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A peak-strength strain energy storage index for rock burst proneness of rock materials
International Journal of Rock Mechanics and Mining Sciences 117 (2019) 76–89
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A peak-strength strain energy storage index for rock burst proneness of rock materials
T
Fengqiang Gong∗, Jingyi Yan, Xibing Li, Song Luo School of Resources and Safety Engineering, Central South University, Changsha, Hunan 410083, China
ARTICLE INFO
ABSTRACT
Keywords: Rock material Rock burst proneness Criterion Peak-strength strain energy storage index Cyclic loading-unloading test Elastic strain energy density
Judgement of rock burst proneness of rock materials is one of the basic problems in the field of rock burst research. In this study, a peak-strength strain energy storage index is proposed for estimating and classifying the rock burst proneness of rock materials. The method for determining this index is also introduced in this paper. The peak-strength strain energy storage index is defined as the ratio of the elastic strain energy density to the dissipated strain energy density corresponding to the peak compressive strength of rock specimen. In order to obtain the elastic strain energy density and the dissipated strain energy density at the peak strength, a series of single cyclic loading-unloading uniaxial compression tests on nine rock materials were conducted. The relationships between the total input energy density and the elastic strain energy density at different unloading stress levels were investigated. The results show that, for every rock material, the elastic strain energy density increases linearly with the increase of the total input energy density. Based on this linear storage energy law, the elastic & dissipated strain energy density at the peak strength can be calculated for each specimen, and the peakstrength strain energy storage index can be obtained accordingly. A new criterion for rock burst proneness of rock materials is proposed. The rock burst proneness of nine rock materials estimated with the proposed criterion agreed well with the laboratory test results.
1. Introduction Rock burst is a kind of geological disaster in deep rock engineering, which is usually caused by sudden and violent release of elastic strain energy stored in rocks.1–3 It generally occurs in hard brittle deep rock masses around highly-stressed underground openings,4–7 and usually causes casualties and damages to facilities as well as delays the project schedule.3,6–12 Rock burst has become a hot topic in the field of rock mechanics and engineering.13–25 In the study of rock burst problems, many researchers have focused on rock burst proneness of rock materials and some discriminant indexes were developed, such as the strain energy storage index,26 the decrease modulus index,27 the potential energy of elastic strain,28 the rock brittleness index,28 and the surplus energy index.29 Among these indexes, the index Wet has been most widely used to assess the rock hardness, brittleness and rock burst proneness in the existing literature.7,14,16,19,28–40 The index Wet is calculated as the ratio of the elastic strain energy density to dissipated strain energy density at the stress level of 80–90% of the peak strength of rock specimen, and the corresponding unloading test needs to conduct (Note: For ease of calculation, strain energy density is used instead of strain energy in this paper).26 In fact, the ∗
indoor rock burst phenomenon of rock materials only emerges and develops when the applied stress reaches the peak strength of rock specimen. Hence, the proportional relationship between the elastic strain energy density and the dissipated strain energy density at the peak strength of rock specimen should be investigated and may be applied to evaluate the rock burst proneness. Due to the heterogeneity and brittleness of natural rock materials, the strength of each rock specimen cannot be pre-determined,41 The unloading test of rock specimen at its peak strength is impossible, so the corresponding ratio of elastic strain energy density to dissipated strain energy density can not be calculated. To solve the problem above, the peak-strength strain energy storage index (Wetp ) is introduced in this study, which is determined as the ratio of the elastic strain energy density to the dissipated strain energy density at the peak strength of rock specimen. A series of single cyclic loading-unloading uniaxial compression tests were conducted under different unloading stress levels for nine rock materials. Based on the linear relationships between the elastic strain energy density and the total input energy density under different unloading stress levels, a method for calculating the elastic strain energy density and the dissipated strain energy density at the peak strength of rock specimen is
Corresponding author. E-mail address: [email protected] (F. Gong).
https://doi.org/10.1016/j.ijrmms.2019.03.020 Received 26 April 2018; Received in revised form 23 January 2019; Accepted 10 March 2019 1365-1609/ © 2019 Elsevier Ltd. All rights reserved.
International Journal of Rock Mechanics and Mining Sciences 117 (2019) 76–89
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proposed, and Wetp can then be obtained. Furthermore, based on the distribution of Wetp and the actual bursting degree of nine rock materials, a new classification criterion for rock burst proneness is suggested and discussed. 2. Brief descriptions of Wet and Wetp The index Wet is defined as the ratio of the elastic strain energy density to the dissipated strain energy density when rock specimen is loaded to u ( u equals the 80–90% of peak strength of rock specimen 26 The formula for c ), and Fig. 1 shows the calculation method for Wet . calculating the index Wet is as follows:
Wet =
ue ud
(1)
where ue , ud are the elastic strain energy density and the dissipated strain energy density corresponding to u and can be expressed by Eqs. (3) and (4), respectively.
Fig. 1. Calculation of the strain energy storage index.26.
u=
u
0 u
ue =
0
ud = u
f ( )d
(2)
f1 ( ) d
(3)
ue
(4)
where u denotes the total input energy density at u and can be expressed by Eq. (2). f ( ), f1 ( ) are the loading curvilinear function and the unloading curvilinear function, respectively. u is the total strain at the unloading point, and 0 is the permanent strain after unloading.42,43 The rock burst proneness is classified into three categories based on Wet , i.e., high rock burst proneness for Wet 5; low rockburst proneness for 2 Wet 4.99; no rockburst proneness for Wet < 2 .26 In this paper, the index Wetp is defined as the ratio of the elastic strain energy density to the dissipated strain energy density at the peak strength of rock specimen. Fig. 2 shows the ideal approach for calculating Wetp , and the formula is given as follows:
Wetp = Fig. 2. The ideal approach for calculating the peak-strength strain energy storage index.
Ue Ud
(5)
where Ue , Ud are the elastic strain energy density and the dissipated strain energy density at the peak strength of rock specimen. Obviously, the key problem in calculating Wetp is to accurately determine Ue and Ud . In this study, a new method for calculating Ue and Ud based on the single cyclic loading-unloading uniaxial compression tests under different unloading stress levels is introduced. 3. Single cyclic loading-unloading uniaxial compression test 3.1. Test procedure Two types of tests were conducted, including the uniaxial compression and single cyclic loading-unloading uniaxial compression tests. The UCS ( c ) of rock material was first obtained by the uniaxial compression test with a rate of 120 kN/min. Subsequently, a series of single cyclic loading-unloading compression tests under different setting unloading stress levels k (the ratio of the stress at the unloading point to c , k = 0.1, 0.3, 0.5, 0.7 and 0.9) were designed and carried out. The loading path of the single cyclic loading-unloading uniaxial compression test is shown in Fig. 3. The rock specimen is loaded with a rate of 120 kN/min until the stress reaches k c , and then unloaded at the same rate until the stress is zero. The specimen is then reloaded until the rock specimen fails ( ck is the actual peak strength of specific rock specimen with unloading stress level k ).
Fig. 3. The loading path of single cyclic loading-unloading uniaxial compression test.
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Fig. 4. Photos of rock specimens.
Table 1 Rock types and basic parameters. Rock type
Red sandstone
Green sandstone
Yellow rust granite
Yueyang granite
Fine granite
Yellow granite
White marble
Leiyang marble
Limestone
Origin
Shandong Province Linyi city
Sichuan Province Zigong city
Hubei Province Yichang city
Hunan Province Yueyang city
Fujian Province Zhangzhou city
Hunan Province Changsha city
Hebei Province Baoding city
Hunan Province Leiyang city
Hunan Province Changsha city
(g/cm3) v ( m/s) E (GPa) c (MPa)
2.43
2.41
2.58
2.60
2.80
2.58
2.70
2.58
2.69
2824
3021
3451
4155
5419
3336
3962
3352
6137
18.84
16.69
17.94
41.78
57.91
36.53
23.40
28.95
45.05
97.56
104.22
75.04
206.96
261.55
194.68
67.66
131.56
169.91
Where , v , E and
c
are the density, P-wave velocity, elastic modulus and UCS, respectively.
3.2. Test machine and specimen preparation
were collected from different quarries in China, as presented in Fig. 4. 50 mm × 100 mm were prepared acCylindrical specimens with cording to the standard of International Society for Rock Mechanics (ISRM).44 The basic parameters including density, P-wave velocity, elastic modulus and UCS are listed in Table 1.
The tests were carried out by the INSTRON 1346 test system, which consist of a control computer, a 2.5 mm displacement extensometer, a loading system and a data acquisition system. The maximum loading range can reach 2000 kN in the quasi-static loading state. The stress is recorded by the data acquisition system and the strain is measured by the 2.5 mm displacement extensometer. In this study, the rock specimens including two types of sandstone, four types of granite, two types of marble and one type of limestone
3.3. Stress-strain curves characteristics Figs. 5 and 6 show the representative stress-strain curves of nine rocks in uniaxial compression tests and single cyclic loading-unloading
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Fig. 5. The representative stress-strain curves of nine rocks in uniaxial compression tests.
Fig. 6. The representative stress-strain curves of nine rocks in uniaxial single cyclic loading-unloading uniaxial compression tests at setting unloading stress level k = 0.5. 79
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Table 2 Strain energy density of nine rock materials under different stress levels. Rock type
Specimen No.
Red sandstone
A–4 A–5 A–6 A–7 A–8 B–2 B–3 B–4 B–5 B–6 C–2 C–3 C–4 C–5 C–6 D–2 D–3 D–4 D–5 D–6 E−4 E−5 E−6 E−7 E−8 E−9 E − 10 E − 11 F–2 F–3 F–4 F–5 F–6 G–2 G–3 G–4 G–5 G–6 H–2 H–3 H–4 H–5 H–6 I–2 I–3 I–4 I–5 I–6
Green sandstone
Yellow rust granite
Yueyang granite
Fine granite
Yellow granite
White marble
Leiyang marble
Limestone
p (MPa)
100.90 102.99 99.89 99.74 99.30 107.38 94.37 95.90 103.64 97.84 74.79 71.26 67.13 71.30 70.96 190.23 194.20 199.53 196.20 201.71 264.71 258.86 281.80 274.78 264.55 288.07 275.74 263.08 171.93 176.59 185.30 179.84 186.83 66.36 64.59 73.14 64.04 71.89 128.55 129.43 137.47 139.39 108.07 154.21 169.04 154.27 171.49 150.26
Unloading stress (MPa)
Actual unloading stress level i
ui (mJ/mm3)
uei (mJ/mm3)
udi (mJ/mm3)
10.18 30.26 49.79 69.02 92.03 10.70 31.32 50.59 77.85 88.51 7.92 22.79 37.61 50.47 64.61 21.20 60.51 99.15 139.12 178.15 55.65 82.85 110.59 131.94 165.23 191.98 220.26 247.25 20.16 55.15 91.52 127.76 164.86 6.90 20.35 33.31 47.31 60.57 11.84 35.81 63.56 84.91 96.27 17.04 48.48 82.70 113.83 147.47
0.10 0.29 0.50 0.69 0.93 0.10 0.33 0.53 0.75 0.90 0.11 0.32 0.56 0.71 0.91 0.11 0.31 0.50 0.71 0.88 0.21 0.32 0.39 0.48 0.62 0.67 0.80 0.94 0.12 0.31 0.49 0.71 0.88 0.10 0.32 0.46 0.74 0.84 0.09 0.28 0.46 0.61 0.89 0.11 0.29 0.54 0.66 0.98
0.0086 0.0336 0.0736 0.1306 0.2396 0.0085 0.0532 0.1055 0.2011 0.2589 0.0042 0.0249 0.0537 0.0880 0.1304 0.0143 0.0641 0.1416 0.2730 0.4071 0.0431 0.0704 0.1181 0.1668 0.2553 0.3228 0.4177 0.5272 0.0146 0.0640 0.1365 0.2567 0.3874 0.0037 0.0142 0.0262 0.0530 0.0824 0.0052 0.0248 0.0562 0.0911 0.1387 0.0080 0.0343 0.0838 0.1510 0.2492
0.0035 0.0253 0.0583 0.0942 0.1855 0.0057 0.0338 0.0728 0.1452 0.1857 0.0027 0.0133 0.0304 0.0517 0.0715 0.0099 0.0508 0.1232 0.2410 0.3508 0.0382 0.0625 0.1047 0.1488 0.2262 0.2865 0.3618 0.4462 0.0092 0.0498 0.1145 0.2131 0.3348 0.0020 0.0068 0.0150 0.0325 0.0501 0.0042 0.0195 0.0466 0.0760 0.1069 0.0067 0.0300 0.0776 0.1377 0.2150
0.0051 0.0083 0.0153 0.0364 0.0541 0.0028 0.0194 0.0327 0.0559 0.0732 0.0015 0.0116 0.0233 0.0363 0.0589 0.0044 0.0133 0.0184 0.0320 0.0563 0.0049 0.0079 0.0134 0.0180 0.0291 0.0363 0.0559 0.0810 0.0054 0.0142 0.0220 0.0436 0.0526 0.0017 0.0074 0.0112 0.0205 0.0323 0.0010 0.0053 0.0096 0.0151 0.0318 0.0013 0.0043 0.0062 0.0133 0.0342
stress greater than k c .42 By calculating the excess energy of each crack and invoking the ‘‘no-interaction’’ approximation,43,45 David et al. have presented a sliding crack model to explain the hysteresis loops, and the model were validated by the experimental data of actual sandstone and granite.45
uniaxial compression tests with unloading stress level k = 0.5. It is observed that the features of stress-strain curves of every rock are different. The plasticity of white marble and yellow rust granite is strong, and their complete stress-strain curves can be measured. Fine granite and limestone show obvious stress drop in the post-peak state, indicating that their brittleness is relatively strong. It also can be seen from Fig. 6 that the hysteresis loops of nine rocks are obvious and various. Just as pointed out in literature [42], irreversible changes will occur in the load-unloading process. The initial loading curve and unloading curve will not overlap, a permanent strain 0 will happen when the unloading stress reaches zero, and the secondary loading curve will lie below the initial loading curve and rejoin the loading curve at a
4. Test results and analysis 4.1. Relationships between three energy density parameters The total input strain energy density and elastic strain energy density of rock specimens at different unloading stress level i (i is the
80
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Fig. 7. Relationships between uei , udi and ui .
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Table 3 Fitting functions between ui and uei of nine rocks.
udi = (1
Rock type
Fitting function for ui and uei
Red sandstone
uei = 0.7720ui
0.0014(R2 = 0. 9982)
uei uei uei uei uei uei uei uei
0.0017(R2 = 0.9994)
Green sandstone Yellow rust granite Yueyang granite Fine granite Yellow granite White marble Leiyang marble Limestone
=
0.7234ui
0.0016(R2 = 0.9995)
0.8542ui
+ 0.0037(R2 = 0.9992)
= 0.8632ui
0.0034(R2 = 0.9994)
= 0.6167ui
0.0007(R2 = 0.9986)
0.7851ui
+ 0.0008(R2 = 0.9972)
=
=
(7)
B
where A and B is the fitting coefficients. Table 3 lists the fitting functions and the correlation coefficient (R2) between ui and uei for nine rocks. The fitting functions between ui and udi can also be derived with Eq. (7). In Table 3, R2 of The fitting functions ranges from 0.9972 to 0.9994, which shows there is a strong linear relationship between the elastic strain energy density and total input energy density, which can be considered to satisfy a linear energy storage law. In addition, the values of |B/A| for nine rock materials are listed in Table 4. From Table 4, it is clear that the ratio of B to A is very small with the maximum being 0.43%, which indicates that the effect of constant B on the linear function is minimal. The A in Eq. (6) can reflect the ratio of the elastic strain energy density to total input energy density. In order to characterize the energy storage performance of rock materials, the energy storage coefficient (ESC) is proposed based on the linear storage energy law, which is defined as A in Eq. (6). The greater the value of ESC is, the higher the capability of elastic strain energy storage is. Among the nine rock materials, the ESC of Yueyang granite (0.8726) is the largest and that of yellow rust granite (0.5580) is the smallest. Hence, Yueyang granite has the highest ability to store elastic strain energy, and the lowest for the yellow rust granite. Similarly, (1A) in Eq. (7) is defined as the energy dissipation coefficient (EDC), which reflects the proportion of dissipated strain energy in the total input energy during the loading process.
= 0.5580ui + 0.0003(R2 = 0.9979) = 0.8726ui
A) ui
= 0.8721ui + 0.0014(R2 = 0.9986)
ratio of preset unloading stress to the peak strength of each rock specimen, and equal to k c / ck .) can be calculated by integrating the stressstrain curves as mentioned in Section 2 and the results are shown in Table 2. Based on the test data listed in Table 2, the relationships between uei , i ud and ui was studied (Fig. 7). The results show that uei and udi increase with the increasing of ui linearly. The relationships between uei , udi and ui can be fit with the linear functions, and the fitting functions are uniformly expressed as follows. (6)
uei = Aui + B
Table 4 |B/A| of nine rock materials. Rock type
Red sandstone
Green sandstone
Yellow rust granite
Yueyang granite
Fine granite
Yellow granite
White marble
Leiyang marble
Limestone
|B1/A1|(%)
0.18
0.24
0.05
0.18
0.43
0.39
0.11
0.10
0.16
Fig. 8. Calculation of the total input energy density at the peak strength: (a) uniaxial compression test; (b) single cyclic loading-unloading uniaxial compression test.
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Table 5 Three energy density parameters at the peak strength and the peak-strength energy storage index of nine rock materials. Rock type
Specimen No.
Test type
U (mJ/mm3)
Ue (mJ/mm3)
Ud (mJ/mm3)
Wetp
Average value of Wetp
Red sandstone
A–1 A–2 A–3 A–4 A–5 A–6 A–7 A–8 B–1 B–2 B–3 B–4 B–5 B–6 C–1 C–2 C–3 C–4 C–5 C–6 D–1 D–2 D–3 D–4 D–5 D–6 E−1 E−2 E−3 E−4 E−5 E−6 E−7 E−8 E−9 E − 10 E − 11
Conventional compression
0.3157 0.3454 0.2601 0.2870 0.3245 0.3019 0.3079 0.3068 0.3657 0.3605 0.3269 0.3215 0.3644 0.3474 0.1813 0.2011 0.1866 0.1659 0.1843 0.1739 0.5508 0.4822 0.5231 0.5576 0.5660 0.5598 0.6584 0.6768 0.6625 0.6811 0.6388 0.7306 0.7739 0.6778 0.7849 0.7271 0.6677
0.2423 0.2652 0.1994 0.2202 0.2491 0.2316 0.2363 0.2354 0.2628 0.2591 0.2348 0.2309 0.2619 0.2496 0.1014 0.1125 0.1044 0.0929 0.1031 0.0973 0.4791 0.4192 0.4549 0.4850 0.4923 0.4869 0.5661 0.5818 0.5696 0.5855 0.5494 0.6278 0.6648 0.5827 0.6742 0.6248 0.5740
0.0734 0.0802 0.0607 0.0668 0.0754 0.0703 0.0716 0.0714 0.1029 0.1014 0.0921 0.0906 0.1025 0.0978 0.0799 0.0886 0.0822 0.0730 0.0812 0.0766 0.0719 0.0630 0.0682 0.0726 0.0737 0.0729 0.0923 0.0950 0.0929 0.0956 0.0894 0.1028 0.1091 0.0951 0.1107 0.1023 0.0937
3.30 3.31 3.28 3.29 3.30 3.30 3.30 3.30 2.56 2.56 2.55 2.55 2.56 2.55 1.27 1.27 1.27 1.27 1.27 1.27 6.67 6.65 6.67 6.68 6.68 6.68 6.13 6.13 6.13 6.12 6.14 6.11 6.09 6.13 6.09 6.11 6.13
3.30
Green sandstone
Yellow rust granite
Yueyang granite
Fine granite
Cyclic loading-unloading compression
Conventional compression Cyclic loading-unloading compression
Conventional compression Cyclic loading-unloading compression
Conventional compression Cyclic loading-unloading compression
Conventional compression Cyclic loading-unloading compression
2.55
1.27
6.67
6.12
Rock type
Specimen No.
Test type
U (mJ/mm3)
Ue (mJ/mm3)
Ud (mJ/mm3)
Wetp
Average value of Wetp
Yellow granite
F–1 F–2 F–3 F–4 F–5 F–6 G–1 G–2 G–3 G–4 G–5 G–6 H–1 H–2 H–3 H–4 H–5 H–6 I–1 I–2 I–3 I–4 I–5 I–6
Conventional compression Cyclic loading-unloading compression
0.5544 0.4447 0.4842 0.4996 0.5065 0.5243 0.1269 0.1365 0.1366 0.1543 0.1441 0.1598 0.2949 0.2544 0.2493 0.2756 0.3053 0.1995 0.3422 0.3440 0.3333 0.2938 0.3415 0.2754
0.4752 0.3805 0.4146 0.4279 0.4338 0.4492 0.0776 0.0835 0.0836 0.0944 0.0882 0.0978 0.2341 0.2021 0.1980 0.2189 0.2424 0.1586 0.2999 0.3014 0.2921 0.2576 0.2993 0.2416
0.0792 0.0642 0.0696 0.0717 0.0727 0.0751 0.0493 0.0530 0.0530 0.0599 0.0559 0.0620 0.0608 0.0523 0.0513 0.0567 0.0629 0.0409 0.0423 0.0426 0.0412 0.0362 0.0422 0.0338
6.00 5.92 5.95 5.96 5.97 5.98 1.57 1.57 1.57 1.58 1.58 1.58 3.85 3.86 3.86 3.86 3.85 3.88 7.08 7.08 7.08 7.12 7.08 7.14
5.96
White marble
Leiyang marble
Limestone
Conventional compression Cyclic loading-unloading compression
Conventional compression Cyclic loading-unloading compression
Conventional compression Cyclic loading-unloading compression
83
1.58
3.86
7.10
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Table 6 Variation coefficients of Rock type
p,
4.3. Calculation of Wetp
Ue , Ud and Wetp for nine rock materials.
Based on Ue and Ud , the index Wetp of each rock specimen can be calculated by using Eq. (5), as summarized in Table 5. It can be found from Tables 2 and 5 that, for the same rock material, the index Wetp is approximately constant although p (peak strength of rock specimen), Ue and Ud of each specimen are significantly different. For example, p , Ue and Ud of red sandstone are in the range of 99.30–102.99 MPa, 0.1994–0.2652 mJ/mm3, 0.0607–0.0802 mJ/mm3, respectively. However, the index Wetp is in the range of 3.28–3.31 with an average value of 3.30. Additionally, this can also be supported by the variation coefficients of p , Ue , Ud and Wetp for nine rock materials listed in Table 6. It can be seen from Table 6 that the maximum variation coefficient of Wetp is 0.46% and the minimum value is zero. Nevertheless, the maximum variation coefficients for p , Ue , Ud are 9.29%, 14.44% and 14.66%, respectively. The variation coefficients of p , Ue , Ud and Wetp for nine rocks are also compared as shown in Fig. 9.
Variation coefficient (%) p
Red sandstone Green sandstone Yellow rust granite Yueyang granite Fine granite Yellow granite White marble Leiyang marble Limestone
2.46 5.20 4.06 2.98 4.44 5.73 9.29 5.93 3.62
Ue
Ud
Wetp
8.31 5.62 6.54 5.95 7.44 8.60 14.44 9.15 6.92
8.10 5.49 6.60 5.81 7.04 8.45 14.66 9.53 7.22
0.22 0.21 0.00 0.18 0.46 0.35 0.28 0.37 0.28
4.2. Calculation of U, Ue and Ud According to the linear relationships between uei and ui presented in Section 4.1, Ue and Ud can be calculated for given U using the following equations:
A) U
Rock bursts are violent disasters that are accompanied by ejection of rock fragments. To evaluate the rock burst proneness of rock materials, it is necessary to analyze the failure and ejection characteristics of rock specimens by means of laboratory tests. On one hand, the ejection processes of rock fragments were captured with high-speed camera during the tests. High-speed camera images of some representative rock specimens in a certain period time during the loading process are shown in Fig. 10. The time interval between adjacent images is 8 ms. It should be noted that small amount of local spalling or several slow ejections may occurred on the specimen surface before the final integrity ejection process. Compared with the red sandstone and green sandstone, the fragment ejection speed of Yueyang granite and yellow granite is much higher and a large amount of rock powder and debris can be observed. It illustrates that the failure of Yueyang granite and yellow granite is more violent.
(8)
Ue = AU + B Ud = (1
5. New criterion for rock burst proneness with index Wetp
B
(9)
In this study, for the two types of tests, U can be obtained by different integration methods (Fig. 8). For the uniaxial compression tests, U is calculated by integrating the initial loading curve before the peak strength. For the single cyclic loading-unloading uniaxial compression tests, the integral process is divided into two steps, i.e., 1) integration of the initial loading curve and 2) integration from the unloading point to the peak strength point along the second loading curve. The three energy density parameters at the peak strength of each rock specimen are listed in Table 5.
Fig. 9. The variation coefficients of
84
p,
Ue , Ud and Wetp for nine rocks.
International Journal of Rock Mechanics and Mining Sciences 117 (2019) 76–89
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Fig. 10. High-speed camera images of typical rock specimens during ejection process.
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Table 7 Ejection characteristics of nine rock materials. Rock type
Failure mode
Ejection speed
Ejection sound
Amount of ejected fragments
Wetp
Rock burst proneness Wetp
Red sandstone Green sandstone Yellow rust granite Yueyang granite Fine granite Yellow granite White marble Leiyang marble Limestone
Shear - Splitting Shear - Splitting Shear Shear - Splitting Shear - Splitting Shear - Splitting Shear - Splitting Shear - Splitting Splitting
Slow Slow – Fast Fast Fast – Slow Fast
Slight Slight – Loud Loud Loud – Slight Loud
Minor Minor – Great Great Great – Minor Great
3.30 2.55 1.27 6.67 6.12 5.96 1.58 3.86 7.10
Low Low No High High High No Low High
On the other hand, some information involving the failure mode, the ejection sound and the number of ejected rock fragments was collected and listed in Table 7. Fig. 11 presents the failure modes and fragments characteristics of nine typical rock specimens. Seven rock materials exhibited ejection features except yellow rust granite and white marble. The yellow rust granite and white marble specimens remained almost intact with a macroscopic crack throughout the specimen after the tests (Fig. 11h and i), which demonstrates that the two rock materials have no rock burst proneness. For the Yueyang granite, fine granite, yellow granite and limestone specimens, a great number of rock fragments, debris, and powder were ejected out of the loading platen with loud blast-like sound during the tests (Fig. 11d and e). These phenomena indicated high rock burst proneness. As for the red sandstone, green sandstone and Leiyang marble specimens, some fragments and debris were ejected with relatively slight sound. Most of the fragments fell over the loading platen (Fig. 11a–c). Therefore, the rock burst proneness of the three rock materials was low. From the above analyses, the rock burst proneness of the nine rock materials can be classified into three grades: no, low and high. The detailed results are given in Table 7. According to the calculated Wetp and the failure pattern of the nine rock materials, a new criterion for rock burst proneness with index Wetp is proposed as follows:
constant and uninfluenced by p , Ue and Ud , which can be regarded as a basic parameter of rock materials. It is feasible and reliable to use Wetp to study the rock burst proneness of rock materials. 7. Conclusions To obtain the strain energy storage index Wetp of rock materials at peak strength, a series of uniaxial compression and single cyclic loading-unloading uniaxial compression tests were designed and conducted on nine rock materials. Based on the experimental results, the following conclusions can be drawn: (1) A strong linear relationship was observed between the elastic strain energy density and the total input energy density for every rock material, which implies that rock materials obey a linear storage energy law during the loading process. Based on this linear storage energy law, the elastic & dissipated strain energy density at the peak strength of rock specimen can be conveniently calculated. This method solves the problem that the elastic & dissipated strain energy density at the peak strength of rock specimen can't be separated accurately. (2) To estimate and classify the rock burst proneness of rock materials, the ejection speed, ejection sound, amount of ejected fragments of rock specimens during failure process were used to evaluate the rock burst proneness of rock materials. Identification results of rock burst proneness of nine rock materials are: The Yueyang granite, fine granite, yellow granite and limestone specimens are high rock burst proneness, the red sandstone, green sandstone and Leiyang marble specimens are low rock burst proneness. Yellow rust granite and white marble specimens have no rock burst proneness. (3) A peak-strength strain energy storage index Wetp was proposed on the basis of the linear storage energy law for rock materials. This index revealed the proportional relation between the elastic strain energy and the dissipated strain energy at the peak strength is almost constant and can well estimate the rock burst proneness of rock materials. Furthermore, based on the actual rock burst proneness of nine rock materials in laboratory and the peak-strength strain energy storage index, a new criterion for rock burst proneness was put forward.
Wetp <2 No rock burst proneness Wetp = 2 5 Low rock burst proneness Wetp >5 High rock burst proneness (10) In Table 7, the results obtained by using the new rock burst criterion are completely consistent with the actual rock burst proneness. 6. Discussion To compare the two indexes, Wet and Wetp , the variation tendency of with i are shown in Fig. 12. It can be observed that, for every rock material, the values of uei / udi are fluctuant and vary irregularly even when i is in the range of 0.8–0.9. It means that Wet obtained in the tests are not unique. As described in Section 4.3, Wetp of the same rock type is nearly
uei /udi
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Fig. 11. Failure mode and pattern of nine typical rock specimens.
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Fig. 12. Variation tendency of uei /udi with i.
Acknowledgements
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This work was supported by the National Natural Science Foundation of China (Grant No. 41877272 and 41472269). Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.ijrmms.2019.03.020. References 1. Cook NGW. Seismicity associated with mining. Eng Geol. 1976;10:99–122. 2. Srinivasan C, Arora SK, Yaji RK. Use of mining and seismological parameters as premonitors of rockbursts. Int J Rock Mech Min Sci. 1997;34:1001–1008. 3. Li SJ, Feng XT, Li ZH, Chen BR, Zhang CQ, Zhou H. In situ monitoring of rockburst nucleation and evolution in the deeply buried tunnels of Jinping II hydropower station. Eng Geol. 2012;137–138(7):85–96. 4. Grodner M. Delineation of rockburst fractures with ground penetrating radar in the Witwatersrand Basin, South Africa. Int J Rock Mech Min Sci. 2001;38:885–891. 5. He MC, Sousa LRE, Miranda T, Zhu GL. Rockburst laboratory tests database — application of data mining techniques. Eng Geol. 2015;185:116–130. 6. Malan DF, Napier JAL. Rockburst support in shallow-dipping tabular stopes at great depth. Int J Rock Mech Min Sci. 2018;112:302–312. 7. Miao SJ, Cai MF, Guo QF, Huang ZJ. Rock burst prediction based on in-situ stress and energy accumulation theory. Int J Rock Mech Min Sci. 2016;83:86–94. 8. Durrheim RJ, Haile A, Roberts MKC, Schweitzer JK, Spottiswoode SM, Klokow JW. Violent failure of a remnant in a deep South African gold mine. Tectonophysics. 1998;289:105–116. 9. Qiu SL, Feng XT, Zhang CQ, Xiang TB. Estimation of rockburst wall-rock velocity invoked by slab flexure sources in deep tunnels. Can Geotech J. 2014;51:520–539. 10. Gong QM, Yin LJ, Wu SY, Zhao J, Ting Y. Rock burst and slabbing failure and its influence on TBM excavation at headrace tunnels in Jinping II hydropower station. Eng Geol. 2012;124:98–108. 11. Mazaira A, Konicek P. Intense rockburst impacts in deep underground construction and their prevention. Can Geotech J. 2015;52:1426–1439. 12. Zhang CQ, Feng XT, Zhou H, Qiu SL, Wu WP. Case histories of four extremely intense rockbursts in deep tunnels. Rock Mech Rock Eng. 2012;45:275–288. 13. Ortlepp WD, Stacey TR. Rockburst mechanisms in tunnels and shafts. Tunn Undergr
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40. Guo WY, Zhao TB, Tan YL, Yu FH, Hu SC, Yang FQ. Progressive mitigation method of rock bursts under complicated geological conditions. Int J Rock Mech Min Sci. 2017;96:11–22. 41. Zhang CQ, Lu JJ, Chen J, Zhou H, Yang FJ. Discussion on rock burst proneness indexes and their relation. Rock Soil Mech. 2017;38:1397–1404. 42. Jaeger JC, Cook NGW, Zimmerman RW. Fundamentals of Rock Mechanics. fourth ed. Oxford: Wiley-Blackwell; 2007. 43. Zimmerman RW. Effect of microcracks on the elastic moduli of brittle solids. J Mater Sci Lett. 1985;4:1457–1460. 44. Fairhurst CE, Hudson JA. Draft ISRM suggested method for the complete stress-strain curve for intact rock in uniaxial compression. Int J Rock Mech Min Sci. 1999;36:281–289. 45. David EC, Brantut N, Schubnel A, Zimmerman RW. Sliding crack model for nonlinearity and hysteresis in the uniaxial stress–strain curve of rock. Int J Rock Mech Min Sci. 2012;52:9–17.
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International Journal of Rock Mechanics and Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
A peak-strength strain energy storage index for rock burst proneness of rock materials
T
Fengqiang Gong∗, Jingyi Yan, Xibing Li, Song Luo School of Resources and Safety Engineering, Central South University, Changsha, Hunan 410083, China
ARTICLE INFO
ABSTRACT
Keywords: Rock material Rock burst proneness Criterion Peak-strength strain energy storage index Cyclic loading-unloading test Elastic strain energy density
Judgement of rock burst proneness of rock materials is one of the basic problems in the field of rock burst research. In this study, a peak-strength strain energy storage index is proposed for estimating and classifying the rock burst proneness of rock materials. The method for determining this index is also introduced in this paper. The peak-strength strain energy storage index is defined as the ratio of the elastic strain energy density to the dissipated strain energy density corresponding to the peak compressive strength of rock specimen. In order to obtain the elastic strain energy density and the dissipated strain energy density at the peak strength, a series of single cyclic loading-unloading uniaxial compression tests on nine rock materials were conducted. The relationships between the total input energy density and the elastic strain energy density at different unloading stress levels were investigated. The results show that, for every rock material, the elastic strain energy density increases linearly with the increase of the total input energy density. Based on this linear storage energy law, the elastic & dissipated strain energy density at the peak strength can be calculated for each specimen, and the peakstrength strain energy storage index can be obtained accordingly. A new criterion for rock burst proneness of rock materials is proposed. The rock burst proneness of nine rock materials estimated with the proposed criterion agreed well with the laboratory test results.
1. Introduction Rock burst is a kind of geological disaster in deep rock engineering, which is usually caused by sudden and violent release of elastic strain energy stored in rocks.1–3 It generally occurs in hard brittle deep rock masses around highly-stressed underground openings,4–7 and usually causes casualties and damages to facilities as well as delays the project schedule.3,6–12 Rock burst has become a hot topic in the field of rock mechanics and engineering.13–25 In the study of rock burst problems, many researchers have focused on rock burst proneness of rock materials and some discriminant indexes were developed, such as the strain energy storage index,26 the decrease modulus index,27 the potential energy of elastic strain,28 the rock brittleness index,28 and the surplus energy index.29 Among these indexes, the index Wet has been most widely used to assess the rock hardness, brittleness and rock burst proneness in the existing literature.7,14,16,19,28–40 The index Wet is calculated as the ratio of the elastic strain energy density to dissipated strain energy density at the stress level of 80–90% of the peak strength of rock specimen, and the corresponding unloading test needs to conduct (Note: For ease of calculation, strain energy density is used instead of strain energy in this paper).26 In fact, the ∗
indoor rock burst phenomenon of rock materials only emerges and develops when the applied stress reaches the peak strength of rock specimen. Hence, the proportional relationship between the elastic strain energy density and the dissipated strain energy density at the peak strength of rock specimen should be investigated and may be applied to evaluate the rock burst proneness. Due to the heterogeneity and brittleness of natural rock materials, the strength of each rock specimen cannot be pre-determined,41 The unloading test of rock specimen at its peak strength is impossible, so the corresponding ratio of elastic strain energy density to dissipated strain energy density can not be calculated. To solve the problem above, the peak-strength strain energy storage index (Wetp ) is introduced in this study, which is determined as the ratio of the elastic strain energy density to the dissipated strain energy density at the peak strength of rock specimen. A series of single cyclic loading-unloading uniaxial compression tests were conducted under different unloading stress levels for nine rock materials. Based on the linear relationships between the elastic strain energy density and the total input energy density under different unloading stress levels, a method for calculating the elastic strain energy density and the dissipated strain energy density at the peak strength of rock specimen is
Corresponding author. E-mail address: [email protected] (F. Gong).
https://doi.org/10.1016/j.ijrmms.2019.03.020 Received 26 April 2018; Received in revised form 23 January 2019; Accepted 10 March 2019 1365-1609/ © 2019 Elsevier Ltd. All rights reserved.
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proposed, and Wetp can then be obtained. Furthermore, based on the distribution of Wetp and the actual bursting degree of nine rock materials, a new classification criterion for rock burst proneness is suggested and discussed. 2. Brief descriptions of Wet and Wetp The index Wet is defined as the ratio of the elastic strain energy density to the dissipated strain energy density when rock specimen is loaded to u ( u equals the 80–90% of peak strength of rock specimen 26 The formula for c ), and Fig. 1 shows the calculation method for Wet . calculating the index Wet is as follows:
Wet =
ue ud
(1)
where ue , ud are the elastic strain energy density and the dissipated strain energy density corresponding to u and can be expressed by Eqs. (3) and (4), respectively.
Fig. 1. Calculation of the strain energy storage index.26.
u=
u
0 u
ue =
0
ud = u
f ( )d
(2)
f1 ( ) d
(3)
ue
(4)
where u denotes the total input energy density at u and can be expressed by Eq. (2). f ( ), f1 ( ) are the loading curvilinear function and the unloading curvilinear function, respectively. u is the total strain at the unloading point, and 0 is the permanent strain after unloading.42,43 The rock burst proneness is classified into three categories based on Wet , i.e., high rock burst proneness for Wet 5; low rockburst proneness for 2 Wet 4.99; no rockburst proneness for Wet < 2 .26 In this paper, the index Wetp is defined as the ratio of the elastic strain energy density to the dissipated strain energy density at the peak strength of rock specimen. Fig. 2 shows the ideal approach for calculating Wetp , and the formula is given as follows:
Wetp = Fig. 2. The ideal approach for calculating the peak-strength strain energy storage index.
Ue Ud
(5)
where Ue , Ud are the elastic strain energy density and the dissipated strain energy density at the peak strength of rock specimen. Obviously, the key problem in calculating Wetp is to accurately determine Ue and Ud . In this study, a new method for calculating Ue and Ud based on the single cyclic loading-unloading uniaxial compression tests under different unloading stress levels is introduced. 3. Single cyclic loading-unloading uniaxial compression test 3.1. Test procedure Two types of tests were conducted, including the uniaxial compression and single cyclic loading-unloading uniaxial compression tests. The UCS ( c ) of rock material was first obtained by the uniaxial compression test with a rate of 120 kN/min. Subsequently, a series of single cyclic loading-unloading compression tests under different setting unloading stress levels k (the ratio of the stress at the unloading point to c , k = 0.1, 0.3, 0.5, 0.7 and 0.9) were designed and carried out. The loading path of the single cyclic loading-unloading uniaxial compression test is shown in Fig. 3. The rock specimen is loaded with a rate of 120 kN/min until the stress reaches k c , and then unloaded at the same rate until the stress is zero. The specimen is then reloaded until the rock specimen fails ( ck is the actual peak strength of specific rock specimen with unloading stress level k ).
Fig. 3. The loading path of single cyclic loading-unloading uniaxial compression test.
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Fig. 4. Photos of rock specimens.
Table 1 Rock types and basic parameters. Rock type
Red sandstone
Green sandstone
Yellow rust granite
Yueyang granite
Fine granite
Yellow granite
White marble
Leiyang marble
Limestone
Origin
Shandong Province Linyi city
Sichuan Province Zigong city
Hubei Province Yichang city
Hunan Province Yueyang city
Fujian Province Zhangzhou city
Hunan Province Changsha city
Hebei Province Baoding city
Hunan Province Leiyang city
Hunan Province Changsha city
(g/cm3) v ( m/s) E (GPa) c (MPa)
2.43
2.41
2.58
2.60
2.80
2.58
2.70
2.58
2.69
2824
3021
3451
4155
5419
3336
3962
3352
6137
18.84
16.69
17.94
41.78
57.91
36.53
23.40
28.95
45.05
97.56
104.22
75.04
206.96
261.55
194.68
67.66
131.56
169.91
Where , v , E and
c
are the density, P-wave velocity, elastic modulus and UCS, respectively.
3.2. Test machine and specimen preparation
were collected from different quarries in China, as presented in Fig. 4. 50 mm × 100 mm were prepared acCylindrical specimens with cording to the standard of International Society for Rock Mechanics (ISRM).44 The basic parameters including density, P-wave velocity, elastic modulus and UCS are listed in Table 1.
The tests were carried out by the INSTRON 1346 test system, which consist of a control computer, a 2.5 mm displacement extensometer, a loading system and a data acquisition system. The maximum loading range can reach 2000 kN in the quasi-static loading state. The stress is recorded by the data acquisition system and the strain is measured by the 2.5 mm displacement extensometer. In this study, the rock specimens including two types of sandstone, four types of granite, two types of marble and one type of limestone
3.3. Stress-strain curves characteristics Figs. 5 and 6 show the representative stress-strain curves of nine rocks in uniaxial compression tests and single cyclic loading-unloading
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Fig. 5. The representative stress-strain curves of nine rocks in uniaxial compression tests.
Fig. 6. The representative stress-strain curves of nine rocks in uniaxial single cyclic loading-unloading uniaxial compression tests at setting unloading stress level k = 0.5. 79
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Table 2 Strain energy density of nine rock materials under different stress levels. Rock type
Specimen No.
Red sandstone
A–4 A–5 A–6 A–7 A–8 B–2 B–3 B–4 B–5 B–6 C–2 C–3 C–4 C–5 C–6 D–2 D–3 D–4 D–5 D–6 E−4 E−5 E−6 E−7 E−8 E−9 E − 10 E − 11 F–2 F–3 F–4 F–5 F–6 G–2 G–3 G–4 G–5 G–6 H–2 H–3 H–4 H–5 H–6 I–2 I–3 I–4 I–5 I–6
Green sandstone
Yellow rust granite
Yueyang granite
Fine granite
Yellow granite
White marble
Leiyang marble
Limestone
p (MPa)
100.90 102.99 99.89 99.74 99.30 107.38 94.37 95.90 103.64 97.84 74.79 71.26 67.13 71.30 70.96 190.23 194.20 199.53 196.20 201.71 264.71 258.86 281.80 274.78 264.55 288.07 275.74 263.08 171.93 176.59 185.30 179.84 186.83 66.36 64.59 73.14 64.04 71.89 128.55 129.43 137.47 139.39 108.07 154.21 169.04 154.27 171.49 150.26
Unloading stress (MPa)
Actual unloading stress level i
ui (mJ/mm3)
uei (mJ/mm3)
udi (mJ/mm3)
10.18 30.26 49.79 69.02 92.03 10.70 31.32 50.59 77.85 88.51 7.92 22.79 37.61 50.47 64.61 21.20 60.51 99.15 139.12 178.15 55.65 82.85 110.59 131.94 165.23 191.98 220.26 247.25 20.16 55.15 91.52 127.76 164.86 6.90 20.35 33.31 47.31 60.57 11.84 35.81 63.56 84.91 96.27 17.04 48.48 82.70 113.83 147.47
0.10 0.29 0.50 0.69 0.93 0.10 0.33 0.53 0.75 0.90 0.11 0.32 0.56 0.71 0.91 0.11 0.31 0.50 0.71 0.88 0.21 0.32 0.39 0.48 0.62 0.67 0.80 0.94 0.12 0.31 0.49 0.71 0.88 0.10 0.32 0.46 0.74 0.84 0.09 0.28 0.46 0.61 0.89 0.11 0.29 0.54 0.66 0.98
0.0086 0.0336 0.0736 0.1306 0.2396 0.0085 0.0532 0.1055 0.2011 0.2589 0.0042 0.0249 0.0537 0.0880 0.1304 0.0143 0.0641 0.1416 0.2730 0.4071 0.0431 0.0704 0.1181 0.1668 0.2553 0.3228 0.4177 0.5272 0.0146 0.0640 0.1365 0.2567 0.3874 0.0037 0.0142 0.0262 0.0530 0.0824 0.0052 0.0248 0.0562 0.0911 0.1387 0.0080 0.0343 0.0838 0.1510 0.2492
0.0035 0.0253 0.0583 0.0942 0.1855 0.0057 0.0338 0.0728 0.1452 0.1857 0.0027 0.0133 0.0304 0.0517 0.0715 0.0099 0.0508 0.1232 0.2410 0.3508 0.0382 0.0625 0.1047 0.1488 0.2262 0.2865 0.3618 0.4462 0.0092 0.0498 0.1145 0.2131 0.3348 0.0020 0.0068 0.0150 0.0325 0.0501 0.0042 0.0195 0.0466 0.0760 0.1069 0.0067 0.0300 0.0776 0.1377 0.2150
0.0051 0.0083 0.0153 0.0364 0.0541 0.0028 0.0194 0.0327 0.0559 0.0732 0.0015 0.0116 0.0233 0.0363 0.0589 0.0044 0.0133 0.0184 0.0320 0.0563 0.0049 0.0079 0.0134 0.0180 0.0291 0.0363 0.0559 0.0810 0.0054 0.0142 0.0220 0.0436 0.0526 0.0017 0.0074 0.0112 0.0205 0.0323 0.0010 0.0053 0.0096 0.0151 0.0318 0.0013 0.0043 0.0062 0.0133 0.0342
stress greater than k c .42 By calculating the excess energy of each crack and invoking the ‘‘no-interaction’’ approximation,43,45 David et al. have presented a sliding crack model to explain the hysteresis loops, and the model were validated by the experimental data of actual sandstone and granite.45
uniaxial compression tests with unloading stress level k = 0.5. It is observed that the features of stress-strain curves of every rock are different. The plasticity of white marble and yellow rust granite is strong, and their complete stress-strain curves can be measured. Fine granite and limestone show obvious stress drop in the post-peak state, indicating that their brittleness is relatively strong. It also can be seen from Fig. 6 that the hysteresis loops of nine rocks are obvious and various. Just as pointed out in literature [42], irreversible changes will occur in the load-unloading process. The initial loading curve and unloading curve will not overlap, a permanent strain 0 will happen when the unloading stress reaches zero, and the secondary loading curve will lie below the initial loading curve and rejoin the loading curve at a
4. Test results and analysis 4.1. Relationships between three energy density parameters The total input strain energy density and elastic strain energy density of rock specimens at different unloading stress level i (i is the
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Fig. 7. Relationships between uei , udi and ui .
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Table 3 Fitting functions between ui and uei of nine rocks.
udi = (1
Rock type
Fitting function for ui and uei
Red sandstone
uei = 0.7720ui
0.0014(R2 = 0. 9982)
uei uei uei uei uei uei uei uei
0.0017(R2 = 0.9994)
Green sandstone Yellow rust granite Yueyang granite Fine granite Yellow granite White marble Leiyang marble Limestone
=
0.7234ui
0.0016(R2 = 0.9995)
0.8542ui
+ 0.0037(R2 = 0.9992)
= 0.8632ui
0.0034(R2 = 0.9994)
= 0.6167ui
0.0007(R2 = 0.9986)
0.7851ui
+ 0.0008(R2 = 0.9972)
=
=
(7)
B
where A and B is the fitting coefficients. Table 3 lists the fitting functions and the correlation coefficient (R2) between ui and uei for nine rocks. The fitting functions between ui and udi can also be derived with Eq. (7). In Table 3, R2 of The fitting functions ranges from 0.9972 to 0.9994, which shows there is a strong linear relationship between the elastic strain energy density and total input energy density, which can be considered to satisfy a linear energy storage law. In addition, the values of |B/A| for nine rock materials are listed in Table 4. From Table 4, it is clear that the ratio of B to A is very small with the maximum being 0.43%, which indicates that the effect of constant B on the linear function is minimal. The A in Eq. (6) can reflect the ratio of the elastic strain energy density to total input energy density. In order to characterize the energy storage performance of rock materials, the energy storage coefficient (ESC) is proposed based on the linear storage energy law, which is defined as A in Eq. (6). The greater the value of ESC is, the higher the capability of elastic strain energy storage is. Among the nine rock materials, the ESC of Yueyang granite (0.8726) is the largest and that of yellow rust granite (0.5580) is the smallest. Hence, Yueyang granite has the highest ability to store elastic strain energy, and the lowest for the yellow rust granite. Similarly, (1A) in Eq. (7) is defined as the energy dissipation coefficient (EDC), which reflects the proportion of dissipated strain energy in the total input energy during the loading process.
= 0.5580ui + 0.0003(R2 = 0.9979) = 0.8726ui
A) ui
= 0.8721ui + 0.0014(R2 = 0.9986)
ratio of preset unloading stress to the peak strength of each rock specimen, and equal to k c / ck .) can be calculated by integrating the stressstrain curves as mentioned in Section 2 and the results are shown in Table 2. Based on the test data listed in Table 2, the relationships between uei , i ud and ui was studied (Fig. 7). The results show that uei and udi increase with the increasing of ui linearly. The relationships between uei , udi and ui can be fit with the linear functions, and the fitting functions are uniformly expressed as follows. (6)
uei = Aui + B
Table 4 |B/A| of nine rock materials. Rock type
Red sandstone
Green sandstone
Yellow rust granite
Yueyang granite
Fine granite
Yellow granite
White marble
Leiyang marble
Limestone
|B1/A1|(%)
0.18
0.24
0.05
0.18
0.43
0.39
0.11
0.10
0.16
Fig. 8. Calculation of the total input energy density at the peak strength: (a) uniaxial compression test; (b) single cyclic loading-unloading uniaxial compression test.
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Table 5 Three energy density parameters at the peak strength and the peak-strength energy storage index of nine rock materials. Rock type
Specimen No.
Test type
U (mJ/mm3)
Ue (mJ/mm3)
Ud (mJ/mm3)
Wetp
Average value of Wetp
Red sandstone
A–1 A–2 A–3 A–4 A–5 A–6 A–7 A–8 B–1 B–2 B–3 B–4 B–5 B–6 C–1 C–2 C–3 C–4 C–5 C–6 D–1 D–2 D–3 D–4 D–5 D–6 E−1 E−2 E−3 E−4 E−5 E−6 E−7 E−8 E−9 E − 10 E − 11
Conventional compression
0.3157 0.3454 0.2601 0.2870 0.3245 0.3019 0.3079 0.3068 0.3657 0.3605 0.3269 0.3215 0.3644 0.3474 0.1813 0.2011 0.1866 0.1659 0.1843 0.1739 0.5508 0.4822 0.5231 0.5576 0.5660 0.5598 0.6584 0.6768 0.6625 0.6811 0.6388 0.7306 0.7739 0.6778 0.7849 0.7271 0.6677
0.2423 0.2652 0.1994 0.2202 0.2491 0.2316 0.2363 0.2354 0.2628 0.2591 0.2348 0.2309 0.2619 0.2496 0.1014 0.1125 0.1044 0.0929 0.1031 0.0973 0.4791 0.4192 0.4549 0.4850 0.4923 0.4869 0.5661 0.5818 0.5696 0.5855 0.5494 0.6278 0.6648 0.5827 0.6742 0.6248 0.5740
0.0734 0.0802 0.0607 0.0668 0.0754 0.0703 0.0716 0.0714 0.1029 0.1014 0.0921 0.0906 0.1025 0.0978 0.0799 0.0886 0.0822 0.0730 0.0812 0.0766 0.0719 0.0630 0.0682 0.0726 0.0737 0.0729 0.0923 0.0950 0.0929 0.0956 0.0894 0.1028 0.1091 0.0951 0.1107 0.1023 0.0937
3.30 3.31 3.28 3.29 3.30 3.30 3.30 3.30 2.56 2.56 2.55 2.55 2.56 2.55 1.27 1.27 1.27 1.27 1.27 1.27 6.67 6.65 6.67 6.68 6.68 6.68 6.13 6.13 6.13 6.12 6.14 6.11 6.09 6.13 6.09 6.11 6.13
3.30
Green sandstone
Yellow rust granite
Yueyang granite
Fine granite
Cyclic loading-unloading compression
Conventional compression Cyclic loading-unloading compression
Conventional compression Cyclic loading-unloading compression
Conventional compression Cyclic loading-unloading compression
Conventional compression Cyclic loading-unloading compression
2.55
1.27
6.67
6.12
Rock type
Specimen No.
Test type
U (mJ/mm3)
Ue (mJ/mm3)
Ud (mJ/mm3)
Wetp
Average value of Wetp
Yellow granite
F–1 F–2 F–3 F–4 F–5 F–6 G–1 G–2 G–3 G–4 G–5 G–6 H–1 H–2 H–3 H–4 H–5 H–6 I–1 I–2 I–3 I–4 I–5 I–6
Conventional compression Cyclic loading-unloading compression
0.5544 0.4447 0.4842 0.4996 0.5065 0.5243 0.1269 0.1365 0.1366 0.1543 0.1441 0.1598 0.2949 0.2544 0.2493 0.2756 0.3053 0.1995 0.3422 0.3440 0.3333 0.2938 0.3415 0.2754
0.4752 0.3805 0.4146 0.4279 0.4338 0.4492 0.0776 0.0835 0.0836 0.0944 0.0882 0.0978 0.2341 0.2021 0.1980 0.2189 0.2424 0.1586 0.2999 0.3014 0.2921 0.2576 0.2993 0.2416
0.0792 0.0642 0.0696 0.0717 0.0727 0.0751 0.0493 0.0530 0.0530 0.0599 0.0559 0.0620 0.0608 0.0523 0.0513 0.0567 0.0629 0.0409 0.0423 0.0426 0.0412 0.0362 0.0422 0.0338
6.00 5.92 5.95 5.96 5.97 5.98 1.57 1.57 1.57 1.58 1.58 1.58 3.85 3.86 3.86 3.86 3.85 3.88 7.08 7.08 7.08 7.12 7.08 7.14
5.96
White marble
Leiyang marble
Limestone
Conventional compression Cyclic loading-unloading compression
Conventional compression Cyclic loading-unloading compression
Conventional compression Cyclic loading-unloading compression
83
1.58
3.86
7.10
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Table 6 Variation coefficients of Rock type
p,
4.3. Calculation of Wetp
Ue , Ud and Wetp for nine rock materials.
Based on Ue and Ud , the index Wetp of each rock specimen can be calculated by using Eq. (5), as summarized in Table 5. It can be found from Tables 2 and 5 that, for the same rock material, the index Wetp is approximately constant although p (peak strength of rock specimen), Ue and Ud of each specimen are significantly different. For example, p , Ue and Ud of red sandstone are in the range of 99.30–102.99 MPa, 0.1994–0.2652 mJ/mm3, 0.0607–0.0802 mJ/mm3, respectively. However, the index Wetp is in the range of 3.28–3.31 with an average value of 3.30. Additionally, this can also be supported by the variation coefficients of p , Ue , Ud and Wetp for nine rock materials listed in Table 6. It can be seen from Table 6 that the maximum variation coefficient of Wetp is 0.46% and the minimum value is zero. Nevertheless, the maximum variation coefficients for p , Ue , Ud are 9.29%, 14.44% and 14.66%, respectively. The variation coefficients of p , Ue , Ud and Wetp for nine rocks are also compared as shown in Fig. 9.
Variation coefficient (%) p
Red sandstone Green sandstone Yellow rust granite Yueyang granite Fine granite Yellow granite White marble Leiyang marble Limestone
2.46 5.20 4.06 2.98 4.44 5.73 9.29 5.93 3.62
Ue
Ud
Wetp
8.31 5.62 6.54 5.95 7.44 8.60 14.44 9.15 6.92
8.10 5.49 6.60 5.81 7.04 8.45 14.66 9.53 7.22
0.22 0.21 0.00 0.18 0.46 0.35 0.28 0.37 0.28
4.2. Calculation of U, Ue and Ud According to the linear relationships between uei and ui presented in Section 4.1, Ue and Ud can be calculated for given U using the following equations:
A) U
Rock bursts are violent disasters that are accompanied by ejection of rock fragments. To evaluate the rock burst proneness of rock materials, it is necessary to analyze the failure and ejection characteristics of rock specimens by means of laboratory tests. On one hand, the ejection processes of rock fragments were captured with high-speed camera during the tests. High-speed camera images of some representative rock specimens in a certain period time during the loading process are shown in Fig. 10. The time interval between adjacent images is 8 ms. It should be noted that small amount of local spalling or several slow ejections may occurred on the specimen surface before the final integrity ejection process. Compared with the red sandstone and green sandstone, the fragment ejection speed of Yueyang granite and yellow granite is much higher and a large amount of rock powder and debris can be observed. It illustrates that the failure of Yueyang granite and yellow granite is more violent.
(8)
Ue = AU + B Ud = (1
5. New criterion for rock burst proneness with index Wetp
B
(9)
In this study, for the two types of tests, U can be obtained by different integration methods (Fig. 8). For the uniaxial compression tests, U is calculated by integrating the initial loading curve before the peak strength. For the single cyclic loading-unloading uniaxial compression tests, the integral process is divided into two steps, i.e., 1) integration of the initial loading curve and 2) integration from the unloading point to the peak strength point along the second loading curve. The three energy density parameters at the peak strength of each rock specimen are listed in Table 5.
Fig. 9. The variation coefficients of
84
p,
Ue , Ud and Wetp for nine rocks.
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Fig. 10. High-speed camera images of typical rock specimens during ejection process.
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Table 7 Ejection characteristics of nine rock materials. Rock type
Failure mode
Ejection speed
Ejection sound
Amount of ejected fragments
Wetp
Rock burst proneness Wetp
Red sandstone Green sandstone Yellow rust granite Yueyang granite Fine granite Yellow granite White marble Leiyang marble Limestone
Shear - Splitting Shear - Splitting Shear Shear - Splitting Shear - Splitting Shear - Splitting Shear - Splitting Shear - Splitting Splitting
Slow Slow – Fast Fast Fast – Slow Fast
Slight Slight – Loud Loud Loud – Slight Loud
Minor Minor – Great Great Great – Minor Great
3.30 2.55 1.27 6.67 6.12 5.96 1.58 3.86 7.10
Low Low No High High High No Low High
On the other hand, some information involving the failure mode, the ejection sound and the number of ejected rock fragments was collected and listed in Table 7. Fig. 11 presents the failure modes and fragments characteristics of nine typical rock specimens. Seven rock materials exhibited ejection features except yellow rust granite and white marble. The yellow rust granite and white marble specimens remained almost intact with a macroscopic crack throughout the specimen after the tests (Fig. 11h and i), which demonstrates that the two rock materials have no rock burst proneness. For the Yueyang granite, fine granite, yellow granite and limestone specimens, a great number of rock fragments, debris, and powder were ejected out of the loading platen with loud blast-like sound during the tests (Fig. 11d and e). These phenomena indicated high rock burst proneness. As for the red sandstone, green sandstone and Leiyang marble specimens, some fragments and debris were ejected with relatively slight sound. Most of the fragments fell over the loading platen (Fig. 11a–c). Therefore, the rock burst proneness of the three rock materials was low. From the above analyses, the rock burst proneness of the nine rock materials can be classified into three grades: no, low and high. The detailed results are given in Table 7. According to the calculated Wetp and the failure pattern of the nine rock materials, a new criterion for rock burst proneness with index Wetp is proposed as follows:
constant and uninfluenced by p , Ue and Ud , which can be regarded as a basic parameter of rock materials. It is feasible and reliable to use Wetp to study the rock burst proneness of rock materials. 7. Conclusions To obtain the strain energy storage index Wetp of rock materials at peak strength, a series of uniaxial compression and single cyclic loading-unloading uniaxial compression tests were designed and conducted on nine rock materials. Based on the experimental results, the following conclusions can be drawn: (1) A strong linear relationship was observed between the elastic strain energy density and the total input energy density for every rock material, which implies that rock materials obey a linear storage energy law during the loading process. Based on this linear storage energy law, the elastic & dissipated strain energy density at the peak strength of rock specimen can be conveniently calculated. This method solves the problem that the elastic & dissipated strain energy density at the peak strength of rock specimen can't be separated accurately. (2) To estimate and classify the rock burst proneness of rock materials, the ejection speed, ejection sound, amount of ejected fragments of rock specimens during failure process were used to evaluate the rock burst proneness of rock materials. Identification results of rock burst proneness of nine rock materials are: The Yueyang granite, fine granite, yellow granite and limestone specimens are high rock burst proneness, the red sandstone, green sandstone and Leiyang marble specimens are low rock burst proneness. Yellow rust granite and white marble specimens have no rock burst proneness. (3) A peak-strength strain energy storage index Wetp was proposed on the basis of the linear storage energy law for rock materials. This index revealed the proportional relation between the elastic strain energy and the dissipated strain energy at the peak strength is almost constant and can well estimate the rock burst proneness of rock materials. Furthermore, based on the actual rock burst proneness of nine rock materials in laboratory and the peak-strength strain energy storage index, a new criterion for rock burst proneness was put forward.
Wetp <2 No rock burst proneness Wetp = 2 5 Low rock burst proneness Wetp >5 High rock burst proneness (10) In Table 7, the results obtained by using the new rock burst criterion are completely consistent with the actual rock burst proneness. 6. Discussion To compare the two indexes, Wet and Wetp , the variation tendency of with i are shown in Fig. 12. It can be observed that, for every rock material, the values of uei / udi are fluctuant and vary irregularly even when i is in the range of 0.8–0.9. It means that Wet obtained in the tests are not unique. As described in Section 4.3, Wetp of the same rock type is nearly
uei /udi
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Fig. 11. Failure mode and pattern of nine typical rock specimens.
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Fig. 12. Variation tendency of uei /udi with i.
Acknowledgements
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