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Dynamic Characterization of Himalayan Quartzite using SHPB
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 191 (2017) 2 – 9
Symposium of the International Society for Rock Mechanics
Dynamic Characterization of Himalayan Quartzite Using SHPB Sunita Mishra*, Tanusree Chakraborty, Vasant Matsagar Department of Civil Engineering, Indian Institute of Technology (IIT) Delhi, Hauz Khas, New Delhi – 110 016, India
Abstract In the present work, dynamic stress-strain response of Himalayan quartzite is tested under high loading rates using split Hopkinson pressure bar (SHPB) device for the first time in the literature. The physical and static mechanical properties of quartzite e.g. dry and saturated density, specific gravity, static compressive strength and elastic modulus values are also determined. Petrological studies of quartzite are carried out through X-ray diffraction (XRD) test and scanning electron microscope (SEM) test. In the SHPB tests, it is observed from the stress-strain response that the dynamic peak stress increases with increasing strain rate whereas the elastic modulus does not show any clear trend with increase in strain rate. Dynamic force equilibrium at the incident and transmission bar ends of the rock samples is attained in all tests till the failure of the rock samples. Dynamic increase factor (DIF) for the rock is determined at a particular strain rate by comparing the dynamic to static peak compressive stress. Correlation equation for dynamic strength increase factor with respect to strain rate has been proposed herein. © Published by Elsevier Ltd. This © 2017 2017The TheAuthors. Authors. Published by Elsevier Ltd. is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of EUROCK 2017. Peer-review under responsibility of the organizing committee of EUROCK 2017 Keywords: Rock dynamic characterization; split Hopkinson pressure bar; quartzite
1. Introduction Design, development and building of civil infrastructure in the mountainous regions involve many complexities in terms of diverse geological and geomorphological features of the region - the Chenab river bridge in the Himalayas, the Gotthard Base tunnel in the Alps are to name a few. The young mountain ranges of the Himalayas and the Alps contain joint planes, shear seams, active fold, and fault zones. Moreover high in-situ stresses and high level of seismicity in these regions pose severe challenges to the construction of infrastructure. In addition to this, unanticipated loads caused by natural hazards, e.g. landslide, earthquake and manmade hazards, e.g. blast and projectile penetration add to the difficulties already existing therein. It may be noted that the loads caused
* Corresponding author. Tel.: +91-9717253877; fax: +91-11-2659-1117. E-mail address: [email protected], [email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of EUROCK 2017
doi:10.1016/j.proeng.2017.05.147
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by hazardous events like earthquake and blast are highly transient in nature generating high strain rates in rock and strain rate caused by blast may reach up to 104ͼs-1 [1, 2] which in turn affects both the stiffness and the strength properties of the rocks. Thus, in order to ensure sustainable design of civil infrastructure in the mountains, it becomes necessary to characterize the rocks under static and dynamic loading conditions. 1.1. Literature review In the present work, only the dynamic compression response of the rock has been discussed and reported. Dynamic compression tests of rocks at low to high strain rates have been performed by several researchers through split Hopkinson pressure bar (SHPB) and dynamic triaxial tests tests [3–18]. Dynamic uniaxial compression tests were performed on three rocks by [3] using SHPB at strain rates from 10-4s-1 to 104s-1 at varying temperatures and the SHPB test data for rocks were reported for the first time in the literature. They observed that the rocks exhibited increased stiffness and higher stress with increasing strain rate and decreasing temperature. Energy absorption in SHPB test in two different rocks, Bohus granite and Solenhofen limestone was reported in [6]. It was observed that the energy absorbed by the rocks increased markedly when the applied load reached the critical value of 1.8 and 1.3 times the static compressive strength for Bohus granite and Solenhofen limestone, respectively. SHPB tests on tuff, which is a hard igneous rock of volcanic origin, was performed in [9] for strain rates varying from 10-6s-1 to 103s-1. It was observed from the results that the strength of the rock was a weak function of the strain rate for strain rates varying from 10-6s-1 to 76s-1; however, for the strain rate above 76s-1, the rate of increase in strength was proportional to the cube root of the strain rate. Dynamic uniaxial compression tests were conducted in [11] on Bukit Timah granite in Singapore at four different loading rates (100 MPaͼs-1, 101 MPaͼs-1, 103 MPaͼs-1 and 105 MPaͼs-1). It was concluded from the tests that, for each log scale increase in loading rate, the compressive strength of the rock increased by 15%. They also observed that there were small changes in the elastic modulus and Poisson’s ratio values with an increase in loading rate. Uniaxial compressive SHPB tests on limestone was conducted in [12] by using a copper disk at the impact end of the incident bar as a pulse shaper, which resulted in dynamic stress equilibrium of the samples and maintained constant strain rates over the test duration. An improved experimental approach for eliminating oscillation that exists in the dynamic stress–strain response of rocks and other brittle materials obtained from SHPB tests was reported in [13]. The tests were conducted on granite, sandstone and limestone and was concluded that the improved method eliminates oscillation in the tests, provides better stability of strain rate and more representative results than those obtained from the conventional rectangular loading waveform shape. The dynamic stress-strain response of Bukit Timah granite loaded at a medium strain rate of 20s-1 – 60s-1 using SHPB testing was reported by [15]. It was observed from the results that the dynamic fracture strength of the granite was directly proportional to the cube root of the strain rate, whereas the elastic modulus remained unchanged with increasing strain rate. At higher strain rates, the rocks showed a higher amount of energy absorption and the particle size of the fragments at the end of the test became smaller. Uniaxial compression tests on Thai sandstones were reported in [17] and they reported an increase in strength and elastic modulus with an increase in strain rate. It was observed that both the strength and the elastic modulus tended to increase exponentially, with the loading rates ranging from 0.001 MPaͼs-1 to 10 MPaͼs-1. A maximum of 71% increase in the modulus of elasticity was observed for sandstone for the increase in loading rate from 0.001 MPaͼs-1 to 10 MPaͼs-1. It may be summarized from the literature review that dynamic compressive strength testing on rocks using SHPB has been carried out on different rock types, e.g., granite, Barre granite, basalt, volcanic tuff, Kawazu tuff, red sandstone, Indiana limestone, porphyritic tonalite, oil shale, granodiorite, coal, kidney stone, Tennessee marble and Akyoshi marble at up to a 2000s-1 strain rate [19] and that strain rate had a significant effect on the mechanical behavior of the rocks. 1.2. Test plan The objective of the present work is to characterize Himalayan quartzite under strain rate dependent loading at different levels of strain rates. The quartzite rock blocks collected herein are of unweathered nature. The rock blocks are collected from a hydropower project site in Vilaspur, India under National Thermal Power Corporation (NTPC) Ltd., India. The rock samples have been tested for both physical and mechanical properties. The physical properties,
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e.g. dry density, saturated density and specific gravity, the petrological studies, e.g. X-ray diffraction test (XRD) and scanning electron microscope test (SEM) and the static mechanical properties, e.g. uniaxial compressive strength, static elastic modulus, and static tensile strength of the rocks are determined in the Indian Institute of Technology (IIT) Delhi laboratories. Static uniaxial compressive strength tests on dry and saturated rock samples have been carried out using the automatic uniaxial compression and splitting test device for rock samples with aspect ratio (L/D) = 2:1. Brazilian and point load tests on dry and saturated rock samples have been carried out to determine the tensile strength values of the rocks. For the Brazilian test, L/D = 0.5:1 and for the point load test, L/D = 1:1 have been used. All static tests have been carried out following the specifications given in [20] and [21]. The strain rate dependent tests are carried out using SHPB in the Rel Inc. laboratory, Michigan, USA. The stress-strain response of the rocks under dynamic loading, force equilibrium at incident and transmission bar ends of rock sample, peak stress and dynamic elastic modulus are studied. The dynamic tests are performed using a 38 mm diameter SHPB for all three rocks at different strain rate levels. In the present study, the low strain rate range is defined for 10 to 100s-1, medium strain rate is defined for 100 to 250s-1, and high strain rate is defined for 250 to 500s-1. For this test, the rock samples are prepared with a diameter of 38 mm and aspect ratio of 0.5:1. The results obtained from the static tests are presented in Table 1 and those from the dynamic tests are presented in Tables 2 for quartzite. The striker bar is propelled using a compressed air gas gun. The strain rate in the dynamic tests is controlled by varying the striker bar length and the striking velocity. The sample number, sample length, length of the striker bar used and the striking velocity applied for a particular strain rate are also reported in the above-mentioned tables. The dynamic increase factor (DIF), i.e. the ratio of dynamic to static peak stress is calculated for each test at different strain rates. Further, a suitable correlation equation is proposed herein for change in DIF with strain rate for quartzite tested for the strain rate range in the present work. 1.3. Test setup for SHPB device The SHPB in the Rel Inc. laboratory in Calumet, Michigan is designed and manufactured by Rel Inc. group. The setup comprises of an incident bar, a transmission bar and striker bars of different sizes. The bars are made up of C300 maraging steel. The incident bar length is 2.59 m and diameter is 38.1 mm. The transmission bar length is 2.43 m and diameter is 38.1 mm. The dimensions of the incident and transmission bars allow one dimensional loading of the sample. In the present work, two different lengths of striker bars are used, e.g. 152.4 mm and 304.8 mm; the diameter of striker bar is 38.1 mm. Compressed air is used to launch the striker bar on the incident bar. The striker bar is propelled by a compressed air gas gun at varying pressure magnitudes which generate stress waves inside the striker bar. The striker bar hits the impact end of the incident bar and remains in contact till the stress wave travels from one end of the striker bar to the other end. The stress wave upon reaching the other end of the striker bar gets reflected back. As a result the contact between the striker bar and the incident bar is lost. The time duration taken by the stress wave to travel from one end of the striker bar to the other is the time duration for loading of the sample. The time duration is given by 't
2 Ls cbar
(1)
where, Ls is the length of the striker bar and cbar is one dimensional longitudinal stress wave velocity in the bar. Thus, using longer striker bar increases the loading time and the rock sample gets time to respond. As a result lower strain rate develops when longer striker bar is used. It may be observed from the Tables 2 that for low strain rate level, the length of the striker bar used is 304.8 mm whereas a striker bar of length 139.7 mm is used for medium and higher strain rates. For smaller striker bars, the time taken by the stress wave to propagate from one end to the other end of the bar is less and hence the loading time duration is also less for the same amount of the load intensity which results in higher strain rate. The strains in the incident and transmission bars are measured using two strain gauges, one mounted on the incident bar and the other mounted on the transmission bar. The strain gauges are 6.35 mm in length, OMEGA 120: in resistance and dynamic gauge with tolerance of ±0.35%. Strain gauges are attached to the incident and the transmission bars with M-bond adhesive. In order to read the strain signal, Vishay
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2310B signal conditioner and amplifier with ¼ wheatstone bridge has been used with a Picoscope5242 having sampling rate of 1 in 8 nanosecond. Momentum trap, though present in the setup, has not been used in the present experiments. The impact of striker bar on the incident bar causes a longitudinal elastic compressive stress wave which propagates through the incident bar. The strain pulse generated within the incident bar is designated as incident strain pulse Hi(t). The strain pulse generated in the incident bar is recorded by the strain gauge mounted on the incident bar. Upon reaching the bar-specimen interface, a part of the pulse, designated as reflected strain pulse Hr(t) is reflected back in the incident bar and the remaining part of the compressive pulse passes through the specimen. Upon reaching the transmission bar end of the specimen, the pulse propagates through the transmission bar and it is then termed as transmitted strain pulse Ht(t). The histories of strain İ(t ) , strain rate İ (t ) and stress ı(t ) within the sample in dynamic compression test are given by İ(t )
C L
İ (t )
C İ i İ r İ t L
(3)
ı(t )
A E İ i İ r İ t 2 A0
(4)
t
³ İ 0
i
İ r İ t dt
(2)
where L is the length of the sample, C is the one dimensional longitudinal stress wave velocity in the bar, E is the elastic modulus of the bar material, A is the cross-sectional area of the bar and A0 is the initial area of the sample. Assuming that stress equilibrium and uniform deformation of the sample prevails during dynamic loading, i.e. İ i İ r İ t , the strain, strain rate and stress are given by İ(t )
t
³ İ dt 0
r
(6)
C İr L
(7)
A Eİ t A0
(8)
İ (t )
ı(t )
C L
The data processing method for the SHPB test is based on two assumptions - one is force equilibrium on both sides of the specimen and the other is the one dimensional uniform deformation of the sample. The force equilibrium is achieved herein by preparing samples with small slenderness ratio, preferably below or equal to 0.5 [22, 23]. The experimental requirement for conducting SHPB test is maintaining good contact between the bars and the sample, reducing the friction between the bars and the supports, keeping the bars coaxial and using pulse shapers [24, 25]. However, for rock samples, maintaining force equilibrium becomes challenging due to the anisotropic nature of the rock and the propagation of the crack inside the rock. Moreover, due to the brittle behavior of the rocks, it becomes important that the equilibrium is achieved before brittle failure of the sample [12]. Researchers have discussed the technique of pulse shaping to achieve force equilibrium in the SHPB tests for brittle materials such as rock [26]. It was shown in [27] that a wide variety of incident pulses can be produced by varying the geometry of the pulse shaper, which can be used for different materials under investigation. To ascertain the force equilibrium in the tests reported herein, mild steel and copper pulse shapers have been used in all tests. The pulse shapers used herein are circular disks of 12.7 mm diameter and different thicknesses. Moreover, the incident and transmission bar ends of the specimen are prepared through rigorous grinding to a smoothness of
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approximately 4.1 μm. Further, a 38 mm diameter SHPB setup has been used to capture the effect of anisotropy on the stress-strain response of the rock samples. The sample thickness has been maintained at half of its diameter to achieve stress uniformity throughout the length of the sample. The velocity of the striker is also recorded automatically using fiber optic speed sensors with a response time of 1 μs. A program has been used for data processing, e.g., obtaining stress-strain plots, strain and strain rate time histories, force equilibrium at the interfaces of the incident bar and sample and the transmission bar and sample. From these plots, peak stress, average strain rate and strain at peak stress are studied. 2. Results and discussion 2.1. Physical properties The dry density value of quartzite is determined for five samples and average values are determined to be 2585.84 kgͼm-3. The saturated density value of quartzite is also measured for five samples and average values are determined to be 2605.57 kgͼm-3. The saturated density value is observed to be smaller than the dry density value. The specific gravity value of quartzite is estimated to be 2.80. The density and specific gravity values are compared with the available data from the literature and observed to be well in agreement [28]. 2.2. Results of petrological study Petrological studies are performed by using XRD and SEM techniques to identify the mineralogical contents of the rocks. The SEM and XRD tests are done in the central research lab facility of Indian Institute of Technology (IIT) Delhi. The SEM Central Facility at IIT Delhi has a ZEISS EVO Series Scanning Electron Microscope EVO 50 which has magnification capacity of 5x to 1,000,000x and field of view of 6 mm at the Analytical Working Distance.The SEM image of quartzite is shown in Fig. 1 (a). The SEM images shows the grain texture of quartzite. Since quartzite is a metamorphic rock, the SEM images confirms the amorphous grain structure of the rock sample collected. The mineral content of quartzite is determined through X-ray diffraction tests. Fig. 1 (b) shows the results of X-ray diffraction graph of quartzite. The Himalayan quartzite is identified by its lustrous red color. The mineralogy of quartzite is ascertained to be quartz (90%) with feldspar and mica. The XRD graphs of quartzite is compared with the available data from the literature and observed to be well in agreement [28].
Fig. 1. (a) Scanning electron microscope images; (b) X-ray diffraction technique graph of quartzite.
2.3. Static uniaxial compressive strength The static uniaxial compressive strength and elastic modulus of the rock is determined under dry conditions following ASTM standards [29, 30]. and presented in Table 1. The tests for each rock type are repeated three times and the average strength values are noted. The static uniaxial compressive strength of quartzite is found to be 108.18 MPa. The elastic modulus from the stress-strain graph at 50% of peak stress value is calculated to be 11.65 GPa [31].
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Table 1. Physical and static properties. Rocks
Dry Density, Ud (kgͼm-3)
Saturated Density, Usat (kgͼm-3)
Specific Gravity, G
Uniaxial Compressive Strength, ıc (MPa)
Modulus of Elasticity, Et (GPa)
Quartzite
2585.841
2605.577
2.80
108.18
11.65
2.4. Dynamic stress-strain and force equilibrium for quartzite Figs. 2(a), 2(b) and 2(c) show the stress-strain plots for quartzite at three different strain rate ranges as obtained from SHPB tests e.g. high, medium and low, respectively. The high level strain rate varies from 260 s-1 to 316 s-1, the medium level strain rate varies from 134 s-1 to 215 s-1 and the low level of strain rate varies from 77 s-1 to 94 s-1. For high strain rate range as seen in Figure 2(a), the peak stress values obtained are 688.511 MPa at 260 s-1, 676.592 MPa at 280 s-1, 617.537 MPa at 295 s-1 and 669.297 MPa at 316 s-1. Fig. 2(b) shows the stress-strain plots for quartzite at different medium strain rate values. For medium strain rate range, the peak stress values obtained are 379 MPa at 134 s-1, 513.166 MPa at 173 s-1, 547.102 MPa at 213 s-1 and 553.981 MPa at 215 s-1. Fig. 2(c) shows the stress-strain plots for quartzite at different low strain rate values. For low strain rate range, the peak stress values obtained are 409.533 MPa at 77 s-1, 390.894 MPa at 86 s-1 and 379.302 MPa at 94 s-1. It may be noted that for all the strain rate values, force equilibrium is achieved nicely up to the peak of the force equilibrium curves and equilibrium is lost after the peak. Thus, the rock sample is under stress equilibrium till the peak stress is reached, however, stress equilibrium is lost once breakage of the sample starts taking place. When there is force equilibrium between the specimen’s incident and transmitted sides, the loading on specimen is one dimensional and the specimen deforms, uniformly. The strain rate in each test is obtained when the sample is under force equilibrium and the strain rate reaches a stable value, i.e., the strain rate-time plot becomes parallel to time axis. Figs. 3(d) and 3(e) show the peak stress and elastic modulus values, respectively obtained from all the stress-strain curves for quartzite at different strain rates. The peak stress clearly increases with strain rate whereas the elastic modulus does not exhibit strain rate dependency. This can be explained by the fact that rock is heterogeneous in nature and till date no standards have been defined for high loading rate dynamic tests of rocks. Hence, the dynamic elastic modulus of the rock is difficult to be ascertained at present. The peak stress, elastic modulus and the strain at peak stress values are also reported in Table 2 for different strain rates. It can be seen that the peak stress increases by almost 22% from low strain rate (77 s-1 to 94 s-1) to medium strain rate (134 s-1 to 215 s-1) and by 37.7% from medium strain rate (134 s-1 to 215 s-1) to high strain rate (260 s-1 to 316 s-1). 2.5. Dynamic increase factor and proposed correlation equation Dynamic increase factor (DIF) for quartzite has been determined by comparing the dynamic peak stress with the static peak stress. The DIF values of the rock with respect to varying strain rate are reported in Table 2. The DIF values are also plotted in Fig. 3(f) for quartzite. From Fig. 3(f), it may be seen that the dynamic strength of quartzite are 3.5 to 6.36 times that of the static strength for strain rates varying from 77 s-1 to 316 s-1. A correlation equation has been developed for the calculated DIF with strain rate by setting a best fitted curve through the obtained DIF values which satisfies the 95% confidence interval. The correlation equation for quartzite with coefficient of determination (R2) = 0.89 is presented in Figure 3(f) and given by
DIF
0.012 İ 2.48 for 77 s -1 d İ d 316 s -1
(8)
It may be noted that the DIF equations proposed herein will be applicable for the strain rate ranges considered in the current work.
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Strain rate and sample number
Peak Stress V MPa)
Stress V (MPa)
600 400 200
0 0.000 0.005 0.010 0.015 0.020 0.025 Strain H
700 600 500 400 100
215/sec, Q2 213/sec, Q14 173/sec, Q12 134/sec, Q9
600
200
200 300 Strain rate İ (/sec)
0.005
0.010 Strain H
0.015
200 0 0.000
150 100 50
400
0.020
200
0 50
94/sec, Q8 86/sec, Q6 77/sec, Q11
600
(f)
250
Strain rate and sample number
800
400
0 0.000
(e)
(c) 1000
Strain rate and sample number
800
Elastic Modulus Et (GPa)
Stress V (MPa)
800
(d)
(b) 1000
316/sec, Q5 295/sec, Q4 280/sec, Q13 260/sec, Q1
Stress V (MPa)
(a) 1000
Dynamic Increase Factor
8
100 150 200 250 300 350 Strain rate İ (/sec)
0.005 0.010 Strain H
0.015
7
DIF for Quartzite Experimental Data Equation Trendline 6
5 4
DIF = 0.012(İ)2.48 R2 = 0.89
3
70
140 210 280 Strain rate İ (/sec)
350
Fig. 2. Quarzite stress-strain curves for (a) high strain rate range; (b) medium strain rate range; (c) low strain rate range; data points for; (d) peak stress-strain rate; (e) elastic modulus-strain rate; (f) dynamic increase factor-strain rate. Table 2. Dynamic properties: quartzite. Strain Rate Range
Rock Type
Low
Quartzite
Medium
High
Sample Number
Sample Length (mm)
Q11 Q6 Q8 Q9 Q12 Q14 Q2 Q1 Q13 Q4 Q5
18.54 18.87 18.34 17.25 18.39 19.14 18.59 18.62 18.01 18.11 18.85
Striker Bar Length (mm)
304.8
139.7
139.7
Striker Bar Velocity (mͼs-1)
21.27 19.46 19.67 19.49 32.69 32.69 32.64 40.60 40.60 39.76 41.60
Strain Rate, İ (s-1)
77 86 94 134 173 213 215 260 280 295 316
Peak Stress, ıdc (MPa) 409.533 390.894 379.302 379.000 513.166 547.102 553.981 688.511 676.592 617.537 669.297
Strain at Peak Stress, İ
Modulus of Elasticity, Et (GPa)
Dynamic Increase Factor, DIF
0.006 0.009 0.01 0.014 0.008 0.012 0.009 0.013 0.013 0.015 0.012
10.44 38.09 35.31 25.03 217.44 35.43 63.52 57.09 50.41 39.63 57.79
3.78 3.61 3.50 3.50 4.74 5.05 5.12 6.36 6.25 5.70 6.18
3. Conclusions
Characterization of Himalayan quartzite has been performed in the present work for a strain rate range varying from 49 s-1 to 316 s-1 through uniaxial compressive 38 mm diameter SHPB test. The dynamic stress-strain response, peak stress, elastic modulus and force equilibrium at the incident and transmission bar ends of the rock specimen are studied. The physical properties and static stress-strain behaviour of the rock are also investigated. The peak stress increases at a rate of 22% from low (77 s-1 to 94 s-1) to medium (134 s-1 to 215 s-1) strain rate and by 37.7% from medium (134 s-1 to 215 s-1) to high (260 s-1 to 316 s-1) strain rate. The elastic modulus of quartzite does not show any particular trend with increasing strain rate. The dynamic compressive stress 669.297 MPa at 316 s-1 is nearly six times of static compressive stress. The dynamic increase factor varies from 3.5 to 6.36.
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Acknowledgement
This work is a part of an ongoing research project funded by Terminal Ballistics Research Laboratory (TBRL), Chandigarh under Defence Research and Development Organisation (DRDO), India. The authors acknowledge the funding provided by TBRL in this work.
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Chona, Dynamic constitutive behavior and fracture initiation toughness of hastelloy x under thermo-mechanical loads. Proceedings of the society of experimental mechanics, implast conference, Wright-Patterson Air Force Base RBSM., 2010. [26] D.J. Frew, M.J. Forrestal, W. Chen, Pulse shaping techniques for testing brittle materials with a split Hopkinson pressure bar, Exp. Mech. 42(1) (2002) 93–106. [27] K. Xia, W. Yao, Dynamic rock tests using split Hopkinson (Kolsky) bar system - A review, J. Rock Mech. Geot. Eng. 7 (2014) 27–59. [28] R. Kumar, Testing and constitutive modelling of the strain-softening behaviour of some rocks. Ph.D. Thesis, Indian Institute of Technology (IIT) Delhi, 2007. [29] ASTM D3148-02, Standard test method for elastic moduli of intact rock core specimens in uniaxial compression, Amer. Soc. Test. Mat., 1998. [30] ASTM D7012-14, Standard Test Methods for Compressive Strength and Elastic Moduli of Intact Rock Core Specimens under Varying States of Stress and Temperatures, Amer. Soc. Test. Mat., 2014. [31] ASTM D4543-08, Standard Practices for Preparing Rock Core as Cylindrical Test Specimens and Verifying Conformance to Dimensional and Shape Tolerances, Amer. Soc. Test. Mat., 2008.
9
ScienceDirect Procedia Engineering 191 (2017) 2 – 9
Symposium of the International Society for Rock Mechanics
Dynamic Characterization of Himalayan Quartzite Using SHPB Sunita Mishra*, Tanusree Chakraborty, Vasant Matsagar Department of Civil Engineering, Indian Institute of Technology (IIT) Delhi, Hauz Khas, New Delhi – 110 016, India
Abstract In the present work, dynamic stress-strain response of Himalayan quartzite is tested under high loading rates using split Hopkinson pressure bar (SHPB) device for the first time in the literature. The physical and static mechanical properties of quartzite e.g. dry and saturated density, specific gravity, static compressive strength and elastic modulus values are also determined. Petrological studies of quartzite are carried out through X-ray diffraction (XRD) test and scanning electron microscope (SEM) test. In the SHPB tests, it is observed from the stress-strain response that the dynamic peak stress increases with increasing strain rate whereas the elastic modulus does not show any clear trend with increase in strain rate. Dynamic force equilibrium at the incident and transmission bar ends of the rock samples is attained in all tests till the failure of the rock samples. Dynamic increase factor (DIF) for the rock is determined at a particular strain rate by comparing the dynamic to static peak compressive stress. Correlation equation for dynamic strength increase factor with respect to strain rate has been proposed herein. © Published by Elsevier Ltd. This © 2017 2017The TheAuthors. Authors. Published by Elsevier Ltd. is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of EUROCK 2017. Peer-review under responsibility of the organizing committee of EUROCK 2017 Keywords: Rock dynamic characterization; split Hopkinson pressure bar; quartzite
1. Introduction Design, development and building of civil infrastructure in the mountainous regions involve many complexities in terms of diverse geological and geomorphological features of the region - the Chenab river bridge in the Himalayas, the Gotthard Base tunnel in the Alps are to name a few. The young mountain ranges of the Himalayas and the Alps contain joint planes, shear seams, active fold, and fault zones. Moreover high in-situ stresses and high level of seismicity in these regions pose severe challenges to the construction of infrastructure. In addition to this, unanticipated loads caused by natural hazards, e.g. landslide, earthquake and manmade hazards, e.g. blast and projectile penetration add to the difficulties already existing therein. It may be noted that the loads caused
* Corresponding author. Tel.: +91-9717253877; fax: +91-11-2659-1117. E-mail address: [email protected], [email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of EUROCK 2017
doi:10.1016/j.proeng.2017.05.147
Sunita Mishra et al. / Procedia Engineering 191 (2017) 2 – 9
by hazardous events like earthquake and blast are highly transient in nature generating high strain rates in rock and strain rate caused by blast may reach up to 104ͼs-1 [1, 2] which in turn affects both the stiffness and the strength properties of the rocks. Thus, in order to ensure sustainable design of civil infrastructure in the mountains, it becomes necessary to characterize the rocks under static and dynamic loading conditions. 1.1. Literature review In the present work, only the dynamic compression response of the rock has been discussed and reported. Dynamic compression tests of rocks at low to high strain rates have been performed by several researchers through split Hopkinson pressure bar (SHPB) and dynamic triaxial tests tests [3–18]. Dynamic uniaxial compression tests were performed on three rocks by [3] using SHPB at strain rates from 10-4s-1 to 104s-1 at varying temperatures and the SHPB test data for rocks were reported for the first time in the literature. They observed that the rocks exhibited increased stiffness and higher stress with increasing strain rate and decreasing temperature. Energy absorption in SHPB test in two different rocks, Bohus granite and Solenhofen limestone was reported in [6]. It was observed that the energy absorbed by the rocks increased markedly when the applied load reached the critical value of 1.8 and 1.3 times the static compressive strength for Bohus granite and Solenhofen limestone, respectively. SHPB tests on tuff, which is a hard igneous rock of volcanic origin, was performed in [9] for strain rates varying from 10-6s-1 to 103s-1. It was observed from the results that the strength of the rock was a weak function of the strain rate for strain rates varying from 10-6s-1 to 76s-1; however, for the strain rate above 76s-1, the rate of increase in strength was proportional to the cube root of the strain rate. Dynamic uniaxial compression tests were conducted in [11] on Bukit Timah granite in Singapore at four different loading rates (100 MPaͼs-1, 101 MPaͼs-1, 103 MPaͼs-1 and 105 MPaͼs-1). It was concluded from the tests that, for each log scale increase in loading rate, the compressive strength of the rock increased by 15%. They also observed that there were small changes in the elastic modulus and Poisson’s ratio values with an increase in loading rate. Uniaxial compressive SHPB tests on limestone was conducted in [12] by using a copper disk at the impact end of the incident bar as a pulse shaper, which resulted in dynamic stress equilibrium of the samples and maintained constant strain rates over the test duration. An improved experimental approach for eliminating oscillation that exists in the dynamic stress–strain response of rocks and other brittle materials obtained from SHPB tests was reported in [13]. The tests were conducted on granite, sandstone and limestone and was concluded that the improved method eliminates oscillation in the tests, provides better stability of strain rate and more representative results than those obtained from the conventional rectangular loading waveform shape. The dynamic stress-strain response of Bukit Timah granite loaded at a medium strain rate of 20s-1 – 60s-1 using SHPB testing was reported by [15]. It was observed from the results that the dynamic fracture strength of the granite was directly proportional to the cube root of the strain rate, whereas the elastic modulus remained unchanged with increasing strain rate. At higher strain rates, the rocks showed a higher amount of energy absorption and the particle size of the fragments at the end of the test became smaller. Uniaxial compression tests on Thai sandstones were reported in [17] and they reported an increase in strength and elastic modulus with an increase in strain rate. It was observed that both the strength and the elastic modulus tended to increase exponentially, with the loading rates ranging from 0.001 MPaͼs-1 to 10 MPaͼs-1. A maximum of 71% increase in the modulus of elasticity was observed for sandstone for the increase in loading rate from 0.001 MPaͼs-1 to 10 MPaͼs-1. It may be summarized from the literature review that dynamic compressive strength testing on rocks using SHPB has been carried out on different rock types, e.g., granite, Barre granite, basalt, volcanic tuff, Kawazu tuff, red sandstone, Indiana limestone, porphyritic tonalite, oil shale, granodiorite, coal, kidney stone, Tennessee marble and Akyoshi marble at up to a 2000s-1 strain rate [19] and that strain rate had a significant effect on the mechanical behavior of the rocks. 1.2. Test plan The objective of the present work is to characterize Himalayan quartzite under strain rate dependent loading at different levels of strain rates. The quartzite rock blocks collected herein are of unweathered nature. The rock blocks are collected from a hydropower project site in Vilaspur, India under National Thermal Power Corporation (NTPC) Ltd., India. The rock samples have been tested for both physical and mechanical properties. The physical properties,
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e.g. dry density, saturated density and specific gravity, the petrological studies, e.g. X-ray diffraction test (XRD) and scanning electron microscope test (SEM) and the static mechanical properties, e.g. uniaxial compressive strength, static elastic modulus, and static tensile strength of the rocks are determined in the Indian Institute of Technology (IIT) Delhi laboratories. Static uniaxial compressive strength tests on dry and saturated rock samples have been carried out using the automatic uniaxial compression and splitting test device for rock samples with aspect ratio (L/D) = 2:1. Brazilian and point load tests on dry and saturated rock samples have been carried out to determine the tensile strength values of the rocks. For the Brazilian test, L/D = 0.5:1 and for the point load test, L/D = 1:1 have been used. All static tests have been carried out following the specifications given in [20] and [21]. The strain rate dependent tests are carried out using SHPB in the Rel Inc. laboratory, Michigan, USA. The stress-strain response of the rocks under dynamic loading, force equilibrium at incident and transmission bar ends of rock sample, peak stress and dynamic elastic modulus are studied. The dynamic tests are performed using a 38 mm diameter SHPB for all three rocks at different strain rate levels. In the present study, the low strain rate range is defined for 10 to 100s-1, medium strain rate is defined for 100 to 250s-1, and high strain rate is defined for 250 to 500s-1. For this test, the rock samples are prepared with a diameter of 38 mm and aspect ratio of 0.5:1. The results obtained from the static tests are presented in Table 1 and those from the dynamic tests are presented in Tables 2 for quartzite. The striker bar is propelled using a compressed air gas gun. The strain rate in the dynamic tests is controlled by varying the striker bar length and the striking velocity. The sample number, sample length, length of the striker bar used and the striking velocity applied for a particular strain rate are also reported in the above-mentioned tables. The dynamic increase factor (DIF), i.e. the ratio of dynamic to static peak stress is calculated for each test at different strain rates. Further, a suitable correlation equation is proposed herein for change in DIF with strain rate for quartzite tested for the strain rate range in the present work. 1.3. Test setup for SHPB device The SHPB in the Rel Inc. laboratory in Calumet, Michigan is designed and manufactured by Rel Inc. group. The setup comprises of an incident bar, a transmission bar and striker bars of different sizes. The bars are made up of C300 maraging steel. The incident bar length is 2.59 m and diameter is 38.1 mm. The transmission bar length is 2.43 m and diameter is 38.1 mm. The dimensions of the incident and transmission bars allow one dimensional loading of the sample. In the present work, two different lengths of striker bars are used, e.g. 152.4 mm and 304.8 mm; the diameter of striker bar is 38.1 mm. Compressed air is used to launch the striker bar on the incident bar. The striker bar is propelled by a compressed air gas gun at varying pressure magnitudes which generate stress waves inside the striker bar. The striker bar hits the impact end of the incident bar and remains in contact till the stress wave travels from one end of the striker bar to the other end. The stress wave upon reaching the other end of the striker bar gets reflected back. As a result the contact between the striker bar and the incident bar is lost. The time duration taken by the stress wave to travel from one end of the striker bar to the other is the time duration for loading of the sample. The time duration is given by 't
2 Ls cbar
(1)
where, Ls is the length of the striker bar and cbar is one dimensional longitudinal stress wave velocity in the bar. Thus, using longer striker bar increases the loading time and the rock sample gets time to respond. As a result lower strain rate develops when longer striker bar is used. It may be observed from the Tables 2 that for low strain rate level, the length of the striker bar used is 304.8 mm whereas a striker bar of length 139.7 mm is used for medium and higher strain rates. For smaller striker bars, the time taken by the stress wave to propagate from one end to the other end of the bar is less and hence the loading time duration is also less for the same amount of the load intensity which results in higher strain rate. The strains in the incident and transmission bars are measured using two strain gauges, one mounted on the incident bar and the other mounted on the transmission bar. The strain gauges are 6.35 mm in length, OMEGA 120: in resistance and dynamic gauge with tolerance of ±0.35%. Strain gauges are attached to the incident and the transmission bars with M-bond adhesive. In order to read the strain signal, Vishay
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2310B signal conditioner and amplifier with ¼ wheatstone bridge has been used with a Picoscope5242 having sampling rate of 1 in 8 nanosecond. Momentum trap, though present in the setup, has not been used in the present experiments. The impact of striker bar on the incident bar causes a longitudinal elastic compressive stress wave which propagates through the incident bar. The strain pulse generated within the incident bar is designated as incident strain pulse Hi(t). The strain pulse generated in the incident bar is recorded by the strain gauge mounted on the incident bar. Upon reaching the bar-specimen interface, a part of the pulse, designated as reflected strain pulse Hr(t) is reflected back in the incident bar and the remaining part of the compressive pulse passes through the specimen. Upon reaching the transmission bar end of the specimen, the pulse propagates through the transmission bar and it is then termed as transmitted strain pulse Ht(t). The histories of strain İ(t ) , strain rate İ (t ) and stress ı(t ) within the sample in dynamic compression test are given by İ(t )
C L
İ (t )
C İ i İ r İ t L
(3)
ı(t )
A E İ i İ r İ t 2 A0
(4)
t
³ İ 0
i
İ r İ t dt
(2)
where L is the length of the sample, C is the one dimensional longitudinal stress wave velocity in the bar, E is the elastic modulus of the bar material, A is the cross-sectional area of the bar and A0 is the initial area of the sample. Assuming that stress equilibrium and uniform deformation of the sample prevails during dynamic loading, i.e. İ i İ r İ t , the strain, strain rate and stress are given by İ(t )
t
³ İ dt 0
r
(6)
C İr L
(7)
A Eİ t A0
(8)
İ (t )
ı(t )
C L
The data processing method for the SHPB test is based on two assumptions - one is force equilibrium on both sides of the specimen and the other is the one dimensional uniform deformation of the sample. The force equilibrium is achieved herein by preparing samples with small slenderness ratio, preferably below or equal to 0.5 [22, 23]. The experimental requirement for conducting SHPB test is maintaining good contact between the bars and the sample, reducing the friction between the bars and the supports, keeping the bars coaxial and using pulse shapers [24, 25]. However, for rock samples, maintaining force equilibrium becomes challenging due to the anisotropic nature of the rock and the propagation of the crack inside the rock. Moreover, due to the brittle behavior of the rocks, it becomes important that the equilibrium is achieved before brittle failure of the sample [12]. Researchers have discussed the technique of pulse shaping to achieve force equilibrium in the SHPB tests for brittle materials such as rock [26]. It was shown in [27] that a wide variety of incident pulses can be produced by varying the geometry of the pulse shaper, which can be used for different materials under investigation. To ascertain the force equilibrium in the tests reported herein, mild steel and copper pulse shapers have been used in all tests. The pulse shapers used herein are circular disks of 12.7 mm diameter and different thicknesses. Moreover, the incident and transmission bar ends of the specimen are prepared through rigorous grinding to a smoothness of
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approximately 4.1 μm. Further, a 38 mm diameter SHPB setup has been used to capture the effect of anisotropy on the stress-strain response of the rock samples. The sample thickness has been maintained at half of its diameter to achieve stress uniformity throughout the length of the sample. The velocity of the striker is also recorded automatically using fiber optic speed sensors with a response time of 1 μs. A program has been used for data processing, e.g., obtaining stress-strain plots, strain and strain rate time histories, force equilibrium at the interfaces of the incident bar and sample and the transmission bar and sample. From these plots, peak stress, average strain rate and strain at peak stress are studied. 2. Results and discussion 2.1. Physical properties The dry density value of quartzite is determined for five samples and average values are determined to be 2585.84 kgͼm-3. The saturated density value of quartzite is also measured for five samples and average values are determined to be 2605.57 kgͼm-3. The saturated density value is observed to be smaller than the dry density value. The specific gravity value of quartzite is estimated to be 2.80. The density and specific gravity values are compared with the available data from the literature and observed to be well in agreement [28]. 2.2. Results of petrological study Petrological studies are performed by using XRD and SEM techniques to identify the mineralogical contents of the rocks. The SEM and XRD tests are done in the central research lab facility of Indian Institute of Technology (IIT) Delhi. The SEM Central Facility at IIT Delhi has a ZEISS EVO Series Scanning Electron Microscope EVO 50 which has magnification capacity of 5x to 1,000,000x and field of view of 6 mm at the Analytical Working Distance.The SEM image of quartzite is shown in Fig. 1 (a). The SEM images shows the grain texture of quartzite. Since quartzite is a metamorphic rock, the SEM images confirms the amorphous grain structure of the rock sample collected. The mineral content of quartzite is determined through X-ray diffraction tests. Fig. 1 (b) shows the results of X-ray diffraction graph of quartzite. The Himalayan quartzite is identified by its lustrous red color. The mineralogy of quartzite is ascertained to be quartz (90%) with feldspar and mica. The XRD graphs of quartzite is compared with the available data from the literature and observed to be well in agreement [28].
Fig. 1. (a) Scanning electron microscope images; (b) X-ray diffraction technique graph of quartzite.
2.3. Static uniaxial compressive strength The static uniaxial compressive strength and elastic modulus of the rock is determined under dry conditions following ASTM standards [29, 30]. and presented in Table 1. The tests for each rock type are repeated three times and the average strength values are noted. The static uniaxial compressive strength of quartzite is found to be 108.18 MPa. The elastic modulus from the stress-strain graph at 50% of peak stress value is calculated to be 11.65 GPa [31].
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Table 1. Physical and static properties. Rocks
Dry Density, Ud (kgͼm-3)
Saturated Density, Usat (kgͼm-3)
Specific Gravity, G
Uniaxial Compressive Strength, ıc (MPa)
Modulus of Elasticity, Et (GPa)
Quartzite
2585.841
2605.577
2.80
108.18
11.65
2.4. Dynamic stress-strain and force equilibrium for quartzite Figs. 2(a), 2(b) and 2(c) show the stress-strain plots for quartzite at three different strain rate ranges as obtained from SHPB tests e.g. high, medium and low, respectively. The high level strain rate varies from 260 s-1 to 316 s-1, the medium level strain rate varies from 134 s-1 to 215 s-1 and the low level of strain rate varies from 77 s-1 to 94 s-1. For high strain rate range as seen in Figure 2(a), the peak stress values obtained are 688.511 MPa at 260 s-1, 676.592 MPa at 280 s-1, 617.537 MPa at 295 s-1 and 669.297 MPa at 316 s-1. Fig. 2(b) shows the stress-strain plots for quartzite at different medium strain rate values. For medium strain rate range, the peak stress values obtained are 379 MPa at 134 s-1, 513.166 MPa at 173 s-1, 547.102 MPa at 213 s-1 and 553.981 MPa at 215 s-1. Fig. 2(c) shows the stress-strain plots for quartzite at different low strain rate values. For low strain rate range, the peak stress values obtained are 409.533 MPa at 77 s-1, 390.894 MPa at 86 s-1 and 379.302 MPa at 94 s-1. It may be noted that for all the strain rate values, force equilibrium is achieved nicely up to the peak of the force equilibrium curves and equilibrium is lost after the peak. Thus, the rock sample is under stress equilibrium till the peak stress is reached, however, stress equilibrium is lost once breakage of the sample starts taking place. When there is force equilibrium between the specimen’s incident and transmitted sides, the loading on specimen is one dimensional and the specimen deforms, uniformly. The strain rate in each test is obtained when the sample is under force equilibrium and the strain rate reaches a stable value, i.e., the strain rate-time plot becomes parallel to time axis. Figs. 3(d) and 3(e) show the peak stress and elastic modulus values, respectively obtained from all the stress-strain curves for quartzite at different strain rates. The peak stress clearly increases with strain rate whereas the elastic modulus does not exhibit strain rate dependency. This can be explained by the fact that rock is heterogeneous in nature and till date no standards have been defined for high loading rate dynamic tests of rocks. Hence, the dynamic elastic modulus of the rock is difficult to be ascertained at present. The peak stress, elastic modulus and the strain at peak stress values are also reported in Table 2 for different strain rates. It can be seen that the peak stress increases by almost 22% from low strain rate (77 s-1 to 94 s-1) to medium strain rate (134 s-1 to 215 s-1) and by 37.7% from medium strain rate (134 s-1 to 215 s-1) to high strain rate (260 s-1 to 316 s-1). 2.5. Dynamic increase factor and proposed correlation equation Dynamic increase factor (DIF) for quartzite has been determined by comparing the dynamic peak stress with the static peak stress. The DIF values of the rock with respect to varying strain rate are reported in Table 2. The DIF values are also plotted in Fig. 3(f) for quartzite. From Fig. 3(f), it may be seen that the dynamic strength of quartzite are 3.5 to 6.36 times that of the static strength for strain rates varying from 77 s-1 to 316 s-1. A correlation equation has been developed for the calculated DIF with strain rate by setting a best fitted curve through the obtained DIF values which satisfies the 95% confidence interval. The correlation equation for quartzite with coefficient of determination (R2) = 0.89 is presented in Figure 3(f) and given by
DIF
0.012 İ 2.48 for 77 s -1 d İ d 316 s -1
(8)
It may be noted that the DIF equations proposed herein will be applicable for the strain rate ranges considered in the current work.
Sunita Mishra et al. / Procedia Engineering 191 (2017) 2 – 9
Strain rate and sample number
Peak Stress V MPa)
Stress V (MPa)
600 400 200
0 0.000 0.005 0.010 0.015 0.020 0.025 Strain H
700 600 500 400 100
215/sec, Q2 213/sec, Q14 173/sec, Q12 134/sec, Q9
600
200
200 300 Strain rate İ (/sec)
0.005
0.010 Strain H
0.015
200 0 0.000
150 100 50
400
0.020
200
0 50
94/sec, Q8 86/sec, Q6 77/sec, Q11
600
(f)
250
Strain rate and sample number
800
400
0 0.000
(e)
(c) 1000
Strain rate and sample number
800
Elastic Modulus Et (GPa)
Stress V (MPa)
800
(d)
(b) 1000
316/sec, Q5 295/sec, Q4 280/sec, Q13 260/sec, Q1
Stress V (MPa)
(a) 1000
Dynamic Increase Factor
8
100 150 200 250 300 350 Strain rate İ (/sec)
0.005 0.010 Strain H
0.015
7
DIF for Quartzite Experimental Data Equation Trendline 6
5 4
DIF = 0.012(İ)2.48 R2 = 0.89
3
70
140 210 280 Strain rate İ (/sec)
350
Fig. 2. Quarzite stress-strain curves for (a) high strain rate range; (b) medium strain rate range; (c) low strain rate range; data points for; (d) peak stress-strain rate; (e) elastic modulus-strain rate; (f) dynamic increase factor-strain rate. Table 2. Dynamic properties: quartzite. Strain Rate Range
Rock Type
Low
Quartzite
Medium
High
Sample Number
Sample Length (mm)
Q11 Q6 Q8 Q9 Q12 Q14 Q2 Q1 Q13 Q4 Q5
18.54 18.87 18.34 17.25 18.39 19.14 18.59 18.62 18.01 18.11 18.85
Striker Bar Length (mm)
304.8
139.7
139.7
Striker Bar Velocity (mͼs-1)
21.27 19.46 19.67 19.49 32.69 32.69 32.64 40.60 40.60 39.76 41.60
Strain Rate, İ (s-1)
77 86 94 134 173 213 215 260 280 295 316
Peak Stress, ıdc (MPa) 409.533 390.894 379.302 379.000 513.166 547.102 553.981 688.511 676.592 617.537 669.297
Strain at Peak Stress, İ
Modulus of Elasticity, Et (GPa)
Dynamic Increase Factor, DIF
0.006 0.009 0.01 0.014 0.008 0.012 0.009 0.013 0.013 0.015 0.012
10.44 38.09 35.31 25.03 217.44 35.43 63.52 57.09 50.41 39.63 57.79
3.78 3.61 3.50 3.50 4.74 5.05 5.12 6.36 6.25 5.70 6.18
3. Conclusions
Characterization of Himalayan quartzite has been performed in the present work for a strain rate range varying from 49 s-1 to 316 s-1 through uniaxial compressive 38 mm diameter SHPB test. The dynamic stress-strain response, peak stress, elastic modulus and force equilibrium at the incident and transmission bar ends of the rock specimen are studied. The physical properties and static stress-strain behaviour of the rock are also investigated. The peak stress increases at a rate of 22% from low (77 s-1 to 94 s-1) to medium (134 s-1 to 215 s-1) strain rate and by 37.7% from medium (134 s-1 to 215 s-1) to high (260 s-1 to 316 s-1) strain rate. The elastic modulus of quartzite does not show any particular trend with increasing strain rate. The dynamic compressive stress 669.297 MPa at 316 s-1 is nearly six times of static compressive stress. The dynamic increase factor varies from 3.5 to 6.36.
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Acknowledgement
This work is a part of an ongoing research project funded by Terminal Ballistics Research Laboratory (TBRL), Chandigarh under Defence Research and Development Organisation (DRDO), India. The authors acknowledge the funding provided by TBRL in this work.
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