal under dynamic loads.

failure of coal is generally associated with the maximum stress drop, it can be deduced from Eq. (16) that the dissipated energy, being equal to strain energy release DU, would reach its maximum at the failure point of coal under both quasi-static and dynamic loads, which agrees well with the experimental results shown in Figs. 3 and 4, respectively. The higher strength of coal indicates that more strain energy is accumulated within coal, and thus, more strain energy would be released during the failure process, favoring a positive association between the energy dissipation rate and coal strength (Fig. 5). In conclusion, the peak value of energy dissipation rate is basically attributed to the internal crack slip, which causes stress drop simultaneously; therefore, the energy dissipation rate is an intrinsic indicator of coal partial fracture and final failure. 4.2. Relationship between energy dissipation rate and fragmentation The strain energy released during coal fracture and failure is converted into several other types of energy, which are as follows: (1) fragmentation energy, WG, associated with the new fracture surface; (2) kinetic energy, WK, carried by ejected fragments; (3) other types of energy, WO, that mainly refer to radiant energy (electromagnetic energy and acoustic energy). The energy balance of coal failure is shown in Eq. (17), which is derived according to the first principle of thermodynamics.

DU ¼ W G þ W K þ W O

ð17Þ

where DU is the released strain energy. The energy partition is to some extent arbitrary and based on the final effects of fracture; however, in most cases, the radiant energy is negligible compared with other forms of energy, and the kinetic energy of ejected fragments accounts only for a comparatively small proportion of the

total released strain energy (<7% in dynamic tests and further less in quasi-static tests) [26]. Therefore, the total released strain energy is approximately equal to the fragmentation energy and can be calculated by Eq. (18) based on the fracture mechanics

DU  W G ¼ GAc

ð18Þ

where Ac is the area of new crack surfaces generated in coal under quasi-static or dynamic loads; and G is the energy release rate. Eqs. (17) and (18) suggest that severe fragmentation of coal indicates generation of larger crack surfaces during a catastrophic failure. Consequently, more fragmentation energy is consumed during the fracture process. Because the fractal dimension is an indicator of fragmentation level, and larger fractal dimension is associated with more and smaller fragments generated after failure, the increasing energy dissipation rate is related to larger fractal dimension. This analytical result agrees well with coal failure under dynamic loads only, the fractal dimension of coal failure under quasi-static loads is distributed disorderly in the measured range of energy dissipation rate, which suggests that association between the energy dissipation rate and the coal failure is more complicated under quasi-static loads. 5. Conclusions In this study, the energy dissipation rate is proposed as a novel indicator of coal deformation and failure under static and dynamic loads. The energy dissipation rate remains steady before the failure point and then reaches its maximum near the peak stress of coal. The uniaxial compressive strength of coal under both quasi-static and dynamic loads is positively associated with the energy dissipation rate at failure. The energy dissipation rate is significantly influenced by displacement rate in static tests and loading rate in

Please cite this article in press as: Feng J et al. Energy dissipation rate: An indicator of coal deformation and failure under static and dynamic compressive loads. Int J Min Sci Technol (2017), https://doi.org/10.1016/j.ijmst.2017.11.006

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J. Feng et al. / International Journal of Mining Science and Technology xxx (2017) xxx–xxx

dynamic tests. In quasi-static tests, higher displacement rate leads to an initial increase and then a decrease in peak stress and energy dissipation rate; whereas in dynamic tests, higher loading rate results in larger peak stress and energy dissipation rate. In addition, the energy dissipation rate peak is closely related to the stress drop during coal deformation. It turns out that the energy dissipation rate peak is basically due to the crack propagation within coal, which causes the stress drop simultaneously. Thus, a sudden increase in energy dissipation rate would suggest partial fracture within coal, which is also considered as a precursor of catastrophic failure in coal mine. Acknowledgments Financial supports for this work, provided by the National Natural Science Foundation of China (No. 51574231) and the Youth Fund of Anhui University of Technology (No. QZ201718), are gratefully acknowledged. References [1] Tao ZG, Zhang HJ, Chen YF, Jiang CC. Support principles of NPR bolt/cable and control techniques of large-deformation disasters. Int J Min Sci Technol 2016;26(6):967–73. [2] Lawson H, Weakley A, Miller A. Dynamic failure in coal seams: implications of coal composition for bump susceptibility. Int J Min Sci Technol 2016;26 (1):3–8. [3] He XQ, Xu CS. Specific energy as an index to identify the critical failure mode transition depth in rock cutting. Rock Mech Rock Eng 2016;49(4):1461–78. [4] Song DZ, Wang EY, Li ZH, Liu J, Xu WQ. Energy dissipation of coal and rock during damage and failure process based on EMR. Int J Min Sci Technol 2015;25(5):787–95. [5] Linzer L, Mhamdi L, Schumacher T. Application of a moment tensor inversion code developed for mining-induced seismicity to fracture monitoring of civil engineering materials. J Appl Geophys 2015;112(1):256–67. [6] Zhang ZH, Gong GF, Gao QF, Sun F. Fragmentation energy-saving theory of full face rock tunnel boring machine disc cutters. Chin J Mech Eng 2017;30 (4):913–9. [7] Xie HP, Li LY, Peng RD, Ju Y. Energy analysis and criteria for structural failure of rocks. J Rock Mech Geotech Eng 2009;1(1):11–20. [8] Wasantha PLP, Ranjith PG, Shao SS. Energy monitoring and analysis during deformation of bedded-sandstone: use of acoustic emission. Ultrasonics 2014;54(1):217–26.

[9] Peng RD, Ju Y, Wang JG, Xie HP, Gao F, Mao LT. Energy dissipation and release during coal failure under conventional triaxial compression. Rock Mech Rock Eng 2015;48(2):509–26. [10] Xia KW, Yao W. Dynamic rock tests using split Hopkinson (Kolsky) bar system – a review. J Rock Mech Geotech Eng 2015;7(1):27–59. [11] Bagher Shemirani A, Naghdabadi R, Ashrafi MJ. Experimental and numerical study on choosing proper pulse shapers for testing concrete specimens by split Hopkinson pressure bar apparatus. Constr Build Mater 2016;125(1):326–36. [12] Hokka M, Black J, Tkalich D, Fourmeau M, Kane A, Hoang NH, et al. Effects of strain rate and confining pressure on the compressive behavior of Kuru granite. Int J Impact Eng 2016;91(1):183–93. [13] Yao W, Xu Y, Yu CY, Xia KW. A dynamic punch-through shear method for determining dynamic Mode II fracture toughness of rocks. Eng Fract Mech 2017;176(1):161–77. [14] Liang MZ, Lu FY, Li XY. Dynamic responses and failure of short glass-fiber reinforced syntactic foams. Int J Appl Mech 2017;9(1):1750002–19. [15] Yao W, Xu Y, Liu HW, Xia KW. Quantification of thermally induced damage and its effect on dynamic fracture toughness of two mortars. Eng Fract Mech 2017;169(1):74–88. [16] Wang M, Zhu ZM, Dong YQ, Zhou L. Study of mixed-mode I/II fractures using single cleavage semicircle compression specimens under impacting loads. Eng Fract Mech 2017;177(1):33–44. [17] Feng JJ, Wang EY, Shen RX, Chen L, Li XL, Xu ZY. Investigation on energy dissipation and its mechanism of coal under dynamic loads. Geomech Eng 2016;11(5):657–70. [18] Zhou YX, Xia K, Li XB, Li HB, Ma GW, Zhao J, et al. Suggested methods for determining the dynamic strength parameters and mode-I fracture toughness of rock materials. Int J Rock Mech Min 2012;49(1):105–12. [19] Liu XH, Dai F, Zhang R, Liu JF. Static and dynamic uniaxial compression tests on coal rock considering the bedding directivity. Environ Earth Sci 2015;73 (10):5933–49. [20] Bieniawski ZT, Bernede MJ. Suggested methods for determining the uniaxial compressive strength and deformability of rock materials. Int J Rock Mech Min 1979;16(4):137–40. [21] Lundberg B. A split Hopkinson bar study of energy absorption in dynamic rock fragmentation. Int J Rock Mech Min 1976;13(6):187–97. [22] Zhang QB, Zhao J. A review of dynamic experimental techniques and mechanical behaviour of rock materials. Rock Mech Rock Eng 2014;47 (4):1411–78. [23] Turcotte DL. Fractals and fragmentation. J Geophys Res 1986;91(B2):1921–6. [24] Atkinson BK. Fracture Mechanics of Rock. London: Academic press; 1987. [25] Venkataraman A, Kanamori H. Observational constraints on the fracture energy of subduction zone earthquakes. J Geophys Res 2004;109(B5):302–22. [26] Zhang ZX, Kou SQ, Jiang LG, Lindqvist PA. Effects of loading rate on rock fracture: fracture characteristics and energy partitioning. Int J Rock Mech Min 2000;37(7):745–62.

Please cite this article in press as: Feng J et al. Energy dissipation rate: An indicator of coal deformation and failure under static and dynamic compressive loads. Int J Min Sci Technol (2017), https://doi.org/10.1016/j.ijmst.2017.11.006