Radial Crater Growing Process in Different Materials with Shaped Charge Jets, CHEMIA I PIROTECHNIKA, Chemia i ...

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Propellants, Explosives, Pyrotechnics 24, 339±342 (1999)
339
Radial Crater Growing Process in Different Materials with Shaped
Charge Jets
Manfred Held
TDW - Gesellschaft f Èr verteidigungstechnische Wirksysteme mbH, D-86523 Schrobenhausen (Germany)
A. A. Kozhushko
A.F. Ioffe Physical Technical Institute, Russian Academy of Sciences, St. Petersburg (Russia)
Durch Hohlladunqsstachel bewirkter radialer Kraterwachstums-
prozeû in verschiedenen Materialien
Die Gleichungen von Szendrei=Held fÈr den radialen Kraterwachs-
tumsprozess als Funktion der Zeit kÈnnen auch zu der Bestimmung der
Zielfestigkeit R
t
invertiert werden, wenn insbesondere der maximale
Kraterradius r
cm
bekannt ist Hiermit wurde die Zielfestigkeit von
Aluminium mit 300 N=mm
2
2
und von glasfaserverstÈrktem Kunststoff
Processus de crat
Â
risation radiale par jet de charge creuse dans
diffÂrents matÂriaux
Les
Â
quations de Szendrei=Held d
Â
crivant le processus de crat
Â
r-
isation radiale en fonction du temps peuvent aussi Ãtre inversÂes pour
d
Â
terminer la r
Â
sistance de la cible R
f
, en particulier lorsqu'on conna
Ã
t
le rayon de cratÁre maximal r
cm
. On a ainsi calcul une rÂsistance de
cible de 300 N=mm
2
pour l'aluminium et de 405 N=mm
2
pour le
composite verre-rÂsine, qui sont au moins des valeurs rÂalistes. En
utilisant ces valeurs de r
Â
sistance, les calculs des valeurs de crat
Â
r-
isation radiale sont en assez bon accord avec les valeurs expÂri-
mentales soigneusement agencÂes et analysÂes par l'institut loffe de
Saint-Petersbourg.
Summary
where at least the difference of the target to the penetrator
strength (R
t
7Y) by the simultaneous measurement of
projectile v to cratering velocities u can be evaluated
(6)
.IfY
is very small to the target strength R
t
, then the above
equation gives the direct value of R
t
.
The term R
t
derived from Eq. (1) describes the target
strength resistance to the penetration, to the crater deepening.
This is roughly a continuous high strain rate process. For the
target strength under this condition should be given the value
R
t(P)
. It seems to be of importance to ®nd the target strength
characteristic determining the crater radius resulted from the
unsteady inertial radial ¯ow of the target material. The radial
velocity starts with the axial cratering velocity u, but is then
decreasing up to zero as the radius is increasing to the
maximum value. The radial crater building process starts
with a high deformation velocity, decreases and is zero at the
end. This mean target strength should be denoted as R
t(R)
.
The equations of Szendrei/Held for the radial crater growing process
as a function of time can also be inverted to get the target strength R
t
if
the maximum crater radius r
cm
is known. With this method the strength
was calculated for an aluminum target to 300 N=mm
2
2
and for glass
1. Introduction
With the formula of Szendrei
(1)
slightly modi®ed by
Held
(2)
the radial crater growth process can be predicted
taking jet characteristics and target strength into account.
This equation was principally well veri®ed with pro®le streak
measurements of the radial crater growth process of shaped
charge copper jets penetrating water
(2,3)
. With this equation,
also the target strength R
t
can be de®ned if the maximum
crater radius r
cm
is measured and all the other parameters are
known. With the target strength R
t
de®ned in such a way, the
crater radius r
c
as a function of time t can be calculated. These
values are experimentally measured and available and give
an interesting comparison. The analysis of two examples will
be described in detail.
A popular or normal way to measure the dynamic target
strength R
t
is using the modi®ed Bernoulli-equation or the
Alekseevskii-Tate
(4,5)
for a steady or quasi-steady ¯ow.
R
t
1=2r
t
u
2
Y 1=2r
J
nÿ u
2
2. Theory
A detailed derivation of the used equations is given in
Refs. 2 and 3. The difference of the Held equation to Szendrei
is only a factor of
p
2
1
± the radial cratering velocity is initially equal to the axial
cratering velocity
± the radial pressure p is decreasing with increasing area A
(pp
0
A
0
=A)
# WILEY-VCH Verlag GmbH, D-69451 Weinheim, 1999
0721-3115/99/0612±0339 $17.50:50=0
mit 405 N=mm
berechnet, was zumindest realistische Werte sind.
Unter Verwendung dieser Festigkeitswerte zeigen die berechneten
radialen Kraterbildungswerte zu den sorgfÈltig ausgef Èhrten und
analysierten experimentellen Werten vom Ioffe-Institut in St. Peters-
burg recht gute
È
bereinstimmung.
®ber reinforced plastic to 405 N=mm
, which are at least very rea-
sonable values. By using these values for R
t
, the comparison of the
radial crater growth process with carefully arranged and analysed
experiments by the Ioffe Institute is showing good agreements.
. The ®rst time Szendrei has given the
fundamental physical ideas to these equations:
340 Manfred Held and A. A. Kozhushko
Propellants, Explosives, Pyrotechnics 24, 339±342 (1999)
± the pressure p is working against the target strength R
t
,
here de®ned as R
t(R)
± the initial pressure p
0
is much higher than the target
strength R
t
; therefore the target strength R
t
can be
neglected (if somebody wants to take this also into
account then he has to use the modi®ed Bernoulli equa-
tion with the impactor strength Y and target strength R
t
).
Under these considerations the following equations are
derived
r
A=Bÿ
n
A=B ÿr
j
q
p
o
2
B
r
c
t
ÿ t
1
with
q
r
t
=r
j
A r
j
u
2
r
j
v
j
=1
2
2
B 2R
tR
=r
t
3
Figure 1. Measured crater growth process in an aluminum target with
the following jet velocities: (1) 7.7±7.5 mm=ms, (2) 6.8±6.5 mm=ms, (3)
6.2±6.0 mm=ms.
The maximum radius r
cm
is desribed with the following
equation
q
r
t
=r
j
p
A=B
p
r
t
=2R
tR
r
cm
r
j
v
j
=1
4
cone angle 2a of 30
and a cone base diameter of 20 mm.
These charges were ®red at a standoff of 1.5 cone base
diameters or 30 mm distance with a few ¯ash X-rays photo-
graphed under different views and various time intervals. The
jet diameter and the crater diameter as a function of time were
measured in different target depths. Therefore, the crater
radius as a function of time in an aluminum target and in a
glass ®ber reinforced plastic material is available. The jet
velocities for the 3 curves of radial crater growth with respect
to the penetration in the aluminum target (Fig. 1) are
The Eq. (4) can be solved to the target strength R
t(R)
.
R
tR
0:5r
j
=r
cm
2
r
t
v
j
=1
q
r
t
=r
j
2
5
This target strength R
t(R)
can be calculated if the following
data are known:
v
j
jet velocity
r
j
jet radius
r
cm
maximum crater radius
r
j
jet density
(1) 7.7±7.5 mm=ms
(2) 6.8±6.5 mm=ms
(3) 6.2±6.0 mm=ms
r
t
target density
If all these data are available then the target strength and
the crater growth process as a function of time (Eq. (1)) can
be calculated, respectively predicted.
The crater is built after the time t
cf
, which can be
calculated with the Eq. (6):
and in the glass ®ber reinforced plastic (GRP) only 7.7±
7.5 mm=ms (Fig. 2).
The jet diameter at the crater bottom was also measured by
¯ash X-ray pictures. The round jets give only rough values,
q
A=B ÿ r
j
t
cf
p
B
6
3. Experiments
The measurement of the crater growing process as a
function of time in a target is not at all an easy task. Also
the ®nal radius r
cm
has to be ``dynamically'' de®ned by ¯ash
X-ray technique. The crater, which is measurable in a target
after a shaped charge ®ring, is often increased by impact of
later arriving jet portions, which were deviating from the
original jet directions or of tumbling jet particles with
transverse velocities. Fortunately, the Ioffe Physical-Tech-
nical Institute, St. Petersburg, has carefully arranged and
analysed a large number of ®rings with a small shaped charge
of 25 mm diameter and a copper cone of 0.8 mm thickness, a
Figure 2. Measured crater growth process in glass ®ber reinforced
plastic at 7.7±7.5 mm=ms jet velocity.
Propellants, Explosives, Pyrotechnics 24, 339±342 (1999)
Radial Crater Growing Process in Different Materials 341
then the rims are penetrated, depending on the X-ray hard-
ness. The jet radii are estimated in the range of 0.70 mm to
0.72 mm.
4. Comparison of Experiments and Theory
4.1 Aluminum Target
The target strength R
t(R)
can be predicted with the Eq. (5).
For this purpose the maximum investigated jet velocity with
the mean value between 7.7 mm=ms and 7.5 mm=ms,
7.6 mm=ms or 7600 m=s is used. For the jet density r
j
the
copper value with 8900 kg=m
3
is used and 2750 kg=m
3
for
the target density r
t
. The mean jet radius r
j
was found to be
0.00071 m. The maximum crater radius r
cm
for curve 1 with
the jet velocity of 7600 m=s is 0.0075 m in Figure 1.
With these data the target strength R
t(R)
of 0.300 GPa or
300 N=mm
2
is calculated. This is a very reasonable value for
the static strength of the used aluminum plates.
The authors have expected a higher value from the fast
deforming processes by shaped charge jet penetration which
typically gives a higher ``dynamic'' yield strength compared
to this more or less typical ``static'' value. It may be attributed
to the fact that the radial velocity is decreasing from
u v
j
=1
Figure 3. Comparison of theoretically (dashed lines) predicted and
experimentally gained crater growth history in an aluminum target.
r
t
=r
p
to u0. This means from high strain
rates to a static value.
With this value, the expected maximum crater radii for the
other two jet velocities with their mean values of 6750 m=s
(curve 2) and of 6100 m=s (curve 3) are calculated and
compared with the experimental data.
There is some remarkable deviation for the curve 2 with
6750 m=s, but the agreement with the lower velocity of
6100 m=s (curve 3) is good (Table 1).
Now the crater radius can be calculated as a function of
time and these curves can be compared with the experimental
results (Fig. 3). The upper experimental curve 1 ®ts relatively
well with this equation, also the beginning of the second
curve 2. Compared to the experiment, the radius is faster
growing in the calculation. In the third curve 3 or lowest
considered jet velocity the end value is surprisingly well
predicted, but in the theory it is also earlier achieved.
Figure 4. Comparison of theoretically predicted (dashed line) to
experimentally gained crater growth history in glass ®ber reinforced
target.
maximum crater radius r
cm
achieved experimentally is
0.00576 m from Figure 2 before the crater collapse process
or reverberation starts. With these data a target strength of
0.405 GPa, respectively 405 N=mm
2
is predicted. This is also
a very reasonable value for such a GRP target material.
If these data are used for the crater growing process, then a
more or less perfect ®t is achieved for the increasing crater
radius r
c
as a function of time t (Fig. 4). Certainly, the further
history is not presented by this set of equations.
4.2 Glass Fiber Reinforced Plastic
The same procedure is made for the penetration in the glass
®ber reinforced plastic (GRP). The jet velocity is again the tip
velocity with 7600 m=s as a mean value. The jet radius is
again 0.00071 m, but the target density is 2000 kg=m
3
. The
5. Conclusion
Table 1. Comparison of the Maximum Crater Radii r
cm
in Aluminum
v
j
(mm=ms)
7.6
6.75
6.1
The Szendrei=Held equations for the crater growth process
as a function of time can also be inverted to calculate the
target strength R
t(R)
by the maximum crater diameter r
cm
.
With this value R
t(R)
the radial crater growth process can be
calculated.
r
cm
(mm) Experiment
7.5
7.25
6.0
r
cm
(mm) Calculations
(7.5)*
6.6
6.0
* is used for R
t
calculation?R
t
0.300 GPa
342 Manfred Held and A. A. Kozhushko
Propellants, Explosives, Pyrotechnics 24, 339±342 (1999)
Fortunately detailed experimental data are available to
examine these equations.
The penetration in an aluminum target gave moderate
agreement between theory and experiments. But the ®rst
opening process in a glass ®ber reinforced plastic showed an
excellent agreement for the crater growing history.
These results allow to conclude that theory and the above
given equations are a good tool to investigate radial crater
growth as a function of time, at least for shaped charge jets.
(2) M. Held, ``Veri®cation of the Equation for Radial Crater Growth
by Shaped Charge Jet Penetration'', Int. J. Impact Engng. 17, 387±
398 (1995).
(3) M. Held, N. S. Huang, D. Jiang, and C. C. Chang, ``Determination
of Crater Radius as a Function of Time of a Shaped Charge Jet that
Penetrates into Water'', Propellants, Explosives, Pyrotechnics 21,
64±69 (1996).
(4) V. P. Alekseevskii, Fizika Goreniya i Vzryva 2, 99±106 (1966);
Combustion, Explosion, Shock Waves 2, 63±66 (1966).
(5) A. Tate, ``A Theory for Deceleration of Long Rods After Impact'',
Journal of Impact Mechanics and Physics in Solids 15, 387±399
(1967).
(6) V. Hohler, A. J. Stilp, and K. Weber, ``Hypervelocity Penetration
of Tungsten Sinter-Alloy Rods into Aluminia'', Int. J. Impact
Engng. 17, 409±418 (1995).
(7) M. Held, ``Jet Observation in Synchro-Streak or Pro®le Streak
Technique'', Proceedings of 17th International Symposium on
Ballistics, South Africa, 1998, Vol. 2, pp. 251±258.
6. References
(1) T. Szendrei, ``Analytical Model of Crater Formation by Jet Impact
and its Application to Calculation of Penetration Curves and Hole
Pro®les'', Proceedings of the 7th International Symposium on
Ballistics, Den Haag, The Netherlands, 1983, pp. 575±583.
(Received July 9, 1998; Ms 24=98)
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