The influence of initial nose shape in eroding penetration

1,~. J. Impucr

Engng

Vol.

IS. No. 2. pp. 139-148, 1994

0734743X/94 $6.00 +O.OO ic 1993 Pergamon Press Ltd

Printed in Great Britain

THE INFLUENCE OF INITIAL NOSE SHAPE IN ERODING PENETRATION JAMES

D.

Southwest

(Received

and

WALKER Research

Institute,

2 February

CHARLES

E.

San Antonio,

1993; in revisedjorm

ANDERSON,

JR

TX 78228,

U.S.A.

26 July

1993)

Summary-The effect of projectile nose shape on penetration is examined numerically using the nonlinear, large deformation wavecode CTH. In particular, the impact of a tungsten alloy, long rod projectile into a 4340 steel target is investigated for three different nose shapes: blunt, hemispherical, and conical. The pressures are sufficiently high that erosion of the projectile begins immediately upon impact. The decays of the shocks are examined, as are the resulting material flow fields. Later-time effects are also explored, such as how the nose shape aRects penetration, and how long each case takes to arrive at a quasi-steady-state flow configuration in which the tungsten-steel interface is completely determined by eroding plastic flow.

NOTATION B, C CO E E” G n, m P P” s T T, T, T* 4 UP K I-0 % i* 6 Y P PO bcq

constitutive constants bulk (adiabatic) sound speed specific internal energy Hugoniot specific energy shear modulus constitutive constants (exponents) pressure Hugoniot pressure slope of u, - up curve temperature melt temperature initial (ambient) temperature homologous temperature [(T - TJ/( T, - T.)] shock velocity particle velocity initial yield stress Griineisen parameter equivalent plastic strain equivalent plastic strain rate (normalized by 1.0 S- ‘) compression (1 - p,/p) Poisson’s ratio density initial (ambient) density von Mises equivalent stress

I. INTRODUCTION

It has long been known that nose shape has a dramatic influence on the ability of nondeforming projectiles to penetrate into targets; examples are provided in Refs [l-3]. Mullin et al. [I], performed experiments to measure the residual velocity of conical- and blunt-nose projectiles after perforation of steel plates. They found that the residual velocities of the conical-nose projectiles were higher than those of the blunt-nose projectiles, even though the conical-nose projectiles were less massive (the projectiles had the same length and diameter, but the conical-nose projectile weighed less because of the material removed to achieve a conical nose). In several of the experiments, the blunt-nose projectile failed to perforate the target, whereas the conical-nose projectile perforated with more than a third of the original impact velocity. However, in the cases where the target was sufficiently hard 139

140

J. D. WALKER

and C. E. ANDERSON,

JR

to cause significant projectile erosion, the conical-nose projectiles did not perform as well as the blunt-nose projectiles. Wilkins [2] demonstrated that the type of target failure for thin plates, i.e. petalling or plugging, depends on nose shape. Hill [3] developed an analytical theory to describe the penetration of rigid-body projectiles in armor targets. In particular, he was interested in the phenomenon of “cavitation” where the crater diameter was larger than the diameter of the projectile. He showed that the resistive load to penetration for certain types of “smooth” ogives was independent of the nose shape, but the critical velocity for cavitation was directly dependent on nose shape. However, for conical-nose projectiles, Hill showed that both the resistive load and the extent of cavitation were sensitive to nose shape. Batra [4] numerically investigated the effect of ellipsoidal nose shapes on steady-state penetration by a nondeforming projectile. By varying the ratio of the major to minor axes of the ellipsoidal nose, he was able to examine differences in penetration by a blunt-, hemispherical-, and an ellipsoidal-nose projectile. Batra found that the nose shape significantly affects the deformations of the target material in the vicinity of the projectile/target interface, and that the axial resisting forces are considerably higher for the blunt-nose projectiles compared to the ellipsoidal-nose projectile. Also, he found that target deformations spread to a larger distance away from the projectile/target interface for the blunt-nose projectile. In contrast to the above work, this study is focused on impacts where the velocity is sufficiently high to result in projectile material failure-generally referred to as erosion-at the projectile/target interface. At first glance, a discussion of the effect of nose shape would seem academic, but further reflection would suggest that the initial impact and early penetration behavior should be different for diverse projectile nose shapes. The question to be answered is: how different? In order to obtain a qualitative and quantitative understanding of the dependence of initial penetration on nose shape, a tungsten, long-rod projectile impacting a 4340-steel target was examined computationally. Three different nose shapes were investigated: a blunt nose, a hemispherical nose, and a conical nose of angle 45”. Specific items of interest were the initial penetration history, including shock pressures and subsequent rarefaction waves; the transitioning of the projectile/target interface into a hydrodynamic-like eroding head; and the length of time (penetration depth) required for the three different nose shapes to converge to nominally the same penetration rate. 2. COMPUTATIONS

Computations were performed for a nominal full-scale kinetic energy tungsten-alloy projectile (2.54-cm diameter with a length-to-diameter L/D ratio of 8) impacting a semi-infinite, hardened (R,30) 4340-steel target at 1.5 km/s. The wave propagation code CTH [S] was used, using the two-dimensional cylindrical-symmetric option. CTH uses a van Leer algorithm for second-order accurate advection that has been generalized to account for a nonuniform and finite grid, and multiple materials. Further, CTH has an advanced material interface algorithm for the treatment of mixed cells [5]. Also, CTH has been modified to account for more realistic constitutive treatment of material response by allowing the flow stress to be functions of strain and strain rate [6,7]. The metals were modeled as elastic-plastic materials, and the Jaumann stress rate was used. The Mie-Griineisen equation of state was used to calculate the thermodynamic response of the metals, and the Johnson-Cook constitutive model [8] was used to calculate the flow stress. Fracture, based on a maximum tensile stress, was allowed. The various values used for the study are given in Tables 1 and 2 (for a discussion of the constitutive values, see Ref. [9]), along with the appropriate expressions relating the material constants. The minimum zoning for results reported in this paper was 20 zones across the radius of the projectile. The zones in the target, radially out to twice the projectile radius, were square. There were 105 zones in the radial direction, of which 40 zones were of 0.0635-cm size, leading into 65 zones with a geometric growth of 5%. Along the axis were 190 zones. Starting from the projectile/target interface, the 55 zones in the projectile grew geometrically

The influence TABLE

I.

of initial

EQUATION

141

OF STATE PARAMETERS

P. (g/cm7 Tungsten 4340 Steel

nose shape

co (km/s)

17.37 7.85

3.85 4.5

A

r. (-)

1.44 1.49

1.58 2.17

u,=c,+su, P = P, +

p,r,(E - Ed) TABLE

r, Tungsten 4340 Steel

2. CONSTITUTIVE

PARAMETERS

(GW

B @Pa)

(1,

C (-)

(“1,

T Inch W)

G GPa)

CL,

Frac. stress (GM

1.350 0.735

0.0 0.473

0.26

0.06 0.14

I .03

1793

119 77.6

0.30 0.29

2.4 2.0

towards the projectile rear at 6% from a size of 0.0635 cm at the front. In the target, the first 80 zones were 0.0635-cm in size (the first 10.16 cm in the target), leading into 55 zones which grew at 6%. The projectile was 20.32-cm long (L/D = 8), but this has little bearing on the results presented here since the rear of the projectile is too far from the interface to affect the penetration process reported here. The radius and height of the target block were 30.48 cm. The target surfaces at the edge of the mesh were given a transmitting boundary condition to make the target appear “semi-infinite”. 3. RESULTS

AND

DISCUSSION

3.1. Impact pressureand release The first obvious effect of nose shape is its influence on impact pressures. Figure 1 shows the pressure versus time on the centerline at the projectile/target interface. The insert to the figure provides an expanded scale of the first 10 ps. For the blunt nose, the early response at the centerline is the same as would be observed in a plate impact. After an initial rise of a few tenths of a microsecond, the numerical simulation has a nearly 2 /JS plateau with 44.2 GPa pressure and a particle velocity of 960 m/s. The computational result is in excellent agreement with the one-dimensional shock jump calculations which give (assuming hydrodynamic behavior and the EOS parameters listed in Table 1) a pressure of 44.2 GPa and a particle velocity of 952 m/s. Thus, for the blunt nose, the early response at the centerline is the same as would be observed for a plate impact, i.e. uniaxial strain conditions prevail. As the penetration proceeds, rarefaction waves emanating from the free surfaces of the projectile and target relieve the high pressure, and the pressure at the interface drops to zero. The stresses become tensile within the projectile, and extensive fracturing occurs in a region just above the projectile/target interface. The fracture algorithm in CTH inserts void into a computational cell to preclude excessive tension, thereby simulating fracture. At 6 ps, a region from the interface extending up the penetrator one projectile radius has fractured. The fracture algorithm has inserted void into nearly 75% of the zones in this region. The oscillations between 10 and 20 ps in the pressure curve of the blunt projectile are caused by the collapse of these voids. Theoretically, the peak pressure of 44.2 GPa should occur from the impact of the hemispherical-nose projectile. However, the peak pressure was found to be zone-dependent in the calculations (see Section 4 for further discussion), and is shown to be approximately 34 GPa in Fig. 1. This peak pressure is of very short duration-less than a 0.1 ps-and

J. D. WALKER and C. E. ANDERSON,JR

142

30 ';; 4 -

25

ki zE

20

B

15

0

10

20

30

40

50

TIME (ps) FIG. 1. Centerline interface pressure versus time.

then the pressure decays (the pressure decay was not sensitive to the zone sizes investigated). Of the three nose shapes, the hemispherical nose is the one that reaches a steady pressure most quickly since the initial projectile shape is the most similar to the hydrodynamic hemispherical-like flow field achieved in steady-state penetration. A small amount of fracturing occurs around 8 ps. The impact of the conical-nose projectile results in the lowest peak pressure (15 GPa) since the conical shape allows immediate release from free surfaces. Fairly large oscillations occur initially in the pressure-time history for the conical-nose projectile. These were found to be numerical oscillations, and are discussed in Section 4. Figure 1 shows that an average pressure of roughly 12.5 GPa is maintained until the conical nose is completely embedded in the target, at which time the pressure rapidly drops to the steady state pressure of 6.2 GPa. No fracturing occurred in the conical-nose case. 3.2. Impulse

The time integrals of the pressure on the centerline, i.e. the specific impulse, at the interface are shown in Fig. 2. It is remarkable that with such different early pressure-time histories these impulses quickly (15 ps) become the same. As with the pressure response, the hemispherical-nose case lies between the two extremes of the blunt and conical noses. There are two regions evident in Fig. 2. The first region is at very early times, although different in specifics for each projectile considered, where the impulse is rapidly increasing because of the higher pressures that occur, and the initial resistance of the target to material motion and flow. Later in time, and after the release from the high pressures resulting from the initial impact, the slope of the impulse-time response is representative of a more pseudo-steady-state penetration in which projectile and target material are flowing. 3.3. Penetration velocity

The velocity of the projectile/target interface is, by definition, the penetration rate. Figure 3 shows the penetration velocities versus time; a detail of the first 20 ps is shown in Fig. 4. The penetration velocity for the blunt-nose projectile has an initial plateau, and then increases as rarefaction waves from the free lateral surface of the projectile and the top surface of the target release the “geometric” confinement of impact; the state of stress changes from uniaxial strain to a more complicated state of stress. For the blunt- and

The influence of initial nose shape

143

400 350 300 z P’ .l:: c0 w a+I 2

250 200 150 100

50 0 20

IO

30

40

50

TIME (ps) FIG. 2. Specific impulse versus time.

I .6 1.4

1.2

penetrationvelocity

0.4

--es s

0.2

Conical Hemispherical Blunt

0.0

1

0

20

1

1

40

I

I

60

I

80

I

100

TIME (ps) FIG. 3. Centerline velocities versus time.

hemispherical-nose cases, stress release waves allow radial motion of target material, thus releasing confinement and allowing easier penetration; hence, the rapid increase in penetration velocity. The penetration velocity subsequently decreases, however, since the lateral motions fall to lower velocities as the high pressures from the shock are attenuated and the nose of the projectile moves into the target and away from the free surface. All three nose cases have an early peak velocity around 1.05-l. 1 km/s. The conical-nose projectile maintains this high, early penetration velocity the longest-for nearly 10 ps. This is due to the ease with which the conical-nose projectile initially produces a crater. The target material, which must be moved out of the way of the projectile, is able to travel along a path close to the conical-nose shape. This path is

J. D. WALKER and C. E. ANDERSON,JR

144

+

Conical

+

Hemispherical

+

Blunt

FIG. 4. Centerline penetration velocities for first 20 ps after impact.

shorter and closer to the target surface, so there is less confinement, giving rise to less resistance to target flow than for either the blunt- or hemispherical-nosed projectiles. However, this advantage is short-lived, for by 30 ps, the conical nose has the lowest penetration velocity. At this point, the conical nose is being eroded away, and is being replaced by a hydrodynamic (more hemispherical) interface. The hydrodynamic flow requires that there be a minimal diameter crater, which the “easy” motion in the early-time conical-like flow did not produce. Thus, the eroding conical-nose projectile is now required to do more work to open up a crater large enough for the quasi-steady-state flow regime, thereby resulting in a decrease in penetration velocity. The early-time plateau in the blunt-nose interface velocity has already been discussed to some extent. The subsequent superposition of rarefaction waves fractures the blunt-nose projectile. As the penetration event proceeds, the projectile motion closes these voids, and by 20 /s, the interface velocity is converging to the quasi-steady-state penetration velocity. The blunt-nose projectile maintains a higher interface velocity from this time on because the initial shock put more target material in motion, which aided in subsequent crater formation. Once again, the hemispherical-nose case is between the other two. The hemispherical nose shape begins moving material in the directions necessary for obtaining steady flow, but it does not have the advantage of the duration of a strong shock initially to begin moving a lot of target material. At 100 ps, the velocities for all three cases are nominally the same. 3.4. Comparison of penetration flows

Figure 5 shows five snapshots for each projectile nose’ shape: one at 2 ps, one at 6 ps, one at 10 /s, one at 30 ps, and finally one at 100 ps. Pressure contours are plotted to the left of the centerline (note that a tensile contour is also plotted), while contours of velocity magnitude (the square root of the sum of the squares of the axial and radial velocities) are plotted to the right of the centerline. The contour values are the same for all figures. Initially, the largest pressures are caused by the blunt projectile, but at 2 ps the rarefactions are clearly evident (rarefaction waves are travelling in towards the axis). The pressures for the hemispherical-nose projectile are slightly higher than for the conical-nose projectile, but in contrast to the blunt projectile case, no large tensile stresses develop in either the hemispherical- or conical-nose cases.

The influence

i

-1

0

-*

7

1

-6

-6

-I

0 1 lcml

1 f-1

Hem~pherical

Conical

FIG. 5. Pressure

of initial

and velocity

contour

nose shape

2

4

145

-6

-1

-2

0 I (cml

BlUlIl

plots at 2, 6, 10, 30 and 100 ps.

I

4

146

J. D. WALKERand C. E. ANDERSON,JR

The pictures at 6 ps are quite revealing. The extent of the high pressure and material flow region is dependent upon the nose shape. The conical nose produces a flow region in the target that is half the radius of the flow region produced by the blunt nose. This is consistent with the arguments presented above for the higher penetration velocity by the conical-nose projectile at early times. The pressure wave traveling up the projectile remains at higher magnitudes for the blunt case (the center pressure is over 10 GPa), with the second highest pressure magnitudes for the hemispherical-nose case, and the lowest pressure in the conical case. However, these very high pressures decay before the wave travels very far. One way to see this is to note that the upper curves in Fig. 3 present the back-end (tail) velocity of the projectiles. As all the’projectiles have virtually the same back-end response, the stress waves arriving at the back end must be essentially the same. Also evident is the tensile region in the blunt projectile; material has fractured in this region. Such a region is absent in both the hemispherical and conical cases. At 10 ps, tensile regions are evident in both the blunt and hemispherical cases, and fracture in the projectile has occurred for both cases. The conical nose is nearly fully embedded in the target, and the oscillations in the pressure and interface velocity curves (Figs 1 and 3) will soon cease. At 30 ps, the projectiles are beginning to erode, as is most clear for the blunt case. The initial nose shapes are still evident, and the flow region for the blunt case is still larger than for the other two cases. The crater diameter is largest for the blunt-nose projectile, with the hemispherical-nose projectile having the next largest diameter, followed by the conical-nose projectile. The shapes of the interface regions are approximately the same at 80 ps (not shown) for all three cases. It takes nearly 80 ps (which is equivalent to approximately 2.4 projectile diameters of penetration for the specific impact conditions investigated here) before the three cases have the same crater diameters near the projectile/target interface. The crater diameters near the interface are about the same, although, of course, the conical- and hemispherical-nose cases have smaller diameter holes nearer the surface. At this point the interface velocities are nearly the same, and the effects of the initial nose shape are over. At 100 ps, the depths of penetration are approximately 3.2 projectile diameters. Specifically, they are 7.95, 8.15, and 8.14 cm for the conical, hemispherical, and blunt case, respectively. These depths differ by less than 3%. 4. NUMERICAL

ISSUES

The numerical results provided some surprises. First, it was found that the maximum pressure resulting from the impact of the hemispherical-nose projectile was sensitive to the zoning. Calculations were performed for 20,40, and 80 zones across the radius; the results are shown in Fig. 6. The maximum pressure seen at the interface was 34.4 GPa (at 0.3 p(s), 38.5 GPa (at 0.13 ps), and 40.94 GPa (at 0.07 ps), respectively. (With such fine zoning, it took many integration time steps to reach these times.) Rarefaction waves from the hemispherical free surface begin unloading the shock at these times. Although the one-dimensional peak shock pressure should be present, the discretization of the continuum equations and the presence of an artificial viscosity, to prevent excessively large numerical oscillations [lo], results in a finite rise time for the shock. This finite rise time is zoning dependent. However, the arrival of the rarefaction waves is considerably less zone size dependent, and thus unloading of the peak pressure occurs near the “theoretical” time of arrival of the rarefaction waves, providing the zoning is sufficient for overall numerical convergence (which is typically 5-7 zones across the radius for this type of problem). During the embedment of the conical nose in the target, the pressure at the interface oscillates considerably for the first 12 ps, as is seen in Fig. 1. The oscillation period depends linearly on zone size, as was discovered by also running the conical case with 10 and 40 zones across the radius. It was found that the oscillation period is roughly that for the material to cross a zone, i.e. the period of the oscillation is roughly given by the zone size divided by the penetration velocity (approximately 1 km/s). Increasing each of the artificial

The influence

of initial

nose shape

147

50 45 40 35 30 25 20 15 10

0

I

2

3 llME

FIG.

6. Centerline

interface

pressure

for

5

4

6

(p)

a hemispherical-nose resolution.

projectile

for

different

zoning

viscosity coefficients by a factor of 10 had little effect on the oscillations (increasing the artificial viscosity coefficients did damp the oscillations a bit more quickly, but for the first 5 ps, they were roughly the same). These oscillations are not due to the strength of the material, or the fact that two different materials are colliding. In calculations of a like-into-like fluid impact with a pointed nose (fluid) projectile, the same oscillations were seen, and although the pressure values differed, the oscillation frequency did not. The oscillations occur in plane strain geometry, proving they are not due to a centerline problem, and also occur if the plane strain zoning is rotated at 45”. The frequency of the oscillations increases with a large angle on the front of the projectile (making the projectile more blunt). The peaks in the oscillations seem to appear with the same frequency at which cells along the interface become involved in the collision process. Velocities are face-centered in CTH. Similar calculations were done with the hydrocode CALE [l 11, where the cell velocities are defined on the corners of the cells; similar results were obtained. (CALE is a two-dimensional arbitrary Lagrangian-Eulerian computer code-the calculation described here was run in the full Eulerian mode). An attempt was made to remove the oscillations by adjusting the constant associated with the anti-hourglass viscosity in CALE. Although this had a large effect on the results (the magnitude of peaks was altered, not always decreasing), the frequency of the oscillations did not change. For a large value of hourglass viscosity, a phase shift was introduced, but the oscillations remained. Whether a more general tensor artificial viscosity would remove these oscillations, which are clearly a numerical artifact, is an open question. 5. CONCLUSIONS

The effects of nose shape (blunt, hemispherical, and conical) on the initial impact flow conditions of a tungsten-alloy projectile into an armor-like steel target were investigated numerically. It was found that the largest differences in early time behavior were the high pressures the blunt-nose projectile delivered to the target and itself, and subsequent effects due to these high pressures. Superposition of rarefaction waves led to extensive fracture in the blunt-nose projectile not seen in the other two projectile cases examined. The larger pressures for the blunt case also gave rise to a larger material flow zone in the target,

J. D. WALKER and C. E. ANDERSON,JR

148

which led to a reversal of which nose shape case had the largest penetration velocity. By 80 ps, the original nose shape effect was gone, and penetration velocity was nearly the same for all three cases. The specific impulse along the centerline was the same in all cases after full nose embedment. In general, the results for the hemispherical-nose projectile are between the two extremes of the blunt and conical noses. Also pointed out in the paper are certain numerical issues, some of which are understood, and some of which are unresolved. Finally, in retrospect, the conclusions by Batra [4] have many things in common with the conclusions drawn in this study. This is somewhat remarkable since this study focused on the transient behavior of an eroding projectile until steady-state penetration, whereas Batra investigated steady-state penetration of a nondeforming projectile. REFERENCES I. S. A. MULLIN, J. P. RIEGEL III, D. A. TENENBAUM and D. W. ERDLEY, Dynamic plasticity modeling of conical and blunt nosed projectiles and dual layer armor. 1991 TACOM Combat Vehicle Suruiuobility Symposium. Gaithersburg, MD, April 15-17 (1991). 2. M. L. WILKINS, Mechanics of penetration and perforation. Int. J. Engng Sci. 16(11), 793-807 (1978). 3. R. HILL, Cavitation and the influence of headshape in attack of thick targets by non-deforming projectiles. J. Me&

Phys.

So/ids

28, 249-263

(1980).

4. R. C. BATRA. Effect of nose shape and strain-hardening on steady state penetration of viscoplastic targets. In Computational Plasticiry, Models, Sofrware and Applications (edited by D. R. J. OWEN, E. HINTON and E. ONATE) 463-475, Pineridge Press, Swansea, U.K. (1987). 5. J. M. MCGLAUN, S. L. THOMPSON and M. G. ELRICK, CTH: a three-dimensional shock wave physics code. lnt. J. Impocr Engng 10(&4), 351-360 (1990). 6. W. W. PREDEBON, C. E. ANDERSON Jr and J. D. WALKER, Inclusion of evolutionary damage measures in Eulerian wavecodes. Comput. Me& 7(4), 221-236 (1991). 7. S. A. SILLING, Stability and accuracy of differencing methods for viscoplastic models in wavecodes. J. Comp. Phys. 104(l), 3&40 (1993). 8. G. R. JOHNSONand W. H. COOK, Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Engng Fracr. Mech. 21(l), 31-48 (1985). 9. C. E. ANDERSON Jr and J. D. WALKER, An examination of long-rod penetration. lnt. J. Impact Engng. 11(4), 481-501 (1991). 10. C. E. ANDERSONJr, An overview of the theory of hydrocodes. Int. J. Impact Engng S(ld), 33-59 (1987). 11. R. TIPTON, CALE Llsers Manual, Version 901101. Lawrence Livermore National Laboratory, Livermore, CA (1990).