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 994.
DEVELOPMENT OF A SIMPLE MODEL OF "HOT-SPOT"
INITIATION IN HETEROGENEOUS SOLID EXPLOSIVES
N.J.Whitworth
AWE, Aldermaston, Reading, United Kingdom
Abstract. Previously we numerically studied "hot-spot" formation in an explosive material as a result
of shock induced pore collapse via microscale one-dimensional hydrocode simulations. Following this
work, a simple model of the shock compaction process, leading to the formation and subsequent ignition of "hot-spots", has been developed for use in macroscale simulations of shock initiation problems
of interest. The simple model is presented, where "hot-spots" are formed as a result of elastic-plastic
and viscous stresses generated in the solid explosive during pore collapse. Results from the model are
compared with corresponding results from the hydrocode simulations to illustrate how well, or otherwise,
the simple model is performing. The model has also been used to help analyse data obtained from single
and double shock initiation experiments on the HMX-based explosive PBX-9404.
INTRODUCTION
lidity of the various assumptions in the simplified
model, and results obtained from the simple "hotspots" model are compared with the corresponding
hydrocode calculations to illustrate how well, or otherwise, the simple model is performing.
Work is in progress to develop a new, physicsbased, explosive ignition and burn model. The requirements of the model are that it should explicity
describe the important physical and chemical processes involved in explosive shock initiation, yet still
be simple enough for use within a hydrodynamics
code to simulate real problems of interest.
In this paper a simple model to explicitly describe
the formation and subsequent ignition of "hot-spots"
is presented. It is assumed that the solid heterogeneous explosive contains pores, and that "hot-spots"
are formed as a result of elastic-plastic and viscous
stresses generated in the solid explosive during the
collapse of the pores under shock wave loading. The
described model is based on very similar assumptions and equations to other published "hot-spot" ignition models [1,2,3].
In a previous paper [4], we numerically studied
the dynamic formation of "hot-spots" in an explosive
material as a result of shock induced pore collapse
using a hydrocode which incorporated an elasticviscoplastic constitutive model. This hydrocode
modelling work is used as a reference to test the va-
SIMPLE "HOT-SPOT" INITIATION MODEL
Physical Model and Assumptions
The explosive is represented by the ID hollow
sphere pore collapse model developed by Carroll and
Holt [5] for compaction of inert porous materials, see
Figure 1. The initial inner radius, #o> represents the
average pore radius in the explosive, and the external
radius, Z>o, is chosen such that the initial porosity and
the measured overall porosity of the explosive material are equal. Ps is the applied stress. The initial
distention ratio, OQ, of the material is the ratio of the
total volume of the porous material to the volume of
the solid material, and OQ is the initial porosity.
In conjunction with the physical model of the
explosive, it is assumed that: ft) pore collapse and
flow of the solid material is treated as ID, spherically
symmetric, fti) the solid explosive material is incom-
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sist the pore collapse, are given by,
ft) Elastic phase (oo > a > cti);
P
s
,=0,
oc 0 =-
Py =
4//(ao-a)
3a(a-l) '
(2)
CCi =
(ii) Elastic-viscoplastic phase (cci > a > 0.2);
FIGURE 1. Hollow sphere configuration used to model "hotspot" formation.
where the time-dependent interface between elastic
and viscoplastic flow, c, is given by,
pressible during its radial motion, (in) the solid explosive is an isotropic elastic-viscoplastic material,
where the shear modulus, //, yield strength, 7, and
viscosity, r\9 are constant, and (iv) the pore is a void.
c—
,
tfo 3 (oo-a)
where BD — ——-——-—-
(4)
OCo — 1
(Hi) Viscoplastic phase ((X2 > a > 1);
Model Equations
(5)
The time evolution of the pore radius is given by,
= Ps + Pv-Py
The hydrocode modelling work [4] has also
shown that the size of a "hot-spot" created in the
vicinity of a collapsing pore is roughly equal to the
size of the pore before it collapsed. These findings
are used to define a domain for the "hot-spot" which
is then defined by a number of Lagrangian points,
where the local temperature, reaction rate, and mass
fraction reacted are computed at each step.
The temperature increase at the Lagrangian positions as a result of the mechanical deformation during pore radial motion is given by,
(1)
where p is the density of the solid explosive, a is
the pore radius, Pv is the viscous (strain-rate dependent) stress, Py is the elastic-plastic (strain dependent) stress, and a dot above a symbol denotes a
derivative with respect to time.
Following [5], the deformation of the pore occurs
in three distinct phases: ft) an initial (elastic) phase
(oto > oc > oci), fti) a transitional elastic-viscoplastic
phase (oci > a > 062), and (Hi) a viscoplastic phase
(062 > a > 1), where (Xi and 0*2 define the distention
ratios at which the transition from one state of stress
to the next occurs. Other "hot-spot" ignition models based on viscoplastic pore collapse [1,2,3] assume that volume changes during the first two phases
of collapse are negligible, and pore collapse occurs
by virtue of viscoplastic flow only in the material.
An important difference between our model and the
other "hot-spot" models, is that all three possible
phases of pore collapse are modelled as hydrocode
calculations [4] have shown that, for weak and moderate shock waves, the spherical shell is usually in
the transitional elastic-viscoplastic state. The viscous
and elastic-plastic stresses, which act together to re-
=«, (£) +27 (H
. „=
(6)
where T is the local temperature, Cv is the specfic
heat capacity at constant volume, r is the radial (Lagrangian) position, and u is the local velocity in the
solid shell. The local reaction rates are calculated using the Arrhenius rate law,
F = (\-F)Ze-
(7)
where F is the local mass fraction of explosive that
has reacted, Z is the frequency factor, E* is the activation energy, and R is the universal gas constant.
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MODEL CALCULATIONS
- hydrocode
•;
- - - - 'hot-spot' model
:
- - - 'hot-spot1 model (viscoplastic) •
To initially test the model, the HMX-based explosive PBX-9404 was chosen as the majority of its material parameters, as required for input to the model,
are to be readily found in the literature, see Table 1.
Sample results from the described simplified
"hot-spots" model, in terms of the time evolution
of the pore radius and pore surface temperature, are
compared with the corresponding hydrocode calculations in Figures 2 and 3 respectively. Good agreement with the hydrocode results are obtained. The
observed small difference in pore surface motion at
low shock pressures is due to the assumption of incompressibility of the solid material in the model.
Also shown in Figures 2 and 3 are the corresponding model results assuming that pore collapse
occurs by virtue of viscoplastic flow only in the material, as assumed in other "hot-spot" ignition models. Significant differences in pore response are observed between the different simplified modelling approaches, particularly for weak shocks, with the results from the model described in this paper being
in closer agreement to the hydrocode results where
we model the problem as accurately as possible. At
higher pressures, very similar results are obtained.
Empirical ignition and burn models cannot adequately describe the response of an explosive to multiple shock loading. Experimentally it is known that
preshocked explosives are less sensitive than virgin
material [6]. Here, the response of the hollow sphere
model to a planar double shock input has been cal-
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
time (us)
FIGURE 2. Time evolution of pore radius.
2800
Ps=2.0 GPa
2300
1800
- hydrocode
- - - - 'hot-spot' mods!
- - - 'hot-spot1 model (viscoplastic):
1300
Ps=1.5GPa
800
Ps=1.0GPa
300,
0.0
0.5
1.0
1.5
2.0
2.5
time (us)
FIGURE 3. Temperature at surface of collapsing pore.
culated. The double shock loading consisted of a
preshock of 1.0 GPa followed a given time later by
a second shock of 2.0 GPa. Computed results from
the "hot-spot" model and hydrocode, corresponding
to a time delay of 0.5 //s between the two shocks are
compared in Figures 4 and 5, where the single shock
results are also shown. Good agreement with the hydrocode results are again obtained, and it is seen that
the temperature at the pore surface in a double shock
process is less than in a single shock at the pressure
of the second (main) shock alone. Preshocking results in a lower temperature "hot-spot" due to the reduction in pore size before arrival of the main shock,
thus making the material less sensitive. In addition,
the calculated "hot-spot" temperatures are also dependent on the time delay between the precursor and
main shocks, with increasing preshock duration resulting in lower temperature "hot-spots".
TABLE1. Material parameters for PBX-9404.
Material parameter
Value
1.84
Initial density, po (g/cc)
Yield strength, 7 (GPa)
0.2
4.54
Shear modulus, // (GPa)
Viscosity, T| (GPa jus)
0.1
Initial temperature, TO (°K)
300.0
Specific heat, Cy (GPa cc/g/°K)
1.512e-03
Frequency factor, Z (jus~l)
1.81e+19
Activation energy, E* (GPa cc/mole)
220.5
10.0
Initial pore radius, #o (/#n)
Initial outer radius, b$ (jam)
46.416
1.0
Initial porosity, OQ (%)
Rise time of shock, T (jus)
0.1
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0.125
0.0
0.5
1.0
1.5
2.0
0.000
-0.5
2.5
0.0
0.5
1.0
1.5
2.0
2.5
time (us)
FIGURE 4. Time evolution of pore radius for a double shock.
FIGURE 6. PBX-9404 double shock experiments vs calculation.
2800
stages of reaction build-up corresponding to the ignition phase, reasonable agreement with experiment is
obtained.
CONCLUSIONS
300
0.0
0.5
1.0
1.5
2.0
Overall, the performance of the simple "hotspots" model is encouraging, and provides a useful
starting point for further development of the physicsbased reactive burn model. Future enhancements will
include taking account of the effects of heat conduction on "hot-spot" initiation. Consideration will also
be given to improving the growth of reaction from the
"hot-spots" as the bulk of the explosive is consumed.
2.5
FIGURE 5. Temperature at pore surface for a double shock.
MODEL IMPLEMENTATION
REFERENCES
The described simple "hot-spot" ignition model
has been implemented in a 2D Lagrangian hydrocode
to enable modelling of shock initiation problems of
interest. The model has been applied to both single
and double shock experimental data on PBX-9404,
where the growth of reaction from the "hot-spots"
is modelled using the Lee and Tarver model growth
terms with published parameters for PBX-9404 [7].
Calculations of sustained single shock inputs
show very similar results to Lee and Tarver ignition
and growth model calculations which are in agreement with experiment eg [7]. The PBX-9404 double shock experiments of Mulford et al. [6], where
a 2.3 GPa preshock was followed 0.65 //s later by a
5.6 GPa shock, have also been calculated. The experimental and calculated particle velocity histories
are compared in Figure 6. Concentrating on the early
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and Flame 89, 117-139 (1992).
2. Bonnett, D. L., and Butler, P. B., Journal of Propulsion
and Power 12, 680-690 (1996).
3. Massoni, JL, Saurel, R., Baudin, G., and Demol, G,
Physics of Fluids 11(3), 710-736 (1999).
4. Whitworth, N. J., and Maw, J. R., Shock Compression
of Condensed Matter-1999, AIP Conference Proceedings 505, New York, 2000, pp. 887-890.
5. Carroll, M. M., and Holt, A. C., Journal of Applied
Physics 43, 1626-1635 (1972).
6. Mulford, R. N., Sheffield, S. A., and Alcon, R. R.,
Proceedings of the Tenth Symposium (International) on
Detonation, 1993, pp. 459-467.
7. Tarver, C. M., and Hallquist, J. O., Proceedings of
the Seventh Symposium (International) on Detonation,
1981, pp. 488-497.
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