Analysis of the Noneroding Penetration of Tungsten Alloy 

Long Rods Into Aluminum Targets 

by Steven B. Segletes 

ARL-TR-3075 September 2003 

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Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5066 

ARL-TR-3075 September 2003 

Analysis of the Noneroding Penetration of Tungsten Alloy 

Long Rods Into Aluminum Targets 

Steven B. Segletes 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited.
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Analysis of the Noneroding Penetration of Tungsten Alloy Long Rods Into 

Aluminum Targets 

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Steven B. Segletes 

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13. SUPPLEMENTARY NOTES 

14. ABSTRACT 

Data concerning the rigid/eroding-rod threshold transition are reported for hemispherical-nosed tungsten rods penetrating 

into thick 5083-aluminum targets. Presented data quantitatively buttress existing explanations. The current analysis suggests

that the penetrator must bring to bear a different “apparent” strength in the noneroding- vs. the eroding-penetration regimes. 

Conventional one-dimensional penetration analysis reveals that the noneroding datum is wholly consistent with the notion of 

treating the rod as if it penetrated in a rigid-body fashion, possessing unrealistically high yield strength. Study of a recovered 

rod fragment reveals that the penetrating rod nonetheless deformed, but did so without erosion. Such an observation for 

hemispherical-nosed rods is consistent with past qualitative explanations posited for ogival-nosed rods. The phenomenon, 

supported by analysis, is that an exaggerated stress was axially applied by the rod to the target interface, composed of both 

the rod’s intrinsic yield strength plus a confining stress caused by a lateral interference fit between the rod and target during 

the penetration event. The lateral interference of the target was kinematically sufficient to prevent an erosive flow field from 

being established in the rod. In such a fashion, the rod was able to employ the target’s lateral resistance to great axial 

advantage. 

15. SUBJECT TERMS

noneroding, penetration, rigid-body, tungsten, aluminum, analysis 

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Steven B. Segletes 

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iii

Contents 

List of Figures iv

Acknowledgments v

1. Introduction 1

2. Observation 4

3. Analysis 8

4. Interpretation 11

5. Conclusions 15

6. References 17

Distribution List 20
iv

List of Figures 

Figure 1. Model predictions and data for eroding WA rods into BHN 97 Al-5083 targets. ..........5 

Figure 2. Residual rod-tip fragment recovered from noneroding test, with hemispherical nose 

intact...........................................................................................................................................6 

Figure 3. Crater cross-section of low-velocity test at a depth ~228 mm below the target 

surface (residual rod fragment shown for scale)........................................................................7 

Figure 4. Model predictions and data of penetration and residual-velocity, for (rigid and Y =

1.1 GPa) WA rods into 344 mm thick BHN 97 Al-5083 target (target rear surface 

modeled with plastic-zone extent: 3.5; spall resistance: HSPALL = H/3 [1, 7])........................12 
v

Acknowledgments 

The author would like to thank Ms. Eleanor Deal and the staff of the Lethal Mechanism Branch’s 

Experimental Facility 110 for performing the experiments documented in this report. Also, the 

author would like to express appreciation to Mr. William Edmanson and Dr. Todd Bjerke of the 

Impact Physics Branch’s Solid Mechanics Team for lending photographic support to this effort. 

The author would like to graciously thank Messrs. Dan Scheffler, Konrad Frank, and Dr. Lee 

Magness for their consultations and pointers to the existing body of literature on the subject. 
vi

INTENTIONALLY LEFT BLANK. 
1

1. Introduction 

This report details a ballistic result observed from the impact of a hemispherically capped 

tungsten alloy rod into a monolithic aluminum-5083 (Al-5083) target. The testing was 

conducted in the summer of 2001. The data span impact velocities over what is known as the 

transitional range, between so-called “rigid-body” penetration and eroding-body penetration, and 

provide corroborative evidence to existing theories on the subject. What makes the data of 

particular interest is the nature of the noneroding phase of penetration, below the threshold 

velocity, which was observed as not truly rigid, yet at the same time noneroding. The three tests 

comprising the presented data were part of a larger test series. This larger test series will be 

addressed by the author in a separate report (manuscript in preparation), whose focus will not be 

solely limited to the issue of threshold velocity. 

We analyze the data presented here in terms of a one-dimensional penetration model (1), itself an 

adaptation of Tate-Alekseevskii (2, 3) and follow-on methodologies (4−6). The model (1) has 

previously been employed (7) to describe, very successfully, the penetration/perforation behavior 

of small-caliber (14.5 mm B32) munitions through laminate targets. 

The ballistic response in the low-velocity regime is uniquely different from the response in the 

higher velocity eroding-penetration regime, where most research attention has been directed in 

recent decades. In the low-velocity regime, interesting behavior has been noted wherein the 

penetration is observed to exceed the penetration of the identical configuration at higher impact 

velocities. Kinslow (8) noted in the 1960s the disparity between the low-velocity and highvelocity penetration behaviors, particularly for the case of hard-projectile/ductile-target 

configurations. Brooks and Erikson (9) studied the low-velocity regime in 1971 and noted a 

profound dependence upon both rod-material as well as rod nose-shape, for hard-rod/ductiletarget combinations. They noted this unique behavior to be a strong function of the cone angle in 

conical-nosed rods and found ogival-nosed rods to exhibit this elevated-penetration behavior 

more profoundly, a response so different from the conical-nosed rods that they deemed the result 

“anomalous.” Their experimental results led them to infer that the ogival rod’s “nose is 

supported hydrostatically by the target material.” This observation will be important to the 

current report, as it attempts to connect the elevated-penetration behavior not only with the nose 

shape, but also with a multi-axial stress state imposed on the rod by the target. 

In 1980, Hill (10) recounted his work dating back to World War II. His interest here concerned 

the nature of penetration prior to and the onset of what he called ballistic “cavitation.” Hill 

defines ballistic cavitation as penetration wherein the crater formed by the penetrating rod 

becomes larger than the diameter of the rod itself, a kinematic condition later noted by 

Wijk (11) as necessary to permit rod erosion. He understood that noncavitating (thus, 

noneroding) penetration, associated with lower velocity penetrations, was more efficient than 
2

cavitating penetration, wherein the extreme case “the missile itself deforms and the entire physics 

is different in kind.” Furthermore, the shape of the rod’s nose was seen as an overarching factor in 

the behavior because the nose shape was primarily responsible for determining the velocity at 

which cavitation might manifest itself. Hill notes that, in noncavitating penetration, the average 

axial stress (axial load per unit area) in the rod is dependent on “the mean work to form unit 

volume of the resultant (target) cavity,” and not upon the inertial stresses of impact (i.e., those 

stresses proportional to ρV2

, where ρ is density and V is velocity). 

Forrestal et al. (12), in 1988, formulated a more detailed description of noneroding penetration 

that accounted for rod’s nose shape and, unlike Hill (10), included inertial terms in addition to 

target strength terms in the formulation of the axial stress component. The inertial component of 

axial stress in the Forrestal model was shown to be proportional to (V/c)

2

, where c is the bulk sound 

speed. However, the proportionality constant is such that the contribution of this term to the axial 

stress is often minor for material combinations and noncavitating impact velocities of interest. In 

either case, neither Hill nor Forrestal et al. address the specific case of current interest, wherein the 

average axial stress at the rod’s nose exceeds the rod’s intrinsic strength, even as the penetration 

remains noneroding. In the same trend, Forrestal’s later coauthored works (13−16), validating the 

merit of the 1988 theory, retained impact conditions that specifically sought to exclude rod 

plasticity. 

In contrast, Woodward was very much interested in the low-velocity penetration regime where 

explicit penetrator plasticity was, nonetheless, evident. His 1980s papers (17, 18) focus solely on 

conical-nosed penetrators, for which he showed the low-velocity penetrating-flow structure to be 

unique. When the rod angle is not so small, such that the narrow tip bends or buckles outright, 

there instead occurs a flow separation of crater material from the rod at the location where the 

conical rod tip merges with the cylindrical rod shank. This flow separation allows the rod shank to 

strain radially, while the conical tip is simultaneously restrained by the crater. If the impact 

velocity is large enough, the resulting shear strain at this location causes a shearing fracture in the 

rod, between the shank and conical nose. Thereafter, the embedded conical tip appears to remain 

rigid and is pushed through the target by an eroding cylindrical shank that bears down upon it. 

Woodward noted the penetration capacity to be dependent upon the included angle of the conical 

rod tip. This cone-angle dependence of penetration capacity can be attributed to the relative ease 

by which a sharp nose embeds in a target material vis-à-vis a blunt nose. From a traditional onedimensional penetration analysis point of view, this dependence would, at minimum, show up in 

the target’s kT “shape factor” parameter of the inertial stress, kTρTU 2, where U is the penetration 

velocity of the event. Segletes et al. (7) noted that the functionality of this variation should 

proceed as 1/2·(1 − cosδ), where δ is the half angle of the rigid conical nose of the rod. 

While much of the cited investigation through the 1980s was experimental and/or analytical in 

nature, some more recent efforts have adopted computational approaches to studying the lowvelocity penetration regime. In some cases, like that of Chen (19), perforation by rigid penetrators 

was of interest, and so rod plasticity was effectively precluded by modeling the projectile with a 
3

small number of coarsely resolved high-strength elements. Yatteau and Dzwilewski (20), in 1995, 

augmented their experiments with code computations to assist in the analytical modeling of 

conical-nosed penetrators at low velocity. Their resulting model appears fully consistent with the 

concepts espoused by Woodward (17, 18) on the phenomenology of low-velocity conical-tip rod 

penetration. Specifically, the model allows for the embedded conical rod tip to fracture from the 

rod shank and remain rigid while the shank bearing upon it is permitted to erode. 

More recently still, Scheffler (21, 22) employed computational modeling to examine the 

noneroding phenomenon for tungsten alloy rods, with ogival and hemispherical nose shapes, upon 

aluminum targets. Interestingly, Scheffler’s stated intent was to test a new computational 

algorithm in the CTH code (the BLINT, or Boundary Layer INTerface algorithm), intended to 

provide needed improvements for modeling noneroding penetration. Scheffler’s computations 

with CTH, using BLINT, compared favorably with the experiments of Magness (23), which were, 

to that point, unpublished. Both Magness’ data and Scheffler’s computations corroborated the 

unusual penetration behavior that Brooks and Erikson (9), decades earlier, had deemed 

“anomalous.” In particular, even as the transition velocity for ogival-nosed rods was significantly 

higher than that for hemispherical-nosed rods, in both cases the penetration levels just below the 

erosion-threshold velocity were observed to be significantly more than double that of experiments 

and computations at striking velocities above this transition velocity. And while Scheffler referred 

to this type of penetration as “rigid” (rather than noneroding), he nonetheless notes that, in his 

computations, the rods indeed deformed. In the case of the hemispherical-nosed rods perforating 

finite plates, Scheffler’s computations (21, 22) generally revealed an increased diameter near the 

front of the rod and a significantly flattened nose profile, resulting from the target interaction. 

Magness and Scheffler (23) most recently addressed this issue in greater detail, for both tungsten 

alloy and depleted uranium penetrators, and formulated a phenomenology for how the process 

occurs for ogival-nosed rods, which we will quote at some length. Just below the erosionthreshold velocity, based upon their computations of ogival-nosed penetrators, they describe the 

penetration phenomenology thus: 

“The rapid [radial] displacement of the armor material initially causes it to lift off 

the surface of the ogival nose. Penetrator material located immediately behind the 

contact surface is not confined by armor material and deforms until contact is reestablished with the wall of the penetration cavity. This process continues as the 

projectile burrows into the target. This deformation appears as a high strain-rate 

region where the shank of the penetrator meets the enlarged head. This high strainrate region propagates toward the rear of the rod as the penetration continues. 

Strain rates drop rapidly in the head of the penetrator once it is constrained by the 

surrounding armor material and loaded under triaxial compression.”
4

Above this erosion-threshold transition, as the penetration levels drop, they note that the 

phenomenology of the ogival-nosed penetration computation seems to follow the 

phenomenology described by Woodward for conical-nosed penetrators, in which the imbedded 

penetrator nose is rigidly pushed ahead of an eroding-rod shank. No simulations are presented 

for the hemispherical-nosed rods, though the data cited for the tungsten alloy hemi-rods appear 

the same as presented previously by Scheffler (21). 

Forrestal and Piekutowski (24) present much data of a similar character. In their work, 6061-

T6511 Al targets are penetrated by hemispherical-nosed steel rods of hardnesses RC 36.6, 39.5, 

and 46.2. While the work here employs tungsten alloy (WA) rods, both rod hardness and target 

hardness were comparable to that employed by Forrestal and Piekutowski. To model their data, 

they use an eroding model above the erosion threshold and a rigid model below it, in order to 

recreate the observed penetration discontinuity at the threshold. Forrestal and Piekutowski also 

note (with radiography of rods embedded in targets) significant rod-bulging deformities below 

the erosion threshold, which is to say, among those data for which rigid-body modeling was 

applied. No hypotheses are, however, offered by Forrestal and Piekutowski as to how and why a 

significantly deforming rod should be treated as “rigid” in an analysis, and more importantly, the 

kinematic justification for why such a rod fails to erode in the first place. We intend to explore 

such issues here. 

Earlier work of Piekutowski et al. (25) using nearly identical materials, but ogival instead of 

hemispherical rods, demonstrates (as have many other authors) the dependence of the transition 

phenomenon on rod-nose shape. There, the transition to deforming and subsequent rod erosion 

was only observed at significantly higher striking velocities. 

Like Brooks and Erikson (9) before them (who noted “hydrostatic” support of the ogival 

rod nose), Magness and Scheffler (23) attribute the presence of a triaxial compression to the 

phenomenology of ogival-nosed penetration below the erosion-threshold velocity. The work of 

this report will be to present data that further corroborates these explanations of Brooks and 

Erikson (9) and Magness and Scheffler (23), while providing additional information about the 

deformations and stress states associated with the noneroding penetration of hemispherical-nosed 

rods. 

2. Observation 

In the present study, the ballistic threat consisted of a hemispherical-nosed, WA rod of the 

following characteristics: 65 g, length to diameter ratio (L/D) of 15, a content of 93% tungsten 

(W), 6.3% nickel (Ni) and 0.7% iron (Fe), and swaged (reduced in cross-sectional area) 8%. The 

W-Ni-Fe rods had a nominal hardness of RC 37, with nominal dimensions of 101.9 mm long 

× 6.79 mm diameter. The target in these tests consisted of a stack of six 8 × 8 × 2.25 in 
5

(nominal) Al-5083 blocks. The blocks were measured at a Brinell hardness of BHN 97 and a net 

target thickness of 344 mm. 

The three experiments were conducted at impact velocities, V, of 1108, 1416, and 1701 m/s 

respectively. The two higher velocity (1416 and 1701 m/s) tests were above the 

erosion-threshold velocity and achieved penetrations, P, of 193.2 and 227.2 mm (Figure 1), 

respectively. These data are in agreement with the penetration model’s prediction, if the rod’s 

strength is characterized (with Tate-Alekseevskii parameters) at Y = 1.1 GPa, with the Al-5083 

target resistance of H = 1.78 GPa. Such a characterization is wholly consistent with an RC 37 

WA penetrator and a BHN 97 Al target, (σULT = 0.31 GPa), when one uses Tate’s formulation 

for estimating H. 

Were the remaining test, conducted at an impact velocity of 1108 m/s, to lie above the 

erosion-threshold velocity, one might have expected a penetration of ~164 mm into the Al-5083 

target in question (Figure 1). However, this low-velocity test fell below the erosion-threshold 

velocity, and the rod in the experiment perforated the 344 mm target with a residual velocity, Vr, 

of 314 m/s. Additionally, a very fortunate circumstance arose in which the leading segment of 

the residual penetrator was recovered intact after the test. At first glance, the tip of the residual 

penetrator fragment, recovered after the test seemed to indicate that the rod had penetrated in a 

V (m/s)

0 500 1000 1500 2000

P (mm)

0

50

100

150

200

250

P(1108 m/s) ~ 164 mm

Datum:

P(1108 m/s) > 344 mm

Figure 1. Model predictions and data for eroding WA rods into 

BHN 97 Al-5083 targets.
6

rigid-body fashion, as the hemispherical nose appeared intact and completely undeformed 

(Figure 2). Radiography behind the target captured two residual rod segments on film: the first, 

61.5 mm long, which included the rod tip, and the second, 18.5 mm in length. There may have 

been additional rod segments to follow, accounting for the remaining 21.9 mm or original rod 

length; unfortunately, the exposure times of the radiographs would have placed them just inside 

the target and hence, not viewable. Note that the recovered fragment of Figure 2 is the leading 

part of the 61.5 mm segment, which subsequently fractured upon impact with the wall of the 

testing facility. 

Figure 2. Residual rod-tip fragment recovered from noneroding test, with hemispherical nose intact. 

Interestingly, a careful measurement of the post-mortem target crater and fortuitously recovered 

residual-rod fragment (Figure 3) from the low-velocity experiment revealed the following data: 

the rod diameter, as a result of the ballistic event, increased ~6% from 6.76 mm prior to the test 

to 7.19 mm; the crater diameter (except at the front entrance and accounting for yaw-induced 

crater ovalness) was ~6.40 mm. While there is some scatter in the crater diameter 

measurements, the final crater size was, nonetheless, observed to form an interference fit with 

not only the recovered residual fragment (7.19 mm), but also an undeformed rod of identical 

nominal dimensions (6.79 mm). While it is more complicated to estimate the actual rod and 

crater diameters during the course of the test, a co-equal value would have been necessary in 

order to permit the penetration to proceed. The fact that the postmortem rod/target diameter 

values differ is indicative of an elastic residual-stress relief (i.e., crater-wall rebound) that has 

occurred following the passage of the rod. And while there is surely a finite time necessary for 

the crater wall to elastically rebound as the deformed rod’s diameter varies along its length, it is 
7

Figure 3. Crater cross-section of low-velocity test at a depth ~228 mm below the target surface 

(residual rod fragment shown for scale). 

highly plausible that, not only the tip, but much of the rear of the rod was also laterally engaged 

with the target during the penetration event. Because of difficulty associated with edge detection 

of radiographic images, it was impossible to estimate, from the exiting residual rod fragment 

images, what length of the rod tip was bulged in diameter. And while the fracture and 

subsequent separation of the two radiographed rod segments is indicative of a tensile fracture 

61.5 mm from the rod tip, association of this dimension with rod bulging would be conjectural, 

since both the rebound dynamics of the crater and the tensile reflections of the axial wave, off the 

rear of the projectile, might both influence this event. 

Comparable results to this in the literature, for hemispherical-nosed rods, would include those 

reported by Scheffler (21, 22). Unfortunately, the only simulation detailed, in the case of the 

deep penetration results (21), was at an impact velocity above the erosion threshold 

(1296 m/s), and so the computationally predicted rod deformation was significantly more 

extensive than observed presently at 1108 m/s. In the case of finite target perforations (22), there 

is a datum at 1147 m/s for the case of a hemispherical-nosed rod against a 76.2 mm thick 

Al-5083 target. In addition to the vast difference in target thickness (76.2 mm vs. 344 mm 

presently), the rods reported on by Scheffler (22) were 95% W, compared to the 93% rods tested 

for this report. The depiction of the radiographically imaged residual penetrator is too small and 

coarse to detect deformations near the rod tip, and Scheffler characterizes the experiment as 

“rigid.” It is difficult to discern whether his corresponding CTH simulations show a visible 

flattening of the rod tip, because of the limited simulation resolution and the presence of Eulerian 
8

mixed-cells. The simulations do, however, show an equivalent plastic strain between 5% and 

12% in the leading one to three rod diameters of rod length (the plastic extent depending upon 

the simulation mixed-cell methodology). Such an observation of strain is compatible with the 

presently recovered fragment of Figure 2, which exceeds two rod diameters in length, with a 

radial strain of 6% (plastic incompressibility would imply an axial strain, here, of −12%). 

The photographs in the work of Forrestal and Piekutowski (24) also corroborate the present 

observation of a noticeably expanded, but noneroded tip diameter, at velocities below the erosion 

threshold. While they don’t explicitly report the diameter expansion of the noneroded rods, they 

do report the associated length compaction, which in one case resulted in a 10.2 mm length 

shrinkage on a 71.1 mm rod. Such strains are on a comparable scale to the diameter expansions 

presently observed. 

The most detailed description of the plastic-yet-noneroding phenomenology is the 

aforementioned extended quotation by Magness and Scheffler (23). However, that description is 

specifically for the case of an ogival-nosed penetrator. Nonetheless, it would appear, on the 

basis of the postmortem data from the present experiment, that the noneroding phenomenology 

described by Magness and Scheffler (23) for ogival penetrators applies identically to that for 

hemispherical-nosed penetrators. This result is striking, given that Magness and Scheffler 

astutely observed that, above the erosion-threshold velocity, ogival-nosed penetration 

phenomenology most closely follows that of conical-nosed penetrators (23), rather than 

hemispherical-nosed penetrators. 

Finally, despite the similarity of penetration phenomenology for noneroding hemispherical- and 

ogival-nosed penetrators, the erosion-threshold velocities for penetrators with these two nose 

shapes is significantly different (21, 22), but understandably so. Hill (10) recognized early that 

the propensity for cavitation was intimately related to the local radius of curvature of the rod 

nose. The geometrical profiles, and thus the radii of curvature and, by deduction, the propensity 

for cavitation (thus, erosion) will be different for hemispherical- and ogival-nosed penetrators. 

3. Analysis 

While the cited works reveal a thoughtful consideration of the low-velocity penetration 

phenomenon, there are still gaps in the understanding of this process, most particularly with 

regard to the analytical modeling of the penetration process below the erosion-threshold velocity, 

when plasticity is nonetheless involved. Much detailed and thorough analysis has been offered 

in the literature by Hill (10) and Forrestal et al. (12−16), but in these cases the analyses limit the 

engagement conditions so as to preclude the possibility of rod plasticity, a priori. So while they 

provide much to draw upon when considering rigid-body penetration, they do not answer all the 

questions associated with plastic, yet noneroding, penetration.
9

Yatteau and Dzwilewski (20) presented an initial outline for an analytical model for the 

penetration of conical-nosed penetration. For the limited condition in which plasticity occurs, 

yet prior to rod erosion, the criterion they use to establish this condition is whether the rod’s 

foreshortening rate, V–U, is less than the rod’s plastic wave speed. If so, the plasticity associated 

with the rod foreshortening is assumed to be “accommodated” by radial expansion at the tip and 

“presumed to become part of the rigid nose piece.” While they are correct in asserting that a 

foreshortening rate above the plastic wave speed cannot be accommodated by plasticity at the 

nose of the rod, there is no guarantee, on the other hand, that a foreshortening rate below the 

plastic wave speed must be accommodated by plasticity at the nose, especially as V–U

approaches the plastic wave speed. 

One interesting aspect that has not been addressed adequately in the literature is the inability of 

traditional penetration methodology (e.g., Tate/Alekseevskii methodology [2, 3] ) to account for 

the observed discontinuity of penetration behavior across the erosion-transition barrier. The 

kinematic statements of rod erosion and penetration (P) are, respectively, 

L& =U& −V& ; (1) 

P& =U , (2) 

where L is rod length and the dot indicates time differentiation. Rod deceleration is obtained by 

way of force/momentum balance on the elastic portion of the rod, 

LV Y ρ R = − / & . (3) 

The other vital component to traditional penetration methodology is the force/momentum 

balance in the rod/target stagnation zone, 

kR R V −U +Y = kT TU + H 2 2 ρ ( ) ρ . (4) 

This latter equation indicates that the stress at the rod/target interface, balanced on both the rod 

and target sides, is composed of an axial strength term superimposed over the stagnation stress of 

the flow field along the centerline. In the traditional penetration methodology (2, 3), the 

parameters kR, Y, kT, and H are considered constants. 

The parameters kR and kT are so-called “shape factors” for the rod and target respectively, 

associated with the manner in which the rod/target stagnation flow is split. With a blunt-nosed 

rod or an eroding interface that naturally establishes a blunt profile, these parameters take on the 

value of 1/2, in accordance with the dictates of Bernoulli stagnation flow. For geometries where 

the stagnation flow experiences a turning angle other than 90°, Segletes et al. (7) noted that the 

functionality of this variation in k should proceed as 1/2 (1−cosδ), where δ is the angle of flow 

turning. In the case, for example, of rigid penetration by a conical-nosed rod, the angle δ
10

corresponds to the half angle of the nose of the rod, in the calculation of kT (note that kR becomes 

irrelevant as V–U becomes zero for rigid penetration). The parameter Y is a strength term that 

characterizes the rod, and is typically associated with the uniaxial compressive strength of the 

rod. On the other hand, H is the target resistance, a strength term associated with resistance to 

penetration. While related to the compressive strength of the target, the value of H is, for a 

ductile target, typically 3 to 5 times the uniaxial compressive strength of the target material, 

compatible with various theories of indentation and crater formation. 

Employing this theory with fixed parameters kR, Y, kT, and H reveals the following: the 

penetration vs. velocity curves arising from the traditional methodology, such as that in Figure 1, 

will always produce a smooth curve with no abrupt renormalization as rigid-body penetration 

transitions to an eroding-rod configuration. 

Rather, in order to capture the sudden penetration dislocation associated with the erosion 

transition, as a minimum, one or more of the traditional, fixed parameters kR, Y, kT, and H needs 

to be varied as a function of the penetration mode. For the rigid penetration of conical-nosed 

rods, it has already been described how the kT parameter is dependent upon the cone angle of the 

nose. Such a variation can account for the penetration disparity of various conical-nosed rods of 

differing nose angles. Furthermore, were the nose to be blunted following the transition to 

eroding penetration, the value of kT could arguably revert back to the 1/2 value. Nonetheless, the 

penetration methodology for conical-nosed rods depicted by Woodward (17, 18) would not seem 

to indicate a sudden reversion to blunt-nosed erosive penetration, at velocities immediately 

above the erosion-transition velocity. 

While the penetration-mode dependence of kT and kR might be reasonably argued for conical or 

other sharp-nosed penetrators (including ogival-nosed rods), it is much harder to justify for 

ductile hemispherical-nosed rods, since the shape of the target’s (hemispherical) flow field is 

self-similar in both noneroding and eroding modes. Such is the case for the current analysis, 

involving hemispherical-nosed W rods impacting Al-5083 targets. This would leave as the only 

option, in an attempt to model the erosion-threshold transition, the penetration-mode variation of 

rod strength and target resistance parameters, Y and H, respectively. 

Frank (26) modeled the penetration behavior across the erosion threshold in the mid 1990s by 

decreasing the target’s surface resistance vis-à-vis the core resistance, H. In this report, however, 

we instead choose to examine and model the noneroding-penetration problem by examining the 

influence of penetration mode upon the “rod strength” parameter, Y. Consider the ballistic-stress 

balance of equation 4 for the case of noneroding penetration, in which U = V: 

Y = TV + H 2

2

1

ρ . (5) 
11

In the case of the noneroding impact presently observed at 1108 m/s, the target’s inertial head 

alone, equal to 1/2ρTV2

, computes to 1.66 GPa, far in excess of the 1.1 GPa rod strength 

associated with an RC 37 tungsten penetrator. The H = 1.78 GPa target resistance coupled with 

this inertial head would require a rod strength of Y ≥ 3.44 GPa in order to retain rigidity. 

Ballistic rod strength of 3.44 GPa is more than triple the 1.1 GPa amount estimated for the rod on 

the basis of both the W’s intrinsic properties as well as the two other similar ballistic tests. Such 

a disparity between actual and apparent rod strength leaves the one-dimensional traditional 

analytical modeling wanting for a credible explanation. 

That the recovered rod fragment of Figure 2 indicates noneroding penetration leads to the 

probing question: if the rod were, in some way, able to feign strength in excess of 3.44 GPa, 

would the analytical model prediction for the penetration of a rod, with this fictitious strength, 

match the noneroding penetration datum? While an affirmative reply to this query would not 

reveal how the feat was accomplished, it would nonetheless indicate that the solution to 

analytically addressing the problem is properly accomplished through an increase in apparent rod 

strength, rather than a decrease in the target resistance. It would also indicate that the rod, for all 

practical purposes, did penetrate as if it were a rigid body and as if it had a yield strength in 

excess of 3.44 GPa. Figure 4 presents the results of the model and data, which strikingly shows 

the validity of this conjecture. But while one may conclude that the noneroding rod penetrated 

as if it were a rigid penetrator with strength three times its nominal value, the question remains as 

to the proper manner and interpretation by which this phenomenon may be incorporated into the 

framework of analytical modeling. 

4. Interpretation 

The cratering phenomenology for noneroding penetration is notably different from that arising as 

a result of eroding penetration. Above the erosion threshold, the crater profile is rough and 

somewhat larger than the rod diameter (more than 1.5× in the current testing). By contrast, the 

crater profile for noneroding penetration is smooth and appears to remain at a size approximately 

equal to the rod-diameter. Indeed, for the noneroding impact observed at 1108 m/s, the crater 

diameter was actually slightly smaller than that of the rod, indicating an interference fit during 

the penetration event. 

This disparity in the crater formation between the eroding and noneroding penetration leads to 

the inference that the reason that the plastic, yet noneroding, rod fails to erode was because the 

penetration cavity being formed was too small to permit rod material to turn away from the rod 

trajectory, so as to deposit itself (in the traditional sense of an eroding rod penetrator) on the 

crater wall. Wijk (11) seems cognizant of this condition wherein the rod is unable to flow 

radially in the eroding-rod sense. He goes so far as to calculate both the minimum crater radius 

needed to permit an eroding-flow field, as well as the corresponding penetration velocity needed
12

V (m/s)

0 500 1000 1500 2000

P (mm)

0

50

100

150

200

250

300

0 500 1000 1500 2000

Vr (m/s)

0

200

400

600

800

1000

1200

1400

Target rear surface

Eroding Rod

Model:

Y = 1.1 GPa

Rigid Rod

Model

Rigid Rod

Model

YEFF > H + 1/2ρ Τ

 Vs

2

 = 3.44GPa

Figure 4. Model predictions and data of penetration and residual-velocity, 

for (rigid and Y = 1.1 GPa) WA rods into 344 mm thick BHN 97 

Al-5083 target (target rear surface modeled with plastic-zone 

extent: 3.5; spall resistance: HSPALL = H/3 [1, 7]).
13

to achieve this crater radius. Wijk’s primary intent seems to be on the end stage of penetration, 

wherein the penetration velocity of an eroding rod decreases to the point where traditional 

eroding penetration is no longer possible, resulting in an alternate phenomenology of penetration. 

Using Wijk’s equations with the current material properties (ρT = 2700 kg/m3, YT = 0.31 GPa, 

target shear modulus GT = 30 GPa, ρR = 17600 kg/m3, YR = 1.1 GPa) yields an estimation of 

erosion-threshold velocity at 866 m/s. The present data show the erosion-threshold velocity to 

be between 1100 and 1400 m/s. Scheffler’s analyses (21, 22) further constrain this value below 

1200 m/s and indicate a rod-nose-shape dependence to even this result. 

Wijk, despite the quantitative discrepancy, deserves credit for understanding certain facets of 

what defines this erosion-threshold velocity, wherein the crater is large enough a diameter to 

permit the establishment of an eroding flow field in the rod. However, it does not appear that his 

comprehension of the altered penetration phenomenology resulting from it extended to the 

present form of noneroding behavior, as he explicitly states that “for maximum penetration 

capacity the rear part of the projectile must be able to pass the lined hole created by the front 

part” (11). The present datum belies this assumption. And while, as a point of reference, the 

noneroding datum achieved a P/L = 3.37 with significant residual velocity (Vr /V = 0.28), Wijk 

dismissed the possibility that a rod would achieve even hydrodynamic levels of penetration 

(P/L = 2.55 for tungsten on aluminum) at low-impact velocities as “of course an unrealistic 

result” (11). 

Having inferred that the rod is unable to erode because the radial flow is constrained does not 

directly answer the question, however, of how a 1.1 GPa rod is able to remain noneroding while 

exerting the necessary 3.44 GPa of stress at the penetrator/target interface. For more quantitative 

insight, consider also prior qualitative explanations of the process. Brooks and Erikson’s (9) 

characterization of the rod’s ogival nose being “supported hydrostatically by the target material,” 

and Magness and Scheffler’s (23) contention that the ogival nose of the rod is eventually 

“constrained by the surrounding armor material and loaded under triaxial compression” are both 

directly supportive of the notion that the rod material is attempting to turn away from the rod’s 

axial trajectory, but is unable to do so fully, for lack of radial inertia. 

The idea that the “hydrostatic” or “triaxial” stress might play a key role for hemispherical-nosed 

rods, as well, is completely corroborated by the interference fit observed between the residual 

rod and the target crater that resulted from the 1108 m/s impact in the current test series. While 

it may not be readily apparent how this realization should be represented in the Bernoulli stress 

balance of equation 4, one may draw upon Segletes and Walters (27), who rederived various 

interpretations of the penetration equations through a derivation and application of a highly 

generalized “extended” Bernoulli equation. 

The characterization of the stress that manifests itself as Y in equation 4 is predicated upon the 

assumed uniaxial stress field within the elastic portion of the rod. The actual term, as indicated 

in the generalized derivation of Segletes and Walters (25), is really σzz and not Y (σzz being the 
14

normal stress in the coordinate direction of penetration). In the presumed absence of a laterally 

induced stress component (and consistent with Tate’s original derivation), the axial stress 

magnitude of σzz in the noneroding rod is thus limited by the rod strength, Y. In the presence, 

however, of a lateral stress component σLAT, induced in the rod by the lateral interference with 

the target, the generalized axial rod-stress term, σzz, will manifest itself as Y + σLAT, in accordance 

with Tresca yielding. This revision will yield a ballistic-stress balance of 

R V −U + Y + = TU + H 2

LAT

2

2

1 ( ) ( ) 21

ρ σ ρ , (6) 

where σLAT is positive in compression. When, as in the current case, the lateral stress provokes 

noneroding penetration (wherein U = V), the magnitude of the stress may be explicitly calculated 

as 

= TV + H − Y 2

LAT 2

1

σ ρ . (7) 

That the hemispherical cap of the deformed (postmortem) rod remained intact indicates the 

absence of erosive flow during the penetration event and leads one to a particular kinematic 

interpretation, consistent with the qualitative description of Magness and Scheffler (23), given 

for ogival rods. Namely, during the initial stages of penetration, the rod’s inertia, under force of 

impact, causes the rod to compress axially and expand radially in a plastic manner. Such 

behavior is not unlike the early stages of a Taylor impact test, excepting the fact that the rod is 

simultaneously embedding itself into the target material. The rod (or the leading portion thereof) 

expands radially in full contact with and against the target, in a controlled manner, until such 

time that sufficient radial expansion has been imposed upon the target to raise the 

lateral-interface stress to a level of 2.34 GPa, acting as a confining stress σLAT upon the rod. At 

this time, the rod’s axial stress, being composed of Y + σLAT, is brought the level of 3.44 GPa, 

sufficient to trigger rigid-body penetration in the target, and arrest further plastic, axial 

compression of the rod. 

Unlike traditional penetration theory, wherein the rod (excepting the tip) penetrates in a state of 

presumed uniaxial stress, the leading portion of the rod in the noneroding case penetrates under 

the condition of triaxial rod stress (σzz = 3.44 GPa, σxx = σyy = 2.34 GPa in this case). This 

interference fit of the rod and target crater becomes the necessary facilitator for the elevated 

penetration behavior, for it allows the axial stress brought to bear, by the rod upon the target, to 

be composed of a uniaxial strength component (traditionally associated with Y) augmented by a 

superimposed pressure component (transmitted laterally from the target). In so doing, it allows 

the utilization of the target’s lateral resistance to great axial effect. 

At higher velocities, Wijk’s (11) explanation holds true qualitatively, if not quantitatively. 

Namely, at a large enough velocity, the radial expansion of the forming crater becomes large 
15

enough to permit room for flow turning in the rod, thereby kinematically allowing for erosive 

flow of the penetrator. Once this flow turning occurs, the rod’s tip is robbed of its laterally 

induced triaxial-stress support. In that event, the stress across the rod/target interface must 

instead be balanced by way of added inertial stress, generated from the stagnation of the erodingrod flow. According to the present data, this threshold velocity would fall between 1108 and 

1416 m/s. Scheffler’s computations (21) would further indicate the erosion threshold to fall 

below 1200 m/s for hemispherical-nosed W rods onto Al-5083, but with a strong nose-shape 

dependence on this threshold. 

It would seem that the nose-shape dependence of the transition velocity, while not addressed 

here, is related more to the local curvature of the nose geometry. An approach that incorporates 

the reasoning of Hill (10) or Forrestal et al. (12), in this regard, might provide the necessary 

insight to adequately predict the effect of nose geometry on threshold velocity. 

5. Conclusions 

This report analyzes ballistic data for WA rods penetrating Al-5083 targets in the vicinity of the 

erosion-threshold velocity. Attention was directed at the one-dimensional modeling of the 

noneroding penetration event. Conventional one-dimensional penetration analysis reveals that 

the penetration and residual velocity in this test was wholly consistent with the notion of treating 

the rod as if it penetrated in a rigid-body fashion, and possessing an unrealistically large yield 

strength in excess of 3.4 GPa. Such strength is more than triple the 1.1 GPa value inferred from 

hardness measurements and other ballistic testing. 

Study of a recovered postmortem rod fragment and the target from the noneroding test revealed 

that, in fact, the penetrating rod deformed plastically, but did so without erosion. The evidence 

of rod plasticity was a ~6% increase in rod diameter between the original rod and the recovered 

postmortem rod fragment. Supported by analysis is the hypothesis that the rod did, in fact, apply 

a 3.44 GPa axial stress to the target interface. However, that stress comprised both the rod’s 

intrinsic 1.1 GPa yield strength plus a 2.34 GPa confining stress caused by a lateral interference 

fit between the rod and target crater during the penetration event. In such a fashion, the rod was 

able to employ the target’s lateral strength to great axial advantage. Higher velocity tests did not 

exhibit the noneroding behavior, precisely because the higher impact velocities created large 

enough craters in the target to remove the lateral interference that had augmented the axial stress 

at lower velocities. In so doing, the flow field in the rod was permitted to turn, creating the 

necessary conditions, according to Wijk (11), to establish an eroding flow field. 

The literature concerning the penetration of rods with different nose shapes (e.g., conical, ogival, 

hemispherical) has shown unique phenomenologies near the erosion threshold. Above the 

rod-erosion threshold, Magness and Scheffler noted that ogival-nosed rods exhibit a morphology 
16

similar to conical-nosed rods (9, 17) in establishing a rigid embedded tip ahead of an eroding rod 

shank. Scheffler’s earlier results (21, 22) would indicate that eroding hemispherical rods do not 

establish this embedded tip. The current work buttresses this literature by helping to establish 

that, below the rod-erosion threshold, the behavior of hemispherical-nosed rods is 

characteristically similar to that of ogival-nosed rods, whose penetration phenomenology was 

clearly laid out by Magness and Scheffler (23). 

Thus, it would seem that smooth transition from rod tip to shank (i.e., when the slope of the rod’s 

geometric profile is continuous), evident in both ogival- and hemispherical-nosed rods helps to 

retard cavitation (i.e., retards the separation of target from rod). In so doing, it establishes the 

capacity for rods of these nose profiles to utilize plastic, yet noneroding, penetration to a greater 

extent than their conical-nosed counterparts. In contrast, above the erosion threshold, the sharp 

point of both ogival- and conical-nosed rods produces a stress field that resists an erosive 

flattening of the rod nose. In so doing, there is a propensity for rods of these nose profiles to 

establish a pointed, rigid, embedded tip ahead of an eroding rod shank, not evident in ductile 

hemispherical-nosed rods.
17

6. References 

1. Segletes, S. B. An Adaptation of Walker-Anderson Model Elements Into the Frank-Zook 

Penetration Model for Use in MUVES; ARL-TR-2336; U.S. Army Research Laboratory: 

Aberdeen Proving Ground, MD, 2000. 

2. Tate, A. A Theory for the Deceleration of Long Rods After Impact. Journal of Mechanics 

and Physics of Solids 1967, 15, 387–399. 

3. Alekseevskii, V. Penetration of a Rod Into a Target at High Velocity. Combustion, 

Explosions and Shock Waves 1966, 2, 63–66. 

4. Zook, J. A.; Frank, K.; Silsby, G. F. Terminal Ballistics Test and Analysis Guidelines for the 

Penetration Mechanics Branch; BRL-MR-3960; U.S. Army Ballistic Research Laboratory: 

Aberdeen Proving Ground, MD, 1992. 

5. Wright, T. W.; Frank, K. Approaches to Penetration Problems; BRL-TR-2957; U.S. Army 

Ballistic Research Laboratory: Aberdeen Proving Ground, MD, 1988. 

6. Walker, J. D.; Anderson, C. E. Jr. A Time-Dependent Model for Long-Rod Penetration. 

International Journal of Impact Engineering 1995, 16 (1), 19−48. 

7. Segletes, S. B.; Grote, R.; Polesne, J. Improving the Rod-Penetration Algorithm for 

Tomorrow's Armors; ARL-RP-23; U.S. Army Research Laboratory: Aberdeen Proving 

Ground, MD, 2001. 

8. Kinslow, R. Collisions at High Velocity. International Science and Technology 1965, 40. 

9. Brooks, P. N.; Erikson, W. H. Ballistic Evaluation of Materials for Armour Penetrators; 

DREV R-643/71; Defence Research Establishment Valcartier, Quebec, Canada, 1971. 

10. Hill, R. Cavitation and the Influence of Headshape in Attack of Thick Targets by NonDeforming Projectiles. Journal of Mechanics and Physics of Solids 1980, 28, 249−263. 

11. Wijk, A. G. High-Velocity Projectile Penetration Into Thick Armour Targets. International 

Journal of Impact Engineering 1999, 22, 45−54. 

12. Forrestal, M. J.; Okajima, K.; Luk, V. K. Penetration of 6061-T651 Aluminum Targets With 

Rigid Long Rods. Journal of Applied Mechanics 1988, 55, 755−760. 

13. Rosenberg, A.; Forrestal, M. J. Perforation of Aluminum Plates With Conical-Nosed 

Rods − Additional Data and Discussion. Transactions of the ASME 1988, 55, 236−238. 

14. Forrestal, M.; Luk, V. K. Perforation of Aluminum Armor Plates With Rigid Conical-Nose 

Projectiles. Mechanics of Materials 1990, 10, 97−105. 
18

15. Forrestal, M.; Brar N. S.; Luk, V. K. Penetration of Strain-Hardening Targets With Rigid 

Spherical-Nose Rods. Journal of Applied Mechanics 1991, 58, 7−10. 

16. Forrestal, M.; Luk, V. K.; Rosenberg, Z.; Brar N. S. Penetration of 7075-T651 Aluminum 

Targets with Ogival-Nose Rods. International Journal of Solids and Structures 1992, 29

(14/15), 1729−1736. 

17. Woodward, R. L. A Note on the Deformation of Conical Penetrators. International Journal 

of Impact Engineering 1984, 2 (4), 325–330. 

18. Woodward, R. L. The Effect of Projectile Tip Geometry on Penetration Into Semi-Infinite 

Metal Targets. Proceedings of 9th International Symposium on Ballistics, Shrivenham, 

U.K., May, 1986. 

19. Chen, E. P. Finite Element Simulation of Perforation and Penetration of Aluminum Targets 

by Conical-Nosed Steel Rods. Mechanics of Materials 1990, 10, 107−115. 

20. Yatteau, J. D.; Dzwilewski, P. T. The Onset of Deformation and Flow in Penetrators With 

Conical Noses. Proceedings of 15th International Symposium on Ballistics, Jerusalem, 

Israel, 21−24 May 1995. 

21. Scheffler, D. R. Modeling the Effect of Penetrator Nose Shape on Threshold Velocity for 

Thick Aluminum Targets; ARL-TR-1417; U.S. Army Research Laboratory: Aberdeen 

Proving Ground, MD, 1997. 

22. Scheffler, D. R. Modeling Threshold Velocity of Hemispherical and Ogival-Nose TungstenAlloy Penetrators Perforating Finite Aluminum Targets; ARL-TR-1583; U.S. Army 

Research Laboratory Report: Aberdeen Proving Ground, MD, 1998. 

23. Magness, L. S.; Scheffler, D. R. The Influence of Penetrator Material and Projectile Nose 

Shape on the Onset of Penetrator Deformation and Erosion. Proceedings of 2nd 

Australasian Conference on Applied Mechanics, Canberra, Australia, February 1999. 

24. Forrestal, M. J.; Piekutowski, A. J. Penetration Experiments With 6061-T6511 Aluminum 

Targets and Spherical-Nose Steel Projectiles at Striking Velocities Between 0.5 and 

3.0 km/s.” International Journal of Impact Engineering 2000, 24, 51–67. 

25. Piekutowski, A. J.; Forrestal, M. J.; Poorman, K. L.; Warren, T. L. Penetration of 

6061-T6511 Aluminum Targets by Ogive-Nose Steel Projectiles With Striking Velocities 

Between 0.5 and 3.0 km/s. International Journal of Impact Engineering 1999, 23, 

723–734. 

26. Frank, K. U.S. Army Research Laboratory: Aberdeen Proving Ground, MD. Private 

communication, citing unpublished results, 2003. 
19

27. Segletes, S. B.; Walters. W. P. A Note on the Application of the Extended Bernoulli 

Equation. International Journal of Impact Engineering 2002, 27, 561−576. 
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