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Obtaining dynamic complete stress–strain curves for rock using the Split Hopkinson Pressure Bar technique
International Journal of Rock Mechanics and Mining Sciences 37 (2000) 983±992
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Obtaining dynamic complete stress±strain curves for rock using the Split Hopkinson Pressure Bar technique Renliang Shan*, Yusheng Jiang, Baoqiang Li Civil Engineering, China University of Mining & Technology, Beijing, 100083, People's Republic of China Accepted 5 April 2000
Abstract The feasibility of using the Split Hopkinson Pressure Bar (SHPB) technique to obtain complete dynamic stress±strain curves for rock is established in the laboratory. The SHPB test system, in conjunction with a mean strain hypothesis, can be used not only for obtaining the rock's constitutive curve before the peak strength but also after the peak strength Ð and so it is possible to analyze and characterize the post-failure behaviour of rock with the SHPB method. Some typical complete dynamic curves for marble and granite are given in this paper, together with an interpretative discussion on the shapes of the curves. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Rock; Dynamic; SHPB method; Complete stress±strain curve
1. Introduction SHPB is the acronym for `Split Hopkinson Pressure Bar', which is an important apparatus for studying the dynamic characteristics of metal, polymers, bone material, concrete, ore, synthetic ®bers and other solids (Fig. 1). The SHPB technique originated from the HPB (Hopkinson Pressure Bar), which was designed by Hopkinson in 1914, and used in testing the shape of a stress pulse wave; the pulse signal was tested by the short column `timer', so it could only read the time and the peak value roughly, but could not measure the pulse shape accurately. Davies recorded continuous longitudinal displacement at the free end by a capacitor displacement sensor in 1948. Kolsky ®rst divided the pressure bar into two parts and put a small sample between them, and measured the pulse signal and displacement by a capacitor displacement sensor in 1949 [1]. The SHPB set, which is widely used today, has no * Corresponding author. E-mail address: [email protected] (R. Shan).
dierence from the 1949 version in principle. In order to overcome the restrictions of the measuring system using a capacitor displacement sensor, Lindholm [2] applied strain gauges in place of the capacitor sensor to measure the signals on the surfaces of two bars in 1963, which played an important role in enhancing the use of the SHPB method. From then on, many scholars have extensively studied a variety of problems using the SHPB, such as: sample size, friction eect of the ends, data processing methods, wave dispersed characters and transverse inertia eect, etc. [2±5]. Compared with metals, polymers and other manmade materials, the study of the dynamic characteristics of using the SHPB apparatus was late. Kumar ®rst researched the in¯uence of loading ratio and temperature on the strength of granite and basalt with a SHPB apparatus in 1968 [6]. An experiment incorporating compressive pressure on the rock sample was carried out by Cristensen in 1972 [7]. In 1976, Goldsmith's study showed that both compressive and tensile moduli and strengths are clearly related to loading rate [8]. Chinese scholars have carried out a great deal of research on both the SHPB experimental technique and the dynamic character of rock since 1980 [9±12].
1365-1609/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 1 3 6 5 - 1 6 0 9 ( 0 0 ) 0 0 0 3 1 - 9
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R. Shan et al. / International Journal of Rock Mechanics and Mining Sciences 37 (2000) 983±992
From the references cited, it is found that most of past studies into the rock dynamic characteristics paid more attention to the elastic modulus and compressive strength. That is to say, only the elastic character before the peak strength was researched: both the non-linear and stepwise characteristics of loading were seldom studied. Moreover, reports on the study of the post-failure region of the dynamic stress±strain curve were rarely found. There are some causes for this scenario. First, in the SHPB method, the calculation of stress, strain and other parameters require that the sample is continuous, uniform and isotropic. Second, the transverse inertial in¯uence of the sample and the eect of interface friction between the sample and the bars can be neglected. Because these assumptions were based on the theory of one-dimensional stress waves, the majority of researchers, intentionally or unconsciously, avoided the post-failure portion of the complete stress±strain curve when they studied the rock's dynamic character. According to strict principles, we almost cannot measure the rock's dynamic character using the SHPB technique. In order to overcome the diculties, some scholars propose that stress wave attenuation should be taken into account as the wave passes through a fractured rock sample Ð if its amplitude exceeds the initial failure stress of the rock sample. Gende Zhang had studied such a mechanical model of stress wave attenuation in detail, and proposed to calculate the stress s at the middle section [10] with stresses s1 and s2 measured at the two ends and using an exponent attenuation model, and with the hypothesis of directly measured strain at the middle section as the true strain E. Also, he suggested that the dynamic constitutive relation should be made up of the above stress s and strain E, but only the part before the peak strength of the rock's stress±strain curve was given in his articles; it appears that his model only suits the initial rock damage. It is shown in this paper, that the SHPB test system and the related mean strain hypothesis can be used not
only for the analysis of the mechanical behavior of rock before the peak strength, but also for that of the post-failure region. This requires the use of two comparisons: the ®rst is the comparison between the `mean-strain' calculated from the strains measured by strain gauges on the incident and transmission bars and the axial strain obtained directly from a strain gauge on rock sample; the second is the comparison between the input bar's end stress s1 and the output bar's end stress s2. Then, the SHPB test system in this more developed form can be used for determination of complete dynamic stress±strain curves for rock. 2. Experimental method and analysis of the tested wave shape 2.1. Experimental system The SHPB apparatus is the core of the testing system (see Fig. 1). It consists of four components. 1. The power supply component, made up of nitrogen bottle and chamber. 2. The generating and transferring loads component, made up of the rock sample, striking bar, input bar and output bar. 3. The striking velocity measuring component, made up of spotlight, photoelectric diode, ampli®er and counter. 4. The strain measuring component, made up of strain gauges and ultra-dynamic apparatus as well as the dynamic test and analysis equipment of CS2092. The diameters of the striking bar, input bar and output bar are 30 mm with their lengths of 500, 1500 and 1400 mm, respectively. Marble and granite were selected as the sample material with static parameters of compressive strengths 35±45 MPa and 186±218 MPa, and elastic moduli of 5.3±6.5 104 MPa and 6±9 104 MPa, and Poisson's ratios of 0.19±0.22 and 0.25±0.30, respectively. Using the experience from previous work, we made samples with three dierent ratios of length L to diameter D: 50 samples for L: D = 1 (L = 30 mm), 40 samples for
Fig. 1. The uniaxial SHPB testing system.
R. Shan et al. / International Journal of Rock Mechanics and Mining Sciences 37 (2000) 983±992
L:D = 1.5 (L = 45 mm) and 10 samples for L:D = 2 (L = 60 mm). 2.2. Experimental principle
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As1 A0 E0 EI ER
5
As2 A0 E0 ET
6
It is supposed that the mean value of stresses from the two ends of rock sample can be regarded as the stress in the whole sample, because the rock sample is short compared with the striking bar. So we can obtain the following:
In the SHPB experiments, high pressure gas provided by a nitrogen bottle expands in a chamber, and pushes and accelerates a striking bar which moves forward to strike the input bar at a certain speed and produces an input wave EI in the input bar. When the wave EI reaches the interface 1-1 (Fig. 2), two waves are produced, one part is re¯ected back along the input bar and produces the re¯ected wave ER (Fig. 2) and the other wave moves forward and reaches the output bar through the sample and produces the transmission wave ET, (Fig. 2). The strain pulse signals are collected via strain gauges and transformed into electric signals through ultra-dynamic strain equipment, and then transferred into dispersed signals and stored in recorded form through the CS2092. All these recorded dispersed signals are analyzed all together on completion of the tests. Based on wave propagation theory and with a onedimensional stress hypothesis, as well as the continuity demands of displacement, we can compute the velocities at the interfaces 1-1 and 2-2 shown in Fig, 2, respectively as:
Now, the rock sample's constitutive relation s ÿ E ÿ E_ can be determined from the input wave and the re¯ected wave as well as the output wave Ð as actually measured in the experiments Ð from Eqs. (3), (4) and (7). According to the homogeneous supposition of stress, there should be s1 s2 or EI ER ET , and substituting them into Eqs. (3), (4) and (7), we can then obtain:
v1 C0 EI ÿ ER
1
s
v2 C0 ET
2
A0, E0 and C0 refer respectively to sectional area, elastic modulus and longitudinal wave velocity of the input bar or output bar, A and L are the sectional area and length of rock sample respectively, and s1,
The strain of the rock sample generated in a unit time (strain rate) is: E_
v1 ÿ v2 C0 EI ÿ ER ÿ ET L L
s
s 1 s2 A0 E0 EI ER ET 2 2A
E_ ÿ
2C0 ER L
2C0 Eÿ L
t 0
8
ER dt 0
A0 E0 ET A
9
10
3
So, the calculated strain of the rock sample in a certain period of time is: t C0 t 0 EI ÿ ER ÿ ET dt 0 4 E E_ dt L 0 0 Also, according to Newton's third law, the following two equations apply at interfaces 1-1 and 2-2, respectively:
Fig. 2. Re¯ected and transmitted waves in the SHPB apparatus.
7
Fig. 3. Tested data processing chart.
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R. Shan et al. / International Journal of Rock Mechanics and Mining Sciences 37 (2000) 983±992
s2, s represent stresses at interfaces 1-1 and 2-2, and the mean stress, respectively. Thus, the transmitted wave describes the change of stress in the sample, and the re¯ected wave determines the change of the sample's strain. 2.3. Data processing Because the signals coming from the strain testing system are voltage signals, not the actual strain signals we need, we must ®nd the relation between the voltage signals and strain signals, i.e. calibrate the testing system. In these experiments, the calibration was adopted for the strain gauges on the sample, and for strain gauges on the input and output bars. After accurate calibration, the electric signal array can be transferred into the strain signal array. And then, we can use the above-mentioned formulae to compute the rock sample's dynamic character. The procedure for the experimental data processing is shown in Fig. 3. 2.4. Typical measured waves and associated analyses Fig. 4 illustrates typical strain waves measured in
the SHPB apparatus. The thick and thin lines in Fig. 4 represent the strain waves from the input bar and output bar respectively. The rock sample for this case was 45 mm in length with a diameter of 30 mm. Fig. 4(a) represents the testing result for marble with a striking velocity of 7.21 m/s, the rock sample remaining whole but with many visual criss-cross crackles on its surface. Fig. 4(b) is also the testing result for marble, this time with a striking velocity of 8.97 m/s, the rock sample being broken into six pieces and some detritus after being struck. Fig. 4(c) shows the testing result for granite with a striking velocity of 10.69 m/s, the sample remaining intact with no visual cracks on its surface. Fig. 4(d) also shows the testing result for granite, this time with a striking velocity of 12.47 m/s, and the sample was broken into six small fragments and some detritus. From Fig. 4, we can see that the input wave is approximately rectangular, in accordance with the onedimensional theory of stress wave propagation. But the re¯ected wave and transmitted wave are closely related to the striking velocity, and especially to the degree of damage of the rock sample. For a complete sample, the re¯ected wave is changed into compressive strain from tensile strain by way
Fig. 4. Typical strain waves measured in the SHPB apparatus.
R. Shan et al. / International Journal of Rock Mechanics and Mining Sciences 37 (2000) 983±992
of the zero strain platform (no re¯ection), the duration of the peak of the transmitted wave is relatively long, and the transmitted wave is similar to the input wave. For a broken sample, the re¯ected wave is only a tensile strain wave with a `W' shape, its two valleys, especially the second one, can indicate the degree of specimen degradation well. The deeper the second valley is, the greater the damage extent is. The transmitted wave will decrease along an approximately straight line after reaching the peak, and the decreas-
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ing rate is related to the loading magnitude and the degree of damage to the rock sample. For a critical sample (integral sample but with many tiny cracks on its surface), the re¯ected wave represents mostly tensile strain with a small fraction of compressive strain at the rear, so it shows a skewed `W' shape. The transmitted wave usually has a short region of small, sharp decline at the main rising stage, indicating that micro-fractures have occurred in the sample.
Fig. 5. Comparison of the strains measured directly on the rock samples with the average strains calculated from the strain waves measured on the input and output bars.
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R. Shan et al. / International Journal of Rock Mechanics and Mining Sciences 37 (2000) 983±992
3. Feasibility of determining the complete stress±strain curve with the SHPB technique 3.1. Comparison between `average strains' and strains measured directly on the rock sample The preliminary results show that the stresses at the two ends of the rock sample with 30 mm or 45 mm length accord with each other, and that of the sample with 60 mm length have a discrepancy. So, experimental results from the 60 mm long samples are excluded. The comparisons of strain (Fig. 5) measured directly on the surface of the rock sample (solid line) with `average strain' calculated from the strain waves measured on the input and output bars (dotted line) can be seen (Fig. 5). The experimental conditions in Fig. 5 are the same as those in Fig. 4 for each plot in the ®gure. We can reach the following conclusions from Fig. 5, plus the related fracture modes of the rock samples. (1) The striking velocities for marble samples are all lower than those for granite samples, while the strain amplitudes for marble samples are larger than those for granite. This shows that marble is easier to deform than granite; that is to say, marble is softer than granite. (2) The amplitude of `average strains' is closely related to the extent of damage of the same rock; the higher the damage extent, the larger is the `average strain' is. While the strain measured directly from the rock sample is completely dierent, it has little relation with the degree of fracturing of the rock. (3) Generally, the rock dynamic deformation behavior can be divided into three stages: . The ®rst stage is before the ®rst `knee point'. During this stage, the two curves coincide with each other, especially when the primary strain is smaller than 4000 mm/m; they can almost be superposed on each other, indicating that rock has good elasticity at this stage. . The second stage is from the ®rst `knee point' to the peak point. The slope of this stage is smaller than that of the ®rst stage. The two curves have a certain divergence, and the `average strain' is usually slightly lower than the directly measured strain, indicating that strain hardening has taken place at this stage. . The third stage is the post-failure stage. In this stage, the two curves for the only granite sample that remained almost intact after the impact have good coincidence, and the other sample's curves are dierent. The `average strain' is much higher than the directly measured strain in this stage. Why did the strains in the post-peak region obtained from these two methods have a such divergence? The following two explanations are possible. One is the
stress wave attenuation theory mentioned earlier. It is considered that the stress wave does not attenuate when its amplitude is low; while, if the amplitude of the stress wave is higher than the primary fracture stress, micro-cracks will be produced in the rock sample by the stress wave, and these micro-cracks lead to stress wave attenuation according to a negative exponential trend along the sample. Therefore, in the rock sample, the stress sr at any section is less than the `average stress s', see Fig. 6. This attenuation theory is reasonable, and it can be deduced that the stress at the output end is less than that at the input end with this idea. The validity was not con®rmed by our experiments (see Fig. 7 and the analyses in the next paragraph). Therefore, we have to consider the other explanation. The reasoning in this second explanation is that the attenuation theory does not consider the increasing stress resulting from the re¯ecting compression when the stress wave reaches the output end through the sample, because the wave resistance of the rock is less than that of steel bar. The failure strain of rock samples under impact often exceeds 6000 mm/m, sometimes even reaching to 10,000 mm/m. Thus, the dynamic failure strain exceeds the quasi-static failure strain for uniaxial compression. For such ultra dynamic strain, together with the micro-structural inhomogeneity as well as the non-linearity of the rock response, there may be some doubt about the result measured from the ordinary strain gauges stuck directly on the sample. Because ®rst, the utmost strain for ordinary strain gauges used in static measuring is from 6000 to 8000 mm/m with the requirement that the error is less than 10%. Second, given that the rock sample is non-linear and non-homogenous, it is possible that the rock at the spot where a strain gauge is located is not broken, while the integrity of the rock sample is actually violated: this makes the whole strain after the peak more than the directly measured strain at some location on the broken sample.
Fig. 6. Stress attenuation model for rock sample.
R. Shan et al. / International Journal of Rock Mechanics and Mining Sciences 37 (2000) 983±992
In fact, the strain gauges glued directly on the surface of a rock sample cannot be used to measure the sample's deformation after the peak, as is the case in quasi-static testing. First, many directly measured strains on broken samples change with time, similar to the solid line in Fig. 5(d), and the compressive strain often suddenly drops to negative values when it slowly goes down to the point about two thirds of the peak value after the peak, causing the strain gauge to be pulled o. In impact compressive experiments for rock, the axial tensile strain takes place only when swelling occurs, which usually occurs with the conditions of high con®ning pressure and friction at the
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end. The experiments described in this paper are of uniaxial impact and the two ends of the sample were smeared with butter, so there were no conditions of swelling, and the axial tensile strain is a false appearance. Second, the axial compressive strain is measured only after the peak is declining and its amplitude is not clearly related to the broken state of the sample. The declining phenomenon is false, because it is apparent that the strain may increase when breakage becomes worse after the peak strength of the rock. Therefore, there is no doubt that the strain gauges glued directly on the sample cannot be used to measure the sample's post-failure deformation.
Fig. 7. Comparison of stress waveforms at the two ends of the rock sample.
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R. Shan et al. / International Journal of Rock Mechanics and Mining Sciences 37 (2000) 983±992
The SHPB experimental method overcomes this dif®culty of directly measuring strain by strain gauges stuck on the rock samples. What the SHPB re¯ects is the deformed and broken character of a whole sample, not that of only a part of a sample, as in quasi-static tests to obtain the complete stress±strain curve. And the SHPB apparatus can indicate large deformations produced in broken rock by processing the small elastic strain waves in the input and output bars. From Eq. (4), it can been seen that the `average strain' of the rock sample is not only dependent upon the strains measured on the input and output bars, but is also dependent upon the steel bars' longitudinal wave velocity and sample length. From our tests, it can be seen that a measured strain of 1000 mm/m on the two steel bars can represent a strain of 8000±10000 mm/m of rock sample. Therefore, the SHPB method is an eective means to test the rock's broken and deformed character in the post-failure region, and hence can be used to establish rock deformation behavior not only before the peak, but also after the peak. In order to demonstrate this viewpoint further, consider a comparison of stress at the input end with that at the output end.
the input end. This indicates that the stress attenuation theory has insucient evidence to support it for the SHPB experimental calculation, and that the `mean stress' method or homogeneous supposition is more in accord with the experimental evidence. Thus, both strain analysis and stress comparison con®rm that the SHPB apparatus and the corresponding `mean stress' calculating model can be used to measure and analyze the complete stress±strain curve obtained under dynamic conditions. 4. Measured dynamic complete stress±strain curves for marble and granite
The change of stresses s1 and s2 at the two ends with time is shown in Fig. 7, the experimental condition for each plot being the same as that in Fig. 4. In Fig. 7, the solid line represents the stress wave at the input interface, and the dotted line represents the stress wave at the output interface. It is clear that, no matter what the rock is and whether the rock sample is broken or not, the stress wave forms at the two ends of each sample have good coincidence, except for deviations near the `leap points'; and, in most cases, the stresses at the output end are slightly larger than at
Figs. 8 and 9 show typical dynamic complete stress± strain curves for marble and granite (L = 30 mm) measured and calculated using the SHPB apparatus and the corresponding mean stress method mentioned above. In our experimental results (for impact speed in the range of 6.2±19.6 m/s with a strain ratio in a range of 100±600 sÿ1), most of the stress±strain curves of granite before utmost points have jagged characteristics. When the curves reach to the ®rst maximal point, the stress ®rstly goes down to a minimal point, and then goes up to the utmost peak. Before the ®rst peak, no matter what the striking speed or load rate is, the curve is approximately a straight line, and the deviation between the lines is small, showing that granite at this strain stage has good linear elasticity. And the line slope (elastic modulus, E ) is in the range of 0.4± 1.0 105 MPa. After the ®rst maximal points, and especially after the utmost peaks, the stress±strain curve is closely related with the striking speed and broken state, and the curve's shape is variable. When the striking speed is low, and the sample remains whole after being struck,
Fig. 8. Complete dynamic stress±strain curves for granite.
Fig. 9. Complete dynamic stress±strain curves for marble.
3.2. Stress comparison at input and output interfaces
R. Shan et al. / International Journal of Rock Mechanics and Mining Sciences 37 (2000) 983±992
the stress±strain curve after the utmost peak will rebound at once, shown by the dot-dash curve in Fig. 8. When the striking speed is high, along the stress±strain curve after the utmost peak, the strain increases continuously with stress and the ability of withstanding load decreases, even when the stress decreases to zero. This shows that the granite sample was broken thoroughly and began to be detached from the two bars at this time. The moving out of the rock sample chips causes the interfaces between sample and steel bar to be free and so the sample's `strain' is the result of re¯ection of the input wave on the free end; therefore, it is an artifact. This situation is shown by the solid line curve in Fig. 8. When the striking speed is in the medium range, and the sample is already damaged after being struck, but not seriously broken into pieces, the strain after the peak increases at ®rst, and then decreases: this indicates that the sample has some rebounding characteristics at the late stage of deformation. This situation is shown by the dotted curve in Fig. 8. The complete stress±strain curve of marble (Fig. 9) usually rises in a line with positive slope in the ®rst part. In the post-failure region, the curve usually descends with a negative slope at ®rst, then falls in a steep line. It diers markedly from that of granite. Firstly, the slope of the ®rst part (elastic modulus, E ) is related with the striking speed or strain rate to some degree. The larger the strain ratio is, the larger the elastic modulus is, and the marble's dynamic elastic modulus in this experiment is in the range of 1.8±5.0 104 MPa. Secondly, there is an approximate horizontal line between the ®rst part and the post failure region, indicating the plastic character of marble after being struck. Finally, the post-failure region of the marble's stress±strain curve only has the ®rst kind of unloading (slope is negative) and `rigid unloading' (slope is in®nite); but the second kind of unloading (slope is positive) does not occur. In a word, and just as for the rock's quasi-static constitutive curve, the post-failure region is more complex than that of the pre-failure part.
5. Conclusions The SHPB apparatus and the corresponding homogeneous stress supposition can be used to measure and analyze not only the ®rst stage but also the post-failure region of a rock's dynamic complete stress±strain curve. The technique can be used to measure the dynamic character for elastic homogeneous materials and fractured rocks, as has been shown by our experiments. So we hope researchers in this ®eld will join
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together to enrich and develop the SHPB experimental method. The dynamic complete stress±strain curve of granite before the utmost peak has a clear jagged character. It is usually is an approximately straight line in the primary stage, and the slope of the straight line is not apparently related to the strain rate. Both Class I (monotonically increasing in strain) and Class II curves (not monotonically increasing in strain) exist in the post-failure region, and the curves in this stage can vary widely. Usually, the dynamic complete stress±strain curve for marble has three stages. The slope of the straight line (elastic modulus) at the beginning is related to the strain rate to some extent; the higher the strain rate is, the larger the modulus is. The post failure curve only decreases with a negative slope or in a steep line and has no rebound elasticity. The approximate horizontal line in the middle of the curve indicates the marble's plastic character after being impacted.
Acknowledgements The project was supported by the National Science Foundation of China (No. 49502041). The experiments were carried out at Beijing University of Science and Technology. The authors would like to acknowledge Professor Z. X. Zhang for helping with the tests.
References [1] Wang Lili, Yu Tongxi, Li Yongci [In Chinese]. In: Development of impact dynamics. Hafei: Publishing House of China University of Science and Technology, 1992. p. 379±413 [In Chinese]. [2] Follansbee PS, Frantz C. Wave propagation in the Split Hopkinson pressure bar. Journal of Engineering Materials and Technology, Transactions of the ASME 1983;105(1):61±6. [3] Briscoe BJ, Nosker RW. The in¯uence of interfacial friction on the deformation of high density polyethylene in a Split Hopkinson pressure bar. Wear 1984;95(3):241±62. [4] Jenkins DA. Dispersion investigation in the Split Hopkinson pressure bar. Journal of Engineering Materials and Technology, Transactions of the ASME 1990;112(3):309±14. [5] Leber H. Data processing in the Split Hopkinson pressure bar tests. International Journal of Impact Engineering 1994;15(6):723±33. [6] Kumar A. Eect of stress rate and temperature on the strength of basalt and granite. Geophysics 1968;33(3):501±10. [7] Cristensen RJ, Swanson SR, Brown WS. Split Hopkinson bar tests on rock under con®ning pressure. Experimental Mechanics 1972;12(11):508±13. [8] Goldsmith W, Sackman JL, Ewert C. Static and dynamic fracture strength of Barre granite. Int J Rock Mech Min Sci, & Geomech Abstr 1976;13:303±9. [9] Li Xibing. The relationship s ÿ E_E of rock under dierent loading waves. The Chinese Journal of Nonferrous Metals 1994;4(3):16±8 [in Chinese].
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[10] Zhang Gende. Dynamic response of rock to impact loading. Explosion and Shock Waves 1982;2(2):1±9 [in Chinese]. [11] Lu Yueping, et al. Measuring fracture stress and Yang's modulus of sandstone and limestone use Hopkinson pressure bar.
Chinese Journal of Geotechnical Engineering 1983;5(3):28±37 [in Chinese]. [12] Yu Yalun, Jing Kexue. Study on characteristics of minerals. Explosion and Shock Waves 1990;10(3):266±71.
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Obtaining dynamic complete stress±strain curves for rock using the Split Hopkinson Pressure Bar technique Renliang Shan*, Yusheng Jiang, Baoqiang Li Civil Engineering, China University of Mining & Technology, Beijing, 100083, People's Republic of China Accepted 5 April 2000
Abstract The feasibility of using the Split Hopkinson Pressure Bar (SHPB) technique to obtain complete dynamic stress±strain curves for rock is established in the laboratory. The SHPB test system, in conjunction with a mean strain hypothesis, can be used not only for obtaining the rock's constitutive curve before the peak strength but also after the peak strength Ð and so it is possible to analyze and characterize the post-failure behaviour of rock with the SHPB method. Some typical complete dynamic curves for marble and granite are given in this paper, together with an interpretative discussion on the shapes of the curves. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Rock; Dynamic; SHPB method; Complete stress±strain curve
1. Introduction SHPB is the acronym for `Split Hopkinson Pressure Bar', which is an important apparatus for studying the dynamic characteristics of metal, polymers, bone material, concrete, ore, synthetic ®bers and other solids (Fig. 1). The SHPB technique originated from the HPB (Hopkinson Pressure Bar), which was designed by Hopkinson in 1914, and used in testing the shape of a stress pulse wave; the pulse signal was tested by the short column `timer', so it could only read the time and the peak value roughly, but could not measure the pulse shape accurately. Davies recorded continuous longitudinal displacement at the free end by a capacitor displacement sensor in 1948. Kolsky ®rst divided the pressure bar into two parts and put a small sample between them, and measured the pulse signal and displacement by a capacitor displacement sensor in 1949 [1]. The SHPB set, which is widely used today, has no * Corresponding author. E-mail address: [email protected] (R. Shan).
dierence from the 1949 version in principle. In order to overcome the restrictions of the measuring system using a capacitor displacement sensor, Lindholm [2] applied strain gauges in place of the capacitor sensor to measure the signals on the surfaces of two bars in 1963, which played an important role in enhancing the use of the SHPB method. From then on, many scholars have extensively studied a variety of problems using the SHPB, such as: sample size, friction eect of the ends, data processing methods, wave dispersed characters and transverse inertia eect, etc. [2±5]. Compared with metals, polymers and other manmade materials, the study of the dynamic characteristics of using the SHPB apparatus was late. Kumar ®rst researched the in¯uence of loading ratio and temperature on the strength of granite and basalt with a SHPB apparatus in 1968 [6]. An experiment incorporating compressive pressure on the rock sample was carried out by Cristensen in 1972 [7]. In 1976, Goldsmith's study showed that both compressive and tensile moduli and strengths are clearly related to loading rate [8]. Chinese scholars have carried out a great deal of research on both the SHPB experimental technique and the dynamic character of rock since 1980 [9±12].
1365-1609/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 1 3 6 5 - 1 6 0 9 ( 0 0 ) 0 0 0 3 1 - 9
984
R. Shan et al. / International Journal of Rock Mechanics and Mining Sciences 37 (2000) 983±992
From the references cited, it is found that most of past studies into the rock dynamic characteristics paid more attention to the elastic modulus and compressive strength. That is to say, only the elastic character before the peak strength was researched: both the non-linear and stepwise characteristics of loading were seldom studied. Moreover, reports on the study of the post-failure region of the dynamic stress±strain curve were rarely found. There are some causes for this scenario. First, in the SHPB method, the calculation of stress, strain and other parameters require that the sample is continuous, uniform and isotropic. Second, the transverse inertial in¯uence of the sample and the eect of interface friction between the sample and the bars can be neglected. Because these assumptions were based on the theory of one-dimensional stress waves, the majority of researchers, intentionally or unconsciously, avoided the post-failure portion of the complete stress±strain curve when they studied the rock's dynamic character. According to strict principles, we almost cannot measure the rock's dynamic character using the SHPB technique. In order to overcome the diculties, some scholars propose that stress wave attenuation should be taken into account as the wave passes through a fractured rock sample Ð if its amplitude exceeds the initial failure stress of the rock sample. Gende Zhang had studied such a mechanical model of stress wave attenuation in detail, and proposed to calculate the stress s at the middle section [10] with stresses s1 and s2 measured at the two ends and using an exponent attenuation model, and with the hypothesis of directly measured strain at the middle section as the true strain E. Also, he suggested that the dynamic constitutive relation should be made up of the above stress s and strain E, but only the part before the peak strength of the rock's stress±strain curve was given in his articles; it appears that his model only suits the initial rock damage. It is shown in this paper, that the SHPB test system and the related mean strain hypothesis can be used not
only for the analysis of the mechanical behavior of rock before the peak strength, but also for that of the post-failure region. This requires the use of two comparisons: the ®rst is the comparison between the `mean-strain' calculated from the strains measured by strain gauges on the incident and transmission bars and the axial strain obtained directly from a strain gauge on rock sample; the second is the comparison between the input bar's end stress s1 and the output bar's end stress s2. Then, the SHPB test system in this more developed form can be used for determination of complete dynamic stress±strain curves for rock. 2. Experimental method and analysis of the tested wave shape 2.1. Experimental system The SHPB apparatus is the core of the testing system (see Fig. 1). It consists of four components. 1. The power supply component, made up of nitrogen bottle and chamber. 2. The generating and transferring loads component, made up of the rock sample, striking bar, input bar and output bar. 3. The striking velocity measuring component, made up of spotlight, photoelectric diode, ampli®er and counter. 4. The strain measuring component, made up of strain gauges and ultra-dynamic apparatus as well as the dynamic test and analysis equipment of CS2092. The diameters of the striking bar, input bar and output bar are 30 mm with their lengths of 500, 1500 and 1400 mm, respectively. Marble and granite were selected as the sample material with static parameters of compressive strengths 35±45 MPa and 186±218 MPa, and elastic moduli of 5.3±6.5 104 MPa and 6±9 104 MPa, and Poisson's ratios of 0.19±0.22 and 0.25±0.30, respectively. Using the experience from previous work, we made samples with three dierent ratios of length L to diameter D: 50 samples for L: D = 1 (L = 30 mm), 40 samples for
Fig. 1. The uniaxial SHPB testing system.
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L:D = 1.5 (L = 45 mm) and 10 samples for L:D = 2 (L = 60 mm). 2.2. Experimental principle
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As1 A0 E0 EI ER
5
As2 A0 E0 ET
6
It is supposed that the mean value of stresses from the two ends of rock sample can be regarded as the stress in the whole sample, because the rock sample is short compared with the striking bar. So we can obtain the following:
In the SHPB experiments, high pressure gas provided by a nitrogen bottle expands in a chamber, and pushes and accelerates a striking bar which moves forward to strike the input bar at a certain speed and produces an input wave EI in the input bar. When the wave EI reaches the interface 1-1 (Fig. 2), two waves are produced, one part is re¯ected back along the input bar and produces the re¯ected wave ER (Fig. 2) and the other wave moves forward and reaches the output bar through the sample and produces the transmission wave ET, (Fig. 2). The strain pulse signals are collected via strain gauges and transformed into electric signals through ultra-dynamic strain equipment, and then transferred into dispersed signals and stored in recorded form through the CS2092. All these recorded dispersed signals are analyzed all together on completion of the tests. Based on wave propagation theory and with a onedimensional stress hypothesis, as well as the continuity demands of displacement, we can compute the velocities at the interfaces 1-1 and 2-2 shown in Fig, 2, respectively as:
Now, the rock sample's constitutive relation s ÿ E ÿ E_ can be determined from the input wave and the re¯ected wave as well as the output wave Ð as actually measured in the experiments Ð from Eqs. (3), (4) and (7). According to the homogeneous supposition of stress, there should be s1 s2 or EI ER ET , and substituting them into Eqs. (3), (4) and (7), we can then obtain:
v1 C0 EI ÿ ER
1
s
v2 C0 ET
2
A0, E0 and C0 refer respectively to sectional area, elastic modulus and longitudinal wave velocity of the input bar or output bar, A and L are the sectional area and length of rock sample respectively, and s1,
The strain of the rock sample generated in a unit time (strain rate) is: E_
v1 ÿ v2 C0 EI ÿ ER ÿ ET L L
s
s 1 s2 A0 E0 EI ER ET 2 2A
E_ ÿ
2C0 ER L
2C0 Eÿ L
t 0
8
ER dt 0
A0 E0 ET A
9
10
3
So, the calculated strain of the rock sample in a certain period of time is: t C0 t 0 EI ÿ ER ÿ ET dt 0 4 E E_ dt L 0 0 Also, according to Newton's third law, the following two equations apply at interfaces 1-1 and 2-2, respectively:
Fig. 2. Re¯ected and transmitted waves in the SHPB apparatus.
7
Fig. 3. Tested data processing chart.
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s2, s represent stresses at interfaces 1-1 and 2-2, and the mean stress, respectively. Thus, the transmitted wave describes the change of stress in the sample, and the re¯ected wave determines the change of the sample's strain. 2.3. Data processing Because the signals coming from the strain testing system are voltage signals, not the actual strain signals we need, we must ®nd the relation between the voltage signals and strain signals, i.e. calibrate the testing system. In these experiments, the calibration was adopted for the strain gauges on the sample, and for strain gauges on the input and output bars. After accurate calibration, the electric signal array can be transferred into the strain signal array. And then, we can use the above-mentioned formulae to compute the rock sample's dynamic character. The procedure for the experimental data processing is shown in Fig. 3. 2.4. Typical measured waves and associated analyses Fig. 4 illustrates typical strain waves measured in
the SHPB apparatus. The thick and thin lines in Fig. 4 represent the strain waves from the input bar and output bar respectively. The rock sample for this case was 45 mm in length with a diameter of 30 mm. Fig. 4(a) represents the testing result for marble with a striking velocity of 7.21 m/s, the rock sample remaining whole but with many visual criss-cross crackles on its surface. Fig. 4(b) is also the testing result for marble, this time with a striking velocity of 8.97 m/s, the rock sample being broken into six pieces and some detritus after being struck. Fig. 4(c) shows the testing result for granite with a striking velocity of 10.69 m/s, the sample remaining intact with no visual cracks on its surface. Fig. 4(d) also shows the testing result for granite, this time with a striking velocity of 12.47 m/s, and the sample was broken into six small fragments and some detritus. From Fig. 4, we can see that the input wave is approximately rectangular, in accordance with the onedimensional theory of stress wave propagation. But the re¯ected wave and transmitted wave are closely related to the striking velocity, and especially to the degree of damage of the rock sample. For a complete sample, the re¯ected wave is changed into compressive strain from tensile strain by way
Fig. 4. Typical strain waves measured in the SHPB apparatus.
R. Shan et al. / International Journal of Rock Mechanics and Mining Sciences 37 (2000) 983±992
of the zero strain platform (no re¯ection), the duration of the peak of the transmitted wave is relatively long, and the transmitted wave is similar to the input wave. For a broken sample, the re¯ected wave is only a tensile strain wave with a `W' shape, its two valleys, especially the second one, can indicate the degree of specimen degradation well. The deeper the second valley is, the greater the damage extent is. The transmitted wave will decrease along an approximately straight line after reaching the peak, and the decreas-
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ing rate is related to the loading magnitude and the degree of damage to the rock sample. For a critical sample (integral sample but with many tiny cracks on its surface), the re¯ected wave represents mostly tensile strain with a small fraction of compressive strain at the rear, so it shows a skewed `W' shape. The transmitted wave usually has a short region of small, sharp decline at the main rising stage, indicating that micro-fractures have occurred in the sample.
Fig. 5. Comparison of the strains measured directly on the rock samples with the average strains calculated from the strain waves measured on the input and output bars.
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3. Feasibility of determining the complete stress±strain curve with the SHPB technique 3.1. Comparison between `average strains' and strains measured directly on the rock sample The preliminary results show that the stresses at the two ends of the rock sample with 30 mm or 45 mm length accord with each other, and that of the sample with 60 mm length have a discrepancy. So, experimental results from the 60 mm long samples are excluded. The comparisons of strain (Fig. 5) measured directly on the surface of the rock sample (solid line) with `average strain' calculated from the strain waves measured on the input and output bars (dotted line) can be seen (Fig. 5). The experimental conditions in Fig. 5 are the same as those in Fig. 4 for each plot in the ®gure. We can reach the following conclusions from Fig. 5, plus the related fracture modes of the rock samples. (1) The striking velocities for marble samples are all lower than those for granite samples, while the strain amplitudes for marble samples are larger than those for granite. This shows that marble is easier to deform than granite; that is to say, marble is softer than granite. (2) The amplitude of `average strains' is closely related to the extent of damage of the same rock; the higher the damage extent, the larger is the `average strain' is. While the strain measured directly from the rock sample is completely dierent, it has little relation with the degree of fracturing of the rock. (3) Generally, the rock dynamic deformation behavior can be divided into three stages: . The ®rst stage is before the ®rst `knee point'. During this stage, the two curves coincide with each other, especially when the primary strain is smaller than 4000 mm/m; they can almost be superposed on each other, indicating that rock has good elasticity at this stage. . The second stage is from the ®rst `knee point' to the peak point. The slope of this stage is smaller than that of the ®rst stage. The two curves have a certain divergence, and the `average strain' is usually slightly lower than the directly measured strain, indicating that strain hardening has taken place at this stage. . The third stage is the post-failure stage. In this stage, the two curves for the only granite sample that remained almost intact after the impact have good coincidence, and the other sample's curves are dierent. The `average strain' is much higher than the directly measured strain in this stage. Why did the strains in the post-peak region obtained from these two methods have a such divergence? The following two explanations are possible. One is the
stress wave attenuation theory mentioned earlier. It is considered that the stress wave does not attenuate when its amplitude is low; while, if the amplitude of the stress wave is higher than the primary fracture stress, micro-cracks will be produced in the rock sample by the stress wave, and these micro-cracks lead to stress wave attenuation according to a negative exponential trend along the sample. Therefore, in the rock sample, the stress sr at any section is less than the `average stress s', see Fig. 6. This attenuation theory is reasonable, and it can be deduced that the stress at the output end is less than that at the input end with this idea. The validity was not con®rmed by our experiments (see Fig. 7 and the analyses in the next paragraph). Therefore, we have to consider the other explanation. The reasoning in this second explanation is that the attenuation theory does not consider the increasing stress resulting from the re¯ecting compression when the stress wave reaches the output end through the sample, because the wave resistance of the rock is less than that of steel bar. The failure strain of rock samples under impact often exceeds 6000 mm/m, sometimes even reaching to 10,000 mm/m. Thus, the dynamic failure strain exceeds the quasi-static failure strain for uniaxial compression. For such ultra dynamic strain, together with the micro-structural inhomogeneity as well as the non-linearity of the rock response, there may be some doubt about the result measured from the ordinary strain gauges stuck directly on the sample. Because ®rst, the utmost strain for ordinary strain gauges used in static measuring is from 6000 to 8000 mm/m with the requirement that the error is less than 10%. Second, given that the rock sample is non-linear and non-homogenous, it is possible that the rock at the spot where a strain gauge is located is not broken, while the integrity of the rock sample is actually violated: this makes the whole strain after the peak more than the directly measured strain at some location on the broken sample.
Fig. 6. Stress attenuation model for rock sample.
R. Shan et al. / International Journal of Rock Mechanics and Mining Sciences 37 (2000) 983±992
In fact, the strain gauges glued directly on the surface of a rock sample cannot be used to measure the sample's deformation after the peak, as is the case in quasi-static testing. First, many directly measured strains on broken samples change with time, similar to the solid line in Fig. 5(d), and the compressive strain often suddenly drops to negative values when it slowly goes down to the point about two thirds of the peak value after the peak, causing the strain gauge to be pulled o. In impact compressive experiments for rock, the axial tensile strain takes place only when swelling occurs, which usually occurs with the conditions of high con®ning pressure and friction at the
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end. The experiments described in this paper are of uniaxial impact and the two ends of the sample were smeared with butter, so there were no conditions of swelling, and the axial tensile strain is a false appearance. Second, the axial compressive strain is measured only after the peak is declining and its amplitude is not clearly related to the broken state of the sample. The declining phenomenon is false, because it is apparent that the strain may increase when breakage becomes worse after the peak strength of the rock. Therefore, there is no doubt that the strain gauges glued directly on the sample cannot be used to measure the sample's post-failure deformation.
Fig. 7. Comparison of stress waveforms at the two ends of the rock sample.
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The SHPB experimental method overcomes this dif®culty of directly measuring strain by strain gauges stuck on the rock samples. What the SHPB re¯ects is the deformed and broken character of a whole sample, not that of only a part of a sample, as in quasi-static tests to obtain the complete stress±strain curve. And the SHPB apparatus can indicate large deformations produced in broken rock by processing the small elastic strain waves in the input and output bars. From Eq. (4), it can been seen that the `average strain' of the rock sample is not only dependent upon the strains measured on the input and output bars, but is also dependent upon the steel bars' longitudinal wave velocity and sample length. From our tests, it can be seen that a measured strain of 1000 mm/m on the two steel bars can represent a strain of 8000±10000 mm/m of rock sample. Therefore, the SHPB method is an eective means to test the rock's broken and deformed character in the post-failure region, and hence can be used to establish rock deformation behavior not only before the peak, but also after the peak. In order to demonstrate this viewpoint further, consider a comparison of stress at the input end with that at the output end.
the input end. This indicates that the stress attenuation theory has insucient evidence to support it for the SHPB experimental calculation, and that the `mean stress' method or homogeneous supposition is more in accord with the experimental evidence. Thus, both strain analysis and stress comparison con®rm that the SHPB apparatus and the corresponding `mean stress' calculating model can be used to measure and analyze the complete stress±strain curve obtained under dynamic conditions. 4. Measured dynamic complete stress±strain curves for marble and granite
The change of stresses s1 and s2 at the two ends with time is shown in Fig. 7, the experimental condition for each plot being the same as that in Fig. 4. In Fig. 7, the solid line represents the stress wave at the input interface, and the dotted line represents the stress wave at the output interface. It is clear that, no matter what the rock is and whether the rock sample is broken or not, the stress wave forms at the two ends of each sample have good coincidence, except for deviations near the `leap points'; and, in most cases, the stresses at the output end are slightly larger than at
Figs. 8 and 9 show typical dynamic complete stress± strain curves for marble and granite (L = 30 mm) measured and calculated using the SHPB apparatus and the corresponding mean stress method mentioned above. In our experimental results (for impact speed in the range of 6.2±19.6 m/s with a strain ratio in a range of 100±600 sÿ1), most of the stress±strain curves of granite before utmost points have jagged characteristics. When the curves reach to the ®rst maximal point, the stress ®rstly goes down to a minimal point, and then goes up to the utmost peak. Before the ®rst peak, no matter what the striking speed or load rate is, the curve is approximately a straight line, and the deviation between the lines is small, showing that granite at this strain stage has good linear elasticity. And the line slope (elastic modulus, E ) is in the range of 0.4± 1.0 105 MPa. After the ®rst maximal points, and especially after the utmost peaks, the stress±strain curve is closely related with the striking speed and broken state, and the curve's shape is variable. When the striking speed is low, and the sample remains whole after being struck,
Fig. 8. Complete dynamic stress±strain curves for granite.
Fig. 9. Complete dynamic stress±strain curves for marble.
3.2. Stress comparison at input and output interfaces
R. Shan et al. / International Journal of Rock Mechanics and Mining Sciences 37 (2000) 983±992
the stress±strain curve after the utmost peak will rebound at once, shown by the dot-dash curve in Fig. 8. When the striking speed is high, along the stress±strain curve after the utmost peak, the strain increases continuously with stress and the ability of withstanding load decreases, even when the stress decreases to zero. This shows that the granite sample was broken thoroughly and began to be detached from the two bars at this time. The moving out of the rock sample chips causes the interfaces between sample and steel bar to be free and so the sample's `strain' is the result of re¯ection of the input wave on the free end; therefore, it is an artifact. This situation is shown by the solid line curve in Fig. 8. When the striking speed is in the medium range, and the sample is already damaged after being struck, but not seriously broken into pieces, the strain after the peak increases at ®rst, and then decreases: this indicates that the sample has some rebounding characteristics at the late stage of deformation. This situation is shown by the dotted curve in Fig. 8. The complete stress±strain curve of marble (Fig. 9) usually rises in a line with positive slope in the ®rst part. In the post-failure region, the curve usually descends with a negative slope at ®rst, then falls in a steep line. It diers markedly from that of granite. Firstly, the slope of the ®rst part (elastic modulus, E ) is related with the striking speed or strain rate to some degree. The larger the strain ratio is, the larger the elastic modulus is, and the marble's dynamic elastic modulus in this experiment is in the range of 1.8±5.0 104 MPa. Secondly, there is an approximate horizontal line between the ®rst part and the post failure region, indicating the plastic character of marble after being struck. Finally, the post-failure region of the marble's stress±strain curve only has the ®rst kind of unloading (slope is negative) and `rigid unloading' (slope is in®nite); but the second kind of unloading (slope is positive) does not occur. In a word, and just as for the rock's quasi-static constitutive curve, the post-failure region is more complex than that of the pre-failure part.
5. Conclusions The SHPB apparatus and the corresponding homogeneous stress supposition can be used to measure and analyze not only the ®rst stage but also the post-failure region of a rock's dynamic complete stress±strain curve. The technique can be used to measure the dynamic character for elastic homogeneous materials and fractured rocks, as has been shown by our experiments. So we hope researchers in this ®eld will join
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together to enrich and develop the SHPB experimental method. The dynamic complete stress±strain curve of granite before the utmost peak has a clear jagged character. It is usually is an approximately straight line in the primary stage, and the slope of the straight line is not apparently related to the strain rate. Both Class I (monotonically increasing in strain) and Class II curves (not monotonically increasing in strain) exist in the post-failure region, and the curves in this stage can vary widely. Usually, the dynamic complete stress±strain curve for marble has three stages. The slope of the straight line (elastic modulus) at the beginning is related to the strain rate to some extent; the higher the strain rate is, the larger the modulus is. The post failure curve only decreases with a negative slope or in a steep line and has no rebound elasticity. The approximate horizontal line in the middle of the curve indicates the marble's plastic character after being impacted.
Acknowledgements The project was supported by the National Science Foundation of China (No. 49502041). The experiments were carried out at Beijing University of Science and Technology. The authors would like to acknowledge Professor Z. X. Zhang for helping with the tests.
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