Propellants, Explosives, Pyrotechnics 24, 339ą342 (1999) 339
Radial Crater Growing Process in Different Materials with Shaped
Charge Jets
Manfred Held
TDW - Gesellschaft fur verteidigungstechnische Wirksysteme mbH, D-86523 Schrobenhausen (Germany)
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A. A. Kozhushko
A.F. Ioffe Physical Technical Institute, Russian Academy of Sciences, St. Petersburg (Russia)
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Durch Hohlladunqsstachel bewirkter radialer Kraterwachstums- Processus de craterisation radiale par jet de charge creuse dans
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proze� in verschiedenen Materialien differents materiaux
Die Gleichungen von Szendrei=Held fur den radialen Kraterwachs- Les equations de Szendrei=Held decrivant le processus de crater-
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tumsprozess als Funktion der Zeit konnen auch zu der Bestimmung der isation radiale en fonction du temps peuvent aussi etre inversees pour
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Zielfestigkeit Rt invertiert werden, wenn insbesondere der maximale determiner la resistance de la cible Rf, en particulier lorsqu'on conna�t
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Kraterradius rcm bekannt ist Hiermit wurde die Zielfestigkeit von le rayon de cratere maximal rcm. On a ainsi calcule une resistance de
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Aluminium mit 300 N=mm2 und von glasfaserverstarktem Kunststoff cible de 300 N=mm2 pour l'aluminium et de 405 N=mm2 pour le
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mit 405 N=mm2 berechnet, was zumindest realistische Werte sind. composite verre-resine, qui sont au moins des valeurs realistes. En
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Unter Verwendung dieser Festigkeitswerte zeigen die berechneten utilisant ces valeurs de resistance, les calculs des valeurs de crater-
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radialen Kraterbildungswerte zu den sorgfaltig ausgefuhrten und isation radiale sont en assez bon accord avec les valeurs experi-
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analysierten experimentellen Werten vom Ioffe-Institut in St. Peters- mentales soigneusement agencees et analysees par l'institut loffe de
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burg recht gute Ubereinstimmung. Saint-Petersbourg.
Summary where at least the difference of the target to the penetrator
strength (Rt 7 Y) by the simultaneous measurement of
The equations of Szendrei/Held for the radial crater growing process
projectile v to cratering velocities u can be evaluated(6). If Y
as a function of time can also be inverted to get the target strength Rt if
is very small to the target strength Rt, then the above
the maximum crater radius rcm is known. With this method the strength
equation gives the direct value of Rt.
was calculated for an aluminum target to 300 N=mm2 and for glass
�ber reinforced plastic to 405 N=mm2, which are at least very rea- The term Rt derived from Eq. (1) describes the target
sonable values. By using these values for Rt, the comparison of the
strength resistance to the penetration, to the crater deepening.
radial crater growth process with carefully arranged and analysed
This is roughly a continuous high strain rate process. For the
experiments by the Ioffe Institute is showing good agreements.
target strength under this condition should be given the value
Rt(P). It seems to be of importance to �nd the target strength
characteristic determining the crater radius resulted from the
1. Introduction unsteady inertial radial Żow of the target material. The radial
velocity starts with the axial cratering velocity u, but is then
With the formula of Szendrei(1) slightly modi�ed by decreasing up to zero as the radius is increasing to the
Held(2) the radial crater growth process can be predicted maximum value. The radial crater building process starts
taking jet characteristics and target strength into account. with a high deformation velocity, decreases and is zero at the
This equation was principally well veri�ed with pro�le streak end. This mean target strength should be denoted as Rt(R).
measurements of the radial crater growth process of shaped
charge copper jets penetrating water(2,3). With this equation,
also the target strength Rt can be de�ned if the maximum
crater radius rcm is measured and all the other parameters are
2. Theory
known. With the target strength Rt de�ned in such a way, the
crater radius rc as a function of time t can be calculated. These
A detailed derivation of the used equations is given in
values are experimentally measured and available and give
Refs. 2 and 3. The difference of the Held equation to Szendrei
p���
an interesting comparison. The analysis of two examples will
is only a factor of 2. The �rst time Szendrei has given the
be described in detail.
fundamental physical ideas to these equations:
A popular or normal way to measure the dynamic target
ą the radial cratering velocity is initially equal to the axial
strength Rt is using the modi�ed Bernoulli-equation or the
cratering velocity
Alekseevskii-Tate(4,5) for a steady or quasi-steady Żow.
ą the radial pressure p is decreasing with increasing area A
Rt � 1=2 rt u2 � Y � 1=2 rJn uą2 1ą
(p � p0 A0=A)
# WILEY-VCH Verlag GmbH, D-69451 Weinheim, 1999 0721-3115/99/0612ą0339 $17.50�:50=0
340 Manfred Held and A. A. Kozhushko Propellants, Explosives, Pyrotechnics 24, 339ą342 (1999)
ą the pressure p is working against the target strength Rt,
here de�ned as Rt(R)
ą the initial pressure p0 is much higher than the target
strength Rt; therefore the target strength Rt 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 Rt).
Under these considerations the following equations are
derived
r��������������������������������������������������������������
q�������������������
n
p���o2
�
rctą � A=B A=B r2 t B 1ą
j
with
q����������
A � r2 u2 � r2 v2=1 � rt=rją2 2ą
j j j
B � 2 RtRą=rt 3ą
Figure 1. Measured crater growth process in an aluminum target with
the following jet velocities: (1) 7.7ą7.5 mm=ms, (2) 6.8ą6.5 mm=ms, (3)
The maximum radius rcm is desribed with the following
6.2ą6.0 mm=ms.
equation
q������������
p��������� p�������������������
�
cone angle 2a of 30 and a cone base diameter of 20 mm.
rcm � A=B � rj vj rt=2RtRą=1 � rt=rją 4ą
These charges were �red at a standoff of 1.5 cone base
The Eq. (4) can be solved to the target strength Rt(R).
diameters or 30 mm distance with a few Żash X-rays photo-
q����������
graphed under different views and various time intervals. The
RtRą � 0:5 rj=rcmą2 rt v2=1 � rt=rją2 5ą
j
jet diameter and the crater diameter as a function of time were
measured in different target depths. Therefore, the crater
This target strength Rt(R) can be calculated if the following
radius as a function of time in an aluminum target and in a
data are known:
glass �ber reinforced plastic material is available. The jet
vj jet velocity velocities for the 3 curves of radial crater growth with respect
rj jet radius to the penetration in the aluminum target (Fig. 1) are
rcm maximum crater radius
(1) 7.7ą7.5 mm=ms
rj jet density
(2) 6.8ą6.5 mm=ms
rt target density
(3) 6.2ą6.0 mm=ms
If all these data are available then the target strength and
and in the glass �ber reinforced plastic (GRP) only 7.7ą
the crater growth process as a function of time (Eq. (1)) can
7.5 mm=ms (Fig. 2).
be calculated, respectively predicted.
The jet diameter at the crater bottom was also measured by
The crater is built after the time tcf , which can be
Żash X-ray pictures. The round jets give only rough values,
calculated with the Eq. (6):
q�������������������
A=B r2
j
�
tcf � p��� 6ą
B
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 rcm has to be ``dynamically'' de�ned by Żash
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
Figure 2. Measured crater growth process in glass �ber reinforced
of 25 mm diameter and a copper cone of 0.8 mm thickness, a plastic at 7.7ą7.5 mm=ms jet velocity.
Propellants, Explosives, Pyrotechnics 24, 339ą342 (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 Rt(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 rj the
copper value with 8900 kg=m3 is used and 2750 kg=m3 for
the target density rt. The mean jet radius rj was found to be
0.00071 m. The maximum crater radius rcm for curve 1 with
Figure 3. Comparison of theoretically (dashed lines) predicted and
the jet velocity of 7600 m=s is 0.0075 m in Figure 1.
experimentally gained crater growth history in an aluminum target.
With these data the target strength Rt(R) of 0.300 GPa or
300 N=mm2 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
p����������
u � vj=1 � rt=rją to u � 0. 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
Figure 4. Comparison of theoretically predicted (dashed line) to
results (Fig. 3). The upper experimental curve 1 �ts relatively
experimentally gained crater growth history in glass �ber reinforced
well with this equation, also the beginning of the second
target.
curve 2. Compared to the experiment, the radius is faster
growing in the calculation. In the third curve 3 or lowest
maximum crater radius rcm achieved experimentally is
considered jet velocity the end value is surprisingly well
0.00576 m from Figure 2 before the crater collapse process
predicted, but in the theory it is also earlier achieved.
or reverberation starts. With these data a target strength of
0.405 GPa, respectively 405 N=mm2 is predicted. This is also
a very reasonable value for such a GRP target material.
4.2 Glass Fiber Reinforced Plastic
If these data are used for the crater growing process, then a
more or less perfect �t is achieved for the increasing crater
The same procedure is made for the penetration in the glass
radius rc as a function of time t (Fig. 4). Certainly, the further
�ber reinforced plastic (GRP). The jet velocity is again the tip
history is not presented by this set of equations.
velocity with 7600 m=s as a mean value. The jet radius is
again 0.00071 m, but the target density is 2000 kg=m3. The
5. Conclusion
Table 1. Comparison of the Maximum Crater Radii rcm in Aluminum
The Szendrei=Held equations for the crater growth process
vj (mm=ms) 7.6 6.75 6.1
as a function of time can also be inverted to calculate the
rcm (mm) Experiment 7.5 7.25 6.0
target strength Rt(R) by the maximum crater diameter rcm.
rcm (mm) Calculations (7.5)* 6.6 6.0
With this value Rt(R) the radial crater growth process can be
* is used for Rt calculation?Rt � 0.300 GPa
calculated.
342 Manfred Held and A. A. Kozhushko Propellants, Explosives, Pyrotechnics 24, 339ą342 (1999)
(2) M. Held, ``Veri�cation of the Equation for Radial Crater Growth
Fortunately detailed experimental data are available to
by Shaped Charge Jet Penetration'', Int. J. Impact Engng. 17, 387ą
examine these equations.
398 (1995).
The penetration in an aluminum target gave moderate
(3) M. Held, N. S. Huang, D. Jiang, and C. C. Chang, ``Determination
agreement between theory and experiments. But the �rst of Crater Radius as a Function of Time of a Shaped Charge Jet that
Penetrates into Water'', Propellants, Explosives, Pyrotechnics 21,
opening process in a glass �ber reinforced plastic showed an
64ą69 (1996).
excellent agreement for the crater growing history.
(4) V. P. Alekseevskii, Fizika Goreniya i Vzryva 2, 99ą106 (1966);
These results allow to conclude that theory and the above
Combustion, Explosion, Shock Waves 2, 63ą66 (1966).
(5) A. Tate, ``A Theory for Deceleration of Long Rods After Impact'',
given equations are a good tool to investigate radial crater
Journal of Impact Mechanics and Physics in Solids 15, 387ą399
growth as a function of time, at least for shaped charge jets.
(1967).
(6) V. Hohler, A. J. Stilp, and K. Weber, ``Hypervelocity Penetration
of Tungsten Sinter-Alloy Rods into Aluminia'', Int. J. Impact
Engng. 17, 409ą418 (1995).
6. References
(7) M. Held, ``Jet Observation in Synchro-Streak or Pro�le Streak
Technique'', Proceedings of 17th International Symposium on
(1) T. Szendrei, ``Analytical Model of Crater Formation by Jet Impact
Ballistics, South Africa, 1998, Vol. 2, pp. 251ą258.
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. 575ą583. (Received July 9, 1998; Ms 24=98)