Data processing in the split Hopkinson pressure bar tests

Pergamon

Int. J. Impact Engng Vol. 15, No. 6, pp. 723-733, 1994 Elsevier Science Ltd Printed in Great Britain 0734-743X(94)-E0002-D 0734-743X/94 $7.00+ 0.00

DATA PROCESSING IN THE SPLIT HOPKINSON PRESSURE BAR TESTS J. M. LIFSHITZ and H. LEBER Material Mechanics Laboratory, Faculty of Mechanical Engineering, Technion-lsrael Institute of Technology, Haifa 32000, Israel (Received 28 May 1993; in revised form 11 November 1993) Summary--The effect on dynamic stress-strain curves of dispersion and shifting of elastic strain pulses travelling in a split Hopkinson pressure bar is reported. The dispersion correction is done in the frequency domain after employing FFT algorithm by adjusting the phase of each Fourier component. The three pulses (incident, reflected and transmitted) are analyzed inside three identical windows that propagate along the time axis with a reference velocity co = v / ~ . The oscillations in the dynamic stress-strain curves are shown to be very sensitive to small variations in the value of co. A calibration procedure for determining the value of Co for each SHPB setup is suggested. The predictions are compared to experimental measurements.

INTRODUCTION

In 1949 Kolsky [1] introduced a method for determining mechanical properties of materials at high rates of loading by the split Hopkinson pressure bar. The method is based on the theory of wave propagation in elastic bars and the interaction between a stress pulse and a short cylindrical specimen of different mechanical impedance. During the years the method was extended to tensile [2,3] and shear [4,5] specimens using various types of specimens, loading and measuring techniques. When shear behavior is determined by torsional waves, the analytical treatment of the loading waves is simple, since torsional waves are non-dispersive. However, when we want to determine compressive or tensile behavior by applying a longitudinal pulse through a loading bar we face a problem that stems from the dispersive nature of the pulse. Although the problem of wave propagation in cylindrical bars was solved by Pochhammer (1876) and independently by Chree (1889), most of the analyses of SHPB results have been based on the assumption that the one-dimensional approximation is adequate and the loading pulse travels in the bar without any distortion. Davies [6] used the Pochhammer-Chree solution to show the dispersion that takes place in a trapezoidal (periodic) wave as it travels along the bar. He expressed the periodic wave by a Fourier series and let each component travel with its phase velocity. Hsieh and Kolsky [7] used a similar approach to predict dispersion of a stress pulse, produced by detonating a small explosive charge at one end of a steel cylinder, that had a shape of an error function. Their prediction agreed well with the measured values and they concluded (as Davis had done) that in the analysis we can consider the waves to belong to the first mode of vibration. Some 25 years later Follansbee and Frantz [8] repeated the calculation made by Davies on the dispersion of a trapezoidal (periodic) wave. Then they used similar calculations to predict dispersion of a measured pulse in a SHPB test in order to improve the accuracy of dynamic stress strain curves (of brass, work hardened OFE copper and annealed iridium). The dispersion correction used by them improved the shape of the stress-strain curve by reducing its oscillations, but not eliminating them. The method for treating the dispersion effects was based on Fourier series of the stress pulse in the time domain. Another, more efficient method, is to consider the pulse as a single pulse (and not a periodic loading) and use FFT technique to transform it to the 723

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J.M. Lifshitzand H. Leber

frequency domain. The shape of the pulse after travelling a distance x is obtained by adding the appropriate phase angle to each component of the pulse and taking the inverse transform. Yew and Chen 1,9,10] suggested using FFT technique for studying dispersion characteristics of waves in rods, plates and thick blocks. They demonstrated that the method is reliable for this purpose by showing that the dispersion relations that they obtained, by measuring a stress pulse at two different locations, agreed with prediction of the Pochhammer-Chree equations. A few years later, Gorham [11] used FFT technique to remove dispersion effects from measured pulses in order to reconstruct the original load-time history. He used Bancroft's 1,12] data for the dispersion relations and showed that the oscillatory load-time curves resulted from the geometrical dispersion. Recently, Gong et al. [13] used FFT technique to remove dispersion effects from measured pulses, in their study of concrete by the SHPB method. Another recent paper that deals with material and geometrical dispersion correction is by Gary et al. [14]. The conclusion from the papers mentioned here is that (a) dispersion correction improves the shape of the dynamic stress-strain curve and (b) FFT technique, which is readily available on PC, improves the efficiency of the calculation. However, even with dispersion correction, the stress-strain curve still contains some oscillations in the plastic region. In this paper we investigate the source of these oscillations and suggest a method to improve the shape of the dynamic stress-strain curve. EXPERIMENTAL The system used in this work is the split Hopkinson pressure bar (SHPB), outlined in Fig. I. A pressure pulse is created in a long input bar when a striker bar is fired by an air gun at the end of the input bar. When the pulse reaches the thin specimen, part of it is reflected and part is transmitted, depending on the impedance mismatch between the specimen and the bars. The incident (e~) and reflected (e,) pulses are picked up by a pair of strain gauges in the middle of the input bar, and the transmitted (Q pulse is picked up by a pair of strain gauges located in the middle of the output bar. The three pulses are recorded on a digital oscilloscope and transmitted by GPIB to PC for data analysis and calculation of stress, strain and strain-rate. The pressure bars in our setup were made from Maraging steel. Each bar was 876 mm long and 12.7 mm in diameter. The striker bar was 200 mm long and had the same diameter. The specimens were short cylinders, 11.5 mm in diameter and about 5 mm long. The basis for selecting these dimensions for the specimens is the work by Davies and Hunter 1,15], where they attempted to establish conditions to minimize effects of radial and axial inertia. The exact length of the specimen is not critical, as long as it lies within the range given in 1-15], as found by Lindholm 1,16]. Three materials were used: aluminium AL-6061-T6; commercial copper; and brass; the first is known to be rate insensitive. DATA ANALYSIS The average strain, strain rate and stress in the specimen are [16], respectively

es(t) = Co "|t (e~ - er - e,)d~ lo 30

(1)

~s(t) = Co (ei - e, - e,)

(2)

lo EA

as(t) = ~ where Co = x / ~

(ei + e, + e,),

(3)

is the phase velocity of infinitely long waves; A and As are cross-sectional

725

Data processing in the split Hopkinson pressure bar tests

Strikerbar \

L.E.D's

Air gun

i i

\

Photo diodes

accPressulreator

~

i i

Input bar

Specimen

/

/

q-i

! i

/

/

S~g

m/ i~

_ AxI

~ Electronic "~ unter

Output bar \ Stopper

J

Ax2 ~_

I Wheastone bridge

Digital oscilloscope

Solenoid v a l v e ~

~ ,m

e

m

~

Q

G.P.I.B

Pressure source

Microcomputer

J

1

FIG. 1. Schematicdiagram of the split Hopkinson pressure bar test apparatus.

areas of the Hopkinson bars and specimen, respectively; l o is the specimen's undeformed length; and e~, e,, e, are the incident, reflected and transmitted strain pulses, respectively, at the specimen-bars interfaces. Since the strains are measured away from the specimen and longitudinal stress waves are known to be dispersive, we must correct the measured pulses to account for the dispersion between the location of the measuring device (strain gauges) and the specimen-bars interfaces. The correction is done using discrete Fourier transform, by means of FFT algorithm. Consider the problem of predicting a pulse shape e(Xo + Ax, t), at a location x o + Ax, when the shape of the pulse at another location, Xo, is known and given by e(x o, t),

(4)

~(Xo, t) = ~ lej(ico)l exp [i(2nfjt - 4,~°~)], j=O

where le~(ico)l and ~o~ are amplitude and phase of the jth Fourier component of the pulse at location Xo. Each component of the pulse has a different phase velocity cj. The shape of the pulse, at the new location Xo + Ax, is given by 8(Xo + Ax, t) =

j=O

I~jlico)l exp i 2rcfj t -

- qS}°)

.

(5)

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J.M. Lifshitz and H. Leber

It is more convenient to express the pulse at x o + Ax using a shifted time scale t'given by t" = t - -

Ax

(6)

Co

then

e(Xo + Ax, t")= ~ lej(ko)lexp [(i(2rtfjT- ¢7' - A~b}a~')],

(7)

j=O

where ~.j

=

2r(jAx

-

\cj

(8)

is the phase increment of the jth fourier component at location x o + Ax, when the time is measured by ~. These increments, that are caused by the difference in phase velocities, are responsible for the dispersion of the pulse as it travels down the bar. Note that when cj = c o there is no dispersion. The calculation of the pulse shape at x 0 + Ax is done in the frequency domain using F F T algorithm. A window is fixed around the original pulse e(x o, t) and the discrete Fourier spectrum co(f) is calculated from the data inside the window s o ( f ) = ~-{e(x o, t)}.

(9)

The spectrum of the pulse e(x o + Ax, t') is then [see Eqns (4) and (7)] ~Ax(f) = ~o(f) exp (-iA~b(a~')).

(lO)

By taking the inverse transform of (10) the pulse shape in the time domain is obtained inside a displaced window [see Eqn (6)] e(x o + Ax, t') = ~ - 1 {aax(f)}.

(11)

The phase velocities cj that are needed to calculate A~b~ax) in Eqn (8) are obtained from Bancroft's data 1-12], after some modification. Bancroft's data are given in the form of

c(o)

co

X '

(12)

where a is the radius of the bar and A is the wave length. For our purpose it is more convenient to express c/c o as a function of a non-dimensional frequencyfa/c o. This is done by observing that the non-dimensional frequency can be obtained by multiplication of two basic parameters in Bancroft's data, namely c/c o and a/A

f a _ f A. a co co A

c a. Co A

(13)

Thus the dispersion relation (12) can be given by a different function, say F1 c Co

,14,

Data processing in the split Hopkinson pressure bar tests

727

Before proceeding v¢ith the description of the method to obtain Co, it is important to remember that the dispersion correction requires knowledge of both c o and Poisson's ratio v, as given by Bancroft's data. However for low values of a/A, which is normally the case in most Hopkinson bar applications, the dispersion correction is insensitive to changes in the value of v. Therefore, to complete the calculation of the phase difference [Eqn (8)] we must know, in addition to Bancroft's dispersion relation, only the value of c o. Normally, this value is determined by calibration. Follansbee 1-17] calibrated his SHPB setup by establishing actual timing between ei and e,, by firing the striker bar at the input bar which is separated from the output bar. Then he ran a calibration test between el and e,, with the input and output bars in direct contact. Y o k o y a m a and Kishida 1-18] obtained the "elastic wave velocity" Co by measuring the transit time in a single bar experiment. None of the above papers explains the details of the calibration procedure and it is likely that the timing between ei and e, was determined by measuring the distance, along the time axis, between the beginning of e~ and the beginning of er. This, however, leads to an error because of two reasons: (a) it is difficult to determine the beginning of each pulse, and particularly that of the reflected pulse; and (b) due to dispersion, er is different from e~ and comparing time between two different pulses cannot be used to calculate Co. We suggest to base the calculation of Co on the corrected pulse in an iterative method, starting with a book value of c o, as follows: e~ and e, are measured by firing the striker bar at the end of the input bar which is separated from the output bar. Two identical windows with a time interval of Ax/co between them (Ax is twice the distance between the 600

(A)

I

I

I

l

I

I

I

I

t

t

b 5OO "5 400 .>_ 30O :

200


100

I

-100 0

(B)

600

20

i

I

40 60 80 100 Time [ microseconds] i

i

I

I

t

t

I

120

140

i

500 400

300 200 ~"~ 100 < 0 -100 0

t

t

20

40

60 80 100 Time [microseconds]

120

140

FIG. 2. Effect of variation in co on a corrected pulse after travelling a distance of Ax = 0.874 m. (A): Measured (a) vs calculated (b). (B): Three calculated pulses.

728

J.M. Lifshitzand H. Leber

strain-gauge and the end of the input bar) are drawn along the time axis such that the incident pulse is positioned inside the middle of the first window and it occupies about 75% of the window time span. If the initial value of Co is approximately correct, then the reflected pulse will be positioned inside the second window. The discrete Fourier spectrum of e~ is determined [Eqn (9)] and this is used to calculate the spectrum of e, from Eqn (10) by taking the book value for Co in Eqn (8). The inverse transform [Eqn (11)], is the calculated er, drawn inside a window which is shifted along the original time axis according to (6). Since the calculated pulse e, and the measured one are plotted inside the same displaced window, they can be compared without any further shifting. If c o has the correct value, the calculated and measured e, should be (almost) identical. If the difference between them is too large, the calculation is repeated with a new value ofc o until the best fit is obtained. RESULTS AND DISCUSSION All the specimens in the present work had the same dimensions, following Davies and Hunter [15] recommendations. Similar results for specimen dimensions were obtained by Bertholf and Karnes [ 19] who performed a two-dimensional numerical analysis and defined the conditions required for a valid SHPB experiment. The purpose of this paper is therefore not to study the effect of varying the ratio of length-to-diameter of the specimen. Given a specimen with optimal dimensions, we want to determine additional sources that introduce oscillations to the dynamic stress-strain curves and suggest a method to improve the results. 600

(A)

I

I

I

I

I

I

I

I

500 "~ 400

~

3oo 200


100

-100

(B)

600

I

I

20

40

I

I

I

I

I

I

I

I

I

60 80 100 Time [microseconds]

I

120

140

I

500 400 .>. •.~ 300 a~

200

/

ca.


-100

I

0

20

40 60 80 100 Time [microseconds]

I

120

140

FIG. 3. Predicted (b) vs measured (a) pulses after different distances of propagation Ax. (A): Ax = 1.748m; (B): Ax = 2.572 m.

Data processingin the split Hopkinson pressure bar tests

729

The quality and accuracy of dynamic stress-strain curves depend not only on the accuracy in predicting the shapes of the individual pulses (e~,er and et) at the specimen-bars interfaces, but also on their position along the time axis. Both requirements depend on using a correct value of Co. A small change (1 to 2%) in the value of c o leads, in a setup where each bar is approximately 1 m long, to a shift of a few microseconds in the location of the calculated pulses. This relative shift between the pulses is the main source of the oscillations in the dynamic stress-strain curves even after performing dispersion corrections. The influence of small changes in c o on the calculated pulse is shown in Figs 2(A) and 2(B). A measured reflected pulse (a) is compared in Fig. 2(A) to a calculated pulse (b) based on a correct value of Co. Then, in Fig. 2(B), three calculated pulses, based on different values of c o, are compared to each other. The small differences of 1 and 2% in the values of c o lead to a relative shift of a few microseconds. The effect of these small shifts on the dynamic stress-strain curve will be shown later in Fig. 7. Although the strain gauges in a SHPB arrangement are located not very far from the specimen, it is worth while showing the accuracy of the present method in predicting dispersion correction after long distances of travel. This is done by measuring some reflections of a stress pulse in the input bar, that is removed from the specimen. The first pulse is used to calculate its Fourier spectrum. From this the shape a n d location of the following reflected pulses are calculated. A comparison of such predictions to measured values after travelling different distances is shown in Fig. 3. The agreement is good even after long distances of travel due to the accuracy of the value of c o. Returning now to Eqns (1)-(3), we have to calculate the three strain pulses at the two interfaces of the specimen-bars, from measurements that are taken Axl and A x z away from the interfaces (see Fig. 1). A typical measured record is shown in Fig. 4 with three windows containing the pulses and the intervals between them. The corrected pulses at the proper interfaces are calculated by the method described earlier, using the appropriate value for Ax in Eqn (8): for e~, Ax = Axl; for er, Ax = - A x l , and for et, Ax = - A x 2. This method assures us that the pulses will be shifted to the proper position inside a single time window, without having to determine the "zero" time for each pulse. In other words, the incident and the reflected pulses will start at the same point in time within the window while the transmitted pulse will start after a short time interval (inside the same window), corresponding to the time required to travel across the specimen. The method presented here does the time adjustment automatically, regardless of small variations in specimen length or material. A typical set of e~, e, and et, after being shifted to the appropriate faces of the specimen, is shown in Fig. 5. Now that the three corrected pulses are drawn at the appropriate place along the time

1200

T

I

T

"5 600

=
0

-600

r

200 Time [microseconds]

400

FIG. 4. A typical record of incident, reflectedand transmitted pulses and the location of the three time windows.

730

J . M . Lifshitz and H. Leber

1200

I

I

I

i

i

600 600 i 0

i

30

60 90 Time [microseconds]

120

FIG. 5. A typical record of incident, reflected and transmitted pulses after being shifted to the specimen's faces.

I

de/dt = 2000 [s- I ] 400

F' b e-,

200 er"

I

0

5 Engineering strain [%]

10

FIG. 6. Dynamic (b) and static (a) stress-strain curves of aluminium AL 6061-T6.

axis, Eqns (1) to (3) can be used to calculate the strain, strain-rate and stress in the specimen. It should be emphasized that these values are meaningful only in the plastic range. The dynamic stress-strain curve derived from Fig. 5 is shown in Fig. 6, together with the static curve. When the value of c o is slightly changed (by 1%), the shape of the dynamic stress-strain curve is changed appreciably and the size of the oscillations is dramatically amplified as shown in Fig. 7. Aluminium 6061-T6 is known to be rate insensitive, as shown in Fig. 6, and this can be used as a verification of the test setup. Other materials were also tested during this study and the dynamic stress-strain curves of commercial copper and brass are shown in Figs 8 and 9. In each figure the "correct" curve is drawn next to an "incorrect" curve that was obtained by taking a value of c o which is 1% higher than the proper calibrated value. It is clear that small variations in the value of Co, which is equivalent to an error in shifting the pulses along the time axis, cause oscillations in the stress-strain curve. When the strain-gauges are located further away from the specimen, the setup is more sensitive to small errors in the value of Co. The strain rate, in the plastic range, in our tests was approximately 2000 I-s-l] for the aluminium and the copper, and 15001-s - l ] for the brass. Strain-time history of the aluminium is shown in Fig. 10.

Data processing in the split Hopkinson pressure bar tests

731

I

de/dt = 2 0 0 0 [s "l] 400

/~ ~ " - a t" .J eX0 ¢--

200 a ... c O

u~

b ... 1.01c 0

I

0

5 Engineering strain [%]

10

FIG. 7. Effects of a small change in the value of co on the shape of the dynamic stress-strain curve of AL-6061-T6.

400

i

i

drddt = 2 0 0 0 Is 1 ]

200

:" ~

,

,,

-

. . . .

a

0

5 10 E n g i n e e r i n g strain [%]

15

FIG. 8. Effects of a small change in the value of c o on the shape of the dynamic stress-strain curve of copper.

I

drr_/dt = 1500 [

s

'

~

400

- -

f

e~

e,a t--r

.r"

**_A,

-

•

200 a ... c 0

¢-,

b ... 1.01c 0

r-,

I

0

5 E n g i n e e r i n g strain [%]

10

FIG. 9. Effects of a small change in the value of c o on the shape of the dynamic stress-strain curve of brass.

732

J.M. Lifshitz and H. Leber 0.1

I

drr./dt = ~ [ s

E

1]

tt~

b

ex0 e.,

0.05

o e., e..,

0

30

Time [microseconds]

60

FIG. 10. Strain-time history of AL-6061-T6.

CONCLUSIONS

1. Data analysis, from SHPB tests, in the frequency domain is convenient and more efficient than in the time domain. 2. The shape of the dynamic stress-strain is very sensitive to small errors in the value of Co and to errors in shifting the pulses along the time axis. 3. It is recommended to calibrate each SHPB setup for a correct value of c o. This is done by considering dispersion correction of a pulse after travelling some distance and comparing the calculated shape to a measured one. 4. Shifting of the pulses along the time axis is done automatically by enclosing the pulses in individual identical windows with time intervals between them as shown in Figs 1 and 4. Acknowledgement--This research was supported by the fund for the promotion of research at the Technion and by Technion V.P.R. Fund--A and B. Greenburg Research Fund (Ottawa).

REFERENCES 1. H. KOLSKY,An investigation of the mechanical properties of materials at very high rates of loading. Proc. Phys. Soc. B. 62, 676-701 (1949). 2. J. HARDING,E. O. WOOD and J. D. CAMPBELL,Tensile testing of materials at impact rates of strain. J. Mech. Engng Sci. 2, 88-96 (1960). 3. T. NICHOLAS,Tensile testing of materials at high rates of strain. Exp. Mech. 21, 177-185 (1980). 4. J. D. CAMPBELLand J. L. LEWIS, The Development and Use of a Torsional Split Hopkinson Bar for Testing Materials at Shear Strain Rates up to 1500 sec -I. Department of Engineering Science Report No. 1080, 69, University of Oxford (1969). 5. J. DUFFY, J. D. CAMPBELLand R. H. HAWLEY,On the use of torsional split Hopkinson bar to study rate effects in 1100-0 aluminium. J. appl. Mech. 38, 83-91 (1971). 6. R. M. DAVIS,A critical study of the Hopkinson pressure bar. Phil. Trans. A 240, 375--457 (1948). 7, D. Y. HSmH and H. KOLSKY, An experimental study of pulse propagation in elastic cylinders. Proc. Phys. Soc. 71, 608-612 (1958). 8. P. S. F•LLANsBEE and C. FRA•`rrz• wave pr•pagati•n in the SHPB.J. En•n• Mater. Techn•l. ••5• 6 •-66 ( •983). 9. E. H. YEW and C. S. CHEN, Experimental study of dispersive waves in beam and rod using FFT. A S M E J. appl. Mech. 45, 940-942 (1978). 10. E. H. YEW and C. S. CHEN, Study of linear wave motion using FFT and its potential application to non-destructive testing. Int. J. Engng Sci. 18, 1027-1036 (1980). 11. D.A. GORHAM,A numerical method for the correction of dispersion in pressure bar signals. J. Phys. E: Sci. Instrum. 16, 477-479 (1983). 12, D. BANCROFT,The velocity of longitudinal waves in cylindrical bars. Phys. Reg. 59, 588-593 (1941). 13. J. C. GONG, L. E. MALVERNand D. A. JENKINS, Dispersion investigation in the split Hopkinson pressure bar. J. Engng Mater. Technol. 112, 309-314 (1990).

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14. G. GARY, J. R. KLEPACZKO and H. ZHAO, Correction for wave dispersion and analysis of small strain with split Hopkinson bar. Proc. Int. Syrap. on Impact Engineering, (edited by MAEKAWA),Nov. 2/4, 1992. Sendai, Japan. 15. E. D. H. DAVIES and S. C. HUNTER, The dynamic compression testing of solids by the method of the split Hopkinson pressure bar. J. Mech. Phys. Solids 11, 155-179 (1963). 16. U.S. LINDHOLM, Some experiments with the split Hopkinson pressure bar. J. Mech. Phys. Solids 12, 317-335 (1964). 17. P. S. FOLLANSBEE, High strain-rate deformation of FCC metals and alloys. In Metallurgical Applications of Shock-Wave and High-Strain-Rate Phenomena, pp. 451-479. Marcel Dekker Inc. (Mechanical Engineering 52), New York 0986). 18. T. YOKOYAMAand K. KISHIDA, A microcomputer-based system for the high-speed compression test by the split Hopkinson pressure bar technique. J. Testing and Evaluation, JTEVA 14, 236-242 (1986). 19. L. D. BERTHOLF and C. H. KARNES, Two-dimensional analysis of the split Hopkinson pressure bar system. J. Mech. Phys. Solids 23, 1-19 0975).