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
MECHANISTIC MODEL OF HOT-SPOT: A UNIFYING
FRAMEWORK
K. Yano, Y. Horie, and D. Greening
Applied Physics Division, Los Alamos National Laboratory, Los Alamos, NM 87545
Abstract. A one dimensional hot-spot model is proposed to consider wide range of ignition
scenarios for solid high explosives. Model components commonly found in typical mechanistic
hot-spot models are included in the model. They are a finite region of enhanced heating in the
solid matrix, a space occupied by gases, heat flow between solid and gas phases, and chemical
reaction in the gas phase, at the solid-gas interface, or both. The model also contains advanced
features of spherical pore-collapse models. Test calculations based on cyclonite (RDX) data show
that the model behavior is comparable to that of spherical models under conditions indicative
of shock initiation. Critical conditions for ignition is summarized in a single chart in terms of
total energy dissipation, the rate of energy dissipation, and the size of hot-spot.
INTRODUCTION
we consider a model that incorporates these
features without introducing the mechanismspecific traits.
In high explosive (HE) science, the existence
of "hot-spot," or the region where energy dissipation is highly localized is generally accepted
as essential to the ignition of explosives. Various
mechanisms for hot-spot have been proposed,
such as pre collapse, shear banding, friction, and
fracture (1-5). Currently, there is no agreement
as to the mechanism(s) by which the dissipation
is localized (6).
To model hot-spot initiation of HE, one typically choose a mechanism that describes mechanical energy dissipation.
The mechanism
tends to imply a specific geometry that may not
be typical of the actual localization. In addition, any single-mechanism approach quickly becomes problematic in a complex system such as
polymer-binder explosives (PBX).
If one examines the various models, setting aside the mechanisms, a common modeling
structures become clear. They are a region of
enhanced heating, creation or existence of the
region occupied by gas, heat flow between the
regions, and chemical reaction in the gas, at
the solid-gas interface, or both. In this study,
MODEL DESCRIPTION
Figure 1 shows the simplified one dimensional model that incorporates the components
commonly found in pore collapse, shear band,
friction models of hot-spot. The model consists of solid mass, gas cavity, and the interface.
The solid phase is assumed to be incompressible. Thus the particle velocity is common, and
it is determined by the force balance on the solid
phase through the equation of motion. Temperature profile is calculated using analytic solution
for a fixed length bar with heat source. The energy localization is represented parametrically by
the size of the heating region, J, with a known
heating rate, <j>8. In this way, the very significant
issue of how to consider the localization as dependent on the state of the material is set aside
for the moment. Instead, attention is focused on
whether a model constructed on this framework
produces results that are consistent with other
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Pore Collapse
Shear Band
=3.3Xl016W/m3
Friction
J_GPa
(a)
1
5|Lim H-h- 0.25 [im
8.6 Jim
external
load
(gas phase)
infcjrface
g is cavity
•
*
solid mass
(solid phase)
2
adiabatic 1>oundary
W/m
IQMPa
(b)
—— heating zone
external
load
90 Jim
lOum
o
(gas pnase)
(solid phase)
FIGURE 1. A schematic of the one dimensional model
FIGURE 2.
structure that consists of the essential components com-
Conditions for ignition simulations, (a)
The heat is deposited over a finite thickness region at the
monly found in typical mechanistic hot-spot models.
rate (f>a. (b) The heat is deposited at the solid-gas inter-
Three major regions are identified: gas cavity, solid-gas
face at the rate (j>ati.
interface, and solid mass with a region of localized heating.
the same features found in the spherical hot-spot
model by Kang, et al. (2), except that in Kang's
model the heating rate and zone width is coupled
with the geometry instead of free parameters as
in this paper. Details of the model are found in
mechanistic models by ranging over the parameter space.
At the solid-gas interface, conditions are imposed for mass, momentum, and energy conservation. There is mass flux, m, into the gas phase,
which is caused by the localized heating in the
solid phase. The flux is determined by the interface temperature. Part of the mass flux may
go through an instantaneous chemical reaction.
Heating at the solid-gas interface can also be considered as a known heating rate, $5j;, depending
on the modeling needs such as friction.
In the gas phase, time evolution of the average density, temperature, and chemical species
are described by a set of ordinary differential
equations. The equations are obtained through
integral form of conservation equations. Heat
flow is considered between the gas phase and
the interface. The equation of state is a modified form of the ideal gas equation of state (Abel
equation of state). The gas-phase reaction is described by a single-step reaction with the Arrhenius kinetics model.
Mathematically, the system consists of 7 ordinary differential equations with 10 algebraic
equations. Runge-Kutta scheme was used for
time integration. The model contains essentially
(7).
MODEL CALCULATIONS
Three simulation results are presented. The
first two are concerned with ignition simulations
for different types of heat deposition as depicted
in Figs. 2 (a) and (b). The third result presents
a parametric study of critical ignition conditions.
Figure 2 (a) shows simulation conditions
where heat is deposited within a finite volume
in the solid matrix. At time t — 0, a confining pressure of 1 GPa was applied to the solid
matrix. At the same time, heat was added over
the thickness of 0.25 /mi at a constant rate of
3.3xl0 16 W/m 3 . The heat deposition was terminated at 150 ns. These conditions are comparable to those in Kang et al. (2), and indicative
of shock initiation. The rate of heat flux was deduced from the results by Kang et al. (2), Massoni et al. (3), and Bonnet and Butler (8). Our
value is close to the average of the three. Lastly,
the duration of heat deposition was selected so
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(a)
(b)
(a)
(b)
FIGURE 3. Results of simulation where heat is added
FIGURE 4. Results of simulation where heat is added
over a finite region as shown in Fig. 2 (a). Time evolu-
at the interface as shown in Fig. 2 (b). Time evolutions
tions are shown for (a) displacement of the interface and
are shown for (a) displacement of the interface and the
the adiabatic solid boundary measured from their initial
adiabatic solid boundary measured from their initial po-
positions, and (b) temperatures of the gas phase and the
sitions, and (b) temperatures of the gas phase and the
interface.
interface.
that the system results in ignition after removal
of the heat. Figures 3 (a) and (b) show time histories of the interface and solid outer boundary
displacements, and temperatures of gas phase
and the interface, respectively. Upon the application of external pressure, the pore starts to
collapse at almost constant velocity during the
period 0 < t < 170 ns. The deviation of interface displacement from that of outer solid boundary reflects the effect of mass flux at the interface. The interface temperature rises steadily
until t = 150 ns, when the heat addition is
turned off. When the temperature exceeds approximately 940 K, the rate of temperature increase slows down because the heat consumed for
sublimation of the solid exceeds the heat evolved
from surface chemical reaction. In the gas phase,
the temperature increases steadily, mainly because of heat conduction from the interface.
At about 170 ns, the conditions in the gas
phase reach a critical state, and the accumulated
reactant gas reacts in a near discontinuous manner. Because of high density and high temperature after the reaction, the pressure of the gas
reaches well above the confining pressure (about
3 GPa) and expands the gas pore (Fig. 3(a)).
Once the expansion occurs, both the gas temperature and pressure start to drop. Although
there are some minor differences, the observed
model behaviors and those presented by Kang et
al. (2) are very similar in general.
In the second simulation, the heat was de-
posited to the interface, as shown in Fig. 2 (b).
The heating rate was made comparable, in estimate, to that caused by sliding solid explosive
over a rough surface at 10 m/s under the imposed
normal stress of 10 MPa. The corresponding energy flux is 2 x 107 W/m 2 . The deposition of
energy was maintained for the duration of 350
/^s.
Figures 4 (a) and (b) show time evolutions of
the interface and solid outer boundary displacements, and temperatures of gas phase and the interface, respectively. The maximum compression
is reached almost instantaneously and the solid
plate remains virtually at rest for 0 < t < 150 /^s.
During this time period, the interface and gasphase temperatures rise steadily. Also, the temperatures are nearly in equilibrium (Fig. 4(b)).
As the interface temperature rises, it starts to deviate from the gas temperature at about 150 /^s
as a result of exothermic reaction at the interface. At the same time, mass flux increases
rapidly and effects a pressure rise in the gas
phase. At about 270 /^s, the gas-phase pressure exceeds the imposed stress, and the pore
starts to expand, as seen in Fig. 2(a). Finally,
at about 395 /^s, the reactant accumulated in the
gas phase reaches a critical temperature and undergoes an explosive reaction, causing an almost
instantaneous jump in temperature and pressure. These jumps are followed by rapid drops
because of pore expansion.
Lastly, Fig. 5 shows the results of critical
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The specific calculations, based on the RDX
data, show ignition behaviors consistent with
those published for a comparable mechanismspecific model (2,5).
Secondly, an initial parametric study was
presented. The results, in the form of an ignition chart, are compatible with observations on
the size range and duration of heating required
for an effective hot-spot ignition source, 0.1 to
10 /im and 10~5 to 10~3 s, respectively (6). The
results also show that ignition thresholds depend
not only on total energy deposition, but also on
the rate of deposition and the size of the zone
where the energy is localized.
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3
<1>. (W/m )
3.3x1 016
3.3x10"
3.3x1 014
Heating zone, 5 (ym)
FIGURE 5. Parametric study of critical ignition con-
ditions in terms of the total energy input to the hot-spot
as a function of the heat deposition rate (f>3 and the size
REFERENCES
of the heating zone S.
1. Davis, W.C, Los Alamos Science 2, 48-75 (1981).
2. Rang, J., Butler, P.B., Baer, M.R, Combustion
and Flame 89, 117-139 (1992).
3. Dienes, J.K., "A unified theory of flow, hot spots,
and fragmentation, with an application to explosive sensitivity," in High-Pressure Shock Compression of Solids II, edited by L. Davison, D.E.
Grady, and M. Shahinpoor, Springer, New York,
1995, pp. 366-398.
4. Bennett, J.G, Haberman, K.S., Johnson, J.N.,
Assay, B.W., and Henson, B.F., J. Mech. Phys.
Solids 46, 2303-2322 (1998).
5. Massoni, J., Saurel, R., Baudin, B., and Demol,
G., Phys. Fluids 11, 710-736 (1999).
6. Field, J.E., Bourne, N.K., Palmer, S.J.P., and
WaUey, S.M., Phil. Trans. R. Soc. Lond. A339,
269-283 (1992).
7. Yano, K, Horie, Y., and Greening, D.R., "A unifying framework for hot spots and the ignition of
energetic materials," Los Alamos National Laboratory Report (in process).
8. Bonnet, D.L. and Butler, P.B., J. Propul. Power
12, 680-690 (1996).
conditions for ignition in terms of total energy
deposition as a function of hot-spot size 6 and
rate of energy dissipation (/)s. The simulation
condition is essentially the same as those shown
in Fig. 2 (a), except that 8 and <j>8 are changed
for the parametric study. For a given S and </> 5 ,
any energy above the profile leads to ignition
and any energy below does not lead to ignition.
The profiles are characterized by the existence
of minima and convergence for large 8. In the
left branch of the minimum point, less energy is
required for larger 8. This is because heat loss at
the interface resulting from conduction becomes
smaller as the size of the hot-spot increases. On
the other hand, in the right branch, more energy is required for larger 8 simply because of
the volume increase. The curve approaches a
linear profile for large values of 8 as the effect
of thermal conduction becomes negligible. The
convergence behavior for large 8 indicates that
the ignition occurs as the interface temperature
reaches the critical value (thermal ignition).
CONCLUSIONS
Two distinct conclusions are highlighted.
Firstly, the model was tested well under conditions indicative of a shock-load/pore-collapse
ignition regime and a frictional ignition regime.
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