DEFLAGRATION AND DETONATION MODELING OF HETEROGENEOUS CONDENSED PI-IASE EXPLOSIVES by Joseph Ryan Peterson A Senior Honors Thesis Submitted to the Faculty of The University of Utah In Partial Fulfillment of the Requirements for the Honors Degree in Bachelor of Science In Chemisty Approved: Dr. Charles A. Wight Supervisor Dr.'Henry S. White Chair, Department of Chemistry Department Honors Advisor Dean, Flonors College May 2012
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DEFLAGRATION AND DETONATION MODELING OFHETEROGENEOUS CONDENSED PI-IASE EXPLOSIVES
by
Joseph Ryan Peterson
A Senior Honors Thesis Submitted to the Faculty of The University of Utah
In Partial Fulfillment of the Requirements for the
Honors Degree in Bachelor of Science
In
Chemisty
Approved:
Dr. Charles A. Wight Supervisor
Dr.'Henry S. White Chair, Department of Chemistry
Department Honors Advisor Dean, Flonors College
May 2012
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u0109130
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ABSTRACT
A modeling approach to the deflagration and detonation phenomena is employed to study
a variety of explosive scenarios. An engineering model for macroscale reaction of
energetic materials over the wide range of explosive reaction from deflagration to
detonation is developed based upon solid foundation of two previously developed
reactive models. The model is calibrated for the representative explosive PBX-9501 with
a number of standard experiments including the strand burner, flyer plate, and rate stick
tests. The model is then validated against a number of more complex experiments
including the cylinder test and Steven test. Resolution effects for both deflagration and
detonation models are explored. Upper bound resolutions are identified for both reaction
regimes and implications for large-scale modeling are presented. Work continues in
extending the model to represent heterogeneous explosive configurations as homogenous
materials. This necessitates implementation of a bulk compaction model, which has yet
to be completed. Validation of the bulk compaction model in various scenarios and the
resulting thermodynamic state necessitated comparisons with mesoscale simulations. A
mesoscale model is implemented from various thermodynamic and material models
posed in literature. Validation of the mesoscale models against experiments led to very
favorable agreement. Results from varied density granular simulations lead to insight into
initiation dependence on initial density, leaving the platform good for comparing bulk
scale scenarios for experimentally intractable problems.
TABLE OF CONTENTS
ABSTRACT ii
I. INTRODUCTION 1
A. DEFLAGRATION 3
B. DETONATION 7
C. MODELING 10
II. METHODS 15
A. SIMULATION METHODOLOGY 15
B. BULK SCALE MODELING 16
1. DDT1 FORMULATION 17
2. VALIDATION EXPERIMENTS 21
C. MESOSCALE MODELING 23
III. RESULTS 24
A. BULK SCALE MODELS 24
1. STRAND BURNER TEST 24
2. POP-PLOT TEST 25
3. RATE STICK TEST 27
4. CYLINDER TEST 27
5. STEVEN TEST 28
6. SLOW COOK-OFF TEST 28
B. MESOSCALE MODELS 29
1. SINGLE CRYSTAL EXPERIMENTS 29
iii
Ij
2. GRANULAR HMX STRESS GAUGE EXPERIMENTS 29
3. VARYING DENSITY GRANULAR BEDS 30
IV. DISCUSSION 31
A. BULK SCALE MODELS 31
B. MESOSCALE MODELS 36
V. CONCLUSIONS 38
VI. REFERENCES 40
A. FIGURES 45
iv
I. INTRODUCTION
A great push occurred in the last century towards understanding the energetics of
explosive materials largely due to the desire to create weapons that are more powerful,
more reliable and safer to handle. What captivated the researchers of explosions was the
rate at which high energetic (explosive) materials react and the ensuing damage from the
energy release. Furthermore, the sensitivity of the explosives could vary significantly
with formulation, temperature, pressure and age. One of the greatest curiosities is that
many explosives can react at two very different rates. The two regimes are separated by
the Mach 1 point, and named deflagration and detonation for reaction in subsonic and
supersonic flow regimes respectively. The shock physics nature of detonation interested
such great scientists as Taylor and ZePDovich, von Neumann and Doring, and great
strides where made in understanding the detonation phenomena [21,55,61]. At the other
end of the spectrum, those such as Kuo and Beckstead applied the principles of gas flame
and thermodynamics of various phases of materials to understand the low rate
combustion of explosives thus brining fine control to practical applications [7].
Of particular interest over the past half century has been the safety of explosives.
Inevitably, the explosive sciences and technologies were used to generate great weapons.
The safety of these weapons in storage and transport has been of great interest as a
number of situations have caused loss of life and property. A number of accidents
involving rocket motors and mining explosives have resulted from what conventionally
was thought would be a safe operating conditions. In general, some thermal or
mechanical insult occurred and the explosive transitioned from the slow reaction to a
detonation. The rate of energy release is much larger in detonations resulting in more
1
damage and increased danger to human. Studies have identified a number of different
types of transitions from slow reaction to detonation including deflagration to detonation
transition (DDT), slow cook-off to detonation (XDT) and shock to detonation transition
(SDT) depending on the type and duration of the initiation. A difficulty exists in
designing, implementing and accurately analyzing large accident scenarios
experimentally. Generally, small tests have been performed on the order of a meter at
maximum. Furthermore, the tests generally involve one or just a handful of the explosive
devices. These tests cannot adequately address the “sympathetic” explosion seen during
real world shipping and storage accidents with any economic feasibility. This fact has
driven the field of explosives modeling.
Computational models provide the potential for a safe and economically feasible
means of exploring these real life scenarios. Advances in explosives modeling, material
modeling and simulation capabilities in parallel with increase in computational capability
in the last two decades has put the predictive simulation within grasp. What remains is
development of bulk scale models capable of capturing relevant physics. The aim of this
paper is to describe the advancements made at the University of Utah towards these ends.
The introduction continues with discussion of subsonic reaction known as
deflagration, followed by discussion of supersonic reaction known as detonation and
concludes with a discussion of general thermodynamics modeling and a brief discussion
of the paradigms chosen in these modeling efforts. Simulation methods are presented in
Section II with discussions of the reaction model developed along with discussion of the
material models and material parameters chosen for various materials in the validation
experiments. Section III begins with results from the calibration experiment results,
2
continues with discussion of validation experiments for the bulk scale reaction model,
and concludes with results from more resent mesoscale modeling of granular compaction.
Section IV concludes with a discussion of the results and implications for current
capabilities. Finally, a critical review of the necessary model advancements that still
must be accurately modeled in order to simulate the large accidents is presented.
A. DEFLAGRATION
Deflagration is a term used to describe subsonic combustion of a material. The
subsonic nature of the reaction is due to the time scale of thermal activation. The actual
burning rate of energetic material is generally temperature and pressure dependent due to
the competition of hot product gases away from the reaction surface and the thermal
diffusion from those products back to the unreacted starting material. Usually, two
modes of deflagration are identified, due to the large discrepancy in the rate of reaction
propagation normal to the surface of the explosive of interest. Conductive burning is the
slower of the two, generally consuming material at rates between a few millimeters per
second and a few centimeters per second at reasonable pressure and temperature. At the
other end of the spectrum is convective burning which is characterized by rates between a
few hundred meters per second to perhaps one or two thousand meters per second that are
usually induced via some mechanical insult or reactant confinment
The nature of conductive burning can easily be understood by striking a match
and watching the flame burn down the surface of the match staff. The propagation of the
flame down the staff can be seen to be slow, and the flame itself is seen to be sheet-like,
or in computational fluid dynamic language the flame is said to be laminar. Conductive
3
burning can then be thought of as surface burning. Convective burning is a little bit more
difficult to visualize, but one can easily think of a scenario that embodies this type of
burning. For example, if one imagines a dust cloud of energetic particles that are burning
it is easy to imagine the flame flowing through this material rapidly due to the large
amount of low-density space where hot gas can diffuse. The increased surface area
allows the reaction front to move rapidly through and thus consume more of the body of
explosive in a given timeframe. In this way, convective burning can be thought of as the
process of pushing hot gas into a material, which differs from surface burning where hot
gas is pushed way form the material surface. Before further discussing conductive and
convective burning, and the transition between the two, it is necessary to understand the
chemical and physical processes involved in deflagration.
Combustion is a thermally activated oxidation reaction between some fuel and an
oxidizer. Solid materials generally have combustion character similar to the schematic
seen in Figure 1. The solid is heated by conduction or radiation until reaching its melting
point or in some cases thermal decomposition temeprautre, after which combustion
generally begins to occur, causing the large molecules making up the solid to split into
smaller reactive intermediates. These reactive intermediates generally “foam” out of the
melt layer in bubbles. Beyond this melt layer, exists a “dark-zone” which is
characterized by a “stand-off distance.” This dark zone is generally a region in which the
reactive intermediates absorb more energy and eventually either react into smaller
molecules or act as radical catalysts for further reaction. Following the decomposition of
large molecules, smaller diatomic and triatomic molecules are formed releasing energy
that causes the luminous flame seen above a burning surface. In “high energetic”
materials, or those that are used as primary, secondary or tertiaiy explosives, this trend of
burning is followed, with some variation in flame nature. For example, some materials
begin to react before they melt or some materials have a double flame structure where a
temperature plateau is hit, prior to further reaction of high activation energy reactants that
increases the temperature to a second plateau. An example of the first is ammonium
perchlorate (AP), a common rocket propellant. These double plateau combustibles are
generally a mixture of two different fuels and are often called double base propellants for
this reason. The explosives studied here are generally made of organic molecules that
show the single plateau nature. Furthermore, the explosives studied are generally mixed
with a small percentage of plastic bonding agent and are known as plastic bonded
explosives (PBX). The “ideal” nature of these explosives, or the tendency to detonate
completely with little effect from inertial confinement, makes them useful in a number of
applications. In addition, this property makes them dangerous thus desirable materials
for computational study.
The organic molecule studied most commonly, and the focus of this study, is
known as octahydro-l,3,5,7-tetranitro-l,3,5,7-tetrazocine (HMX), and can be seen in
Figure 2 along with other common organic high energetics such as pentaerythritol
The plug-and-play style model interface employed by Uintah allows for a high
degree of complexity in problem formulation, while at the same time facilitating fast
prototyping of new simulations. Once a good problem setup is found, activating more
advanced models entails implementing sources/sinks, writing an input specification and
adding the input parameters to the input file. Using a material model already
implemented merely requires having correct input parameters. For example steel, copper
and aluminum already have well validated hypo-elastic behavior and parameterization
available in Uintah [2]. Each validation simulation performed herein went through a
variety of stages, consisting of increased complexity at each iteration step. The following
subsections describe the models used for the final validation experiments.
Of particular importance to modeling are the spatial resolution dependences of the
various fundamental metrics of interest. Because MPMICE is a mixed
Eulerian/Lagrangian code, studies over cell and particle convergence spaces are required.
Quantification of resolution dependences allows assignment of reasonable uncertainty
outside the ranges of model calibration. Resolution studies were performed for all
simulated observables, including detonation velocity, deflagration velocity, CJ pressure
and case expansion velocities.
B. BULK SCALE MODELING
Bulk scale modeling came in two phases, calibration of the reaction models,
followed by validation experiments. During calibration, a consistent set of EOSs and
reaction parameters, as well as a material model were selected for the explosive. The
explosive chosen for all simulations was the HMX based plastic bonded explosive (PBX)
known as PBX 9501, which contains 5 percent plasticized Estane™. These were then
used in validation experiments, with a number of other materials such as aluminum, steel
and copper. Material models for non-reactive materials will be mentioned in name only.
For a complete description of the models used for bulk scale modeling as well as
comprehensive set of citations, see the paper by Peterson and Wight [41].
1. DDT1 FORMULATION
Three EOSs were used for representing the high energetic reactant and product
states. Two of these are JWL equations, the standard
equation of state chosen for SDT simulations by a number of scientific groups [14,46,48].
The JWL EOS takes the relative volume, v, the temperature, T, and specific heat, Cv, as
well as five fitting parameters, A, B, Ri, R2 and to, and calculates the bulk average
pressure. Numerous fits for product and reactant materials axe available in literature.
Most of which have standardized Ri, R2 and 0) parameters, making the use of the JWL
EOS efficient when prototyped with a different explosive material. EOS parameters for
detonation products and reactant behavior of PBX 9501 were taken from Vandersall et al.
[54],
A second product equation of state is necessary for representing products from
surface and bulk burning, as they have fundamentally different behavior than detonation
products. The TST EOS, F m {f « !)CrF/(K J was used for its!>
particularly good representation of species relevant to combustion [53] Parameters used
for product gases are -260.1 Pa m3 for a, 7.955 xlO"4 m3 for b and y is 1.63. Variable V,
T and Cv are the specific volume, temperature and specific heat respectively.
17
Mass and energy is either transferred to the burn products or detonation products
depending on the mode of combustion. Surface burning and convective burning products
were represented by the TST EOS and detonation products were represented by the JWL
EOS. Regardless of the reaction mode, the reactants were represented by the JWL
reactant EOS. Bennett presented a model for statistical treatment of cracks in a bulk
volume of viscoelastic material and called it ViscoSCRAM [9]. The JWL EOS was
implemented in the ViscoSCRAM material model. Advantages of this model include the
ability to evolve average cracking without predefined boundaries or advancement of the
state of such boundaries as other crack models do, making it suitable for use in Uintah
where boundaries are not well defined. ViscoSCRAM takes an initial crack size,
maximum crack growth rate, several other crack rate parameters and five Maxwell
elements, each including a modulus and related relaxation time. From these, average
crack size is evolved from a balance between crack coalescence and separation. PBX
9501 calibrated parameters from Bennet et al were used [9]. Average crack size is used
in the threshold criteria for onset of convective burning into cracks, as will be discussed
shortly. A bulk modulus of 11.4 GPa was used as it is an often cited literature value [91].
Two previously validated models for the two limiting modes of combustion were
used in DDT1. Combustion in both surface burning and bulk convective burning regimes
were represented by the WSB reaction model [56]. This model had previously been
implemented in Uintah as a 3D model, named Steady Burn, by including a surface area
calculation based on the direction of density gradient in the cells and the total mass of
reactant in that cell [59]. Surface detection for surface burning and density gradient
require MPM materials, hence the reason MPMICE is used. PBX 9501 parameters can
be found in literature [56,59]. Transition from surface burning, where only the top cell of
the surface is allowed to burn, to convective burning, where the burning front penetrates
into the material, is determined by Eq. (3), a relationship described and fit by Berghout et
al. and proposed by Belyaev et al. [8,10].
m IQ* (3)
An extent of cracking in the cell, w, is calculated by ViscoScram. The critical pressure
for penetration of combustion, pc, is then calculated. When the pressure in a cell exceeds
the critical pressure, pc, bulk burning occurs. In addition to this requirement, the gas in
the cell must be above the melting and thermal runaway temperature for the explosive,
550 K in the case of PBX 9501 [27]. This allows burning in damaged material (i.e.
cracked or porous) ahead of the deflagration wave without any sub-grid scale models for
hot spots or shear heating. Good agreement has been obtained with a variety of
experiments that exhibit convective burning. This threshold is applicable in cook-off
scenarios, where the material is heated to its decomposition point over large areas of the
bulk.
Detonation is carried out by a modified version of the JWL++ model that was
originally formulated by Souers et al. [46]. Model fitting simplicity is attained by using
the first-order, single-term rate model, seen in Equation (4), as only two parameters need
to be fit.
19
— -exa-Fjxi®* a
(4)
The pressure exponent was chosen to be 1.2 and the rate coefficient 2.33 jxPa“1,2s-1. As
described elsewhere, these parameters control the size effect curvature and infinite radius
detonation velocity, respectively [45,46,48]. Two differences exist between the JWL++
implementation by Souers et al and DDT1. First, DDT1 uses a JWL EOS for reactants
instead of the Murnaghan EOS. The decision to use the JWL equation of state was made
as a consequence of the zero experienced by the Murnaghan EOS at relative volume
equal to 1.0 and subsequent “negative pressure” during expansion beyond initial state—a
common occurrence for viscoelastic materials undergoing large relaxation. Negative
pressures cause exceptions to be thrown in Uintah. As mentioned, parameters from
Vandersall et al were used as they had been found to give good detonation and CJ
behavior with Ignition and Growth [54]. Second, instead of additive pressure, the built in
iterative pressure solve in ICE is used to correct for the simple Dalton's law assumption
in its original formulation.
A transition mechanism in the form of a pressure threshold acts as the link
between burning and detonating. The models are mutually exclusive, only burning or
detonating is allowed in any given cell at a time. Furthermore, burning is prevented in
cells adjacent to detonation, as this sort of burning was found to accelerate the shock
wave non-physically. Once the pressure threshold is exceeded, the detonation begins.
Equilibrium cell pressure has mixed contributions from stresses of all MPM materials and
the cell centered pressures for each ICE material. However, the primary functionality of
ViscoSCRAM is turned off in a cell undergoing detonation to minimize computation,
leaving stress to be simply calculated from the reactant JWL equation of state. For PBX
9501, 5.3 GPa was chosen as the pressure threshold to match Pop-plot data.
The deflagration parameters were chosen to match the strand burner tests done by
a number of experimenters [1,56]. This required validation against both pressure and
temperature dependence, as well as a thorough assessment of errors. Detonation
parameters were chosen to match the size-effect curve [47]. The detonation velocity is
known to decrease in small explosive samples, due to reduced inertial effects, causing
larger transverse motion. This effect is also seen in curved explosive devices and hence
the need to calibrate against. Schematics of these experiments can be seen in Figure 6.
2. VALIDATION EXPERIMENTS
Three validation tests for the models general performance were chosen, including
the confined cylinder test [29,42], the modified Steven test [30] and an annular confined
slow cook-off test [20]. Schematics of these experiments can be seen in Figure 7. The
first is to test the detonation properties, including reaction zone length and product
expansion behavior by comparing the velocity of the expanding case, post-detonation.
The second is for low velocity impact where a detonation transition occurred on
timescales longer than a normal SDT or DDT and tests both the reaction pressure
threshold and the deflagration behavior. The final test is of reaction in a heated sample
that eventually undergoes a transition to detonation. Models used to represent the non
reactive materials in these simulations are discussed below. Generally, the best available
models available in Uintah were chosen to reduce uncertainty due to reactive material
interactions with non-reactive materials.
Oxygen-free, high conductivity copper was used as the encasing material in the
Cylinder tests. Since the velocity of the copper is the data of interest in these simulations,
accurate representation of the material is paramount. A compressible Neo-Hookean
stress relationship was used, with standard bulk and shear moduli. A yield stress of 70
MPa and a hardening modulus of 4.38 p.Pa were used along with a thermal conductivity
of 400 W/tnK and a specific heat of 386 J/kgK.
During Pop-plot simulations, the impactor and cover plate was made of
Aluminum 6061-T6, and was the only non-reactant material type of relevance in the
simulation. Standard shear and bulk moduli were used along with a Mie-Griineisen EOS
with common parameters. A shear modulus model developed by Nadal-LePoac (NP)
was used. Melting was modeled with the Steinberg-Cochran-Guinan (SCG) model. A
Hancock-MacKenzie damage model in conjunction with a Gurson yield condition
allowed plastic flow of the material post failure.
The final material used in simulating the validation experiments was steel. A
hypoelastic-plastic constitutive model with a Mie-Griineisen EOS was used with common
parameters. A John son-Cook plasticity model was used in conjunction with a Johnson-
Cook damage model, and a Gurson yield condition with the same parameters as those for
copper. The specific heat model described by Lederman et al was used. A full
description of the steel, copper and aluminum material representations as well as citations
for material parameters with assessment of accuracy and error can be found in the works
of Banerjee [2]. Finally, a friction model developed for MPM was used for interaction
between materials, generally with a frictional coefficient of 0.3.
22
C. MESOSCALE MODELING
Mesoscale simulations of the explosives have become desirable while model
development on DDT1 continues. Specifically, it will be necessary to validate bulk-scale
models against finely resolved simulations, as experimental data is difficult to obtain on
the length scales of interest. As such, a model has been implemented in Uintah and
validated against experiments for granular compaction. The HMX material was
represented by the SCG viscoelasticity model with parameters from literature [15].
Melting behavior and specific heat behavior was modeled using models of Menikoff [39].
Validation simulations were based on rise time measurements from Sheffield [43], using
randomly generated granular beds with the appropriate grain size distribution [4,18], and
single crystal measurements [19]. Schematics of these experiments can be seen in Figure
8. Some of the experimental cases from Sheffield indicated reaction of the material, and
both the WSB model and a Prout-Tompkins model [49] have been applied to attempt to
elucidate the process of ignition and reaction in granular beds that have been shocked.
The models were then applied to impact scenarios with varying initial granular
bed densities from 42 to 77 percent theoretical max density (TMD). Wave profiles and
temperature profiles were averaged transverse to the impact. Results were compared
qualitatively with the works of Barua [5]. Randomly generated spheres from measured
particle size distributions where used [18].
23
24
III. RESULTS
A. BULK SCALE MODELS
1. STRAND BURNER TEST
Surface regression rate normal to the gas/solid interface is used as the primary
metric for validation of combustion in both the convective burning and surface burning
regimes. Surface regression rates were found to generally agree in value and trend for
pressures greater than 1 MPa and less than 70 MPa. As seen in Figure 9, outside the 2.3
to 9.2 MPa range, the simulated regression rate was found to deviate by as much as 2
times, though this large a magnitude difference is only seen at very low pressures. At
large pressures, simulations overestimate the burn rate by approximately 10%. Simulated
burn rate dependence on bulk solid temperature, seen in Figure 10, is overestimated for
low and high temperatures. Overestimated rates are especially pronounced at high
pressures.
Convergence results for the burn rate with varied cell resolution can be seen in
Figure 11. A representative mid-range simulation was used for convergence, that of the
6.21 MPa pressure and 298 K bulk temperature. The bum rate converges at higher
zonings per millimeter. As can be seen in the figure, the convergence is fast for
resolutions higher than 1 zone/mm, though the converged value is about 7% larger than
the averaged experimental value of 1.089±0.077 cm/s.
Relaxation to steady burning state is a function of the cell size. Figure 11 shows
time to steady state as a function of resolution. Particle density was varied from 2 cubed
to 6 cubed per cell and had negligible effect on the burn rate. Because the Steady Burn
implementation only uses cell centered values for computation of the actual burn rate, no
paxticle density dependence was expected. At 2, 3, 4 and 6 cubed particles per cell, the
burn rates differed by a maximum of 0.02% over a factor of 27 times higher particle
resolution.
A weighted average of the uncertainties in Figure 10 gives a value of 6.04%. At
low bulk temperatures reasonable error is seen, with reasonable error being quantified as
less than 10%. Errors calculated over the full pressure range at 273, 298 and 423 K were
computed with a weighted average of 4.42% error. These results indicate that
simulations performed with bulk temperature near room temperature should give less
than 7% uncertainty during burning.
25
2. POP-PLOT TEST
Figure 12 shows the Pop-plot for simulations and experiments. Though the
simulations were run with 1.832 g/cm3 density, the Pop trend is found to lie closer to the
1.844 g/cm3 experiments from LASL in magnitude [42]. Of particular note, is the slope
of the trend, which is in between the two density trends cited for experiments. Detonation
characteristics were determined from the Pop-plot simulations.
3. RATE STICK TEST
Parameters for Equation (4) were found that match both detonation velocity and
detonation pressure at the resolution used for calibration with values of 1.2 for b and 2.33
HPa"1,2s-1 for G. The detonation velocity at this resolution was found to be 8843 m/s and
the detonation pressure was found to be 37.2 GPa. Resolution effects were explored for
the same resolutions studied for the Strand Burner simulations. Particle resolution
dependence was studied at 4 zones/mm cell resolution, as this was the calibration
resolution. Results are presented in Figure 13. Detonation pressure error was found to
increased with greater particle resolution, relative to the 37.2 GPa at 8 particles per cell.
However this effect is small, with the error decreasing by only a factor of two over 27
times higher particle density. Detonation velocity error, however, increased with higher
particle resolution relative to the 8.870 mm/jus cited in literature. In fact, increasing the
density of particles by 27 times increases the error from about 0.04 mm/|us to as high as
0.221 mm/|iis. The relative error in all cases studied for both velocity and pressure were
less than 6%. Cell resolution has a considerably more noticeable effect on detonation
pressure and velocity as demonstrated in Figure 14. Convergence in detonation velocity
was found to be strong, while convergence of detonation pressure less so. The values are
converged to within 5% relative simulation error at resolutions lower than the calibration
resolution 4 zone/mm but greater than 0.5 zones/mm. Detonation velocities quickly
diverge at coarser zoning. Chapman-Jouguet pressure for the explosions was found to
follow a less favorable trend, with unexpected behavior at low zoning. At low resolution,
the pressures and velocities were underestimated, with normal convergence behavior
found at finer zoning.
Simulated detonation velocity versus inverse radius can be seen in Fig. 15
extending to infinite radius detonation velocity already discussed. The general behavior
of explosives has been found to follow: D(f) ** ^ where A and rc are
fit parameters. A fit to the simulation data yields 8.843 mm/fxs for D(r), 0.028
mm±0.009mm for A and 0.29 mm±0.19 mm for rc. Extrapolating to intersection with the
inverse radius axis gives a failure diameter (df) of 0.70 mm. This is consistent with the
26
experimental value cited in LASL Explosive Property Data, df smaller than 1.52 mm,
which is of little certainty, as they never achieved failure with PBX 9501 charges [42].
Experimental fit parameters are D(oo) equal to 8.802 mm/us, A equal to 0.019 mm±0.001
mm and rc equal to 0.48 mm±0.02 mm and lie on the edge of uncertainty of the
simulation fit parameters. Error between experimental and simulated had a standard
deviation of 0,063 mm/|AS, which is equivalent to about 0.7% total velocity.
4. CYLINDER TEST
Case expansion velocity as a function of case expansion distance from initial
radial position is the primary metric. Measurements were taken at least 9 times the
diameter down the axis of the tube from the initiation point, as suggested by Souers et al
[47]. Case expansion velocity at 2 zones/mm resolution was used as a benchmark and a
screenshot of the test can be seen in Fig. 16. This simulation consisted of 5 .3 million grid
cells and 12 million particles. During simulation the CJ pressure at the front reached
37.12 GPa, in good agreement with detonation pressure presented in the previous section.
Comparison of simulated data and an experimental data set from Gibbs et al are seen in
Figure 17 [42]. Standard deviation between experimental and simulated velocity profiles
was between 6.3 and 7.8%. The detonation velocity varied by only about 0.2% and the
detonation pressure varied by about only 1.3%. Case expansion velocity varied by a
maximum of 3.7% over the range of particle resolutions studied.
27
5. STEVEN TEST
In simulation, pressure in the vessel never exceeds 5.3 GPa indicating detonation
was not achieved, in good agreement with analysis of experiments where explosive
energy releases are much lower than fully formed detonation [30], Experiments show a
threshold for large, 1-inch thick targets of approximately 72-75 m/s with cracking
occurring at lower velocities, and target consumption at higher velocities than the
threshold. A stress threshold of approximately 250 MPa follows from 75 m/s
experimental case for rapid reaction, indicating a lower threshold to dissipation of energy.
Figure 18 shows simulated versus experimental go/no-go results. Good agreement was
found between DDT1 and experiments, indicating decent burn initiation behavior.
Pinducer traces of the experiments qualitatively match simulated pressure profiles at the
same point in the explosive as can be seen in Figure 19. This is especially true with
regards to timescales of material response, but stress magnitudes differ by as much as
three times. Inconsistencies have been attributed to MPMICE's failure to handle negative
pressures. Figure 20 shows reaction beginning in the 75 m/s case.
6. SLOW COOK-OFF TEST
A five-millimeter thick annularly confined disk of PBX 9501 heated to 573 K was
simulated. The experimental setup can be seen in Figure 8. Of particular interest is the
existence of detonation, as well as the time to detonation. In the experiments [20], one
case was thought to have detonated, due to the luminosity measured. Of interest was the
long timescale of reaction prior to detonation. It is estimated from the frames taken in the
experiment that the explosive underwent detonation somewhere between 15 and 20
microseconds. A simulation of the same scenario transitioned to detonation in roughly 25
microseconds, which shows very favorable results with compared with experiments. A
frame of the detonated explosive can be seen in Figure 21. Also, the material damage,
represented as cracking in simulation, looks similar to that in experiments as evidenced
by Figure 22.
B. MESOSCALE MODELS
1. SINGLE CRYSTAL EXPERIMENT
Comparison of single crystal shock experiments with simulations can be seen in
Figure 23. The magnitudes can be seen to be similar, however, due to the fact that an
interface was not simulated, no elastic precursor is seen at the points of measurement in
experiment. These results can be seen to be generally within about 10% of the value.
2. GRANULAR HMX STRESS GAUGE EXPERIMENTS
In granular bed experiments of large diameter HMX crystals, generally good
agreement was found between models and experiments. Figure 24 how stress propagates
through the bed as well as how the temperature is localized in boundaries and in plastic
flow regions. Figures 25 and 26 shows the comparison of stress and velocity profiles
respectively. Front gauge measurements are in very good agreement, while back gauge
measurements differ in event initiation. An elastic precursor wave appears to traverse the
bed too quickly and prematurely cause stress to be felt at the back boundary.
29
3. VARYING DENSITY GRANULAR BEDS
Varying the initial density of the explosive was seen to affect both compaction
rate and average temperature. Temperature minimum and maximums were similar in
each case. Mean temperature through the compaction zone was found to increase with
porosity. The standard deviation highlights this trend, as can be seen in Figure 27.
31
IV. DISCUSSION
A. BULK SCALE MODELS
Due to overestimation in bum rates at high temperatures, slow cook-off scenarios
will be accelerated. In room temperature scenarios, such as those in which DDT
experiments are performed, the agreement in trend is decent over a large pressure range
[58]. The agreement is especially good over the range of pressures that surface burning
transitions to convective burning in damaged materials which is imperative for the
convective burning, crack-size switching criteria. At low zoning per millimeter the trend
of increased error in the burn rate could cause a DDT timescales to be skewed due to
overestimated burn rate during ignition and subsequent move towards stabilization. An
effect that may require the model pressure threshold for transition to detonation to be
larger than those experimentally determined [44]. However, as long as the resolution is
close to convergence the error is dominated by the temperature dependence.
Run distance to detonation is an important metric of shock sensitivity that is
generally studied using a flyer plate test [42]. Buildup to detonation is due largely to
nucleation and growth of hotspots, as well as other dissipative mechanisms such as
frictional heating between cracks and resultant bulk decomposition due to expansion of
hot product gasses through cracks; effectively increasing burning surface area and in turn
the rate of reaction. Support and strengthening of the lead front results, eventually
providing enough available energy for rapid transition to detonation. A pressure
threshold of 5.3 GPa was used for transition from deflagration to detonation in
simulations, a value that roughly represents the amount of work necessary for rapid
reaction in the bulk explosive. The results of the Pop-plot are favorable, however, the
slope of line is incorrect indicating the density dependence of the model is incorrect.
Detonation characteristics where also studied using the flyer-plate tests. The converged
CJ pressure is overestimated; however this has already been identified by Menikoff as a
consequence of the EOS fits [37]. Particle resolution dependences of the pressure and
velocity have opposite behaviors. Therefore, the particle resolution should be limited to
between 2 cubed and 4 cubed particles per cell for the reactant. Simulations run must be
at least 4 zone/mm resolution for accurate detonation.
Size effect curves are indicative of explosive performance as they give an idea of
the reaction zone length. Larger reaction zones cause failure on larger length scales (of
the order of millimeters to centimeters). On length scales slightly larger than failure, the
detonation velocity is greatly reduced due to curvature of the reaction front, essentially;
the explosive violence is dampened due to inertial effects in the case of unconfined
explosives. The size effect was well represented by the parameters chosen for Equation
(4). The difference in infinite radius fit parameters, D(o°), accounts for much of the
difference in lit parameters.
The first of the validation tests, the cylinder test, gave good agreement, with a few
exceptions. Of note is the overestimation of case expansion at low expansion volumes
(early times), potentially due to the over estimation in CJ pressure. This is consistent with
the overestimation of CJ pressure due to a fitting form that is too stiff for PBX 9501 [37].
The agreement increases with radial distance. Overestimation of explosive violence
should be expected in packed arrays of explosives and other sympathetic scenarios.
Slow cook-off simulations proved to be qualitatively correct. The favorable
comparison of cracking behavior is encouraging, as is the fact that detonation was
32
achieved for the experimental case that was suspected to undergo the transition [20].
Simulations were only performed on the 5-mm, copper confined, thick-disk experiment,
and should not be considered conclusive. Differences should be studied with thick and
thin confinements, as well as thick and thin disks in order to fully quantify slow cook-off
behavior.
The Steven test is perhaps the most complicated experiment designed for
explosive model validation [30]. Similarly, in simulation, this is the most expensive test,
consisting of 66.5 million cells and 15.6 million particles. A lightly confined disk of
PBX 9501 is impacted with a round nosed projectile and stress histories are measured.
From this test, a velocity threshold is determined for go/no-go behavior where “go”
implies rapid, sustained reaction. Go/no-go is a basic metric for explosive impact
assigned based on reaction of the sample. Sometimes another set of terms: nonviolent,
semi-violent and violent, are used to describe the test behavior. Cracking and damage to
the material are exhibited in all velocity impacts and are not used as the go/no-go criteria.
Instead, consumption of large amounts of target material along with melting and
scorching are used to assign the go/no-go label. Simulations of the go/no-go threshold
was in good agreement within a 10 m/s window, and the stress measurements prior to
ignition were shown to be at least qualitatively correct. This indicates that a good
resolution was chosen and that the reactant material model adequately represents damage.
Furthermore, the fact that the device did not undergo detonation helps affirm that the
behavior of our burn model and the transition to detonation pressure are reasonable.
Of largest uncertainty in the model is the pressure threshold chosen to transition
from fast deflagration to detonation. Thermodynamically, the activation of ITMX
reaction requires average energy in the range of 140 to 165 kJ/mol [12]. Bulk reaction
should occur when the entire material within a given volume has exceeded this energy,
which will result in fast reaction and large energy buildup. The pressure threshold of 5.3
GPa in this model was chosen for two reasons. First, the threshold was chosen so that the
351 m/s aluminum impact experiment gave a reasonable run distance to detonation.
Overestimated burn rate likely increases the threshold pressure above what it might be in
reality. Second, a 2003 paper by Esposito et al shows an increase in the pressure
dependence (e,g, Vielle’s Law and JWL++) of HMX burn rate from 0.924 to 1.27 at
about 5 GPa. WSB has a pressure dependence of approximately 0.9 at low pressures and
the pressure dependence of the detonation model calibrated herein show reasonable size
effect at a pressure dependence of 1.2, indicating that there is good agreement with this
experimental data.
A simple mathematical analysis of the energy available under compression to 5.3
GPa leads to further understanding. Using transition state theory and assuming reversible
work a value for the energy imparted can be calculated. Using a simple expression
P=IC(l/v - v)/2, where v is the relative volume, the bulk modulus, K, is 11.4 GPa and
FIMX has an approximate molar volume of 6200 mols/m , one comes to an average
energy of compression of 138.9 kJ/mol. Surprisingly this value is close to the
experimental activation energies. Using an upper bound of experimental bulk moduli for
HMX reported in literature of 15.7 GPa the same analysis leads to a slightly lower value
of 119.4 kJ/mol available energy [33]. Since the bulk modulus of PBX 9501 is lower
than HMX due to the plasticized binder, the higher energy result first discussed may be
more reasonable in light of the energy uptake during plastic flow. As a check we apply
34
the same analysis with the shock EOS. Using the JWL equation of state for reactants,
which fits the experimental data up to relative volume of 0.8 fairly well, and the afore
mentioned molar volume, the available energy is much lower, around 45.2 kJ/mol.
Some drawbacks and uncertainties are associated with using DDT1. Firstly, the
calibration of JWL++ rate parameters are required after a factor of 8 coarser zoning, as
the reaction zone becomes over-represented and the errors in detonation pressure and
velocity are greater than 5%, which seems to be the reasonable value for simulations.
Souers et al has already covered this issue in several of their papers [46,48]. The
recommended resolution for correct detonation is 2 times the converged resolution, or 0.5
zones/mm for PBX 9501. Secondly, a constant value for a pressure threshold is subject
to much uncertainty, due in part to the fact that this initiation pressure is dependent on
explosive density [44],
A third drawback comes from the WSB model. Few explosives parameters are
available in literature. Such values as chemical heat release and the specific heat of both
gas and solid, and condensed phase activation energy have been identified as the sensitive
parameters of the model and thus need to be known to high certainty from experiments
[56]. In addition, the time to steady state burning likely has a large effect on transition to
detonation. The initial burn rate instability is due to pressure oscillations in the domain.
These are inherent to the ignition means and the boundary conditions.
Along a similar note, the final drawback is the limited parameters in literature for
the ViscoSCRAM model cracking behavior. Recent work in some LX explosives and
RX explosives (two PBX formulations from different national laboratories) could allow
the parameterization of the mechanical model for of explosives [57], but to the author’s
35
knowledge no data regarding propagation of surface flames into cracks is available for
explosives other than PBX 9501 [56]. Fortunately, flame stand-off and crack penetration
are probably much more similar between explosives than burning parameters, and thus
the models should be applicable within a quantitative sense, for other explosives.
However, lack of complete sets of ID, 2D and 3D experiments for many explosives other
than PBX 9501 make validating the model for other explosives difficult.
B. MESOSCALE MODELS
The use of the SCG model for the HMX is suspect, as it was originally designed
to be a material model for metals. The oscillations seen in the flow stress level are likely
due to this fact. However, the parameters seem to be give reasonable behavior, especially
when put into the granular simulation, as was evidenced by the agreement in Figures 25
and 26. Of largest suspect in these simulations is the early arrival of the wave at the back
boundary. Two things contribute to this simulation trace. The first is that the stress
traces were taken in the HMX grains instead of the receiving material. This
preferentially weights any precursor waves that might have averaged to a lower value
when considering the actual surface area of HMX that is contacting the receiving
material. Secondly, no cut-off distance was used in the frictional contact algorithm,
which could increase the elastic wave’s speed through the bed as nearby grains “feel” the
stresses more quickly [3].
Temperature comparisons of the model with other modeling efforts showed
similar averages and maximums [5]. Additionally, all features that are expected in
granular compaction can be seen in the temperature field demonstrated in Figure 24.
36
These include frictional heating at contact surfaces, heating in the plastic flow region, as
well as in fracture paths. Assuming correct behavior, it then becomes interesting that the
distribution of temperatures is wider in the low-density case. This can be understood to
be due to the decreased mass to shock energy ratio. While there are fewer contact points
in the lower packing, once compaction yields to flow, there is a smaller amount of overall
mass in which the energy is localized.
This supports the general observation that shock initiation pressure decreases with
increased porosity as evidenced in a compilation of experimental data [44]. Energy
localization, along with the slower compaction wave likely contributes to the observation
that run-distance to detonation in DDT of granular beds decreases with increased
porosity. Further study is needed in the area to apply trends seen on the mesoscale that
may be applied as sub-grid scale models in the bulk scale.
38
V, CONCLUSIONS
While the goal of macroscale simulations of hazard scenarios has not yet been
achieved, great strides have been made. A generalized engineering model capable of
deflagration and detonation, and the transition from one to the other in energetic materials
at a variety of length scales has been created, and validated against a number of well-
posed experiments. These include the cylinder test, the Steven test and the annularly
confined slow cook-off test. In addition, models have been implemented that allow
simulation of mesoscale simulation scenarios with high fidelity. This creates the
potential to validate via simulation macroscale models of experimentally intractable
problems posed on the micrometer length scale. However, the road to predictive
simulation is long. In order to move to larger length scales, sub-grid scale models based
on multiscale modeling results will need to be incorporated to capture enough of the
physics in length scales associated with accident scenarios. If the current work is carried
forward, a few key steps must be taken on the road to predictive simulation.
The first of those steps is the implementation of a reaction model in the mesoscale
simulations. The WSB model would likely suffice, and could easily be compared with
experimental reactive granular results. The next step would be the inclusion of plastic
binder in mesoscale simulations, as well as inclusion of interface debonding models.
These would allow simulation of PBXs over a range of length scales. Heterogeneous
repeatable scenarios such as a box full of explosive rocket motors (modeled as cylinders)
must be modeled via homogenous mixture models with sub-grid scale behaviors inspired
by results from resolved mesoscale simulations. Sub-grid scale models will likely need to
incorporate surface area effects, permeability effects, and damage effects [40]. Only then
can real predictive simulations of accident scenarios be simulated. Finally, it becomes
necessary to actually extend the model to work for other explosives for it to truly achieve
the goal of being a general engineering model.
1
40
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45
APPENDIX A. FIGURES
Figure 1. A temperature profile through the reaction zone in deflagration of a solid
explosive is depicted. Between the initial temperature, To, and the melt temperature, Tm,
is an exponential temperature profile. In the melt (also called “foam” zone) the
temperature increases slightly to the surface temperature, Ts. Beyond this exists a zone of
endothermic reaction where the N-N bond cleavage occurs and reactive radicals are
formed [28]. Exothermic reaction increases, and near the luminous flame, which begins
roughly at the stand-off distance, Xg, the reaction tends to completion.
PETNRDX
OpN>N 0
IIN-
TATB
T etryl
N+
\ r \ /'N+— N N---N+/ \
H M X
Figure 2. Chemical structures of common high explosives are depicted. The common
motif of fuel and oxidizer built into the same molecule can be seen in the high oxygen to
carbon ratios.
47
A.C4N80 8H8 --------- ^ 4(CH2NN02)
il u
4HONO + 4HCN CH20 + N20
B .c h 2o + n o 2
n 2o + h _
HONO -----
HCO + HONO
-£*•- N2 + OH
■>- NO + OH
C.H CN--------- — --------- — HNC
HNC + O _ — --------------- ^ NCO + H
NO + N ---------------------- — ► N2 + O
NO + NH ----------------- — ► n + OH
Figure 3. Commonly cited main decomposition steps for HMX are depicted [11,28]. A.
Solid phase reactions. B. Endothermic gas phase reactions. C. Exothermic gas phase
reactions. The rate-limiting step is thought to be the isomerization of HCN to HNC [34].
48
V
Figure 4. A representation in the pressure-volume plane of the equilibrium state of
reaction in both the deflagration and detonation regimes is depicted. The lower and
upper curves represent the locus of states for the Hugoniots of the reactants and products
respectively. The line tangent to the each product curve is known as the upper Rayleigh
line and represents the equilibrium state for reaction. The CJ point and the VN point are
both marked 0 11 the Rayleigh line. The zones denoted in numerals are: I) strong
Figure 6. Schematics of the three simulation setups run in code. The top shows the strand
burner simulation setup. In the middle is a ID simulation of an explosive cylinder either
45 or 90 mm in length. On top a 5 mm thick cover plate provides inertial confinement
(experiments also have a 5 mm bottom cover plate that was not included in the
simulations). A 10 mm aluminum impactor contacts the cover plate at the beginning of
the simulation with ail initial velocity of interest. The bottom simulation is used for the
rate stick test, and is the same as the cylinder test minus the confinement. ‘S’ denotes a
symmetry boundary and ‘N 5 denotes a Neumann boundary.
51
-135 inra-
N
_25.4 mm_ HMX
5 mm Cu Confinement
Figure 7. Validation experiments schematics are depicted. The top simulation setup is
that for the Cylinder test, where a copper confinement sits around a cylinder with a 1-inch
radius. The middle shows the Steven test with the steel impactor. The third shows the
setup for the annularly confined slow cook-off test. ‘S’ denotes a symmetry boundary and
‘N* denotes a Neumann boundary.
52
j
N
Gas
Kel-F 80018881
o o o o o o o o oo „ o „ o „ o „ o „ o „ o " o „ o „ o „ o „ o „ o „ o „ oz. o „ o „ o „ o „ o „ o „ o „ o „ o „ o „ oo ° o 0 o 0 o 0 n 0 n ° o ° o ° o° o ° o ° G r a n u l a n 0 o 0 o
§ o ° o ° o ' H M X > g o g o ° oo o _ O _ O — O — o o „ o o o o o o o o o o o „ o „ o „ o ^ o „ o „ o ) : o „ o O „„ O. O „ O. ... Q . Q - Q . Q
:TPX
N
Figure 8. Granular compaction schematic is depicted. The granular bed was about 4 mm
thick and the TPX and Kel-F 800 inert materials varied in thickness depending on the
simulation. ‘S’ denotes a symmetry boundary and ‘N’ denotes a Neumann boundary.