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ISSN 0940-5151 Vol 17 (2002), No. 1, 113 – 122

Modelling of Explosives Sensitivity

Part 1: The Bruceton Method

Roland Wild and Elart von Collani

Abstract: Considerable work has been done to investigate the sensitivity of explosive materi-

als. Various methods are developed for determining statistically the “explosion characteristics”.

Any statistical method is based on a stochastic model and, clearly, an inappropriate model

yields generally meaningless results, independently of the applied method. The most commonly

applied method in the ﬁeld of explosive material is known as Bruceton method. In this paper

the Bruceton method and the underlying model are analyzed critically and the conclusion is

drawn that the Bruceton method cannot be recommended for determining the sensitivity of

explosive material. In a subsequent paper a diﬀerent approach to solve the problem will be

suggested.

1 Introduction

Scientiﬁc investigation of real phenomena yielding quantiﬁed statements must necessarily

be based on a mathematical model of the aspects of interest. Mathematical modelling

consists of translating known information about the behavior of the phenomenon into

mathematical relations, e.g., in equations. In the following the proceeding is illustrated

by means of explosive sensitivity.

The ﬁrst question which has to be addressed is how sensitivity of an explosive material

shall be described and the next how the actual sensitivity of a given material shall be

determined.

The sensitivity of an explosive material shall reﬂect its tendency to explode under stress.

Thus, for describing sensitivity one has to include a variable “stress” and a function

“tendency” deﬁned on the stress. On the basis of stress and tendency a random experiment

and an estimating procedures must be developed for determining the unknown actual

values of the material in question.

The stress is given by energy and the amount of stress by the applied energy level. The

explosion tendency of a material is appropriately modelled by the probability to explode,

if it is exposed to a certain stress level. There is no tendency, if the stress falls below

a lower limit and there might be a maximum explosion probability if the stress level

exceeds an upper limit. Thus, the lower and upper limit and the run of the probability

curve between these two limits are of interest.

114 Roland Wild and Elart von Collani

2 Modelling

The stress level E may be measured in Joule (or any other unit) and its actual value is

denoted by e ∈ IR

+

0

. Each time the material is exposed to the stress level e it may explode

or not, i.e., has an outcome which occurs at random. For describing the future outcome

of the experiment a random variable is needed whose probability distribution depends on

e. Therefore, the random variable is denoted by X|e and deﬁned by

X|e =

1 if an explosion occurs

0 if no explosion occurs

(1)

X|e is a simple Bernoulli variable with the following probability distribution:

P

X|e

{1}

= p(e)=1− P

X|e

{0}

(2)

The explosion probability p(e) as a function of the variable E with actual value e describes

the tendency, i.e., sensitivity of the material to explode.

Next, relevant features of p(e) have to be formulated:

• p(e)=p

min

for e<e

(3)

• p(e)=p

max

for e>e

u

(4)

• 0 ≤ p(e) ≤ p

max

for e

≤ e ≤ e

u

(5)

Very often, we have p

min

=0andp

max

= 1 and from experience there are rather good

information about the two limits e

and e

u

.

Moreover, the explosion tendency and, hence, the explosion probability is monotonously

increasing with the exposed stress:

• e

1

<e

2

⇒ p(e

1

) ≤ p(e

2

)(6)

For investigating the sensitivity of an explosive material at energy level e

0

, one would

deﬁne a sample of size n by

X

1

|e

0

,...,X

n

|e

0

(7)

and assume that (7) consists of independent and identically distributed Bernoulli vari-

ables. The underlying assumption is that the specimen of the explosive material exhibit a

more or less identical sensitivity. Consequently, the random process starts with exposing

the specimen to stress under identical conditions. The fact that sometimes the outcome

is a reaction and sometime no reaction is due to omnipresent randomness.

3 The Bruceton Method

3.1 The Model

The model used as basis of the Bruceton method diﬀers greatly from the approach outlined

in the previous section. The approach described e.g. in [5] uses the Bernoulli experiments

Modelling of Explosives Sensitivity 115

only implicitly. Instead of regarding the Bernoulli variables X|e, a quantity called critical

energy level is introduced and a random process is regarded assigning to each ﬁctitious

specimen of explosive material one speciﬁc critical energy level.

Let Ω denote the set of all ﬁctitious specimen and e

ω

be the critical energy level of

specimen ω ∈ Ω. The assigning process is deﬁned by a random variable, say ξ with

ξ :Ω→ IR

+

(8)

For a given specimen ω with critical energy level e

ω

the Bernoulli variable (2) degenerates:

P

X(e)

({1})=

1ife ≥ e

ω

0ife<e

ω

(9)

Next, it is assumed that ξ or a known function of ξ has a normal distribution N (µ, σ

2

).

Consequently,

F

ξ

(e)=

1

√

2πσ

2

e

−∞

e

−

(e−µ)

2

2σ

2

dx (10)

gives the probability that the critical energy level assigned to the specimen in question is

less than or equal to e.Ofcourse,µ is the expectation of ξ and σ its variance.

Next consider the two-stage experiment of ﬁrstly assigning a critical energy level to spec-

imen ω and then exposing this specimen to an energy level e. The event of explosion is

equivalent to the event of assigning a critical energy level to ω which is less or equal to e.

Hence, we obtain:

P

X|e

({1})=p(e)=F

ξ

(e)=

1

√

2πσ

2

e

−∞

e

−

(e−µ)

2

2σ

2

dx (11)

The approach used as a basis of the Bruceton method diﬀers fundamentally from the one

described in the previous section. The explosion experiments are thought to be purely

deterministic. Randomness comes in when assigning the critical energy levels to each

specimen. In contrast to the previous model which assumes more or less homogenous

specimen, the actual model assumes comparatively large inhomogeneity among the spec-

imens.

However, one should note that either approach leads to a function p(e) which can be in-

terpreted as explosion probability. We conclude that the Bruceton up-and-down method,

as described in [5], is based on an explosion probability p(e) which is given by the normal

distribution integral. The parameter µ can be identiﬁed as the 50%-point of p(e), i.e.:

p(e

0.5

)=0.5=p(µ) (12)

and the parameter σ

2

determines how fast the explosion probability increases. A small σ

2

means a steep rise of p(e) and a large value of σ

2

a slow rise. The latter eﬀect is illustrated

by Figure 1 (small σ

2

) and Figure 2 (large σ

2

).

As to the features listed in the previous section, the model (10) assumed for the Bruceton

method meets the requirements (5) and (6), but violates (3) and (4).

116 Roland Wild and Elart von Collani

The run of the explosion probability p(e) as function of the energy level e is illustrated in

Figure 1 and 2.

5

10 15 20 25 30

e

0.2

0.4

0.6

0.8

1

explosion probability

Figure 1: The explosion probability (Bruceton method) for µ =10andσ

2

=1.

5

10 15 20 25 30

e

0.2

0.4

0.6

0.8

1

explosion probability

Figure 1: The explosion probability (Bruceton method) for µ =10andσ

2

=6.

Evidently, model (10) should only be applied,

• if it is known that the explosion probability has a symmetric S-shaped course with

symmetry-center at (µ, 0.5) and

• if the analysis does not refer to small or large values of p(e).

Thus, if the behavior of the explosive material is of importance for small values of p(e)

(which is always true in the case of explosives), model (10) is worthless, because require-

ment (3) is not met and because the assumed model is not capable of reﬂecting suﬃciently

well a possible steep (concave) rise of the explosion probability in an neighborhood of e

.

Modelling of Explosives Sensitivity 117

Moreover, if it is to be expected that the explosion probability is not symmetric with

respect to (µ, 0.5), then again it should not be applied, as the conclusions drawn could be

wrong.

Because of the limited usefulness of (10), it is proposed in [5] to use transformations

h(e) instead of e aiming in coming closer to a symmetric S-shaped explosion probability

p(h(e)). Particularly, the use of the transformation y =log(e) is recommended and

investigated.

However, a reasonable selection of the transformation must be based on some knowledge

about the actual course of the explosion characteristic p(e). But, if there are information

available then one could use them directly for developing a more realistic model p(e)than

to try to ﬁnd a model by rather obscure means. The transformation itself represents

another potential source of error without guaranteeing any essential advantages.

3.2 The Method

Following [5] we assume that the original energy level e is transformed into y =loge and

the explosion probability refers to p(y).

The Bruceton method aims at determining the explosion probability p(y) by estimating

the actual values of the two parameters µ and σ

2

of model (10) based on a sample which

is deﬁned by a sampling plan, i.e., an instruction how many experiments have to be made

on which energy levels.

The Bruceton method refers to a so-called adaptive sampling plan, where each subsequent

experiment is determined by the outcome of the actual experiment. First, a starting

energy level, say y

0

=loge

0

, and the total sample size N are chosen.

Next, a sample must be deﬁned. To this end consider the random variable

Y

i

=

1 if the experiment on trensformed energy level y

i

yields a reaction

0 if the experiment on transformed energy level y

i

yields no reaction

(13)

with p

i

=Φ

y

i

−µ

σ

being the probability of a reaction and 1 − p

i

the probability of no

reaction.

Assume that the following energy levels are ﬁxed:

...,y

−k

,y

−k+1

,...,y

−1

,y

0

,y

1

,...,y

k

,y

k+1

,... (14)

with

y

−k

= y

0

− kd and y

k

= y

0

+ kd (15)

and d suitably chosen.

Then a Bruceton sample of size N is deﬁned in the following way:

Y =

Y

i

0

,Y

i

1

,...,Y

i

N−1

(16)

118 Roland Wild and Elart von Collani

where

i

0

=0 and i

j

= i

j−1

+(−1)

Y

i

j−1

for j =1,...,N − 1 (17)

Note that each sample element depends on the outcome of the previous sample element.

The sampling plan given by

Y (16) has the following meaning:

• Start with an experiment on level y

0

.

• If a reaction is observed (Y

0

= 1), then perform the second experiment on level

y

0+(−1)

1

= y

−1

. If no reaction is observed (Y

0

= 0), then perform the second experi-

ment on level y

0+(−1)

0

= y

1

.

• Let the experiment number j − 1, j =1,...,N − 2, be performed on level y

i

j−1

.

If a reaction is observed (Y

i

j−1

= 1), then perform experiment number j on level

y

i

j−1

+(−1)

1

= y

i

j−1

−1

. If no reaction is observed (Y

i

j−1

= 0), then perform experiment

number j on level y

i

j−1

+(−1)

0

= y

i

j−1

+1

.

In other words, in case of a reaction the energy level for the next experiment is decreased

by d, and in case of no reaction the level is increased by d.

Deﬁne

Z

i

(Y

i

j

)=i

j

(18)

then the sequence of random variables Z

i

, i =0, 1,...,N − 1 deﬁnes a ﬁnite Markov

chain. For k

1

,k

2

=0, ±1, ±2,..., the transition probabilities are given by

P (Z

i+1

= k

2

|Z

i

= k

1

)=

1 − p

k

1

for k

2

= k

1

+1

p

k

1

for k

2

= k

1

− 1

0otherwise

(19)

and the initial state probabilities by

P (Z

i

= k)=

1 for k =0

0 for k = ±1, ±2,...

(20)

For illustration, consider a Bruceton sample (16) of size N = 5 and the following event

with respect to the 5-dimensional random vector

Y :

E

Y

= {(1, 1, 0, 1, 0)} (21)

The event E

Y

with respect to

Y corresponds to the following event with respect to

Z

E

Z

= {(0, −1, −2, −1, −2, −1)} (22)

The probability for E

Y

or likewise for E

Z

is given by:

P

Y

(E

Y

)=p

0

p

−1

(1 − p

−2

)p

−1

(1 − p

−2

)

=Φ

y

0

− µ

σ

Φ

y

−1

− µ

σ

2

1 − Φ

y

−2

− µ

σ

2

(23)

Modelling of Explosives Sensitivity 119

Consider a Bruceton sample

Y of size N with N

i

being the number of reactions and M

i

the

number of no reactions at level y

i

implying that N =

i

(N

i

+ M

i

), where the summation

goes over all possible levels y

i

. Note that the number of possible energy levels is bounded

by the sample size.

The log-likelihood function for µ and σ is given by:

log L(

Y )=

i

Φ

y

i

− µ

σ

N

i

+

i

1 − Φ

y

i

− µ

σ

M

i

(24)

The log-likelihood equations for µ and σ are obtained by equating the derivatives of

log L(

Y ) with respect to µ and σ with zero.

i

N

i

φ

y

i

−µ

σ

Φ

y

i

−µ

σ

− M

i

φ

y

i

−µ

σ

1 − Φ

y

i

−µ

σ

= 0 (25)

i

N

i

(y

i

− µ)

φ

y

i

−µ

σ

Φ

y

i

−µ

σ

− M

i

(y

i

− µ)

φ

y

i

−µ

σ

1 − Φ

y

i

−µ

σ

= 0 (26)

where

...,(N

−k

,M

−k

), (N

−k+1

,M

−k+1

),...,(N

o

,M

o

),...,(N

k

,M

k

), (N

k+1

,M

−k+1

),...

are the random numbers of the occurrences of reactions and no reactions at the trans-

formed energy levels y

i

, i =0, ±1, ±2,....

For determining a maximum-likelihood estimator for (µ, σ

2

) the two equations (25) and

(26) must be solved simultaneously for µ and σ

2

. Note that this is a purely numerical

problem. If not the normal integral but a diﬀerent explosion probability would be selected,

the numerical problem would change, but the stochastic problem would be in principle

the same, namely to determine the actual values of the distribution parameters of the

assumed model by means of a sample.

The maximum-likelihood estimators of µ and σ

2

are functions of the random numbers N

i

and M

i

. The joint probability distribution of the N

i

and M

i

has been used for specifying

the log-likelihood function (24). Note that N

i

and M

i

are conditional random variables

under the condition of y

i

and that the normal integral is used only for parameterizing the

probability of the corresponding Bernoulli experiments.

Because the random variables (N

i

and M

i

) linked to the estimators for µ and σ are not

normally distributed, it is not at all justiﬁed to perform statistical analysis using methods

based on the normal distribution. Unfortunately this is done in data analysis referring to

the Bruceton method.

120 Roland Wild and Elart von Collani

4 Conclusion

In sensitivity analysis of explosive materials, specimen of the same explosive mixture

are exposed to energy and observed whether or not a reaction takes place. Evidently

each of these experiments is appropriately modelled by a Bernoulli variable, adopting

either the value 1 with a probability depending on the applied energy or the value 0 with

complementary probability.

Consequently, any series of such experiments is appropriately modelled by a sequence of

Bernoulli variables and these are the only random variables which can be analyzed and

exploited.

The aim of performing such experiments is generally to determine the unknown probability

of reaction or explosion p as a function of the energy level e. It makes sense to use a

parametric representation of p(e) and to determine the actual values by means of the

outcomes of the Bernoulli experiments. In order to obtain an appropriate estimate of the

explosion probability function p(e) it is of utmost importance to observe p(e) for energy

levels e covering the whole interval [e

,e

u

].

The approach used by the Bruceton method is based on the normal integral:

p(e)=

1

√

2πσ

2

e

−∞

e

−

(e−µ)

2

2σ

2

dx (27)

which violates two of the four necessary conditions. More over, the normal integral is

rather inﬂexible. To overcome this inﬂexibility additional transformations have to be

used leading to an increased complexity of the model and the subsequent analysis.

Besides these more strategic weaknesses, the indirect modelling approach used for ana-

lyzing Bruceton samples leads to some additional technical shortcomings.

• The Bruceton method concentrates on energy levels with explosion probability p(e)

close to 0.5 and does not yield any observations for small or large energy levels.

Thus, any implication referring to energy levels wich are not in a neighborhood of

e

0.5

with p(e

0.5

)=0.5 can be looked upon as a matter of faith without almost any

rational basement.

• It is numerically diﬃcult to obtain the solutions of the log-likelihood equations

(25) and (26). The recommended approximate solutions are based on a number of

simpliﬁcations making a quality assessment of the resulting estimators more or less

impossible.

• The derivations of conﬁdence intervals (see e.g. [5]) are based on the wrong assump-

tion that the applied (observed) energy levels visited during the up-and-down exper-

iments are the realizations of normally distributed random variables and, therefore,

the corresponding statements are doubtful.

Modelling of Explosives Sensitivity 121

In all it occurs that the up-and-down method may have merits for determining the ac-

tual value e

0.5

of explosive material, but cannot be used for making any statement on

the explosion probability p(e) as function of the energy level e. More over, the analysis

of the estimator for e

0.5

as described in literature is based on too many simpliﬁcations

and, therefore, doubtful. If the Bruceton method is good for determining e

0.5

, but not

appropriate for determining the function p(e), then the normal model for p(e) is unneces-

sary and can be omitted. Abandoning the normal integral would, of course, result in an

entirely diﬀerent estimator for e

0.5

.

In summary we conclude that from a scientiﬁc and even more from a practical point

of view the Bruceton method cannot be recommended for studying the sensitivity of

explosive material.

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122 Roland Wild and Elart von Collani

Roland Wild

Wehrwissenschaftliches Institut f¨ur Werk-, Explosiv- und Betriebsstoﬀe (WIWEB)

Großes Cent

D-53912 Swisttal

RolandWild@bundeswehr.org

Elart von Collani

University of W¨urzburg

Sanderring 2

D-97070 W¨urzburg

collani@mathematik.uni-wuerzburg.de