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
USING THE PENETRATION-VELOCITY RELATIONSHIP TO
CORRECT FOR VARIATIONS IN TARGET HARDNESS
S. J. Bless1 and J. Cazamias1'2
1
Institute for Advanced Technology, The University of Texas at Austin, 3925 W. Broker Ln., Suite 400, Austin,
TX 78759
2
currently at Lawrence Livermore National Laboratory, L-414, PO Box 808, Livermore, CA 94551
Abstract. There is a correlation between the variation of penetration with impact velocity and the
variation with target strength. This is because penetration is dependent on the ratio of inertial to
strength forces. By using a Taylor series expansion, one can use penetration-velocity data to predict
penetration-strength relationships.
BACKGROUND
analytical function of velocity around the velocity
of interest.
It frequently happens that penetration data must
be adjusted for relatively small changes in target
hardness. Sometimes corrections for variations in
penetrator density are also required. Past treatment
of this problem include references [1] and [2].
Many penetration equations take the form
THEORY
We can find the effects of hardness, H, and
density variations by Taylor expansion. First we
work out the velocity partial derivative.
(D
2R
x=-
pv 2
3/ = 4CR dg
dv~
(3)
(2)
Rewriting this as
where P is penetration, L is projectile length, p is
penetrator density, v is impact velocity, R is target
cavity expansion stress, C is an empirical constant,
and g(x) is a function to be determined. Examples
of formulae with this form include the Poncelet
equation and high velocity forms of the Tate
equations and Odermatt equation.
That penetration must depend primarily on x
also follows from dimensional arguments, for x
represents the target strength non-dimensionalized
by the projectile stagnation pressure. We assume
that we know the variation of P/L as an empirical or
_
dx
-pvj
dv
(4)
and assuming that we know 3//3vfrom empirical
data, we can then write the other partial derivatives
as
dp
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2p
2p dv
(5)
dR~
with R-Y in equation (2) and AH / H with
A(R-Y)/(R0-Y) in equation (8). However, for
the rigid projectile case, one would again expect
equations (1) and (2) to be valid.
(6)
2Rdv
The 1st order Taylor expansion at constant v is
p
—=/
+—
y
L
AP+-M M
(7)
COMPARISON WITH DATA
In order to evaluate this technique, we used the
data from Hohler and Stilp for L/D=10 tungsten
rods. They have data for p = 17 and 17.6 g/cm3
tungsten alloys penetrating into steel of BHN
hardness 180, 255, 295, and 388 (taken from [3],
also quoted in [2]). We fit the BHN 255 and 388
data between 1.4 and 2.6 km/s with an Odermatt
type equation,
In ductile cavity expansion theory, R is proportional
to strength. Strength is proportional to hardness.
Hence, R is proportional to hardness, and
AR/R0 = M f / H 0 .
After substitution and
rearranging, this becomes
T fp 0 +Ap,// 0 +A//,v;
Ll
= Aexp(~B/v)2
(9)
The correction for density was negligible. The best
fit parameters are given in Table 1. The data are
plotted in Fig. 1.
> 0 ,// 0 ,v;JvljAp_A//
#oj
For extremely hard projectiles in which Y and R
are commensurate, R should probably be replaced
117gm/cc WHA in Various BHN Steels)
+
•
BHN 180
BHN 255
BHN 388
- Odermatt A=1.931 b=1.561 km/s
--- BHN 388-> BHN 180
- • Odermatt A=1.917 b=1.349 km/s
0.8-
- - - BHN 388 -> BHN 255
0.6-
1.4
1.6
1.8
2.0
2.2
2.6
FIGURE 1. Points represent data. Heavy line is Odermatt fit to BHN 388 data. Two light dashed lines are "predictions" using equation (1)
for BHN 180 and BHN 255 data. Heavy dashed line is Odermatt fit to 255 data.
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Based on the Odermatt fit for BHN 388, using
equation (8), "predictions" were made for BHN
255 and 180. Equation (8) predictions fit the BHN
180 data as well as a direct empirical fit. Equation
(8) predicts the BHN 255 data to be about 5%
higher than a direct Odermatt fit, but the data
scatters by almost this much about the fit. We also
found that errors were a little larger extrapolating
from higher hardness to lower hardness, apparently
because of the asymmetry in the second order
Taylor terms.
REFERENCES
Rapacki, Jr., E. J., Frank, K., Leavy, R. Brian, Keele,
M. J., and Prifti, J. J., "Armor Steel Hardness
Influence on Kinetic Energy Penetration," in 15th
Int'L Symp. Ballistics, Vol. 1, 1995, 323-330.
Littlefield, D. L., Anderson, C. E., Partom, Y., and
Bless, S. J., "The Penetration of Steel Targets Finite
in Radial Extent," Int. J. Impact Energy 19, 49-62
(1997).
Anderson, C. L., Morris, B. L., and Littlefield, D. L.,
A Penetration Mechanic Database, SWRI Report
3593/001, Southwest Research Institute, San
Antonio, Texas, 1992.
TABLE 1. Odermatt Fits
BHN
a
b (ktn/s)
388
1.931
1.561
255
1.917
1.349
CONCLUSION
We conclude that using equation (8) with
empirical data is a relatively safe procedure to
adjust for small changes in target hardness with
rod-like projectiles. Naturally, however, one should
be alert to threshold conditions when small changes
in hardness can result in a change in penetration
mode.
ACKNOWLEDGMENTS
This work was supported by the U.S. Army
Research Laboratory (ARL) under contract
DAAA21-93-C-0101.
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