D effect for long-rod penetrators

Int. J. lmpactEngng, Vol. 17, pp. 13 24, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0734-743X/95 $9.50+0.00

Pergamon

ON THE VELOCITY DEPENDENCE OF THE L/D EFFECT FOR LONG-ROD PENETRATORS

Charles E. Anderson, Jr.*,James D. Walker*, Stephan J. Bless*, and T. R. Sharron" *Southwest Research Institute, P.O. Drawer 28510, San Antonio, TX 78228 *Institute for Advanced Technology, The University of Texas at Austin, Austin, TX 78759

S u m m a r y - - A t ordnance velocities (1.0 - 1.9 km/s), there is a pronounced decrease in penetration efficiency, as measured by P/L, when projectiles of larger L/D are used. The influence of L/D on penetration is referred to as the L/D effect. We numerically examine the L/D effect at higher velocities, from 2.0 km/s to 4.5 krn/s. It is found that as the velocity increases, there is a change in mechanism for the L/D effect. At ordnance velocities the L/D effect is mostly due to the decay in penetration velocity during the "steady-state" region of penetration. At higher velocities, the steady-state region of penetration shows no L/D dependence, and the L/D effect is due primarily to the penetration of the residual (non-eroding) rod at the end of the penetration event. This change in mechanism is related to the change in slope of the penetration-versus-impact velocity "S-shaped" curve for eroding projectiles. NOTATION a l , a2

bE, bL co D g k l

l,

regression coefficients regression coefficients bulk sotmd speed projectile diameter proportionality coefficient for terminal phase penetration equation of state constant instant~uaeous projectile length instan~aeous eroded length of projectile [L-/]

t,

-l

l g, Lo L,

projectile erosion rate [-(v - u)] initial projectile length eroded length of projectile at beginning of temainal penetration phase residual projectile length instantaneous depth of penetration

Lr P

P P

U

depth of penetration

P u v V Vo f'

regression coefficient of determination penetration velocity projectile tail velocity impact velocity 1.0 km/s V/Vo

ep ~p Fo Po pp Pt 0 0m

plastic strain plastic strain rate Griineisen coefficient initial (ambient) density projectile density target density temperature melt temperature

0°

homologous temperature

INTRODUCTION A measure of penetration efficiency is the depth of penetration P into a semi-infinite target, normalized by the initial length of the projectile L. Typically, P/L is plotted as a function of impact velocity, for example, see Hohler and Stilp [1]. The L/D effect--the decrease in penetration efficiency with increasing aspect ratio--has been studied fairly extensively for projectiles with small aspect ratios [2-4]. Recent studies have examined the penetration efficiency of long-rod projectiles--usually defined as projectiles that have L/D > 10--to quantify the magnitude of the L/D effect [5-7]. In particular, Anderson, et al. [7], examined a wide variety of data from various sources for tungsten-alloy projectiles into armor steel. The penetration efficiency as a function of L/D is shown for three impact velocities (1.2, 1.5, and 1.8 km/s) in Fig. 1. The open symbols represent experimental data from various sources. The results from numerical simulations, also shown in the figure as the solid symbols, agree very well with the experimental data. This last observation is important since it implies that no new physics must be invoked to explain 13

14

C.E. ANDERSON, JR et al.

,,,I,,,,I,,,,I,,,,I,,,,I,,,,I,''1''11 1.55

1.6

''''1''''1''''

I''''1'''' I-: ~

Bar Target

1.4

'~1.8 km/I s

1.2

4e

1.50

1.0

_..e..__

•

_-. -- -e--

L/D=20 Regression F i t "

0.8

°.6I

1.45

0.4

0.2

, , . t . . . . a . . . . t . . . . i . . . . i . . . . i . . . . i . . . . I,q

0.0 0

5

10

15

20

25

30

35

40

1.40 I , 2.85

•

IJD = 15

•

L/D = 20

Im

L/D = 30

,,,I,,,,I,,,,I,,,,I,,, 2.90

L/D

2.95

3.00

3.05

3.10

V(km/s)

Fig. 1. P/L versus L/D at ordnance velocities (from Ref. [7])

Fig. 2. Data from Sorensen, et al. [8]

the L/D effect. The solid lines represent a two-parameter regression fit for P/L as a function of normalized impact velocity and/_/D: P L

-

0.212+1.044!2-0.194In(L1

~D)

(1)

where 12=V/V o, and 1/o is 1.0 km/s; the root mean square error between the data and Eqn. (1) is 0.034, with an re = 0.996. Equation (1) is valid for tungsten-alloy projectiles into roiled homogeneous armor (RHA), or very similar armor-type steels, in the conventional ordnance velocity range (- 0.75 < V < - 1.9 km/s). However, virtually all the studies to date have only examined the/JD effect for long-rod projectiles at ordnance velocities. Sorensen, et al. [8], obtained experimental data for L/D 15, 20, and 30 projectiles in the 2-3 km/s range. They performed statistical tests on their data to investigate the dependence on L/D for L/D's 15 to 30, and reported: "...the difference between the L/D 15 and the L/D 30 data, where such a difference would be the most observable, was insignificant." A subset of the data--only those data with impact velocities of approximately 3.0 km/s--from Ref. [8] are plotted in Fig. 2. The legend in the figure distinguishes the different projectile aspect ratios. A dashed line, which represents a least-squares polynomial curve fit through a fairly large set ofL/D = 20 data [9] is also shown. One of the/_/D 30 data points, denoted by the open symbol, lies above the other two L/D 30 data points; however, this point is associated with a slightly different target. The bar target consisted of a RHA bar, 150 mm square by 600-mm long, cut from RHA plate; penetration was in the rolling plane rather than normal to it. The remainder of the targets consisted of eight 75-mm RHA plates welded together at the comers. Neglecting the data point from the bar target l, the trend in the data suggests that a slight L/D effect persists at higher impact velocities. Bless and Hamburger [10], in experiments with 96-g, L/D 15 rods at 2.63 and 2.67 km/s, found P/L = 1.44. This value is 3% greater than P/L measured at these velocities for 383-g, L/D 30 rods in Ref. [9]. It was decided to extend the analysis performed at ordnance velocities [7] to investigate if the L/D effect persists at higher impact velocities, and if it does, to quantify the magnitude of the effect. Numerical simulations provide the basis for this study. NUMERICAL S I M U L A T I O N S - - I N I T I A L CONDITIONS AND P/L RESULTS The nonlinear, large deformation Eulerian wavecode CTH [11] was used to investigate penetration by projectiles of varying aspect ratios. The 2-D cylindrically symmetric option of CTH was used to simulate the projectile-target interaction. CTH uses a van Leer algorithm for second-order accurate advection that has been generalized to account for a non-uniform and finite grid, and multiple materials; CTH has an advanced material interface algorithm for the treatment of mixed cells. A Mie-Grtineisen equation of state

1 Anotherdifferenceis alsonoted. The nominalmassof the projectilefiredat the bar targetwas 250 g. The otherdata displayed in Fig. 2 wereobtainedfromnominal 125-gprojectiles.

Onthe velocity dependence of the L/D effect

15

Table 1. EOS Parameters

Tungsten Steel

17.0 7.85

3.85 4.50

1.44 1.49

1.58 2.17

0* ( 0 - 0 o ] =~0,-0o)

0° = 300°K

Table 2. Constitutive Parameters

O,q = (Yo + Be"p) [1 + C ln(+p/~o)](1 - 0 TM)

~o = 1.0s -1

was used, with the constants given in Table 1. CTH allows the flow stress to be a function of strain, strain rate, and temperature [ 12-13]. The Johnson-Cook model [ 14] with parameters for 4340 steel and a tungsten alloy were used for the computations. Table 2 lists the parameters used for this study. Seven zones were used to resolve the projectile radius; the zoning was square in the interaction region. Four aspect ratios were considered for the computations: /_/D = 5, 10, 20, and 30. It was demonstrated by Anderson, et al. [7], that P/L is a function of L/D, and not L and D separately, except for strain-rate effects which have been shown to be small for typical scale sizes [15]. This is an important observation since the results of constant length projectiles can be scaled to constant diameter projectiles, with the same aspect ratios, by multiplying the time and penetration depth by the geometric increase in projectile length. Since P/L depends only on the aspect ratio (for specified materials and impact velocity), and is independent of actual projectile dimensions, the numerical simulations reported herein were performed for constant length projectiles (L = 3.0 cm). The primary advantage to using constant length projectiles is that the results for different L/D's have nominally the same time scale and penetration depth. Simulations using the four aspect: ratios were performed at five impact velocities: 1.5, 2.0, 2.5, 3.0, and 4.5 km/s. Normalized penetrations as a function of impact velocity are plotted in Fig. 3 for the four different aspect ratios. The dashed lines are spline curves that have been drawn through the points with a common L/D to assist in visualization of the results. The dotted line represents the classical, steady-state, hydrodynamic limit of (Pc/P, )~a, using the densities of the projectile and target materials, respectively. The normalized penetration results are also summarized in Table 3. Table 3. P/L from Numerical Simulations

5 10 20 30

1.073 0.945 0.781 0.680

1.398 1.284 1.197 1.163

1.568 1.448 1.367 1.336

1.680 1.533 1.445 1.412

1.921 1.681 1.539 1.490

The L/D = 10 simulations are similar to those discussed in Ref. [16] with the exception of the constitutive treatment of the tungsten alloy projectile. In Ref. [16], thermal softening was ignored for the projectile (but included for the target); in the work discussed here, thermal softening effects are included for the projectile. The primary differences in P/L are at 1.5 km/s. Without thermal softening, P/L = 0.823; this is 13% less than the P/L shown in Table 1. At the other impact velocities, the differences in P/L are only 1%. The effects of projectile softening on penetration efficiency is discussed in some detail by Partom [17].

16

C.E. ANDERSON,JR et al. NUMERICAL SIMULATIONS--ANALYSIS

A nice feature of numerical simulations is the ability to follow the projectile through the penetration process (see Ref. [ 18] for a comparison of computational and test results of time-dependent velocities and projectile nose and tail positions). We begin our analysis by examining the penetration (nose) and tail velocities (the tail velocities are approximately equal to the impact velocity until late in the penetration history), along the projectile-target centerline, as a function of normalized penetration depth p/L for the five impact velocities, Fig. 4a-d. Examining Figs. 4a-d, it is convenient to divide the penetration history into three phases: 1) initial penetration phase, 2) the quasi-steady-state penetration phase, and 3) the terminal penetration phase 2. The first and third phases are highly transient. The end of the initial transient phase is not too difficult to estimate, but the beginning of the terminal phase is considerable more difficult to estimate, particularly for the lower impact velocities. The results at 3.0 km/s, Fig. 4c, are used to obtain semi-quantitative estimates of the relative importance of each of the three phases to overall penetration. The percentage contribution to total normalized penetration is given in Table 4 for each phase. Although the numbers in Table 4 change as a function of impact velocity, so that they must be considered approximate values, it is clear that as the aspect ratio increases, the more dominant the quasi-steady-state phase. Table 4. Contribution of the Phases of Penetration to Total Penetration (%)

5 10 20 30

11% 6% 3% 2%

61% 83% 91% 94%

28% 11% 6% 4%

The initial penetration histories differ in Figs. 4a-d because the diameters are different; this initial transient phase of penetration scales with projectile diameter [7]. The tail velocity scales with projectile length until the final deceleration phase, which again scales with diameter [7]. These observations are independent of the impact velocity, and are reconfirmed with further analysis of the computational results in this paper. For the remaining discussion in this section, the initial transient phase will be ignored. 2.0 • ' ' ' ' 1

....

I ....

I''''

I ~ ' /

1.8 1.6

~.:~..-

. H~droLim.it

1.4

/i

.

1.2

J

1.0 0.8

"°°=!:1

0.6

0.4 0.2 0.0

• •

UD = UD =

3.0

4.0

i

0.0

1.0

2.0

V(km/s) Fig. 3. Results of numerical simulations: P/L versus V

5.0

The character of the penetration velocity changes rather remarkably with impact velocity. At 1.5 krn/s, Fig. 4a, penetration velocity decreases noticeably with penetration depth. Additionally, the penetration velocities are considerably different as a function of aspect ratio. The average penetration velocity decreases with increasing aspect ratio; the average penetration velocity is highest for the L/D 5 projectile, and lowest for the L/D 30 projectile. Ignoring the final deceleration phase of the .projectiles for the moment, as the impact velocity increases, Figs. 4b-d, the penetration velocities become more steady state (u = constant), and they converge; at 3.0km/s (Fig. 4c), the penetration velocities for the four aspect ratios are essentially equal (they are also approximately equal at 2.5 km/s). At 4.5 krn/s (Fig. 4d), there are two subtle changes in the penetration velocities compared to the lower impact velocity cases. First, there is a reversal in which aspect ratio has the highest average penetration velocity (the larger aspect ratio rods now penetrate faster than the shorter L/D rods). Second, there is a slight, but noticeable, increase in penetration velocity with increasing depth.

2 If the velocitieshad been plottedversustime, a fourthpenetrationphase--elastic recovery--couldhave been identified.

On the velocity dependence of the L/D effect

1"61'

J'

J'

~

'

~

'

z/

::\ -] o8

~.

2.01.'1

' I'

I'

I ' I ' I ' I_

1.4

¢ 10

,'._"---. i "L"<

0. 6 ~ -

\-'

'1"-

o.e .....

~-I----~-,_ .... ~-'~1ol "~.. "d i~,l__..~ X! ".~ \/-I ~-1

o4

17

\]

02

'

0.0 I' 0.0

,

::

o.,

I

I , I , I l, II , I II 0.2 0.4 0.6 0.8 !.0

0.0 0.0

p/L

[i.....I _.

UD=

5

--

uD~o

0.2

0.4

:.

0.6

0.8

1.0

Ill

1

1.2

1.4

p/L

Fig. 4a. Penetration and tail velocities versus depth Fig. 4b. Penetration and tail velocities versus of penetration: 1.5 km/s depth of penetration: 2.0km/s 3.0

4.04"5f' I ' I ' I ' I' " ~J ' I

' I ' I ' L.

2.5 3.5 ~"

0

2.0

3.0 2.5

1.5

o >

2.0 1.0 0.5 0.0

L~

1.s

l

L/D=20

1.0

I

L/D=30

0.5

--~

L/D=

.......

L/D=10

5

--'-....

-r, I , I I , I,I I I 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

p/L

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

p/L

Fig. 4c. Penetration and tail velocities versus depth Fig. 4d. Penetration and tail velocities versus of perLetration: 3.0 km/s depth of penetration: 4.5 km/s Most of the penetration occurs in the quasi- or pseudo-steady-state regime. At the end of the steady-state regime, during which there is considerable erosion of the projectile, the projectile enters the terminal phase of penetration. This is characterized by a rapid decay in both penetration and back-end (tail) velocities. There is even a small amount of rigid penetration, where the front and tall velocities are the same. For the 1.5-km/s case, the projectiles enter the terminal phase at a lesser depth for increasing /JD. For the higher velocity cases, the steady-state velocities tend towards the same constant value, with little or no decay. In these cases, the projectile enters the terminal phase at roughly the same p/L for/_J'D's of 10-30. (As a reminder, the hydrodynamic limit is given by a value ofp/L -- 1.47. This value can be compared to the end of the quasi-steady-state phase in the figures.) At the end of penetration, there is a small amount of the projectile remaining at the bottom of the penetration crater. Figure 5 shows these residual lengths, normalized by the respective projectile diameters, as a function of projectile aspect ratio for the five impact velocities investigated. The solid lines represent linear least squares curve fits of Lr/D

C.E. ANDERSON,JR et al.

18

Table 5. Residual Projectile Length, Lr (cm) [Lo = 3.0 cm]

5 10 20 30

0.456 0.273 0.205 0.151

0.242 0.157 0.118 0.107

0.192 0.134 0.109 0.103

0.152 0.123 0.107 0.102

0.112 0.107 0.102 0.101

Table 6. Changes in Penetration Efficiency (%)

L/D L/D L/D L/D

5---> 10 10 --->20 20 --->30 10 --->30

-12.1 -17.3 -12.9 -28.0

-8.6 -6.8 -2.9 -9.5

-7.6 -5.6 -2.2 -7.7

-8.9 -5.7 -2.3 -7.9

-12.5 -8.4 -3.2 -11.4

at constant impact velocity. Although Eulerian wavecodes do not model well the transition from eroding to rigid-body penetration [19], the overall trend seen in Fig. 5 should be correct. That is, the normalized length of the residual projectile remaining at the bottom of the crater decreases with increasing impact velocity, but at a given impact velocity, increases with aspect ratio. Table 5 gives Lr ; these results are used later in the paper. Since the diameter is a function of aspect ratio (for our constant initial length projectiles), the actual lengths of the residual projectiles decrease with increasing aspect ratio. QUANTIFYING T H E L/D E F F E C T AS A FUNCTION OF IMPACT VELOCITY The results of the numerical simulations are now used to quantify the effect of L/D on penetration as a function of impact velocity and changes in L/D. The results are summarized in Table 6. Examining the results in Table 6, the/_/D effect is quite pronounced at 1.5 km/s; however, the differences in P/L versus L/D decrease with increasing impact velocity, until approximately 2.5 km/s. Above 2.5 km/s, differences in P/L with L/D increase with further increases of impact velocity. Thus, the following conclusion is made: the L/D effect is manifested at all impact velocities, and is a minimum at approximately 2.5 krn/s. 1.6

"

I .... I .... I .... I .... I .... I ....

l

l

I 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

5

10

15

20

25

30

/./D Fig. 5. Normalized residual projectile length versus L/D

35

There is another observation that is important. Between 2.0 and 4.5 krn/s, the decrease in penetration efficiency is on the order of 2.5% as the L/D changes from 20 to 30. Variations of P/L of+ 2.5% are typical of experimental variability (scatter) for these types of projectile and target materials, such that the L/D effect would be extremely difficult to document experimentally in the velocity regime of 2.0-4.5 km/s for L/D 20 and 30 projectiles. Note however, that the differences between L/D 10 and 30, and probably L/D 10 and 20, should be observed experimentally. Returning to Fig. 2, and ignoring the L/D = 30 data point of the bar target, there is a decrease of 2.65% in penetration efficiency between the L/D 15 (or 20) and the L/D 30 projectiles at 3.0 kin/s, which is nominally consistent with the computational results. Likewise, the data of Bless and Hamburger[10] are consistent with these observations.

On the velocity dependence of the L/D effect

19

THEORY We now consider an instantaneous measure of penetration efficiency for eroding penetration. The amount of penetration per unit projectile eroded, given by u/(v-u), is a measure of the efficiency of penetration. Figure 6 displays u/(v-u) for the four L/D's for four of the five impact cases. A qualitative transition is evident as the impact velocity increases. For the 1.5-km/s impact velocity case, the penetration efficiency decrease~, nearly linearly with p/D for all four DD's, while for the higher velocity cases, the penetration efficiency is nearly a constant. The vertical lines in the figures occur at the terminal phase of penetration, when the projectile ceases eroding, i.e., v = u. 2.0

''''il''''

I'''

?.

2.0

'11 ' ' .

'~1

"l':"l.. : :

..'.. 1.5

•

.... i,,r,l,,,,,,,,l,r,

ii

i

J

~!

1.5 /

.

,

..~i 1.0

1.0

0.5

0.5

i i

:

I-

: :

I I

itat): s

i'

....... !L/D= 10 ---~', , , ~ If, , , ,:I , , , , II, , , , I I I

0.0

5

0

10

15

0.0

20

, , I , , ; , I•, , , , I ~ , , L I , , , , I , , , ,

,,,I 0

j i

iL/D=20 ! L/D=30

5

10

15

p/D

20

2.0

ULIIIJ

I"!'l'.~"i'"'l'N"lr'"l

I

I

...., s '.... .... ....J i l

1.5

1.0

-

0.5

I I ]'

-

IUD= 5

~-

,,,I,,,,I,,,,h,,,

0

5

10

15

20

25

30

35

p/D Fig. 6c. Penetration rate normalized by erosion rate veraus normalized depth of penetration: 3.0 km/s

45

i

f I

•

"

/ ,.¢.

.~,,,,,. :v,.

,J

i L/D= 5 ....... i I / D = 1 0 ----'i L/D=20 - - - - :: L / D = 3 0

0.0 40

I

.... I I ' " ' p "

1.0

o.s

I,,,,I,,,il,,,,I,,,,ll~,

J

;.;'~..~-,,,

....... ~UD=10 - - - - IUD-.=20 ~'~L / D = 3 0 0.0

35

Fig. 6b. Penetration rate normalized by erosion rate versus normalized depth of penetration: 2.0 km/s

2.0

-.

30

p/D

Fig. 6a. Penetration rate normalized by erosion rate versus normalized depth of penetration: 1.5 km/s

1.5

i

25

,,,I,,,,

0

5

. . . . I , ~ , , I . . . . I . . . . I L , , , I . . . . I . . . . i,

10

15 20

25

30

35

40

45

p/D Fig. 6d. Penetration rate normalized by erosion rate versus normalized depth of penetration: 4.5 km/s

C.E. ANDERSON, JR et al.

20

The fact that the penetration efficiencies overlay each other when plotted against penetration depth normalized by projectile diameter suggests that there may exist, for a given velocity, a function that relates the cumulative penetration to the cumulative erosion of the projectile, both normalized by the diameter of the projectile. Figure 7 is a plot ofp/D versus the normalized eroded length of the projectile le/D = (Lo-l)/D for four impact velocities with all four L/D' s at each specific impact velocity (the results for 3.0 km/s were not included in the plot to enhance the clarity of the figure). What is immediately apparent is that for most of the penetration, these curves overlay each other (for constmlt impact velocity). The primary component of these curves occurs in the pseudo-steady-state phase of penetration. (The bending up of the p/D versus [e/Dcurve at the end of an specific L/D trace occurs during the terminal phase of penetration, and represents essentially rigid-body penetration; this facet of the penetration curves will be discussed a little later.) Thus, the p/D curve for quasi-steady-state penetration can be written as a function of l,/D:

(2)

D where

V--U

since p = u and l, = v - u, and the prime means differentiation with respect to the argument. As the terminal phase deviates from the quasi-steady-state portion of penetration, it suggests that the final penetration depth can be divided into two phases: P

=

P~teadystate+ Pterminsl

(4)

It may be argued that there should be an initial phase in addition to the quasi-steady-state phase and terminal phase. However, the fact that the p/D versus Ie/D curves (Fig. 7) overlay at early times suggests the initial transient does not greatly affect the penetration depth, and specifically, that it need not be considered separately from the quasi-steady-state penetration phase. In Fig. 7, the nominal steady-state phase of penetration has two qualitatively different behaviors, depending on the impact velocity. For the low velocities (1.5 km/s and mildly for 2.0 km/s), thep/D versus eroded length curve is concave down, indicating a diminishing return on penetration as the projectile is consumed. Thus, the low impact velocity cases manifest an/_]D effect during the quasi-steady-state portion of penetration. For the higher velocities (2.5, 3.0 and 4.5 km/s) the steady-state portion of the penetration is a straight line (actually, it is slightly concave upwards at 4.5 km/s). Thus, there is no L/D effect in the steady-state portion of the calculation for these higher velocities. Although Eqn. (2) introduced a generic or functional form for the p/D curve, to be more specific for the calculations here, the p/D versus eroded length curve can be written as p

=

a l ~ + a 2(~l ,) y

(5)

where at and a2 depend on the impact velocity, but not on the length or diameter of the rod. For low velocities, a2 < 0, corresponding to the curve being concave down. At higher velocities, a2 -- 0. Least squares fits to the quasi-steady-state portions of penetration provide the values of al and a2 given in Table 7. These values are heading towards the hydrodynamic limit of al = ~p/p, = 1.47, a2 = 0. For tungsten into steel, where the target is stronger than the projectile, the hydrodynamic limit is an upper bound for a~. (We mention that in cases where the projectile is much stronger than the target, the pseudo-steady-state penetration efficiency can exceed the hydrodynamic limit.) Table 7. Model Constants for Eqns. (5), (6), and (8)

1.5 2.0 2.5 3.0 4.5

1.091 1.236 1.318 1.37 1.41

-0.0135 -0.00249 m

0.0317 O.0268 0.0286 0.0307 0.0328

0.622 0.261 0.168 0.101 0.0242

0.82 1.54 1.74 2.05 3.02

On the velocity dependence of the L/D effect

~.

'''' I'''' L/D= 5

40

I''''

I''''

2.5

I'''~

'~'1'"'1'"'1 I\

i

:.-.

uD:,o I

; ..i..%.-

4

.... I " ' 1 ' " ' 1 " ' • l's km/sl I-I 2.0 km/s I • 2.5 km/,I V 3.0 km/$

\

~\ •

2.0

~\~

uo:3o

30 -

21

1.5

°o°" o, • °~,

M

o

20

1.0

i ,

/

..........

10

0.5

0

,,I

0

....

5

I ....

10

I ....

15

....... I ....

20

I,,,

0.0

25

30

"

i

0

'

Eq. (14) 1 |

....

5

10

le/O

15

20

, ....

25

,,,.

30

35

UO

Fig. 7. Normalized depth of penetration versus normalized length of eroded rod

Fig. 8. Comparison of model results of P/L versus L/D to computational results

Another interesting observation concerning the results in Fig. 7 is that at the higher velocities, the final penetrations an~ nearly given by a constant offset from the p/D versus eroded length curve. Since the terminal phase of penetration scales with D, it is therefore possible to write this observation as: Pt~i,~a = gD

(6)

where g depends on impact velocity, but not the length or diameter of the rod. (Actually, g slightly increases with L/D for the higher impact velocities; for 1.5 km/s, as L/D increases, g decreases. But a reasonably good approximation is to consider g independent of L/D for the higher impact velocities.) The values for g are given in Table 7. A surprising result from the calculations is that the residual length, or remaining length after the penetration pro?ess is over, is strongly related to the initial length of the projectile, and only mildly dependent on the diameter of the projectile. Figure 5 shows the normalized residual lengths for all the computations. It can be seen that all but the 1.5-km/s case can be fit very well by a straight line of (and even the 1.5-krn/s results can be approximated by) the form Lr

L bt.-~ + bo

(7)

Lr = bLL +boD

(8)

--D

=

or

Table 7 gives the values of b Land bo for the five impact velocity cases using the computed residual lengths from Table 5. ]it is seen that the slope bL is relatively constant (approximately equal to 0.03), and that the intercept bo decreases with increasing velocity. Thus, for high velocity impacts, the residual length is nearly a constmat times the initial length, and is almost independent of the diameter. This information about the residual length implies another conclusion regarding the penetration depth for the terminal phase. First, we note that g is a function of L/D indirectly. The penetration depth for the terminal phase depends on two things: the penetration velocity at the beginning of the terminal phase, and the residual length of the projectile. Since the residual length is nearly constant, g therefore mainly depends on the penetratiLon velocity upon entering the terminal phase. This penetration velocity can be obtained from Eqns. (3) and (5) as u

-

vf "(LJD) 1 +f'(LJD)

-

v

al + 2a2(LJD) 1 + al + 2a2(LJD)

(9)

C.E. ANDERSON,JR et al.

22

Table 8. Estimate of Penetration Velocity at Beginning of Terminal Phase (from Eqn. (9))

0.741 0.688 0.554 0.367

5 10 20 30

1.10 1.09 1.07 1.04

1.42 1.42 1.42 1.42

1.73 1.73 1.73 1.73

2.63 2.63 2.63 2.63

where L, is the eroded length upon entering the terminal phase. Using the constants from Table 7, estimates for the penetration velocity upon entering the terminal phase are provided in Table 8. The values in Table 8 must be considered estimates of the penetration velocity at the on-set of the terminal phase since there are several assumptions in the derivation of Eqn. (9); as the impact velocity or L/D increases, the assumptions become more valid. Nevertheless, the trends given by the results in Table 8 are informative. For the 1.5-km/s impact velocity case, where the penetration velocity decreases as the target is penetrated, the penetration velocity is less at the beginning of the terminal phase (because a2 < 0) as L/D increases; therefore, g decreases as L/D increases. For the higher velocity impacts, the penetration velocity is nearly constant, and so the penetration velocity at the beginning of the terminal phase is independent of L/D, and therefore g is also nearly constant and independent of L/D. Values ofg are given in Table 7 and are discussed later. Equations (4), (5) and (6) allow the final penetration to be written as: P

=

P,,~ly,==-t-P=,.m~,,,

=

,~[ L< (L<'~2] --~a,-~-t-a~-~)f+gD.

(10)

If the length of the projectile eroded during quasi-steady-state penetration is approximated by the original length less the final residual length (this is an approximation since some erosion occurs during the terminal phase), then:

_P = ~L-L, L alL+a:

(L -Lr)2+ g....~ LD L/D

(11)

Inserting the relation, Eqn. (8), for residual length, Eqn. (11) becomes: P

=

a1 1-b L

-~--~

+a 2

(1 -bL) e

2 --2(1--bL)bD+~--~f+~.

(12)

Differentiating this expression with respect to L/D explicitly identifies the parts that give rise to the L/D effect:

d(P/L) I ~ 1 g-a~bo 1 d(g) d(L/D) - a z{(1-bL) 2- (L/D)2j (L/D)2 ~-L/Dd(L/D).

(13)

A derivative of g is included since g decreases for increasing/_/D at 1.5 krn/s. For low impact velocities (V ~ 1.5 km/s), this last term and the fact that a2 is nonzero give rise to the L/D effect; for large L/D's the oa term dominates. For the higher velocities, a2 ~- 0, and the P/L expression simplifies to: - albo

PL = al(1- bL) + gL/D

(14)

As bo tends towards zero for higher velocities, the L/D effect for these cases is due nearly entirely to the residual penetration term, as measured by g. To demonstrate that Eqn. (14) reflects the correct behavior, Fig. 8 provides the P/L versus L/D for the various impact velocities, with Eqn. (14) plotted for each impact velocity. The values ofai, bo and bL are from Table 7. The values ofg are obtained by performing a least squares fit to the computational points in Fig. 8; it is these values ofg that are given in Table 7. The values of g in the table are close to those measured directly from Fig. 7. The agreement between Eqn. (14) and the numerical simulations is excellent for all impact velocities save 1.5 km/s. This demonstrates that the L/D effect is due to the terminal phase of penetration for the higher impact velocities. Lastly, at the higher

On the velocity dependence of the L/D effect

23

impact velocities where the product a~bo is less than 10% of g, the differentiation of Eqn. (14) with respect to/_/D gives: d(P/L) d(L/D)

g (L/D) 2

---

(15)

This is the same dependence on/JD found in Ref. [7] for the terminal phase of penetration. The 1.5-km/s impact velocity case displays two curves. The solid line used Eqn. (14) and a different al (to go with a2 = 0) to fit the p/D versus le/D curve. The agreement is not good, and it is seen that the qualitative behavior of the decrease in P/L for increasing L/D is not reflected in the curve. The broken curve is from Eqn. (12) with al and a2 from Table 7. This does a much better job fitting the data. This demonstrates that the L/D effect for the 1.5-krn/s case arises from the quasi-steady-state penetration phase. Thus, it has been shown that there is a change in mechanism for the L/D effect as velocities transition from ordnance velocities to higher velocities. The regime where the pseudo-steady-state penetration has a great deal of influence on the final P/L is in the regime of 0.8 to 1.9 km/s, i.e., the linear portion of the "S-shaped" P/L versus V curve where there is a large increase in P/L for increasing impact velocities. For lower impact velocities, the terminal phase of penetration is not great, so the terminal phase has only a minor influence on the/_/D effect. Once the impact velocity is high enough that the penetration velocity is nearly constant throughout the quasi-steady-state penetration phase, then the/_/D effect is due almost entirely to residual penetration, since the steady-state phase of penetration gives rise to nearly the same depth of the penetration for the same length rod. The magnitude of the L/D effect therefore increases for increasing impact velocities above 2.5 km/s since residual penetration is greater for higher impact velocities. Note that the L/D effect for the higher impact velocities is a consequence of the terminal phase simply making less and less of a contribution to total penetration as the/_279 increases. HYDRODYNAMIC CALCULATIONS A limited number of purely hydrodynamic calculations were performed. In particular, results for all four L/D's were obtained at 1.5 krn/s and 4.5 km/s. Analysis of these calculations is made difficult by the fact that the projectile does not come to rest. Once the rear of the projectile is within a couple of diameters of the projectil,:-target interface, the projectile behaves as a "rigid" body (nose and tail moving at the same velocity). Therefore, the equation of motion is given approximately by: 1

2

Lrfl = --~pt v

(16)

Thus, the velocity decelerates as 1/t, and so the penetration depth behaves as In(t), never coming to rest. However, ,;ome insight can still be gleaned from the calculations. For both impact velocities, the slope of the p/D versus le/D curve is the hydrodynamic limit; that is, a~ = 1.47. Also, the residual length of the projectile (the length when u = v) was found to depend on the initial length of the projectile, and not on the diameter (Fig. 9), as was seen for the higher velocity calculations with strength. These computations provide insight into the importance of strength effects at the lower impact velocities. ""I

....

I .... l""l'"'l

....

1.0

I'"~ o

0.8

I•

"

0

"

0.6

.J

0.4 0

C)

0.2

0

1.5 km/s

r-I

4.5 kmls

m

0.0

....

0

I,,,,I,,,,I,,,,I,,,,I,,,,I,,,

5

I0

15

20

25

30

35

/./D

Fig. 9. Normalized residual projectile length versus L/D: hydrodynamic calculations

24

C.E. ANDERSON,JR et al. SUMMARY

Numerical calculations have been performed to examine the L/D effect as a function of impact velocity. The mechanism for the L/D effect changes as the velocity increases. For lower and ordnance velocities, where penetration is strongly dependent on velocity, the pseudo-steady-state phase dominates penetration, and the majority of the L/D effect originates during this penetration phase because of the decay of the penetration velocity. However, as the velocity increases, the L/D effect, which is still present, now shifts to depending on the terminal phase of penetration. The arguments presented show that one would expect a large/_,/D effect from the pseudo-steady-state phase of penetration if the impact velocity is in the reg!on of the steep slope of the "S"-shaped P/L versus V curve. Once beyond the rapid increase in penetranon depth with impact velocity, and into the more slowly increasing "saturated" portion of the P/L curve, the /_/D effect is due mainly to the terminal phase of penetration. ACKNOWLEDGEMENTS A portion of this work was performed under Contract No. DAAL03-92-K-001, administered by the U. S. Army Research Office; and under Contract No. UT/SWRI-0009 with the Institute for Advanced Technology. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

V. Hohler and A. Stilp, Hypervelocity impact of rod projectiles with L/D from 1 to 32. Int. J. Impact Engng., 5, 323-331 (1987). D.R. Christman and J. W. Gehring, Analysis of high-velocity projectile penetration mechanics. J. Appl. Phys., 37(4), 1579-1587 (1966). W. Herrmann and J. S. Wilbeck, Review ofhypervelocity penetration theories. Int. J. Impact Engng., 5, 307-322 (1987). C.E. Anderson Jr, D. L. Littlefield, N. W. Blaylock, S. J. Bless and R. Subramanian, The penetration performance of short L/D projectiles. High-Pressure Science and Technology--1993 (edited by S. C. Schmidt, J. W. Shaner, G. A. Samara and R. Ross), pp. 1809-1812, AIP Press, New York (1994). Y. Partom and D. Yaziv, Penetration of L/D = 10 and 20 tungsten alloy projectiles into RHA. High-Pressure Science and Technology-- 1993 (edited by S. C. Schmidt, J. W. Shaner, G. A. Samara and R. Ross), pp. 1801-1804, AIP Press, New York (1994). Z. Rosenberg and E. Dekel, The relation between the penetration capability of long rods and their length to diameter ratio. Int. J. Impact Engng., 15(2), 125-129 (1994). C.E. Anderson Jr, J. D. Walker, S. J. Bless and Y. Partom, On the L/D effect for long-rod penetration. Int. J. Impact Engng., 18(1), in publication (1996). B.R. Sorensen, K. D. Kimsey, G. F. Silsby, D. R. Scheffler, T. M. Sherrick and W. S. deRosset, High velocity penetration of steel targets. Int. J. Impact Engng., 11(1), 107-119 (1991). Co E. Anderson Jr, S. J. Bless, D. L. Littlefield and R. Subramanian, Prediction of Large Scale Impact Experiments on Steel Targets. 14th Int. Syrup. on Ballistics, Vol. 2, 459-468, Qu6bec, Canada (1993). S.J. Bless and C. Hamburger, In preparation (1994). J.M. McGlaun, S. L. Thompson and M. G. Elrick, CTH: A three-dimensional shock wave physics code. Int. J. Impact Engng., 10, 351-360 (1990). W. Wo Predebon, C. E. Anderson Jr and J. D. Walker, Inclusion of evolutionary damage measures in Eulerian wavecodes. Comput. Mech., 7(4), 221-236 (1991). S. Silling, Stability and accuracy of differencing methods for viscoplastic models in wavecodes. J. Comp. Phys., 104, 30-40 (1993). G.R. Johnson and W. H. Cook, Fracture characteristics of three metals subjected to various strains, strain rates, temperatures, and pressures. Engng. Fracture Mech., 21(1), 31-48 (1985). C.E. Anderson Jr, S. A. Mullin and C. J. Kuhlman, Computer simulations of strain-rate effects in replica scale model penetration experiments. Int. J. Impact Engng., 13(1), 35-52 (1993). C.E. Anderson Jr, D. L. Littlefield and J. D. Walker, Long-rod penetration, target resistance, and hypervelocity impact. Int. J. Impact Engng., 14, 1-12 (1993). Y. Partom, Projectile-flow effect for long rod penetration. IAT.R 0036, Institute for Advanced Technology, The University of Texas at Austin, Austin, TX (1994). C.E. Anderson Jr, V. Hohler, J. D. Walker and A. J. Stilp, Time-resolved penetration of long rods into steel targets. Int. J. Impact Engng., 16(1), 1-18 (1995). J.D. Walker and C. E. Anderson Jr, Multi-material velocities for mixed cells. High-Pressure Science and Technology--1993 (edited by S. C. Schmidt, J. W. Shaner, G. A. Samara and R. Ross), pp. 1773-1776, AIP Press, New York (1994).