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A technique for obtaining compressive strength at high strain rates using short load cells
Int. J. Mech. Sci. Vol. 20. pp. 553-560 © PergamonPress Ltd., 1978. Printedin Great Britain
0020-7403/78/0901~0553/$02.0(t/0
A T E C H N I Q U E FOR OBTAINING COMPRESSIVE STRENGTH AT HIGH STRAIN RATES USING SHORT LOAD CELLS A. J. HOLZER Department of Mechanical Engineering, Monash University, Clayton, Victoria 3168, Australia (Received 29 N o v e m b e r 1977; in revised f o r m 21 February 1978)
Summary--A technique for obtaining stress-strain data at high strain rates is given. The method involves use of a short load cell and a fibre optics displacement transducer. Computer aided analysis of the force-time data allows correction of the signal for the effect of the dynamic characteristic of the force measurement system. The method involves transformation to the frequency domain of the force data by means of a Fast Fourier Transform algorithm, and is shown to be able to make effective corrections for the dynamic ringing of the load cell and the phase lag of the recording system. The compression mode of deformation is used, lubrication being effected by means of a Teflon film. The results of preliminary tests performed on a medium carbon resulphurised steel are given.
NOTATION F Fi(t) G,to,,~ K R Vo(t) [(D) Io m v v0 a 8t,~
Hertzian Force input (force) system constants constant radius output (voltage) system transfer function initial specimen length mass velocity of tup impact velocity approach of colliding bodies material property mean natural strain rate ~i initial natural strain rate r contact duration.
INTRODUCTION
Experimental work aimed at evaluating stress-strain characteristics of materials deformed at high rates of strain has received considerable attention since Kolsky 1 introduced the concept of the split Hopkinson Pressure Bar. The resulting work of many experimenters is quite well known and has been reviewed by several authors. 2-7 The method for experimental evaluation of material properties given in the current paper differs from most previous work in that a short load cell is used for measurement of short duration force pulses. A method for analysing the force data generated by such a load cell is described. The following brief review is therefore confined to previously used methods of force measurement. P R E V I O U S M E T H O D S OF F O R C E M E A S U R E M E N T Many early experimenters, attempting to obtain material properties at high rates of deformation, used energy absorption techniques for calculation of forces and stresses. Indeed Habib 8 has used a technique of this type as recently as 1948. Methods of this type can of course only yield mean or average forces for a certain time period. Instantaneous force-time histories cannot be obtained. Warnock and Taylor 9 and Taylor j°'" used electrical resistance strain gauges mounted directly on the flange of a tensile specimen for stress measurement at high strain rates. They point out 9 that vibration of the stress recording dynamometers is often severe in the neighbourhood of the yield stress and may mask the yield process in high strain rate investigations. Energy methods were also used in the early work of Baraya et al. n using an experimental drop hammer. The same drop hammer was subsequently instrumented with direct reading force and displacement transducers ~3 and investigations of the compressive strength of pure lead ~3 and (0.55%C) steel 14a~ performed. The force transducer used in these investigations was a short dumbell solid 553
554
A.J. HOLZER
cell using four strain gauges. Some minor vibration is evident on the force-time osciliograms presented. No dynamic calibration was performed. Jain and Amini t~ examined the short ring type load cell, and concluded that it is superior to the short solid types. Lengyel and Mohitpour t7 used a cell of the type described by Jain and Amini, and dismiss a small peak in their load-time curve as being due to reflected stress waves, concluding that the results were unaffected. No dynamic calibration of the load cell was attempted. Suzuki et aL la used a capacitor strain meter, a cell of the short type, for load measurement. The authors themselves concede that vibration of the load measuring instrument causes the greatest difficulty in interpreting the results. Samanta ~9 used an accelerometer mounted on the tup of an impacting drop hammer to obtain force-time records. Vibration due to dynamic ringing of the tup is evident on the oscillograms presented. Assumptions are also made regarding the rigidity of the mass in order to calculate forces from accelerations. Woodward and Brown 2° used a short piezo-electric load cell to measure force in high strain rate investigations, and showed that the maximum strain rate for which a reliable force record can be obtained is determined by the frequency response of the load cell used. They obtained instantaneous force-time records by drawing a mean line through the oscillations of the force transducer output, The split Hopkinson Pressure Bar has been used extensively to record short duration force pulses, This method is limited in that the input and output bars must, or are assumed to, remain elastic and the force values obtained are average values for the time of the pulse in the specimen. Hence most work using this technique has been restricted to comparatively small values of total strain. Complete stress-strain characteristics have usually been obtained by superposition of the results from each impact. Thus an essentially incremental approach is used. Samanta z~ has oversome the problem to some extent by use of a long (3-5 m) pressure bar in a modified Hopkinson apparatus. The technique of the split Hopkinson Pressure bar is extremely well documented. ~'~2'23'24 It is clear that all of the above methods of force measurement are limited in one respect or another. While each method discussed is valid in certain applications, there seems to be no universal method for obtaining instantaneous force-time histories during experiments where deformation is carried out to large strains at high rates of strain. When measuring forces in dynamic situations it is possible that a force frequency and a natural frequency of the force measuring system will correspond. This situation is even more likely if the material being studied exhibits a yield point phenomenon. Thus. any yield point phenomenon is likely to be camouflaged by ringing and the results generally will be uninterpretable. It is therefore necessary to correct the static calibration of a force measuring system to account for its dynamic behaviour. In the following sections a method of performing this correction is described. T H E B A S I S OF THE T E C H N I Q U E Description o f f o r c e transducer and mounting A gravity drop forge has been developed for the purpose of high strain rate investigations and has been described previously.2s This apparatus differs from the drop hammer of Siater et al. ~3-~5 in that the tup is arrested by means of stop blocks, the excess energy being absorbed by a mass air-suspension system. Thus the strain rate can be made to remain nearly constant or to increase throughout the major part of a deformation test. Two alternative force measuring systems have been developed, one of medium frequency response but large capacity (400 kN) and the other of higher frequency response and smaller capacity (30 kN). Each consists of a piezo-electric force transducer fitted beneath a compression die. The large load cell and its associated mounting is illustrated in Fig. 1. The arrangement for the small cell is of an identical nature except that space has not allowed mounting of the displacement feeler integral with the bottom die. In each case the mounting consists of the die, an elastic preload stud, a locking washer and, beneath the load cell, a TUP
1
|
" I ''~ TOP DIE
1
STOP |LOCK DISPLACEMENT GAUGE
LOAD
CELL
- -
SPACEr WASH| It ,r
..
~|oCu
5000
Kg
N IbOLT
MASS
FIG. 1. Large force measurement system, capacity 400 kN, natural frequency 8 kHz.
A technique for obtaining compressive strength
555
spacer washer. The surfaces of all components are lapped together to ensure minimum dynamic ringing. Both the top and bottom dies are hardened to about 60 Rockwell C. The elastic bolt (beryllium copper) is used to apply a preload to the load cell. This ensures that the cell is operating in the linear portion of its range and that sufficient load is on all the components so that extraneous vibration is not introduced. The load cell and beryllium copper bolt are equivalent to a parallel spring mass damper system. Thus, as the applied force is not actually that measured by the load cell, it is necessary to make a correction to the static calibration of the free load cell. The preload, the bolt material and its dimensions are chosen to ensure that the bolt remains in tension up to the maximum load used.
Outline of signal correction method It will be appreciated that the load cell and associated mounting of Fig. I is analogous to a second order linear spring mass damper system. Considering such a second order system, the transfer function f(D) can be written f(D) = Vo(t) F/(t)
G m 2D2 + 2_~ D + 1
(I)
¢O n
where F~ is the disturbance or input force, V0 is the output, in the case of the load measurement system a voltage, and G, to, and £ are system constants. It is evident from (i) that if V0 and f(D) are known, then F~ can be obtained, i.e.
Vo(t) F~(t) = f(D)"
(2)
The simplest method of performing this manipulation with real data is to obtain the Fourier transform of V(t) and then divide this by the frequency domain transfer function or frequency response function. Inverse Fourier transformation then yields the actual force input to the system as a function of time. The main problem is obtaining a reliable frequency response function for the system that represents the actual behaviour of the system during a high speed compression test.
Obtaining the dynamometer frequency response function There are several methods of measuring system frequency response functions. The three basic methods are: (i) The frequency sweep method. (ii) The impulse response method. (iii) The wide band random noise method. These are well known and have been described by Broch? 6 In the current study, the Fourier transform of the system output voltage was obtained by means of a Fast Fourier Transform TechniqueY This produced real and imaginary components at discrete frequency values up to a maximum frequency. Clearly, both real and imaginary components of the values of the frequency response are required at these same frequencies if the manipulation discussed above is to be carried out. All of the above methods have shortcomings. However, the impulse response method is both the most conveninet and accurate method for obtaining the required data. The frequency sweep method, while theoretically unlimited in dynamic range, is in fact limited to a maximum frequency somewhat less than the impulse response method, by the availability of exciters. Both methods (i) and (iii) are very time consuming. Method (ii) has the additional advantage that rechecking of the frequency response function is a simple matter which can be easily performed on site, thus ensuring that the system calibration does not alter. In the present work the impulse was supplied to the dynamometer by means of a free falling hardened steel ball bearing. Different size ball bearings were dropped from different heights, and the repeatability of the results was checked. Fig. 2 shows a typical time history obtained in this way using a 0-25 in. diameter ball bearing and Fig. 3 shows the corresponding frequency response function for the large capacity load cell. For both cells the frequency response function was obtained by averaging in the frequency domain
-
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, AAAAAAAA ,
ti i/
Vv
VVvvvvv';
,
i
,
J
TIME (rn sec}
FIG. 2. Time response of the large load cell to impact by 0.25 in. diameter ball bearing. Drop height 0.3 m.
556
A.J. HOLZER
several sets of data. It should be noted that only the modulus, calculated from the real and imaginary parts is shown. The data was recorded in a digital transient recorder and transmitted to a computer. The results were then obtained directly in the form of a plot from the computer. The hardware for signal handling and manipulation has been described in more detail previously 25.~ and is illustrated in Fig. 4. For the examples illustrated the recorder was set for a 2msec sweep and the capacity was 1024 words. Thus the first frequency obtained was 488Hz and the maximum was 250kHz. Therefore the problem of aliasing is avoided for this and all shorter duration signals. The resolution of the transient recorders is approximately 0.4%. These instruments have a pretrigger facility which allows recording of both the test impulses and actual deformation impulses to be repeatably commenced just prior to commencement of the signal. This is achieved by means of a continuously recirculating memory. Therefore the need for an external trigger signal is avoided. Although the impulse from the bali is not a perfect delta function, an idea of its effect on the frequency response function can be obtained by calculating the duration of contact of the ball with the bottom die. Goldsmithz9 has given the expression for contact duration r,
r = 4-53
I-(,s, + 8~)m]~'~ L-TE~--oJ
(3)
where 8~ and 82 are material properties and m, R and v0 are the mass, radius and impact velocity of a sphere colliding with a massive plane surface according to the Hertzian Law F = Ka
3/2
(4)
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RG. 3. Frequency response of large load cell for time data of Fig. 2. LOAD
CELL
S
Force Signal
ICHARGE L i L
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FIG. 4. Layout of measuring and analysing hardware.
A technique for obtaining compressive strength
557
where F is the Force and a is the approach of the colliding bodies. Evaluation of (3) for the 0.25 in. diameter ball bearing used in obtaining Figs. 2 and 3 yields a value of contact duration a little less than 20/zsec. The Fourier transforms for a rectangular shock pulse and a half-sine shock pulse are well known 3° and illustrated in Fig. 5 for the frequency range 1-100kHz. The ordinate shows the normalised Fourier transform in dB. The duration of the pulse used in calculating the Fourier transforms of Fig. 5 was 20 t~sec for both curves. In both cases it can be seen that the Fourier transform of the shock is flat to about 6 kHz and not more than 2dB down at 18 kHz. It is concluded that although the nature of the impulse used for the force measurement system frequency response determination is not exactly known, the response is substantially unaffected up to 20 kHz. For smaller balls this frequency value increases. That this is the case was verified experimentally by the use of several different size balls as mentioned above. The frequency response function of the system as shown by a modulus vs frequency plot was almost identical for 0.125, 0.25 and 0-5 in. balls. A difference in the frequency response function was evident when l and 2in. balls were used. This may possibly be due to the plastic deformation which occurred on the bottom die when impacted with the larger balls, and also to the fact that the duration of the contact was longer, and therefore the impulse was not as good as that obtained with the smaller balls. It was concluded that the frequency response functions of both load cells was represented sufficiently accurately by the functions obtained from the 0.25 in. ball. As the technique described involves a complex division, there is a tendency for high frequency noise amplification to occur. It was found necessary to employ low pass filters at both the analogue and digital stages of the signal manipulation. This aspect has been discussed more fully by Holzer and Koss 2~. Nevertheless it should be pointed out here that the first natural frequency of the large load cell occurs at about 8 kHz and of the small load cell at about 25 kHz. However, the technique presented has extended the usable range of these devices to about 15 kHz in the former case and 32 kHz in the latter.
The displacement-time signal Displacement is used here to refer to the separation of the top and bottom dies of Fig. 1. The displacement-time record is obtained from a fibre-optic device. This type of transducer was used by Woodward and Brown. ~° The response of this transducer is non-linear and displacement time records are obtained automatically by interpolation into a calibration table held on file in the processing computer. Cubic spline methods are used for this interpolation and have also been used sucessfuily for obtaining instantaneous natural strain rates from the displacement-time record with an accuracy of better than 2%. As indicated in Fig. 4, the raw displacement signal is handled in a similar fashion to the force signal. The response of the transducer and associated electronics is greater than 50 kHz and therefore no correction for dynamic characteristics is necessary. As the transducer is mounted in the bottom die of the large load cell, deflection of this die is automatically compensated. This type of mounting was not possible with the smaller die, however calculation of deflection at maximum load indicated that errors introduced were not significant. R E S U L T S O B T A I N E D BY T H E T E C H N I Q U E Fig. 6 shows a typical raw force signal and the corresponding corrected force signal plotted on the same axes. It can be seen that the ringing of the raw signal has been removed, and a phase correction has also been made. The intermediate steps showing the force data in the frequency domain have been given previously.2s Accuracy of the phase correction can be obtained by ascertaining the commencement time of deformation from the displacement-time record (Fig. 7) of the test. The specimen in this case was 8.03 mm high. Thus from Fig. 7 deformation commenced 0.28 msec. after the commencement of recording. This is also the commencement time for the corrected force. The raw force trace commences approximately 0.36 msec. after time equal to zero. The two signal recorders are synchronised to within 1 psec.
F(f) (dB)
0
-5.1
- 101
-1~
I
i
=
t
,
k ~
I
J
11
10
I
I I
100
FREQUENCY
(kHz)
FIG. 5. Fourier transform of a rectangular shock pulse and a half-sine shock pulse.
MS VoL 20. No. 9--B
558
A.J.
HOLZER
25-
20-
CORRECTED SIGNAL~ , . . . . j , ~ . 15-
nO U- I0-
t"
AWSGNAL :/ t
TIME (m set:)
FIG. 6. Raw force and corrected force vs time. Material :commercial aluminium. 9
B--
t---
Z i,i 7-IM _d
5--
4
I
1
I
I
I
I
I
I
I
TIME (m sec) F I G . 7.
D i s p l a c e m e n t vs t i m e r e c o r d for test o f Fig.
6.
Some true stress-natural strain curves for a medium carbon re-sulphurised steel are given in Fig. 8. The composition of this steel was C--0.14%, Mn-l.10%, Si-0.01%, S-0.28%, P-0.01% Ni-0.02%, Cr-0.02%, Mo-0-01%. T h e s e curves were drawn by a computer driven Calcomp plotter. The values of true stress and natural strain were computed directly by the computer from force and displacment data such as that illustrated earlier. A study of the displacement-time and natural strain-time results indicated that providing the force of deformation was below a certain threshold value, the velocity of the tup could be considered constant. Thus it was a simple matter to calculate mean natural strain rate as the deformation was interrupted at a natural strain of about 0.7 or 50% reduction. Therefore the mean natural strain rate is given by
~; = 1.5i, = 1.5 £Io
(5)
where di is the initial natural strain rate, v the constant test velocity and 10 the initial specimen height. The mean natural strain rate for each curve is indicated in Fig. 8. Friction effects at the specimen tooling interfaces are important in the large deformation tests described. For the results given in Fig. 8, the specimen sizes were nominally the same ( 4 . 0 0 × 3 . 0 0 ± 0 . 0 1 mm). Lubrication was effected by means of a Teflon film as has been done by Woodward and Brown. 2° It w a s noted that "barrelling" of the specimens increased very marginally as the strain rate was increased.
A technique for obtaining compressive strength
559
BOO
~400~lfS/ -\~i~2~! C\....._IVI~~t .~~ Ro(s-l te ) \\
~oolI~
\\
. 3.o~,o-~
\ ,X--3o~,0", \\ 3°~'°° 3ox,o'
\\
1.0 x 103 El
o
I
I
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PiG. 8. Some typical true stress natural strain curves at different mean natural strain rates. Material: medium carbon resulphurised steel. Nevertheless, it was small compared with that which occurred when specimens were deformed dry. It was noted also that the results were affected by variation of height/diameter ratio and size. Further work investigating this aspect is being concluded at present and it is hoped to publish the results shortly. Referring to Fig. 8, curve 1 was obtained using the experimental drop hammer and the techniques described. Curves 2-5 were obtained using the same recording and computing methods and an hydraulic testing machine. The Fourier transform technique was unnecessary as the strain rates were intermediate only, Curve 6 was obtained using a conventional screw test machine. The normal compliance corrections were made automatically by computer. It must be pointed out that these true stress-natural strain characteristics are so few in number that no firm conclusions can be drawn regarding material behaviour at this stage. A more complete study of material behaviour including a comparison with other published work and using the techniques described here will be published shortlyfl A number of points regarding this data can be made however. The first is that the material seems to obey the now accepted phenomena of increasing stress with increasing strain rate. Both the yield stress and flow stress are shown to increase, indeed the yield stress seems to have increased greatly from the "static" value. The second point is that although the absolute value of stress increases with increasing strain rate, the rate of work hardening decreases with increasing strain rate. The third point concerns the yield point resolution of the measurement technique described. Clearly the material tested exhibits upper and lower yield points. This effect is evident in curve 1 Fig. 8 however, it is not as clearly resolved aswould be expected. It is also pointed out that the value of Young's modulus would be greatly underestimated using curve 1. Nevertheless, this curve highlights the fact that the yield point phenomenon can be isolated from a ringing dynamic signal of the type illustrated in Fig. 6, It is hoped to improve the system response by further refinement of the data handling software. In particular improvements in the analogue and digital filtering methods are expected to improve the results obtained to date.
CONCLUSIONS A v i a b l e s y s t e m for m e a s u r e m e n t of s t r e s s - s t r a i n c h a r a c t e r i s t i c s at low, m e d i u m a n d high s t r a i n r a t e s has b e e n d e s c r i b e d . T h e t e c h n i q u e s p r e s e n t e d h a v e i n d i c a t e d that reliable d a t a c a n be o b t a i n e d w i t h o u t r e s o r t i n g to m o r e c o m p l e x a p p a r a t u s s u c h as split H o p k i n s o n P r e s s u r e Bar m e t h o d s . T h e s e t e c h n i q u e s are able to isolate the f o r c e s b e i n g m e a s u r e d f r o m the d y n a m i c effects of the m e a s u r e m e n t s y s t e m . T h e y h a v e also i n d i c a t e d a n a b i l i t y to m a k e p h a s e c o r r e c t i o n s a n d h a v e s h o w n that " e y e i n g i n " m a y n o t be a n a c c u r a t e m e t h o d of i n t e r p r e t a t i o n . T h e s y s t e m a n d t e c h n i q u e s d e s c r i b e d are efficient in that c o m p r e s s i o n s t r e n g t h d a t a has p r o v e d a m e n a b l e to h a n d l i n g a n d a n a l y s i s b y c o m p u t e r . H e n c e , the a d v e n t of simplified a u t o m a t e d t e s t i n g at higher s t r a i n rates is at h a n d .
Acknowledgements--The author is grateful in particular to Prof. R. H. Brown of the University of Western Australia for supervision and help during the course of the above work and also to Drs. H. Nolle and L. L. Koss of Monash University for much assistance. Thanks is also extended to Mr. R. R. Peach for help with the experimental program.
560
A.J. HOLZER
REFERENCES 1. H. KOLSKY, Proc. Phys. Soc. (London), B62, 676 (1949). 2. U. S. LINDHOLM, Proc. Conf. on Mech. Properties at High Rates of Strain, pp. 3-21. Inst. of Phys., Oxford (1974). 3. H. NOLLE, Int. Met. Rev. 19, 223 (1974). 4. J. O. CAMPBELL, Mater. Sci. & Eng. 12, 3 (1973). 5. K. B1TANSand P. W. WHI~'rON, Int. Met. Rev. 17, 66 (1972). 6. C. M. SELEARS and W. J. McG. TE6ART, Int. Met. Rev. 17, i (1972). 7. H. G. HOPKINS, Applied Mechanics Surveys, pp. 847-867. Spartan, Washington (1966). 8. E. T. HABIB, J. Appl. Mech. Trans A S M E 15, 248 (1948). 9. F. V. WARNOCK and D. B. C. TAYLOR. Proc. Instn. Mech. Engrs. 161, 165 (1949). 10. D. B. C. TAYLOR, J. Mech. Phys. Solids 3, 38 (1954). I I. D. B. C. TAYLOR, Proc. Conf. on the Properties of Materials at High Rates of Strain, pp. 229-238. Inst. Mech. Engrs., London (1957). 12. G. L. BARAYA, W. JOHNSON and R. A. C. SLATER, Int. J. Mech. Sci. 7, 621 (1965). 13. R. A. C. SEATER, W. JOHNSON and S. Y. AKU, Int. Z Mech. Sci. 10, 169 (1968). 14. R. A. C. SEATER, W. JOHNSON and S. Y. AKU, Proc. 9th Int. M.T.D.R. Con[, pp. 115-133 (1968). 15. R. A. C. SEATER,S. Y. AKU and W. JOHNSON, Annals of C.I.R.P. 19, 513 (1971). 16. S. C. JAIN and E. AMINI, Proc. 9th Int. M.T.D.R. Conf. pp. 229-237 (1968). 17. B. LENGYEE and M. MOH1TPOUR, J. Inst. Metals 100, I (1972). 18. H. SUZUKI, S. HASIZUMA, Y. YAKUB1, Y. ICHIHARA, S. NAKAJIMAand K. KENMOCHI, Rep. Inst. Ind. Sci. Tokyo Univ. 18, l (1968). 19. S. K. SAMANTA,Int. J. Mech. Sci. 10, 613 (1968). 20. R. L. WOODWARDand R. H. BROWN, Proc. Instn. Mech. Engrs. 189, 107 (1975). 21. S. K. SAMANTA,J. Mech. Phys. Solids 19, !17 (1971). 22. R. M. DAVIES, Phil. Trans. Roy. Soc. A240, 375 (1949). 23. F. E. HAUSER, Exp. Mech. 6, 395 (1966). 24. W. E. JAHSMAN, J. Appl. Mech A S M E 38, 75 (1971). 25. A. J. HOLZER and R. H. BROWN, 2nd Int. Conf. Mech. Behaviour of Materials pp. 1574-1578. Fed. Materials Soc. Boston (1976). 26. J. T. BROCH, Briiel Kjaer tech. Rev. No. 4, 3 (1975). 27. J. W. COOLEY and J. W. TURKEY, Maths. Comput. 19, 297 (1965). 28. A. J. HOllER and L. L. Koss, Proc. Aust. Conf. Manufacturing Engng pp. 167-171. Inst. Eng. Aust. (1977) 29. W. GOLDSMITH, Impact. The Theory and Physical Behaviour of Colliding Solids, p. 90. Arnold, London (1960). 30. J. T. BROCK, Mechanical Vibrations and Shock Measurements, p. 28. Briiel & Kjaer, Denmark (1976). 31. A. J. HOLZER and R. H. BROWN, 28th C.I.R.P. General Assembly, Eindhoven (1978) (To be published).
0020-7403/78/0901~0553/$02.0(t/0
A T E C H N I Q U E FOR OBTAINING COMPRESSIVE STRENGTH AT HIGH STRAIN RATES USING SHORT LOAD CELLS A. J. HOLZER Department of Mechanical Engineering, Monash University, Clayton, Victoria 3168, Australia (Received 29 N o v e m b e r 1977; in revised f o r m 21 February 1978)
Summary--A technique for obtaining stress-strain data at high strain rates is given. The method involves use of a short load cell and a fibre optics displacement transducer. Computer aided analysis of the force-time data allows correction of the signal for the effect of the dynamic characteristic of the force measurement system. The method involves transformation to the frequency domain of the force data by means of a Fast Fourier Transform algorithm, and is shown to be able to make effective corrections for the dynamic ringing of the load cell and the phase lag of the recording system. The compression mode of deformation is used, lubrication being effected by means of a Teflon film. The results of preliminary tests performed on a medium carbon resulphurised steel are given.
NOTATION F Fi(t) G,to,,~ K R Vo(t) [(D) Io m v v0 a 8t,~
Hertzian Force input (force) system constants constant radius output (voltage) system transfer function initial specimen length mass velocity of tup impact velocity approach of colliding bodies material property mean natural strain rate ~i initial natural strain rate r contact duration.
INTRODUCTION
Experimental work aimed at evaluating stress-strain characteristics of materials deformed at high rates of strain has received considerable attention since Kolsky 1 introduced the concept of the split Hopkinson Pressure Bar. The resulting work of many experimenters is quite well known and has been reviewed by several authors. 2-7 The method for experimental evaluation of material properties given in the current paper differs from most previous work in that a short load cell is used for measurement of short duration force pulses. A method for analysing the force data generated by such a load cell is described. The following brief review is therefore confined to previously used methods of force measurement. P R E V I O U S M E T H O D S OF F O R C E M E A S U R E M E N T Many early experimenters, attempting to obtain material properties at high rates of deformation, used energy absorption techniques for calculation of forces and stresses. Indeed Habib 8 has used a technique of this type as recently as 1948. Methods of this type can of course only yield mean or average forces for a certain time period. Instantaneous force-time histories cannot be obtained. Warnock and Taylor 9 and Taylor j°'" used electrical resistance strain gauges mounted directly on the flange of a tensile specimen for stress measurement at high strain rates. They point out 9 that vibration of the stress recording dynamometers is often severe in the neighbourhood of the yield stress and may mask the yield process in high strain rate investigations. Energy methods were also used in the early work of Baraya et al. n using an experimental drop hammer. The same drop hammer was subsequently instrumented with direct reading force and displacement transducers ~3 and investigations of the compressive strength of pure lead ~3 and (0.55%C) steel 14a~ performed. The force transducer used in these investigations was a short dumbell solid 553
554
A.J. HOLZER
cell using four strain gauges. Some minor vibration is evident on the force-time osciliograms presented. No dynamic calibration was performed. Jain and Amini t~ examined the short ring type load cell, and concluded that it is superior to the short solid types. Lengyel and Mohitpour t7 used a cell of the type described by Jain and Amini, and dismiss a small peak in their load-time curve as being due to reflected stress waves, concluding that the results were unaffected. No dynamic calibration of the load cell was attempted. Suzuki et aL la used a capacitor strain meter, a cell of the short type, for load measurement. The authors themselves concede that vibration of the load measuring instrument causes the greatest difficulty in interpreting the results. Samanta ~9 used an accelerometer mounted on the tup of an impacting drop hammer to obtain force-time records. Vibration due to dynamic ringing of the tup is evident on the oscillograms presented. Assumptions are also made regarding the rigidity of the mass in order to calculate forces from accelerations. Woodward and Brown 2° used a short piezo-electric load cell to measure force in high strain rate investigations, and showed that the maximum strain rate for which a reliable force record can be obtained is determined by the frequency response of the load cell used. They obtained instantaneous force-time records by drawing a mean line through the oscillations of the force transducer output, The split Hopkinson Pressure Bar has been used extensively to record short duration force pulses, This method is limited in that the input and output bars must, or are assumed to, remain elastic and the force values obtained are average values for the time of the pulse in the specimen. Hence most work using this technique has been restricted to comparatively small values of total strain. Complete stress-strain characteristics have usually been obtained by superposition of the results from each impact. Thus an essentially incremental approach is used. Samanta z~ has oversome the problem to some extent by use of a long (3-5 m) pressure bar in a modified Hopkinson apparatus. The technique of the split Hopkinson Pressure bar is extremely well documented. ~'~2'23'24 It is clear that all of the above methods of force measurement are limited in one respect or another. While each method discussed is valid in certain applications, there seems to be no universal method for obtaining instantaneous force-time histories during experiments where deformation is carried out to large strains at high rates of strain. When measuring forces in dynamic situations it is possible that a force frequency and a natural frequency of the force measuring system will correspond. This situation is even more likely if the material being studied exhibits a yield point phenomenon. Thus. any yield point phenomenon is likely to be camouflaged by ringing and the results generally will be uninterpretable. It is therefore necessary to correct the static calibration of a force measuring system to account for its dynamic behaviour. In the following sections a method of performing this correction is described. T H E B A S I S OF THE T E C H N I Q U E Description o f f o r c e transducer and mounting A gravity drop forge has been developed for the purpose of high strain rate investigations and has been described previously.2s This apparatus differs from the drop hammer of Siater et al. ~3-~5 in that the tup is arrested by means of stop blocks, the excess energy being absorbed by a mass air-suspension system. Thus the strain rate can be made to remain nearly constant or to increase throughout the major part of a deformation test. Two alternative force measuring systems have been developed, one of medium frequency response but large capacity (400 kN) and the other of higher frequency response and smaller capacity (30 kN). Each consists of a piezo-electric force transducer fitted beneath a compression die. The large load cell and its associated mounting is illustrated in Fig. 1. The arrangement for the small cell is of an identical nature except that space has not allowed mounting of the displacement feeler integral with the bottom die. In each case the mounting consists of the die, an elastic preload stud, a locking washer and, beneath the load cell, a TUP
1
|
" I ''~ TOP DIE
1
STOP |LOCK DISPLACEMENT GAUGE
LOAD
CELL
- -
SPACEr WASH| It ,r
..
~|oCu
5000
Kg
N IbOLT
MASS
FIG. 1. Large force measurement system, capacity 400 kN, natural frequency 8 kHz.
A technique for obtaining compressive strength
555
spacer washer. The surfaces of all components are lapped together to ensure minimum dynamic ringing. Both the top and bottom dies are hardened to about 60 Rockwell C. The elastic bolt (beryllium copper) is used to apply a preload to the load cell. This ensures that the cell is operating in the linear portion of its range and that sufficient load is on all the components so that extraneous vibration is not introduced. The load cell and beryllium copper bolt are equivalent to a parallel spring mass damper system. Thus, as the applied force is not actually that measured by the load cell, it is necessary to make a correction to the static calibration of the free load cell. The preload, the bolt material and its dimensions are chosen to ensure that the bolt remains in tension up to the maximum load used.
Outline of signal correction method It will be appreciated that the load cell and associated mounting of Fig. I is analogous to a second order linear spring mass damper system. Considering such a second order system, the transfer function f(D) can be written f(D) = Vo(t) F/(t)
G m 2D2 + 2_~ D + 1
(I)
¢O n
where F~ is the disturbance or input force, V0 is the output, in the case of the load measurement system a voltage, and G, to, and £ are system constants. It is evident from (i) that if V0 and f(D) are known, then F~ can be obtained, i.e.
Vo(t) F~(t) = f(D)"
(2)
The simplest method of performing this manipulation with real data is to obtain the Fourier transform of V(t) and then divide this by the frequency domain transfer function or frequency response function. Inverse Fourier transformation then yields the actual force input to the system as a function of time. The main problem is obtaining a reliable frequency response function for the system that represents the actual behaviour of the system during a high speed compression test.
Obtaining the dynamometer frequency response function There are several methods of measuring system frequency response functions. The three basic methods are: (i) The frequency sweep method. (ii) The impulse response method. (iii) The wide band random noise method. These are well known and have been described by Broch? 6 In the current study, the Fourier transform of the system output voltage was obtained by means of a Fast Fourier Transform TechniqueY This produced real and imaginary components at discrete frequency values up to a maximum frequency. Clearly, both real and imaginary components of the values of the frequency response are required at these same frequencies if the manipulation discussed above is to be carried out. All of the above methods have shortcomings. However, the impulse response method is both the most conveninet and accurate method for obtaining the required data. The frequency sweep method, while theoretically unlimited in dynamic range, is in fact limited to a maximum frequency somewhat less than the impulse response method, by the availability of exciters. Both methods (i) and (iii) are very time consuming. Method (ii) has the additional advantage that rechecking of the frequency response function is a simple matter which can be easily performed on site, thus ensuring that the system calibration does not alter. In the present work the impulse was supplied to the dynamometer by means of a free falling hardened steel ball bearing. Different size ball bearings were dropped from different heights, and the repeatability of the results was checked. Fig. 2 shows a typical time history obtained in this way using a 0-25 in. diameter ball bearing and Fig. 3 shows the corresponding frequency response function for the large capacity load cell. For both cells the frequency response function was obtained by averaging in the frequency domain
-
AAA
, AAAAAAAA ,
ti i/
Vv
VVvvvvv';
,
i
,
J
TIME (rn sec}
FIG. 2. Time response of the large load cell to impact by 0.25 in. diameter ball bearing. Drop height 0.3 m.
556
A.J. HOLZER
several sets of data. It should be noted that only the modulus, calculated from the real and imaginary parts is shown. The data was recorded in a digital transient recorder and transmitted to a computer. The results were then obtained directly in the form of a plot from the computer. The hardware for signal handling and manipulation has been described in more detail previously 25.~ and is illustrated in Fig. 4. For the examples illustrated the recorder was set for a 2msec sweep and the capacity was 1024 words. Thus the first frequency obtained was 488Hz and the maximum was 250kHz. Therefore the problem of aliasing is avoided for this and all shorter duration signals. The resolution of the transient recorders is approximately 0.4%. These instruments have a pretrigger facility which allows recording of both the test impulses and actual deformation impulses to be repeatably commenced just prior to commencement of the signal. This is achieved by means of a continuously recirculating memory. Therefore the need for an external trigger signal is avoided. Although the impulse from the bali is not a perfect delta function, an idea of its effect on the frequency response function can be obtained by calculating the duration of contact of the ball with the bottom die. Goldsmithz9 has given the expression for contact duration r,
r = 4-53
I-(,s, + 8~)m]~'~ L-TE~--oJ
(3)
where 8~ and 82 are material properties and m, R and v0 are the mass, radius and impact velocity of a sphere colliding with a massive plane surface according to the Hertzian Law F = Ka
3/2
(4)
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0
|
i
w n," 121 w -20-
Z -40-
-
- ~ I )3
• 2
• 4
!
3
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2'
)4
5'
'~
;
;
{89
5
FREQUENCY (Hz)
RG. 3. Frequency response of large load cell for time data of Fig. 2. LOAD
CELL
S
Force Signal
ICHARGE L i L
°, s, Ii
AMP.
IvoL,,G
I,ECO,DE, T~ANS,EMTI
?
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I
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I ----- I lC'OI
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R
C 0 MPUTER
TERMINAL
FIG. 4. Layout of measuring and analysing hardware.
A technique for obtaining compressive strength
557
where F is the Force and a is the approach of the colliding bodies. Evaluation of (3) for the 0.25 in. diameter ball bearing used in obtaining Figs. 2 and 3 yields a value of contact duration a little less than 20/zsec. The Fourier transforms for a rectangular shock pulse and a half-sine shock pulse are well known 3° and illustrated in Fig. 5 for the frequency range 1-100kHz. The ordinate shows the normalised Fourier transform in dB. The duration of the pulse used in calculating the Fourier transforms of Fig. 5 was 20 t~sec for both curves. In both cases it can be seen that the Fourier transform of the shock is flat to about 6 kHz and not more than 2dB down at 18 kHz. It is concluded that although the nature of the impulse used for the force measurement system frequency response determination is not exactly known, the response is substantially unaffected up to 20 kHz. For smaller balls this frequency value increases. That this is the case was verified experimentally by the use of several different size balls as mentioned above. The frequency response function of the system as shown by a modulus vs frequency plot was almost identical for 0.125, 0.25 and 0-5 in. balls. A difference in the frequency response function was evident when l and 2in. balls were used. This may possibly be due to the plastic deformation which occurred on the bottom die when impacted with the larger balls, and also to the fact that the duration of the contact was longer, and therefore the impulse was not as good as that obtained with the smaller balls. It was concluded that the frequency response functions of both load cells was represented sufficiently accurately by the functions obtained from the 0.25 in. ball. As the technique described involves a complex division, there is a tendency for high frequency noise amplification to occur. It was found necessary to employ low pass filters at both the analogue and digital stages of the signal manipulation. This aspect has been discussed more fully by Holzer and Koss 2~. Nevertheless it should be pointed out here that the first natural frequency of the large load cell occurs at about 8 kHz and of the small load cell at about 25 kHz. However, the technique presented has extended the usable range of these devices to about 15 kHz in the former case and 32 kHz in the latter.
The displacement-time signal Displacement is used here to refer to the separation of the top and bottom dies of Fig. 1. The displacement-time record is obtained from a fibre-optic device. This type of transducer was used by Woodward and Brown. ~° The response of this transducer is non-linear and displacement time records are obtained automatically by interpolation into a calibration table held on file in the processing computer. Cubic spline methods are used for this interpolation and have also been used sucessfuily for obtaining instantaneous natural strain rates from the displacement-time record with an accuracy of better than 2%. As indicated in Fig. 4, the raw displacement signal is handled in a similar fashion to the force signal. The response of the transducer and associated electronics is greater than 50 kHz and therefore no correction for dynamic characteristics is necessary. As the transducer is mounted in the bottom die of the large load cell, deflection of this die is automatically compensated. This type of mounting was not possible with the smaller die, however calculation of deflection at maximum load indicated that errors introduced were not significant. R E S U L T S O B T A I N E D BY T H E T E C H N I Q U E Fig. 6 shows a typical raw force signal and the corresponding corrected force signal plotted on the same axes. It can be seen that the ringing of the raw signal has been removed, and a phase correction has also been made. The intermediate steps showing the force data in the frequency domain have been given previously.2s Accuracy of the phase correction can be obtained by ascertaining the commencement time of deformation from the displacement-time record (Fig. 7) of the test. The specimen in this case was 8.03 mm high. Thus from Fig. 7 deformation commenced 0.28 msec. after the commencement of recording. This is also the commencement time for the corrected force. The raw force trace commences approximately 0.36 msec. after time equal to zero. The two signal recorders are synchronised to within 1 psec.
F(f) (dB)
0
-5.1
- 101
-1~
I
i
=
t
,
k ~
I
J
11
10
I
I I
100
FREQUENCY
(kHz)
FIG. 5. Fourier transform of a rectangular shock pulse and a half-sine shock pulse.
MS VoL 20. No. 9--B
558
A.J.
HOLZER
25-
20-
CORRECTED SIGNAL~ , . . . . j , ~ . 15-
nO U- I0-
t"
AWSGNAL :/ t
TIME (m set:)
FIG. 6. Raw force and corrected force vs time. Material :commercial aluminium. 9
B--
t---
Z i,i 7-IM _d
5--
4
I
1
I
I
I
I
I
I
I
TIME (m sec) F I G . 7.
D i s p l a c e m e n t vs t i m e r e c o r d for test o f Fig.
6.
Some true stress-natural strain curves for a medium carbon re-sulphurised steel are given in Fig. 8. The composition of this steel was C--0.14%, Mn-l.10%, Si-0.01%, S-0.28%, P-0.01% Ni-0.02%, Cr-0.02%, Mo-0-01%. T h e s e curves were drawn by a computer driven Calcomp plotter. The values of true stress and natural strain were computed directly by the computer from force and displacment data such as that illustrated earlier. A study of the displacement-time and natural strain-time results indicated that providing the force of deformation was below a certain threshold value, the velocity of the tup could be considered constant. Thus it was a simple matter to calculate mean natural strain rate as the deformation was interrupted at a natural strain of about 0.7 or 50% reduction. Therefore the mean natural strain rate is given by
~; = 1.5i, = 1.5 £Io
(5)
where di is the initial natural strain rate, v the constant test velocity and 10 the initial specimen height. The mean natural strain rate for each curve is indicated in Fig. 8. Friction effects at the specimen tooling interfaces are important in the large deformation tests described. For the results given in Fig. 8, the specimen sizes were nominally the same ( 4 . 0 0 × 3 . 0 0 ± 0 . 0 1 mm). Lubrication was effected by means of a Teflon film as has been done by Woodward and Brown. 2° It w a s noted that "barrelling" of the specimens increased very marginally as the strain rate was increased.
A technique for obtaining compressive strength
559
BOO
~400~lfS/ -\~i~2~! C\....._IVI~~t .~~ Ro(s-l te ) \\
~oolI~
\\
. 3.o~,o-~
\ ,X--3o~,0", \\ 3°~'°° 3ox,o'
\\
1.0 x 103 El
o
I
I
I
I
I
I
I
I
STRAIN
PiG. 8. Some typical true stress natural strain curves at different mean natural strain rates. Material: medium carbon resulphurised steel. Nevertheless, it was small compared with that which occurred when specimens were deformed dry. It was noted also that the results were affected by variation of height/diameter ratio and size. Further work investigating this aspect is being concluded at present and it is hoped to publish the results shortly. Referring to Fig. 8, curve 1 was obtained using the experimental drop hammer and the techniques described. Curves 2-5 were obtained using the same recording and computing methods and an hydraulic testing machine. The Fourier transform technique was unnecessary as the strain rates were intermediate only, Curve 6 was obtained using a conventional screw test machine. The normal compliance corrections were made automatically by computer. It must be pointed out that these true stress-natural strain characteristics are so few in number that no firm conclusions can be drawn regarding material behaviour at this stage. A more complete study of material behaviour including a comparison with other published work and using the techniques described here will be published shortlyfl A number of points regarding this data can be made however. The first is that the material seems to obey the now accepted phenomena of increasing stress with increasing strain rate. Both the yield stress and flow stress are shown to increase, indeed the yield stress seems to have increased greatly from the "static" value. The second point is that although the absolute value of stress increases with increasing strain rate, the rate of work hardening decreases with increasing strain rate. The third point concerns the yield point resolution of the measurement technique described. Clearly the material tested exhibits upper and lower yield points. This effect is evident in curve 1 Fig. 8 however, it is not as clearly resolved aswould be expected. It is also pointed out that the value of Young's modulus would be greatly underestimated using curve 1. Nevertheless, this curve highlights the fact that the yield point phenomenon can be isolated from a ringing dynamic signal of the type illustrated in Fig. 6, It is hoped to improve the system response by further refinement of the data handling software. In particular improvements in the analogue and digital filtering methods are expected to improve the results obtained to date.
CONCLUSIONS A v i a b l e s y s t e m for m e a s u r e m e n t of s t r e s s - s t r a i n c h a r a c t e r i s t i c s at low, m e d i u m a n d high s t r a i n r a t e s has b e e n d e s c r i b e d . T h e t e c h n i q u e s p r e s e n t e d h a v e i n d i c a t e d that reliable d a t a c a n be o b t a i n e d w i t h o u t r e s o r t i n g to m o r e c o m p l e x a p p a r a t u s s u c h as split H o p k i n s o n P r e s s u r e Bar m e t h o d s . T h e s e t e c h n i q u e s are able to isolate the f o r c e s b e i n g m e a s u r e d f r o m the d y n a m i c effects of the m e a s u r e m e n t s y s t e m . T h e y h a v e also i n d i c a t e d a n a b i l i t y to m a k e p h a s e c o r r e c t i o n s a n d h a v e s h o w n that " e y e i n g i n " m a y n o t be a n a c c u r a t e m e t h o d of i n t e r p r e t a t i o n . T h e s y s t e m a n d t e c h n i q u e s d e s c r i b e d are efficient in that c o m p r e s s i o n s t r e n g t h d a t a has p r o v e d a m e n a b l e to h a n d l i n g a n d a n a l y s i s b y c o m p u t e r . H e n c e , the a d v e n t of simplified a u t o m a t e d t e s t i n g at higher s t r a i n rates is at h a n d .
Acknowledgements--The author is grateful in particular to Prof. R. H. Brown of the University of Western Australia for supervision and help during the course of the above work and also to Drs. H. Nolle and L. L. Koss of Monash University for much assistance. Thanks is also extended to Mr. R. R. Peach for help with the experimental program.
560
A.J. HOLZER
REFERENCES 1. H. KOLSKY, Proc. Phys. Soc. (London), B62, 676 (1949). 2. U. S. LINDHOLM, Proc. Conf. on Mech. Properties at High Rates of Strain, pp. 3-21. Inst. of Phys., Oxford (1974). 3. H. NOLLE, Int. Met. Rev. 19, 223 (1974). 4. J. O. CAMPBELL, Mater. Sci. & Eng. 12, 3 (1973). 5. K. B1TANSand P. W. WHI~'rON, Int. Met. Rev. 17, 66 (1972). 6. C. M. SELEARS and W. J. McG. TE6ART, Int. Met. Rev. 17, i (1972). 7. H. G. HOPKINS, Applied Mechanics Surveys, pp. 847-867. Spartan, Washington (1966). 8. E. T. HABIB, J. Appl. Mech. Trans A S M E 15, 248 (1948). 9. F. V. WARNOCK and D. B. C. TAYLOR. Proc. Instn. Mech. Engrs. 161, 165 (1949). 10. D. B. C. TAYLOR, J. Mech. Phys. Solids 3, 38 (1954). I I. D. B. C. TAYLOR, Proc. Conf. on the Properties of Materials at High Rates of Strain, pp. 229-238. Inst. Mech. Engrs., London (1957). 12. G. L. BARAYA, W. JOHNSON and R. A. C. SLATER, Int. J. Mech. Sci. 7, 621 (1965). 13. R. A. C. SEATER, W. JOHNSON and S. Y. AKU, Int. Z Mech. Sci. 10, 169 (1968). 14. R. A. C. SEATER, W. JOHNSON and S. Y. AKU, Proc. 9th Int. M.T.D.R. Con[, pp. 115-133 (1968). 15. R. A. C. SEATER,S. Y. AKU and W. JOHNSON, Annals of C.I.R.P. 19, 513 (1971). 16. S. C. JAIN and E. AMINI, Proc. 9th Int. M.T.D.R. Conf. pp. 229-237 (1968). 17. B. LENGYEE and M. MOH1TPOUR, J. Inst. Metals 100, I (1972). 18. H. SUZUKI, S. HASIZUMA, Y. YAKUB1, Y. ICHIHARA, S. NAKAJIMAand K. KENMOCHI, Rep. Inst. Ind. Sci. Tokyo Univ. 18, l (1968). 19. S. K. SAMANTA,Int. J. Mech. Sci. 10, 613 (1968). 20. R. L. WOODWARDand R. H. BROWN, Proc. Instn. Mech. Engrs. 189, 107 (1975). 21. S. K. SAMANTA,J. Mech. Phys. Solids 19, !17 (1971). 22. R. M. DAVIES, Phil. Trans. Roy. Soc. A240, 375 (1949). 23. F. E. HAUSER, Exp. Mech. 6, 395 (1966). 24. W. E. JAHSMAN, J. Appl. Mech A S M E 38, 75 (1971). 25. A. J. HOLZER and R. H. BROWN, 2nd Int. Conf. Mech. Behaviour of Materials pp. 1574-1578. Fed. Materials Soc. Boston (1976). 26. J. T. BROCH, Briiel Kjaer tech. Rev. No. 4, 3 (1975). 27. J. W. COOLEY and J. W. TURKEY, Maths. Comput. 19, 297 (1965). 28. A. J. HOllER and L. L. Koss, Proc. Aust. Conf. Manufacturing Engng pp. 167-171. Inst. Eng. Aust. (1977) 29. W. GOLDSMITH, Impact. The Theory and Physical Behaviour of Colliding Solids, p. 90. Arnold, London (1960). 30. J. T. BROCK, Mechanical Vibrations and Shock Measurements, p. 28. Briiel & Kjaer, Denmark (1976). 31. A. J. HOLZER and R. H. BROWN, 28th C.I.R.P. General Assembly, Eindhoven (1978) (To be published).