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
For special copyright notice, see page 1030.
LAGRANGIAN ANALYSIS OF EDC37 SHOCK INITIATION DATA
J. R. Maw
AWE Aldermaston, Reading, RG7 4PR, U.K.
Previous papers have presented experimental and theoretical studies of shock initiation in EDC37.
Multiple embedded electromagnetic gauges were used to provide particle velocity histories over a
range locations in the explosive but attempts to simulate these data by adjusting reaction rate
parameters in an ignition and growth model were only partially successful. In an attempt to improve
the modelling, the Lagrangian Analysis technique has been applied to the data to obtain direct
information on the reaction history in EDC37. Results are presented illustrating the sensitivity of the
inferred pressure, density and energy fields to uncertainties introduced in the application of the
technique. Reaction histories are derived corresponding to different assumptions in the treatment of
the Equation of State of the partially reacted mixture and these are compared with those obtained from
a number of commonly used reaction rate formulations.
technique. This paper describes an attempt to follow
this approach.
INTRODUCTION
LAGRANGIAN ANALYSIS
Two earlier papers (1,2) described experimental
and theoretical studies of shock initiation in
EDC37, an HMX based explosive. In (1) planar
shock wave initiation of EDC37 was studied using
embedded electromagnetic gauges to measure
particle velocities over a range of depths. These
data, together with measurements of the trajectory
of the leading shock, clearly demonstrated the build
up to detonation. In (2) the experiments were
modelled using the Lee and Tarver ignition and
growth model (3). Reasonable agreement with the
data was obtained by suitable adjustment of the
model parameters but it was recognised that there
was room for improvement in both the predicted
run distances to detonation and the detailed
modelling of the particle velocity data.
The process of refining the model by parameter
adjustment is time consuming and there is also
some doubt that the true reaction rates can indeed
be represented within the confines of the reaction
rate formulation. An alternative approach is to
extract information on the reaction rates directly
from the data using the Lagrangian analysis
There is extensive literature on the theory of the
Lagrangian Analysis and its application to inert and
reacting materials (e.g. 4-6). In this paper we
present only the basis of the technique restricting
attention to the analysis of particle velocity gauge
data. Integrated equations for the conservation of
the mass, momentum and energy are written in the
form
(1)
= p(h 0 ,t)-—
(2)
e(h,t)-e(h,t 0 )-v 0Jf t pf^-ldt
(3)
t0 l^3hjt
Where v is the specific volume, p the pressure
and u the particle velocity, t and h denote time and
Lagrangian position and subscript 0 denotes an
initial condition. Equations (1) and (3) give the
specific volume and internal energy as functions of
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time at a given Lagrangian position, while
equation (2) gives the pressure as a function of
Lagrangian position at a given time. Ideally the
integration of the latter equation should start from
initial pressure conditions at a specified Lagrangian
position provided by pressure gauge data. In the
absence of such data the starting point of the
integration is on the shock front along which all
conditions must be evaluated.
Us-C0+sup
(6)
enables the particle velocity to be calculated. The
pressure, specific volume and internal energy are
then calculated using the Rankine Hugoniot
relations.
Particle velocity
Figure 2 shows the corresponding particle
velocity records. Measurements were made at
~0.5 mm intervals at depths between 6 and 11 mm
but for clarity the figure shows only the records at
6,7,8,9 and 10 mm depths.
APPLICATION TO EDC37 DATA
The Shock Front
Figure 1 shows the shock trajectory obtained
from the gauge records and a "shock tracker" gauge
(1) in an experiment where the HE was subjected to
a 3.5 Gpa shock.
FIGURE 2. Particle velocity data.
6
8
These data were fitted by an equation of the
form
u = u s (h) + [umax (h) - u s (h)]u(T)
(7)
10
Distance (mm)
FIGURE 1. Shock trajectory.
where us is the particle velocity at the shock front
and umax is the maximum particle which is fitted by a
polynomial in h. The non dimensional time coordinate T is
In applying the technique to these data we have
fitted the shock arrival time at position h by
ts =
from which the shock velocity follows as
Us =
(4)
(t max (h)-t s (h))
(8)
Where tmax (h) is the time of maximum velocity
at each gauge location also fitted by a polynomial
inh. The non-dimensional velocity u, is
represented by a polynomial with coefficients
which are also functions of h.
(5)
dt
Figure 1 shows that the fit is good up to 13 mm
where the shock develops into a detonation wave.
Assuming a linear shock velocity-particle
velocity relation for the unreacted explosive
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The fitted velocities obtained using this
functional form are compared with those measured
in fig 2. No attempt has been made to precisely fit
individual records but it can be seen that the overall
fit is good and the main features of the data are well
reproduced.
of the EOS of the partially reacted mixture of
unreacted HE and reaction products. In the first the
unreacted HE is assumed to lie on the Hugoniot
while the products lie on the isentrope passing
through the Chapman Jouguet state. Writing the
mixture equation for the volume in the form
v = (l-A,)v s (p) + A,v p (p)
RESULTS
(9)
Gives the following relation for X as a function
of the pressure and volume
Pressure - Volume Paths
Pressure specific volume and internal energy
have been calculated using equations (l)-(3)
together with the above representations of the
experimental data.
Figure 3 shows the inferred pressure — volume
paths at a number of Lagrangian positions. Also
shown is the unreacted shock Hugoniot. The
general behaviour is as expected with the HE
initially undergoing some compression behind the
shock. At the smaller depths the paths lie above the
Hugoniot indicating that significant reaction is
occurring. At greater depths the paths lie initially
below the Hugoniot indicating isentropic
compression with little initial reaction. At later
stages the HE begins to expand and the pressure
starts to decrease due to further reaction.
(10)
v
V
p
—v
V
s
In the second method the unreacted HE lies on
the isentrope through the shocked state and the
products are assumed to satisfy an EOS of the form
= F(v p ,e p +Q)
(11)
Note that the chemical energy Q is added to ep
since the latter, deduced from the Lagrangian
analysis, is only the energy of compression.
Equation (9) together with the EOS and the energy
of the mixture in the form
e = (l-X)e s (p) + Xep(p)
(12)
enables X to be obtained by an iterative procedure.
0.4
0.35
0.3
0.25
310
n
0.2
0.15
0.1
—— P - V paths
---•
Hugoniot
0,35
0,4
0.05
0,45
0
0,5
Volume (cc/g)
-0.05
FIGURE 3. Pressure - Volume paths.
2
2.5
Timefual
FIGURE 4. Reaction histories
Figure 4 plots the variation of A, with time
obtained by each method at depths 6,7 and 8 mm.
The two methods give very similar results at 6 and
7 mm but there is some disagreement at 8 mm and
Degree of Reaction
Two methods have been used to calculate the
degree of reaction, X, based on different treatments
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9mm. Closer inspection shows that at these depths
method I gives negative reaction initially, a
consequence of the assumption that the unreacted
material compresses along the Hugoniot. Note that
it has only been possible to calculate the degree of
reaction up to about 30%.
CONCLUSIONS
We have shown that the Lagrangian Analysis
technique can be successfully applied to deduce
sensible reaction rates from shock initiation data.
This offers the prospect in the short term of
improving the modelling of experimental data using
existing reactive burn models and, in the longer
term, will allow assessment of the value of more
physically based models. It should be noted that
with particle velocity data alone the method gives
information only on the early stages of the reaction.
It would be useful to have additional pressure gauge
data to extend the technique to later times
Reaction Rates
Figure 5 shows reaction rates at the 6,7 and
8mm locations obtained by differentiating the
reaction calculated using the second method. These
appear reasonably smooth apart from some jitter
due to the numerical differentiation at the 8 mm
depth. This figure also shows reaction rates
calculated using the Lee and Tarver model with the
parameters for EDC37 obtained in (2). The overall
agreement is surprisingly good bearing in mind the
very different ways in which the reaction rates were
obtained. More detailed examination shows that the
Lagrangian Analysis gives reaction rates higher
than those of the model at the 6 and 7mm depths
but somewhat lower at 8 mm. Further work is now
needed to determine whether the model parameters
can be further adjusted to give an improved fit to
the reaction rates deduced from this analysis.
ACKNOWLEDGEMENTS
My thanks to Rick Gustavsen for providing the
experimental data in digital form.
REFERENCES.
Gustavsen, R.L. et al., "Initiation of EDC37 Measured with
Embedded Electromagnetic Particle Velocity Gauges," in
Shock Compression of Condensed Matte?"-1999, edited by
M. D. Furnish et al., AIP Conference Proceedings 505,
New York, 2000, pp. 879-882.
Winter, R.E. et al., "Modelling Shock Initiation of HMXBased Explosive," in Shock Compression of Condensed
Matter-1999, edited by M. D. Furnish et al., AIP
Conference Proceedings 505, New York, 2000, pp. 883886.
Lee, E. L. and Tarver, C. M., "Phenomenological Model of
Shock Initiation in Heterogeneous Explosives", Physics of
Lagrangian analysis
Ignition and Growth model
Fluids, 23(12), 1980.
Fowles, F. and Williams R. F., "Plane Stress Wave
Propagation in Solids", /. Appl. Phys., 41(1), 1970.
Cowperthwaite, M. and Rosenberg, J. T., "Lagrange Gage
Studies in Ideal and Non-Ideal Explosives", Seventh
Symposium on Detonation.
1.5
Forest, C.A., "Lagrangian Analysis, Data Covariance, and
the Impulse Time Integral," in Shock Compression of
Condensed Matter-1991, edited by S. C. Schmidt et al.,
North-Holland, 1992, pp. 317-324.
2.5
Time(u9)
FIGURE 5. Reaction rates
© British Crown Copyright 200 I/MOD
Published with the permission of the Controller
of Her Britannic Majesty's Stationery Office.
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