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European Journal of Applied Sciences – Vol. 11, No. 1
Publication Date: February 25, 2023
DOI:10.14738/aivp.111.14053.
Partom, Y. (2023). Steven Test Modeling. European Journal of Applied Sciences, Vol - 11(1). 576-583.
Services for Science and Education – United Kingdom
Steven Test Modeling
Y. Partom
Retired, 18 HaBanim, Zikhron Ya'akov 3094017, ISRAEL
Abstract
The Steven test was introduced for studying low velocity impact initiation of
explosives. The original diagnostics of the Steven test were blast gauges at a
distance of about 3m. Trying to use these diagnostics to characterize the response
of the explosive after its ignition, we realized that they’re not informative enough.
We therefore replaced the blast gauges by velocity gauges, looking at the free
surface of the back plate of the test sample. Our explosive is similar to LX07, and we
performed several tests with impact velocities from 30m/s to 122m/s. The velocity
histories we obtained from the gauges show the following: 1) there is a rather long
delay between impact and ignition (or gauge response), 50μs for the highest impact
velocity and around 250 μs for the 36m/s impact; 2) there’s no ignition at the impact
velocity of 30m/s; 3) the gauge velocity histories rise gradually to a maximum and
then continue with elastic oscillations. We model the response of the explosive
assuming that it reacts through shear initiation. The projectile impact causes shear
flow in the explosive, which leads to strain localization and formation of shear
bands. The shear bands heat up and reach ignition temperature and deflagration
waves expand out of them, similar to deflagration waves out of hot spots for shock
initiation. Shear initiated reaction rate is rather slow as: 1) shear bands are surfaces
(and not spots); and 2) the average distance between shear bands is of the order of
1mm, and not 0.1mm (as for hot spots). Also, shear-initiated reaction rate depends
on pressure and not on reactant temperature (as for hot spot initiation). Here we
use our PDSR (= Pressure Dependent Shear Reaction) model to reproduce our
Steven test data. We get good agreement with delay times and amplitudes of the
velocity gauge outputs.
INTRODUCTION
The Steven test was introduced for studying low velocity impact initiation and reaction of
explosives [1, 2]. Here we report on a set of Steven tests with LX07 like explosive, which we use
to calibrate the low velocity impact response of that explosive. Finding that the original Steven
test diagnostics (blast gauges at a distance of about 3m) is not reliable and not accurate enough
for such a calibration, we use instead velocity gauges looking at the free surface of the back
plate of the sample (see later). Following others [3, 4], we assume that the process of low
velocity impact initiation begins with shear localization. Accordingly, we call our low velocity
impact reaction model PDSR (= Pressure Dependent Shear Reaction). Low velocity impact may
cause localizations of plastic flow in the explosive, and these localizations may develop into
adiabatic shear bands (ASB). A fully developed ASB is usually between 1 and 100μm thick, and
flow variables that localize are strain, strain rate and temperature. In the presence of ASBs the
macroscopic plastic strain rate is not evenly distributed, but is instead concentrated inside the
ASBs. As a result, the rate of plastic work deposited in an ASB is also very high. As heat
generated in an ASB does not have enough time to flow out, the explosive there heats up very
quickly to the ignition temperature. When ASBs are ignited, the chain of events that follows is
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Partom, Y. (2023). Steven Test Modeling. European Journal of Applied Sciences, Vol - 11(1). 576-583.
URL: http://dx.doi.org/10.14738/aivp.111.14053
similar to the chain of events that results from the formation of hot spots formed by a passing
shock. Deflagration fronts spread out from the ASBs into the unreacted explosive around them.
It follows that ASBs play for shear initiation a similar role to what hot spots play for shock
initiation. But there are two main differences: 1) the reactant temperature threshold for
formation of deflagration waves out of ASBs is much lower than that of deflagration waves out
of hot spots (because of their different shape and the larger size of ASBs); and 2) the speed of
deflagration waves out of ASBs and the resulting reaction rate are much lower than that of
deflagration waves out of hot spots (as the average distance between ASBs is much higher). It
therefore turns out that except for the reaction rate equation, all other equations of PDSR (=
Pressure Dependent Shear Reaction) are identical to those of our shock initiation and
detonation model for heterogeneous explosives TDRR (= Temperature Dependent Reaction
Rate). We do not repeat those equations here, and they are outlined in detail in [5]. Contrary to
the reaction rate of TDRR which depends on reactant temperature, we assume that the reaction
rate (
W
) of PDSR depends on pressure up to a limit:
( )
( )
sb impact m
sb impact m m
W R u P P P
W R u P P P
=
=
(1)
where Rsb (sb for shear bands) is a function to be calibrated from the tests (see later), uimpact is
the impact velocity, and Pm is a limit pressure for pressure dependence of the reaction rate. This
reaction rate equation is based on numerous measurements of deflagration velocities for
various explosives at low pressures [6, 7], and on analysis of the rate of heat conduction at the
front of a deflagration wave, as described in [8, 9]. The main points of this analysis are: 1)
deflagration wave speed increases with the rate of heat conduction at its front; 2) the explosive
at the wave front reacts after it gasifies; and 3) pressure, as long as it is low enough (below Pm),
decreases the gas dimensions proportionally, and by doing so increases the heat conduction
rate and thus the deflagration wave speed.
STEVEN TESTS
We performed 7 Steven tests at different projectile velocities. Our projectile and target
configurations are similar to those of the original Steven test projectile and target [2]: Stainless
steel (SS) projectile of 60mm diameter and 60mm length with a spherical head, an LX07 like
explosive disc of 110mm diameter and 13mm thickness, sitting in a SS shallow pot of 19mm
base thickness and 120mm diameter, and a SS front plate 4mm thick. We use a 64mm powder
gun, and the target is at a distance of about 3m from the gun, for safety reasons. Since the
ignition limit for this arrangement is at a relatively low projectile velocity, we had to make some
changes to the original gun-target system. First, we reduced the gun combustion chamber to
improve the combustion process for small amounts of gun powder. Second, we lowered the
target as function of projectile velocity to take into account the projectile trajectory. Our
diagnostics include: 1) a high-speed camera to record projectile velocity and its pitch and yaw
angles; and 2) a Photon Doppler Velocimetry (PDV) diagnostic to measure the target back plate
velocity history on the axis. We secure the velocity gauge system in place using an additively
manufactured stand made of Nylon 12, which enables a 500μs long measurement. We show the
main test results in Table 1 and in Fig. 1.
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TABLE 1. Ignition results
Test # Projectile velocity
(m/s)
Ignition
YES/NO
Ignition delay time
(μs)
1 25.6±0.1 NO
2 25.9±0.1 NO
3 25.9±0.1 NO
4 36.3±0.2 YES 251±14
5 45.2±0.2 YES 152±5
6 94.0±2. YES 76±5
7 122.0±2. YES 51±4
Figure 1. Target back plate velocity histories on axis.
From Table 1 we see that: 1) the ignition threshold is between 30 and 36m/s projectile velocity,
and in what follows we adopt 30m/s as the ignition threshold velocity; and 2) there’s a
substantial delay (or incubation time) between projectile impact and ignition (start of reaction
or start of back plate measurable motion, (see later about this ignition delay)). Also, ignition
delay decreases as impact velocity increases. From Fig. 1 we see that: 1) the velocity gauge
diagnostics of the two high impact velocity tests failed at an early time, because they were
placed too close to the back plate. We corrected this problem in the following tests; 2) the back
plate moves elastically at a velocity of about 10m/s before the explosive reaction starts; 3)
reaction starts after a substantial delay which increases with decreasing impact velocity; 4) the
back plate velocity reaches a substantial maximum of about 300m/s, and then continues to
oscillate elastically with a period of 150 to 200μs. Our PDSR (shear reaction) model needs
therefore to include the following: 1) a procedure to calibrate Rsb(uimpact) (see Eq. (1) and its
following comment); and 2) a procedure to calibrate the ignition delay (for each computational
cell) from test results. We outline these procedures in the next section.
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Partom, Y. (2023). Steven Test Modeling. European Journal of Applied Sciences, Vol - 11(1). 576-583.
URL: http://dx.doi.org/10.14738/aivp.111.14053
CALIBRATION OF IGNITION DELAY AND REACTION RATE
We performed Steven test simulations with the PISCES2D old commercial hydrocode in
cylindrical symmetry. We use in the simulations 1mm cells in both directions. We do not check
for convergence, as our aim is only to outline the shear reaction model and to suggest a
procedure for its calibration. We recognize that convergence may be an issue with our PDSR
model, because ignition delay equations depend on effective plastic strain rate. We leave the
convergence issue to future work when we would have more data from different test
configurations. In the past we calibrated the shear localization threshold of our explosive using
mesoscale simulations [10]. We express there the shear localization threshold in terms of the
product PD, where P is pressure and D is effective plastic strain rate. We found that:
( )L
PD 0.01GPa / s = (2)
where (PD)L=localization threshold. In what follows we use this value to start accumulating
ignition delay (for each computational cell). The localization threshold is related to yield stress
under dynamic loading. Upon passing the yield surface the material starts to flow plastically.
This may cause strain hardening on one hand and thermal softening on the other hand. Strain
localization may start when the softening rate overcomes the hardening rate. As pressure
causes hardening of soft materials like explosives, and plastic strain rate (through plastic
heating) causes softening, we get a hyperbolic-like dependence of the localization threshold on
the product of these two quantities. It turns out that this value of (PD)L is quite low and there’s
no need to be precise with it. Even at the impact velocity of 30m/s below which there’s no
ignition (see Fig. 1), PD at the impact face is still above (PD)L, and failure to ignite is caused by
an extremely long ignition delay.
When PD in any computational cell goes above (PD)L, we assume that: 1) one or more shear
localizations may start to form there; 2) these shear localizations become shear bands that heat
up as they develop; and 3) if there’s enough time (before rarefactions take over) these shear
bands may ignite. The time delay to ignition (for each cell) we denote by h (= ignition delay).
On the macroscale we assume that h for each cell is inversely dependent on the PD value there,
and that it is a decreasing function of the impact velocity:
h A u PD = h impact ( ) (3)
where the coefficient Ah(uimpact) is to be calibrated from the tests (see later). Upon crossing the
localization threshold, we start to accumulate (for each cell) the ignition delay time th. We do
this by computing the integral I, where I=1 represents ignition:
( )
h
t
0
I dt h PD =
(4)
We see that for the special case of h=const. we get th=h at I=1. We assume that this is also the
case for the general case of non-constant h, and we postpone the examination of this
assumption to future work, to be done on the mesoscale. To calibrate the reaction rate
coefficient Rsb(uimpact) and the ignition delay coefficient Ah(uimpact), we run our code for the four
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tests with impact velocities of 122, 94, 45 and 36m/s (see Eq. (1)). We calibrate Rsb from the
maxima of the back plate velocities, and Ah from the ignition delays (see Table 1). In all of these
runs we use Pm=1.GPa (see Eq. 1). In Fig.2 we show an example from these runs for the impact
velocity of 45m/s. In these runs the explosive is defined in a Lagrange mesh and the metals in
a Euler mesh. We see in Fig. 2 the deformed configuration every 50μs and lines of equal reaction
parameter every 0.2.
Figure 2. Simulation results for impact velocity of 45m/s. Target and projectile configuration
every 50μs, and lines of equal W (reaction parameter).
We see from Fig. 2 that: 1) the explosive ignites between 150 and 200μs after impact; 2) after
ignition, the reaction wave moves at an average velocity of about 1mm/μs, somewhat less than
the speed of sound. For each of the four impact velocities we change the values of the
parameters Ah and Rsb to get agreement with the test data. We show the results in Figs. 3 and 4.
We refer to the values of Ah(uimpact) and Rsb(uimpact) as global calibration. We use global
calibration in our modeling as we don’t have enough diagnostics to perform local calibration.
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Partom, Y. (2023). Steven Test Modeling. European Journal of Applied Sciences, Vol - 11(1). 576-583.
URL: http://dx.doi.org/10.14738/aivp.111.14053
Figure 3. Ignition delay parameter Ah as function of impact velocity, calibrated from test data.
Figure 4. Reaction rate coefficient Rsb as function of impact velocity, calibrated from test data.
In Fig. 5 we show the simulated back plate velocity histories as obtained with the values of Ah
and Rsb shown in Figs. 3 and 4. We see from Fig. 5 that: 1) ignition delays are in agreement with
test results shown in Fig. 1; 2) rise times to the maxima are also in agreement, little less than
100 μs for the high impact velocities, and little more than 100 μs for the low impact velocities;
3) the shapes of the velocity signals is not smooth and is unlike those in Fig. 1. We intend to
improve that in future work; 4) the simulated signals stop a little beyond the maxima because
of numerical difficulties. We assume that this has to do with the breakout of the explosive from
its casing, as shown in Fig. 2; and 5) as can be seen from Fig. 3, the ignition delay interpolation
curve is very steep below the impact velocity of 40m/s, and at 30m/s it goes to infinity and
there’s no ignition.
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Figure 5. Back plate velocity histories on axis for the four tests as obtained from the
simulations, with the values of Ah and Rsb shown in Figs. 3 and 4.
SUMMARY
We performed 5 low velocity impact Steven tests with a LX07 like explosive and a range of
velocities, from a relatively high impact velocity of 122m/s down to a low impact velocity of
30m/s, which is below the ignition threshold. In our Steven test we do not use the original
diagnostics of blast gauges, because of their low temporal precision. Instead, we use PVD gauges
to measure the velocity history of the back plate on the axis. We use a hydrocode with our PDSR
(= Pressure Dependent Shear Reaction) reactive flow model to simulate the Steven tests. From
the test data we calibrate two parameters of our PDSR model as function of impact velocity.
These are the ignition delay coefficient Ah and the reaction rate coefficient Rsb. Running
simulations of the tests, we get good agreement of ignition delays and of the back plate velocity
peaks. The shapes of the rising part of the velocity signals need improvement, which we
postpone to future work. We also reproduce the ignition threshold, which is not determined by
the localization threshold, but rather by a very long ignition delay. Finally, we recognize that
because PD may depend on strain rate, our modeling may have a mesh convergence issue. We
did not check for convergence, and we postpone the convergence issue to future work.
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Partom, Y. (2023). Steven Test Modeling. European Journal of Applied Sciences, Vol - 11(1). 576-583.
URL: http://dx.doi.org/10.14738/aivp.111.14053
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