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Hypervelocity penetration of plate targets by rod and rod-like projectiles
Int.1. Impact Engng VoL 5, pp. 101-110,1987 PrintedinGreatBritain
0734-743X/8"/$3.00+ 0.00 Pergamon JournalsLtd.
HYPERVELOCITY PENETRATION OF PLATE TARGETS BY ~OD AND ROD-LIKE PROJECTILES
J. Baker and A. Williams U.S. Naval Research Laboratory, Washington, D.C. INTRODUCTION
The a b i l i t y of an elongated p r o j e c t i l e to o b t a i n deep p e n e t r a t i o n in t h e h y p e r v e l o c i t y regime i s what makes i t p a r t i c u l a r l y i n t e r e s t i n g . This paper summarizes the r e s u l t s of r e s e a r c h performed at NRL on the impact of h y p e r v e l o c i t y rod and r o d - l i k e p r o j e c t i l e s over the s i x y e a r s from 1964 to 1969. Semi-empirical c o r r e l a t i o n s are r e p o r t e d which r e l a t e the p r o j e c t i l e and t a r g e t c h a r a c t e r i s t i c s and the impact c o n f i g u r a t i o n to the r e s u l t a n t impact damage.
ROD PROJECTILES
SinGle P l a t e P e r f o r a t i o n Rod l e n g t h l o s s for l i k e - m a t e r i a l impact. Baker (1969) r e p o r t e d the following e q u a t i o n f o r the ~ ~ - o ' s ' ~ , ~ , as a f u n c t i o n ~ impact parameters: A1
t -
d
-
d
At +--
v~ +--
d
/£l/d + t / d + ~t/d { i
- exp [ -
(
d
)
] }
(1)
fv~/d
where a l l l e n g t h s have been normalized by the rod diameter, dp t i s the t a r g e t t h i c k n e s s , v i s the impact v e l o c i t y , and f, T, and At/d are e m p i r i c a l c o n s t a n t s . Note t h a t Eq.(1) must be solved I t e r a t i v e l y s i n c e the unknown parameter Al/d appears on both s i d e s of the e q u a t i o n . The f i r s t term in E q . ( 1 ) , t / d , i s simply the l e n g t h l o s s due to steady s t a t e hydrodynamic flow. The second term, &t/d, i s a small a d d i t i o n a l l o s s due to a non-steady s t a t e p r o c e s s t h a t i s p o s s i b l y a s s o c i a t e d with the i n i t i a l shock phase of the impact p r o c e s s . For most c a s e s of p r a c t i c a l i n t e r e s t (t/d>O.05), t h i s term can be taken as a c o n s t a n t , however, as noted below, t h i s i s not s t r i c t l y t r u e . The t h i r d term, involving v~/d, i s the primary n o n - s t e a d y s t a t e l o s s a s s o c i a t e d with the f i n a l breakout phase of the p e r f o r a t i o n p r o c e s s . The value of v~/d can be computed from the equation v~Id - 1.98 v /
/ [c2 - (us-v/2)2 ]
(2)
where c is the sound speed in the shocked rod material and us is the shock velocity.
These
quantities can be obtained using either published equation-of-state data or coefficients the approximation that the shock velocity is a linear function of the particle velocity.
for
It should be noted that for thick target plates (tld>2) the rod loss approaches a linear dependence on t/d given by Al/d.tld+Atld+vzld. For very thin target plates (t/d<0.1) the term At/d in gq.(1) should be replaced by the expression (At/d)[1-exp(-~t/d)] so that the rod loss approaches zero as t/d approaches zero. The effect of this modification is however n e g l i g i b l e for t/d>O.05.
Rod ~ l o s s f o r u n l i k e - m a t e r i a l impact. For an impact c o n f i g u r a t i o n i n which the p'r'oJectile ~ t-'a~get p l a t e are of d i f f e r e n t m a t e r i a l s , the rod l o s s can be s t i l l be computed u s i n g a m o d i f i c a t i o n of E q . ( 1 ) . By analogy with the theory of steady s t a t e , incompressible f l u i d flow which p r e d i c t s that f o r unlike f l u i d s , the p e n e t r a t i o n v a r i e s as the square root of the d e n s i t y r a t i o of the two f l u i d s , so too, does the l e n g t h of rod l o s t depend on the rod and target materials, i.e. (/11/d)i~2 = / ( p 2 / O l ) (A1/d)l~l
(3)
101
102
J. Baker and A. Williams
Figure i shows the result of applying Eq.(3) to Eq.(1) for a wide material combinations.
range
of
projectile/target
R_~od l e n _ ~ loss for ~ i~act. For oblique impa~ts in which the angle between the rod axls anu ~ ~a'{get surface (8) is less than 90 , the impact process behaves as if the target thickness is greater thaff its nominal value. This additional thickness arises because of the greater distance which is traversed by the rod in passing through the target and is given by a factor of llsin(8). Also, because of the finite diameter of the rod, the point of contact between the rod and the target (and therefore the effective thickness of the target) is extended by a distance d/2tan(O). Therefore Eq.(1) can still be applied to oblique impacts if the target thickness, t/d, is replaced by its effective thickness given by (4)
(t/d)ef f = (t/d)/sin(e) + 1/[2tan(e)]
A yawed rod is one in which there is an angle between the velocity vector o~ the rod and its axis. For the most part, the NgL data was limited to yaw angles below 10 and it is within that range that the results given here are valid and independent o f yaw angle. The NRL experiments also showed that, for yaw angles greater than 15v, the rod was completely destroyed. See Bertke et al (1973) for additional information on yawed rod craterlng. Hole diameter for normal impact. The target plate hole diameter approaches the diameter of the ro-'6"~a~ target pa-l"~'e'-become"'----'s"-very thin, and, conversely, the diameter of the hole approaches the'dlameter of a crater in a semi-lnfinlte target as the target plate becomes very thick. The following analytical expression provides such a dependence: D/d = 1 ÷ (D~/d - i) [1 - exp (-kt/d)]
(5)
where k is a constant that depends on the rod and target materials and D./d, the corresponding crater diameter, has been shown by Baker (1969) to be, very nearly, a llnear function of velocity, D d d = 1 + or, so that Eq.(5) becomes D/d = 1 ÷ a~v [I - exp (-kt/d)]
(6)
where o= is a constant which depends on the rod and target materials. for an aluminum rod and a variety of target plate materials.
Figure 2 shows Eq.(6)
More recently, Baker (1974) analyzed some crater diameter data which was better expression of the form
fitted
by
S~/d = I + Our [i - (re/V)[1 - exp (-V/re)]}
(7)
where ~o and v o are constants for a given combination of projectile and target materials. alternate equation for crater diameter is especially appropriate difference in the densities of the target and projectile materials.
an
when
there
is
a
This large
Hole diameter for ~ impact. For an oblique impact, the hole in the target plate is ~Is'~orte--"eq"'-~to-"-an a p p r o ~ y elliptical shape in which the major diameter lies along the trajectory of the rod. The equation for the major diameter is given by purely geometrical considerations while the minor diameter was shown experimentally to have a weak dependence on the impact angle, 8. Dmaj0 r
D/d
Dmino r
d
sin(8)
d
D/d
(B)
[sin(8)] 1/5
Multiple Plate Perforation Rod length loss for spaced, multiple platetargets. For a t a r g e t consisting of a series of s--~g~pac-'6"~ p-lates,----'~-~rod length Yossa, tdtafget plate hole diameter equations can simply be applied in a sequential manner to each plate individually using the residual rod from the p r e c e d i n g p l a t e . Note t h a t , because of a lack of s u f f i c i e n t experimental data to e s t a b l i s h an anlytical correlation, the velocity of the rod is assumed to remain undlmlshed throughout the impact process.
Rod l e n g t h l o s s form m u l t i l a y e r t a r g e t s . In 1970 Baker and Zalesak extended the s i n g l e p l a t e a"~lysis t~ ~ multilayer target using the concept of "corresponding layers." This concept asserts that the rod length lost in a particular layer of a multilayer target is given by the incremental loss that would occur, according to Eq.(1), in passing through the corresponding
103
H)q~erveloclty penetration of plate targets
p o r t i o n ( i . e . , t h a t p o r t i o n l o c a t e d in the same r e l a t i v e p o s i t i o n ) in a t a r g e t which has a t h i c k n e s s equal to t h a t of the l a y e r being c a l c u l a t e d plus a l l the preceding l a y e r s and which i s composed e n t i r e l y o f the same m a t e r i a l as the l a y e r being c a l c u l a t e d . This approximation assumes t h a t the rod p e n e t r a t e s each l a y e r e n t i r e l y under steady s t a t e c o n d i t i o n s which v i l i be n e a r l y t r u e as long as each subsequent l a y e r i s of l e s s e r d e n s i t y than the preceding one. When a subsequent l a y e r has a higher d e n s i t y than the preceding one, then the p r e s s u r e a t the r o d - t a r g e t i n t e r f a c e must undergo a t r a n s i t i o n from one steady s t a t e p r e s s u r e i n the f i r s t l a y e r to a new, h i g h e r p r e s s u r e in the second. The s t r e n g t h of t h i s e f f e c t v i i i depend on the r e l a t i v e magnitudes of the two s t e a d y s t a t e p r e s s u r e s . The g r e a t e r the d i f f e r e n c e in p r e s s u r e s , the more n e a r l y the rod l o s s in the subsequent l a y e r w i l l approach t h a t of a s i n g l e p ] a t e r a t h e r than t h a t of a corresponding l a y e r . The following equation e x p r e s s e s t h i s relationship: ~l/d = (A1/d)c 1 + [(A1/d)s p - ( ~ l / d ) c l ] (1 - P s l 2 / P s l 3 )
(9)
vhere ( ~ l / d ) c l i s the corresponding l a y e r rod l o s s , ( ~ l / d ) s p i s the s i n g l e p l a t e rod l o s s , and Psnm i s the i n c o m p r e s s i b l e steady s t a t e s t a g n a t i o n p r e s s u r e f m a t e r i a l n p e n e t r a t i n g i n t o m a t e r i a l m and m=2 r e f e r s to the preceding l a y e r and m=3 r e f e r s c a l c u l a t e d . The s t e a d y s t a t e p r e s s u r e i s given by
to
the
current
layer
Psnm " ~°mv2 1 [I + /(0mlPn)] 2
being (I0)
vhere o i s the m a t e r i a l d e n s i t y . In a d d i t i o n , i f the preceding l a y e r i s r e l a t i v e l y t h i n (t/d
(II)
the e f f e c t i v e t h i c k n e s s of the t a r g e t , i s the d i s t a n c e from the c u r r e n t l a y e r to
the l o c a t i o n of the l a s t s t r o n g shock ( e i t h e r the i n i t i a l impact point or a p r e c e d i n g i n t e r f a c e with a l a r g e d e n s i t y change, nominally a f a c t o r of two or more, whichever d i s t a n c e i s l e s s ) and (A1/d) e i s the rod l o s s for a s i n g l e p l a t e of t h i c k n e s s ( t / d ) e of the same m a t e r i a l as the rod. Denoting the s t e a d y s t a t e f a c t o r by H123, then the rod l o s s f o r each l a y e r of a m u l t i l a y e r t a r g e t i s given by ~111d = ( ~ l l d ) c I + [ ( b l l d ) s p - ( ~ l l d ) c l ] ~123A
for 03 > p2
(12)
•l/d = (Alld)c I
f o r 03 < o2
(13)
and The t o t a l rod l o s s i s given by sununlng the l o s s e s f o r each l a y e r . Rod l e n g t h l o s s f o r obllque impact i n t o m u l t l l a y e r t a r g e t s . As for s i n g l e p l a t e s , an o b l i q u e ~acT--i-nt~'mu~t-[layer target--~--'s~ou-l~--useEqs.(12)~'-'~--~) vith an effective target thickness similar to that of Eq.(4). It must be recognized, however, that the second term in Eq.(4), 1/[2tan(e)], is independent of target thickness and arises from the geometry of impacting the front surface of the target. Therefore this term is modified so that it enters into the calculation only to the degree that the transition between layers is llke a new impact as expressed by the steady state factor ~2~, and is furthermore modlfied by the factor A for similar reasons. The equation for th@ ~ffectlve thickness of an oblique, multllayer target is therefore given by (t/d)ef f = (t/d)/sin(e) + H123 A /
[2tan(e)]
for o3 > p2
(14)
for p3 ~ p2
(15)
and (t/d)ef f = (tld)/sln(8)
Hole d i a m e t e r f o r m u l t i l a ~ e r t a r g e t s . Lacking s u f f i c i e n t experimental data to determine the e--le'~c~da~ent l a y e r s of a m u l t i l a y e r t a r g e t on the h o l e diameter, the s i n g l e p l a t e h o l e d i a m e t e r e q u a t i o n s a r e used as an approximate value f o r the hole diameter in each l a y e r of the m u l t i l a y e r t a r g e t . There i s some evidence that the m u l t i l a y e r hole diameters tend to be l a r g e r than the s i n g l e p l a t e v a l u e s , perhaps because of the p e t a l l i n g and o t h e r s t r u c t u r a l e f f e c t s t h a t occur i n the m u l t i l a y e r t a r g e t . The s i n g l e p l a t e hole diameter should t h e r e f o r e b e . c o n s i d e r e d to be a minimum e s t i m a t e of hole s i z e .
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J. Baker and A. Williams
S i n g l e P l a t e Craterin@ An impact will create a crater'in the target when the rod projectile either comes to rest or is totally consumed before it perforates the target. The impact analysis is then primarily concerned with the depth and ~iameter of the crater and 0nly secondarily with any residual rod length. ~
state craterin~ model. Tate (1967 and 1969) proposed a model of rod cratering in which i~1"~-Bernoulli'se"~'~ion for incompressible, steady state flow by adding, a dynamic
strength, for each material to both sides of the equation, i.e., ~Op(V-U)2+ao-~ ~Ptu2+at , where v is the projectile velocity, u is the velocity of the interface, pp is the projectile density, and Pt is the target density.
The final target penetration, p, consists of a
hydrodynamic phase (if the impact velocity is high enough) which is calculated by integrating the term u/(u-v) over the initial length of the projectile, 1 (in general this must be done numerically) plus a rigid projectile phase (if the dynamic strength of the rod is greater than that of the target). The Tate model shows good agreement with experimental data below about 2 km/s, however in the hypervelocity regime, the crater depth of the model becomes independent of velocity and asymptotes to the steady state limit given by the initial rod length times the square root of the density ratio. This does not agree with the hypervelocity data presented by Christman and Gehring (1965) in which the penetration continues to rise above the steady state limit with increasing impact velocity. This divergence from Tate's one-dimensional, steady state model is most likely due to the two-dimensional, non-steady effects which dominate the final phase of the crater formation. Modification of the Tate model. Christman and Gehring (1966) proposed an empirical model of rod cratering'~n--~ic-h"~he penetration occurs hydrodynamically for all but the last diameter of rod length and that that last diameter of length creates a hemispherical crater charateristic of a chunky fragment impact. Their model of crater penetration depth is given by p/d = /(pp/Pt)(i/d - 1) + ~ D~/d.
(16)
The factor ~ implies that the chunky fragment portion of the crater is always hemispherical, but that is only true if the projectile and target are of the same material. In general, the f i n a l p o r t i o n of the c r a t e r v i i i be e i t h e r g r e a t e r than or l e s s than h e m i s p h e r i c a l depending on the relative densities of the target and projectile. Therefore the factor ~ should be replaced by an expression such as ~(pp/Pt )m.
If it is assumed that this dependence on the density
ratio also goes as the square root (m=~) then Eq.(16) for rod penetration depth becomes p/d = /(Op/Pt)(i/d - 1 + ~ DJd)
(17)
Baker (1974) observed that the difference betweep the simple empirical model of Eq.(17) and the hydrodynamic limit ~o which the Tate model asymptotes is the second factor in parentheses which multiplies the square root of the density ratio. He therefore suggested that the Tate model could be modified to have the correct dependence at high velocity by simply multiplying the square root of the material density ratio, /(pp/Pt ), by the factor ~ = 1 - ~ (1-a~v)/(lld). When this modification is introduced into Tate's analysis, it not only has the desired effect of having the penetration continue to increase at the higher velocities to values above the steady state limit but also leaves the lower velocity behavior and the residual rod length virtually unchanged. In addition, the modification introduces a dependence on the initial rod length such that the shorter the initial rod, the greater viii be the deviation of the penetration from the original Tare model at high velocities. Figure 3 compares the penetration (normalized by the inital rod length) versus velocity calculated from the original Tare model with that calculated from the empirically modified model. The values of the dynamic strengths of the rod and target materials differ for each combination off materials and have been selected so that the modified model best fits the experimental data points (also shown in the figure) which are taken from Christman and Gehring (1965) for a C1015 steel rod with an i/d of three impacting an 1100-0 aluminum target.
Hypervelocity penetration.of plate targets
I05
ROD-LIKE PROJECTILES Having developed a reasonably good understanding of the penetration process of a rod projectile, the next step was to investigate the possibility of designing a projectile which would be an even more e f f i c i e n t p e n e t r a t o r than a rod. The f i v e d e s i g n s which were examined a r e i l l u s t r a t e d i n F i g . 4 and a r e d e s i g n a t e d as TYPES 1 t h r o u g h 5. The b a s e l i n e rod p r o j e c t i l e i s d e s i g n a t e d as TYPE 0. Since the s h a p e s of the r o d - l i k e p r o j e c t i l e s a r e a l l v e r y d i f f e r e n t , t h e common measure of t h e i r p e n e t r a t i n g a b i l i t y was taken to be the amount of mass ( r a t h e r t h a n l e n g t h ) l o s t by each p r o j e c t i l e i n p e r f o r a t i n g the same t a r g e t . Application of Rod Analysis to Rod-like Projectiles The rod analysis can be applied to the rod-like projectile designs by using the following procedures (I) Calculate the steady state projectile length loss for penetrating a plate of t h i c k n e s s , t , tXlss - t / ( P t / P p ) ; (2) C a l c u l a t e the s t e a d y s t a t e mass l o s s , &~ss' c o r r e s p o n d i n g to the above l e n g t h l o s s , which i s a s i m p l e g e o m e t r i c problem t h a t depends on the p r o j e c t i l e d e s i g n ; . ( 3 ) C a l c u l a t e the d i a m e t e r a t the f r o n t of the r e s i d u a l p r o j e c t i l e , d r , a f t e r removing
the amount of projectile lost by steady state; (4) Use t/d r in Eq.(1) without the tld term to determine the non-steady state mass loss, ~mnss, that a rod projectile of diameter d r would experience.
To do this, calculate the total length loss for a rod of diameter d r impacting a
plate of thickness t and then subtract the steady state length loss obtained in step 1, i.e., (AI - Alss )nppd:14; (5) The total mass loss is then ~m = ~mss + hnnss; (6) Calculate ~mns s the total length loss corresponding to the total mass loss and then compare it to the measured experimental length loss. The key to this procedure is the assumption in step 4 that the diameter of the projectile after perforating the target determines the amount of continued plastic flow which produces the additional projectile mass loss. This is a reasonable assumption and a comparison of the test results with the length loss calculations shows good agreement. D i s c u s s i o n of Each R o d - l i k e P r o j e c t i l e
Design
TYPE I projectile. The TYPE i projectile design, a truncated cone, is subdivided into two categories designated as TYPE 1A with small values of the nose diameter and TYPE 1B vlth larger values of the nose diameter. TYPE IA projectile. The projectile-target material combinations were aluminum-aluminum, s-'~'e"el--"aluminum, steel-titanium, and tungsten carbide-steel. In general, only for the steel-alumlnum tests with small yaw angles did the projectile survive the severe lateral forces of the impact. A p l o t o f the c a l c u l a t e d f r a c t i o n a l mass l o s s v e r s u s t a r g e t p l a t e t h i c k n e s s f o r b o t h the TYPE 1A and rod p r o j e c t i l e s i s shown in F i g . 5 . The r a t i o o f cone mass l o s s to rod mass l o s s i s o b v l o u s l y s m a l l f o r v a l u e s o f t / d l e s s than about two and the r a t e of mass l o s s f o r t h e cone r e m a i n s l e s s than t h a t f o r the rod f o r v a l u e s of t / d up to a p p r o x i m a t e l y s e v e n , above which t h e r a t e of l o s s f o r b o t h a r e a p p r o x i m a t e l y e q u a l .
For a spaced target configuration, the TYPE IA projectile becomes a TYPE iB after impacting the first target plate. The TYPE IA projectile would therefore be particularly effective against a target configuration in which the first plate is thick and made of a low density material. TYPE 1B projectile. Two of the impacts which failed with the TYPE IA projectile were successful w i t h the TYPE 1B p r o j e c t i l e which has a l a r g e r nose d i a m e t e r . I f the t a r g e t h o l e d i a m e t e r i s g r e a t e r than the d i a m e t e r of the p r o j e c t i l e b a s e t h e n t h e r e s h o u l d be no s i d e c o n t a c t between the p r o j e c t i l e and the t a r g e t . A p l o t o f the c a l c u l a t e d f r a c t i o n a l mass l o s s versus target plate thickness for a steel TYPE IB and rod projectile impacting a titanium target is shown in Fig.6.
TYPE 2 p r o j e c t i l e . The p r i m a r y a d v a n t a g e of the TYPE 2 p r o j e c t i l e , a pointed rod, is its s-'~ITclty. T h i s d e s i g n was i n t e n d e d f o r l i k e - m a t e r i a l impact i n the e x p e c t a t i o n t h a t t h e p o i n t e d n o s e reduce the e f f e c t o f the impact shock which would more than compensate f o r t h e s i d e c a v i t a t i o n mass l o s s . A p l o t of the c a l c u l a t e d f r a c t i o n a l mass l o s s v e r s u s ~ a r g ~ t p ~ a t e t h i c k n e s s f o r s t e e l rod and TYPE 2 p r o j e c t i l e s w i t h t h r e e cone t i p a n g l e s , ~ = 15v,30v,60 v, i m p a c t i n g a s t e e l t a r g e t i s shown i n P i g . ? . The c a l c u l a t i o n o f the s t e a d y s t a t e mass l o s s f o r the p o i n t e d rod w i l l eha~ge from a cone volume e v a l u a t i o n to i n c l u d e a rod volume e v a l u a t i o n i f the s t e a d y s t a t e l e n g t h l o s s exceeds the l e n g t h of the c o n i c a l t i p . T h i s change in mass l o s s r a t e i s the knee in the g r a p h o f F i g . ? .
106
J. Baker and A. Williams
TYPE 3 p r o j e c t i l e . The c o n c e p t o f t h e TYPE 3 p r o j e c t i l e i s t h a t t h e l e a d i n g s m a l l e r diameter ~portion ( s n o u t rod) v i l I produce a hole in the target plate sufficient to p a s s t h e remaining larger d i a m e t e r rod v i t h o u t any a d d i t i o n a l projectile loss. Tvo s n o u t rod configurations v e r e u s e d . One had a s n o u t d i a m e t e r t h a t v a s h a l f o f t h e main rod d i a m e t e r and a s n o u t l e n g t h t h a t v a s f o u r d i a m e t e r s lonE. The s e c o n d c o n f i g u r a t i o n had a s n o u t d i a m e t e r that vas tvo-thlrds o f t h e ~ a t n rod d i a m e t e r and a s n o u t l e n g t h t h a t v a s f o u r and a h a l f d i a m e t e r s l o n g . A p l o t o f t h e c a l c u l a t e d f r a c t i o n a l mass l o s s f o r a s t e e l rod and t h e t v o TYPE 3 p r o j e c t i l e s i m p a c t i n g a s t e e l t a r g e t i s s h o r n in F I E . 8 . TYPE 4 p r o j e c t i l e . The TYPE 4 p r o j e c t i l e i s a c o m b i n a t i o n o f a h i g h s t r e n g t h TYPE 1A t r u n c a t e d cone p r o j e c t i l e and a l o v ~ e n s i t y s l e e v e . The f i v e s u c c e s s f u l t e s t s had a 4340 s t e e l cone h a r d e n e d to Rockwell "C" 55 v i t h a Z e l u x ( p o l y c a r b o n a t e plastic) sleeve with a density of 1.19 g / c c . The r a t i o o f t h e s t e e l cone n o s e to b a s e d i a m e t e r s v a s 0 . 0 8 v h i l e t h e r a t i o s o f Z e l u x s l e e v e d i a m e t e r to cone b a s e d i a m e t e r v e r e 1 . 0 , 0 . 6 6 , and 0 . 5 8 . The i m p a c t e f f e c t o f t h i s c o m b i n a t i o n p r o j e c t i l e i s c a l c u l a t e d a s t h e sum o f t h e impact part. The s l e e v e i s modeled a s a s h o r t rod w i t h a d i a m e t e r and a l e n g t h s u c h t h a t t h e t h e s h o r t rod i s e q u i v a l e n t to t h e s l e e v e m a s s . The d e s i r e d effect of the sleeve creation o f a h o l e i n t h e t a r g e t w i t h s u f f i c i e n t d i a m e t e r and d e p t h to p r e v e n t c o n t a c t t h e h o l e w a l l and t h e c o n i c a l p r o j e c t i l e b a s e .
of each mass o f is the betveen
A p l o t o f t h e c a l c u l a t e d f r a c t i o n a l mass l o s s f o r a s t e e l rod and t v o TYPE 4 p r o j e c t i l e s impacting a steel target is shorn in Pig.9. Due to t h e low s t r e n g t h o f t h e s l e e v e m a t e r i a l , t h e c a l c u l a t i o n s a s s u m e t h a t t h e e n t i r e s l e e v e i s d e s t r o y e d by t h e i m p a c t , e v e n f o r v e r y t h i n targets. The s t e e l cone l o s e s mass a c c o r d i n g to t h e p r o c e d u r e f o r t h e TYPE 1A p r o j e c t i l e . The mass l o s s f o r t h e TYPE 4 p r o j e c t i l e i s t h e r e f o r e t h e s l e e v e mass p l u s t h e cone mass l o s s . TYPE 5 p r o j e c t i l e . The TYPE 5 p r o j e c t i l e c o n s i s t s o f two m a s s e s : a l e a d i n g "bumper" and a l o n g ~'~oving rod v h i c h a r e c o n n e c t e d by a s m a l l e r d i a m e t e r r o d , o r s t e m . T h i s p r o j e c t i l e c o n f i g u r a t i o n i s an e f f i c i e n t one f o r p e r f o r a t i n g a t / d - 2 t a r g e t o f l i k e - m a t e r i a l . The i m p a c t characteristics o f t h e bumper ( 1 / d - l ) a r e e s s e n t i a l l y t h e same a s t h a t o f a s p h e r e o f e q u a l mass.
For v e r y t h i c k t a r g e t s , t h e t a r g e t damage due to bumper impact i s t h e same a s c r a t e r formation in a semi-inifinlte target. The bumper v I I l be d e s t r o y e d by t h e impact s h o c k i n a l l c a s e s e x c e p t f o r low p r o j e c t i l e v e l n c i t l e s o r v e r y t h i n t a r g e t s ( t / d - O . 0 1 ) . The f r a g i l e connecting stem is also likely to be d e s t r o y e d i n most impact s i t u a t i o n s . The mass l o s s o f t h e TYPE 5 projectile i s t h e r e f o r e t h e sum o f t h e bumper and s t e m m a s s e s p l u s t h e mass l o s s o f t h e following rod. A p l o t o f t h e c a l c u l a t e d f r a c t i o n a l mass l o s s f o r a s t e e l rod and TYPE 5 p r o j e c t i l e s impacting a steel t a r g e t i s shown i n F i g . l O . Note t h e g r e a t e r e f f e c t i v e n e s s o f t h e bumper rod V l t h t h e larger diameter of the connecting stem.
CONCLUSIONS Ro__.ddP r o j e c t i l e s The t e r m i n a l b a l l i s t i c r e s u l t s o f a h y p e r v e l o c i t y rod i m p a c t i n g a s i n g l e t a r g e t p l a t e h a s been successfully q u a n t i f i e d w i t h a f e v , r e l a t i v e l y s i m p l e , a n a l y t i c e x p r e s s i o n s from v h i c h one c a n c a l c u l a t e t h e l e n g t h o£ t h e r e s i d u a l rod and t h e d i a m e t e r o f t h e h o l e i n t h e t a r g e t p l a t e g i v e n the value of a fev empirically determined constants v h i c h depend on t h e rod and t a r g e t materials. T h e s e a n a l y t i c c o r r e l a t i o n s have been shown to be v a l i d f o r m a t e r i a l combinations r a n g i n g from a v e r y d e n s e u r a n i u m rod i m p a c t i n g a low d e n s i t y aluminum t a r g e t p l a t e to a l o v d e n s i t y aluminum rod i m p a c t i n g a h i g h d e n s i t y u r a n i u m t a r g e t p l a t e . The e f f e c t o f t a r g e t p l a t e obliquity h a s a l s o been q u a n t i f i e d . The s i n g l e p l a t e e q u a t i o n s can be r e a d i l y a p p l i e d , i n a s e q u e n t i a l manner, to a t a r g e t c o n s i s t i n g o f a s e r i e s o f s p a c e d p l a t e s . F u r t h e r m o r e , t h e s i n g l e t a r g e t p l a t e a n a l y s i s h a s been e x t e n d e d to a m u l t l l a y e r e d t a r g e t using a c o n c e p t c a l l e d " c o r r e s p o n d i n g l a y e r s " a l o n g v i t h some e a s i l y computed f a c t o r s v h i c h c o n t r o l t h e e x t e n t to v h i c h t h e p e n e t r a t i o n process diverges from t h a t f o r a s i n g l e plate and approaches that of a "corresponding layer." The rod p r o j e c t i l e a n a l y s i s c o n c l u d e s v i t h an a n a l y t i c a l model o f c r a t e r p e n e t r a t i o n d e p t h and residual rod l e n g t h . The p r o p o s e d model s l i g h t l y m o d i f i e s t h a t o f A. T a r e by i n t o d u c i n g a factor vhich multiplies the density ratio. This factor, vhich is velocity dependent, permits the calculated crater penetration d e p t h to r i s e above t h e s i m p l e h y d r o d y n a m i c l i m i t a t v e r y h i g h v e l o c i t i e s i n a g r e e m e n t v i t h t h e e x p e r i m e n t a l d a t a o f C h r i s t m a n and G e h r i n g .
Hypervelocity penetration of plate targets
107
Rod-like Projectiles The five rod-like projectile designs, illustrated in Fig.4, were chosen to provide a significant reduction in the projectile mass loss, compared to that of a rod projectile, for particular impact configurations. This improved performance is due to one or more of the following mechanisms: (1) a smaller nose diameter having less mass per unit length than a rod but which s t i l l c r e a t e s a l a r g e enough c l e a r a n c e h o l e ; (2) a p r o j e c t i l e of two independent s e c t i o n s with the l e a d i n g , l o v e r mass one sacrificed to create a clearance hole for the other; (3) a projectile which is strong enough to create a hole in the target without suffering any damage to itself. The most effective of the new projectile designs is the TYPE IA sharp steel cone. For all but very thick targets the projectile mass losses are n e g l i g i b l e . Hoverer, t h e impact configuration is somewhat limiting as it requires a weak, low density target and a h i g h strength projectile. The TYPE 1B t r u n c a t e d cone and the TYPE 3 s n o u t rod p r o j e c t i l e s s h o u l d be p a r t i c u l a r l y effective against thin targets since the mass loss would be a function of a small projectile nose diameter. When the t a r g e t t h i c k n e s s i s such t h a t the s t e a d y s t a t e l e n g t h l o s s i s g r e a t e r than t h e l e n g t h of the c o n i c a l p o i n t , the r e d u c t i o n in mass l o s s f o r the TYPE 2 p r o j e c t i l e over t h a t f o r t h e rod will simply be trice the mass of the conical point. For thinner targets the TYPE 2 projectile suffers an unknown amount of side erosion to the conical point during target penetration and the modified rod loss theory will therefore predict low values of mass loss. The target penetration efficiency of the TYPE 5 bumper rod projectile is due to the decoupling of the shock and plastic flow in the bumper from the following rod. The connecting stem must provide a margin of time to allow for sufficient flow of the target debris such that the following rod does not encounter any significant amount of target material. The connecting stem may also act like the snout of a TYPE 3 projectile, punching a clearance hole in any remaining target thickness. The TYPE 4 s l e e v e / c o n e p r o j e c t i l e combines the p e n e t r a t i o n mechanisms of the TYPE 1 and TYPE 5 projectiles. For t h i n t a r g e t s the low d e n s i t y s l e e v e simply punches a c l e a r a n c e h o l e in t h e target while for thick targets the sleeve creates a crater of sufficient diameter and depth to g i v e h o l e c l e a r a n c e f o r the p r o j e c t i l e base and to allow the s t e e l cone to o p e r a t e a s a TYPE 1B p r o j e c t i l e which p e n e t r a t e s the r e m a i n i n g t a r g e t t h i c k n e s s . The s t e e l cone must be s t r o n g enough to withstand the shock raves generated by the impact of the low density sleeve. Summary T h i s paper h a s summarized the r e s u l t s of e x p e r i m e n t a l t e s t s and a n a l y t i c a l s t u d i e s of t h e h y p e r v e l o c i t y impact of rod and r o d - l i k e p r o j e c t i l e s which were conducted a t the Naval R e s e a r c h L a b o r a t o r y . The r e s u l t s p r e s e n t e d h e r e provide r e l a t i v e l y s i m p l e a n a l y t i c e x p r e s s i o n s from which one can c a l c u l a t e the r e s u l t s of a h y p e r v e l o c i t y impact of a rod or r o d - l i k e p r o j e c t i l e even i n t o complex t a r g e t s under most impact c o n f i g u r a t i o n s of i n t e r e s t . The methodology does r e q u i r e a knowledge o f c e r t a i n e m p i r i c a l c o n s t a n t s which depend on t h e p r o j e c t i l e and t a r g e t materials. For those cases where the values of these constants have not been provided, they can easily be determined by performing a relatively few experimental impacts.
108
J. Baker and A. Ni111ams
REFERENCES
Baker, J. (1969).
See author for complete reference.
Baker, J. and Zalesak, S. (1970). Baker, J. (1974).
See author for complete reference.
See author for complete reference.
Bertke, R., Seyda, J., Svift, S. and Lehman, H. (1974). A Study of Yaved Rod Impacts. University of Dayton Research Institute, U.S. Army Ballistic Research Laboratory Report No. BRL-CR-182. Christman, D. and Gehrlng, J. (1966). Analysis of High-Velocity Projectile Penetration Hechanics. J...cAppl. Phys.p 3.77,1579-1587.' Tare, A. (1967). A Theory for the Deleceratlon of Long Rods after Impact.
J. Mech. Phys. Solids., 15, 387-399. Tate, A. (1969). Further Results in the Theory of Long Rod Penetration. J. Hech. Phys. Solids., 17, 141-150.
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0734-743X/8"/$3.00+ 0.00 Pergamon JournalsLtd.
HYPERVELOCITY PENETRATION OF PLATE TARGETS BY ~OD AND ROD-LIKE PROJECTILES
J. Baker and A. Williams U.S. Naval Research Laboratory, Washington, D.C. INTRODUCTION
The a b i l i t y of an elongated p r o j e c t i l e to o b t a i n deep p e n e t r a t i o n in t h e h y p e r v e l o c i t y regime i s what makes i t p a r t i c u l a r l y i n t e r e s t i n g . This paper summarizes the r e s u l t s of r e s e a r c h performed at NRL on the impact of h y p e r v e l o c i t y rod and r o d - l i k e p r o j e c t i l e s over the s i x y e a r s from 1964 to 1969. Semi-empirical c o r r e l a t i o n s are r e p o r t e d which r e l a t e the p r o j e c t i l e and t a r g e t c h a r a c t e r i s t i c s and the impact c o n f i g u r a t i o n to the r e s u l t a n t impact damage.
ROD PROJECTILES
SinGle P l a t e P e r f o r a t i o n Rod l e n g t h l o s s for l i k e - m a t e r i a l impact. Baker (1969) r e p o r t e d the following e q u a t i o n f o r the ~ ~ - o ' s ' ~ , ~ , as a f u n c t i o n ~ impact parameters: A1
t -
d
-
d
At +--
v~ +--
d
/£l/d + t / d + ~t/d { i
- exp [ -
(
d
)
] }
(1)
fv~/d
where a l l l e n g t h s have been normalized by the rod diameter, dp t i s the t a r g e t t h i c k n e s s , v i s the impact v e l o c i t y , and f, T, and At/d are e m p i r i c a l c o n s t a n t s . Note t h a t Eq.(1) must be solved I t e r a t i v e l y s i n c e the unknown parameter Al/d appears on both s i d e s of the e q u a t i o n . The f i r s t term in E q . ( 1 ) , t / d , i s simply the l e n g t h l o s s due to steady s t a t e hydrodynamic flow. The second term, &t/d, i s a small a d d i t i o n a l l o s s due to a non-steady s t a t e p r o c e s s t h a t i s p o s s i b l y a s s o c i a t e d with the i n i t i a l shock phase of the impact p r o c e s s . For most c a s e s of p r a c t i c a l i n t e r e s t (t/d>O.05), t h i s term can be taken as a c o n s t a n t , however, as noted below, t h i s i s not s t r i c t l y t r u e . The t h i r d term, involving v~/d, i s the primary n o n - s t e a d y s t a t e l o s s a s s o c i a t e d with the f i n a l breakout phase of the p e r f o r a t i o n p r o c e s s . The value of v~/d can be computed from the equation v~Id - 1.98 v /
/ [c2 - (us-v/2)2 ]
(2)
where c is the sound speed in the shocked rod material and us is the shock velocity.
These
quantities can be obtained using either published equation-of-state data or coefficients the approximation that the shock velocity is a linear function of the particle velocity.
for
It should be noted that for thick target plates (tld>2) the rod loss approaches a linear dependence on t/d given by Al/d.tld+Atld+vzld. For very thin target plates (t/d<0.1) the term At/d in gq.(1) should be replaced by the expression (At/d)[1-exp(-~t/d)] so that the rod loss approaches zero as t/d approaches zero. The effect of this modification is however n e g l i g i b l e for t/d>O.05.
Rod ~ l o s s f o r u n l i k e - m a t e r i a l impact. For an impact c o n f i g u r a t i o n i n which the p'r'oJectile ~ t-'a~get p l a t e are of d i f f e r e n t m a t e r i a l s , the rod l o s s can be s t i l l be computed u s i n g a m o d i f i c a t i o n of E q . ( 1 ) . By analogy with the theory of steady s t a t e , incompressible f l u i d flow which p r e d i c t s that f o r unlike f l u i d s , the p e n e t r a t i o n v a r i e s as the square root of the d e n s i t y r a t i o of the two f l u i d s , so too, does the l e n g t h of rod l o s t depend on the rod and target materials, i.e. (/11/d)i~2 = / ( p 2 / O l ) (A1/d)l~l
(3)
101
102
J. Baker and A. Williams
Figure i shows the result of applying Eq.(3) to Eq.(1) for a wide material combinations.
range
of
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R_~od l e n _ ~ loss for ~ i~act. For oblique impa~ts in which the angle between the rod axls anu ~ ~a'{get surface (8) is less than 90 , the impact process behaves as if the target thickness is greater thaff its nominal value. This additional thickness arises because of the greater distance which is traversed by the rod in passing through the target and is given by a factor of llsin(8). Also, because of the finite diameter of the rod, the point of contact between the rod and the target (and therefore the effective thickness of the target) is extended by a distance d/2tan(O). Therefore Eq.(1) can still be applied to oblique impacts if the target thickness, t/d, is replaced by its effective thickness given by (4)
(t/d)ef f = (t/d)/sin(e) + 1/[2tan(e)]
A yawed rod is one in which there is an angle between the velocity vector o~ the rod and its axis. For the most part, the NgL data was limited to yaw angles below 10 and it is within that range that the results given here are valid and independent o f yaw angle. The NRL experiments also showed that, for yaw angles greater than 15v, the rod was completely destroyed. See Bertke et al (1973) for additional information on yawed rod craterlng. Hole diameter for normal impact. The target plate hole diameter approaches the diameter of the ro-'6"~a~ target pa-l"~'e'-become"'----'s"-very thin, and, conversely, the diameter of the hole approaches the'dlameter of a crater in a semi-lnfinlte target as the target plate becomes very thick. The following analytical expression provides such a dependence: D/d = 1 ÷ (D~/d - i) [1 - exp (-kt/d)]
(5)
where k is a constant that depends on the rod and target materials and D./d, the corresponding crater diameter, has been shown by Baker (1969) to be, very nearly, a llnear function of velocity, D d d = 1 + or, so that Eq.(5) becomes D/d = 1 ÷ a~v [I - exp (-kt/d)]
(6)
where o= is a constant which depends on the rod and target materials. for an aluminum rod and a variety of target plate materials.
Figure 2 shows Eq.(6)
More recently, Baker (1974) analyzed some crater diameter data which was better expression of the form
fitted
by
S~/d = I + Our [i - (re/V)[1 - exp (-V/re)]}
(7)
where ~o and v o are constants for a given combination of projectile and target materials. alternate equation for crater diameter is especially appropriate difference in the densities of the target and projectile materials.
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a
This large
Hole diameter for ~ impact. For an oblique impact, the hole in the target plate is ~Is'~orte--"eq"'-~to-"-an a p p r o ~ y elliptical shape in which the major diameter lies along the trajectory of the rod. The equation for the major diameter is given by purely geometrical considerations while the minor diameter was shown experimentally to have a weak dependence on the impact angle, 8. Dmaj0 r
D/d
Dmino r
d
sin(8)
d
D/d
(B)
[sin(8)] 1/5
Multiple Plate Perforation Rod length loss for spaced, multiple platetargets. For a t a r g e t consisting of a series of s--~g~pac-'6"~ p-lates,----'~-~rod length Yossa, tdtafget plate hole diameter equations can simply be applied in a sequential manner to each plate individually using the residual rod from the p r e c e d i n g p l a t e . Note t h a t , because of a lack of s u f f i c i e n t experimental data to e s t a b l i s h an anlytical correlation, the velocity of the rod is assumed to remain undlmlshed throughout the impact process.
Rod l e n g t h l o s s form m u l t i l a y e r t a r g e t s . In 1970 Baker and Zalesak extended the s i n g l e p l a t e a"~lysis t~ ~ multilayer target using the concept of "corresponding layers." This concept asserts that the rod length lost in a particular layer of a multilayer target is given by the incremental loss that would occur, according to Eq.(1), in passing through the corresponding
103
H)q~erveloclty penetration of plate targets
p o r t i o n ( i . e . , t h a t p o r t i o n l o c a t e d in the same r e l a t i v e p o s i t i o n ) in a t a r g e t which has a t h i c k n e s s equal to t h a t of the l a y e r being c a l c u l a t e d plus a l l the preceding l a y e r s and which i s composed e n t i r e l y o f the same m a t e r i a l as the l a y e r being c a l c u l a t e d . This approximation assumes t h a t the rod p e n e t r a t e s each l a y e r e n t i r e l y under steady s t a t e c o n d i t i o n s which v i l i be n e a r l y t r u e as long as each subsequent l a y e r i s of l e s s e r d e n s i t y than the preceding one. When a subsequent l a y e r has a higher d e n s i t y than the preceding one, then the p r e s s u r e a t the r o d - t a r g e t i n t e r f a c e must undergo a t r a n s i t i o n from one steady s t a t e p r e s s u r e i n the f i r s t l a y e r to a new, h i g h e r p r e s s u r e in the second. The s t r e n g t h of t h i s e f f e c t v i i i depend on the r e l a t i v e magnitudes of the two s t e a d y s t a t e p r e s s u r e s . The g r e a t e r the d i f f e r e n c e in p r e s s u r e s , the more n e a r l y the rod l o s s in the subsequent l a y e r w i l l approach t h a t of a s i n g l e p ] a t e r a t h e r than t h a t of a corresponding l a y e r . The following equation e x p r e s s e s t h i s relationship: ~l/d = (A1/d)c 1 + [(A1/d)s p - ( ~ l / d ) c l ] (1 - P s l 2 / P s l 3 )
(9)
vhere ( ~ l / d ) c l i s the corresponding l a y e r rod l o s s , ( ~ l / d ) s p i s the s i n g l e p l a t e rod l o s s , and Psnm i s the i n c o m p r e s s i b l e steady s t a t e s t a g n a t i o n p r e s s u r e f m a t e r i a l n p e n e t r a t i n g i n t o m a t e r i a l m and m=2 r e f e r s to the preceding l a y e r and m=3 r e f e r s c a l c u l a t e d . The s t e a d y s t a t e p r e s s u r e i s given by
to
the
current
layer
Psnm " ~°mv2 1 [I + /(0mlPn)] 2
being (I0)
vhere o i s the m a t e r i a l d e n s i t y . In a d d i t i o n , i f the preceding l a y e r i s r e l a t i v e l y t h i n (t/d
(II)
the e f f e c t i v e t h i c k n e s s of the t a r g e t , i s the d i s t a n c e from the c u r r e n t l a y e r to
the l o c a t i o n of the l a s t s t r o n g shock ( e i t h e r the i n i t i a l impact point or a p r e c e d i n g i n t e r f a c e with a l a r g e d e n s i t y change, nominally a f a c t o r of two or more, whichever d i s t a n c e i s l e s s ) and (A1/d) e i s the rod l o s s for a s i n g l e p l a t e of t h i c k n e s s ( t / d ) e of the same m a t e r i a l as the rod. Denoting the s t e a d y s t a t e f a c t o r by H123, then the rod l o s s f o r each l a y e r of a m u l t i l a y e r t a r g e t i s given by ~111d = ( ~ l l d ) c I + [ ( b l l d ) s p - ( ~ l l d ) c l ] ~123A
for 03 > p2
(12)
•l/d = (Alld)c I
f o r 03 < o2
(13)
and The t o t a l rod l o s s i s given by sununlng the l o s s e s f o r each l a y e r . Rod l e n g t h l o s s f o r obllque impact i n t o m u l t l l a y e r t a r g e t s . As for s i n g l e p l a t e s , an o b l i q u e ~acT--i-nt~'mu~t-[layer target--~--'s~ou-l~--useEqs.(12)~'-'~--~) vith an effective target thickness similar to that of Eq.(4). It must be recognized, however, that the second term in Eq.(4), 1/[2tan(e)], is independent of target thickness and arises from the geometry of impacting the front surface of the target. Therefore this term is modified so that it enters into the calculation only to the degree that the transition between layers is llke a new impact as expressed by the steady state factor ~2~, and is furthermore modlfied by the factor A for similar reasons. The equation for th@ ~ffectlve thickness of an oblique, multllayer target is therefore given by (t/d)ef f = (t/d)/sin(e) + H123 A /
[2tan(e)]
for o3 > p2
(14)
for p3 ~ p2
(15)
and (t/d)ef f = (tld)/sln(8)
Hole d i a m e t e r f o r m u l t i l a ~ e r t a r g e t s . Lacking s u f f i c i e n t experimental data to determine the e--le'~c~da~ent l a y e r s of a m u l t i l a y e r t a r g e t on the h o l e diameter, the s i n g l e p l a t e h o l e d i a m e t e r e q u a t i o n s a r e used as an approximate value f o r the hole diameter in each l a y e r of the m u l t i l a y e r t a r g e t . There i s some evidence that the m u l t i l a y e r hole diameters tend to be l a r g e r than the s i n g l e p l a t e v a l u e s , perhaps because of the p e t a l l i n g and o t h e r s t r u c t u r a l e f f e c t s t h a t occur i n the m u l t i l a y e r t a r g e t . The s i n g l e p l a t e hole diameter should t h e r e f o r e b e . c o n s i d e r e d to be a minimum e s t i m a t e of hole s i z e .
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J. Baker and A. Williams
S i n g l e P l a t e Craterin@ An impact will create a crater'in the target when the rod projectile either comes to rest or is totally consumed before it perforates the target. The impact analysis is then primarily concerned with the depth and ~iameter of the crater and 0nly secondarily with any residual rod length. ~
state craterin~ model. Tate (1967 and 1969) proposed a model of rod cratering in which i~1"~-Bernoulli'se"~'~ion for incompressible, steady state flow by adding, a dynamic
strength, for each material to both sides of the equation, i.e., ~Op(V-U)2+ao-~ ~Ptu2+at , where v is the projectile velocity, u is the velocity of the interface, pp is the projectile density, and Pt is the target density.
The final target penetration, p, consists of a
hydrodynamic phase (if the impact velocity is high enough) which is calculated by integrating the term u/(u-v) over the initial length of the projectile, 1 (in general this must be done numerically) plus a rigid projectile phase (if the dynamic strength of the rod is greater than that of the target). The Tate model shows good agreement with experimental data below about 2 km/s, however in the hypervelocity regime, the crater depth of the model becomes independent of velocity and asymptotes to the steady state limit given by the initial rod length times the square root of the density ratio. This does not agree with the hypervelocity data presented by Christman and Gehring (1965) in which the penetration continues to rise above the steady state limit with increasing impact velocity. This divergence from Tate's one-dimensional, steady state model is most likely due to the two-dimensional, non-steady effects which dominate the final phase of the crater formation. Modification of the Tate model. Christman and Gehring (1966) proposed an empirical model of rod cratering'~n--~ic-h"~he penetration occurs hydrodynamically for all but the last diameter of rod length and that that last diameter of length creates a hemispherical crater charateristic of a chunky fragment impact. Their model of crater penetration depth is given by p/d = /(pp/Pt)(i/d - 1) + ~ D~/d.
(16)
The factor ~ implies that the chunky fragment portion of the crater is always hemispherical, but that is only true if the projectile and target are of the same material. In general, the f i n a l p o r t i o n of the c r a t e r v i i i be e i t h e r g r e a t e r than or l e s s than h e m i s p h e r i c a l depending on the relative densities of the target and projectile. Therefore the factor ~ should be replaced by an expression such as ~(pp/Pt )m.
If it is assumed that this dependence on the density
ratio also goes as the square root (m=~) then Eq.(16) for rod penetration depth becomes p/d = /(Op/Pt)(i/d - 1 + ~ DJd)
(17)
Baker (1974) observed that the difference betweep the simple empirical model of Eq.(17) and the hydrodynamic limit ~o which the Tate model asymptotes is the second factor in parentheses which multiplies the square root of the density ratio. He therefore suggested that the Tate model could be modified to have the correct dependence at high velocity by simply multiplying the square root of the material density ratio, /(pp/Pt ), by the factor ~ = 1 - ~ (1-a~v)/(lld). When this modification is introduced into Tate's analysis, it not only has the desired effect of having the penetration continue to increase at the higher velocities to values above the steady state limit but also leaves the lower velocity behavior and the residual rod length virtually unchanged. In addition, the modification introduces a dependence on the initial rod length such that the shorter the initial rod, the greater viii be the deviation of the penetration from the original Tare model at high velocities. Figure 3 compares the penetration (normalized by the inital rod length) versus velocity calculated from the original Tare model with that calculated from the empirically modified model. The values of the dynamic strengths of the rod and target materials differ for each combination off materials and have been selected so that the modified model best fits the experimental data points (also shown in the figure) which are taken from Christman and Gehring (1965) for a C1015 steel rod with an i/d of three impacting an 1100-0 aluminum target.
Hypervelocity penetration.of plate targets
I05
ROD-LIKE PROJECTILES Having developed a reasonably good understanding of the penetration process of a rod projectile, the next step was to investigate the possibility of designing a projectile which would be an even more e f f i c i e n t p e n e t r a t o r than a rod. The f i v e d e s i g n s which were examined a r e i l l u s t r a t e d i n F i g . 4 and a r e d e s i g n a t e d as TYPES 1 t h r o u g h 5. The b a s e l i n e rod p r o j e c t i l e i s d e s i g n a t e d as TYPE 0. Since the s h a p e s of the r o d - l i k e p r o j e c t i l e s a r e a l l v e r y d i f f e r e n t , t h e common measure of t h e i r p e n e t r a t i n g a b i l i t y was taken to be the amount of mass ( r a t h e r t h a n l e n g t h ) l o s t by each p r o j e c t i l e i n p e r f o r a t i n g the same t a r g e t . Application of Rod Analysis to Rod-like Projectiles The rod analysis can be applied to the rod-like projectile designs by using the following procedures (I) Calculate the steady state projectile length loss for penetrating a plate of t h i c k n e s s , t , tXlss - t / ( P t / P p ) ; (2) C a l c u l a t e the s t e a d y s t a t e mass l o s s , &~ss' c o r r e s p o n d i n g to the above l e n g t h l o s s , which i s a s i m p l e g e o m e t r i c problem t h a t depends on the p r o j e c t i l e d e s i g n ; . ( 3 ) C a l c u l a t e the d i a m e t e r a t the f r o n t of the r e s i d u a l p r o j e c t i l e , d r , a f t e r removing
the amount of projectile lost by steady state; (4) Use t/d r in Eq.(1) without the tld term to determine the non-steady state mass loss, ~mnss, that a rod projectile of diameter d r would experience.
To do this, calculate the total length loss for a rod of diameter d r impacting a
plate of thickness t and then subtract the steady state length loss obtained in step 1, i.e., (AI - Alss )nppd:14; (5) The total mass loss is then ~m = ~mss + hnnss; (6) Calculate ~mns s the total length loss corresponding to the total mass loss and then compare it to the measured experimental length loss. The key to this procedure is the assumption in step 4 that the diameter of the projectile after perforating the target determines the amount of continued plastic flow which produces the additional projectile mass loss. This is a reasonable assumption and a comparison of the test results with the length loss calculations shows good agreement. D i s c u s s i o n of Each R o d - l i k e P r o j e c t i l e
Design
TYPE I projectile. The TYPE i projectile design, a truncated cone, is subdivided into two categories designated as TYPE 1A with small values of the nose diameter and TYPE 1B vlth larger values of the nose diameter. TYPE IA projectile. The projectile-target material combinations were aluminum-aluminum, s-'~'e"el--"aluminum, steel-titanium, and tungsten carbide-steel. In general, only for the steel-alumlnum tests with small yaw angles did the projectile survive the severe lateral forces of the impact. A p l o t o f the c a l c u l a t e d f r a c t i o n a l mass l o s s v e r s u s t a r g e t p l a t e t h i c k n e s s f o r b o t h the TYPE 1A and rod p r o j e c t i l e s i s shown in F i g . 5 . The r a t i o o f cone mass l o s s to rod mass l o s s i s o b v l o u s l y s m a l l f o r v a l u e s o f t / d l e s s than about two and the r a t e of mass l o s s f o r t h e cone r e m a i n s l e s s than t h a t f o r the rod f o r v a l u e s of t / d up to a p p r o x i m a t e l y s e v e n , above which t h e r a t e of l o s s f o r b o t h a r e a p p r o x i m a t e l y e q u a l .
For a spaced target configuration, the TYPE IA projectile becomes a TYPE iB after impacting the first target plate. The TYPE IA projectile would therefore be particularly effective against a target configuration in which the first plate is thick and made of a low density material. TYPE 1B projectile. Two of the impacts which failed with the TYPE IA projectile were successful w i t h the TYPE 1B p r o j e c t i l e which has a l a r g e r nose d i a m e t e r . I f the t a r g e t h o l e d i a m e t e r i s g r e a t e r than the d i a m e t e r of the p r o j e c t i l e b a s e t h e n t h e r e s h o u l d be no s i d e c o n t a c t between the p r o j e c t i l e and the t a r g e t . A p l o t o f the c a l c u l a t e d f r a c t i o n a l mass l o s s versus target plate thickness for a steel TYPE IB and rod projectile impacting a titanium target is shown in Fig.6.
TYPE 2 p r o j e c t i l e . The p r i m a r y a d v a n t a g e of the TYPE 2 p r o j e c t i l e , a pointed rod, is its s-'~ITclty. T h i s d e s i g n was i n t e n d e d f o r l i k e - m a t e r i a l impact i n the e x p e c t a t i o n t h a t t h e p o i n t e d n o s e reduce the e f f e c t o f the impact shock which would more than compensate f o r t h e s i d e c a v i t a t i o n mass l o s s . A p l o t of the c a l c u l a t e d f r a c t i o n a l mass l o s s v e r s u s ~ a r g ~ t p ~ a t e t h i c k n e s s f o r s t e e l rod and TYPE 2 p r o j e c t i l e s w i t h t h r e e cone t i p a n g l e s , ~ = 15v,30v,60 v, i m p a c t i n g a s t e e l t a r g e t i s shown i n P i g . ? . The c a l c u l a t i o n o f the s t e a d y s t a t e mass l o s s f o r the p o i n t e d rod w i l l eha~ge from a cone volume e v a l u a t i o n to i n c l u d e a rod volume e v a l u a t i o n i f the s t e a d y s t a t e l e n g t h l o s s exceeds the l e n g t h of the c o n i c a l t i p . T h i s change in mass l o s s r a t e i s the knee in the g r a p h o f F i g . ? .
106
J. Baker and A. Williams
TYPE 3 p r o j e c t i l e . The c o n c e p t o f t h e TYPE 3 p r o j e c t i l e i s t h a t t h e l e a d i n g s m a l l e r diameter ~portion ( s n o u t rod) v i l I produce a hole in the target plate sufficient to p a s s t h e remaining larger d i a m e t e r rod v i t h o u t any a d d i t i o n a l projectile loss. Tvo s n o u t rod configurations v e r e u s e d . One had a s n o u t d i a m e t e r t h a t v a s h a l f o f t h e main rod d i a m e t e r and a s n o u t l e n g t h t h a t v a s f o u r d i a m e t e r s lonE. The s e c o n d c o n f i g u r a t i o n had a s n o u t d i a m e t e r that vas tvo-thlrds o f t h e ~ a t n rod d i a m e t e r and a s n o u t l e n g t h t h a t v a s f o u r and a h a l f d i a m e t e r s l o n g . A p l o t o f t h e c a l c u l a t e d f r a c t i o n a l mass l o s s f o r a s t e e l rod and t h e t v o TYPE 3 p r o j e c t i l e s i m p a c t i n g a s t e e l t a r g e t i s s h o r n in F I E . 8 . TYPE 4 p r o j e c t i l e . The TYPE 4 p r o j e c t i l e i s a c o m b i n a t i o n o f a h i g h s t r e n g t h TYPE 1A t r u n c a t e d cone p r o j e c t i l e and a l o v ~ e n s i t y s l e e v e . The f i v e s u c c e s s f u l t e s t s had a 4340 s t e e l cone h a r d e n e d to Rockwell "C" 55 v i t h a Z e l u x ( p o l y c a r b o n a t e plastic) sleeve with a density of 1.19 g / c c . The r a t i o o f t h e s t e e l cone n o s e to b a s e d i a m e t e r s v a s 0 . 0 8 v h i l e t h e r a t i o s o f Z e l u x s l e e v e d i a m e t e r to cone b a s e d i a m e t e r v e r e 1 . 0 , 0 . 6 6 , and 0 . 5 8 . The i m p a c t e f f e c t o f t h i s c o m b i n a t i o n p r o j e c t i l e i s c a l c u l a t e d a s t h e sum o f t h e impact part. The s l e e v e i s modeled a s a s h o r t rod w i t h a d i a m e t e r and a l e n g t h s u c h t h a t t h e t h e s h o r t rod i s e q u i v a l e n t to t h e s l e e v e m a s s . The d e s i r e d effect of the sleeve creation o f a h o l e i n t h e t a r g e t w i t h s u f f i c i e n t d i a m e t e r and d e p t h to p r e v e n t c o n t a c t t h e h o l e w a l l and t h e c o n i c a l p r o j e c t i l e b a s e .
of each mass o f is the betveen
A p l o t o f t h e c a l c u l a t e d f r a c t i o n a l mass l o s s f o r a s t e e l rod and t v o TYPE 4 p r o j e c t i l e s impacting a steel target is shorn in Pig.9. Due to t h e low s t r e n g t h o f t h e s l e e v e m a t e r i a l , t h e c a l c u l a t i o n s a s s u m e t h a t t h e e n t i r e s l e e v e i s d e s t r o y e d by t h e i m p a c t , e v e n f o r v e r y t h i n targets. The s t e e l cone l o s e s mass a c c o r d i n g to t h e p r o c e d u r e f o r t h e TYPE 1A p r o j e c t i l e . The mass l o s s f o r t h e TYPE 4 p r o j e c t i l e i s t h e r e f o r e t h e s l e e v e mass p l u s t h e cone mass l o s s . TYPE 5 p r o j e c t i l e . The TYPE 5 p r o j e c t i l e c o n s i s t s o f two m a s s e s : a l e a d i n g "bumper" and a l o n g ~'~oving rod v h i c h a r e c o n n e c t e d by a s m a l l e r d i a m e t e r r o d , o r s t e m . T h i s p r o j e c t i l e c o n f i g u r a t i o n i s an e f f i c i e n t one f o r p e r f o r a t i n g a t / d - 2 t a r g e t o f l i k e - m a t e r i a l . The i m p a c t characteristics o f t h e bumper ( 1 / d - l ) a r e e s s e n t i a l l y t h e same a s t h a t o f a s p h e r e o f e q u a l mass.
For v e r y t h i c k t a r g e t s , t h e t a r g e t damage due to bumper impact i s t h e same a s c r a t e r formation in a semi-inifinlte target. The bumper v I I l be d e s t r o y e d by t h e impact s h o c k i n a l l c a s e s e x c e p t f o r low p r o j e c t i l e v e l n c i t l e s o r v e r y t h i n t a r g e t s ( t / d - O . 0 1 ) . The f r a g i l e connecting stem is also likely to be d e s t r o y e d i n most impact s i t u a t i o n s . The mass l o s s o f t h e TYPE 5 projectile i s t h e r e f o r e t h e sum o f t h e bumper and s t e m m a s s e s p l u s t h e mass l o s s o f t h e following rod. A p l o t o f t h e c a l c u l a t e d f r a c t i o n a l mass l o s s f o r a s t e e l rod and TYPE 5 p r o j e c t i l e s impacting a steel t a r g e t i s shown i n F i g . l O . Note t h e g r e a t e r e f f e c t i v e n e s s o f t h e bumper rod V l t h t h e larger diameter of the connecting stem.
CONCLUSIONS Ro__.ddP r o j e c t i l e s The t e r m i n a l b a l l i s t i c r e s u l t s o f a h y p e r v e l o c i t y rod i m p a c t i n g a s i n g l e t a r g e t p l a t e h a s been successfully q u a n t i f i e d w i t h a f e v , r e l a t i v e l y s i m p l e , a n a l y t i c e x p r e s s i o n s from v h i c h one c a n c a l c u l a t e t h e l e n g t h o£ t h e r e s i d u a l rod and t h e d i a m e t e r o f t h e h o l e i n t h e t a r g e t p l a t e g i v e n the value of a fev empirically determined constants v h i c h depend on t h e rod and t a r g e t materials. T h e s e a n a l y t i c c o r r e l a t i o n s have been shown to be v a l i d f o r m a t e r i a l combinations r a n g i n g from a v e r y d e n s e u r a n i u m rod i m p a c t i n g a low d e n s i t y aluminum t a r g e t p l a t e to a l o v d e n s i t y aluminum rod i m p a c t i n g a h i g h d e n s i t y u r a n i u m t a r g e t p l a t e . The e f f e c t o f t a r g e t p l a t e obliquity h a s a l s o been q u a n t i f i e d . The s i n g l e p l a t e e q u a t i o n s can be r e a d i l y a p p l i e d , i n a s e q u e n t i a l manner, to a t a r g e t c o n s i s t i n g o f a s e r i e s o f s p a c e d p l a t e s . F u r t h e r m o r e , t h e s i n g l e t a r g e t p l a t e a n a l y s i s h a s been e x t e n d e d to a m u l t l l a y e r e d t a r g e t using a c o n c e p t c a l l e d " c o r r e s p o n d i n g l a y e r s " a l o n g v i t h some e a s i l y computed f a c t o r s v h i c h c o n t r o l t h e e x t e n t to v h i c h t h e p e n e t r a t i o n process diverges from t h a t f o r a s i n g l e plate and approaches that of a "corresponding layer." The rod p r o j e c t i l e a n a l y s i s c o n c l u d e s v i t h an a n a l y t i c a l model o f c r a t e r p e n e t r a t i o n d e p t h and residual rod l e n g t h . The p r o p o s e d model s l i g h t l y m o d i f i e s t h a t o f A. T a r e by i n t o d u c i n g a factor vhich multiplies the density ratio. This factor, vhich is velocity dependent, permits the calculated crater penetration d e p t h to r i s e above t h e s i m p l e h y d r o d y n a m i c l i m i t a t v e r y h i g h v e l o c i t i e s i n a g r e e m e n t v i t h t h e e x p e r i m e n t a l d a t a o f C h r i s t m a n and G e h r i n g .
Hypervelocity penetration of plate targets
107
Rod-like Projectiles The five rod-like projectile designs, illustrated in Fig.4, were chosen to provide a significant reduction in the projectile mass loss, compared to that of a rod projectile, for particular impact configurations. This improved performance is due to one or more of the following mechanisms: (1) a smaller nose diameter having less mass per unit length than a rod but which s t i l l c r e a t e s a l a r g e enough c l e a r a n c e h o l e ; (2) a p r o j e c t i l e of two independent s e c t i o n s with the l e a d i n g , l o v e r mass one sacrificed to create a clearance hole for the other; (3) a projectile which is strong enough to create a hole in the target without suffering any damage to itself. The most effective of the new projectile designs is the TYPE IA sharp steel cone. For all but very thick targets the projectile mass losses are n e g l i g i b l e . Hoverer, t h e impact configuration is somewhat limiting as it requires a weak, low density target and a h i g h strength projectile. The TYPE 1B t r u n c a t e d cone and the TYPE 3 s n o u t rod p r o j e c t i l e s s h o u l d be p a r t i c u l a r l y effective against thin targets since the mass loss would be a function of a small projectile nose diameter. When the t a r g e t t h i c k n e s s i s such t h a t the s t e a d y s t a t e l e n g t h l o s s i s g r e a t e r than t h e l e n g t h of the c o n i c a l p o i n t , the r e d u c t i o n in mass l o s s f o r the TYPE 2 p r o j e c t i l e over t h a t f o r t h e rod will simply be trice the mass of the conical point. For thinner targets the TYPE 2 projectile suffers an unknown amount of side erosion to the conical point during target penetration and the modified rod loss theory will therefore predict low values of mass loss. The target penetration efficiency of the TYPE 5 bumper rod projectile is due to the decoupling of the shock and plastic flow in the bumper from the following rod. The connecting stem must provide a margin of time to allow for sufficient flow of the target debris such that the following rod does not encounter any significant amount of target material. The connecting stem may also act like the snout of a TYPE 3 projectile, punching a clearance hole in any remaining target thickness. The TYPE 4 s l e e v e / c o n e p r o j e c t i l e combines the p e n e t r a t i o n mechanisms of the TYPE 1 and TYPE 5 projectiles. For t h i n t a r g e t s the low d e n s i t y s l e e v e simply punches a c l e a r a n c e h o l e in t h e target while for thick targets the sleeve creates a crater of sufficient diameter and depth to g i v e h o l e c l e a r a n c e f o r the p r o j e c t i l e base and to allow the s t e e l cone to o p e r a t e a s a TYPE 1B p r o j e c t i l e which p e n e t r a t e s the r e m a i n i n g t a r g e t t h i c k n e s s . The s t e e l cone must be s t r o n g enough to withstand the shock raves generated by the impact of the low density sleeve. Summary T h i s paper h a s summarized the r e s u l t s of e x p e r i m e n t a l t e s t s and a n a l y t i c a l s t u d i e s of t h e h y p e r v e l o c i t y impact of rod and r o d - l i k e p r o j e c t i l e s which were conducted a t the Naval R e s e a r c h L a b o r a t o r y . The r e s u l t s p r e s e n t e d h e r e provide r e l a t i v e l y s i m p l e a n a l y t i c e x p r e s s i o n s from which one can c a l c u l a t e the r e s u l t s of a h y p e r v e l o c i t y impact of a rod or r o d - l i k e p r o j e c t i l e even i n t o complex t a r g e t s under most impact c o n f i g u r a t i o n s of i n t e r e s t . The methodology does r e q u i r e a knowledge o f c e r t a i n e m p i r i c a l c o n s t a n t s which depend on t h e p r o j e c t i l e and t a r g e t materials. For those cases where the values of these constants have not been provided, they can easily be determined by performing a relatively few experimental impacts.
108
J. Baker and A. Ni111ams
REFERENCES
Baker, J. (1969).
See author for complete reference.
Baker, J. and Zalesak, S. (1970). Baker, J. (1974).
See author for complete reference.
See author for complete reference.
Bertke, R., Seyda, J., Svift, S. and Lehman, H. (1974). A Study of Yaved Rod Impacts. University of Dayton Research Institute, U.S. Army Ballistic Research Laboratory Report No. BRL-CR-182. Christman, D. and Gehrlng, J. (1966). Analysis of High-Velocity Projectile Penetration Hechanics. J...cAppl. Phys.p 3.77,1579-1587.' Tare, A. (1967). A Theory for the Deleceratlon of Long Rods after Impact.
J. Mech. Phys. Solids., 15, 387-399. Tate, A. (1969). Further Results in the Theory of Long Rod Penetration. J. Hech. Phys. Solids., 17, 141-150.
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