- Email: [email protected]
Characterization of three Himalayan rocks using a split Hopkinson pressure bar
International Journal of Rock Mechanics & Mining Sciences 85 (2016) 112–118
Contents lists available at ScienceDirect
International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
Technical Note
Characterization of three Himalayan rocks using a split Hopkinson pressure bar Tanusree Chakraborty a,n, Sunita Mishra a, Josh Loukus b, Brent Halonen b, Brady Bekkala b a b
Department of Civil Engineering, Indian Institute of Technology (IIT) Delhi, India Rel Inc., Calumet, MI 49913, USA
art ic l e i nf o Article history: Received 12 May 2015 Received in revised form 3 March 2016 Accepted 9 March 2016 Available online 31 March 2016 Keywords: Dynamic increase factor Force equilibrium Split Hopkinson pressure bar Stress–strain response
1. Introduction The design of the civil infrastructure in mountainous regions involves many complexities related to the diverse geological and geomorphological features of the region – the Chenab river bridge in the Himalayas and the Gotthard Base tunnel in the Alps, for example. The young mountain ranges of the Himalayas and the Alps contain joint planes, shear seams, active folds and fault zones. Moreover, high in-situ stresses and high levels 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., landslides and earthquakes, and by manmade hazards, e.g., blasts and projectile penetration, add to the difficulties already existing therein. It may be noted that the loads caused by hazardous events such as an earthquake or a blast are highly transient in nature, generating high strain rates in the rock, and the strain rate caused by a blast may reach up to 104/sec1,2, which in turn affects both the stiffness and the strength properties of the rocks. Thus, to ensure sustainable design of civil infrastructure in the mountains, it is necessary to characterize the rocks under both static and dynamic loading conditions. In the present work, the dynamic compression response of the rocks has been discussed and reported. Dynamic compression tests of rocks at different strain rates have been performed by n Correspondence to: Department of Civil Engineering, Indian Institute of Technology (IIT) Delhi, Hauz Khas, New Delhi 110 016. E-mail address: [email protected] (T. Chakraborty).
http://dx.doi.org/10.1016/j.ijrmms.2016.03.005 1365-1609/& 2016 Elsevier Ltd. All rights reserved.
several researchers using the split Hopkinson pressure bar (SHPB) and dynamic triaxial tests.3–18 Dynamic uniaxial compression tests were performed on three rocks by3 using SHPB at strain rates from 10 4/sec to 104/sec 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 significantly 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 in9 for strain rates varying from 10 6/sec to 103/sec. 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 6/sec to 76/sec; however, for the strain rate above 76/sec, the rate of increase in strength was proportional to the cube root of the strain rate. Dynamic uniaxial compression tests were conducted by11 on Bukit Timah granite in Singapore at four different loading rates (100 MPa/sec, 101 MPa/sec, 103 MPa/sec and 105 MPa/sec). 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 by12 by using a copper disk at the impact end of the incident bar as a pulse
T. Chakraborty et al. / International Journal of Rock Mechanics & Mining Sciences 85 (2016) 112–118
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 in13. 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 20–60/sec. 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 in17 and they found 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 MPa/sec to 10 MPa/sec. A maximum of 71% increase in the modulus of elasticity was observed for sandstone with the increase in loading rate from 0.001 MPa/sec to 10 MPa/sec. 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 2000/sec strain rate19 and that strain rate had a significant effect on the mechanical behavior of the rocks. In the Indian sub-continent, the young and diverse rock formations of the Himalayas are often devastated by high intensity earthquakes, blast loads due to cross-border insurgency and blasting activities necessary for roadway construction, which creates high loading rates in the rocks. The objectives of the present work are to characterize three Himalayan rocks, i.e., quartzite, limestone and dolomite, under strain rate dependent loading at different levels of strain rates, i.e., low strain rates varying from 49/ sec to 94/sec depending on rock type, medium strain rates varying from 91/sec to 135/sec depending on rock type and high strain rates varying from 174/sec to 316/sec depending on rock type. The quartzite and limestone rocks collected herein are of unweathered nature and the dolomite is slightly weathered. The rock blocks are collected from a hydropower project site in Vilaspur, India. In the present work, blocks of the three rock types, quartzite, limestone and dolomite, are collected from the site. The rock samples have been tested for both physical and mechanical properties. The physical properties, e.g., dry density, saturated density and specific gravity; and the static mechanical properties, e.g., uniaxial compressive strength, static elastic modulus and static tensile strength of the rocks, are determined. The strain rate dependent tests are carried out using the SHPB. The stress–strain response of the rocks under dynamic loading, force equilibrium at the incident and transmission bar ends of the rock sample and peak stress and
113
dynamic elastic modulus are studied. The dynamic increase factor (DIF), i.e., the ratio of dynamic to static peak stress, is calculated for each test at different strain rates. The rock tested by using SHPB device should fail under the upper limit of strain rate. The stress– strain response of the rock if tested above the upper limit of strain rate would give erroneous result which can be explained by nonachievement of force equilibrium due to inertia effect. It has been reported in20 that the upper limit of the strain rate for a material to fail should be
ε̇1 =
εf c αL
(1)
where c is the elastic wave speed of the specimen, L is the length of the specimen and α is a non-dimensional parameter which depends on the shape of the incident pulse. Further, suitable correlation equations are proposed herein for changes in DIF with strain rate for all three rocks.
2. Laboratory tests performed on rocks In the present work, three rock types, i.e., quartzite, limestone and dolomite, are collected from the Vilaspur site for testing. The rock samples are prepared with a diamond bit core cutter of 38 mm. For all three rocks, the physical properties, e.g., both dry and saturated densities and specific gravity have been obtained. Static uniaxial compressive strength tests on dry and saturated rock samples have been carried out using the CONTROLS uniaxial compression and splitting test device for rock samples with aspect ratio (L/D) ¼2:1. 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 ¼0.5:1 and for the point load test, L/D ¼1:1 have been used. All static tests have been carried out following the specifications given in21,22. The dynamic tests are performed using a 38-mm diameter SHPB for all three rocks at different strain rate levels. 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, 3 and 4 for quartzite, limestone and dolomite, respectively. 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.
3. SHPB test setup Fig. 1 shows the schematic diagram of the compression SHPB test setup in the RelInc laboratory, Calumet, Michigan, USA. The setup is composed of an incident bar, a transmission bar and striker bars of different sizes. The bars are made of C300 maraging steel. The incident bar length is 2.59 m and the diameter is 38.1 mm. The transmission bar length is 2.43 m and the diameter is 38.1 mm. The dimensions of the incident and transmission bars
Table 1 Physical and static properties. Rocks
Dry density, ρd (kg/m3)
Saturated density, ρsat (kg/m3)
Specific gravity, G
Uniaxial compressive strength, σc (MPa)
Modulus of elasticity, Et (GPa)
Quartzite Limestone Dolomite
2585.841 2630.601 2731.820
2605.577 2664.206 2723.202
2.80 2.71 2.70
108.18 51.21 38.59
11.65 2.62 3.75
114
T. Chakraborty et al. / International Journal of Rock Mechanics & Mining Sciences 85 (2016) 112–118
Table 2 Dynamic properties: quartzite. Rock type Sample number
Sample length Striker bar (mm) length (mm)
Striker bar velocity (m/sec)
Strain rate, ε̇ (/sec)
Peak stress, σdc (MPa)
Strain at peak stress, ε
Modulus of elasticity, Et (GPa)
Dynamic increase factor, DIF
Quartzite
18.54 18.87 18.34 17.25 18.39 19.14 18.59 18.62 18.01 18.11 18.85
21.27 19.46 19.67 19.49 32.69 32.69 32.64 40.60 40.60 39.76 41.60
77 86 94 134 173 213 215 260 280 295 316
409.533 390.894 379.302 379.000 513.166 547.102 553.981 688.511 676.592 617.537 669.297
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
Sample length Striker bar (mm) length (mm)
Striker bar velocity (m/sec)
Strain rate, ε̇ (/sec)
Peak stress, σdc (MPa)
Strain at peak stress, ε
Modulus of elasticity, Et (GPa)
Dynamic increase factor, DIF
15.82 18.59 19.10 17.86 18.05 18.81 18.36 18.59 18.16 18.49 18.39
12.27 12.19 12.96 24.38 24.05 13.45 13.85 36.56 38.09 34.51 36.56
60 60 73 89.7 91 95 115 185 213 245 263
245.889 246.780 264.919 302.410 307.340 263.418 272.670 335.684 336.839 391.640 398.121
0.007 0.006 0.008 0.008 0.008 0.010 0.010 0.011 0.012 0.012 0.012
46.39 41.15 39.30 46.81 46.42 24.71 26.42 35.33 27.93 37.65 47.50
4.81 4.80 5.17 5.90 6.00 5.14 5.32 6.55 6.57 7.64 7.77
Q11 Q6 Q8 Q9 Q12 Q14 Q2 Q1 Q13 Q4 Q5
304.8
139.7
Table 3 Dynamic properties: limestone. Rock type
Sample number
Limestone L22 L17 L7 L11 L12 L9 L10 L6 L4 L5 L2
304.8
139.7 139.7 304.8 304.8 139.7
Table 4 Dynamic properties: dolomite. Rock type Sample number
Sample length Striker bar (mm) length (mm)
Striker bar velocity (m/sec)
Strain rate, ε̇ (/sec)
Peak stress, σdc (MPa)
Strain at peak stress, ε
Modulus of elasticity, Et (GPa)
Dynamic increase factor, DIF
Dolomite
19.02 18.90 19.05 18.31 19.09 19.08 18.58 18.29 19.23 18.38 18.15
20.54 20.55 20.32 20.31 29.50 27.71 24.70 29.50 30.47 35.19 38.10
49 69 76 76 125 132 132 144 174 176 201
395.882 398.760 313.150 396.700 342.170 403.280 326.366 433.338 434.888 403.388 421.930
0.006 0.008 0.006 0.008 0.008 0.008 0.007 0.009 0.014 0.008 0.014
80.79 63.29 65.51 77.78 47.39 45.51 46.75 47.51 31.46 57.46 30.97
10.25 10.33 10.27 8.11 8.86 10.45 8.45 11.22 11.26 10.45 10.93
D15 D3 D14 D12 D8 D10 D9 D5 D7 D1 D11
304.8
139.7
allow one-dimensional loading of the sample. In the present work, two different lengths of striker bars are used, 152.4 mm and 304.8 mm; the diameter of the striker bar is 38.1 mm. Compressed air is used to launch the striker bar onto the incident bar. The striker bar is propelled by compressed air from a gas gun at varying pressure magnitudes that 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
Δt =
2L s cbar
(2)
where Ls is the length of the striker bar, and cbar is the one-
dimensional longitudinal stress wave velocity in the bar. Thus, using a longer striker bar increases the loading time, and the rock sample gets more time to respond. As a result, a lower strain rate develops when the longer striker bar is used in comparison to a shorter striker bar for a particular pressure. It may be observed from the Tables 2, 3 and 4 that, for low strain rate levels, 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 shorter striker bars, the time taken by the stress wave to propagate from one end to the other end of the bar is less; hence, the loading time duration is also less for the same amount of the load intensity as compared to a longer striker bar, which results in a higher strain rate. When the diameters of the striker and input bar are same and pulse shaper is not used, then the shorter or longer striker bars will produce the same average strain rate if loaded under equivalent loading intensity as generated by the pressure released from gas gun. The strains in the incident and transmission bars are measured using two strain gauges, one mounted on the incident
T. Chakraborty et al. / International Journal of Rock Mechanics & Mining Sciences 85 (2016) 112–118
Trigger
115
Launcher/Accelerator
Striker Bar Speed Sensor Momentum Trap
Incident Bar
Specimen Impact Area
Control Cabinet
Control Panel Laptop Vishay
Transmission Bar Bar Stand
Data Acquisition Area
Deceleration Unit
Air/Storage Cabinet
Fig. 1. 3D Schematic diagram of split Hopkinson bar apparatus.
bar and the other mounted on the transmission bar. The strain gauges are 6.35 mm in length and 120 Ω in resistance, and the dynamic gauge has a tolerance of 70.35%. The strain gauges are attached to the incident and the transmission bars with M-bond adhesive. To read the strain signal, a Vishay 2310B signal conditioner and amplifier with a ¼ Wheatstone bridge has been used with a Picoscope5242 having sampling rate of 1 in 8 ns. A momentum trap, though included in the setup, has not been used in the present experiments. The impact of the striker bar on the incident bar causes a longitudinal elastic compressive stress wave that propagates through the incident bar. The strain pulse generated within the incident bar is designated as the incident strain pulse εi (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 the reflected strain pulse εr(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 designated as the transmitted strain pulse εt(t). The histories of the strain ε(t ), strain rate ε̇ (t ) and stress σ(t ) within the sample in the dynamic compression test are given by t
ε(t ) =
C L
ε̇ (t ) =
C ( εi − εr − εt ) L
(4)
σ(t ) =
A E ( εi + εr + εt ) 2A 0
(5)
∫0
( εi − εr − εt ) dt
(3)
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 prevail during dynamic loading, i.e., ϵi + ϵr = ϵt , the strain, strain rate and stress are given by t
ε(t ) = −
C L
ε̇ (t ) = −
C εr L
(7)
A E εt A0
(8)
σ(t ) =
∫0
εr dt
(6)
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 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 0.5.23,24 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.25,26 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.27 It was shown in28 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.
116
T. Chakraborty et al. / International Journal of Rock Mechanics & Mining Sciences 85 (2016) 112–118
Moreover, the incident and transmission bar ends of the specimen are prepared through rigorous grinding to a smoothness of approximately 4.1 mm. 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 ms. 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.
4. Results and discussion 4.1. Physical properties The dry density values of the quartzite, limestone and dolomite rocks are determined for five samples of each rock; the average values are determined to be 2585.84 kg/m3, 2630.60 kg/m3 and 2731.82 kg/m3, respectively. The saturated density values of quartzite, limestone and dolomite are also measured for five samples of each rock; the average values are determined to be 2605.57 kg/m3, 2664.21 kg/m3 and 2723.20 kg/m3, respectively. For all three rocks, the saturated density values are observed to be higher than the dry density values. The specific gravity values of quartzite, limestone and dolomite are estimated to be 2.80, 2.71 and 2.70, respectively. The density and specific gravity values of all three rocks are compared with the available data from the literature and observed to be in good agreement.29 4.2. Static uniaxial compressive strength The static uniaxial compressive strength of the rocks is determined under dry and saturated conditions and presented in Table 1. The tests for each rock type are repeated three times and the average strength values are compared.29 The static uniaxial compressive strength of quartzite, limestone and dolomite are found to be 108.18 MPa, 51.21 MPa and 38.59 MPa, respectively. The elastic modulus from the stress–strain graph at 50% of the peak stress value is calculated to be 11.65 GPa, 2.62 GPa and 3.75 GPa, respectively.30 It is to be noted that the static uniaxial compressive strength of quartzite is much more than that of limestone and dolomite because quartzite is a metamorphic rock and limestone and dolomite are sedimentary rocks.
Voltage V (V)
6 4
4.3. Dynamic stress–strain and force equilibrium The incident and transmission waveforms for quartzite, limestone and dolomite are presented in Fig. 2. The voltage in these waveforms is converted to strain by using the standard equation of Wheatstone bridge that converts the voltage output to strain using the gauge factor of the strain gauge.31 4.3.1. Stress–strain response for quartzite Fig. 3 shows the stress–strain plot for quartzite as obtained from SHPB tests for different strain rates that vary from 77/sec to 316/sec. For the strain rate range tested herein as seen in Fig. 3, the peak stress values obtained are 669.297 MPa at 316/sec and 409.533 MPa at 77/sec. Table 2 presents the values of peak stress, elastic modulus and the strain at peak stress obtained from all of the stress–strain curves for quartzite at different strain rates. It can be seen that the peak stress increases by almost 22% from strain rate 77/sec to 215/sec and by 37.7% from strain rate 134/sec to 316/ sec. 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 the rock is heterogeneous in nature and that, until this time, no standards have been defined for high loading rate dynamic tests of rocks. Hence, the dynamic elastic modulus of the rock is difficult to ascertain at present. 4.3.2. Stress–strain response for limestone Fig. 4 shows the stress–strain plots for limestone at different strain rates that vary from 60/sec to 263/sec. For the strain rate range tested herein as seen in Fig. 4, the peak stress values obtained are 398.121 MPa at 263/sec and 245.889 MPa at 60/sec. The peak stress, elastic modulus and strain rate at peak stress values obtained from the stress–strain curves for limestone are given in Table 3. It can be seen that the peak stress increases with an increase in the strain rate. The peak stress of limestone increases by 17.43% from strain rate 60/sec to 135/sec and by 16.14% from strain rate 135/sec to 263/sec. It may be observed from Table 3 that the elastic modulus does not show any clear trend under strain rate dependent loading. 4.3.3. Stress–strain response for dolomite Fig. 5 shows the stress–strain plots for dolomite at different strain rates that vary from 49/sec to 201/sec. For the strain rate range tested herein as seen in Fig. 5, the peak stress values obtained are 421.930 MPa at 201/sec and 395.882 MPa at 49/sec. Table 4 shows the peak stress, elastic modulus and strain at peak stress obtained from the stress–strain curves for dolomite. It can be seen that, similar to quartzite and limestone, the peak stress increases with an increase in the strain rate. The peak stress increases by 0.04% from strain rate 49/sec to 125/sec and by 11.6% from strain rate 125/sec to 201/sec. It can be observed from Table 4
Limestone: L15 Incident wave Transmission wave
Quartzite: Q14 Incident wave Transmission wave
Dolomite: D8 Incident wave Transmission wave
2 0 -2 -4 -6 0.0
0.4
0.8
1.2
1.6
0.0
0.4
0.8 1.2 Time t (ms)
1.6
Fig. 2. Waveform for incident and transmission bar.
0.0
0.4
0.8
1.2
1.6
T. Chakraborty et al. / International Journal of Rock Mechanics & Mining Sciences 85 (2016) 112–118
117
D y n a m i c In c re a s e F a c t o r
12 9 6 3 0
Fig. 3. Stress–strain response for quartzite.
DIF for Quartzite Equation trendline for Quartzite, DIF = 0.012(ε) + 2.48, R = 0.89 DIF for Limestone Equation trendline for Limestone, DIF = 0.013(ε) + 4.19, R = 0.89 DIF for Dolomite 0
100
200
Strain rate ε (/sec)
300
Fig. 7. Dynamic increase factor with strain rate for all the three rocks.
specimen's incident and transmitted sides, the loading on the specimen is one-dimensional and the specimen deforms uniformly which satisfies one dimensional wave propagation theory.
Fig. 4. Stress–strain response for limestone.
Fig. 5. Stress–strain response for dolomite.
that the elastic modulus does not show any clear trend with changes in dynamic strain rates. 4.3.4. Force equilibrium Fig. 6 presents the force equilibrium achieved at the incident and the transmission bar ends of the specimen in the SHPB tests for typical specimens of quartzite, limestone and dolomite, respectively. It may be noted that for all of the strain rate values, force equilibrium is achieved nicely and the peak strength is achieved while the sample is in stress equilibrium signifying the peak compressive strength of the rock to be correct for the particular strain rate tested. Thus, the dynamic increase factor is calculated specifying the correct increase in strength at a particular strain rate.32 When there is force equilibrium between the
4.3.5. Dynamic increase factors and proposed correlation equations The dynamic increase factors (DIF) for all three rocks have been determined by comparing the dynamic peak stress with the static peak stress. The DIF values of the rocks are reported in Tables 2, 3 and 4. The DIF values are also plotted in Fig. 7 for quartzite, limestone and dolomite. From Fig. 7, it may be seen that the dynamic strength of quartzite is 3.5 to 6.36 times that of the static strength for strain rates varying from 77/sec to 316/sec. The dynamic strength of limestone is approximately 4.8 to 7.77 times that of the static strength for a strain rate range of 60–263/sec. And the dynamic strength of dolomite varies from 8.11 to 11.26 times that of the static strength for a strain rate range from 49/sec to 201/sec. Two correlation equations for quartzite and limestone have been developed for the calculated DIF with respect to the strain rate by calculating a best-fit curve through the obtained DIF values that satisfies the 95% confidence interval. The correlation equation for quartzite with a coefficient of determination (R2) ¼0.89 is presented in Fig. 7 and given by
DIF = 0.012 ( ε̇) + 2.48 for 77/sec ≤ ( ε̇) ≤ 316/sec
(9)
The correlation equation for limestone with a coefficient of determination (R2)¼0.89 is presented in Fig. 7 and given by
DIF = 0.013 ( ε̇) + 4.19 for 60/sec ≤ ε̇ ≤ 263/sec
(10)
The dynamic increase factor for dolomite is observed to be approximately constant for the particular strain range considered herein. Hence, the correlation equation with respect to strain rate does not show any particular trend. It may be noted that the DIF equations proposed herein will be applicable for the strain rate ranges considered in the current work. The DIF correlations thus developed for the given strain rate range can be used as material model in dynamic analysis of underground structures and developing blast resistant designs.
5. Conclusions
Force F (kN)
800 Quartzite
600 Limestone
400 200
Transmission bar end 213/sec, Q14, Quartzite 135/sec, L16, Limestone 144/sec, D5, Dolomite
Dolomite
0 0
40 80 Time t (µ s)
Incident bar end 213/sec, Q14, Quartzite 135/sec, L16, Limestone 144/sec, D5, Dolomite
120
Fig. 6. Force equilibrium achieved for all the three rocks tested.
Characterizations of three Himalayan rocks, quartzite, limestone and dolomite, have been performed in the present work for a strain rate range varying from 49/sec to 316/sec through uniaxial compression using a 38 mm diameter SHPB test setup. 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 behavior of the rocks are also investigated. The following conclusions are drawn from the tests. For quartzite, the peak stress increases with increase in strain rate. The elastic modulus of quartzite does not show any particular
118
T. Chakraborty et al. / International Journal of Rock Mechanics & Mining Sciences 85 (2016) 112–118
trend with increasing strain rate. The dynamic increase factor varies from 3.5 to 6.36. For limestone, the peak stress increases with increase in strain rate. The elastic modulus does not show any clear trend with an increase in strain rate. The dynamic increase factor varies from 4.8 to 7.77. For dolomite, the peak stress increases with increase in strain rate. The elastic modulus does not show any clear trend with an increase in strain rate. The dynamic increase factor varies from 8.11 to 11.26 times of the static compressive stress. Due to clay content in limestone and dolomite, its dynamic increase factor is more than quartzite.
Acknowledgments This work is a part of an ongoing research project with research grant number ARMREB/CDSW/2013/151 funded by the Terminal Ballistics Research Laboratory (TBRL), Chandigarh under Defense Research and Development Organization (DRDO), India. The authors acknowledge the funding provided by TBRL in this work.
References 1. Ngo T, Mendis P, Gupta A, Ramsay J. Blast loading and blast effects on structures - an overview. Electron J Struct Eng. 2007:76–91. 2. Dusenberry DO. Handbook for blast resistant design of buildings. New York: Wiley; 2010. 3. Perkins RD, Green SJ. Macroscopic description low and medium strain rates. Int J Rock Mech Min Sci. 1970;7:527–535. 4. Christensen RJ, Swanson SR, Brown WS. Split Hopkinson bar tests on rock under confining pressure. Exp Mech. 1972;12:508–541. 5. Lindholm US, Yeakley LM, Nagy A. The dynamic strength and fracture properties of dresser basalt. Int J Rock Mech Min Sci Geomech Abstr. 1974;11:181–191. 6. Lundberg B. A split Hopkinson bar study of energy absorption in dynamic. Int J Rock Mech Min Sci Geomech Abstr. 1976;13:187–197. 7. Blanton TL. Effect of strain rates from 10 2 to 10 s 1 in triaxial compression tests on three rocks. Int J Rock Mech Min Sci Geomech Abstr. 1981;18(1):47–62. 8. Klepaczko JR. Behavior of rock-like materials at high strain rates in compression. Int J Plast. 1990;6:415–432. 9. Olsson WA. The compressive strength of tuff as a function of strain rate from 10-6 to 103/sec. Int J Rock Mech Min Sci Geomech Abstr. 1991;28(1):115–118. 10. Grady DE. Shock wave properties of brittle solids. In: Schmidt S, ed. Shock Compression of Condensed Matter. New York: AIP Press; 1995:9–20. 11. Zhao J, Li HB, Wu MB, Li TJ. Dynamic uniaxial compression tests on a granite. Int J Rock Mech Min Sci. 1999;36:273–277.
12. Frew DJ, Forrestal MJ, Chen W. A split Hopkinson pressure bar technique to determine compressive stress-strain data for rock materials. Exp Mech. 2000;41 (1):40–46. 13. Li XB, Lok TS, Zhao J, Zhao PJ. Oscillation elimination in the Hopkinson bar apparatus and resultant complete dynamic stress-strain curves for rocks. Int J Rock Mech Min Sci. 2000;37(7):1055–1060. 14. Fukui K, Okubo S, Ogawa A. Some aspects of loading-rate dependency of sanjome andesite strengths. Int J Rock Mech Min Sci. 2004;41(7):1215–1219. 15. Li XB, Lok TS, Zhao J. Dynamic characteristics of granite subjected to intermediate loading rate. Rock Mech Rock Eng. 2005;38(1):21–39. 16. Qi CZ, Wang MY, Qihu Q. Strain-rate effects on the strength and fragmentation size of rocks. Int J Imp Eng. 2009;36:1355–1364. 17. Fuenkajorn K, Kenkhunthod N. Influence of loading rate on deformability and compressive strength of three Thai sandstones. Geotech Geol Eng. 2010;28:707– 715. 18. Ming L, Xian-biao M, Li-li C, Rong-rong M, Guang-hui Z. High strain rate effects on mechanical behavior of coal-serial sandstone. Electron J Geol Eng. 2014;19:6035–6046. 19. Lu YB, Li QM, Ma GW. Numerical investigation of the dynamic compressive strength of rocks based on split Hopkinson pressure bar tests. Int J Rock Mech Min Sci. 2010;47(5):829–838. 20. Ravichandran G, Subhash G. Critical appraisal of limiting strain rates for compression testing of ceramics in a split Hopkinson pressure bar. J Am Ceram Soc. 1994;77:263–267. 21. ISRM. International society of rock mechanics commission on testing methods, suggested methods for determining the uniaxial compressive strength and deformability of rock materials. Int J Rock Mech Min Sci Geomech Abstr. 1979;16 (2):138–140. 22. ISRM. International society of rock mechanics commission on testing methods, suggested method for determining point load strength. Int J Rock Mech Min Sci Geomech Abstr. 1985;22:51–60. 23. Davis EDH, Hunter SC. The dynamic compression testing of solids by the method of the split Hopkinson bar. J Mech Phys Solids. 1963;11:155–179. 24. Zhou Z, Li X, Liu A, Zou Y. Stress uniformity of split Hopkinson pressure bar under half-sine wave loads. Int J Rock Mech Min Sci. 2011;48(4):697–701. 25. Wang ZG, Meyer LW. On the plastic wave propagation along the specimen length in SHPB test. Exp Mech. 2010;50:1061–1074. 26. Abotula S., Shukla A., Chona R. Dynamic constitutive behavior and fracture initiation toughness of hastelloy x under thermo-mechanical loads. In: Proceedings of the society of experimental mechanics, implast conference. WrightPatterson Air Force Base RBSM. 2010. 27. Frew DJ, Forrestal MJ, Chen W. Pulse shaping techniques for testing brittle materials with a split Hopkinson pressure bar. Exp Mech. 2002;42(1):93–106. 28. Xia K, Yao W. Dynamic rock tests using split Hopkinson (Kolsky) bar system – a review. J Rock Mech Geotech Eng. 2014;7:27–59. 29. Kumar R. Testing and constitutive modelling of the strain-softening behaviour of some rocks [Ph.D. Thesis]. Delhi: Indian Institute of Technology (IIT); 2007. 30. ASTM D3148-02. Standard test method for elastic moduli of intact rock core specimens in uniaxial compression. Am Soc Test Mat; 1998. 31. Chen W, Song B. Split Hopkinson (Kolsky) Bar – Design, Testing and Applications. Berlin: Springer; 2011. 32. Li XB, Zou Y, Zhou ZL. Numerical simulation of the rock SHPB test with a special shape striker based on the discrete element method. Rock Mech Rock Eng. 2014;47(5):1693–1709.
Contents lists available at ScienceDirect
International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
Technical Note
Characterization of three Himalayan rocks using a split Hopkinson pressure bar Tanusree Chakraborty a,n, Sunita Mishra a, Josh Loukus b, Brent Halonen b, Brady Bekkala b a b
Department of Civil Engineering, Indian Institute of Technology (IIT) Delhi, India Rel Inc., Calumet, MI 49913, USA
art ic l e i nf o Article history: Received 12 May 2015 Received in revised form 3 March 2016 Accepted 9 March 2016 Available online 31 March 2016 Keywords: Dynamic increase factor Force equilibrium Split Hopkinson pressure bar Stress–strain response
1. Introduction The design of the civil infrastructure in mountainous regions involves many complexities related to the diverse geological and geomorphological features of the region – the Chenab river bridge in the Himalayas and the Gotthard Base tunnel in the Alps, for example. The young mountain ranges of the Himalayas and the Alps contain joint planes, shear seams, active folds and fault zones. Moreover, high in-situ stresses and high levels 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., landslides and earthquakes, and by manmade hazards, e.g., blasts and projectile penetration, add to the difficulties already existing therein. It may be noted that the loads caused by hazardous events such as an earthquake or a blast are highly transient in nature, generating high strain rates in the rock, and the strain rate caused by a blast may reach up to 104/sec1,2, which in turn affects both the stiffness and the strength properties of the rocks. Thus, to ensure sustainable design of civil infrastructure in the mountains, it is necessary to characterize the rocks under both static and dynamic loading conditions. In the present work, the dynamic compression response of the rocks has been discussed and reported. Dynamic compression tests of rocks at different strain rates have been performed by n Correspondence to: Department of Civil Engineering, Indian Institute of Technology (IIT) Delhi, Hauz Khas, New Delhi 110 016. E-mail address: [email protected] (T. Chakraborty).
http://dx.doi.org/10.1016/j.ijrmms.2016.03.005 1365-1609/& 2016 Elsevier Ltd. All rights reserved.
several researchers using the split Hopkinson pressure bar (SHPB) and dynamic triaxial tests.3–18 Dynamic uniaxial compression tests were performed on three rocks by3 using SHPB at strain rates from 10 4/sec to 104/sec 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 significantly 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 in9 for strain rates varying from 10 6/sec to 103/sec. 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 6/sec to 76/sec; however, for the strain rate above 76/sec, the rate of increase in strength was proportional to the cube root of the strain rate. Dynamic uniaxial compression tests were conducted by11 on Bukit Timah granite in Singapore at four different loading rates (100 MPa/sec, 101 MPa/sec, 103 MPa/sec and 105 MPa/sec). 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 by12 by using a copper disk at the impact end of the incident bar as a pulse
T. Chakraborty et al. / International Journal of Rock Mechanics & Mining Sciences 85 (2016) 112–118
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 in13. 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 20–60/sec. 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 in17 and they found 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 MPa/sec to 10 MPa/sec. A maximum of 71% increase in the modulus of elasticity was observed for sandstone with the increase in loading rate from 0.001 MPa/sec to 10 MPa/sec. 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 2000/sec strain rate19 and that strain rate had a significant effect on the mechanical behavior of the rocks. In the Indian sub-continent, the young and diverse rock formations of the Himalayas are often devastated by high intensity earthquakes, blast loads due to cross-border insurgency and blasting activities necessary for roadway construction, which creates high loading rates in the rocks. The objectives of the present work are to characterize three Himalayan rocks, i.e., quartzite, limestone and dolomite, under strain rate dependent loading at different levels of strain rates, i.e., low strain rates varying from 49/ sec to 94/sec depending on rock type, medium strain rates varying from 91/sec to 135/sec depending on rock type and high strain rates varying from 174/sec to 316/sec depending on rock type. The quartzite and limestone rocks collected herein are of unweathered nature and the dolomite is slightly weathered. The rock blocks are collected from a hydropower project site in Vilaspur, India. In the present work, blocks of the three rock types, quartzite, limestone and dolomite, are collected from the site. The rock samples have been tested for both physical and mechanical properties. The physical properties, e.g., dry density, saturated density and specific gravity; and the static mechanical properties, e.g., uniaxial compressive strength, static elastic modulus and static tensile strength of the rocks, are determined. The strain rate dependent tests are carried out using the SHPB. The stress–strain response of the rocks under dynamic loading, force equilibrium at the incident and transmission bar ends of the rock sample and peak stress and
113
dynamic elastic modulus are studied. The dynamic increase factor (DIF), i.e., the ratio of dynamic to static peak stress, is calculated for each test at different strain rates. The rock tested by using SHPB device should fail under the upper limit of strain rate. The stress– strain response of the rock if tested above the upper limit of strain rate would give erroneous result which can be explained by nonachievement of force equilibrium due to inertia effect. It has been reported in20 that the upper limit of the strain rate for a material to fail should be
ε̇1 =
εf c αL
(1)
where c is the elastic wave speed of the specimen, L is the length of the specimen and α is a non-dimensional parameter which depends on the shape of the incident pulse. Further, suitable correlation equations are proposed herein for changes in DIF with strain rate for all three rocks.
2. Laboratory tests performed on rocks In the present work, three rock types, i.e., quartzite, limestone and dolomite, are collected from the Vilaspur site for testing. The rock samples are prepared with a diamond bit core cutter of 38 mm. For all three rocks, the physical properties, e.g., both dry and saturated densities and specific gravity have been obtained. Static uniaxial compressive strength tests on dry and saturated rock samples have been carried out using the CONTROLS uniaxial compression and splitting test device for rock samples with aspect ratio (L/D) ¼2:1. 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 ¼0.5:1 and for the point load test, L/D ¼1:1 have been used. All static tests have been carried out following the specifications given in21,22. The dynamic tests are performed using a 38-mm diameter SHPB for all three rocks at different strain rate levels. 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, 3 and 4 for quartzite, limestone and dolomite, respectively. 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.
3. SHPB test setup Fig. 1 shows the schematic diagram of the compression SHPB test setup in the RelInc laboratory, Calumet, Michigan, USA. The setup is composed of an incident bar, a transmission bar and striker bars of different sizes. The bars are made of C300 maraging steel. The incident bar length is 2.59 m and the diameter is 38.1 mm. The transmission bar length is 2.43 m and the diameter is 38.1 mm. The dimensions of the incident and transmission bars
Table 1 Physical and static properties. Rocks
Dry density, ρd (kg/m3)
Saturated density, ρsat (kg/m3)
Specific gravity, G
Uniaxial compressive strength, σc (MPa)
Modulus of elasticity, Et (GPa)
Quartzite Limestone Dolomite
2585.841 2630.601 2731.820
2605.577 2664.206 2723.202
2.80 2.71 2.70
108.18 51.21 38.59
11.65 2.62 3.75
114
T. Chakraborty et al. / International Journal of Rock Mechanics & Mining Sciences 85 (2016) 112–118
Table 2 Dynamic properties: quartzite. Rock type Sample number
Sample length Striker bar (mm) length (mm)
Striker bar velocity (m/sec)
Strain rate, ε̇ (/sec)
Peak stress, σdc (MPa)
Strain at peak stress, ε
Modulus of elasticity, Et (GPa)
Dynamic increase factor, DIF
Quartzite
18.54 18.87 18.34 17.25 18.39 19.14 18.59 18.62 18.01 18.11 18.85
21.27 19.46 19.67 19.49 32.69 32.69 32.64 40.60 40.60 39.76 41.60
77 86 94 134 173 213 215 260 280 295 316
409.533 390.894 379.302 379.000 513.166 547.102 553.981 688.511 676.592 617.537 669.297
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
Sample length Striker bar (mm) length (mm)
Striker bar velocity (m/sec)
Strain rate, ε̇ (/sec)
Peak stress, σdc (MPa)
Strain at peak stress, ε
Modulus of elasticity, Et (GPa)
Dynamic increase factor, DIF
15.82 18.59 19.10 17.86 18.05 18.81 18.36 18.59 18.16 18.49 18.39
12.27 12.19 12.96 24.38 24.05 13.45 13.85 36.56 38.09 34.51 36.56
60 60 73 89.7 91 95 115 185 213 245 263
245.889 246.780 264.919 302.410 307.340 263.418 272.670 335.684 336.839 391.640 398.121
0.007 0.006 0.008 0.008 0.008 0.010 0.010 0.011 0.012 0.012 0.012
46.39 41.15 39.30 46.81 46.42 24.71 26.42 35.33 27.93 37.65 47.50
4.81 4.80 5.17 5.90 6.00 5.14 5.32 6.55 6.57 7.64 7.77
Q11 Q6 Q8 Q9 Q12 Q14 Q2 Q1 Q13 Q4 Q5
304.8
139.7
Table 3 Dynamic properties: limestone. Rock type
Sample number
Limestone L22 L17 L7 L11 L12 L9 L10 L6 L4 L5 L2
304.8
139.7 139.7 304.8 304.8 139.7
Table 4 Dynamic properties: dolomite. Rock type Sample number
Sample length Striker bar (mm) length (mm)
Striker bar velocity (m/sec)
Strain rate, ε̇ (/sec)
Peak stress, σdc (MPa)
Strain at peak stress, ε
Modulus of elasticity, Et (GPa)
Dynamic increase factor, DIF
Dolomite
19.02 18.90 19.05 18.31 19.09 19.08 18.58 18.29 19.23 18.38 18.15
20.54 20.55 20.32 20.31 29.50 27.71 24.70 29.50 30.47 35.19 38.10
49 69 76 76 125 132 132 144 174 176 201
395.882 398.760 313.150 396.700 342.170 403.280 326.366 433.338 434.888 403.388 421.930
0.006 0.008 0.006 0.008 0.008 0.008 0.007 0.009 0.014 0.008 0.014
80.79 63.29 65.51 77.78 47.39 45.51 46.75 47.51 31.46 57.46 30.97
10.25 10.33 10.27 8.11 8.86 10.45 8.45 11.22 11.26 10.45 10.93
D15 D3 D14 D12 D8 D10 D9 D5 D7 D1 D11
304.8
139.7
allow one-dimensional loading of the sample. In the present work, two different lengths of striker bars are used, 152.4 mm and 304.8 mm; the diameter of the striker bar is 38.1 mm. Compressed air is used to launch the striker bar onto the incident bar. The striker bar is propelled by compressed air from a gas gun at varying pressure magnitudes that 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
Δt =
2L s cbar
(2)
where Ls is the length of the striker bar, and cbar is the one-
dimensional longitudinal stress wave velocity in the bar. Thus, using a longer striker bar increases the loading time, and the rock sample gets more time to respond. As a result, a lower strain rate develops when the longer striker bar is used in comparison to a shorter striker bar for a particular pressure. It may be observed from the Tables 2, 3 and 4 that, for low strain rate levels, 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 shorter striker bars, the time taken by the stress wave to propagate from one end to the other end of the bar is less; hence, the loading time duration is also less for the same amount of the load intensity as compared to a longer striker bar, which results in a higher strain rate. When the diameters of the striker and input bar are same and pulse shaper is not used, then the shorter or longer striker bars will produce the same average strain rate if loaded under equivalent loading intensity as generated by the pressure released from gas gun. The strains in the incident and transmission bars are measured using two strain gauges, one mounted on the incident
T. Chakraborty et al. / International Journal of Rock Mechanics & Mining Sciences 85 (2016) 112–118
Trigger
115
Launcher/Accelerator
Striker Bar Speed Sensor Momentum Trap
Incident Bar
Specimen Impact Area
Control Cabinet
Control Panel Laptop Vishay
Transmission Bar Bar Stand
Data Acquisition Area
Deceleration Unit
Air/Storage Cabinet
Fig. 1. 3D Schematic diagram of split Hopkinson bar apparatus.
bar and the other mounted on the transmission bar. The strain gauges are 6.35 mm in length and 120 Ω in resistance, and the dynamic gauge has a tolerance of 70.35%. The strain gauges are attached to the incident and the transmission bars with M-bond adhesive. To read the strain signal, a Vishay 2310B signal conditioner and amplifier with a ¼ Wheatstone bridge has been used with a Picoscope5242 having sampling rate of 1 in 8 ns. A momentum trap, though included in the setup, has not been used in the present experiments. The impact of the striker bar on the incident bar causes a longitudinal elastic compressive stress wave that propagates through the incident bar. The strain pulse generated within the incident bar is designated as the incident strain pulse εi (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 the reflected strain pulse εr(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 designated as the transmitted strain pulse εt(t). The histories of the strain ε(t ), strain rate ε̇ (t ) and stress σ(t ) within the sample in the dynamic compression test are given by t
ε(t ) =
C L
ε̇ (t ) =
C ( εi − εr − εt ) L
(4)
σ(t ) =
A E ( εi + εr + εt ) 2A 0
(5)
∫0
( εi − εr − εt ) dt
(3)
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 prevail during dynamic loading, i.e., ϵi + ϵr = ϵt , the strain, strain rate and stress are given by t
ε(t ) = −
C L
ε̇ (t ) = −
C εr L
(7)
A E εt A0
(8)
σ(t ) =
∫0
εr dt
(6)
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 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 0.5.23,24 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.25,26 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.27 It was shown in28 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.
116
T. Chakraborty et al. / International Journal of Rock Mechanics & Mining Sciences 85 (2016) 112–118
Moreover, the incident and transmission bar ends of the specimen are prepared through rigorous grinding to a smoothness of approximately 4.1 mm. 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 ms. 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.
4. Results and discussion 4.1. Physical properties The dry density values of the quartzite, limestone and dolomite rocks are determined for five samples of each rock; the average values are determined to be 2585.84 kg/m3, 2630.60 kg/m3 and 2731.82 kg/m3, respectively. The saturated density values of quartzite, limestone and dolomite are also measured for five samples of each rock; the average values are determined to be 2605.57 kg/m3, 2664.21 kg/m3 and 2723.20 kg/m3, respectively. For all three rocks, the saturated density values are observed to be higher than the dry density values. The specific gravity values of quartzite, limestone and dolomite are estimated to be 2.80, 2.71 and 2.70, respectively. The density and specific gravity values of all three rocks are compared with the available data from the literature and observed to be in good agreement.29 4.2. Static uniaxial compressive strength The static uniaxial compressive strength of the rocks is determined under dry and saturated conditions and presented in Table 1. The tests for each rock type are repeated three times and the average strength values are compared.29 The static uniaxial compressive strength of quartzite, limestone and dolomite are found to be 108.18 MPa, 51.21 MPa and 38.59 MPa, respectively. The elastic modulus from the stress–strain graph at 50% of the peak stress value is calculated to be 11.65 GPa, 2.62 GPa and 3.75 GPa, respectively.30 It is to be noted that the static uniaxial compressive strength of quartzite is much more than that of limestone and dolomite because quartzite is a metamorphic rock and limestone and dolomite are sedimentary rocks.
Voltage V (V)
6 4
4.3. Dynamic stress–strain and force equilibrium The incident and transmission waveforms for quartzite, limestone and dolomite are presented in Fig. 2. The voltage in these waveforms is converted to strain by using the standard equation of Wheatstone bridge that converts the voltage output to strain using the gauge factor of the strain gauge.31 4.3.1. Stress–strain response for quartzite Fig. 3 shows the stress–strain plot for quartzite as obtained from SHPB tests for different strain rates that vary from 77/sec to 316/sec. For the strain rate range tested herein as seen in Fig. 3, the peak stress values obtained are 669.297 MPa at 316/sec and 409.533 MPa at 77/sec. Table 2 presents the values of peak stress, elastic modulus and the strain at peak stress obtained from all of the stress–strain curves for quartzite at different strain rates. It can be seen that the peak stress increases by almost 22% from strain rate 77/sec to 215/sec and by 37.7% from strain rate 134/sec to 316/ sec. 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 the rock is heterogeneous in nature and that, until this time, no standards have been defined for high loading rate dynamic tests of rocks. Hence, the dynamic elastic modulus of the rock is difficult to ascertain at present. 4.3.2. Stress–strain response for limestone Fig. 4 shows the stress–strain plots for limestone at different strain rates that vary from 60/sec to 263/sec. For the strain rate range tested herein as seen in Fig. 4, the peak stress values obtained are 398.121 MPa at 263/sec and 245.889 MPa at 60/sec. The peak stress, elastic modulus and strain rate at peak stress values obtained from the stress–strain curves for limestone are given in Table 3. It can be seen that the peak stress increases with an increase in the strain rate. The peak stress of limestone increases by 17.43% from strain rate 60/sec to 135/sec and by 16.14% from strain rate 135/sec to 263/sec. It may be observed from Table 3 that the elastic modulus does not show any clear trend under strain rate dependent loading. 4.3.3. Stress–strain response for dolomite Fig. 5 shows the stress–strain plots for dolomite at different strain rates that vary from 49/sec to 201/sec. For the strain rate range tested herein as seen in Fig. 5, the peak stress values obtained are 421.930 MPa at 201/sec and 395.882 MPa at 49/sec. Table 4 shows the peak stress, elastic modulus and strain at peak stress obtained from the stress–strain curves for dolomite. It can be seen that, similar to quartzite and limestone, the peak stress increases with an increase in the strain rate. The peak stress increases by 0.04% from strain rate 49/sec to 125/sec and by 11.6% from strain rate 125/sec to 201/sec. It can be observed from Table 4
Limestone: L15 Incident wave Transmission wave
Quartzite: Q14 Incident wave Transmission wave
Dolomite: D8 Incident wave Transmission wave
2 0 -2 -4 -6 0.0
0.4
0.8
1.2
1.6
0.0
0.4
0.8 1.2 Time t (ms)
1.6
Fig. 2. Waveform for incident and transmission bar.
0.0
0.4
0.8
1.2
1.6
T. Chakraborty et al. / International Journal of Rock Mechanics & Mining Sciences 85 (2016) 112–118
117
D y n a m i c In c re a s e F a c t o r
12 9 6 3 0
Fig. 3. Stress–strain response for quartzite.
DIF for Quartzite Equation trendline for Quartzite, DIF = 0.012(ε) + 2.48, R = 0.89 DIF for Limestone Equation trendline for Limestone, DIF = 0.013(ε) + 4.19, R = 0.89 DIF for Dolomite 0
100
200
Strain rate ε (/sec)
300
Fig. 7. Dynamic increase factor with strain rate for all the three rocks.
specimen's incident and transmitted sides, the loading on the specimen is one-dimensional and the specimen deforms uniformly which satisfies one dimensional wave propagation theory.
Fig. 4. Stress–strain response for limestone.
Fig. 5. Stress–strain response for dolomite.
that the elastic modulus does not show any clear trend with changes in dynamic strain rates. 4.3.4. Force equilibrium Fig. 6 presents the force equilibrium achieved at the incident and the transmission bar ends of the specimen in the SHPB tests for typical specimens of quartzite, limestone and dolomite, respectively. It may be noted that for all of the strain rate values, force equilibrium is achieved nicely and the peak strength is achieved while the sample is in stress equilibrium signifying the peak compressive strength of the rock to be correct for the particular strain rate tested. Thus, the dynamic increase factor is calculated specifying the correct increase in strength at a particular strain rate.32 When there is force equilibrium between the
4.3.5. Dynamic increase factors and proposed correlation equations The dynamic increase factors (DIF) for all three rocks have been determined by comparing the dynamic peak stress with the static peak stress. The DIF values of the rocks are reported in Tables 2, 3 and 4. The DIF values are also plotted in Fig. 7 for quartzite, limestone and dolomite. From Fig. 7, it may be seen that the dynamic strength of quartzite is 3.5 to 6.36 times that of the static strength for strain rates varying from 77/sec to 316/sec. The dynamic strength of limestone is approximately 4.8 to 7.77 times that of the static strength for a strain rate range of 60–263/sec. And the dynamic strength of dolomite varies from 8.11 to 11.26 times that of the static strength for a strain rate range from 49/sec to 201/sec. Two correlation equations for quartzite and limestone have been developed for the calculated DIF with respect to the strain rate by calculating a best-fit curve through the obtained DIF values that satisfies the 95% confidence interval. The correlation equation for quartzite with a coefficient of determination (R2) ¼0.89 is presented in Fig. 7 and given by
DIF = 0.012 ( ε̇) + 2.48 for 77/sec ≤ ( ε̇) ≤ 316/sec
(9)
The correlation equation for limestone with a coefficient of determination (R2)¼0.89 is presented in Fig. 7 and given by
DIF = 0.013 ( ε̇) + 4.19 for 60/sec ≤ ε̇ ≤ 263/sec
(10)
The dynamic increase factor for dolomite is observed to be approximately constant for the particular strain range considered herein. Hence, the correlation equation with respect to strain rate does not show any particular trend. It may be noted that the DIF equations proposed herein will be applicable for the strain rate ranges considered in the current work. The DIF correlations thus developed for the given strain rate range can be used as material model in dynamic analysis of underground structures and developing blast resistant designs.
5. Conclusions
Force F (kN)
800 Quartzite
600 Limestone
400 200
Transmission bar end 213/sec, Q14, Quartzite 135/sec, L16, Limestone 144/sec, D5, Dolomite
Dolomite
0 0
40 80 Time t (µ s)
Incident bar end 213/sec, Q14, Quartzite 135/sec, L16, Limestone 144/sec, D5, Dolomite
120
Fig. 6. Force equilibrium achieved for all the three rocks tested.
Characterizations of three Himalayan rocks, quartzite, limestone and dolomite, have been performed in the present work for a strain rate range varying from 49/sec to 316/sec through uniaxial compression using a 38 mm diameter SHPB test setup. 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 behavior of the rocks are also investigated. The following conclusions are drawn from the tests. For quartzite, the peak stress increases with increase in strain rate. The elastic modulus of quartzite does not show any particular
118
T. Chakraborty et al. / International Journal of Rock Mechanics & Mining Sciences 85 (2016) 112–118
trend with increasing strain rate. The dynamic increase factor varies from 3.5 to 6.36. For limestone, the peak stress increases with increase in strain rate. The elastic modulus does not show any clear trend with an increase in strain rate. The dynamic increase factor varies from 4.8 to 7.77. For dolomite, the peak stress increases with increase in strain rate. The elastic modulus does not show any clear trend with an increase in strain rate. The dynamic increase factor varies from 8.11 to 11.26 times of the static compressive stress. Due to clay content in limestone and dolomite, its dynamic increase factor is more than quartzite.
Acknowledgments This work is a part of an ongoing research project with research grant number ARMREB/CDSW/2013/151 funded by the Terminal Ballistics Research Laboratory (TBRL), Chandigarh under Defense Research and Development Organization (DRDO), India. The authors acknowledge the funding provided by TBRL in this work.
References 1. Ngo T, Mendis P, Gupta A, Ramsay J. Blast loading and blast effects on structures - an overview. Electron J Struct Eng. 2007:76–91. 2. Dusenberry DO. Handbook for blast resistant design of buildings. New York: Wiley; 2010. 3. Perkins RD, Green SJ. Macroscopic description low and medium strain rates. Int J Rock Mech Min Sci. 1970;7:527–535. 4. Christensen RJ, Swanson SR, Brown WS. Split Hopkinson bar tests on rock under confining pressure. Exp Mech. 1972;12:508–541. 5. Lindholm US, Yeakley LM, Nagy A. The dynamic strength and fracture properties of dresser basalt. Int J Rock Mech Min Sci Geomech Abstr. 1974;11:181–191. 6. Lundberg B. A split Hopkinson bar study of energy absorption in dynamic. Int J Rock Mech Min Sci Geomech Abstr. 1976;13:187–197. 7. Blanton TL. Effect of strain rates from 10 2 to 10 s 1 in triaxial compression tests on three rocks. Int J Rock Mech Min Sci Geomech Abstr. 1981;18(1):47–62. 8. Klepaczko JR. Behavior of rock-like materials at high strain rates in compression. Int J Plast. 1990;6:415–432. 9. Olsson WA. The compressive strength of tuff as a function of strain rate from 10-6 to 103/sec. Int J Rock Mech Min Sci Geomech Abstr. 1991;28(1):115–118. 10. Grady DE. Shock wave properties of brittle solids. In: Schmidt S, ed. Shock Compression of Condensed Matter. New York: AIP Press; 1995:9–20. 11. Zhao J, Li HB, Wu MB, Li TJ. Dynamic uniaxial compression tests on a granite. Int J Rock Mech Min Sci. 1999;36:273–277.
12. Frew DJ, Forrestal MJ, Chen W. A split Hopkinson pressure bar technique to determine compressive stress-strain data for rock materials. Exp Mech. 2000;41 (1):40–46. 13. Li XB, Lok TS, Zhao J, Zhao PJ. Oscillation elimination in the Hopkinson bar apparatus and resultant complete dynamic stress-strain curves for rocks. Int J Rock Mech Min Sci. 2000;37(7):1055–1060. 14. Fukui K, Okubo S, Ogawa A. Some aspects of loading-rate dependency of sanjome andesite strengths. Int J Rock Mech Min Sci. 2004;41(7):1215–1219. 15. Li XB, Lok TS, Zhao J. Dynamic characteristics of granite subjected to intermediate loading rate. Rock Mech Rock Eng. 2005;38(1):21–39. 16. Qi CZ, Wang MY, Qihu Q. Strain-rate effects on the strength and fragmentation size of rocks. Int J Imp Eng. 2009;36:1355–1364. 17. Fuenkajorn K, Kenkhunthod N. Influence of loading rate on deformability and compressive strength of three Thai sandstones. Geotech Geol Eng. 2010;28:707– 715. 18. Ming L, Xian-biao M, Li-li C, Rong-rong M, Guang-hui Z. High strain rate effects on mechanical behavior of coal-serial sandstone. Electron J Geol Eng. 2014;19:6035–6046. 19. Lu YB, Li QM, Ma GW. Numerical investigation of the dynamic compressive strength of rocks based on split Hopkinson pressure bar tests. Int J Rock Mech Min Sci. 2010;47(5):829–838. 20. Ravichandran G, Subhash G. Critical appraisal of limiting strain rates for compression testing of ceramics in a split Hopkinson pressure bar. J Am Ceram Soc. 1994;77:263–267. 21. ISRM. International society of rock mechanics commission on testing methods, suggested methods for determining the uniaxial compressive strength and deformability of rock materials. Int J Rock Mech Min Sci Geomech Abstr. 1979;16 (2):138–140. 22. ISRM. International society of rock mechanics commission on testing methods, suggested method for determining point load strength. Int J Rock Mech Min Sci Geomech Abstr. 1985;22:51–60. 23. Davis EDH, Hunter SC. The dynamic compression testing of solids by the method of the split Hopkinson bar. J Mech Phys Solids. 1963;11:155–179. 24. Zhou Z, Li X, Liu A, Zou Y. Stress uniformity of split Hopkinson pressure bar under half-sine wave loads. Int J Rock Mech Min Sci. 2011;48(4):697–701. 25. Wang ZG, Meyer LW. On the plastic wave propagation along the specimen length in SHPB test. Exp Mech. 2010;50:1061–1074. 26. Abotula S., Shukla A., Chona R. Dynamic constitutive behavior and fracture initiation toughness of hastelloy x under thermo-mechanical loads. In: Proceedings of the society of experimental mechanics, implast conference. WrightPatterson Air Force Base RBSM. 2010. 27. Frew DJ, Forrestal MJ, Chen W. Pulse shaping techniques for testing brittle materials with a split Hopkinson pressure bar. Exp Mech. 2002;42(1):93–106. 28. Xia K, Yao W. Dynamic rock tests using split Hopkinson (Kolsky) bar system – a review. J Rock Mech Geotech Eng. 2014;7:27–59. 29. Kumar R. Testing and constitutive modelling of the strain-softening behaviour of some rocks [Ph.D. Thesis]. Delhi: Indian Institute of Technology (IIT); 2007. 30. ASTM D3148-02. Standard test method for elastic moduli of intact rock core specimens in uniaxial compression. Am Soc Test Mat; 1998. 31. Chen W, Song B. Split Hopkinson (Kolsky) Bar – Design, Testing and Applications. Berlin: Springer; 2011. 32. Li XB, Zou Y, Zhou ZL. Numerical simulation of the rock SHPB test with a special shape striker based on the discrete element method. Rock Mech Rock Eng. 2014;47(5):1693–1709.