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European Journal of Applied Sciences – Vol. 11, No. 6
Publication Date: December 25, 2023
DOI:10.14738/aivp.116.15700
Partom, Y. & Lindenfeld, A. (2023). Dynamic Response of Ductile Materials from Split Hopkinson Tension Bar Tests. European
Journal of Applied Sciences, Vol - 11(6). 23-39.
Services for Science and Education – United Kingdom
Dynamic Response of Ductile Materials from Split Hopkinson
Tension Bar Tests
Y. Partom
18 HaBanim, Zikhron Ya'akov 3094017, Israel
A. Lindenfeld
18 HaBanim, Zikhron Ya'akov 3094017, Israel
ABSTRACT
We use a SHTB (= Split Hopkinson Tension Bar) system to test the response of
aluminum 6061-T651 samples under dynamic tension, and to calibrate a strength
model for them. Interpretation of SHTB tests cannot be done with the so-called
classical analysis, because: 1) the samples are long (relative to compression
samples) and are not in dynamic equilibrium; 2) a neck is usually formed towards
the end of the test; and 3) the strain to failure is quite high (around 50%). We
therefore interpret the tests (calibrating from them a strength model and validating
it), using computer simulations of the entire test system. Our entire system
simulations are able to reproduce with good fidelity: 1) the response of the SHTB
gauges on the two bars; 2) the time evolution, shape and location of the neck; and
3) the strain to failure measured after the test.
INTRODUCTION
High strain rate material characterization in compression, for strain rates of the order of 102-
104/s, is commonly carried out with a SHPB (also called Kolsky bar) [1]. With this method the
specimen is compressed between input and output bars, and the resulting stress-strain curve
is deduced from strain measurements on both bars, at some distance from the sample. The
equationsused are known as the classical analysis. It has been shown [1,2], that the validity of
the classical analysis depends on a uniform stress distribution along the specimen (called stress
equilibrium).
Several modifications to the SHPB configuration have been developed for tests in tension, and
they are all called SHTB (= Split Hopkinson Tension Bar). Some of these use special specimen
geometries [3], and others use special loading arrangements [4-6]. With all these configurations
it’s not possible to use the classical analysis because: 1) the samples are usually long (relative
to compression samples), and are generally not in dynamic equilibrium; 2) a neck is usually
formed towards the end of the test; and 3) the strain to failure is quite high (around 50%) [7-
9].
Various techniques have been suggested to overcome this difficulty. Rajendran and Bless [10]
use Bridgman’s approximation [11] to obtain a stress-strain curve in dynamic tension beyond
the point of necking. To this end they measure the geometry of the neck, using high speed
photography. They find this approach to have different levels of accuracy for different
materials. Rodriguez et al. [7], using finite element simulations, suggest using an ‘effective
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European Journal of Applied Sciences (EJAS) Vol. 11, Issue 6, December-2023
length’ for which uniform strain can be assumed to hold in the specimen, and they determine
this effective length from the simulations. Rotbaum et al. [12] suggest using long specimen, to
keep the stress and strain constant in a large part of the specimen prior to necking. In this way
they are able to use the classical analysis with minimal error, but only up to 3% plastic strain.
Mirone [13] uses a correction to the classical analysis, which enables to reconstruct the stress- strain curve up to failure, but with an error of 15-25%.
Here we abandon the classical analysis and use instead computer simulations of the entire
SHTB system. With nowadays computer capabilities it’s not difficult to run simulations of the
entire test system including the bars, all the way to the specimen failure. From the simulations
we obtain time histories of particle velocities, and stresses and strains everywhere in the test
system. This includes details of neck formation, and strains (or particle velocities) measured on
the bars.
As shown in [14-17] and in [26], the final neck location depends on the boundary (or loading)
conditions, and is not sensitive to imperfections in the sample material or geometry. We
therefore use the final neck location as additional diagnostics of the test.
As mentioned above, to reach specimen failure in a dynamic tension test, we need a relatively
long loading pulse. For an existing Hopkinson bar system, this is not generally possible by using
a long striker bar, as these systems are usually designed for compression tests. As shown later,
we achieve a long enough loading pulse by using a technique that allows for wave reflections
from the input and output bars.
In section 2 we describe the experimental setup of our SHTB system, using circular DB (= Dog
Bone) specimens. In section 3 we describe our computational scheme. In section 4 we calibrate
a strength model in both compression and tension. Finally, in section 5 we extract the strain to
failure relation from the tests and from the simulations.
EXPERIMENTAL
The material we use in the tests is aluminum 6061-T651, which fails in tension at an average
strain of about 55%. We first calibrate the strength model for this material in compression. We
then use this calibration as the starting point for the calibration in tension.
For the compression tests we use our standard SHPB system with hardened C300 maraging
steel bars. The striker and the input and output bar lengths are 40, 399 and 250cm respectively,
and their diameter is 25.4mm (Fig. 1a). In both compression and tension tests we use our PDV
(= Photonic Doppler Velocimetry) diagnostics system [19], instead of the traditional strain
gauges diagnostics, at 1m from the specimen on the input and output bars. This PDV diagnostics
system, developed by our colleagues, was shown in [19] to be advantageous over the traditional
strain gauges diagnostics.
For the tension tests we use aluminum 7075 bars, to reduce the impedance mismatch between
the bars and the specimen. The striker and the input and output bar lengths are 75, 300 and
220cm, respectively. The input pulse goes mainly through the sleeve into the output bar. A small
part of it goes through the first thread into the specimen (which stays elastic), and then through
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Partom, Y. & Lindenfeld, A. (2023). Dynamic Response of Ductile Materials from Split Hopkinson Tension Bar Tests. European Journal of Applied
Sciences, Vol - 11(6). 23-39.
URL: http://dx.doi.org/10.14738/aivp.116.15700
the other thread back into the output bar. The input pulse is reflected from the far end of the
output bar, and then travels back to load the specimen in tension through the thread shown at
the bottom part of Fig. 1. This loading arrangement in tension was suggested by Nicholas [4].
When the first reflected pulse in not long enough to fail the specimen, the loading in tension
continues by a similar second reflection of the same pulse. In our tests the specimen has always
failed before the end of the second pulse. In Fig. 2 we show an x-t diagram of the various waves
in the input and output bars, including the second reflected wave. To recover the specimen after
it fails without additional damage, we use a sand box located behind the output bar.
Figure 1: Experimental setup. Top: Compression and tension. Bottom: Tension.
Figure 2: x-t diagram of the SHTB test including the first and second pulse reflections from the
far end of the output bar; blue=compression, red=tension.
Interferometer 2 Interferometer 1
Output bar
Specime
n
Input
bar
30
o
Striker
Region shown below
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For the compression tests we use disk specimens with L0/D0=0.5, with diameters, striker
velocities, and average strain rates, as shown in Table 1.
Table 1: Summary of the experimental conditions for the compression tests.
Specimen no. L0/D0(mm/mm) Vstriker (m/sec) (1/sec)
1 6/12 8.6 560
2 5/10 11.0 1500
3 7/14 18.5 1700
4 6/12 19.0 2500
5 7/14 30.0 4000
Both compression and tension specimens are made of the inner core of a 25.4mm diameter rod
with an average grain size of 30μm.
In Fig. 3 we show our tension specimen. We see that the diameter is 7.5mm and the gauge
lengths are 22.5, 30 and 37.5mm, so that their L0/D0 are 3, 4 and 5, respectively.
Figure 3: Geometry and dimensions of the SHTB specimens.
COMPUTER SIMULATIONS
As outlined in the introduction, we use here computer simulations of the entire SHTB system
to calibrate the strength model parameters in tension from test diagnostics, and we do this also
for the compression tests with the SHPB system. This enables us to: 1) to take into account the
early part of the output signals, when the specimen is not yet in dynamic equilibrium; and 2) to
take into account the effect of TS (=Thermal Softening), which is small but not negligible, and is
usually ignored by the classical analysis.
We perform the computer simulations with an in-house 2D Lagrangian hydrocode [20-22]. Our
hydrocode uses J2 plasticity with Wilkins’ radial return method for elastic-viscoplastic flow
[23]. There are many strength models in the literature to represent elastic-viscoplastic
response, and many of them show ability to reproduce test results. We could choose any of
them to demonstrate our approach to strength model calibration from SHTB tests via computer
simulations. We choose to use the JC (= Johnson-Cook) strength model [24], which is widely
used by far. The JC model is given by:
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Partom, Y. & Lindenfeld, A. (2023). Dynamic Response of Ductile Materials from Split Hopkinson Tension Bar Tests. European Journal of Applied
Sciences, Vol - 11(6). 23-39.
URL: http://dx.doi.org/10.14738/aivp.116.15700
(1)
where A, B, n, C, m are material parameters, is a reference strain rate taken as 1/s, T0 is the
initial temperature (room temperature here) and Tm is the melting temperature.
For the compression tests, the striker and bars properties are: (density)=8.1g/cc, G (shear
modulus) =71.8GPa and K (bulk modulus) =128.3GPa, resulting in a bar velocity of 4734m/s.
The cell size along the bars is 1.5mm, and it reduces to 0.3mm toward the specimen. The
average cell size in the specimen is 0.1mm, and the average computing time is several hours on
four processors.
The geometry of the tension tests computations is shown in Fig. 4. The striker and bars
properties are: =2.8g/cc, G=26.7GPa, and K=75.8GPa, resulting in a bar velocity of 5056m/s.
The threaded connection between the specimen and the bars consists of two toroidal rings,
with end angles of 60o on both sides (see Fig. 4). This simplified representation of the threads
is adequate, as it enables radial motion between the specimen and the bars, and keeps the
threads in the elastic range. The meshing is similar to that of the compression tests. We define
two computational Lagrangian gauges at the ends of the specimen gauge section, for the
evaluation of the average strain rate in the specimen.
Figure 4: Geometry of the tension tests simulations: a) specimen region; b) thread region
(enlarged).
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TEST RESULTS AND STRENGTH MODEL CALIBRATION
In Fig, 5 we show the experimental and computed signals from test 5 of Table 1. We focus on
the transmitted pulse as it includes less noise. The calibrated parameters of the JC model are
shown in Table 2, with Tm=900K and m=1. We see from Fig. 5-bottom (red line) that a better
agreement is obtained when the simulation includes thermal softening (TS). A similar level of
agreement is obtained for all other tests (not shown here).
Figure 5: Compression tests, comparison between test and simulation for specimen 5 in Table
1. Top: all signals. Bottom: transmitted signal. (The red line drop of the incident wave signal is
an artifact of the recording system and should be ignored. Similar drops appear in the output
signals of several other figures).
Table 2: Material parameters of the JC model, calibrated from the compression tests.
A (MPa) B (MPa) C n Tm (°K) m
339 180 0.015 0.36 900 1
We check the sensitivity of the above calibration by changing the value of A by ±7MPa, which is
about ±2%, and we show the comparison in Fig. 6. We see from Fig. 6 that the simulations are
rather sensitive to small changes in the parameter A. We find a similar sensitivity (not shown
here) to the other strength model parameters.
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Partom, Y. & Lindenfeld, A. (2023). Dynamic Response of Ductile Materials from Split Hopkinson Tension Bar Tests. European Journal of Applied
Sciences, Vol - 11(6). 23-39.
URL: http://dx.doi.org/10.14738/aivp.116.15700
In Table 3 we summarize the specimen geometries, loading conditions, and average strain rates
of the tension tests. The average strain rate is defined for the gauge section as the ratio of the
velocity difference of its two ends to its current length.
Table 3: Summary of the tension tests. D0=7.5mm.
Specimen no. L0/D0 Vstriker (m/sec) (1/sec) Failure
1 3 10.0 230 No
2 3 13.8 400 No
3 3 19.6 670 Yes
4 3 20.4 700 Yes
5 3 26.5 950 Yes
6 3 27.0 1000 Yes
7 4 15.5 350 No
8 4 15.0 340 No
9 4 20.5 500 Yes
10 4 19.5 470 Yes
11 4 28.0 800 Yes
12 4 28.0 800 Yes
13 5 10.6 150 No
14 5 9.3 120 No
15 5 22.5 470 Yes
16 5 19.5 400 Yes
17 5 29.5 670 Yes
18 5 28.0 620 Yes
Specimens 1-6: L0=22.5mm. Specimens 7-12: L0=30mm. Specimens 13-18: L0=37.5mm.
Table 4: Summary of experimental and simulation results of the tension tests.
Specimen
No.
Final gauge length
(elongation), mm
Loading wave
failure
Normalized neck
location
Strain to
failure
Exp., ±0.125 Sim., ±0.1 Exp. Sim. (in the neck)
1 24.25 (1.75) 23.85 (1.35) N/A N/A N/A N/A
2 26.50 (4.0) 26.67 (4.17) N/A 0.46 0.46 N/A
3 27.50 (5.0) 27.23 (4.73) 2
nd 0.3 0.44 0.6
4 27.50 (5.0) 27.44 (4.94) 1
st 0.3 0.44 0.6
5 28.0 (5.5) 27.53 (5.03) 1
st 0.31 0.4 0.6
6 28.25 (5.75) 27.23 (4.73) 1
st 0.34 0.39 0.7
7 35.0 (5.0) 34.77 (4.77) N/A 0.32 0.41 N/A
8 34.5 (4.5) 34.75 (4.75) N/A 0.28 0.39 N/A
9 n.m. 35.99 (5.99) 2
nd n.m. 0.33 0.5
10 n.m. 35.90 (5.90) 2
nd n.m. 0.36 0.55
11 36.25 (6.25) 35.88 (5.88) 1
st 0.23 0.27 0.6
12 35.75 (5.75) 35.88 (5.88) 1
st 0.22 0.27 0.6
13 39.50 (2.0) 39.59 (2.09) N/A N/A N/A N/A
14 38.75 (1.25) 38.65 (1.15) N/A N/A N/A N/A
15 44.0 (6.5) 44.43 (6.93) 2
nd 0.28 0.23 0.5
16 44.25 (6.75) 44.65 (7.15) 2
nd 0.27 0.26 0.55
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17 44.25 (6.75) 43.67 (6.17) 1
st 0.18 0.2 0.6
18 44.25 (6.75) 43.82 (6.32) 1
st 0.17 0.2 0.55
Neck location is normalized by the gauge length. n.m. means not measurable.
In Fig. 6 we show the recorded signals for specimens 8, 10 and 11 of Table 3.
a) No failure
b) Second wave failure
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Partom, Y. & Lindenfeld, A. (2023). Dynamic Response of Ductile Materials from Split Hopkinson Tension Bar Tests. European Journal of Applied
Sciences, Vol - 11(6). 23-39.
URL: http://dx.doi.org/10.14738/aivp.116.15700
c) Third wave failure
Figure 6: Recorded signals from the tension tests with specimen number: a) 8, b) 10, and c) 11.
The wave for which failure occurs is circled on the output bar signal. (See bracketed remark
under Fig. 5).
In Fig. 7 we compare the experimental and computed signals for specimens 8, 10 and 11, and it
can be seen that agreement is good. Similar agreement is obtained for the other tests (not
shown here). Another parameter that we use to validate the simulation results (beside the neck
location, see section 5) is the elongation of the gauge section. This parameter represents the
integral behavior of the specimen during the whole deformation process, including necking. In
Table 4 we compare the measured and computed gauge section elongation, and see that except
for specimen 1, agreement is within 9%. As we’ve been using in tension the same strength
model parameters calibrated for compression, we conclude that for aluminum 6061-T651
there’s no significant difference between the strength model parameters in uniaxial tension and
compression.
Input bar Output bar
a)
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b)
c)
Figure 7: Comparison between recorded and computed signals for tension tests: a) test 8, b)
test 10 and c) test 11. (See bracketed remark under Fig. 5).
In Fig. 8 we show the neck and the EPS (= Effective Plastic Strain) evolution in specimen 10
from our simulation, all the way until just before failure.
In Fig. 9 we show EPS profiles along the gauge section from simulations of tests 6, 11 and 18 at
four times before failure.
We see that EPS profiles become non-uniform as the neck develops, in accordance with
previous experimental [8, 9] and computational work [7, 15]. Also, maximum EPS is on the axis,
in accordance with previous numerical work [7, 24].
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Partom, Y. & Lindenfeld, A. (2023). Dynamic Response of Ductile Materials from Split Hopkinson Tension Bar Tests. European Journal of Applied
Sciences, Vol - 11(6). 23-39.
URL: http://dx.doi.org/10.14738/aivp.116.15700
Figure 8: Time evolution of the neck, and color maps of effective plastic strain, in specimen 10;
a) to d): first tension wave, e) to h): second tension wave.
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Fig. 9/a
Fig.9/b
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Partom, Y. & Lindenfeld, A. (2023). Dynamic Response of Ductile Materials from Split Hopkinson Tension Bar Tests. European Journal of Applied
Sciences, Vol - 11(6). 23-39.
URL: http://dx.doi.org/10.14738/aivp.116.15700
Fig. 9/c
Figure 9: Effective plastic strain profiles at different times from the simulations of tests: a) 6, b)
11 and c) 18. Specimens loading starts at t1500μs.
FAILURE PREDICTION
When the specimen fails, a release wave is generated at the failure location, which then travels
in both directions. This release wave is represented in both bar signals as a velocity change, as
seen in Figs. 6b and 6c. It decreases the velocity of the input bar, and increases the velocity of
the output bar. The velocity change in the bars caused by specimen failure is:
spec spec
failure
bar bar bar bar
F .A
V
Z .C .A
= =
(2)
where F and spec are the force and the longitudinal stress in the specimen prior to failure
respectively, Zbar is the bar mechanical impedance, bar is the bar density, Cbar is the elastic bar
velocity, and Aspec, Abar, are the cross-section areas of the specimen and the bar, respectively. For
a typical value of this velocity change, we use spec ≈ 450MPa from Fig. 7, bar = 2800kg/m3,
Cbar=5056m/s, Abar 5.0710-4m2 and Aspec = 2.8310-5m2 , for a representative neck diameter of
6 mm (measured from the failed specimens), and we get vfailure = 1.8 m/sec. Hence, an abrupt
velocity increase of ~2 m/sec in the experimental signals marks the time of failure.
We use the time of this velocity change to calibrate the EPS (effective plastic strain) value at
failure. In Fig. 10 we show an example of such a calibration for specimen 11. We see there that
for higher values of EPS at failure, the abrupt velocity change occurs later. We show the
calibrated EPS at failure (StF) in Table 4, and we see there that the average StF is 0.58, with a
standard deviation of 0.05. There is a significant difference between the StF in the neck and the
average strain in the gauge section at failure, as also pointed out in [15].
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In Table 4 we compare neck location between tests and simulations, and find good agreement.
We conclude, in agreement with [15, 17] that: 1) neck location is insensitive to small
perturbations; 2) neck location depends on geometry and loading conditions; and 3) neck
location moves towards the impacted side for a higher impact velocity. In Fig. 11 we show the
effect of loading velocity on neck location.
a)
b)
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Partom, Y. & Lindenfeld, A. (2023). Dynamic Response of Ductile Materials from Split Hopkinson Tension Bar Tests. European Journal of Applied
Sciences, Vol - 11(6). 23-39.
URL: http://dx.doi.org/10.14738/aivp.116.15700
c)
Figure 10: StF (strain to failure) calibration for test 11; a) StF=0.4, b) StF=0.5, c) StF=0.6.
(See bracketed remark under Fig. 5).
a) b)
Figure 11: Specimens before and after impact. a) specimen 6 after failure; b) effect of impact
velocity on neck location. (Impact is on the left, and u=velocity difference between specimen
two sides).
SUMMARY
We focus here on the response of ductile materials to uniaxial dynamic tensile loadings by
means of a SHTB system. We use a combined experimental and numerical approach, our
representative material is aluminum 6061-T651, we use circular dog-bone specimens with
gauge length to diameter ratio of 3, 4 and 5, and we use the JC strength model.
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We do not use the so-called classical analysis to interpret our tests and to calibrate the strength
model. Instead, we use direct computer simulations of the entire test system.
From the tests and simulations, we get: 1) high level overall accuracy, predictions and test data
differ by no more than 2%; 2) accurate prediction of stress-strain curves even for less than 5%
strain; 3) inherent and simple prediction of thermal softening; 4) accurate prediction of StF
(strain to failure) in the neck; 5) accurate prediction of the neck location; and 6) accurate
prediction of the elongation of the gauge section, which can serve as an additional calibration
parameter.
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Sciences, Vol - 11(6). 23-39.
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