02th International Symposium on Ballistics, Orlando, Florida, 23-27 September 2002

Yawed Rod Penetration

Gabi Luttwak

RAFAEL, P.O. Box 2250, Haifa, 31021, Israel

INTRODUCTION

A high speed rod will penetrate into a target made of the same material to a depth,

approximately equal to its length, L. Long rods, with large length to diameter ratio

(L/D>10), can perforate thick armor plates. Any mechanism, which will increase the

region of the interaction, will reduce the penetration depth. For optimal penetration, the

projectile moves along its axis of symmetry and hits the target head-on. Oblique or yawed

impact decreases the penetration depth. The yaw is the angle , between the velocity v

of a projectile and its axis of symmetry (called total yaw by Bjerke [4]). The obliquity is

the angle between the velocity vector and the normal to the target. Oblique and yawed

impact can occur together. In this work, the normal impact of a yawed rod into a thick

target is investigated. A projectile can get a small yaw during its flight in the air toward

the target. After a primary impact with an oblique or moving plate, the residual penetrator

will hit the main target with a large yaw greatly reducing its penetration. There is a

considerable interest to understand and to estimate the effect of yaw on the penetration.

Experiments have been carried out [1-7]. The results have been analyzed both by

analytical models [3-7] and in the recent years more and more by numerical simulations

[11]. The normal impact of a rod with yaw into a thick target produces high deformations

especially in the rod. Parts of the projectile which where located at separate places along

the rod get into contact which each other and with the target as they form the crater.

Multi-material Eulerian calculations are best fit to handle such a problem. The

The impact and penetration of long rods with yaw is investigated.

Numerical simulations are carried out and the results are compared to

experimental data. The feasibility of assuming two-dimensional

plane-strain symmetry is considered. Some of the calculations were

done several years ago in order to benchmark the multi-material Euler

with strength processor in the 3D code MSC/DYTRAN. Now similar

calculations are carried out to check the second order Godunov

processor in Autodyn 3D-V4. Assuming the crater volume to be

independent of the yaw and taking into account the crater shape as

seen in the simulations, a new analytic model is formulated.

available experimental data (Yaziv [1-2]) makes this solution also a good candidate for

code validation. In 1994, after the author has participated in the development of the

multi-material Euler with strength processor in the 3D code MSC-DYTRAN[8,14],

numerical simulations were done in order to study this problem and to serve as another

benchmark for code-validation. For the same purposes, similar calculations were now

carried out , using the second order Godunov, multi-material Euler processor of the code

AUTODYN-3D[9-10,15]. Making the assumption that the crater volume is independent

of the yaw angle and looking at the results of the simulations we formulate a new yaw

model which does reproduce well the experimental data of Yaziv [2] .

THE YAW MODEL

Today it is the role of the numerical simulations to provide quantitative prediction or

reproduction of the experimental results. Still, simple analytical models has a definite

role, to provide additional physical insight and help determine the most important

parameters which influence the results. Several analytical models for penetration with

yaw have been published [3-7]. The present model is based on one basic assumption. For

thick target penetration, all the kinetic energy of the rod is transformed into plastic work

to form the hole in the target. Therefore, we assume that the crater volume is independent

of the yaw angle. To refine the model, we look at the results of the experiments and

simulations and examine the shape of the hole formed for different yaw angles.

Let L,D be the rod length and diameter. The hole diameter, Dh , formed in the target

by a rod penetrating without yaw will be kDDh , with 1k . Let look at the plane

defined by the rod axis and its velocity v. For a rod with a yaw , as seen in Fig.1, the

size of the crater will be )sin(L . The width of the crater in a perpendicular cross

section (plane AA1 in Fig.1) will remain of the order of Dh. If the volume of the hole has

to remain the same, the depth of the hole in the target will decrease by a factor of about:

1

sin

)0()(

DL

kPP

Here, P(0), P(α) are respectively the penetration of the rod without yaw and with a yaw

of α . This sin/1 dependence obtained is similar to Bless[7]. From (1), the effect of yaw

increases with L/D. The effect of yaw is to decrease the penetration, thus (1) holds only

for impact at a yaw above a critical value, αC:

)2(sinsin 11

L

Dk

L

DhC

As the yaw gets smaller, 0)sin( L , but the crater size for penetration without yaw

should be Dh. The effect of yaw increases the crater size and at larger yaws it should

behave like )sin(L . Therefore in the denominator of (1) , we take

C

hDL

exp)sin( , obtaining:

)3(

exp)sin(

)0()(

C

kD

L

kPP

For small yaws, the rod will penetrate through the crater formed by the penetrating head

without further interacting with the walls behind it. The effect of yaw increases

dramatically at angles larger αC . The magnitude of the hole diameter Dh, can be

estimated using the hydrodynamic theory of penetration. In a frame moving with the

penetration velocity up, the kinetic energy of the in-flowing material is 2

2

42

1pT u

D

forming a hole with diameter Dh and a volume per unit length of ,225.0 hD working

against the yield strength Y of the target. Therefore, using the relation from the

hydrodynamic theory of penetration between the impact speed v and penetration velocity,

we get for k:

4

21Tp

T

h

Y

v

D

Dk

Figure1 L=80mm, L/D=10, 30

0 yaw, 1.4mm/μs impact of Tungsten rod on steel

target

The hole entrance diameter increases linearly with the impact velocity. This is consistent

with the data of Bjerke[4]. In the data of Yaziv[2,3], the penetration depth decreases from

30mm to 8mm, while the yaw angle increases from 300 to 90

0. Using (1) would predict a

decrease to only 15mm. In deriving (1) we neglected the changes in the crater shape with

the yaw. For a yaw of 310, the crater profile as seen both in Fig. 1 and in the simulations

(Fig.3-5) is almost triangular in shape, thus to get the same crater volume an increase of

factor of ~2 in the penetration depth is required. For 00

or 900

yaw, a cylindrical crater is

obtained and this factor should be ~1. Therefore, we multiply (1) a yaw angle dependent

shape factor of )2sin(1 . Taking Y=0.7GPa for the yield stress of RHA yields k=2.1.

On the other hand, k=1.34 would make (1) reproduce the 900 yaw penetration depth data

of Yaziv [2,3] . This corresponds to Y=1.4 GPa. The yield stress increases with strain

hardening and Y=1.4 Gpa might be a reasonable value for the average dynamic yield

stress. But, as the experimental crater entrance data for normal impact (Bjerke[3]) also

yields k~2.1, we use the static yield strength to find k and multiply it in our model by a

factor of 3

2 to finally get:

5

exp3

2)sin(

))2sin(1)(0(

3

2)(

C

kD

L

kPP

From (2) we get αC=12.12, reasonable fitting the data of Yaziv[2,3]. We finally chose

αC=11.2 for a best fit. In Table I. and in Fig. 2 we see a good agreement between the

new model and the experimental data for all angles.

Figure 2. Penetration depth for 1.4mm/μs normal impact with yaw of a

L=80,D=8mm,Tungsten rod on a RHA steel target

TABLE I. PENETRATION DEPTH (MM) AS A FUNCTION OF YAW ANGLE

Yaw 00 4

0 7.5

0 19

0 31

0 90

0

Experimental 60 57 52 40.5 30 8

This model 60 57.1 52.3 38.7 30.2 8.

THE NUMERICAL SIMULATIONS

We consider the normal impact of a Tungsten rod moving with 1400m/sec and a yaw

angle of 310

into a steel target as shown in Fig.1. The rod is 80mm long and has a

L/D=10. The plane defined by the symmetry axis of the rod and the normal to the target

is a symmetry plane for the problem. A fixed, 1mm zone size was use in all the

calculations.

The material model and the parameters of the Tungsten rod and the steel target used in

the DYTRAN simulations are described in Luttwak[8]. A polynomial Equation of State

(EOS) was used, with the parameters obtained by fitting them to the shock EOS. A

simple fixed yield strength model was used. The shock equation of state was used in the

Autodyn simulation. The strength calculations utilized the Steinberg [12] model which

allows for strain hardening and take into account the dependence of the yield strength on

pressure and temperature. No failure model was defined in the simulations. The default

parameters from Autodyn's library were taken for Tungsten and for SS-21-6-9 steel .

The 2D Plane Strain Simulations

Oblique or yawed rod impact and penetration are three-dimensional in character. Many

of the features of yawed rod impact can be quickly investigated assuming 2D plane-strain

symmetry. The similarity between oblique rod impact and plate impact has already been

considered[8]. Two-dimensional plane strain calculations were carried out in order to

quickly gain insight on the angle dependence and characteristic time tables of the three-

dimensional problem. A similar approach is taken here. Indeed, there are some

similarities between oblique impact and impact with yaw. Galilean invariance under a

change of frame implies that the normal impact with yaw is equivalent to a corresponding

oblique impact into a moving target. This might explain the reason why the effect of yaw

on penetration is more dramatic than oblique impact alone.

Fig. 2 The plane-strain Dytran calculations at T=20,30,50μs

In Fig. 2 we see the 2D-plane strain calculation carried out in the 3D code

MSC/DYTRAN. In fig. 3 we can see the same calculation done with the second order

Godunov processor in the code Autodyn 3D V4.0. The numerical results in both codes

look similar. The shape of the crater is in agreement with the experimental data in fig.1.

The penetration depth at 50μs is about 32mm. The experimental result for the rod impact

was 27mm .

The analytic yaw model (Equation (5) ) should hold by similar arguments also for plate

impact, but perhaps not with the same factor of 2/3. However, instead of (4) we will get:

62

1

1 2

2

Tp

T

h

Y

v

D

Dk

The lateral size of the crater is larger for plate impact, and increases with the square of

the impact velocity. The critical yaw angle will be larger and the effect of the yaw will be

smaller leading to a larger penetration depth.

The Three Dimensional Simulations

The results of the three dimensional Autodyn simulations are shown in Figs.4,5. The

evolution of the shape of the crater as seen on the symmetry plane (Fig.5) is quite similar

to the result of the 2D plane strain simulations (Figs. 2,3). The final shape is also similar

to the experimental data (Fig.1). The depth of penetration is 22mm at 50μs and 22.2mm

at 60μs. The value is smaller that the 27mm experimental value in Fig. 1. The differences

might be caused also by the material parameters used. This value is smaller than the

32mm obtained with the 2D plate impact simulations. In Fig.4 we see the cross section

of the crater in several planes perpendicular to and cutting the symmetry plane. Of

course, this kind of information can be obtained only from the 3D simulation. The shape

and the sizes of the crater compare reasonable well to the experimental data in Fig.1. We

do not show here the older, Dytran simulation. In 1995, we had to make a variable mesh,

to keep the problem in the available workstation memory. Therefore, we limited the 1mm

Fig. 3 The plane-strain Autodyn calculations at T=20,30,50μs

resolution to the actual interaction region. Early time result were satisfactory, but after

40μs the penetrating head got to a coarser mesh region. Due to the larger memories of

today's computers and taking advantage of the virtual memory scheme [9-10] in

Autodyn

V4, there were no similar problems in the current calculation.

CONCLUSIONS

The numerical simulations carried out in either Dytran or Autodyn reproduce the

essential features of yawed rod penetration. Fine tuning the material model will improve

the quantitative comparisons. The solution of two-dimensional plane strain calculations

provide quickly many of the essential features of the 3D problem. At higher impact

speeds, the larger hole diameter of plate impact makes it less sensitive to the effect of

yaw relative to the case of rod impact.

We formulated a new model for yaw impact. It is based on the reasonable assumption

of the hole volume being independent of the yaw. When taking into account the typical

shape of the hole, the model could be fitted to reproduce excellently well the

experimental data. The real test of this model will be to predict other configurations with

different materials, impact speed or (L/D). We consider using similar reasoning to model

oblique rod penetration.

ACKNOWLEDGEMENT

The author thanks Dr. Dani Yaziv for providing the experimental data shown in Fig.1

Without it, perhaps parts of this work might have not been carried out.

Figure 4. The crater cross section at T=60 for the Autodyn

calculation

REFERENCES

[1] D. Yaziv, personal communication, (1994)

[2] D. Yaziv, Z. Rosenberg, J.P. Riegel III, "Penetration Capability of Yawed Long

Rod Penetrators",12th

Int. Symp. on Ballistics, San Antonio, TX, (1990)

[3] D. Yaziv , J.D. Walker, J.P. Riegel III, ”Analytical Model of Yawed Penetration

in the 0 to 90 Degrees range”, 13th

Int. Symp. on Ballistics, Stockholm, Sweden,

(1992)

[4] T.W. Bjerke, G.F. Silby, D.R. Schefffler, R.M. Mudd, 13th

Int. Symp. on

Ballistics, Stockholm, Sweden, (1992)

[5] F. Jamet, M. Giraud, G. Weirauch, "Experimental Simulation of the Terminal

Ballistics of the EFP",10th

Int. Symp. on Ballistics, San Diego, CA, (1987) [6] M. Lee and S.J. Bless, ”A Discreet Impact Model for Effect on Yaw Angle on

Penetration by Rod Projectiles”, Shock Compression of Condensed Matter-

1997,Schmidt et al. eds. AIP CP429,(1998) ,pp. 929 [7] S.J. Bless, J.P. Barber, R.S. Bertke, H.F. Swift, Int.J.Engn.Sci., 16, (1987), pp.

829 [8] G. Luttwak, H. Lenselink, ”Three Dimensional Numerical Simulation of

Oblique Impact and Penetration of Thin Plates by Long Rods”, HDP IV, Fourth

Int. Symp. of High Dynamic Pressures, Tours, France, (1995) [9] G. Luttwak, M. S. Cowler, “Advanced Eulerian Techniques for the Numerical

Simulation of Impact and Penetration, using AUTODYN3D”, Proc. 9th

Int.

Symposium on “Interaction of the Effects of Munitions with Structures”, Berlin,

(1999) [10] G. Luttwak, M. S. Cowler, N. K. Birnbaum ,”Virtual Memory Techniques in

Eulerian Calculations”, in Shock Compression of Condensed Matter-1999, M.

D. Furnish et.al. eds. AIP CP505, NewYork, (2000), pp. 363

[11] D.J. Gee, D.L. Littlefield, Int.J.Imp. Engn. 26, p211, (2001)

[12] D. J. Steinberg, S.G. Cochran, M.W.Guinan, J.Appl.Phys.,51(3), (1980), pp.

1498

[13] "MSC/Dytran V3.0,User Manual", The MacNeal Schwendler Corp. (1996)

[14] “AUTODYN, Interactive Non-Linear Dynamic Analysis Software”, Theory

Manual, Rev 4.0, 101 Galaxy Way, Suite 325,Concorde, CA 94520, USA,

(1997)

Figure 5. The 3D Autodyn simulations at T=20,50μs