A Modified Criterion for the Prediction of Shock Initiation Thresholds for Flyer

Plate and Rod Impacts

Peter J. Haskins and Malcolm D. Cook

QinetiQ, Fort Halstead

Sevenoaks, Kent, TN14 7BP, UK

Abstract. In this study we propose a modified P

nτ criterion for the prediction of shock

initiation thresholds under 1D impact conditions. We introduce an additional term, based

on single step Arrhenius kinetics, and which we interpret as an ignition delay. This term is

shown to dominate for homogeneous explosives whilst the Pnτ term dominates for

sensitive heterogeneous materials. For insensitive heterogeneous explosives such as

TATB both terms appear to be required. Additionally, we show that by adopting a

rarefaction­overtake condition to define the shock duration for heterogeneous explosives

we are able to fit the data from flyer plate and flat­ended rod impacts with the same

parameters. However, for homogeneous explosives, where detonation is known to break

out at, or close to, the projectile­explosive interface it appears logical to define the shock

duration by the arrival time of the rarefaction at this interface. This is also shown to lead

to unification of the plate and flat­ended rod data.

Introduction

From the early pioneering work of Walker and

Wasley1 to the more recent developments by

James2,3 there has been a considerable interest in

the formulation of simple criteria to predict shock

initiation thresholds for impact stimuli. Such

criteria have been, and remain, of considerable

value, enabling data to be interpolated and

extrapolated to predict initiation thresholds for a

wide range of impact stimuli.

Despite the considerable success of these

approaches there has remained a certain

arbitrariness and lack of rationale. Indeed, at the

6th Detonation Symposium (1976) a number of

papers and a discussion session were devoted to

exploring the basis of such methods. In this paper

we propose a modified criterion and offer a

phenomenological interpretation based on the

simple concept of an ignition delay and a time for

reaction growth.

Shock Duration

In the initial formulation of the critical energy

criterion1 the relevant time was assumed to be the

shock duration. For flyer plates this is the time for

the release wave to arrive at the plate/explosive

interface, and is given by:

τ = 2L / wp, (1)

where L is the plate thickness and wp the shock

velocity in the plate.

However, for flat­ended rods the shock

duration is controlled by release waves originating

on the periphery of the shock and for a cylindrical

rod of diameter D the duration is given by:

τ = D / 2ce, (2)

where ce is the sound speed in the shocked

explosive. James2 showed that in the case of flat­

ended rod impacts into heterogeneous explosives

the use of shock duration led to different values for

the critical energy than that obtained by fitting

plate impact data. He was able to overcome this

problem and fit both rod and plate data, with

approximately the same critical energy value, if he

changed the definition of time to be that at which

the shocked volume of explosive was a maximum.

This did not alter the time for plate impacts (where

both definitions are equivalent) but led to the time

for flat­ended rod impacts being 1/3 of the shock

duration. In this work we propose a different

solution to the unification of the plate and flat­

ended rod data, which is summarised below.

Here we postulate, as suggested by Ramsay4,

that for heterogeneous explosives the time of

relevance is the time at which the release wave

reaches the front of the shock in the explosive. Our

reasoning here is that for heterogeneous materials

the shock is reactive and consequently reaction

will only be quenched if the release reaches this

reactive front before sufficient reaction growth has

occurred to produce a self­sustaining reaction. For

flat­ended rod impacts this time remains that used

in the original criterion (i.e. given by equation 2),

but for plate impacts it is the time for the release to

arrive at the interface plus the time for this release

to catch the shock wave in the explosive. This

leads to an increased time for plate impacts which,

on making the approximation of a constant

velocity shock, is given by:

τ = ce / (ce + ue ­ we) . 2L / wp, (3)

where we and ue are the shock and particle

velocities in the explosive respectively. If we now

assume a linear shock velocity ­ particle velocity

relationship of the form:

we = a + bue, (4)

and Jacob’s approximation for the sound speed in

the shocked explosive:

ce = (we – ue) . (we + bue) / we , (5)

we can re­write equation 3 in the following simple

form:

τ = 2 + a / bue . 2L / wp (6)

It can be seen that whilst the James approach

was based on a threefold reduction in the rod

shock duration, in this new theory rod duration is

unaltered, but the duration for plate impacts is

increased by a factor of 2 + a / bue. Clearly,

when a / bue = 1 these criteria are numerically

equivalent except for a factor of 3 in the empirical

fitting constant. We show that this new definition

of time also enables the plate and flat­ended rod

data for heterogeneous explosives to be fitted to a

good approximation with the same parameters.

It is important when interpreting experimental

data to know whether the projectile being used is

behaving as a plate or a rod. This will be

determined by whichever of the plate or rod

duration is the smaller. For a circular cross­

section, equations 2 and 6 can be seen to give the

critical L / D ratio as:

L / D = wp / 4ce (2 + a / bue). (7)

Hence, when L / D is greater than the above

expression the projectile will act as a rod, and

equation 2 should be used to determine the shock

duration. When L / D is less than this critical value

the projectile will behave as a plate and equation 6

should be used. In the theory developed by James

the critical L / D ratio has a fixed value of 1/12,

whereas equation 7 indicates that even for given

explosive and projectile materials the critical

condition is dependent on the impact velocity.

Consequently, a given projectile may behave as a

rod at some impact velocities and as a plate at

others, and care must therefore be taken to assess

each experimental point from this perspective.

For homogeneous explosives it is known that

when detonation occurs it breaks out at, or close

to, the interface. Consequently, we expect the

relevant time for plate impacts into homogeneous

explosives to be simply the time for the release to

arrive at the interface (i.e. equation 1) as in the

original criterion, and this is borne out by the

results for nitromethane, as we show later.

Ignition & Growth

James3 showed that by adding an additional

constant term to the original critical energy

criterion it was possible to fit the data from a much

wider range of explosives than had previously

been possible. In particular, relatively insensitive

materials such as TATB and homogeneous

explosives5 could now be accommodated with the

new criterion. Here we extend this idea by

replacing the constant term with an ignition delay

term based on the assumption of a single step

Arrhenius reaction rate. Additionally, we replace

the critical energy expression Pueτ = C with the

alternative form P2τ = C, which we generalise to

Pnτ = C. We rearrange this to give τ = C / P

n, and

interpret this as the time required for sufficient

reaction growth to produce a self­sustaining

reaction which will run to detonation. In the

remainder of this paper we refer to this growth

time as τg. The required shock duration for

initiation (τ) is then given by the sum of this

reaction growth time and the ignition delay time,

which we designate as τi. Hence:

τ = τg + τi (8)

For the ignition delay time we adopt the

formula of Hubbard and Johnson6:

τi = υ­1 (Qε)

­1 E2 exp (ε / E), (9)

where E is the shock energy, Q the heat of

reaction, υ the frequency factor and ε is given by:

ε = Cv Ea / R, (10)

in which Ea is the activation energy, Cv the specific

heat and R the gas constant.

For homogeneous explosives the shock energy

E (= ue2 / 2) determines the bulk temperature in the

shocked material. However, for heterogeneous

materials we need to recognise that it is the hot

spots which drive ignition and they have a higher

energy / temperature than the bulk. We make the

assumption that the energy of the hot spots is a

multiple of the bulk energy, i.e.:

Ehs = α E (11)

Substitution of the hot spot energy (Ehs) into

equation 9 now enables us to write:

τi = A E2 exp (B / E) (12)

Where A and B can be regarded as empirical

fitting constants, defined by:

A = α υ­1 (QB)

­1 (13)

B = ε / α (14)

The most general form of the new criterion is

therefore given by:

τ = τg + τi = C / Pn + A E

2 exp (B / E) (15)

In the following section we show that the new

criterion gives excellent fits for a wide range of

explosives. We also show that for homogeneous

explosives the ignition time dominates, whereas

for sensitive heterogeneous materials it is the

growth time. For these materials only τi or τg respectively are required to fit the data. For less

sensitive heterogeneous materials, particularly at

low impact pressures, both terms are significant.

Application of the New Criterion

Heterogeneous Explosives

It is clear from the above that τi is a more

strongly varying function of the shock strength

than τg. As a consequence, we expect that τi will

increase in significance at low shock strengths and

for less sensitive explosives where hot spots may

be less efficient. However, for sensitive explosives

that are known to fit the critical energy criterion in

one of its original forms, it seems likely that τi

may be extremely small and can be neglected.

Consequently, we begin by testing the new model

against the data for such explosives.

Firstly, we consider the Comp B3 (RDX 60%

/ TNT 40%) flyer plate data obtained by de

Longueville et al.7. Figure 1 shows the Go­No/Go

experimental data and a fit using τ = τg = C / P2,

with C = 38 GPa2ms. It can be seen that the new

model fits the Comp B3 data extremely well. In

obtaining this fit the new definition of shock

duration for plate impacts has been applied. The

experiments were carried out using explosive

charges with a diameter of 64mm, and aluminium

flyer plates of varying thickness, and with a

diameter greater than that of the charge. The

effective rod diameter was therefore the charge

diameter (64mm), and when the plate–rod

crossover condition (equation 7) was considered

we found a switch from plate to rod behaviour for

the two lowest pressure points.

Figure 1. Comp B3 experimental flyer plate data

and fit using the new criterion.

Next we analyse the data obtained by

Moulard8 for ISL­Comp B (65­35). Similar flyer

plate experiments were carried out on this

explosive to those on Comp B3 except that 3

different charge diameters were used, namely 20,

35 and 64mm. In the original analysis the data

from the 64mm diameter charges appeared to

show a good fit to a constant critical energy (Pueτ)

criterion, but the results for the 20 and 35mm

charges appeared to show a cut­off to a constant

pressure threshold regardless of the flyer thickness

(and therefore apparent shock duration). However,

when the relevant shock durations are calculated,

using equations 2 and 6, we find that all the 20 and

35mm diameter charge data, as well as the lowest

pressure point in the 64mm diameter data, fall in

the rod regime. It therefore becomes clear that the

apparent constant pressure threshold observed with

the smaller diameter charges is simply an artefact

of considering the plate shock duration as opposed

to the rod duration, which is nearly constant for

each of the 20 and 35mm diameter data sets. With

the new definitions of shock duration we can now

obtain a good fit to the data using τ = τg = C / P2,

with C = 54 GPa2ms. This is illustrated in figure 2,

which also displays the data for 3 different

diameter (5, 10 and 15mm) flat­ended steel rods.

Figure 2. ISL­Comp B (65­35) experimental flyer

plate data for 3 diameters of charge and 3 flat­

ended steel projectiles.

The experimental data for PBX 9404

undoubtedly provides the best comparison

between flyer plate and flat­ended rod impacts. In

figure 3 we display the data from aluminium flyer

plate experiments by Gittings9 and Trott & Jung

10,

Figure 3. PBX 9404 flyer plate and flat­ended rod data, and theoretical fit.

the mylar flyer plate data from Weingart et al.11,

and the flat­ended rod data from Bahl et al.12.

Again we show that an excellent fit can be

obtained to all these data using the most simple

form of the criterion, namely; τ = τg = C / P2, with

C = 17 GPa2ms. The new definition of the shock

duration clearly unifies the flyer plate and rod data

enabling a single parameterisation to be used for

all 1D impacts.

Out of six explosives studied in the classic

paper by de Longueville et al.7 only Comp B3

(discussed above) showed good agreement with

the original critical energy criterion. Of the other

five explosives two were liquids, which we discuss

later under homogeneous explosives. The

remaining three consisted of a cast PBX (RDX

86% / Polybutadiene 14%), a pressed PBX (HMX

89.5% / Nylon 10.5%), and a low density granular

RDX. The granular RDX data appear anomalous

and are not fit by any known criterion. This is

discussed by James3 in terms of changes in the

slope of the Hugoniot, and we do not consider this

further here. We return to discuss the two PBX

compositions after first considering the pressed

TATB data reported by Honodel et al.13.

The plastic flyer data for a range of pressed

TATB powders show extreme deviation from the

original criterion, and this led James to develop his

modified criterion3. In particular, the data show a

marked decrease in sensitivity at the longer shock

durations. To fit these data we now need to employ

the general form of the new criterion, as given in

equation 15. In figure 4 we show the flyer velocity

– shock duration data for the superfine TATB

powder pressed to 1.8 Mg/m3 (the data for the

production and fine grade powders at the same

density are very similar, but have been omitted for

clarity). Figure 4 also shows that the new criterion

can produce an excellent fit to these data with n =

2, C = 37 GPa2ms, A = 4.2 10

­14 s MJ

­2kg

2 and B =

10.3 MJ/kg.

In figure 5 we show the same TATB data and

fit as in figure 4 but on a log­log plot in P­τ space.

We also plot τ = 37 / P2, which can be seen to

match the high pressure data but to deviate

increasingly at low pressures. In this format it can

be clearly seen that the data cannot be fit using τg =

C / Pn alone, regardless of the value of n.

Figure 4. TATB superfine powder, flyer plate

velocity – shock duration data.

Figure 5. TATB superfine powder data, and curve

fits with the new criterion and P2τ = 37.

It is interesting to consider the values of the

parameters A and B used in obtaining the above

fit. Using realistic values for Q and Cv of 4.598

MJ/kg and 1.13 kJ/kg K respectively, the values of

A and B imply that Ea ~ 76α kJ/mol and υ ~

0.5α1012 s­1. Given that α (the factor by which the

hot spot temperature exceeds the bulk) might be

expected to be in the range 1 – 10 these implied

Arrhenius parameters appear reasonable.

Similar fits can be obtained to the data for the

other grades and densities of TATB studied by

Honodel et al.13. Here we discuss only the ultrafine

grade at the same density (1.8 Mg/m3). It is

interesting to note that this grade, which has a

significantly smaller particle size, is more sensitive

at high pressures / short shock durations, but less

sensitive at lower pressures / long shock durations.

This suggests a faster reaction growth (i.e. smaller

τg) and longer ignition delay (i.e. larger τi). We

find that the data for this composition can be fit

very well (see figure 6) with n = 2, C = 18 GPa2ms,

A = 2.6 10­14 s

MJ

­2kg

2 and B = 13 MJ/kg. The

reduced value of C clearly reflects the faster

reaction growth expected from smaller particles.

Interestingly, the values of A and B are consistent

with the same Arrhenius parameters (Ea, υ) as

found for the superfine grade, but with α reduced

by a factor of 10.3 / 13 = 0.79, suggesting less

efficient hot spots from the finer grade material.

Figure 6. TATB ultrafine powder, flyer plate

velocity – shock duration data.

We now consider the two PBX formulations

studied by de Longueville et al.7 which failed to

conform to the original critical energy criterion.

The experimental arrangement for these tests was

the same as for the Comp B3 discussed earlier. On

applying the new shock duration criterion we find

that the HMX/Nylon composition data all lie in the

flyer plate regime, but the two lowest pressure

points in the RDX/Polybutadiene data are

governed by rod behaviour. With these corrections

for the shock duration we find that good fits can be

obtained to both sets of data using τ = τg = C / P2.3,

with C = 47 GPa2.3

ms for HMX/Nylon and C = 110

GPa2.3

ms for RDX/Polybutadiene. The data and

these fits are shown in figure 7.

Figure 7. RDX/Polybutadiene and HMX/Nylon

flyer plate data and theoretical fits.

The necessity to use a burn rate pressure

exponent different from 2 (i.e. n = 2.3) is of

interest, but is not surprising when τg is viewed as

proportional to the inverse of a Vielle’s law

reaction rate. However, it is also of interest to note

that similar quality fits to the data for these two

compositions can be obtained with n = 2 if we

introduce a τi term. It is therefore not entirely clear

whether it is the existence of a significant ignition

delay or different burn rate characteristics which

differentiates these compositions from Comp B3,

ISL­Comp B (65­35) and PBX 9404. However, the

sensitivity of these compositions is closer to Comp

B3 than to TATB, and this suggests that the burn

rate explanation is the more plausible.

Homogeneous Explosives

The response of homogeneous explosives to

flyer plate and rod impacts was discussed by

James et al.5 and interpreted in terms of the

modified two­term critical energy criterion3. Of

particular interest was the finding that it was

necessary to use the original shock duration times

(i.e. equations 1 and 2) to unify the plate and rod

data. Here we suggest that the data can be

modelled by use of τi only (i.e. a thermal explosion

model) and that with our new definition of shock

duration there is a natural explanation of the need

to use equation 1, as opposed to 6, to define the

critical duration for flyer plates.

Whilst there is a small amount of flyer plate

data for liquid TNT7 the largest amount of

homogeneous explosive data is for nitromethane.

This is also the only homogeneous explosive for

which plate and rod data appear to exist. In figure

8 we essentially reproduce the data from figure 1

of James et al.5, but plot the results in P­ τ space.

The data consist of thin flyer results from de

Longueville et al.7, long duration pulse

experiments by Hardesty14, Voskoboinikov et al.

15

and Yoo & Holmes16, and flat­ended rod results

from James et al.5. The flyer plate data are

displayed as actual Go and No/Go points (as in the

previous charts) the other data consist of estimated

threshold values. For the long duration

experiments the luminous induction time has been

taken as corresponding to the duration of the

initiation threshold5.

The theoretical fit shown in figure 8 is given

by τ = τi, with A = 4 10­11 s MJ

­2kg

2 and B = 14.5

MJ/kg. Using literature values17 for Q and Cv of

5.35 MJ/kg and 1.71 kJ/kg K respectively, the

values of A and B imply that Ea ~ 70.5 kJ/mol and

υ ~ 3.2 108 s­1. Whilst this activation energy

appears rather low it must be remembered that this

refers to the overall decomposition process rather

than a single elementary reaction step.

Figure 8. Nitromethane flyer plate, rod and long duration pulse data.

Whilst there is some scatter in the data it can

be seen that the criterion gives a good overall fit

and that the new definition / interpretation of the

shock duration enables the plate and rod data to

be unified.

Discussion and Conclusions

We have proposed a new and simple

empirical criterion for the prediction of 1D

impact initiation thresholds, encompassing both

flyer plate and rod data. The new method differs

from previous approaches in two main aspects.

Firstly, we have redefined the shock duration

time for flyer plate impacts. This is now the time

at which the rarefaction catches the reactive

shock for heterogeneous materials, whilst for

homogeneous materials it remains the time at

which the release wave reaches the flyer­

explosive interface. Secondly, we have

introduced a two­component expression for the

threshold shock duration based on the concept of

an ignition delay time and a reaction growth /

burn time.

We have demonstrated that the new

approach to shock duration enables the flyer

plate and flat­ended rod data to be unified, in a

single model, for both heterogeneous and

homogeneous explosives. Additionally, the new

two­component criterion provides a flexible

function capable of giving excellent fits to data

for a wide range of explosives.

Acknowledgement

The authors would like to thank Mr. Hugh James

of AWE for many helpful discussions on this

topic over a number of years.

References

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