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
MICROSTRUCTURAL MODEL OF IGNITION FOR TIME VARYING
LOADING CONDITIONS
Richard V. Browning and Richard J. Scammon
Los Alamos National Laboratory, PO Box 1663, Los Alamos, NM 87545
Abstract. A micro-mechanical based model of ignition was developed about five years ago based on a
simple inter-granular friction model of mechanical dissipation coupled with a fit to extensive direct
numerical simulations of the resulting thermally induced decomposition. The chemical model used
was the McGuire-Tarver ODTX based model for HMX decomposition. The resulting power law type
model has been reasonably successful in predicting threshold conditions for Steven type experiments.
The final power law form was obtained by assuming a constant time history for both the pressure and
shear strain rate, resulting in time independent loading conditions for the chemical model.
Here we propose to extend the model to handle time varying loading conditions. This is done using
a linear operator that models reactive heat transfer simulations done for a wide variety of loading
conditions. The linear operator is represented by a convolution integral with Prony series kernel form
for efficient numerical implementation. To complete the model the same inter-granular friction model
used previously is employed. Comparisons are made with results of numerical simulations and
experiments.
The technique used here is based on the notion of linearizing the reactive heat transfer problem.
Although the chemical model involves four reactions and is highly nonlinear, we effectively linearize
the problem around ignition conditions with a linear operator fit. We use a simple power law
approximation that gives useful accuracy over at least 4 orders of magnitude in time and fluence. A
non-dimensional scaling method is used to determine the final form. We believe the techniques used
here could also be used with more detailed chemical models and with other types of mechanical
dissipation models.
INTRODUCTION
Models for ignition of reactive materials are needed
for safety studies. They need to incorporate an
energy dissipation mechanism, heat conduction, and
chemical reaction. In this paper we focus on
modeling the heat conduction and chemical reaction
mechanisms in a concise form. Browning (1)
developed a microstructural based model of ignition
based on intergranular friction coupled with fits to
numerical simulations of thermal ignition based on
McGuire-Tarver (2) kinetics and finite element
numerical techniques in Chemical Topaz (3). The
result was a very simple power law relation
between time of ignition, pressure and shear strain
rate, however a basic assumption was that the
pressure and shear strain rate was constant up to the
time of ignition. The simplicity of the model
allowed easy implementation and use compared
with using direct numerical simulation of sub-scale
reactive heat transfer models (4), and produced
reasonable agreement with results from certain
classes of experiments (5,6,7).
In this paper we investigate a technique for
dealing with non-constant time history loading by
using a linear operator to model the heat conduction
and chemistry involved in the ignition process.
Effectively, we linearize about the ignition
conditions, incorporating the nearly linear heat
987
conduction process and effectively hiding the
severe non-linearities of the detailed chemistry.
model, and then verify the accuracy by comparisons
with the results.
THERMO-CHEMICAL MODELING
1.5-
1.071 \
y= 0.776SX2
^
R = 0.9743
We start with the thermo-chemical process as
defined by the McGuire-Tarver kinetics model
implemented in Chemical Topaz. By running the
code on simple axi-symmetric problems for varying
spot sizes we can calculate ignition conditions for
varying spot sizes and flux time histories. To
transform the computed results into a more compact
numerical model we first develop a nondimensional representation of the results following
Barenblatt(S).
We assume seven dimensional variables, the
critical flux cpcr, critical time tcn spot radius /?, heat
capacity cp, thermal conductivity k, density p, and a
heat of reaction q. This is simplified from the
actual system, where we have three reactions in the
chemical model each with separate q values, for
example. These variables have five dimensional
units, so we can choose two independent nondimensional parameters. We choose a flux related
parameter 7CV = (pcrcpRlqk and a time related
#/
| 0.5.^
>^
5" -0.5 •
^
! -1.
W
4^X
-1.5^
-21
0
1
2
3
A
Log10(Scaled Time)
Figure 1. Scaled results from original 2-D calculations. A
constant flux is applied over spot sizes 0.25 to 250 micron
diameter. The linear fit to logarithmic values is equivalent to a
power law fit.
To express the relation in terms of real time, we
substitute the definitions of the non-dimensional
parameters back into the integral representation,
and after changing the integration variable obtain a
representation,
t
F2(t) = \K(s2t - s2x)sl<p(x)s2dx ,
parameter 71'2 — tcrh I R where we used thermal
0
diffusivity X^ — kl pcp as a derived quantity.
where we have used scaling factors defined by
^(^2) = S^(t) and 7T2=S2t. Figure 1 shows
the scaled results of a series of calculations on spot
sizes 0.25, 2.5, 25 and 250 |Lim. These are given in
terms of scaled fluence 7Tlb — S^O = 3> I pqR .
On this log-log plot the straight line fit is equivalent
to a power law function and so we take
Our basic assumption is that the non-dimensional
parameters are related by a linear operator,
Fcr = L(^rj(/r 2 )) . We implement the linear
operator as a convolution integral eventually using
a Prony series representation for the kernel
function. Considering KI a function of 7i2 we can
write,
7Tlh (jc) = axn and
then
the
kernel
becomes
n
K(x) = x~ (\-ri)la . Given the power law
form for the kernel, we can factor out the scaling
terms to obtain the thermo-chemical model in real
time variables,
and then Fcr =L(^(^ 2 )) . This assumption is
valid for linear heat transfer problems as the
temperature at any point of the body can be
obtained from the response of a linear operator
acting on the time dependent boundary conditions.
In the case of non-linear reactive heat transfer, the
ignition condition is not exactly defined by a linear
operator, but we fit computational results to the
o
The critical time is then defined
Fcr = F2(tcr) , where Fcr is a fitting constant.
988
by
We use a Prony series kernel in the convolution
integral to simplify the integration process. The
Prony terms are taken as constant ratios, with the
ratios selected to cover the range of times involved,
and to generate a good fit to the power law
function. Fig. 2 shows a comparison of the Prony
model results and the original calculations for
various spot radii. The scaling gives a good
approximation for the various spot sizes, and the
slope of the fluence time curve is accurate, however
we do miss the time-to-ignition depending on the
distance to the calibration point. This is basically
the result of the scatter in the original scaled timeto-ignition data. Some of this scatter is inherent in
the approximation, however some also results from
numerical effects in the Chemical Topaz
calculations.
excellent. Other more complex problems are in
progress, with good results emerging.
OVERALL MODEL
To create an overall model we use our previous
results(l) on inter-granular friction to complete the
model. The spot radius and flux are expressed in
terms of the macroscopic pressure P and shear
strain rate y , as well as the material properties
grain radius RG, coefficient of friction /3 and grain
elastic compliance C = 2(l — V)/E, where v i s
Poisson's ratio and E is Young's modulus. We use,
R = 0.721 • 21/6 - RGCEPm = C2Pl/3
2/3
v
and
1
= Cf^Y* where C, =1.031J3C~ R- .
/s
10-
*** F *
0.25,
1-
*
*
X
0.1 •
•Original Calculations
X Model calibrated at 2.5b
V
0.01 •
0.2
X*
0.1
10
1000
100000
10000000
0.2
Time to ignition (microsec)
0.4
0.6
0.8
Fraction of Time On
Figure 3. Comparison of Prony model and Chemical Topaz
calculations for one-dimensional geometry. The flux is on for an
interval, then off, then back on. The total on time is split evenly
between the first and second segment. The agreement is very
good indicating the ability to deal with widely separated flux
pulses.
Figure 2. Comparison of Prony model with results from
original model.
COMPARISON WITH OTHER HISTORIES
To verify the linear operator technique we ran a
series of problems with non-constant flux
conditions. The numerical reactive heat transfer
results are compared with the predictions from the
linear operator model. We first ran a series of
calculations on a one-dimensional problem. In this
series the time history was an on-off-on sequence
where the total on time was split equally between
the first and second heating pulse. The results are
compared in Fig. 3 in terms of the reduction of the
ignition criterion. In this case the agreement is
After combining all these expressions, most of
the constants are merged into the fitting constant
Fcr, but we retain the explicit dependence of R
within the integral, to give the final expression
t
= J(r -
We have tried fitting this with n=0.7768, and the
critical value of F taken at several different times.
The fit works well in the vicinity of the selected
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time and radius, with increasing discrepancies
further from the fitting point. The overall model
results are shown in Fig. 4 compared with our
previous results and experimental results from the
Steven experiments done at Los Alamos National
Laboratory and Lawrence Livermore National
Laboratory (9).
I
Experiment - New HE |
Experiment - Aged HE [J
g Power Law I
H Power Law II
Geometry 3
1 inch HE
ACKNOWLEDGEMENTS
The patient and enthusiastic experimental work of
Deanne Idar and Jim Straight was critical to the
continued modeling. We thank Steve Chidester for
sharing experimental details and results from his
tests. This work is supported by the United States
Department of Energy under contract W-7405ENG-36.
cxpenmeiu
Experiment ||
Power Law Calibration
REFERENCES
Geometry 2
1/2 inch HE
1.
I Experiment
Power Law I
Power Law II
2.
Experiment - New HE - p =1.84
Experiment - New HE - p =1.83
I Experiment - Aged HE - p =1.83
K^SSS^ Power Law I
Power Law II
40
45
50
55
60
65
70
3.
75
Velocity Required For Reaction (m/s)
4.
Figure 4. Comparison of models with experimental results from
Steven tests. The horizontal bars show the difference between
the highest no-go test and the lowest go condition for both the
experiments and the models. Geometry 1 is used to calibrate the
model. Geometry 2 involved a thin section of HE and there was
evidence of contact between the cover plate and back support
plate. The Geometry 4 experiments were done at LLNL by
Steve Chidester.
5.
6.
CONCLUSIONS
The incorporation of the linear operator gives
improved consistency in applying the ignition
model, even though it does not appear to have a
major effect on the results for the Steven test. We
hope to expand on the model by including other
dissipation mechanisms such as binder flow and
void collapse. The most important need is to
develop better constitutive equations for the
macroscale behavior of the material.
The
localization behavior is key to correctly predicting
the ignition behavior of real HEs. None of the
currently used constitutive models correctly capture
the localization effects at failure conditions.
7.
8.
9.
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Mechanical Initiation of Energetic Materials," in
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Conference Proceedings 505, New York, 2000, pp.
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