Further results in the theory of long rod penetration∗

J. Me&. Phys. Solids, 1969, Vol. 17, pp. 141 to 150. PergamonPress. Printedin Great Britain.

FURTHER

RESULTS

IN THE

THEORY

OF

LONG ROD PENETRATION* By A.

TATE

Royal Armament Research and Development Establishment Fort Halstead, Sevenoaks, Kent

(Received 2nd January 19f39)

SUMMARY of long rod penetration as given in a previous paper by the author is extended to take account of the deformation of a soft rod against a rigid target and the penetration of a rigid projectile into a soft target. It is shown that it is theoretically possible to have a decrease in depth of penetration with increasing impact velocity, and a method for deducing the average radius of the hole is given. The theory is compared with experimental results. THE

THRORY

1.

INTRODUCTION

IN A PREVIOUS paper (TATE, 1967), here denoted by (I), a modified hydrodynamic theory, which took some account of strength effects, was used to predict the deceleration of a long rod after striking a target. Two strength parameters were introduced, namely stresses YP and Rt, above which the projectile and target respectively behave as a fluid. If Rt is greater than Y, penetration ceases when the rod velocity drops below [Z (Rt - YP)/pp]f, where pP is the density of the projectile material. Below this velocity the target behaves in a rigid manner and we extend our previous analysis to estimate the amount of undeformed material when the motion ceases. This result is then compared with the theory of plastic deformation (TAYLOR, 1948). If Rt is less than YP the rod behaves as a rigid body if the penetration velocity drops below [2 (YP - Rt)/pt]*, where it is the density of the target material. In this paper we estimate the extra penetration of the rigid bit of the rod in the target and it is found that in this case it is theoretically possible for the depth of penetration to decrease as the impact velocity increases. Assuming that the volume of the crater is proportional to the kinetic energy of the rod just before impact, the average diameter of the hole can be calculated. Finally, the theory is compared with experimental results for steel rods striking aluminium targets and rods of aluminium alloy striking lead targets.

2.

THEORY

We begin by briefly recapitulating the principal equations given in (I). Let the velocity of the rod be v, the velocity of penetration be u and the pressure at the *British CrownCopyright Reserved. Publishedwith the permissionof the Controllerof Her Britannic Majesty’s Stationery Office.

141

A. TATE

142

rod-target interface be P, then assuming that Bernoulli’s equation is approximately satisfied gives P = Jpt u2 + Rt = &pp (v - u)2 + Y,, (2.‘) so that u = [v -

CL(n2 +

m

z-p2

(2.2)

’

where /JL=

pt

?

and

A = 2 (Rt -

YP) (1 -

( PP ) While

P2)

(2.3)

Pt

the rod is being consumed

its retardation

is given (2.4)

The rate of decrease in 1 is then

where Eis the length of rod that is not consumed. given by dl - = dt In (I) we determined ing and being consumed

-

(v -

U).

(2.5)

the motion only for the case when the rod was both penetratand the form of the solution depended upon the relative Let us now investigat.e what happens when the above

magnitudes of Rt and Yp. conditions cease to apply. Case 1 (Rt > Yp) When the velocity penetrate

of the rod falls below

zlC= [2 (&

the target which thus begins to behave

and (2.5) remain valid, however,

-

Yp)/pp]* it ceases to

as a rigid body.

Equations

(2.4)

and thus

Yp = pp Evf

(2.6)

.

If the impact velocity V is greater than vC, and the length of rod when v equals v, is denoted by ZC,then, integrating (2.6) gives for the final length ISof undeformed rod, 1, = 1, exp [(YP From

(2.7)

WY&

(2.9) in (I), Rt-Yp

k = L

(

[A (1 + PM1 - P)lf 1 v + (V2 + A)” > exp

I

!-J YP! 9qqyp

FLPP

(p/l

-

[V (772 + A)& -

pvy}

)

(2.8)

1

where L is the initial length of the rod. If the impact velocity V is less than v,, so that the rod never penetrates the target, then (2.6) can be integrated with the initial condition that 1 == L when v = V to give 1s - = exp (L The problem

of the plastic deformation

pp V2/2YP).

(2.9)

of a rod striking a rigid target was solved

Further results in the theory of long rod penetration

143

by TAYLOR (1948). He gives the length of undeformed rod, re-written in our notation, as 1 (2.10) i = exp E- pi0Y (Y + 2C3/2Y271t where C is the velocity of the elastic-plastic boundary. Comparing (2.9) and (2.10) it can be seen that the hydrodynamic theory can only apply to perfectly-plastic rod material in which the velocity of plastic wave propagation is zero. In an experimental paper accompanying Taylor’s theoretical paper, WRIFFIN (1948)estimates C for steel to vary from about 2000 ft/sec for an impact velocity of 400 ftlsec to about 240 ft/sec for an impact velocity of 2500 ft/sec. The impact velocity is comparable with 2C at about 1500 ft/sec and for these low velocities the hydrodynamic theory will clearly break down. However, for velocities of impact which are much greater then 2C the two theories will give almost identical results. Case 2 (Rt < Yp)

When the velocity of the rod falls below the value [2 (Y, - &)/pt] 4 it ceases to deform and proceeds to penetrate as a rigid body. In this case the pressure on the end of tile rod will be (2.11) P = +pt o2 + ftt , and consequently the deceleration will be approximately given by

$f= -

pp 1s

(2.12)

(ipt v2 + Rt).

If we denote by d, the extra penetration in bringi~ from a velocity ~0, then integrating (2.12) gives

the rod of length Zsto rest

(2.13)

At sufheiently low impact velocities for the rod not to deform, the penetration is given by (2.14)

The above formula is of limited applicability, for at sufficiently low velocities, Rc will be determined predominately by free surface effects and the motion will be of a plastic rather than hydrodynamic nature. When the impaet velocity Y is greater than [2 (YP - Rt)fpJ* there is an extra penetration de of the rod as a rigid body. It starts at this velocity and with a length of undeformed rod ISgiven by G-Y,

L [A (CL+ l)ilP _= .L

y +

VI& HcIyp

(Y2 + A)*

> exp

see (2.10)

in (I).

- /-“’ [v (772 + A)* - pvz]}, 2(1 --/AZ)Yp

(2.15)

144

A, %aTE

SubstituLing [2 (YP - &)/pt]+ for uo in (2.18) gives (2.16)

In (I) we described experiments by D. F. T. Winter (private communication) on the penetration of a durah~min rod into a polythene target. If we substitute the data given there, namely ~2 = 2+4, 11t,= 34 ton/ins, Rk = 2’i ton/inz, and IS = CT in., into f2.16), we find that the extra penetration is 0.8 in,, which fits the rest&s of the experiments as given in Pig. 4 of (I) quite well. Let us denote the velocity [Z (TP - &),!ptf* by z’,. It is shown in (I) that the penetration d, up to .thc point at which the rod behaves rigidly is given by V (2.17)

-----Pure

hydrodynamic

theory

Impact verocifyV, IO3ftlwc FIG.

1. Variation of penetration vrith velocity for different vahes of Y, and I& (lneesured in ton/ing).

Further

145

results in the theory of long rod penetration

The total depth of penetration D is obtained by adding d, as given by (2.16) and d, as given by (2.17). For most practical cases the integral in (2.17)must be evaluated nllmeri~ally and computer solutions have been obtained. Some of these results are reproduced in Figs. 1 and 2. Figure z shows clearly that it is theoretically possible to reach a striking velocity beyond which the penetration decreases, tending ultimately to the hydrodynamic limit. The low velocity region of the curves is determined by

0

f+~*----__--___---

p=O.565

p, =I68

Ib/cu

ft

4

6 Impact

FM. 2.

velocity

8

K

V, IO3 ft/sec

Variation of undeformed length of rod with velocity for different values of Y, and & (measured in ton/in2).

Rt only, whilst the peak penetration velocity seems to be predominateIy determined by Y,, and the high velocity region depends mostly on the difference (UP - Rg). Figure z shows that increasing YP makes the rate of decrease of the Llndeformed length of rod with increase of impact velocity less severe. Curves having the same value of YP are nearly similar in shape, the variation of Rt merely shifting the curves along the velocity axis. It has been noted, for instance by CHRISTMAS and GERRING (1966),that the volume of the crater is proportional to the kinetic energy of the projectile and a similar conclusion can be drawn from recent experiments as shown in Fig. 3. Using this fact we can make some estimate of the crater radius as follows. Let the crater volume be W, and then

&L~,~~~V~=KW

(2.19)

where r, is the projectile radius and K is a constant having the dimensions of pressure. If the craters all have a similar shape, and if the average radius of a crater is r,, then (2.20) W = k T DY-2,

146

A. TATE

FIG. 3. Crater volurr~c compared with kinetic energy of projectile. Aluminium alloy projectile ; lead target.

where k is a non-dimensional

constant.

The recent experiments

show that such a

relation does indeed hold, as can be seen from Fig. -c. For these experiments it was found that k equals 1.05.At low velocity we may expect that the size of the hole will be just sufficient for the rod to turn back thus

on

itself without

spreading

Substituting (2.21)and (2.20) into (2.19) and using the approximate for D given by (2.14) we find that

K;;

out, and

equations

(2.22)

3

5 c=3 -.i

40-

;

0 /

FIG. 4. Crater volume compared with penetration x (crater radius)‘. Aluminium alloy projectile; lead target.

Further

results in the theory of long rod penetration

Using this value of K in (2.19) and re-arranging

1-&7

gives (2.23)

3. Two

COMPARISON

sets of experiments

have

WITH EXPERIMENT

been

performed

to see if the predicted

peak

penetration could be observed. In the first set of experiments, IL W. Billington and D. J. Carley (private communication) fired ten-calibre steel rods into aluminium targets.

Taking

given by DUVALL

the Hugoniot

elastic limit of the steel (Vibrac)

(1961),we can estimate

the length of undeformed

as 150 ton/ins,

the value of Rt for the aluminium

rod using (2.15).

The length of undeforrned

as

from

rod is often

rather dificult to measure because the rod appears to buckle severely, especiall!The experimental results are shown in Fig. 5 and at velocities of about 5000 ft/sec.

Y=l50

R=lOO

o Experiment

I

2

4

Impact

velocity

FIG.5. Comparison of theory with experiments

I

I 6

V,

IO3 ft/sec

on projectiles minium targets.

would seem to indicate

of Vibrac steel striking alu-

a value for Rt of about 125 ton/ins.

Duv_ALL (1961) quotes

a value for the Hugoniot elastic limit of strong aluminium of about 35 ton/ins, and if R is about 3.5 Y as in the case of steel, this gives an Rt of about 123 ton/in2 which is quite close to the previous estimate. The penetration results are plotted in Fig. 6 and would seem to fit better for a value of Rt of 100 ton/ins. The experiments do not in this instance show any tendency for the penetration to decrease with increasing impact velocity. This may be due to another penetration mechanism known as ’ residual ’ or ‘ secondary ’ penetration masking the effect. Residual penetration was observed by ALLEN and ROGERS (1961) and depends essentially on the density of the projectile being much

;‘I. TA~‘P:

148

greater than the density of the target. To remox-e this ambiguity to fire strong light projectiles into weak dense targets.

/__ 0

I.._.~ -.-_____J 4

2 Impact

velocity

V,

IO”

it is preferable

___I 6

8

ft /set

FIG. 6. Comparison of theory with experiments on projectiles OCVibrac steel striking aluminiwn targets.

In recent experiments, and ten calibres

aluminium

alloy

HE NWP)

(B.S.1476:

long, were fired into lead targets

at velocities

rods,

0.25

between

in. dia. 2000 and

~OOO ft/sec.

Preliminary experiments had shown that the lead targets, 4 in. dia. and 1 ft long, suffered scT’erc lateral distortion because of thtir finite size. To overcome infinite

this difficulty

and approximate

target of the theory,

and volume

more closely

the lead cylinders

of the craters were measured

the behaviour

were encased

in steel.

of the semiThe depth

relative to the top of the steel casing as

o.s-

0 CI

os_

Y=lOO

R=24

i 0 6 ‘Z e f : a

0.4-

0

FIG. 7.

2 Impact

velocity

V,

I 4 IO3 ft/sec

I 6

1 8

Comparison of theory with experiments on projectiles of aluminium alloy striking lead targets.

Further results in the theory of long rod penetration

this provided

a good

undistorted

results and the variation

surface

of penetration

level.

Table

with velocity

l-h9

1 gives the experimental is shown in Fig. ‘7. As can

be seen the penetration peaks at a velocity just under 5000 ftlsec. and the best theoretical curve corresponds to Yp = 100 ton/ins for the aluminium alloy and Rt = 24 tonins

for the lead.

From (2.19) and the data given in Fig. 3 we find that Then using (2.22), Ilt = 23-R ton/in2

in these experiments 2K = 23.3 ton/ins. which is in satisfactory agreement with

TABLE 1. Lorg rod penetration Velocity (ft/sec)

the penetration (aluminium

Penetration D/L

1640

ulloy ido

The

ultimate

lcnd)

Crater vol. W/LTpZ

Crater radius

4.2 5.1 6.8 7.8 7.3 10.2 13.5 14.4 12.5 17.0 16.6 18.1 20.0 21.1 21.7 25.3 26.3 29.5 27.9 36.6 34.3 34.5

2.6 2.7 3.2 3.6 3.2 3.6 4.1 4.5 4.4 5.0 4.6 5.1 5.2 5.2 5.7 5.3 6.0 5.9 6.3 7.1 6.8 7.2

0.38 0.55 0.60 0.65 0.66 0.69 0.75 070 0.67 0.74 0.75 0.77 0.75 0.69 0.70 0.71 0.70 0.71 0,65 0.65 0.66 0.65

3260 3340 3400 3946 4460 4570 4600 4800 4840 5240 5500 5540 5780 6050 6240

result.

T&d

Projectile is cylinder of length L = 2.5 in and radius r, = 0,125 in, and the material is aluminium alloy of density 2.7 g/ems. Target material is lead of density 11.2 g/cme.

tensile

stress,

(TV, of the aluminium

alloy

is about

I:! ton/ins

while that of the

aluminium is about 4 ton/ins. Taking Poisson’s ratio v to be O&5 and using the formula o’y = [(l- b)/(l - v)]Y given by DUVALL (1961), we find that the equivalent

dynamic

strengths

are 46 ton/ins

and

16 ton/ins

respectively.

Thus,

the

aluminium appears to show a pronounced strain-rate effect. However, if a Young’s modulus of 6.96 x 1011dyn/cms and modulus of rigidity of 2.37 dynlcms, given in the Handbook

of Chemistry

and Physics

(4lst

edition),

are used to compute

Y

these give a value of 0.47. Using this value the equivalent dynamic yield strengths would be 11.3ton/ins for the aluminium alloy and 3.96ton/ins for the aluminium, which agree closely with the static results. It should be noted that at the higher impact velocities the speed of penetration is greater than the longitudinal speed of sound in lead which, because of its weak

150

A. TATE

structure, is very low. The effect of an attached of the cavity tip is to modify (2.1) to read

shock wave running just in front

(2.24) where o is the ratio of the density of the lead ahead of the shock wave to that behind. At the pressure levels attained in the present experiments this ratio is negligibly

different

from unity and the theory should apply in its present form.

cOh-(‘L~wop*‘s

4.

The theory de\-clopcd.

of long rod penetration

In the case Rt > Y,,

target begins to bellave as a rigid body. and compared

set out previously

when the impact

with T~YLOR’S (INS)

velocity

in (I) has been further drops low enough,

The amount of undeformed theory

of plastic

the

rod is estimated

deformation.

This shows

that the present theory approa,chcs Taylor’s theory when the impact velocity exceeds twice the speed of propagation of the elastic-plastic boundary. In the case Kt < YP, when the impact velocity falls low enough, the rod begins to behave as a rigid body and the rxtra penetration bcforc being brought to rest is given by (2.1:%), (2.11) and (2.16). possible

Computer

solutions

show that iu this case it is theoretically

for an increase in \-elocity of pen&ration

pen&ration

but a greater

diameter

of the crater is proportional

of hole.

to the kinetic

to lead to a decrease in depth of

Experiment

shows

that the volume

energy of the projectile

and this fact

allows an estimatt of the crater radius to bc made. Hxperiments by E. IV. Rillington and D. J. Carley on t.he penetration calibre steel rods into alumininm and it is suggested (1961)

might

ahlmininm

that residual

be masking

is grateful

penetration

the el’fect.. Recent

alloy projectiles

The author

did not show the predicted as observed experiments

by ,~LI,EN and ROGERS on the penetration

into lcad targets do show the predicted

to Dr. E. W. BILLINGTON

discussions and making arailablc their experimental

of ten-

decrease in penetration, of

decrease.

and Mr. D. J. CARLEY of R.A.H.L).E. results before publication.

for

zhI.KN, \v. :\.

and Itocr~as, CHRISTMAN, D.

.J. \V.

R.

and GEIIRIN~~, J. TV. G.E.

J. appl. Phys. 37, 1579. Response of Metals to High Velocity Deformation. (Edited by SIIEWMON, P. G. and ZACKAY, V. F.), p. 165. Inter-

l)UVALL,

science, New York. 1067

J.Mech.

Php.

TAYLOR, G. I.

1048

PTOC.

,%c.

wIIIFFIx,

1048

TATI?,

‘1. A. C.

Solids 15, 387. A 194, 280. Proc. IL Sot. A 194, 300. Ii’.