CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
NUMERICAL SIMULATIONS OF THE INFLUENCE OF LOADING
PULSE SHAPE ON SHPB MEASUREMENTS
A.D. Resnyansky1 and G. T. Gray, III2
1
Weapons Systems Division, Aeronautical and Maritime Research Laboratory,
DSTO, PO Box 1500, Salisbury SA 5108, Australia
2
Material Science and Technology Division, Los Alamos National Laboratory,
Los Alamos, New Mexico 87545
Abstract. Direct numerical simulation of a Split Hopkinson Pressure Bar (SHPB) system is utilized
to assess the effect of varying the loading pulse on the stress-strain response of a material.
Calculated incident, reflected, and transmitted pulses are correlated with the stress-strain response
of a generic sample using a number of classic methods (1- and 2-wave analyses (1)); the stressstrain curves obtained in the simulations are compared with the input modeled stress-strain
response. Simulation of deformation in the sample is described using a strain-rate sensitive model.
The influence of ramping the input pulse, via variations in the method of loading the incident
pressure bar, on the sample stress-strain response is presented..
Experimental methodological studies are not
readily conducted (see (1)) because they involve a
real SHPB experimental set-up and a real sample
response such that factors influencing the test
records cannot be simply removed or added.
Theoretical and numerical simulation analysis of
the SHPB however offers such a opportunity. Direct
numerical simulation of the SHB 1) uses input data,
such as the stress-strain curves, fit to a constitutive
equation at constant strain-rates; 2) allows choice of
the experimental SHB device using idealised or
'real' components or configurations; 3) simulates
the behaviour of the SHB including the components
and sample; 4) affords processing of the measured
pulses with the use of classical SHB analysis to
study the effect on the calculated stress-strain
response, and 5) allows numerical simulation of the
effect of loading pulse shape on the sample stressstrain response.
Direct simulation of several SHPB loading
methodologies has been conducted previously (3)
for a strain-rate sensitive generic material with an
'ideal1 (strain-insensitive) stress-strain diagram.
INTRODUCTION
The Split Hopkinson Pressure Bar (SHPB) is one
of the most popular testing tools to study materials
at high strain rates. The relatively simple device is
being enhanced by a number of modifications
which are intended to improve the resolution of the
pressure-bar signals, spectrum of the materials
tested, and thereby overall performance of the
device (for details, see review (1)). Along with
more detailed methods of interpretation of the
signals introduced (for example, 2-wave analyses,
methods of correction of wave dispersion, FEMmethods for better interpretation of the signals, etc),
the SHPB continues to receive broader usage.
The Pochammer-Chree dispersion effect has
been studied for many years (see, eg, recent
numerical study (2)). In addition to the PochammerChree oscillations a number of other factors exist
which may result in divergence of the stress-strain
response recorded from the true response of the
material under investigation.
315
midpoint Gl and G2 during a numerical simulation
run): e i - input strain pulse due to impact by the
striker bar (recorded by the first measuring station
Gl), e t - transmitted strain pulse (recorded by the
second station G2), e r -reflected strain pulse
(recorded by the first gauge Gl with some delay).
The first formula for stress is called the 1-wave
analysis, the second one - 2-wave analysis, their
average is referred to as the 3-wave analysis (1).
The formula for strain rate is deduced with the
assumption of the force equilibrium.
In the present paper we consider two basic
theoretical loading situations: i) impact of the
incident bar by a striker bar with a constant velocity
of 40 ms"1; and ii) impact of the incident bar by a
striker bar accelerated by a simulated gas-dynamic
device up to approximately the same velocity. The
latter allows us to consider the situation when the
impactor has a stress and velocity profile resulting
from the acceleration of the striker. This loading
state allows us to assess how a more complex
loading pulse can affect the stress-strain response
calculated for the sample.
We employ a strain-rate sensitive viscoelastic
material model (4) to describe the constitutive
response of the pressure bars and the sample. The
material constants for the pressure bars correspond
to a hard steel (constitutive parameters of the model
are chosen such that the material would be in the
elastic state in the range of the loads), elastic
constants for the sample material correspond to
aluminium. The yield limit characteristics are
generic and tailored to the needs of the current
numerical simulations.
One-dimensional system of equations of the model
includes conservation laws and constitutive
equation for relaxation of shear stress along with
the assumption of instantaneous lateral release
The present paper reports the results of
numerical simulations of the influence of loading
pulse shape on the calculated stress-strain response
of a strain-hardening material.
METHOD AND MODEL
A schematic of a SHPB is shown in Fig. 1. In the
present paper we simulate the following
components of a SHPB set-up (Fig. 1): (1) - a gas
chamber enabling us to accelerate the striker bar (2)
up to a velocity of approximately 40 ms"1, input bar
(3) with the gauge Gl at the midpoint, output bar
(5) with the gauge G2 at the midpoint, and the
sample (4).
(1)
(5)
(3)
(2)
G2
a
m
(4)
(S)
FIGURE 1. Schematic of the SHPB simulated.
The initial cross-sectional areas of the pressure
bars and sample are A and As, respectively. For the
one-dimensional numerical simulation we do not
need the actual diameters of the bars and sample,
we only have to give the ratio of the bar's to
sample's cross-sectional area^4//4s, which has been
taken in the present case to be 2. The lengths of
input and output bars are 100 cm and the length of
the striker bar is 20 cm.
Conditions of the force equilibrium at the two
ends of the sample (FL and FR in Fig. 1) and
solution of the wave equation yield the formulas of
the classical SHPB analysis (1):
AE
AE (
\
(a 2=0).
The constitutive equation has been taken in the
following form assuming a dislocation dominated
mode of plastic deformation (5):
2c
= — s , a = — fe + £r / I, e = —
A t
A **
I
J^=£ +£ =£ = JL.
AE
i
r
t AE
da
a
——
—
dt
T
0
N + MB
Here e is plastic deformation, a is the shear stress,
D, H, N, M, TO are material constants. The
evolutionary equation for the shear stress is shown
Here E is the Young modulus of the pressure bar
material, c=(Elp)m. ls - is the length of the sample.
The following data are 'measured' (taken at the
316
for the portion describing the irreversible
deformation. The elastic portion of the description
is traditional and is not expanded here.
nonequilibrium in the sample, that the sample's
behaviour is in accordance with the idealized
sample model response. One source of the deviation
might be an asynchronization when processing the
input and reflected pulses with the 2-wave analysis.
Deliberate asynchronization in Fig. 3(b) however
demonstrates that the deviation is related to
something else.
RESULTS OF NUMERICAL SIMULATIONS
We start by considering a generic sample
material exhibiting strain hardening.. The first case
involves the impact of the striker bar with a
constant velocity. In this case the incident pulse has
a perfect square-wave shape and the actual strain
rate being elaborated at the impact is approximately
2-4-10V1. The prescribed constitutive response,
based on the material model of the sample for this
strain rate, is given by curve 3 in Fig. 2. The output
curves corresponding to 1-, and 2-wave analyses
which the SHPB simulation yields are seen to be
very close to the idealized modeled sample
response.
^,
£=
V
^
ss*****
****
ft\l
(b)
2
2
1^
3* ^
****
^
(
3
^
f*
'3
6
12
e,{
6
12
8,\
FIGURE 3. Results of modelling for the loading by a gasdynamic device of a hardening material; (a) - the idealized
material response and traces of stress at the left OL (1) and right
OR (2) sides of the sample; (b) — influence of asynchronization of
the reflected pulse on the 2-wave analysis: (1) is taken from Fig.
2, (2) - the reflected pulse asynchronized 1 jxsec ahead, (3) - the
reflected pulse asynchronized 1 (J,sec back.
400
-ft)'
-(<*)-
\l
(°)
400
12
£,(
12
G, 1
A
FIGURE 2. Stress-strain output curves obtained with 1- and 2wave analyses (curves / and 2, respectively), 3 - the modeled
sample constitutive response (strain rate 4000 s"1); (a) - impact
with the constant velocity; (b) - impact of the striker bar
accelerated by a gas-dynamic device.
^'""
\
.*-*""
^
^
^e**~ -^
3^. ^*
Acceleration by a gas-dynamic device (a gas
gun) results in complex wave circulation along the
striker bar prior to the impact with the incident bar.
This results in the ramping of the incident pulse and
in a non-constant velocity profile for the input pulse
(b)
\
V
"^4
80
\
(^
A
\
^^
^^«
^
N^
/
ri
^-1
=s?=
^< ^r^
^4
f^
12
£,(
12
FIGURE 4. Stress-strain output curves obtained with 1- and 2wave analyses (curves 1 and 2, respectively), 3 - the idealized
sample response, 4 - the traced stress cr#; (a) - results of the
SHB simulations of the data above with the reduction to the
current cross-sectional area of sample; (b) the output curves for a
calculation with reduced length of sample (0.5 cm).
(3).
For the case of the non-constant input pulse we
observe a noticeable deviation between the results
of 1- and 2-wave analyses. In order to elucidate if
this result is due to improper response of the
sample material, the traces of stress magnitude for
the two ends of the sample have been tracked
numerically and are presented in Fig. 3(a) along
with the idealized sample response . It is seen,
ignoring the dispersion associated with the initial
Detailed analysis with the reflection of the
ramping pulse from the pressure bar-sample
interface shows that in the given case errors in the
estimation of the sample's cross-sectional area is
the major reason for the deviation. Unlike during
actual experimental testing, numerical simulations
of a SHPB allows one to readily trace the sample's
length during loading. For a plastically
incompressible material the cross-sectional area of
317
the sample can be calculated from the density
conservation rule. The result of such a calculation is
shown in Fig. 4(a). It is seen that it is in good
agreement with the 2-wave analysis.
A final simulation for a reduced sample length of
5 mm is shown in Fig. 4(b) and demonstrates the
same features as in Fig. 2(b).
DISCUSSION
In this paper numerical simulations are reported
describing the importance of the loading pulse
profile to the calculation of the stress-strain
response of a sample during SHPB testing. A model
system has been used to illustrate how complex
wave behaviour in the SHPB may result in errors in
the calculated stress-strain response. Introduction of
oscillations, especially in the incident bar carrying
high stress pulses, has been shown to be particularly
effective at disrupting the achievement of a valid
SHPB test. Finally, based on our simulations, it can
be concluded that real pulses in real SHPB tests
require additional records (such as auxiliary gauges
at the sample) and additional fixtures
(confinements, indirect information about material)
to obtain understanding the SHPB test results for
solids with complex structure.
REFERENCES
1. Gray III, G.T., "Classic Split-Hopkinson Pressure
Bar Technique", in Mechanical Testing and
Evaluation, ASM - Handbook Volume 8 edited by
H. Kuhn and D. Medlin, Metals Park, OH,, ASM
International,, 2000, pp. 462-476.
2. Bertholf, L.D. and Karnes, C.H., J. Mech. Phys.
Solids, 23, 1-19(1975).
3. Resnyansky, A.D., "Study of Influence of
Loading Method on Results of the Split Hopkinson
Bar", in Proc 7th Int Symp on Structural Failure and
Plasticity (IMPLAST 2000), edited by X.L. Zhao
and R.H. Grzebieta, Pergamon, Amsterdam, 2000,
pp. 597-602.
4. Godunov, S.K. and Romensky, E.I., Elements of
Continuum Mechanics and Conservation Laws,
Nauchnaya Kniga Publ., Novosibirsk, 1998 (in
Russian).
5. Merzhievsky, L.A. and Resnyansky, A.D., J de
Physique Colloq. C5, 45, 67-72 (1985).