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Two-dimensional analysis of the split hopkinson pressure bar system
3. Mech. Phys. Solids, 1975, Vol. 23. pp. 1 to 19. Pergamon Press.
Printed in Great Britain.
TWO-DIMENSIONAL ANALYSIS OF THE SPLIT HOPKINSON PRESSURE BAR SYSTEM? By L, D. BERTHOLF and C. H. KARNES Code Development Division and Mechanics of Materials Division, Sandia Laboratories, Albuquerque, New Mexico 87115, U.S.A. (Rrceived
19th Septembrr
1974)
SUMMARY THE SPLIT Hopkinson
pressure bar is widely used to measure the dynamic properties of solid materials. This paper presents the results of the first comprehensive two-dimensional numerical analysis of the technique, and quantitatively describes the effects of realistic friction and of variations in both the specimen geometry and the imposed strain-rate on the validity of the assumptions used in analyzing experimental data. A two-dimensional axisymmetric numerical analysis is used to compute all components of the stress, strain and strain-rate tensors at each mesh point within the specimen and the elastic bars. The calculated response of the pressure bars is used to reconstruct the stress-strain behavior of the specimen and this is compared to both the input stress-strain curve and the actual calculated stress-strain states in the specimen. Thus, the validity of the assumptions and the corrections used in the analysis of the data is determined. Inertia and friction between the specimen and the elastic bars affect the response of the specimen differently for different Ien~h-todiameter ratios. Inertia effects produce stress waves propagating radialty and axially in the specimen and may result in an oscillating reconstructed stress-strain curve. If the ends of the specimen are well lubricated and care is taken to minimize the effects of inertia, the reconstructed stress-strain curve agrees with the input. However, serious stress and strain nonuniformity exists when the ends are not lubricated and this results in a reconstructed stress-strain curve where, for any given strain, the stress magnitude is larger than the correct value. A comparison of the calculations with experiment shows excellent agreement for various interface conditions. Finally, the inertia correction of E. D. H. Davies and S. C. Hunter (1963) is found to be reasonable.
1. INTRODLJCTICW SINCE the original paper of KOLSKY (1949) describing the use of the split Hopkinson pressure bar (SHPB) system to determine dynamic mechanical properties of materials, many other investigators have used the technique in a more or less routine manner. The technique is used to obtain compression properties in the following way. A sample of material to be studied is placed between and in contact with two elastic rods of approximately the same but slightly larger diameter. A compression stress wave is initiated in one of the bars by impact with a suitable projectile rod. The compression wave propagates along the rod causing multiple reflections in the sample. This results in distorted waves reflected from the sample traveling back along the first rod and transmitted through the sample and into the second rod. Measurements of strain in the two elastic rods on either side of the sample can lead to quantitative information concerning the mechanical behavior of the sample if care is used in conducting the experiment and if the following two assumptions are maintained by
$ This work was supported by the U.S. Atomic Energy Commission.
2
L. D. BERTHOLF and C. H. KARNES
the conditions of the experiment: (i) the specimen is in a state of one-dimensional stress, and (ii) the stress and the strain are uniform throughout the specimen (for detailed discussions of the technique see KOL~KY(1949), HAUSER,SUMMONS and DORN (1960), and LINDHOLM(1964)). The above assumptions can be violated by radial and axial inertia effects and by friction between the specimen and the elastic bars. Approximate corrections have been derived and applied to the data to account for the effects of radial and axial inertia and friction. It is the objective of this paper to present the results of a comprehensive two-dimensional numerical study of the SHPB including the effects of inertia and friction. More specifically, the two-dimensional wave propagation analysis is used (i) to examine in detail the validity of the assumptions that must be made in analyzing the experimental data, (ii) to determine the limitations on the experimental parameters, (iii) to determine the adequacy of approximate corrections given by previous investigators, and (iv) to determine under what conditions vioIation of the assumptions gives erroneous indications of material strain-rate dependence.
2. REVIEWOF PREVIOUSCORRECTIONS AND ANALYSES KOLSKY(1949, 1963) described the use of the SHPB technique and introduced a correction for radial inertia. Radial inertia was assumed to cause a larger stress than that which would have resulted in its absence. The difference, 5,-a,, was evaluated approximately by equating the additional strain energy caused by the increased stress to the radial kinetic energy. Assuming small strains, he obtained 1 d2& cr, = 5, - - vs2d2p, -2, 8 dt
(1)
where 5, is the axial stress determined by the average of the stresses measured in the two bars; 5, is the axial stress required to deform the sample in a one-dimensional stress state; and v,, ps, d and E are Poisson’s ratio, density, diameter and axial strain, respectively, of the sample. Kolsky’s analysis assumed a very thin sampIe that remained in stress equilibrium in the axial direction, and because of sample thinness, was particularly susceptible to friction. DAVIESand HUNTER(1963) gave a more general analysis. In order to minimize friction effects in their experiments, Davies and Hunter used samples whose length-to-diameter ratios were approximately O-5, whereas Rolsky used a ratio of 0.05. Davies and Hunter chose their dimensions as a result of an analysis by SEBEL (1923) which led to ~d/3~ 4 I
(2)
as a criterion for neglecting friction effects, where ~1is the Coulomb friction coefficient and I is the length of the specimen. Davies and Hunter employed an analysis considering kinetic energy due to both axial and radial motions. Their analysis resulted in
where ps12(d2Jdt2)/6 is the axial inertia correction term and 5b is the stress at the interface between the specimen and the output bar.
Two-dimensional analysis of the split Hopkinson pressure bar system
3
RAND (1967) presented an analysis based on the work of JACKSONand WAXMAN (1963) which results in a nonuniform axial stress distribution due to radial friction. The analysis is based on static Coulomb friction and gives an apparent stress
O’a= 2a,o!-‘(e”-cr-
l),
(4)
where (rOis the true one-dimensional stress and cc=
d (1+&)-3’2. 7
DHARAN and HAUSER (1970) have employed a modified version of the SHPB technique by impacting a thin sample directly with a massive, high-strength projectile. The sample is mounted on an elastic pressure bar which is instrumented and serves as the output bar. Because strain-rates as high as 1.2 x lo5 s-r were desired, relatively thin samples were used which result in an amplification of the effects of friction and radial inertia. They employed an analysis to correct the stress-time measurements for the effects of radial inertia. However, the effects of friction were neglected. The analysis of SAMANTA(1971) is similar to that of DAVIES and HUNTER (1963) except that he included the material derivative and assumed incompressibility (v, = 4). His analysis as published included an error, but when corrected reduces to
(6) The first two terms on the right side of (6) are identical to those of Davies and Hunter if incompressibility is assumed. The term proportional to (d.s/&)* results from the convective portion of the material derivative. It can be shown for the calculations presented herein that this last term is no more than a few percent of the second term. Numerous investigators have attempted to analyze the wave propagation response of the SHPB technique. To the writers’ knowledge, all published analyses have been one-dimensional in which only axial stress waves and axial inertia are considered. For examples, see HAUSER,SIMMONS,and DORN (1960) CONN (1965), JAHSMAN(1971), and CHIU and NEUBERT(1967). 3. COMPUTATIONAL TECHNIQUE A description of the problem to be examined is illustrated in Fig. 1. The experimental parameters which are normally recorded are initial specimen length IO,initial specimen diameter d,,, bar diameter D, position of strain gauges from ends of bars (5D for these calculations), and the strain-time records obtained from the strain gauges. The TOODY two-dimensional elastic-plastic wave propagation computer code as developed by BERTHOLFand BENZLEY(1968) is used to simulate, numerically, INPUT BAR
OUTPUT BAR do
5
L/R
G2
FIG. 1. Schematic of SHPB showing pertinent dimensional parameters.
L. D. BERTHOLF and C. H. KARNE~
4
all pertinent features of an experimental SHPB. It is a second-order accurate, artificial viscosity, finite-difference code. It solves the finite difference representations to the equations of motion in Lagrangian coordinates. The stress and stretching tensors are decomposed into spherical and deviatoric components and it is assumed that the stretching can be expressed as the sum of elastic and plastic parts. For the constitutive relation it is assumed that the co-rotational stress-rate is a function of density and energy and also is a linear isotropic function of the stretching. The second invariant of the deviatoric stress tensor is limited by the von Mises yield criterion and the flow rule is chosen such that the plastic stretching is orthogonal to the yield surface. Shock waves are smoothed by artificial viscosity. Smoothing facilitates the numerical solution because shock fitting techniques can be avoided and conventional difference analogs to continuous differential equations can be used. The accuracy of TOODY in calculating two-dimensional elastic and elastic-plastic wave propagation effects has been previously demonstrated (see STANTON (1968), BERTHOLF and KARNES (1969), KARNES and BERTHOLF (1970), MEYER and BLEWETT (1972), STEVENS and JONES (1972), and BERTHOLF (1974)). In addition to the strain-time variations at the two strain-gauge stations, the numerical computations result in computed values of all components of the stress, strain and strain-rate tensors for each time increment at each radial and axial position of the numerical grid defining the specimen and elastic bars. Thus, the time required to attain stress equilibrium can be determined and the degree of stress and strain uniformity throughout the sample can be continuously monitored. Sliding is permitted between the specimen end faces and the two elastic bars with friction coefficients which are constant over the interfaces and controllable between zero and the no-slip interface condition (infinite friction coefficient). The results of computations are presented using elastic bar dimensions of 2.54 cm dia. and 50.8 cm length. The specimens are slightly smaller in diameter (typically 2.12 cm) so that radial expansion to the maximum strain of interest does not cause the diameter of the specimen to exceed that of the bars. The specimen is represented by 12 finite-difference meshes along the radius and 6, 12 and 24 meshes in the axial direction for an initial length-to-diameter ratio of 0.3, O-6 or 1.2, respectively. The elastic bars are represented by 12 meshes along the radius and 160 meshes in the axial direction. The specimen meshes are nearly square and the elastic bar meshes have an axial to radial dimension ratio of 3 : 1. The finite-difference time step is approximately 0.1 us. Properties of steel (p = 7.8 g cm -3, Young’s modulus E = 2 Mb, v = 0.3) are assumed for the elastic bars and the specimen is assumed to be annealed 1100 Al stress-strain curve (p = 2-7 g cmV3, E = 0.724 Mb, v = l/3) having a parabolic with a linear elastic limit of 0.0689 kbar.? The parabolic portion of the stress-strain curve is that given by BELL (1964). Strain-hardening of the aluminum specimen is modeled using the isotropic hardening assumption where the flow stress is given by the accumulative plastic work generated during deformation and the Bauschinger effect is ignored. Since one purpose of this investigation is to determine under what conditions violations of the assumptions can result in apparent strain-rate effects that do not represent real material behavior, the specimen material is modeled as a strainrate independent material. t 1000 psi =
0.0689 kbar = 6.89 MPa.
Two-dimensional
analysis of the split Hopkinson
5
pressure bar system
4. SAMPLECOMPUTATION In a laboratory experiment the information recorded during a test is the input and output elastic bar strain-time variations obtained at positions G1 and G,, respectively, in Fig. 1. The computed strain-time variations at these positions are shown in Fig. 2 for the properties given above for the steel bars and aluminum specimen, for a specimen whose length-to-diameter ratio IJd is 0.3, for zero friction between the specimen and bars, and for an input stress which is Heaviside in time and 1 kbar in amplitude. The input stress is applied at the left end of the input bar in Fig. 1 I
I
.‘. .*
..
I
i -
_a
;.
.
u'
.
-II
I 20
I 40
I 60
I
r% PO.3
%a:?
.
.
.
.
I
INPUT EAR x OUTPUT BAR
l
. . . . . '.
.
I
I
s
.
I
120 1W TIME - JIS
80
I
.
**
l
I
I
I
I
I
I
140
160
l&l
LOI
FIG. 2. Computed input and output bar strain histories at the strain gauge locations. from the initiation of stress in the input bar.
220
Time measured
and the computer code computes all pertinent stress and strain histories throughout the system. The computed strain histories at the two strain gauge locations (G, and G2 in Fig. 1) are treated as experimental data to obtain the stress, strain and strain-rate histories in the following manner. The change in particle velocity of the elastic bars is given by A u,(t) = + (~%)~Asa(0, (7) where A.q,(f) is the change in strain in the elastic bar. The particle velocity history, U,, of the input bar/specimen interface (position L) is determined by the input bar strain history given by the dots in Fig. 2 where time translation is used to account for the wave propagation transit time between G1 and the interface L. The velocity of the interface at position R, U,, is likewise determined by the output bar strain history (crosses in Fig. 2) and the same time translation. The velocities of the specimen/elastic bar interfaces, which are obtained from the elastic bar strain histories through (7) are shown in Fig. 3. Superimposed on the interface velocity histories is the average specimen strain history obtained from &AVG
=
j
(u,--
u&/lo
dt,
(8)
6
L.D. BERTHOLF and C.H. KARNES $
I
I
I
t
I
I
,
6
FIG. 3. Predicted specimen strain and interface velocity histories.
where (U, - U,)/l, is the average strain-rate. There is a slight uncertainty in average specimen strain obtained from (8) because of various assumptions that can be made concerning the rise-time of U,. The assumption that is made in this study is that the rise-time is equal to the initial rise-time of the strain record at G, plus the time due to additional dispersion from having propagated a distance of 5 diameters from an axial position of 15-20 dia (from G, to L). Specimen stress history is obtained from the output bar strain history in Fig. 2 by converting the output bar strain history to stress, using the aforementioned time translation and compensating for the difference in area of the specimen compared to the bar. Only the output bar strain history is used, instead of averaging the input and output bar strain histories because the input bar is subjected to large Pochhammer-Chree oscillations due to short rise-time initial impact conditions and dispersion phenomena. These oscillations almost totally reflect at the first interface and essentially the only stress transmitted into the specimen is that which is proportional to the output bar strain record as shown in Fig. 2. Averaging the responses of both bars retains the input bar oscillations even though they are not transmitted by the specimen (see BERTHOLF (1974) for more detail). RAND (1967) has concluded that one must average the two bar responses to minimize the oscillations in the specimen stress history. However, Rand’s analysis considers only one-dimensional stress behavior and therefore the oscillations he obtained
Two-dimensional
analysis of the split Hopkinson
pressure bar system
7
are due to axial inertia only. Furthermore, any short rise-time disturbance will propagate with a velocity nearly equal to the bulk sound speed {bulk modulus~density)~, which can be at least an order of magnitude greater than the one-dimensional stress, plastic wave propagation velocity [(~~(s)~~&)/~~~predicted by the elementary strainrate independent theory. Therefore, the axial stress should reach equilibrium much faster than any one-dimensional stress analysis can predict. The resulting stress-strain curves for the specimen in the sample computation are shown in Fig. 4. The solid line represents the stress-strain model used to describe the specimen material for the wave propagation calculations. Solid circles represent the resulting stress-strain curve predicted by treating the elastic input and output bar
0.8
VI D F
-
*0.4
1NPUT CURVE 0 l
STRESS PREDlCTED FROM AVERAGE OF ELASTIC BARS STRESS PREDICTED FRWYI OUTPUT BAR
J- AVERAGE AND RANGE OF 1 CALCU~TED VALUES 0.2
0.0
I
3 STRA12N-percent
4
FIG.4. Predicted, computed
and input stress-strain paths for no friction, a specimen Iength-todiameter ratio of O-3, and an average strain-rate of about 400 s-l.
strains as experimental data and using only the output bar to determine the specimen stress. Open circles represent the stress-strain path which would be predicted using the average of the input and output bars to determine the specimen stress. Crosses represent the average of the instantaneous axial stress and strain values of all the numerical meshes within the specimen and show the range of values of the calculated axial stress and strain. The length of the vertical lines along the abscissa represents the magnitude of the largest nonaxial component of the stress tensor (Max (a,,, cr8,, cr,,) where Y,8 and z are the radial, circumferential and axial coordinates, respectively). These lines indicate the degree of deviation from a one-dimensional stress state. It is apparent, for the conditions of these calculations, viz. bar diameter of 2.54 cm, specimen length-to-diameter ratio of 0.3, zero friction and average strain-rate of approximately 400 s- ‘, that (i) the elastic bar strains predict very well the actual stress
8
L. D. BERTHOLP and C. H. KARNES
and strain history experienced by the specimen, (ii) specimen represents the true stress-strain behavior of strain are uniform throughout the specimen within a state experienced by the specimen is one-dimensional
the stress-strain history of the the material, (iii) the stress and few percent, and (iv) the stress within a few percent.
5. EFFECTS OF FRICTION AND LENGTH-TO-DIAMETER RATIO A series of calculations was made in order to determine the effect of realistic friction on (i) the stress-strain path followed by the material, (ii) the deviation of the stress state from one-dimensionality, (iii) the uniformity of the stress and strain throughout the specimen, and (iv) the accuracy with which the elastic bar response predicts the specimen behavior. The results of calculations with l/d = O-3 and for friction coefficients of @05, 0.15, and 0.25 are shown in Figs. 5, 6 and 7, respectively.
l
INPUT CURVE STRESS PREDICTEO FROM @LITPUT BAR
i CALCULATED VARIATION
N =0.05 ,NON-AXIAL STRESS-,
FIG. 5. Effect of friction on specimen stress-strain
response (p = 0*05).
is apparent for this small length-to-diameter ratio that a friction coefficient of O*OS+f produces stress and strain variations throughout the sample resulting in actual stressstrain states and a stress-strain path predicted by the elastic bar strains which are
It
2-3 percent higher than the true one-dimensional Even though higher friction coefficients can
stress-strain
behavior
of the material.
produce totally unacceptable stress and
t This is a reasonable value (see O'CONNOR and BOYD (1968), for exampIe). Hydrodynamic lubrication conditions can be obtained momentarily by the use of liquid lubricants producing coefficients of friction on the order of lO-a-1O-3. One must be cautioned that using this type of hrbricant and comparing SHPB data with quasi-static data could lead to errors unless a greater l/d ratio is used quasi-statically to approach the same friction effect. Use of solid film lubricants is recommended to minimize this problem.
Two-dimensional
analysis of the split Hopkinson
-INPUT l
pressure bar system
9
CURVE
STRESS PREDICTED FROM OUTPUT BAR
STRAIN - percent
FIG. 6. Effect of friction on specimen stress-strain
response (p = 0.15).
l.o-
D.8-
L i?O.6-
.
STRESS PREDICTED FROM OUTPUT BAR
I
I
p .-0.25
l__l-_ 3
I
STRAIN - percent
FIG. 7. Effect of friction on specimen stress-strain
L
4
response (p = 0.25).
10
L. D. BERTHOLF and C. H. KARNES
strain variations in the specimen, accompanied by stress-strain states significantly different from one-dimensional stress, the elastic bar strains still predict correctly the average behavior of the specimen as influenced by friction. That is, the intersection points of the crosses are accurately predicted by the dots in Fig. 7. In order to determine if these friction effects could be reproduced experimentally, Professor W. N. SHARPE (unpublished work, 1971) performed a series of experiments at Lawrence Livermore Laboratory (University of California) using lubricated, dry and bonded specimens. Calculations were made with various degrees of friction for the same bar and specimen geometry and the same initial conditions as the experiments. The resulting stress-strain behavior obtained experimentally and those obtained from the two-dimensional calculations are shown in Fig. 8. For comparison, the parabolic
-l
STRAIN
INPUT CURVE PREDICTED LUBRICATED
-
peicent
FIG. 8. Comparison of calculated response for various friction coefficients and experimental data for different interface conditions.
stress-strain model used in the calculations is shown as the dashed curve. The experimental data includes two experiments for each of the three different friction conditions. Since it was shown in Figs. 4-7 that friction results in a reconstituted stress-strain curve having, for a given strain, a larger stress magnitude than the input curve, thz fact that Sharpe’s data for the lubricated specimens fall slightly below the input stress-strain curve indicates that the input curve is incorrect for the annealed 1100 Al used in the experiments. However, the input curve used is entirely adequate for demonstrating the effect that friction has on the reconstituted stress-strain curve. As shown in Fig. 9, friction has a much less pronounced effect for a length-todiameter ratio of 1,2. This is expected because much less volume of material is constrained by friction for this geometry. However, axial inertia, which will be discussed
Two-dimensional
T
analysis of the split Hopkinson I
I
pressure bar system
11
f
.8 .6
- -
INPUi PREDICTED
.a
.2
.O
830.9.
1 1
I
I
2 STRAIN -
3
percent
Effects of friction for a specimen length-to-diameter
ratio of l-2.
Section 6, results in a reconstituted stress-strain curve significant@ different from the true behavior even without friction. The effect of friction is summarized in Fig. 10 and compared to (4) which is RAND’S (1967) approximation for the ratio of the apparent to the correct one-dimensional stress. The points shown are taken from the twodimensional calculations for various geometries, strain-rates, loadings, and friction coefficients. The calculations indicate that about an eight percent error due to friction can be expected from a friction coefficient of 0.05 for a length-to-diameter ratio of O-1. Therefore, extreme care must be exercised if thin specimens are used. in
FIG. 10. Comparison
of calculated and approximate
corrections
for friction eff&zts.
L. D. BERTHOLF and C. H.
12
KARNFS
The most dramatic effect of friction can be seen in the nonuniformity of the stress and strain throughout the sample and the degree of multidimensionality in the stress state. Figure 11 shows the range of values of the axial stress component throughout the specimen expressed as a fraction of the average axial stress component for various values of the coefficient of friction and for a length-to-diameter ratio of 0.3. The degree to which the stress state in the specimen is multidimensional is illustrated in Fig. 12 where the maximum nonaxial stress component (Max (err, cBO,err) expressed as a fraction of the average axial stress component) is shown for various friction It is seen from these two Figures that even a friction coefficient of O-05 coefficients. produces approximately a ten percent variation in axial stress as well as a ten percent deviation from a one-dimensional stress state for this length-to-diameter ratio.
1.0
I
I
‘I
/ ’
I 0.8 -
I’id - 0.3 = 4005-l w
”Y =b-N 0.6 .E c-N ;I a c.4
1 II - 0.25
2 6 -
I-
0.15
0.20.0 ml
I 120
I 140
1 160 TIME
FIG. 11. Effects of
1.0,
0.8
I
il- 0.05 1 180
I MO
Ifi’0 220
240
- LIS
friction on specimen axial stress uniformity.
,
I
I
1
'1
--
0.6
1W
120
140
100 TIME
FIG. 12.
180
200
220
240
- ps
Effects of friction on specimen nonaxial stress components.
Two-dimensional
analysis of the split Hopkinson
13
pressure bar system
6. EFFECTS OFINERTIA Reconstituted stress-strain curves for three different length-to-diameter ratios, three different friction coefficients, and a Heaviside step function input are shown in Fig. 13. For zero friction, the reconstituted behavior for a length-to-diameter ratio of 0.3 and, to a lesser extent, O-6 represents the actual stress-strain behavior very well. However, a ratio of 1.2 requires such a high initial stress in the input bar to produce an average strain-rate of approximately 400 s - ’ that very large stress and strain gradients are produced which cause the effects of inertia to become important. 1.0
I
I
I
I
I
. . . . . .
h/d = 0.3 L’/d s 0.6
.-.-.-
!/d = 1.2 _. ._
---
o..o’
1
I
I
STRtilN
FIG. 13. Effects of specimen
geometry
+
I
I
INPUI I
I
I
1 PERCENT k
on specimen stress-strain coefficients.
behavior
for various friction
The effects of inertia become more and more pronounced as the strain-rate is increased. A series of computer calculations was made with zero friction and a O-3 initial length-to-diameter ratio to show the effects of inertia on the reconstituted stressstrain curve for various strain-rates. The strain-rate was varied by changing the magnitude of the initial Heaviside stress applied to the input pressure bar. Results are shown in Fig. 14 for average strain-rates of approximately 400, 800, 1600 and 3300 s-l. Higher strain-rate results show the increasing effects of radial inertia on the nonuniformity of the stress distribution. As the average strain-rate is increased, the variation in stress and strain throughout the specimen seriously increases. This results in oscillations in the reconstituted stress-strain curves and large variations in stress and strain. The oscillations are caused by radial waves resulting from the short rise-time initial compression wave. The calculated stress histories in the specimen show that the frequency of the oscillations corresponds closely to radially propagating bulk waves. In order to minimize these oscillations and resulting variations in stress and strain, a series of calculations was performed using a 40 us rise-time ramp compression wave
L. D.
O.OL'
BERTHOLF and
+--4
C. H.
STRAIN- percent
KARNES
i
Ii
4
FIG. 14. Effect of strain-rate on specimen stress-strain behavior for Heaviside input. The groups of numbers represent the calculated instantaneous average strain-rates for the four different curves at the indicated strain levels.
instead of a Heaviside step functiont at the impact end of the input elastic bar. Figure 15 shows a comparison between the resulting average specimen strain-rate histories for the ramp and Heaviside input functions. It is seen that the ramp input effectively eliminates the oscillations without significantly reducing the average strain-rate. The reconstituted stress-strain paths using the ramp input wave are shown in Fig. 16 for average strain-rates of 400, 800, 1600 and 3300 s-i. The effects of inertia are summarized in Fig. 17 where the calculated results are compared to (3) which is Davies and Hunter’s approximate correction for axial and radial inertia, By rewriting (3),
and by assuming v, = +, one obtains the lines in Fig. 17 using the l/d ratios of interest. Points shown in this Figure correspond to the calculated results for step inputs, For each configuration one p = 0, three l/d ratios, and various strain-rates. point is obtained by choosing the time at which d2c/dt2is a maximum. For this value of time the stress and strain predicted by the SHPB (ab and .sb) are obtained. The input stress-strain curve and sb are used to determine a,. The positive portion of Fig. 17 corresponds to l/d > v,J3/4 and indicates the region in which axial inertia predominates. Conversely, the negative portion indicates the region of radial inertia domination. For the specific configurations studied herein, t The Heaviside step function input condition results in a rise-time of approximately propagating 20 diameters (see Figs. 2 and 3).
10 fls after
Two-dimensional
analysis of the split Hopkinson
pressure bar system
15
RAMP INPtiI If
1 RANGE OF CALCULATED / VALUES WITHIN SPECIMEN
‘/
fld = 0.3 p-o.0
: I I
FIG. IS. Calculated specimen average strain-rate histories for ramp and Heaviside inputs.
RAMP INPUT I/d - 0.3 84 = 0.0
STRAIN - percent
FIG. 16. Effect of strain-rate on specimen stress-strain behavior for ramp input. The groups of num&x-S represent the calculated instant~eous average strain-rates for the four different curves at the indicated strain levels.
L. D. BERTHOLF and C. H. KARNES
16 .1.2
I
I
I
I
I
I
STEP INPUT
1.0
/
i/d - 1.2
t
a f/d
= 0.6
o i/d = 0.3
0.2
0.4
0.6
0.8
1.0
i
1. ?
1.4
-1 -2 maxlub -kbar S
;i
FIG. 17. Comparison of the magnitude of the approximate inertia correction to the calculated errors due to inertia.
it is clear that the approximations of Davies and Hunter result in a reasonable correction for inertia effects. While different situations may result in different conclusions, these results lend strong credence to their approximate expression. 7. EFFECTS OF STRAIN-RATE For small values of l/d, it has been shown that friction produces an error when the reconstituted stress-strain curve is compared to the input stress-strain curve. This error could be interpreted as a strain-rate effect in an experiment if one simply compares a single ‘dynamic’ stress-strain curve to a stress-strain curve obtained under quasi-static, one-dimensional stress conditions. In order to determine if this error is dependent on strain-rate, which would invalidate the basic premise of the SHPB technique, a series of calculations was made using a friction coefficient of 0.05. The results for ramp loadings are shown in Fig. 18 in which the reconstructed stress-strain curves for strain-rates of 400, 800, 1600 and 3300 s-l for a friction coefficient of 0.05 are It is seen that the compared to results for a strain-rate of 400 s-l and zero friction. error due to the interface friction coefficient of 0.05 is independent of strain rate. One can conclude, considering the fact that step loadings give similar results, that a reasonably small interface friction alone does not produce an apparent strain-rate effect when stress-strain curves obtained from the SHPB technique are compared. However, one must be certain that equivalent interface friction conditions are attained in quasistatic tests when results from such studies are compared to results from SHPB studies.
8. SCALING All calculations were performed with a constitutive equation which is rate independent in order to separate the effects of intrinsic material rate dependence from apparent rate dependence due to friction or inertia. Because of the absence of rate
Two-dimensional
analysis of the split Hopkinson
I
I
pressure bar system
17
I
STRAIN _ percent
FIG. 18. Indication
of the friction-induced material strain-rate ef%ct. No rate-effect present when comparing many rates for the same interface conditions.
dependence in the constitutive equation used, the solutions obtained can be applied directly to different diameter systems. All the results presented are for 2.54 cm dia bars and all the specimens are initially 2.12 cm dia except those used for comparison with Sharpe’s data (see Section 5) which are 2.438 cm dia. If one were to retain the mechanical properties of the bars and the specimen, retain the amplitude of the input compressive wave, and scale each linear dimension of the components proportionately by a factor n, then the results presented herein are valid if one scales time and strain-rate linearly and inversely with the dimension scaling factor, respectively. For example, if t is the time variable in these calculations and D is the bar diameter, then the time variable t’ for cakulations of a different diameter nD is given by t’ = nt. In
(10)
like manner, the strain-rate 8 for the corresponding configuration is given by E’ = E/n.
(11) Therefore, for a bar diameter of 0635 cm (specimen diameter equaIs 0.53 cm), Figs. 2 and 3 give the calculated responses of the input and output bars at the strain gauge locations and the resulting interface velocity and specimen strain history if one reduces time by a factor of four. Figure 4 becomes the stress-strain behavior as predicted by the elastic bar responses, but for an average strain-rate of 1600 s-l instead of 400 s-l. If it is concluded from Fig. 16 that employing a 40 ps rise-time ramp allows reliable stress-strain paths to be obtained up to strain-rates of about 2000 s-l for the 2.54 cm dia system, then it is apparent that a 10 ps rise-time ramp can be used to obtain reliable data to strain-rates of about 8000 s-l with a 0.635 cm dia system. The critical parameter, which allows one to assess the validity of experimental data as it might be 2
18
L. D. BERTHOLF and C. H. KARNES
influenced by inertia, becomes L?dMAX=5x103cms-‘,
(12) provided that a ramp input compressive wave is used where the ratio of its rise-time to the bar diameter is given by t,/D 2 16 ps cm-‘.
(13)
9. DHX.,JSSION While a large computer program and long running times were required, a comprehensive two-dimensional analysis of the SHPB has been a~compJished. Calculated and experimental results show excehent agreement for a variety of different interface conditions. The experimental conditions at the interface included bonded, dry and lubricated. Inertia and especially friction produce additional constraints and result in multiaxial stress states which might be misconstrued as additional strengthening due to rate effects. However, with a little caution the SHPB experiment is very accurate and can be reliably used for the measurement of mechanical properties at high strainrates. The correction for friction (as predicted by RAND (1967) and verified here) depends upon p/(l/d). While the degree of friction-induced over-stress is somewhat smaller than that predicted by Rand, this ratio is a reliable indicator of the friction effect in the configuration studied herein. As predicted by DAVIESand HUNTER(1963), results of the numerical calculations indicate that the correction required for the over-stress due to inertia is proportional to the second time-derivative of the strain. The sign of this correction changes depending upon the dominance of radial inertia or of axial inertia and I/d = v$@ fi: l/2 has been shown to be a good criterion for SHPB specimen design. Friction has been shown to increase greatly the degree of specimen stress and strain nonuniformity. SHARPEand HOGE(1972) have observed considerable strain variations on the surface of specimens without lubricated ends. The calculations presented here agree with their results and give the stress and strain variation throughout the specimen. BELL’S(1966) conclusion “. . . the aariability in such split Hopkinson bar data . . ., demonstrates that the theoretical as.~umption of uniform stress and strain in short specimen, i.e. the quasi-static h~~othesjs, is ph~sica~~~incorrect.” is correct for the bonded specimens he used; however, it is false if reasonable care is taken to lubricate the ends of the specimen. Since time-dependent material properties were not considered in these calculations, the results for the given SHPB system can be directly scaled to other systems. For the same bar and specimen materials and the same Z/d values, the results of these calculations for D = 2.54 cm and a given strain-rate IJare the same as would be obtained for a system of size nD at a strain-rate of S/n. For the elastic bar properties and configuration considered herein, the limiting maximum value for the DB product exceeds 1000 cm s- 1 for the step input and is considerably larger when d’sldt’ is minimized by ramping. This means that a O-635 cm SHPB system with a ramped input can accuratefy measure material properties to strain-rates of at least 8000 s- ’ as long as care is taken to minimize the effects of friction and inertia.
Two-dimensional analysis of the split Hopkinson pressure bar system
19
ACKNOWLEDGMENT The writers are indebted to Professor W. N. Sharpe, (Michigan State University) who graciousiy provided us with his unpublished SHPB data. REFERENCES BELL,
J. F.
BERTHOLF,L. D. BERTHOLF,L. D. and BENZLEY,S. E. BERTHOLF,L. D. and KARNES,C. H. CHIV, S. S, and NEUBERT,V. H. CONN, A. F. DAVIES,E. D. H. and HUNTER, S. C. DHARAN,C. I(. H. and HAUSER,F. E. HAUSER,F. E., SIMMONS,J. A. and DORN, J. E.
1964
1966 1974 1968 1969 1967 196.5 1963 1970 1960
JACKSON,J. W. and WAXMAN,M.
1963
JAHSMAN,W. E. KARNES,C. H. and BERTHOLF,L. D.
1971 1970
KOLSKY,H.
1949 1963
LINDHOLM,U. S, MEYER,K. A. and BLEWETT,P. J. O’CONNOR.J. J. and BOYD,J.
1964 1972 1968
RAND, J. L.
1967
&MANTA,S. K. SHARPE,W. N. and HOGE, K. G. SIEBEL,E. STANTON,P. L.
1971 1972 1923 1968
STEVENS,A. L. and JONES,0. E.
1972
Stress Waves in Aneiastic Solids (edited by KOLSKY, H. and PRAGER,W.), p. 166. Springer-Verlag, Berlin. J. Me&. Phys. So&& 14, 324. J. appf. Mech. 41, 137. Sandia Laboratories Report SC-RR-68-41. J. appl. Mech. 36, 533. J. Mech. Phys. Solids 15, 177. Ibid. 13,311. Ibid. 11, 155. Exp. Mech. 10, 370. Response of Metals to High Velocity Deformation(Proceedings Metallurgica Society Conferences 9) (edited by SHEWMON,P. 6. and ZACKAY, V. F.), p. 93. Interscience, New York. High Pressure Measurement, p. 39. Butterworths, London. J. appl. Me& 38, 75. Inelastic Behavior of Solids (edited by KANNINEN, M. I;., ADLER, W. F., ROS~NFIELD,A. R. and JAFFEE,R. I.), p. 501. McGraw-Hill, New York. Proc. Phys. Sot. B62,676. Stress Waves in Solids, p. 153. Dover Publications, New York. J. Mech. Phys. Solids 12, 317. Phys. Fluids 15, 753. Standard ~and~oa~ of L~ricatjo~ Engineering. McGraw-Hill, New York. U.S. Naval Ordnance Laboratory Report NOLTR67-156. .7. Mech. Phys. Solids 19, 117. Exp. Mech. 12, 570. Stahl u. Eisen 43, 1295. Sandia Laboratories Report SC-CR-683672. J. appi. Meek 39, 359.
Printed in Great Britain.
TWO-DIMENSIONAL ANALYSIS OF THE SPLIT HOPKINSON PRESSURE BAR SYSTEM? By L, D. BERTHOLF and C. H. KARNES Code Development Division and Mechanics of Materials Division, Sandia Laboratories, Albuquerque, New Mexico 87115, U.S.A. (Rrceived
19th Septembrr
1974)
SUMMARY THE SPLIT Hopkinson
pressure bar is widely used to measure the dynamic properties of solid materials. This paper presents the results of the first comprehensive two-dimensional numerical analysis of the technique, and quantitatively describes the effects of realistic friction and of variations in both the specimen geometry and the imposed strain-rate on the validity of the assumptions used in analyzing experimental data. A two-dimensional axisymmetric numerical analysis is used to compute all components of the stress, strain and strain-rate tensors at each mesh point within the specimen and the elastic bars. The calculated response of the pressure bars is used to reconstruct the stress-strain behavior of the specimen and this is compared to both the input stress-strain curve and the actual calculated stress-strain states in the specimen. Thus, the validity of the assumptions and the corrections used in the analysis of the data is determined. Inertia and friction between the specimen and the elastic bars affect the response of the specimen differently for different Ien~h-todiameter ratios. Inertia effects produce stress waves propagating radialty and axially in the specimen and may result in an oscillating reconstructed stress-strain curve. If the ends of the specimen are well lubricated and care is taken to minimize the effects of inertia, the reconstructed stress-strain curve agrees with the input. However, serious stress and strain nonuniformity exists when the ends are not lubricated and this results in a reconstructed stress-strain curve where, for any given strain, the stress magnitude is larger than the correct value. A comparison of the calculations with experiment shows excellent agreement for various interface conditions. Finally, the inertia correction of E. D. H. Davies and S. C. Hunter (1963) is found to be reasonable.
1. INTRODLJCTICW SINCE the original paper of KOLSKY (1949) describing the use of the split Hopkinson pressure bar (SHPB) system to determine dynamic mechanical properties of materials, many other investigators have used the technique in a more or less routine manner. The technique is used to obtain compression properties in the following way. A sample of material to be studied is placed between and in contact with two elastic rods of approximately the same but slightly larger diameter. A compression stress wave is initiated in one of the bars by impact with a suitable projectile rod. The compression wave propagates along the rod causing multiple reflections in the sample. This results in distorted waves reflected from the sample traveling back along the first rod and transmitted through the sample and into the second rod. Measurements of strain in the two elastic rods on either side of the sample can lead to quantitative information concerning the mechanical behavior of the sample if care is used in conducting the experiment and if the following two assumptions are maintained by
$ This work was supported by the U.S. Atomic Energy Commission.
2
L. D. BERTHOLF and C. H. KARNES
the conditions of the experiment: (i) the specimen is in a state of one-dimensional stress, and (ii) the stress and the strain are uniform throughout the specimen (for detailed discussions of the technique see KOL~KY(1949), HAUSER,SUMMONS and DORN (1960), and LINDHOLM(1964)). The above assumptions can be violated by radial and axial inertia effects and by friction between the specimen and the elastic bars. Approximate corrections have been derived and applied to the data to account for the effects of radial and axial inertia and friction. It is the objective of this paper to present the results of a comprehensive two-dimensional numerical study of the SHPB including the effects of inertia and friction. More specifically, the two-dimensional wave propagation analysis is used (i) to examine in detail the validity of the assumptions that must be made in analyzing the experimental data, (ii) to determine the limitations on the experimental parameters, (iii) to determine the adequacy of approximate corrections given by previous investigators, and (iv) to determine under what conditions vioIation of the assumptions gives erroneous indications of material strain-rate dependence.
2. REVIEWOF PREVIOUSCORRECTIONS AND ANALYSES KOLSKY(1949, 1963) described the use of the SHPB technique and introduced a correction for radial inertia. Radial inertia was assumed to cause a larger stress than that which would have resulted in its absence. The difference, 5,-a,, was evaluated approximately by equating the additional strain energy caused by the increased stress to the radial kinetic energy. Assuming small strains, he obtained 1 d2& cr, = 5, - - vs2d2p, -2, 8 dt
(1)
where 5, is the axial stress determined by the average of the stresses measured in the two bars; 5, is the axial stress required to deform the sample in a one-dimensional stress state; and v,, ps, d and E are Poisson’s ratio, density, diameter and axial strain, respectively, of the sample. Kolsky’s analysis assumed a very thin sampIe that remained in stress equilibrium in the axial direction, and because of sample thinness, was particularly susceptible to friction. DAVIESand HUNTER(1963) gave a more general analysis. In order to minimize friction effects in their experiments, Davies and Hunter used samples whose length-to-diameter ratios were approximately O-5, whereas Rolsky used a ratio of 0.05. Davies and Hunter chose their dimensions as a result of an analysis by SEBEL (1923) which led to ~d/3~ 4 I
(2)
as a criterion for neglecting friction effects, where ~1is the Coulomb friction coefficient and I is the length of the specimen. Davies and Hunter employed an analysis considering kinetic energy due to both axial and radial motions. Their analysis resulted in
where ps12(d2Jdt2)/6 is the axial inertia correction term and 5b is the stress at the interface between the specimen and the output bar.
Two-dimensional analysis of the split Hopkinson pressure bar system
3
RAND (1967) presented an analysis based on the work of JACKSONand WAXMAN (1963) which results in a nonuniform axial stress distribution due to radial friction. The analysis is based on static Coulomb friction and gives an apparent stress
O’a= 2a,o!-‘(e”-cr-
l),
(4)
where (rOis the true one-dimensional stress and cc=
d (1+&)-3’2. 7
DHARAN and HAUSER (1970) have employed a modified version of the SHPB technique by impacting a thin sample directly with a massive, high-strength projectile. The sample is mounted on an elastic pressure bar which is instrumented and serves as the output bar. Because strain-rates as high as 1.2 x lo5 s-r were desired, relatively thin samples were used which result in an amplification of the effects of friction and radial inertia. They employed an analysis to correct the stress-time measurements for the effects of radial inertia. However, the effects of friction were neglected. The analysis of SAMANTA(1971) is similar to that of DAVIES and HUNTER (1963) except that he included the material derivative and assumed incompressibility (v, = 4). His analysis as published included an error, but when corrected reduces to
(6) The first two terms on the right side of (6) are identical to those of Davies and Hunter if incompressibility is assumed. The term proportional to (d.s/&)* results from the convective portion of the material derivative. It can be shown for the calculations presented herein that this last term is no more than a few percent of the second term. Numerous investigators have attempted to analyze the wave propagation response of the SHPB technique. To the writers’ knowledge, all published analyses have been one-dimensional in which only axial stress waves and axial inertia are considered. For examples, see HAUSER,SIMMONS,and DORN (1960) CONN (1965), JAHSMAN(1971), and CHIU and NEUBERT(1967). 3. COMPUTATIONAL TECHNIQUE A description of the problem to be examined is illustrated in Fig. 1. The experimental parameters which are normally recorded are initial specimen length IO,initial specimen diameter d,,, bar diameter D, position of strain gauges from ends of bars (5D for these calculations), and the strain-time records obtained from the strain gauges. The TOODY two-dimensional elastic-plastic wave propagation computer code as developed by BERTHOLFand BENZLEY(1968) is used to simulate, numerically, INPUT BAR
OUTPUT BAR do
5
L/R
G2
FIG. 1. Schematic of SHPB showing pertinent dimensional parameters.
L. D. BERTHOLF and C. H. KARNE~
4
all pertinent features of an experimental SHPB. It is a second-order accurate, artificial viscosity, finite-difference code. It solves the finite difference representations to the equations of motion in Lagrangian coordinates. The stress and stretching tensors are decomposed into spherical and deviatoric components and it is assumed that the stretching can be expressed as the sum of elastic and plastic parts. For the constitutive relation it is assumed that the co-rotational stress-rate is a function of density and energy and also is a linear isotropic function of the stretching. The second invariant of the deviatoric stress tensor is limited by the von Mises yield criterion and the flow rule is chosen such that the plastic stretching is orthogonal to the yield surface. Shock waves are smoothed by artificial viscosity. Smoothing facilitates the numerical solution because shock fitting techniques can be avoided and conventional difference analogs to continuous differential equations can be used. The accuracy of TOODY in calculating two-dimensional elastic and elastic-plastic wave propagation effects has been previously demonstrated (see STANTON (1968), BERTHOLF and KARNES (1969), KARNES and BERTHOLF (1970), MEYER and BLEWETT (1972), STEVENS and JONES (1972), and BERTHOLF (1974)). In addition to the strain-time variations at the two strain-gauge stations, the numerical computations result in computed values of all components of the stress, strain and strain-rate tensors for each time increment at each radial and axial position of the numerical grid defining the specimen and elastic bars. Thus, the time required to attain stress equilibrium can be determined and the degree of stress and strain uniformity throughout the sample can be continuously monitored. Sliding is permitted between the specimen end faces and the two elastic bars with friction coefficients which are constant over the interfaces and controllable between zero and the no-slip interface condition (infinite friction coefficient). The results of computations are presented using elastic bar dimensions of 2.54 cm dia. and 50.8 cm length. The specimens are slightly smaller in diameter (typically 2.12 cm) so that radial expansion to the maximum strain of interest does not cause the diameter of the specimen to exceed that of the bars. The specimen is represented by 12 finite-difference meshes along the radius and 6, 12 and 24 meshes in the axial direction for an initial length-to-diameter ratio of 0.3, O-6 or 1.2, respectively. The elastic bars are represented by 12 meshes along the radius and 160 meshes in the axial direction. The specimen meshes are nearly square and the elastic bar meshes have an axial to radial dimension ratio of 3 : 1. The finite-difference time step is approximately 0.1 us. Properties of steel (p = 7.8 g cm -3, Young’s modulus E = 2 Mb, v = 0.3) are assumed for the elastic bars and the specimen is assumed to be annealed 1100 Al stress-strain curve (p = 2-7 g cmV3, E = 0.724 Mb, v = l/3) having a parabolic with a linear elastic limit of 0.0689 kbar.? The parabolic portion of the stress-strain curve is that given by BELL (1964). Strain-hardening of the aluminum specimen is modeled using the isotropic hardening assumption where the flow stress is given by the accumulative plastic work generated during deformation and the Bauschinger effect is ignored. Since one purpose of this investigation is to determine under what conditions violations of the assumptions can result in apparent strain-rate effects that do not represent real material behavior, the specimen material is modeled as a strainrate independent material. t 1000 psi =
0.0689 kbar = 6.89 MPa.
Two-dimensional
analysis of the split Hopkinson
5
pressure bar system
4. SAMPLECOMPUTATION In a laboratory experiment the information recorded during a test is the input and output elastic bar strain-time variations obtained at positions G1 and G,, respectively, in Fig. 1. The computed strain-time variations at these positions are shown in Fig. 2 for the properties given above for the steel bars and aluminum specimen, for a specimen whose length-to-diameter ratio IJd is 0.3, for zero friction between the specimen and bars, and for an input stress which is Heaviside in time and 1 kbar in amplitude. The input stress is applied at the left end of the input bar in Fig. 1 I
I
.‘. .*
..
I
i -
_a
;.
.
u'
.
-II
I 20
I 40
I 60
I
r% PO.3
%a:?
.
.
.
.
I
INPUT EAR x OUTPUT BAR
l
. . . . . '.
.
I
I
s
.
I
120 1W TIME - JIS
80
I
.
**
l
I
I
I
I
I
I
140
160
l&l
LOI
FIG. 2. Computed input and output bar strain histories at the strain gauge locations. from the initiation of stress in the input bar.
220
Time measured
and the computer code computes all pertinent stress and strain histories throughout the system. The computed strain histories at the two strain gauge locations (G, and G2 in Fig. 1) are treated as experimental data to obtain the stress, strain and strain-rate histories in the following manner. The change in particle velocity of the elastic bars is given by A u,(t) = + (~%)~Asa(0, (7) where A.q,(f) is the change in strain in the elastic bar. The particle velocity history, U,, of the input bar/specimen interface (position L) is determined by the input bar strain history given by the dots in Fig. 2 where time translation is used to account for the wave propagation transit time between G1 and the interface L. The velocity of the interface at position R, U,, is likewise determined by the output bar strain history (crosses in Fig. 2) and the same time translation. The velocities of the specimen/elastic bar interfaces, which are obtained from the elastic bar strain histories through (7) are shown in Fig. 3. Superimposed on the interface velocity histories is the average specimen strain history obtained from &AVG
=
j
(u,--
u&/lo
dt,
(8)
6
L.D. BERTHOLF and C.H. KARNES $
I
I
I
t
I
I
,
6
FIG. 3. Predicted specimen strain and interface velocity histories.
where (U, - U,)/l, is the average strain-rate. There is a slight uncertainty in average specimen strain obtained from (8) because of various assumptions that can be made concerning the rise-time of U,. The assumption that is made in this study is that the rise-time is equal to the initial rise-time of the strain record at G, plus the time due to additional dispersion from having propagated a distance of 5 diameters from an axial position of 15-20 dia (from G, to L). Specimen stress history is obtained from the output bar strain history in Fig. 2 by converting the output bar strain history to stress, using the aforementioned time translation and compensating for the difference in area of the specimen compared to the bar. Only the output bar strain history is used, instead of averaging the input and output bar strain histories because the input bar is subjected to large Pochhammer-Chree oscillations due to short rise-time initial impact conditions and dispersion phenomena. These oscillations almost totally reflect at the first interface and essentially the only stress transmitted into the specimen is that which is proportional to the output bar strain record as shown in Fig. 2. Averaging the responses of both bars retains the input bar oscillations even though they are not transmitted by the specimen (see BERTHOLF (1974) for more detail). RAND (1967) has concluded that one must average the two bar responses to minimize the oscillations in the specimen stress history. However, Rand’s analysis considers only one-dimensional stress behavior and therefore the oscillations he obtained
Two-dimensional
analysis of the split Hopkinson
pressure bar system
7
are due to axial inertia only. Furthermore, any short rise-time disturbance will propagate with a velocity nearly equal to the bulk sound speed {bulk modulus~density)~, which can be at least an order of magnitude greater than the one-dimensional stress, plastic wave propagation velocity [(~~(s)~~&)/~~~predicted by the elementary strainrate independent theory. Therefore, the axial stress should reach equilibrium much faster than any one-dimensional stress analysis can predict. The resulting stress-strain curves for the specimen in the sample computation are shown in Fig. 4. The solid line represents the stress-strain model used to describe the specimen material for the wave propagation calculations. Solid circles represent the resulting stress-strain curve predicted by treating the elastic input and output bar
0.8
VI D F
-
*0.4
1NPUT CURVE 0 l
STRESS PREDlCTED FROM AVERAGE OF ELASTIC BARS STRESS PREDICTED FRWYI OUTPUT BAR
J- AVERAGE AND RANGE OF 1 CALCU~TED VALUES 0.2
0.0
I
3 STRA12N-percent
4
FIG.4. Predicted, computed
and input stress-strain paths for no friction, a specimen Iength-todiameter ratio of O-3, and an average strain-rate of about 400 s-l.
strains as experimental data and using only the output bar to determine the specimen stress. Open circles represent the stress-strain path which would be predicted using the average of the input and output bars to determine the specimen stress. Crosses represent the average of the instantaneous axial stress and strain values of all the numerical meshes within the specimen and show the range of values of the calculated axial stress and strain. The length of the vertical lines along the abscissa represents the magnitude of the largest nonaxial component of the stress tensor (Max (a,,, cr8,, cr,,) where Y,8 and z are the radial, circumferential and axial coordinates, respectively). These lines indicate the degree of deviation from a one-dimensional stress state. It is apparent, for the conditions of these calculations, viz. bar diameter of 2.54 cm, specimen length-to-diameter ratio of 0.3, zero friction and average strain-rate of approximately 400 s- ‘, that (i) the elastic bar strains predict very well the actual stress
8
L. D. BERTHOLP and C. H. KARNES
and strain history experienced by the specimen, (ii) specimen represents the true stress-strain behavior of strain are uniform throughout the specimen within a state experienced by the specimen is one-dimensional
the stress-strain history of the the material, (iii) the stress and few percent, and (iv) the stress within a few percent.
5. EFFECTS OF FRICTION AND LENGTH-TO-DIAMETER RATIO A series of calculations was made in order to determine the effect of realistic friction on (i) the stress-strain path followed by the material, (ii) the deviation of the stress state from one-dimensionality, (iii) the uniformity of the stress and strain throughout the specimen, and (iv) the accuracy with which the elastic bar response predicts the specimen behavior. The results of calculations with l/d = O-3 and for friction coefficients of @05, 0.15, and 0.25 are shown in Figs. 5, 6 and 7, respectively.
l
INPUT CURVE STRESS PREDICTEO FROM @LITPUT BAR
i CALCULATED VARIATION
N =0.05 ,NON-AXIAL STRESS-,
FIG. 5. Effect of friction on specimen stress-strain
response (p = 0*05).
is apparent for this small length-to-diameter ratio that a friction coefficient of O*OS+f produces stress and strain variations throughout the sample resulting in actual stressstrain states and a stress-strain path predicted by the elastic bar strains which are
It
2-3 percent higher than the true one-dimensional Even though higher friction coefficients can
stress-strain
behavior
of the material.
produce totally unacceptable stress and
t This is a reasonable value (see O'CONNOR and BOYD (1968), for exampIe). Hydrodynamic lubrication conditions can be obtained momentarily by the use of liquid lubricants producing coefficients of friction on the order of lO-a-1O-3. One must be cautioned that using this type of hrbricant and comparing SHPB data with quasi-static data could lead to errors unless a greater l/d ratio is used quasi-statically to approach the same friction effect. Use of solid film lubricants is recommended to minimize this problem.
Two-dimensional
analysis of the split Hopkinson
-INPUT l
pressure bar system
9
CURVE
STRESS PREDICTED FROM OUTPUT BAR
STRAIN - percent
FIG. 6. Effect of friction on specimen stress-strain
response (p = 0.15).
l.o-
D.8-
L i?O.6-
.
STRESS PREDICTED FROM OUTPUT BAR
I
I
p .-0.25
l__l-_ 3
I
STRAIN - percent
FIG. 7. Effect of friction on specimen stress-strain
L
4
response (p = 0.25).
10
L. D. BERTHOLF and C. H. KARNES
strain variations in the specimen, accompanied by stress-strain states significantly different from one-dimensional stress, the elastic bar strains still predict correctly the average behavior of the specimen as influenced by friction. That is, the intersection points of the crosses are accurately predicted by the dots in Fig. 7. In order to determine if these friction effects could be reproduced experimentally, Professor W. N. SHARPE (unpublished work, 1971) performed a series of experiments at Lawrence Livermore Laboratory (University of California) using lubricated, dry and bonded specimens. Calculations were made with various degrees of friction for the same bar and specimen geometry and the same initial conditions as the experiments. The resulting stress-strain behavior obtained experimentally and those obtained from the two-dimensional calculations are shown in Fig. 8. For comparison, the parabolic
-l
STRAIN
INPUT CURVE PREDICTED LUBRICATED
-
peicent
FIG. 8. Comparison of calculated response for various friction coefficients and experimental data for different interface conditions.
stress-strain model used in the calculations is shown as the dashed curve. The experimental data includes two experiments for each of the three different friction conditions. Since it was shown in Figs. 4-7 that friction results in a reconstituted stress-strain curve having, for a given strain, a larger stress magnitude than the input curve, thz fact that Sharpe’s data for the lubricated specimens fall slightly below the input stress-strain curve indicates that the input curve is incorrect for the annealed 1100 Al used in the experiments. However, the input curve used is entirely adequate for demonstrating the effect that friction has on the reconstituted stress-strain curve. As shown in Fig. 9, friction has a much less pronounced effect for a length-todiameter ratio of 1,2. This is expected because much less volume of material is constrained by friction for this geometry. However, axial inertia, which will be discussed
Two-dimensional
T
analysis of the split Hopkinson I
I
pressure bar system
11
f
.8 .6
- -
INPUi PREDICTED
.a
.2
.O
830.9.
1 1
I
I
2 STRAIN -
3
percent
Effects of friction for a specimen length-to-diameter
ratio of l-2.
Section 6, results in a reconstituted stress-strain curve significant@ different from the true behavior even without friction. The effect of friction is summarized in Fig. 10 and compared to (4) which is RAND’S (1967) approximation for the ratio of the apparent to the correct one-dimensional stress. The points shown are taken from the twodimensional calculations for various geometries, strain-rates, loadings, and friction coefficients. The calculations indicate that about an eight percent error due to friction can be expected from a friction coefficient of 0.05 for a length-to-diameter ratio of O-1. Therefore, extreme care must be exercised if thin specimens are used. in
FIG. 10. Comparison
of calculated and approximate
corrections
for friction eff&zts.
L. D. BERTHOLF and C. H.
12
KARNFS
The most dramatic effect of friction can be seen in the nonuniformity of the stress and strain throughout the sample and the degree of multidimensionality in the stress state. Figure 11 shows the range of values of the axial stress component throughout the specimen expressed as a fraction of the average axial stress component for various values of the coefficient of friction and for a length-to-diameter ratio of 0.3. The degree to which the stress state in the specimen is multidimensional is illustrated in Fig. 12 where the maximum nonaxial stress component (Max (err, cBO,err) expressed as a fraction of the average axial stress component) is shown for various friction It is seen from these two Figures that even a friction coefficient of O-05 coefficients. produces approximately a ten percent variation in axial stress as well as a ten percent deviation from a one-dimensional stress state for this length-to-diameter ratio.
1.0
I
I
‘I
/ ’
I 0.8 -
I’id - 0.3 = 4005-l w
”Y =b-N 0.6 .E c-N ;I a c.4
1 II - 0.25
2 6 -
I-
0.15
0.20.0 ml
I 120
I 140
1 160 TIME
FIG. 11. Effects of
1.0,
0.8
I
il- 0.05 1 180
I MO
Ifi’0 220
240
- LIS
friction on specimen axial stress uniformity.
,
I
I
1
'1
--
0.6
1W
120
140
100 TIME
FIG. 12.
180
200
220
240
- ps
Effects of friction on specimen nonaxial stress components.
Two-dimensional
analysis of the split Hopkinson
13
pressure bar system
6. EFFECTS OFINERTIA Reconstituted stress-strain curves for three different length-to-diameter ratios, three different friction coefficients, and a Heaviside step function input are shown in Fig. 13. For zero friction, the reconstituted behavior for a length-to-diameter ratio of 0.3 and, to a lesser extent, O-6 represents the actual stress-strain behavior very well. However, a ratio of 1.2 requires such a high initial stress in the input bar to produce an average strain-rate of approximately 400 s - ’ that very large stress and strain gradients are produced which cause the effects of inertia to become important. 1.0
I
I
I
I
I
. . . . . .
h/d = 0.3 L’/d s 0.6
.-.-.-
!/d = 1.2 _. ._
---
o..o’
1
I
I
STRtilN
FIG. 13. Effects of specimen
geometry
+
I
I
INPUI I
I
I
1 PERCENT k
on specimen stress-strain coefficients.
behavior
for various friction
The effects of inertia become more and more pronounced as the strain-rate is increased. A series of computer calculations was made with zero friction and a O-3 initial length-to-diameter ratio to show the effects of inertia on the reconstituted stressstrain curve for various strain-rates. The strain-rate was varied by changing the magnitude of the initial Heaviside stress applied to the input pressure bar. Results are shown in Fig. 14 for average strain-rates of approximately 400, 800, 1600 and 3300 s-l. Higher strain-rate results show the increasing effects of radial inertia on the nonuniformity of the stress distribution. As the average strain-rate is increased, the variation in stress and strain throughout the specimen seriously increases. This results in oscillations in the reconstituted stress-strain curves and large variations in stress and strain. The oscillations are caused by radial waves resulting from the short rise-time initial compression wave. The calculated stress histories in the specimen show that the frequency of the oscillations corresponds closely to radially propagating bulk waves. In order to minimize these oscillations and resulting variations in stress and strain, a series of calculations was performed using a 40 us rise-time ramp compression wave
L. D.
O.OL'
BERTHOLF and
+--4
C. H.
STRAIN- percent
KARNES
i
Ii
4
FIG. 14. Effect of strain-rate on specimen stress-strain behavior for Heaviside input. The groups of numbers represent the calculated instantaneous average strain-rates for the four different curves at the indicated strain levels.
instead of a Heaviside step functiont at the impact end of the input elastic bar. Figure 15 shows a comparison between the resulting average specimen strain-rate histories for the ramp and Heaviside input functions. It is seen that the ramp input effectively eliminates the oscillations without significantly reducing the average strain-rate. The reconstituted stress-strain paths using the ramp input wave are shown in Fig. 16 for average strain-rates of 400, 800, 1600 and 3300 s-i. The effects of inertia are summarized in Fig. 17 where the calculated results are compared to (3) which is Davies and Hunter’s approximate correction for axial and radial inertia, By rewriting (3),
and by assuming v, = +, one obtains the lines in Fig. 17 using the l/d ratios of interest. Points shown in this Figure correspond to the calculated results for step inputs, For each configuration one p = 0, three l/d ratios, and various strain-rates. point is obtained by choosing the time at which d2c/dt2is a maximum. For this value of time the stress and strain predicted by the SHPB (ab and .sb) are obtained. The input stress-strain curve and sb are used to determine a,. The positive portion of Fig. 17 corresponds to l/d > v,J3/4 and indicates the region in which axial inertia predominates. Conversely, the negative portion indicates the region of radial inertia domination. For the specific configurations studied herein, t The Heaviside step function input condition results in a rise-time of approximately propagating 20 diameters (see Figs. 2 and 3).
10 fls after
Two-dimensional
analysis of the split Hopkinson
pressure bar system
15
RAMP INPtiI If
1 RANGE OF CALCULATED / VALUES WITHIN SPECIMEN
‘/
fld = 0.3 p-o.0
: I I
FIG. IS. Calculated specimen average strain-rate histories for ramp and Heaviside inputs.
RAMP INPUT I/d - 0.3 84 = 0.0
STRAIN - percent
FIG. 16. Effect of strain-rate on specimen stress-strain behavior for ramp input. The groups of num&x-S represent the calculated instant~eous average strain-rates for the four different curves at the indicated strain levels.
L. D. BERTHOLF and C. H. KARNES
16 .1.2
I
I
I
I
I
I
STEP INPUT
1.0
/
i/d - 1.2
t
a f/d
= 0.6
o i/d = 0.3
0.2
0.4
0.6
0.8
1.0
i
1. ?
1.4
-1 -2 maxlub -kbar S
;i
FIG. 17. Comparison of the magnitude of the approximate inertia correction to the calculated errors due to inertia.
it is clear that the approximations of Davies and Hunter result in a reasonable correction for inertia effects. While different situations may result in different conclusions, these results lend strong credence to their approximate expression. 7. EFFECTS OF STRAIN-RATE For small values of l/d, it has been shown that friction produces an error when the reconstituted stress-strain curve is compared to the input stress-strain curve. This error could be interpreted as a strain-rate effect in an experiment if one simply compares a single ‘dynamic’ stress-strain curve to a stress-strain curve obtained under quasi-static, one-dimensional stress conditions. In order to determine if this error is dependent on strain-rate, which would invalidate the basic premise of the SHPB technique, a series of calculations was made using a friction coefficient of 0.05. The results for ramp loadings are shown in Fig. 18 in which the reconstructed stress-strain curves for strain-rates of 400, 800, 1600 and 3300 s-l for a friction coefficient of 0.05 are It is seen that the compared to results for a strain-rate of 400 s-l and zero friction. error due to the interface friction coefficient of 0.05 is independent of strain rate. One can conclude, considering the fact that step loadings give similar results, that a reasonably small interface friction alone does not produce an apparent strain-rate effect when stress-strain curves obtained from the SHPB technique are compared. However, one must be certain that equivalent interface friction conditions are attained in quasistatic tests when results from such studies are compared to results from SHPB studies.
8. SCALING All calculations were performed with a constitutive equation which is rate independent in order to separate the effects of intrinsic material rate dependence from apparent rate dependence due to friction or inertia. Because of the absence of rate
Two-dimensional
analysis of the split Hopkinson
I
I
pressure bar system
17
I
STRAIN _ percent
FIG. 18. Indication
of the friction-induced material strain-rate ef%ct. No rate-effect present when comparing many rates for the same interface conditions.
dependence in the constitutive equation used, the solutions obtained can be applied directly to different diameter systems. All the results presented are for 2.54 cm dia bars and all the specimens are initially 2.12 cm dia except those used for comparison with Sharpe’s data (see Section 5) which are 2.438 cm dia. If one were to retain the mechanical properties of the bars and the specimen, retain the amplitude of the input compressive wave, and scale each linear dimension of the components proportionately by a factor n, then the results presented herein are valid if one scales time and strain-rate linearly and inversely with the dimension scaling factor, respectively. For example, if t is the time variable in these calculations and D is the bar diameter, then the time variable t’ for cakulations of a different diameter nD is given by t’ = nt. In
(10)
like manner, the strain-rate 8 for the corresponding configuration is given by E’ = E/n.
(11) Therefore, for a bar diameter of 0635 cm (specimen diameter equaIs 0.53 cm), Figs. 2 and 3 give the calculated responses of the input and output bars at the strain gauge locations and the resulting interface velocity and specimen strain history if one reduces time by a factor of four. Figure 4 becomes the stress-strain behavior as predicted by the elastic bar responses, but for an average strain-rate of 1600 s-l instead of 400 s-l. If it is concluded from Fig. 16 that employing a 40 ps rise-time ramp allows reliable stress-strain paths to be obtained up to strain-rates of about 2000 s-l for the 2.54 cm dia system, then it is apparent that a 10 ps rise-time ramp can be used to obtain reliable data to strain-rates of about 8000 s-l with a 0.635 cm dia system. The critical parameter, which allows one to assess the validity of experimental data as it might be 2
18
L. D. BERTHOLF and C. H. KARNES
influenced by inertia, becomes L?dMAX=5x103cms-‘,
(12) provided that a ramp input compressive wave is used where the ratio of its rise-time to the bar diameter is given by t,/D 2 16 ps cm-‘.
(13)
9. DHX.,JSSION While a large computer program and long running times were required, a comprehensive two-dimensional analysis of the SHPB has been a~compJished. Calculated and experimental results show excehent agreement for a variety of different interface conditions. The experimental conditions at the interface included bonded, dry and lubricated. Inertia and especially friction produce additional constraints and result in multiaxial stress states which might be misconstrued as additional strengthening due to rate effects. However, with a little caution the SHPB experiment is very accurate and can be reliably used for the measurement of mechanical properties at high strainrates. The correction for friction (as predicted by RAND (1967) and verified here) depends upon p/(l/d). While the degree of friction-induced over-stress is somewhat smaller than that predicted by Rand, this ratio is a reliable indicator of the friction effect in the configuration studied herein. As predicted by DAVIESand HUNTER(1963), results of the numerical calculations indicate that the correction required for the over-stress due to inertia is proportional to the second time-derivative of the strain. The sign of this correction changes depending upon the dominance of radial inertia or of axial inertia and I/d = v$@ fi: l/2 has been shown to be a good criterion for SHPB specimen design. Friction has been shown to increase greatly the degree of specimen stress and strain nonuniformity. SHARPEand HOGE(1972) have observed considerable strain variations on the surface of specimens without lubricated ends. The calculations presented here agree with their results and give the stress and strain variation throughout the specimen. BELL’S(1966) conclusion “. . . the aariability in such split Hopkinson bar data . . ., demonstrates that the theoretical as.~umption of uniform stress and strain in short specimen, i.e. the quasi-static h~~othesjs, is ph~sica~~~incorrect.” is correct for the bonded specimens he used; however, it is false if reasonable care is taken to lubricate the ends of the specimen. Since time-dependent material properties were not considered in these calculations, the results for the given SHPB system can be directly scaled to other systems. For the same bar and specimen materials and the same Z/d values, the results of these calculations for D = 2.54 cm and a given strain-rate IJare the same as would be obtained for a system of size nD at a strain-rate of S/n. For the elastic bar properties and configuration considered herein, the limiting maximum value for the DB product exceeds 1000 cm s- 1 for the step input and is considerably larger when d’sldt’ is minimized by ramping. This means that a O-635 cm SHPB system with a ramped input can accuratefy measure material properties to strain-rates of at least 8000 s- ’ as long as care is taken to minimize the effects of friction and inertia.
Two-dimensional analysis of the split Hopkinson pressure bar system
19
ACKNOWLEDGMENT The writers are indebted to Professor W. N. Sharpe, (Michigan State University) who graciousiy provided us with his unpublished SHPB data. REFERENCES BELL,
J. F.
BERTHOLF,L. D. BERTHOLF,L. D. and BENZLEY,S. E. BERTHOLF,L. D. and KARNES,C. H. CHIV, S. S, and NEUBERT,V. H. CONN, A. F. DAVIES,E. D. H. and HUNTER, S. C. DHARAN,C. I(. H. and HAUSER,F. E. HAUSER,F. E., SIMMONS,J. A. and DORN, J. E.
1964
1966 1974 1968 1969 1967 196.5 1963 1970 1960
JACKSON,J. W. and WAXMAN,M.
1963
JAHSMAN,W. E. KARNES,C. H. and BERTHOLF,L. D.
1971 1970
KOLSKY,H.
1949 1963
LINDHOLM,U. S, MEYER,K. A. and BLEWETT,P. J. O’CONNOR.J. J. and BOYD,J.
1964 1972 1968
RAND, J. L.
1967
&MANTA,S. K. SHARPE,W. N. and HOGE, K. G. SIEBEL,E. STANTON,P. L.
1971 1972 1923 1968
STEVENS,A. L. and JONES,0. E.
1972
Stress Waves in Aneiastic Solids (edited by KOLSKY, H. and PRAGER,W.), p. 166. Springer-Verlag, Berlin. J. Me&. Phys. So&& 14, 324. J. appf. Mech. 41, 137. Sandia Laboratories Report SC-RR-68-41. J. appl. Mech. 36, 533. J. Mech. Phys. Solids 15, 177. Ibid. 13,311. Ibid. 11, 155. Exp. Mech. 10, 370. Response of Metals to High Velocity Deformation(Proceedings Metallurgica Society Conferences 9) (edited by SHEWMON,P. 6. and ZACKAY, V. F.), p. 93. Interscience, New York. High Pressure Measurement, p. 39. Butterworths, London. J. appl. Me& 38, 75. Inelastic Behavior of Solids (edited by KANNINEN, M. I;., ADLER, W. F., ROS~NFIELD,A. R. and JAFFEE,R. I.), p. 501. McGraw-Hill, New York. Proc. Phys. Sot. B62,676. Stress Waves in Solids, p. 153. Dover Publications, New York. J. Mech. Phys. Solids 12, 317. Phys. Fluids 15, 753. Standard ~and~oa~ of L~ricatjo~ Engineering. McGraw-Hill, New York. U.S. Naval Ordnance Laboratory Report NOLTR67-156. .7. Mech. Phys. Solids 19, 117. Exp. Mech. 12, 570. Stahl u. Eisen 43, 1295. Sandia Laboratories Report SC-CR-683672. J. appi. Meek 39, 359.