CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
OBTAINING THE GURNEY ENERGY CONSTANT FOR
A TWO-STEP PROPULSION MODEL
Joseph E. Backofen1 and Chris A. Weickert2
1
2
BRIGS Co., 2668 Petersborough St., Herndon, VA 20171, USA
Defence Research Establishment Suffield, P.O. 4000 Station Main,
Medicine Hat, Alberta T1A 8K6, Canada
Abstract. A "Gurney Energy" - reduced to represent only the propulsion imparted during the
expansion of an explosive's gaseous products - is obtained so that it can be used in a two-step
propulsion model which separately accounts for the initial propulsion contributed by an explosive's
brisance. The derivation of this parameter from conventional Gurney Energy "constants" shows that
the original "constants" are also affected by geometrical factors which previously were unknown.
INTRODUCTION
successfully employed to approximately describe
the velocity, which could be imparted when using
the same explosive in different geometrical
arrangements. However, as shown in reference 1,
only about 50% of the final velocity representing
75% of the final kinetic energy is imparted to the
cylinder wall by the gas-push process during
experiments having the geometry ratios employed
during typical cylinder tests.
This paper presents work related to improving
the BRIGS analytical package for explosive
charges by separating explosive detonation-driven
propulsion into a two-step process: 1) initial motion
imparted by a brisant shock-dominated process, and
2) subsequent acceleration by a gas-push process.
Initial motion is envisioned as being caused by the
higher-pressure region of a detonation front (i.e.
envision the von Neumann spike or reaction zone
region as being a finite thickness of solid material
squeezed at high pressure). The gas-push process is
envisioned similar to that assumed by Gurney,
wherein the gas volume expands from a "static"
homogeneous "all-burned" high-pressure state into
one wherein the velocities of the gases at the
boundaries match those of inert boundary materials.
ANALYSIS AND DISCUSSION
Reference 2 - well known worldwide for
presenting the usefulness of the Gurney model to
describe explosive-driven propulsion - provides the
following Gurney formulas for the final "steadystate" velocity imparted in cylindrical and
symmetric sandwich geometry tests:
Vfcyl = (2Eg)
The Gurney model can be and has been used
extensively with various tests, such as the cylinder
expansion test and the symmetric sandwich test, to
obtain a parameter representing the conversion of a
portion of an explosive's chemical energy into
kinetic energy of the gaseous products and
propelled non-reacting materials. Historically, a
single "Gurney Energy" or "Gurney Velocity" has
been obtained during cylinder testing and then
1/2
-1/2
Vfplate= (2Eg)1/z [M/C+1/3]I -1/2
(2Eg)1/2 = 0.605 D / [F - 1]
l/2
(Roth's formula)
Where (2Eg) is the Gurney Velocity (km/sec)
form of the Gurney Energy (Eg), D is the
detonation velocity (km/sec), F is the adiabatic
coefficient, and M and C usually represent the per
958
also to the 2nd propulsion stage of the BRIGS
model, the following formula can be used to
represent the total energy transfer process:
unit length masses of the inert material and the
explosive of an arrangement wherein the boundary
losses are deemed to be un-important When using
Ro, Ri, tpi, and Tex to represent a cylinder's outer
and inner radii, a plate's thickness and half the
thickness of the explosive in a symmetric sandwich,
respectively, these formulas can be rewritten as:
Vf2 = Vi2 + Vgp2
Where the same geometry is used for Vf and Vgp,
which are the final "steady-state" velocity and the
velocity imparted by the gas-push process,
respectively. This formula can be solved for an
"equivalent geometry copper cylinder" locally in
order to obtain a reduced Gurney Velocity (2Egp)1/2
representing only the 2nd propulsion stage by
substituting the initial velocity formula and by
rearranging the terms to yield the following:
Vfcyl = (2Eg)l/2 [(pcyi (Ro2- Ri2) /pex Ri2 ) + 1/2]
Vfplate = (2Eg)
tpi
1/3]
Where pcyi, ppi, and pex are the cylinder, plate, and
explosive densities in g/cm3, respectively.
The BRIGS two-step detonation propulsion
model also is based on the transfer of energy from
an explosive to the propelled materials. As
described in ref.l, initial motion can be represented
by the following Energy Transference Ratio (ETR)
for grazing (side-on) propulsion, which has been
found to be approximately half the ETR for normal
(head-on) impact of a detonation front with a plate.
(2Egp)1/2 = [ ((2Eg)1/2)2 - (Vi / [test geometry] )2 ]1/2
This formula has been solved numerically for
Comp.B explosive, in both cylindrical and
symmetric sandwich geometry, to provide the
Gurney Velocity values plotted in Fig.l as
functions of the ratio of inert material thickness to
explosive thickness.
ETRi = (Vi/D)( P c y l /p e x ) 1 / 2
= 0.2085 [ 3.75 / (F+l)] ((Ro-Ri) / Ri ) "3/4°
As shown in Fig.l, the gas-push Gurney Velocity
is a decreasing function of the thickness ratio. This
was anticipated from the experimental data graphed
in ref.l showing that the relative contribution of the
1st propulsion stage increased as the thickness ratio
increased.
However, Fig.l also highlights a
geometrical dependence in detonation-driven
propulsion. For example, if the total Gurney
Velocity derived by experiment is the same for both
a cylinder and a symmetric sandwich having a
thickness ratio of 0.205, then the gas-push Gurney
Velocities for these would be 85.6% and 93.2 % of
the total Gurney Velocity, respectively.
Where Vj is the initial free-surface velocity
expressed in km/sec. (For plates, tpi and Tex are
used in the formula.)
As mentioned in ref.3, the initial velocity
formulas for grazing and normal impact propulsion
can be combined to account for an angle of
incidence between the detonation front and the
plate by using the following formula:
If
Then Vi = V> (side.on)
Else Vi = ( 1+ sin C ) Vi (side.on)
Figure 2 presents a graph of the final "steadystate" velocities calculated for Comp.B filled
cylinders using the original conventional Gurney
formula and the BRIGS two-step propulsion model,
as well as separate calculations for the 1st and 2nd
propulsion stages. The calculations for copper,
aluminum, and steel cylinders were performed for
the same cylinder thickness to radius ratios and
used the Gurney Velocity found during copper
cylinder expansion tests at a ratio of 0.205 as a
baseline. A factor of 1.9 was used to decrease the
steel cylinder initial velocities to approximate the
energy absorbed by the a to s phase transition.
Where £ is the angle between the normal to the
plate or cylinder wall and the detonation wave. As
described in ref.3, 21° is an approximation for the
angle at which the detonation jumps to a Machwave / Triple-Point flow in explosives such as
Comp.B, Dupont's Detasheet, and various RDXbased PBX explosives.
During cylinder tests, the detonation wave can be
considered as a grazing wave at the position where
measurements are usually taken. By employing the
Gurney assumption both to the entire process and
959
.Gurney Velocity
. . . Ggp_plate
.Ggp_cylinder
Cylinder Wall Thickness to Charge Radius Ratio or
Plate Thickness to Half the Explosive Layer Thickness Ratio
FIGURE 1. Variation of the Gurney Velocity Representing the 2nd Propulsion Stage's Gas-Push Process
3.5 T-:
2.5 --
1.5 .-
0.5 ._ _ «"_•
»-T-a---^."<r».
1.5
2
2.5
Inert Mass to Explosive Charge Ratio
3.5
FIGURE 2. Cylinder Wall Velocities Calculated Using the Gurney Formula and the BRIGS Two-Step Model
960
CONCLUSIONS
As one can expect from ref.2, the velocities
calculated using the conventional Gurney cylinder
equation overlay one another for all three materials.
The two-step propulsion model, however, increases
the velocities for aluminum cylinders due to faster
velocities from the 1st propulsion stage. However,
the velocities for steel cylinders are decreased due
to the effect of the a to 8 phase transition which
absorbs energy during the 1st propulsion stage.
This paper has presented a means by which an
explosive-filled detonation-driven device can be
modeled by a two-step system of equations. These
equations can be used to account for the effects of
phase changes driven by an explosive's brisance
during the 1st propulsion stage as well as the effects
of detonation wave-shaping. These equations also
separately model the gas-push propulsion stage in a
manner consistent with previous propulsion
concepts. Hopefully, this paper will stimulate
additional experimental work to broaden the
available data base so that explosive coupling
effects in both propulsion stages might be more
fully revealed.
Unfortunately, there is currently a paucity of
experimental data for same-material / sameexplosive cylinder tests or symmetric sandwich
tests conducted at multiple mass-to-charge ratios
and/or
multiple
thickness-to-radius
ratios.
Experimental data taken at large ratios of these
parameters is also lacking. Thus, comparisons of
experimental data to the two-step propulsion
model's 2nd stage formulas employing reduced
Gurney Velocity constants is somewhat hindered at
present. The very limited data from ref.4 plotted in
Fig.2 show that the steel cylinder data generally
match the two-step model7s calculations while the
aluminum cylinder data also appear to be affected
by an energy loss mechanism that reduces their
values below both the conventional Gurney and
two-step model calculations.
REFERENCES
1. J.E. Backofen and C. Weickert, "Initial Free-Surface
2.
The question naturally arises "Why have the
Gurney equations generally worked so well?".
The answer may lay in the fact that these equations
are based upon initial and end state assumptions
that:
1) all the Gurney energy is released
instantaneously (and uniformly) within the gaseous
products, and 2) this energy only goes into the
kinetic energy of the explosive products and the
solid body parts being propelled during the gasexpansion-push process. Gurney constants derived
under these assumptions have generally been able
to hide the initial propulsion effects - some of
which cancel each other out - because they are used
to calculate only the final velocity imparted to
similar materials. However, the Gurney equations
still have been known to have problems at low and
high ratios of inert mass to explosive charge mass
as well as with various explosive compositions to
the extent that there has always been argument over
the values of the Gurney "constants" and whether
they have been properly employed in practice.
3.
4.
961
Velocities Imparted by Grazing Detonation Waves",
in Shock Compression of Condensed Matter - 1999,
edited by M.D. Furnish, L.C. Chhabildas, and R.S.
Hixon, American Institute of Physics., Part 2, pp.
919-922
J.E. Kennedy, "Explosive Output for Driving
Metal", in Behavior and Utilization of Explosives in
Engineering Design, Proc. 12th Annual Symp. of the
ASME New Mexico Section, edited by L. Davison,
J.E. Kennedy and F. Coffey, Albuquerque, NM,
1972, pp. 109-124
J.E. Backofen and C.A. Weickert, "A 'Gurney'
Formula for Forward Projection From the End of an
Explosive Charge", Proc. 14th Int. Symp. Ballistics,
Quebec,
American
Defense
Preparedness
Association, 1993, Vol.2, pp.59-68
SJ. Jacobs, "The Gurney Formula: Variations of a
Theme by Lagrange", NOLTR 74-86, Naval
Ordnance Laboratory, Silver Spring, MD, 21 June
1974 (Public release, distribution unlimited)