Analysis of Erosion Transition in Tungsten-Alloy Rods Into 

Aluminum Targets 

by Steven B. Segletes 

ARL-TR-3153 March 2004 

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Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5066 

ARL-TR-3153 March 2004 

Analysis of Erosion Transition in Tungsten-Alloy 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 Erosion Transition in Tungsten-Alloy Rods Into Aluminum Targets 

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Steven B. Segletes 

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14. ABSTRACT 

This work extends and refines the phenomenological understanding of ballistic penetration in the vicinity of the erosionthreshold velocity, for the case of hemispherical-nosed tungsten rods striking ductile targets. Analysis, supported by 

experimentation, indicates a period of noneroding penetration for these configurations, which results from lateral support 

exerted by the target crater upon the deforming, yet noneroding, penetrator. Experiments indicate that the magnitude of the 

lateral support, the direct result of an interference fit between rod and crater, must be on the order of the target’s ballisticpenetration resistance, and does not vary with the impact velocity over the range studied. Analysis suggests that the duration

of the noneroding portion of the ballistic event is neither governed by a fixed time, nor by a fixed depth of penetration, but 

rather by a fixed, permissible level of deformation in the penetrator. 

15. SUBJECT TERMS

erosion, noneroding, transition, threshold, aluminum, long rod, hemispherical nose, ballistics 

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Steven B. Segletes 

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iii

Contents 

List of Figures iv

List of Tables v

1. Introduction 1

2. Observations 3

3. Modeling 6

3.1 Trigger for Erosion..........................................................................................................10 

3.2 HLAT Magnitude...............................................................................................................13 

3.3 Impact Pressure ...............................................................................................................13 

3.4 Plastic Zone Size in Rod..................................................................................................15 

3.4.1 Equation of Noneroding Rod Motion..................................................................15 

3.4.2 Extended Bernoulli Equation ..............................................................................16 

3.5 Integration of Rod Deformation ......................................................................................19 

4. Modeling Results 20

5. Comparison to FZ Model 25

6. Conclusions 33

7. References 35

Distribution List 38
iv

List of Figures 

Figure 1. Penetration and residual-velocity data into 344 mm monolithic Al (BHN 97). For 

comparison, model results for rigid and eroding rods are included...........................................2 

Figure 2. Schematic showing how lateral confinement may augment the axial stress brought 

to bear by the rod, in light of plastic yield condition.................................................................7 

Figure 3. Data for Finite-Target Series tests, with comparison to rigid-model and erodingmodel calculations, for (a) residual rod length and (b) residual velocity. .................................9 

Figure 4. Radiographic image of residual rod from test MV7, depicting intact nose and 

fractured tail (note: rod is moving left to right on image). .......................................................9 

Figure 5. Data for Interply Series tests, with comparison to rigid-model and eroding-model 

calculations, for (a) residual rod length and (b) residual velocity. .........................................10 

Figure 6. Back-calculated “effective” velocity-dependent rod strength, to accurately model 

residual length in Finite-Target Series tests, assuming effective strength was to act over 

full duration of ballistic event..................................................................................................11 

Figure 7. Photograph depicting residual-rod fragment recovered from test MV1. While rod 

tip appears undeformed, measurement reveals a 6% increase in rod diameter. ......................12 

Figure 8. Schematic depicting the axial-stress levels at three locations, A, B, and T along the 

rod centerline, along with two versions of the rod-deceleration equation (one requiring 

knowledge of s, the other not)..................................................................................................17 

Figure 9. Data for Finite-Target Series tests, with comparison to revised-methodology 

calculation, for (a) residual rod length and (b) residual velocity.............................................21 

Figure 10. Data for Semi-Infinite Series I tests, with comparison to rigid-rod, eroding-rod, 

and revised-methodology calculations, for (a) residual rod length and (b) penetration and 

residual velocity.......................................................................................................................22 

Figure 11. Data for Interply Series tests, with comparison to revised-methodology 

calculation, for (a) residual rod length and (b) residual velocity.............................................23 

Figure 12. Datum for Semi-Infinite Series II test, with comparison to rigid-rod, eroding-rod, 

and revised-methodology calculations, for (a) residual rod length and (b) penetration 

(note: revised-methodology calculation shown for two values of parameter ∆LCRIT)............24 

Figure 13. Rod-deformation-to-erosion-onset model parameter, ∆LCRIT/D0, shown as a 

function of target resistance, for the Al alloys modeled in the current test series, with 

anticipated interpolation and extrapolation to other Al alloys.................................................25 

Figure 14. Penetration and residual-velocity data for Semi-Infinite Series I tests, with 

comparison to FZ-model results, for rigid and eroding rods. ..................................................27 

Figure 15. Data for Finite-Target Series tests, with comparison to FZ-rigid-model and FZeroding-model calculations, for (a) residual rod length and (b) residual velocity...................28 

Figure 16. Data for Interply Series tests, with comparison to FZ-rigid-model and FZeroding-model calculations, for (a) residual rod length and (b) residual velocity...................28 
v

Figure 17. Penetration datum for Semi-Infinite Series II test, with comparison to FZ-model 

results, for rigid and eroding rods............................................................................................29 

Figure 18. Penetration and residual-velocity data for Semi-Infinite Series I tests, with 

comparison to FZ-model results, for rigid and eroding rods and FZ-S model with soft 

target surface............................................................................................................................30 

Figure 19. Data for Finite Target Series tests, with comparison to FZ-S (soft surface) model 

calculations, for (a) residual rod length and (b) residual velocity. ..........................................31 

Figure 20. Data for Interply Series tests, with comparison to FZ-S (soft surface) model 

calculations, for (a) residual rod length and (b) residual velocity. ..........................................31 

Figure 21. Target resistance and axial rod stress, through the depth of the Finite Target 

Series target at 1250 m/s striking velocity, as characterized by (a) the FZ-S (soft surface) 

model and (b) the revised model..............................................................................................32

Figure 22. Target resistance and axial rod stress, through the depth of the Finite Target 

Series target at 1000 m/s striking velocity, as characterized by the FZ-S/I (soft surface 

and interfaces) model...............................................................................................................32 

List of Tables 

Table 1. Target descriptions for experimental series......................................................................4 

Table 2. Ballistic results for experimental series............................................................................5 

Table 3. Model properties for target materials................................................................................6 
vi

INTENTIONALLY LEFT BLANK. 
1

1. Introduction 

The subject of this report is the ballistic interactions of rod and target in the vicinity of the roderosion threshold velocity. Its scope is both experimental and analytical. Recently, the author 

(1) detailed the results of several interesting ballistic tests that straddled the rod-erosion threshold 

velocity for the case of hemispherical-nosed tungsten-alloy (WA) rods onto thick aluminum 

(Al)-5083 targets. This report extends that work with added testing and analysis of several 

different target configurations, including several materials. As a result of incorporating these 

additional data into the analysis, the understanding of the phenomenology of hemisphericalnosed WA penetration near the erosion-threshold velocity into Al is extended and refined. 

The analysis of the tests was performed with a one-dimensional (1-D) penetration model, 

developed by the author as an extension to a model by Frank and Zook (2, 3). The current model 

(4, 5) extends the Frank-Zook (FZ) model primarily by incorporating aspects of a theory by 

Walker and Anderson (6) and adapting them for use in a multiply target configuration. Both 

models get their pedigree from original penetration theories by Tate (7) and Alekseevskii (8). In 

response to analysis of the currently presented data, further model revisions are offered to help 

address the specific issues of deforming, yet noneroding, penetration and erosion transition. 

Comparison with the current model and experimental data is offered as well, against the FZ 

model (2, 3), on which the current model’s development was based. 

The literature (9–17) has long noted a disparity of penetration capability in the vicinity of the 

erosion threshold. This disparity appears to be very nose-shape dependent. While conical-nosed 

penetrators appear to gain performance from the altered flow field around the sharp vis-à-vis a 

blunt tip (10, 12, 13, 17), both conical- and ogival-nosed penetrators have been observed to 

penetrate above the erosion threshold with a detached, embedded rod tip (10, 12, 13, 16, 17). 

Below the erosion threshold, both ogival- and hemispherical-nosed rods can exhibit a fascinating 

form of noneroding behavior in which the rod nonetheless deforms, but conforms to the cavity 

profile (1, 14–16). In addition to experimental observation, hydrocode analysis has been brought 

to bear on the problem (14–18). 

In recent work involving hemispherical-nosed WA penetrators onto Al targets (1), the author was 

able to experimentally validate, from post-mortem recovery of test materials, that the leading end 

of the noneroding rod, in fact, deformed by expanding in diameter >6%. More importantly, it 

was experimentally determined that the lateral conformance of rod material to target material 

during the course of this noneroding penetration constitutes an interference fit between rod and 

target. Based on this very limited experimental result, it was hypothesized that the lateral 

resistance developed as a result of this interference, and in accordance with plasticity 

considerations like the Tresca yielding condition, permits an augmentation of the axial stress that 

the rod can bring to bear upon the rod/target interface. In such a manner, the deforming-yet-
2

constrained rod is able to penetrate as if it were a much stronger rigid rod. In the case reported 

(figure 1), the “apparent” or effective strength exhibited by the noneroding rod was roughly 

triple the rod’s intrinsic material strength. According to the hypothesis, it is this effective rod 

strength that produces this penetration-performance dislocation across the rod-erosion threshold 

because, following the onset of erosion, this lateral confinement and the resultant axial-stress 

augmentation is kinematically removed by the flow of the erosion products. In addressing this 

type of noneroding, yet deforming, penetration, similarities are shared with traditional rigidbody-penetration modeling (19–23); nonetheless, the modeling is made distinct, by way of the 

rod’s deformation and the nature of the rod/target interference fit. It should be noted that this 

noneroding yet deforming, circumstance is somewhat analogous (in reverse) to the ballistic 

phenomenon known as “penetrator dwell,” in which the target deforms and may yield (or 

fracture), but does not ballistically erode. 

0 500 1000 1500 2000

Vr (m/s)

0

200

400

600

800

1000

1200

1400

Vs (m/s)

0 500 1000 1500 2000

P (mm)

0

50

100

150

200

250

300

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 1. Penetration and residual-velocity data into 

344 mm monolithic Al (BHN 97). For 

comparison, model results for rigid and eroding 

rods are included.
3

Forrestal and Piekutowski (24) presented much data that can be used to support the subsequent 

hypothesis by this author (1). In Forrestal and Piekutowski’s work, 6061-T6511 Al targets are 

penetrated by hemispherical-nosed steel rods of hardnesses RC 36.6, 39.5, and 46.2. While the 

recently reported work of the author (1) employed 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. This approach was also 

employed by the author, albeit with a simpler model for rigid penetration, as opposed to the 

Forrestal model. Finally, like the author, who noted a bulging rod diameter in the noneroding 

penetrator, 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. 

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. 

Regardless of nose-shape influence on the process, the crux of the issue is this: the stagnation 

pressure (1/2ρU2

) from penetrating Al at 1100 m/s is 1.6 GPa. Adding to that, the target 

resistance of the particular Al alloy (even accounting for variation in what might constitute an 

appropriate resistance) invariably places the axial stress level at the rod/target interface well 

beyond the yield strength of an RC 40 penetrator. These configurations, for hemispherical-nosed 

rods, are nonetheless observed by many researchers (1, 14–16, 24) to penetrate in a noneroding 

fashion. The primary contribution of the recent work (1) is in establishing a kinematic 

hypothesis to explain this discrepancy: that the target’s lateral interference against the deforming 

rod both inhibits the rod erosion while permitting a significantly elevated axial stress to be 

brought to bear by the projectile upon the rod/target interface. The notion of erosion being 

inhibited by lateral target interference is not novel, but was posited by Wijk (26), though in that 

case only as specifically pertains to the final stages of so-called afterflow penetration. No 

mention is made there, however, of an elevated axial stress that might accompany this lateral 

confinement and aid the penetration efficiency. 

2. Observations 

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

following characteristics: 65 g, length-to-diameter (L/D) ratio of 15, a content of 93% tungsten 
4

(W) and 6.3% Nickel (Ni), and swaged (reduced in cross-sectional area) 8%. Because of 

available supply, there were actually two variations of WA rod employed: one with the 

remaining 0.7% composition of iron (Fe) and the other of cobalt (Co). Nonetheless, these two 

rod variations share nearly identical properties, and because all ballistic engagements were at 

normal incidence, influence due to any variation in, e.g., fracture properties, was minimal. The 

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

long × 6.79 mm diameter, while the W-Ni-Co rods had a nominal hardness of RC 37.5, with 

nominal dimensions of 101.7 long × 6.78 mm diameter. The dimensions and mass of the rods 

yield a penetrator density of ~ 17,600 kg/m3. A hardness of RC 37 correlates roughly with a 

strength of 1.1 GPa. The W-Ni-Fe rods were used exclusively in the test series dubbed “SemiInfinite” and “Finite-Target,” while the W-Ni-Co rods were used exclusively in the test series 

dubbed “Interply.” Several of the Semi-Infinite Series test results were reported on in the 

preceding report (1). The target descriptions for the various test series discussed in this report 

are provided in table 1. All target elements were square or rectangular in shape, between 200- 

and 305-mm on a side, in length. Because the pedigree of all target plates was not known or 

available, Brinell hardness testing was done in all cases to provide estimates of material strength 

properties. These hardness data are incorporated into the targets’ functional descriptions in 

table 1. While the so-called Semi-Infinite Series targets (which are, nonetheless, finite stacks of 

Al block) were intended purely as a baseline, it may be noted from table 1 that both the FiniteTarget and Interply test series each have two variations on their target “recipe,” respectively. 

Table 1. Target descriptions for experimental series. 

Rod Employed Nominal Target Description Functional Target Descriptiona

Semi-Infinite Series I 

W-Ni-Fe 6 × 2.25 in Al 5083 344 mm Al BHN 97 

Semi-Infinite (Single Datum) II 

W-Ni-Fe 7 × 2 in Al 362 mm Al BHN 190 

Finite-Target Series (Without Backing) 

W-Ni-Fe 

2 × 1.3 in Al 5083 

1.25 in Al 5083 

1.75 in Al 5083 

65.5 mm Al BHN 97 

32.0 mm Al BHN 99 

45.2 mm Al BHN 112 

Finite-Target Series (With Backing) 

W-Ni-Fe 

2 × 1.3 in Al 5083 

1.25 in Al 5083 

1.75 in Al 5083 

0.0625 in stainless 304 

65.5 mm Al BHN 97 

32.0 mm Al BHN 99 

45.2 mm Al BHN 112 

1.58 mm steel RB 80.2 (~BHN 72) 

Interply Series (Without Interply) 

W-Ni-Co 

5 × 0.5 in Al 7079 (?) 

2 × 1.75 in Al 5083 

0.375 in high-hard steel 

63.5 mm Al BHN 143 

90.4 mm Al BHN 112 

9.5 mm steel BHN 512 

Interply Series (With Interply) 

W-Ni-Co 

5 × 0.5 in Al 7079 (?) 

2 × 1.75 in Al 5083 

0.1 in acrylic 

0.375 in high-hard steel 

63.5 mm Al BHN 143 

90.4 mm Al BHN 112 

2.54 mm Acrylic 

9.5 mm steel BHN 512 a

Target hardness specified in Brinell hardness number (BHN) or Rockwell B (RB) scales.
5

The ballistic results for the test series are given in table 2. There were a total of four tests in the 

Semi-Infinite Series, 10 in the Finite-Target Series (5 each with and without a thin-steel backing) 

and 20 in the Interply Series (10 each with and without the Plexiglas interply). 

Table 2. Ballistic results for experimental series. 

Experiment 

Designation Vs

(m/s) 

Vr

(m/s) 

P 

(mm) 

Lr

(mm) 

Comments 

Semi-Infinite Series I 

MV1 1108 314 344.0 81+ Perforation, noneroding, rod tail missing 

MV3 1416 — 193.2 14 — 

MV2 1701 — 228.1 12 — 

Semi-Infinite (Single Datum) II 

MV4 1413 — 130.4 ~0–3 — 

Finite-Target Series (Without Backing) 

MV7 916 696 — 57.1+ Noneroding, rod tail missing 

MV5 1055 540 — 26.3+ High yaw, noneroding(?), rod tail missing 

MV16 1196 892 — 62.5 — 

MV28 1394 1077 — 48.6 — 

MV22 1628 1310 — 33.3 — 

Finite-Target Series (With Backing) 

MV10 906 593 — 60.1+ Noneroding, rod tail missing 

MV14 1053 776 — 57.8+ Noneroding, rod tail missing 

MV19 1197 934 — 80.3 — 

MV31 1384 1035 — 50.8 — 

MV25 1595 1217 — 40.9 — 

Interply Series (Without Interply) 

MV24 1251 488 — ~0 Test at ballistic limit, small residual debris 

MV15 1300 294+ — — Vr is debris (rod exited from film laterally) 

MV8 1351 791 — 18.5 — 

MV37 1391 588 — 8.5 — 

MV11 1455 1015 — 37.8 — 

MV43 1521 934 — 30.1 — 

MV30 1536 841 — 20.8 — 

MV45 1625 1198 — 25.5 — 

MV34 1641 1039 — 21.6 — 

MV40 1722 1184 — 20.1 — 

Interply Series (With Interply) 

MV21 1261 0 — — Bulge in rear plate, near ballistic limit 

MV12 1281 570 — 30.9 — 

MV39 1360 801 — 28.6 — 

MV18 1381 801 — 27.0 — 

MV42 1455 848 — 28.5 — 

MV27 ? 415 — 21.6 X-rays not triggered, no Vs obtained 

MV46 1501 779 — 19.3 Redo of MV27 

MV44 1547 972 — 21.6 — 

MV33 1597 905 — 20.1 — 

MV36 1708 1128 — 14.7 — 
6

3. Modeling 

From these and other data were deduced the targets’ material properties that would be employed 

in the penetration model. These properties are given in table 3 and include density (ρT), elastic 

modulus (E), yield and ultimate strengths (σ), and resultant target resistance (H). For the softer 

alloys of Al, the properties are drawn from handbook alloys of comparable hardness. For the 

hard Al and the steels, hardness conversion tables were used. The acrylic strength, being highly 

rate dependent, was estimated from scant data. The target resistance was calculated from Tate’s 

formula, except that ultimate rather than yield strength was employed therein. This formula 

takes the following form: 





 



 



 



 = +

σ

σ

3

2 ln

3

2 E H . (1) 

Table 3. Model properties for target materials. 

Material ρT 

(kg/m3

) 

E 

(GPa) 

σY 

(GPa) 

σULT 

(GPa) 

H 

(GPa) 

Al BHN 97 2700 75 0.276 0.310 1.78 

Al BHN 99 2700 75 0.276 0.310 1.78 

Al BHN 112 2700 75 0.345 0.372 2.07 

Al BHN 143 2700 75 0.476 0.531 2.77 

Al BHN 190 2700 75 — 0.627 3.16 

Steel RB 80.2 (~BHN 150) 7850 205 — 0.496 3.12 

Steel BHN 512 7850 205 — 1.896 9.38 

Acrylic 1190 2.55 0.14 (est.) 0.235 (est.) 0.62 

In all cases, a material’s spall resistance, used to characterize a target material’s resistance in the 

limit as a free surface is approached, has been characterized by the value H/3. Because the rod’s 

hemispherical nose shape is blunt, all Bernoulli shape factors, kR and kT, are retained at a value of 

0.5. The plastic zone extent (PZE) for all materials is retained at a model-default value of 3.5 

rod diameters. The PZE, an estimated quantity, represents the size, in rod diameters, that the 

plastic zone extends in front of and away from the moving rod/target interface (4, 5). Any target 

plies entrained in this plastic zone impart a contribution to the aggregated density and resistance 

of the target, as instantaneously perceived by the rod. Results are somewhat insensitive to 

changes in the value of PZE, unless it is made small enough to approach the lower limit of zero. 

The target recipe variations within a test series that are described in table 1, while designed to 

have minimal ballistic influence, were intended to challenge the model’s ability to not over- or 

under-compensate for the influence of target layers with significantly disparate material 
7

properties. This challenge to modeling accuracy can be traced to the manner in which multiple 

plies are treated within a target and have been demonstrated (5) with several earlier penetration 

models. For example, both the original Tate (7) and the FZ (2, 3) models suffer this problem in 

different circumstances and to different extents, due to the inability (in the case of Tate’s) or the 

limited ability (in the case of FZ) to “look ahead” of the rod/target interface and ascertain the 

influence of upcoming density and resistance dislocations (i.e., target interfaces) upon the 

target’s instantaneous resistance. 

The testing program was originally designed for this (now secondary) purpose of validating the 

modeling algorithm for multi-ply target laminates. The much larger experimental effect actually 

observed and the concomitant modeling challenge, not appreciated at program outset, was in 

understanding the modeling ramifications of noneroding, yet deforming, penetration, and any 

associated transitions to a fully eroding configuration. 

The preceding report (1) began this process of modeling noneroding, yet deforming, penetration 

as it pertains to hemispherical-nosed rods into thick monolithic targets (Al-5083). It 

hypothesized the stress field and kinematic behavior of rod deformation to be compatible with 

Magness and Scheffler’s qualitative description of the event for ogival-nosed rods (16), Forrestal 

and Piekutowski’s experimental observations (24), as well as the post-mortem data from the 

current Semi-Infinite Series of tests. Under the hypothesized behavior, the lateral contact 

between rod and crater, established during penetration, takes the form of an interference fit. The 

resulting lateral stress, HLAT, established by this interference fit permits, in accordance with the 

Tresca yield condition, the axial stress in the rod to exceed the rod’s yield strength, Y, taking on 

an value equal to Y + HLAT (figure 2). With a low enough striking velocity, Vs, this augmented 

axial stress (an “effective” rod strength, as it were) is sufficient to deter rod erosion, with 

noneroding penetration as the result. 

HLAT

HLAT

Y + HLAT

Y + HLAT

Tresca Yield 

Condition:

|σzz − σrr| = Y

r

z

Figure 2. Schematic showing how lateral confinement may 

augment the axial stress brought to bear by the 

rod, in light of plastic yield condition. 
8

A primary effect of this stress augmentation, in modeling the problem, is the revision of the 

extended Bernoulli equation. Solved as part of the penetration analysis, this equation balances 

the stress across the rod/target interface by considering the stagnation effects of both the eroding 

rod and eroding target. As pointed out by Segletes and Walters (27), the axial stress term given 

by Y in the equation actually arises from a σzz stress term, where z is the axial coordinate. When 

the stress is uniaxial, σzz is limited to the yield strength, Y. However, for the multiaxial condition 

described by figure 2, the axial stress component will rise above the uniaxial strength to an 

“effective strength” value, YEFF, equal to Y + HLAT. In its simplest (slender-rod assumption) form, 

the extended Bernoulli equation becomes 

kR R V −U + Y = kT TU + H 2

EFF

2 ρ ( ) ρ . (2) 

The observed dislocation in penetration-vs.-striking velocity curve is then a direct result of the 

reduction of YEFF from Y + HLAT back to Y, following the onset of eroding penetration and the 

concomitant loss of lateral interference with the target wall. 

While the preceding report (1) estimated the HLAT value necessary to retain rod rigidity for one 

datum striking at 1108 m/s, it did not establish a general rule or methodology that would permit 

one to calculate or estimate the conditions under which the erosion transition might commence. 

Furthermore, based solely on the limited test results into 344 mm thick monolithic Al-5083 

(BHN 97), as shown in figure 1, one might be tempted to infer from the preceding report that this 

erosion threshold is a function of striking velocity alone, so that once above the erosionthreshold striking velocity, classical eroding penetration is the immediate result from the impact 

outset. If such an assumption has merit, then one should expect that ballistic data would fall 

either upon the curve modeled as noneroding (with effective strength HLAT + Y), or upon that 

modeled as fully eroding (with strength Y), as it does in figure 1. 

However, the data from the other test series prove more complicated than this simple assumption 

allows. Figure 3a shows residual-length-vs.-striking velocity for the so-called Finite-Target 

Series of tests, along with rigid-rod and eroding-rod predictions. In this figure, one may observe 

that the experimentally observed residual length transitions from the noneroding value to the 

fully eroding curve, not abruptly, but over a 500 m/s striking-velocity range from roughly 1100 

to 1600 m/s. The residual-length data at striking velocities below 1100 m/s were measured at 

less than the original rod length. However, this was not because of erosion, but rather because of 

rod fracture in which the tail of the rod remained entrained in the target at the moment of 

radiography, due to lateral interference with the target. The leading edge of the rod, observed 

from radiography, clearly retained its original hemispherical-nosed shape, leading to the 

conclusion that these rods were, in fact, noneroding. Further, the fractured midsection of these 

rods was clearly observable, further confirming that the missing rod length was from the tail, 

rather than the tip of the rod (figure 4). In any event, the notion that the erosion transition is 

governed solely by the striking velocity is wholly dispelled by figure 3a. 
9

Vs (m/s)

0 500 1000 1500 2000

L

r /L0

0.0

0.2

0.4

0.6

0.8

1.0

Rigid Rod Model:

with or w/o

backing

Eroding Rod

Model:

w/o backing

w/ backing

Data:

w/o backing

w/ backing

High-Yaw

Impact

Noneroding Rod:

Measured

Incomplete

 Lr

{

Apparent Rigidto-Eroding

Transition

Vs (m/s)

0 500 1000 1500 2000

Vr (m/s)

0

200

400

600

800

1000

1200

1400

Rigid Rod

Model:

w/o backing

w/backing

Eroding Rod

Model:

w/o backing

w/ backing

Data:

w/o backing

w/ backing

High-Yaw

Impact

 (a) (b) 

Figure 3. Data for Finite-Target Series tests, with comparison to rigid-model and eroding-model 

calculations, for (a) residual rod length and (b) residual velocity. 

Figure 4. Radiographic image of residual rod from test MV7, depicting intact nose and fractured tail (note: 

rod is moving left to right on image). 

The results for the Interply Series of tests, given in figures 5a and 5b, prove equally 

disappointing when the tests are modeled as purely rigid or purely eroding. These results 

reinforce the conclusion that the simple interpretation, defining the penetration mode as a 

function of striking velocity alone, is not sufficient to describe the phenomenon of erosion 

transition. 
10

Vs (m/s)

0 500 1000 1500 2000

L

r /L0

0.0

0.2

0.4

0.6

0.8

1.0

Rigid Rod Model:

with or w/o

interply

Eroding Rod Model:

w/ and w/o interply

Data:

w/o interply

w/ interply

Vs (m/s)

0 500 1000 1500 2000

Vr (m/s)

0

200

400

600

800

1000

1200

1400

Rigid Rod Model:

w/ and w/o interply

Eroding Rod Model:

w/ and w/o interply

Data:

w/o interply

w/ interply

 (a) (b) 

Figure 5. Data for Interply Series tests, with comparison to rigid-model and eroding-model 

calculations, for (a) residual rod length and (b) residual velocity. 

A further deficiency may be noted in figure 3b, which depicts the experimental and predicted 

residual velocities resulting from the Finite-Target Series. Below 1100 m/s, the data are captured 

well by the rigid-rod curve, as is expected. However, for the higher striking velocities, not only does 

the eroding-rod curve overestimate the residual velocities, but the rigid-rod curve does as well. This 

is disconcerting, as one would, in any event, expect the data to fall between the two curves. We will 

endeavor to address this issue later (see subsection 3.3 entitled “Impact Pressure”). Consider first, 

however, the issue of the residual length discrepancy noted in figure 3a and its implications for the 

erosion trigger. 

3.1 Trigger for Erosion 

Figure 3a makes clear that the erosion transition is not triggered by impact velocity alone. There are 

many ways to attempt to address this issue. The simplest, and one first tried, is to assume that the 

specific manner of the rod/target interference fit (and thus the lateral resistance, HLAT) is a function of 

the striking velocity. In this manner, an effective yield strength may be back-calculated, as a function 

of striking velocity, in order to match the experimentally observed residual lengths from the FiniteTarget Series. Figure 6 shows the result of such a back-calculation for YEFF = Y + HLAT. For striking 

velocities lower than 1108 m/s, the Semi-Infinite Series test datum had indicated that noneroding 

penetration would result, and so the rod’s minimum effective strength is governed by the parabolic 

relation derived from the rigid-penetration stress balance, YEFF = H + 1/2ρTVs

2

. The back-calculated 

effective strength covers the striking velocity range between 1108 and 1760 m/s. Per figure 6, the 

back-calculations represent an averaged effective yield strength for the rod (i.e., exerted over the full 

event duration), in order to produce the residual lengths noted in the rigid-to-eroding transition of 

figure 3a.
11

Vs (m/s)

0 500 1000 1500 2000

YEFF (GPa)

0

1

2

3

4

"Effective" Rod Strength 

for noneroding penetration

Average

Rod Strength

to match

experimental Lr

Nominal Uniaxial

Rod Stength

Figure 6. Back-calculated “effective” velocitydependent rod strength, to accurately model 

residual length in Finite-Target Series tests, 

assuming effective strength was to act over 

full duration of ballistic event. 

However, this approach can only have merit if it can be used to accurately predict results from 

other test series beyond the Finite-Target Series. But, in fact, such an approach does a very poor 

job if applied to, for example, the Semi-Infinite Series of tests, and severely overpredicts ballistic 

performance into monolithic Al for striking velocities covered by the back-fit (i.e., between 1108 

and 1760 m/s). The logical conclusion is that HLAT cannot be solely a function of Vs. A more 

likely scenario is that HLAT is a material parameter and no more an intrinsic function of striking 

velocity than is the target resistance, H, itself. Considering HLAT fixed for a given material, the 

remaining possibility to explain the data is that the transition from noneroding to eroding occurs 

at some point during the course of the ballistic event so that, at some intermediate stage, the 

effective rod strength transitions from a value of HLAT + Y to the nominal value of Y. Under such 

a hypothesis, the principal modeling issue becomes the trigger that precipitates the erosion 

transition. 

One may effectively rule out elapsed time as the erosion trigger (i.e., the trigger to remove lateral 

interference from the target), as it would allow, without limit, for progressively greater depth of 

noneroding penetration as the striking velocity was progressively raised. One may also rule out 

penetration depth as the erosion trigger, as this would imply, regardless of striking velocity, that 

all impacts would achieve the noneroding penetration to the same depth before commencing 

erosion. Such a notion is incompatible with the Semi-Infinite Series data of figure 1. 

Consider then the possibility that erosion is triggered by a change in rod length; call it ∆LCRIT. 

While it is normal, in the context of penetration mechanics, to consider any change in rod length 
12

to be precipitated by and following the onset of rod erosion, it is known in the current context 

that the noneroding penetration is accompanied by rod deformation. Forrestal and Piekutowski 

(24) observed this fact definitively. In the current study, the noneroding datum from the SemiInfinite Series (Vs = 1108 m/s) experienced a 6% diameter increase over at least the leading 

25 mm of rod (1) (figure 7). Continuity dictates for that portion of the rod experiencing a 6% 

diameter increase that the length decreases by 11%. Depending on what portion of the overall 

rod length (original length of 101.9 mm) experienced this diameter expansion, the rod 

diminished somewhere between 3% and 11% of its overall length without the onset of erosion. 

Figure 7. Photograph depicting residual-rod fragment recovered 

from test MV1. While rod tip appears undeformed, 

measurement reveals a 6% increase in rod diameter. 

Thus, it seems quite feasible that the amount of noneroded rod foreshortening, ∆LCRIT, being 

indicative in some way of lateral rod strain or perhaps of irreversible nose-geometry 

deformation, be the principal factor to govern the onset of erosion. The observed 

phenomenology is, at casual glance, compatible with this explanation. For if the striking 

velocity is small enough that the noneroding deformation remains limited, the penetration may 

wholly proceed through the target in a noneroding manner. If, on the other hand, the stagnation 

pressure from a high striking velocity is large enough to overwhelm the augmentation of axial 

stress by a value HLAT, then the noneroding phase of the event will quickly pass with the rapid 

rod deformation, and the eroding phase will dominate the event. 

The challenge to understanding this phenomenon in general will be the fact that the tendency to 

establish an eroding flow field is a function of many factors, at a minimum including rod 

material, target material, and most importantly, the rod’s nose geometry. In the current test 

series, both rod material (WA) and the rod’s nose geometry (hemispherical) vary negligibly. 

And for the target material (Al) where this phenomenon might be anticipated, the density is 

constant, and only the target resistance varies with the particular alloy. It is possible that impact 
13

velocity will play a limited role, but only at striking velocities higher than those examined in this 

study. For as the striking velocity is increased toward the hydrodynamic range, with the 

concomitant increase in target-crater size, a point will be reached where the crater size grows 

large enough to preclude the lateral interference between rod and target. When this happens, 

∆LCRIT will necessarily diminish to zero. But in the present study, striking-velocity dependence 

will not be reflected in the modeling of ∆LCRIT. Thus, it is hoped that for the purposes of 

modeling the erosion transition for these test series, the foreshortening-to-erosion-onset 

parameter, ∆LCRIT, may be solely correlated with the target alloy’s resistance to penetration, H. 

In this regard, there are two limiting cases to look to for guidance. For a target with vanishingly 

small resistance, one would expect a negligible phase of interference fit between rod and target 

because there is nothing but inertia to resist the lateral expansion of the target. At the other 

extreme, one of very large target resistance, one would also expect a negligible phase of 

interference fit with the target because a very hard target would quickly force the rod into an 

erosive flow field configuration, before an embedded interference fit could even be established. 

There would thus appear to be only a limited range of intermediate target hardness at which this 

unique interference fit might reasonably be established. This correlation will be sought, once the 

remaining modeling groundwork is established. 

3.2 HLAT Magnitude 

In the preceding report (1), the magnitude of HLAT was estimated so as to provide for a fully 

rigid-body penetration for the Vs = 1108 m/s datum. Requiring rigidity necessitated the use of an 

HLAT value 1.31 × the target’s penetration resistance. The experimental datum was, however, not 

fully rigid, as the recovered rod fragment was observed to have expanded 6% in rod diameter 

during the course of the penetration. This is one reason to believe the actual value of HLAT to be 

less than this 1.31 value. While one might expect the resistance-to-radial crater expansion to be 

of a similar magnitude to the resistance-to-axial penetration, there is reason to suspect that the 

two values are not identical. For example, the classical indentation literature (28) indicates that 

both steel and Al’s resistance to deep indentation (penetration) is from 2.8 to 3 × the yield 

strength, while the derived expressions presented by others (29–31) for spherical cavity 

expansion can range from 3.5 to 4.1 × the yield strength, depending on material stiffness and 

strength for the range of steels and alloys of Al in the current study. On this basis, one might 

expect the ratio HLAT/H to lie in the range of 1.17 to 1.46. In the end, an empirically selected 

value of HLAT = 1.22H proved effective, as will be shown. 

3.3 Impact Pressure 

The rod deceleration equation, for the traditional case, is given by 

(L s)

V

R − = − ρ

σ & , (3) 
14

where V is the rod velocity, σ is the decelerative stress applied at the elastic-plastic boundary in 

the rod or, in the case of a rigid body penetrator, the stress applied at the rod’s nose, ρR, is the 

rod’s density; L is the rod length, and s is the size of the small plastic zone at the leading edge of 

the rod. In Tate’s original derivation (7), a slender rod is assumed so that s is a negligibly small 

fraction of L. In the case of an eroding rod, the rod’s strength limits the stress that composes the 

decelerative force, and thus σ = Y. In the case of rigid penetration, there is no plastic zone, so 

that s = 0. Further, the target’s resistance plus the inertial pressure of the target’s stagnation flow 

upon the penetrator tip, being <Y, composes the decelerative stress. Thus, σ = H + kTρTU2

represents the decelerative stress for rigid-body deceleration in the traditional case. 

The issue was raised in reference to figure 3b that the modeled residual velocity from the FiniteTarget Series was too large, falling above the data in both the case of eroding-rod and rigid-rod 

modeling. The implication is that the actual deceleration of the rod is larger than that being 

modeled. Because the eroding and rigid models should straddle the experimental data, one 

would have to account for a greater decelerative stress in the rigid-penetration case, beyond 

σ = H + kTρTU2, in order to achieve this goal. While raising the target resistance is a simple and 

direct way of increasing this decelerative force on the rigid rod, it suffers the same flaw as 

modeling the erosion transition with a velocity-dependent target resistance; namely, it 

substantially worsens the modeling predictions for the Semi-Infinite Series. 

If the resolution does not come by way of target resistance, then the stagnation pressure is the 

only other quantity at hand. The pressure term in the rigid-body deceleration equation is kTρTU2

. 

While the Bernoulli pressure is more than just a steady-state pressure, it nonetheless does not 

account for any wave mechanics in the flow field. At the moment of actual impact, however, the 

transient impact pressure, given by ρTUsup (Us is the shock-wave velocity and up is the particle 

velocity behind the shock wave), can be more than an order of magnitude larger than the 

Bernoulli pressure. The calculation of this impact pressure requires the knowledge of several 

additional material parameters for the rod and target, namely the bulk sound speed C0 and the 

empirically derived slope of the shock-vs.-particle velocity curve, given by S. These values are 

widely published equation-of-state parameters, available for a wide variety of materials. 

With such a large disparity between Bernoulli and transient shock pressures, the much larger 

shock pressure need not act long in order for it to have a measurable effect on the decelerative 

impulse imparted to the rigid-rod. And while the reality is that this large transient pulse decays 

rapidly to the Bernoulli pressure value, the implementation chosen here, for simplicity, is a step 

function. The full impact stress is arbitrarily applied for a duration equal to the time required for 

a rarefaction (stress-release) wave to traverse the rod diameter three times, i.e., an impact 

pressure differential ∆P is added to the Bernoulli stress during the period of time t < 3D0/C0 (note 

that a Poisson’s ratio of 1/3 has been assumed, making the longitudinal wave speed of the release 

wave equal to the bulk sound speed C0). In the current context of a 6.79 mm WA rod 

(C0 = 4030 m/s), this translates to ∆P being applied for a duration of ~5 μs. 
15

Note that this pressure differential, while present, does not affect the Bernoulli balance of equation 2 

between rod and target (which governs the relation of U and V), because this elevated impact 

pressure is present in both rod and target, thus canceling on balance. Rather, this pressure differential 

serves to bolster the driving stress, σ, for rigid-rod deceleration (equation 3). This pressure 

differential will also play a role in the deceleration of deforming, yet noneroding, rods, to which this 

report presently addresses itself. 

3.4 Plastic Zone Size in Rod 

3.4.1 Equation of Noneroding Rod Motion 

The equation that governs the decelerative motion of a penetrating rod was given as equation 3, valid 

for the cases when the penetrator is an eroding or a purely rigid rod. A vital question to ask is how 

this governing relation may be altered in the current context, where the rod is neither eroding nor 

purely rigid. In the case of a rigid rod, the issue is that the decelerating stress on the rod’s elastic 

nose is below the magnitude of the rod’s strength; nonetheless, its value is still obtainable from 

consideration of the target’s flow around the rigid rod. For eroding rods, however, the issue differs. 

Traditional methodology, espoused by all from Tate (7) to Walker and Anderson (6), is to consider 

only the elastic portion of the rod in the control volume used to derive the rod-deceleration (i.e., 

momentum) equation. The reason for so doing is threefold: (1) because the axial stress magnitude at 

the elastic/plastic boundary is accurately known to be the rod’s strength, (2) it avoids the need to 

calculate the radial stress gradient in the erosion zone, and (3) it avoids the need to tabulate the 

momentum efflux from the products of rod erosion, the velocity of which have changed not only in 

their magnitude, but also their direction of flow. These gradient and flux terms, difficult to estimate, 

would become an important part of the momentum equation, were the eroding rod tip considered as 

part of the control volume. 

The problem with adopting this traditional methodology to the present case of deforming, yet 

noneroding, penetration is in accurately ascertaining the time-dependent location, s, of the 

elastic/plastic interface from the rod’s nose. In the case of a traditionally eroding rod, s may be 

estimated from the consideration of residual length data in semi-infinite penetration tests. But the 

plastic zone size in the traditional eroding rod case bears little resemblance to that currently observed 

in the noneroding tests, nor those noneroding data reported by Forrestal and Piekutowski (24). From 

the nonrigid form of equation 3, given as 

(L s)

Y V

R − = − ρ

& , (4) 

it may be deduced that, as s becomes a significant portion of rod length L, its influence on the 

deceleration result becomes vital. It has already been established with the recovered noneroding rod 

fragment from the Semi-Infinite Series of tests (figure 7) that a significant percentage (25% or more) 

of the noneroding rod becomes plastic, and not just a small zone near the rod tip, as in the case of 

traditional eroding penetration. The radiographic data of Forrestal and Piekutowski (24) show 
16

upwards of 50% of the uneroded rod within the plastic zone. And yet, no immediate guidance 

suggests itself as to how to model a time-dependent value of s for the noneroding case. 

This quandary suggests that an alternative control volume might prove more profitable. If the 

complete rod is considered in developing the rod-deceleration equation, the need to know the 

value of s, in order to solve it, evaporates. Additionally, there is arguably no radial gradient of 

stresses within, nor efflux of erosion products from, the noneroding rod tip, thereby mitigating a 

primary difficulty in considering the plastic portion of the rod as part of the control volume for 

rod deceleration. What is required, however, is knowledge of the velocity and axial stress at the 

plastic tip of the noneroding rod. The rod-tip velocity is, by definition, U. The differential 

between velocity V in the elastic portion of the rod and U in the plastic portion leading up to the 

tip is accommodated by the diameter expansion in the plastic portion of the rod. The axial stress 

at the tip comports with prior discussions on interference fit between the rod and the target 

during the noneroding phase, as well as impact pressure considerations. Thus, the roddeceleration equation becomes in the noneroding phase, in lieu of equation 4, 

L

Y H k V U P V

R

R R

ρ

+ + ρ − + ∆ = −

2

LAT ( ) & , (5) 

where the impact pressure differential ∆P is conditionally applied only during the period of time 

specified by t <3D0/C0. When noneroding penetration ceases and erosion commences, the 

deceleration equation should revert back to the traditional form of equation 4. 

Figure 8 depicts the situation for the case of noneroding penetration. In it, the equations for both 

viable approaches to rod deceleration are given, alternately requiring knowledge of the stress 

magnitude at the rod tip, or the size of the noneroding plastic zone, respectively. Equation 5 

depicts the former. In the simplest of cases where the penetration is steady (and time-rates of 

change may be neglected), the Bernoulli stress balance between rod and target is achieved 

through equating the stresses at locations B and T, as shown in the figure. 

3.4.2 Extended Bernoulli Equation 

While equation 5 effectively removes the need to know how s behaves for the noneroding case, 

its behavior for the traditional eroding case still needs to be specified for use in equation 4. 

Furthermore, while the extended Bernoulli equation 2 for the slender-rod condition is not a 

function of s, the full extended Bernoulli equation actually solved in the model comports with the 

analysis of Walker and Anderson (6) to include nonlinear effects, with extensions by the author 

for laminate targets (4, 5). It is given as 

R

X X R

U

X U V U k U H s V U Y

L

s k U R

T

R

R R

& & & & & &

&  − + − + = + + + + 

 



 − α

α

ρ ρ ρ 2 α

EFF

2 ( ) 2 ( ) 2 , (6) 
17

A

B

s

−σzz

(A)

 = Y

−σzz

(B)

 = Y + HLAT + kR ρ R(V − U)

2 HLAT + ∆P

−σzz

(T)

 = H + kT ρ TU 2 T + ∆P

V

U

dV/dt = −[Y + HLAT + kR ρ R(V − U)

2

 + ∆P] / (ρ RL)

= −Y / [ρ R(L−s)]

z

Figure 8. Schematic depicting the axial-stress levels at three locations, A, B, and T

along the rod centerline, along with two versions of the rod-deceleration 

equation (one requiring knowledge of s, the other not). 

where ρ and H are the appropriately aggregated density and resistance in the target’s plastic 

zone, and XU, Xα , and XR are aggregated factors from the target’s plastic zone associated with the 

nonsteady effects of penetration rate U, Walker-Anderson’s plastic zone size parameter α, and 

crater radius R, respectively. Equation 6 would seem to require knowledge of s for both the 

eroding and noneroding cases. Given the presence of time derivatives of s therein, the response 

of equation 6 can be sensitive to arbitrarily selected forms for s. Therefore extreme care must be 

taken in its specification. First, consider the case of the traditionally eroding rod, where the 

presence of s is made known in both equations 4 and 6. 

One of the primary benefits in properly accounting for s is in modeling rod deceleration in the 

final stages of penetration (where L often shrinks to the order of s in magnitude). In the absence 

of modeling s, the residual rod length is always underestimated, a point that has been noted by 

many regarding the original Tate model (which assumes L>>s, a valid slender-rod assumption 

except in the final stages of rod erosion). To handle s for the traditionally eroding case, the 

current model draws upon and adapts an ad hoc algorithm from its FZ model roots (2). The FZ 

approach chose not to deal with the vagaries of modeling s explicitly, but instead increased Y as a 

decreasing function of L so as to match the experimental record of residual length. From 

equation 3, it can be seen that these two approaches are mathematically analogous (i.e., 

s/L ≈ 1 − Y0/YFZ) in their effect on rod deceleration. But while the YFZ(L) approach remedied the 

residual length calculation, it did introduce a deleterious effect, in that it forced arbitrarily larger 

values of target resistance H to be used in compensation for a rod strength function, YFZ that 

increased value with rod erosion. 
18

Modeling the effect as a finite plastic zone s instead of a rod-strength effect YFZ(L) provides a 

more faithful model of reality and removes grounds for criticism unrelated to the erosionthreshold modeling at hand. Therefore, in lieu of modeling Y as a function of L, a function for s

was selected to closely match the dV/dt behavior of the FZ model’s YFZ(L) function at small L, 

where the influence of s is large. The function to model s, for the case of a traditionally eroding 

rod, is empirically selected as 















































 = = + −

3

0 0

1

ERODE 0 2

tan

2

D

L B

D

L

A

A

s s D π

π

(traditionally eroding rod), (7) 

where A and B have both been set to 0.8. This function asymptotes to a constant value of 

s = AD0 when the rod length is large, and to s = L as the rod length shrinks to zero. The 

parameter B affects the rate of asymptote. Penetration results with this model revision have been 

verified to closely follow the FZ model results for semi-infinite penetration, which in turn, match 

the residual length data for semi-infinite penetration into rolled homogeneous armor (RHA) (2). 

The time derivative of s, for use in equation 6 is obtained by way of the derivative of s with 

respect to L, as 

2 3

0 0

2

0

2

1

1 3

















































+ +

















+

=

D

L B

D

L

A

D

L B

L

s

π &

& (traditionally eroding rod). (8) 

For the deforming, yet noneroding, condition, there is no clear guidance on how s should be 

formulated. It was for this reason that equation 5 was developed to remove its considerable 

influence from the rod deceleration equation. But it’s retention in the extended Bernoulli 

equation 6 remains. Yet if one hypothesizes a noneroding form for s as a nontrivial perturbation 

of equation 7, in the form of 

s = s + δs ERODE (deforming, noneroding rod), (9) 

where δs is the noneroding augmentation to s, then it is possible to select a reasonable form for 

δs that has a zero net contribution to the extended Bernoulli equation (equation 6). From the 

extended Bernoulli equation, it is seen that such a zero-sum form on δs must satisfy the equation 

( ) 0

2 ( ) 2

2 − + V +U = s V U

L

s R

R & & &

& ρ δ

ρ

δ . (10) 

The solution to this equation, in light of the kinematic constraint dL/dt = −(V − U), is 
19















 = − ∫

V

V L

dV

L

L

s s

0

exp 2 0

0 & &

& δ δ . (11) 

While equation 11, like equation 7 before it, ignores the initial transient creation of the plastic 

zone s, they both treat the subsequent decay of s as a function of changes in L, dL/dt, and V. As 

dL/dt approaches zero, the noneroding δs also vanishes, as would be expected. Because the δs

form satisfies equation 10, all contributions to the extended Bernoulli equation (equation 6) for 

nonrigid, noneroding penetration are identical to those arising solely from sERODE, which at least 

has the virtue of being fit to data for residual length for semi-infinite penetration data into armor. 

So while the proposed form for a noneroding s, given by equation 9 remains ad hoc and subject 

to step-wise dislocations as the penetration mode transitions to and from eroding, noneroding, 

and/or rigid-body, it nonetheless possesses several virtues. First, it accommodates arbitrarily 

large values of noneroding s. Second, it does not allow for arbitrary tailoring of results through 

wanton specification of an s function. Most importantly, however, in conjunction with equation 

5, it dispatches the need to know the magnitude of the noneroding s function (i.e., the size of the 

rod’s plastic zone) explicitly in order to solve the governing equations for noneroding 

penetration. 

3.5 Integration of Rod Deformation 

It was proposed that the foreshortening-to-erosion-onset parameter, ∆LCRIT, may be respectively 

correlated with each target alloys’ resistance to penetration. An implementation issue, in this 

regard, is how to tabulate the change in rod length vis-à-vis ∆LCRIT, if the noneroding phase of the 

penetration traverses target elements of different resistance (each with a unique value of ∆LCRIT). 

If one envisions the ∆LCRIT parameter as a measure of the allowable noneroding deformation or 

damage that the rod may sustain prior to the onset of erosion, then the amount of length change 

that the rod sustains in any one element must be prorated by that element’s ∆LCRIT, in order to 

ascertain the fractional damage from that element. Thus, define a nondimensional deformation 

parameter D as 

∑= ∆

∆ =

n

i i

i

L

L D

1 CRIT( )

, (12) 

summed over each target element i of the n successive target elements being penetrated in 

noneroding fashion. With this definition, noneroding penetration ceases when the value of 

deformation D reaches unity.
20

4. Modeling Results 

The framework has been laid out by which it is proposed to model the phenomenon of 

deforming, yet noneroding, penetration, along with its subsequent transition to fully eroding 

penetration. The basic premise is that, during a perhaps limited phase of noneroding penetration, 

the rod nonetheless deforms (expanding in diameter and thus shortening in length), but is 

prevented from eroding because of the prevailing interference fit between rod and target. The 

duration of this noneroding phase of the penetration is dependent upon a permissible level of rod 

deformation as measured by ∆LCRIT, which itself is dependent upon rod and target material 

properties and nose geometry. A method was devised to account for this deformation, should it 

occur through the course of several target elements, each with its own value of ∆LCRIT. 

Equation 2 provides the Bernoulli balance, wherein the effective rod strength, YEFF, equal to Y in 

the fully eroding case, becomes Y + HLAT in the case of noneroding penetration where the rod is 

laterally constrained by the target wall with a resistance HLAT. The value of HLAT, the lateral 

target resistance, is surmised to be in the vicinity of a material’s axial resistance, H, and is taken 

as 1.22H in the current study. 

Further, it is believed that the large impact pressure pulse that occurs during the several 

microseconds after initial impact provides a notable decelerative impulse to rigid and noneroding 

rods (above and beyond the Bernoulli pressure) and should not be ignored. Its magnitude may 

be easily calculated from material-shock properties and, for simplicity, its form is taken as a 

square wave with a duration of application in the current study set to the time required for a 

rarefaction (stress-release) wave to traverse the rod diameter three times, such that t<3D0/C0. In 

the current context of L/D = 15 WA rods of 101.9 mm length, this duration is ~5 μs. 

While equation 3 provides the archetypal rod-deceleration equation form for both rigid and 

eroding rods, the complications with using this form for deforming, yet noneroding, rods proves 

daunting because of the unknown nature of the plastic zone size s. As such, equation 5 was 

derived as an alternative deceleration equation. Unlike the traditional form, which is derived by 

considering the forces and momentum in the elastic portion of the rod, equation 5 is solved by 

considering the forces and momentum in the complete rod, which is not fully elastic. Rather 

than requiring a time-dependent estimate of the plastic-zone size in the rod, equation 5 instead 

requires an accurate knowledge of axial stress at the rod’s noneroding tip. 

Further, a method was devised to allow the extended Bernoulli equation 6, detailing the stress 

balance across the rod/target interface, to be solved without knowing the magnitude of the 

nonrigid, noneroding component to s. 

With this revised methodology, and the material parameters specified in table 3, the FiniteTarget Series of tests, previously modeled in figure 3, were modeled again. A permissible level 
21

of rod deformation, ∆LCRIT, set equal to 1.4D0 for the BHN 97 and BHN 99 Al target elements 

(resistance H = 1.78 GPa), was employed to produce the results shown in figure 9. While this 

value of ∆LCRIT was fitted to the data, its reasonableness is validated by the results of Forrestal 

and Piekutowski (24), who observed noneroding rod foreshortening of 1.43D0 just below the 

erosion threshold for a very comparable RC 39.7 hemispherical-nose steel rod into Al-6061-

T6511. 

Vs (m/s)

0 500 1000 1500 2000

L

r /L0

0.0

0.2

0.4

0.6

0.8

1.0

Data:

w/o backing

w/ backing

High-Yaw

Impact

Noneroding Rod:

Measured

Incomplete

 Lr

{

Revised Model:

w/o backing

w/ backing

Vs (m/s)

0 500 1000 1500 2000

Vr (m/s)

0

200

400

600

800

1000

1200

1400

Revised Model:

w/o backing

Revised Model:

w/ backing

Data:

w/o backing

w/ backing

High-Yaw

Impact

 (a) (b) 

Figure 9. Data for Finite-Target Series tests, with comparison to revised-methodology calculation, for 

(a) residual rod length and (b) residual velocity. 

While the modeled transition from noneroding to fully eroding, as shown in figure 9a is still a bit 

more sudden than the experimental data reflect, the model nonetheless captures the experimental 

trend extremely well, especially when taken in direct comparison to figure 3, which modeled the 

result as an either/or situation: either noneroding or fully eroding penetration. The presence of 

the impact-pressure differential, ∆P, has permitted a better match to the residual velocities in 

figure 9b, for high striking velocities where the event is largely erosive. The methodology of 

rod-deformation-as-trigger for the erosion transition, embodied in the parameter ∆LCRIT, has the 

primary bearing upon the residual length behavior modeled in figure 9a. 

To ascertain whether the rod-deformation-as-trigger methodology adversely affects the 

previously considered Semi-Infinite Series test result that was so well captured in the preceding 

report (1) (shown in figure 1), the revised model is applied to that test series. Because the Al in 

both the Finite-Target Series and the Semi-Infinite Series I of tests are of the same hardness, the 

identical properties are retained for ∆LCRIT; namely, equal to 1.4D0. The result is shown in 

figure 10. While the residual velocity and penetration curves are slightly affected, the important 

aspect to note is that, for deep penetration, the erosion transition has an appearance of being an 

abrupt function of striking velocity, even though the results presented, for all impact velocities, 
22

Vs (m/s)

0 500 1000 1500 2000

L

r 

/L0

0.0

0.2

0.4

0.6

0.8

1.0

Eroding Rod

Model

Rigid Rod

Model

Noneroding Rod:

Measured

Incomplete

 Lr

Revised Model

Vs (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

Rigid Rod

Model

Rigid Rod

Model

Revised Model

 (a) (b) 

Figure 10. Data for Semi-Infinite Series I tests, with comparison to rigid-rod, eroding-rod, and 

revised-methodology calculations, for (a) residual rod length and (b) penetration and 

residual velocity. 

undergo a noneroding phase at the onset of penetration. The transition point agrees with the 

current data set, falling between 1100 and 1400 m/s of striking velocity. The actual transition 

value, ~1200 m/s, also agrees well with the experimental observations reported by Scheffler (14) 

for WA rods striking Al-5083 targets. 

Like the Finite-Target Series, the results of modeling the Interply Series data as either a purely 

rigid or purely eroding phenomenon did not match well with the experimental record, both for 

residual rod length (figure 5a) and residual velocity (figure 5b). The two varieties of Al target 

plies in the Interply Series of tests are substantially stronger than those in the Finite-Target Series 

and Semi-Infinite Series I tests. Based on the experimental record reported by Scheffler (14), 

who reported different erosion-transition velocities in two different Al alloys, it was anticipated 

that the propensity to sustain noneroding penetration, as captured by the ∆LCRIT parameter, would 
23

vary with target resistance. With no quantitative guidance on its selection, this parameter was 

fitted to obtain the best results for the Interply Series. The one self-imposed constraint enforced, 

for simplicity, was to keep ∆LCRIT linear with target resistance, over the range of resistance from 

H = 1.78–2.77 GPa. This range of target resistances covers all data for the Semi-Infinite-Series 

I, the Finite-Target Series, and the Interply Series tests. Within this constraint, the selected 

values of ∆LCRIT were 1.4D0 for H = 1.78 GPa, 1.65D0 for H = 2.07 GPa, and 2.25 D0 for H = 2.77 

GPa, all of which are collinear in the plane of ∆LCRIT vs. H. 

With these values for the erosion-transition parameter ∆LCRIT, the results of the revised model for 

the Interply Series are given in figure 11. Despite the scatter in the test data, the results appear to 

be a vast improvement over figure 5, again supporting the contention that an allowable rod 

deformation, as captured by ∆LCRIT, is the relevant trigger for the onset of rod erosion. 

Vs (m/s)

0 500 1000 1500 2000

L

r /L0

0.0

0.2

0.4

0.6

0.8

1.0

Revised Model:

w/ and w/o interply

Data:

w/o interply

w/ interply

Vs (m/s)

0 500 1000 1500 2000

Vr (m/s)

0

200

400

600

800

1000

1200

1400 Data:

w/o interply

w/ interply

Revised Model:

w/ and w/o interply

 (a) (b) 

Figure 11. Data for Interply Series tests, with comparison to revised-methodology calculation, for 

(a) residual rod length and (b) residual velocity. 

There was a single datum categorized as Semi-Infinite Series II, which involved the same rod 

impacting a much stronger BHN 190 target at 1413 m/s. The test result proves to be above the 

erosion threshold, having penetrated a mere 130.4 mm. However, this result is important, as 

figure 12a reveals that the parameter ∆LCRIT cannot exceed a value of 1.2D0 while simultaneously 

predicting the ballistic result above the erosion threshold. For comparison, a result is plotted on 

the same graph that assumes ∆LCRIT equal to 2.25D0 that shows the unlikelihood of this 

possibility. This datum shows the anticipated trend reversal in behavior of ∆LCRIT with increasing 

H. As H increases, common sense dictates that ∆LCRIT must eventually return to zero, and the 

datum here indicates that that trend reversal has already begun by the time H reaches 3.16 GPa. 
24

Vs (m/s)

0 500 1000 1500 2000

L

r 

/L0

0.0

0.2

0.4

0.6

0.8

1.0

Eroding Rod

Model

Rigid Rod

Model

Revised Model

(Lcrit = 1.1 D0)

Vs (m/s)

0 500 1000 1500 2000

P (mm)

0

50

100

150

200

250

300

Eroding Rod

Model

Rigid Rod

Model

Revised Model:

∆Lcrit = 2.25D0

∆Lcrit = 1.1D0

 (a) (b) 

Figure 12. Datum for Semi-Infinite Series II test, with comparison to rigid-rod, eroding-rod, and 

revised-methodology calculations, for (a) residual rod length and (b) penetration (note: 

revised-methodology calculation shown for two values of parameter ∆LCRIT). 

A plot of the value of ∆LCRIT with increasing H is shown in figure 13. Recall that these values for 

∆LCRIT are empirical, but are nonetheless buttressed by various experimental data. For example, 

Forrestal and Piekutowski (24) witnessed a length foreshortening of 1.43D0 just below the 

erosion threshold for a configuration very comparable to the BHN 97 configuration in which 

∆LCRIT = 1.4D0 is currently employed. And while the Semi-Infinite Series II datum cannot 

indicate precisely the value of ∆LCRIT, it nonetheless serves to place an upper bound on its value 

for that rod/target configuration. 

Modeling ∆LCRIT as a function of H alone was permitted here only because of the invariant rod 

material and rod-nose shape across all the test series, as well as the limited impact-velocity span 

over which the tests were conducted. Unlike hemispherical-nosed (and conical- and ogivenosed) rods, flat-nosed rods will have no kinematic tendency for an applied pressure at the 

contact surface to resist the penetrator’s radial flow that brings about erosion. For this reason, 

one would expect flat-nosed rods to have severely retarded, if not absent, values of ∆LCRIT. 

Similarly, while conical-nosed rods have a geometry that serves to restrain rod-nose material 

from flowing radially outward, their sudden transition in slope between the rod’s nose and shank 

promotes, during penetration, a separation of target material from the rod at that point. 

Preventing this flow separation is essential to inhibiting the onset of erosion because a target not 

in contact with the rod can offer no lateral resistance to the emerging radial rod flow that is a 

precursor to erosion. It is thus no surprise that the phenomenology of erosion transition for 

conical-nose rods occurs not at the rod tip, but where the tip meets at the shank, as noted by 

Brooks and Erikson (10) and Woodward (12, 13). 
25

Hcore (GPa)

01234

∆Lcrit/D0

0.0

0.5

1.0

1.5

2.0

2.5

slope = 0.858/GPa

?

?

Revised Model

∆Lcrit < 1.2D0

Figure 13. Rod-deformation-to-erosion-onset model 

parameter, ∆LCRIT/D0, shown as a function 

of target resistance, for the Al alloys 

modeled in the current test series, with 

anticipated interpolation and extrapolation 

to other Al alloys. 

Exploring the model’s behavior has revealed several parameter sensitivities in general. The 

magnitude of HLAT will primarily control the striking velocity at which significant deformation 

occurs. The impact-pressure transient ∆P only affects the residual velocity to any extent, and not 

the residual rod length. In semi-infinite penetration, the addition of a transient ∆P may depress 

the depth of penetration very slightly. The allowable rod deformation (i.e., length foreshortening 

parameter, ∆LCRIT) that precedes the onset of erosion will primarily control the rapidity of the 

transition from a noneroding to a largely eroding event, and is therefore a significant factor in the 

residual length calculation. 

5. Comparison to FZ Model 

Comparison with the current model and experimental data is now offered against the FZ model 

(2, 3), on which the current model’s development was originally based. Note, however, that the 

FZ model has not been updated to reflect the current thinking on the transition-velocity regime 

phenomenology. Rather, the comparison is performed against the existing FZ model using the 

preexisting modeling options to address this very issue of threshold-velocity penetration 

mechanics. 
26

According to Frank (32), the FZ model is capable of modeling the dislocation in penetration 

associated with the erosion transition by utilizing a feature of the model, wherein the surface 

resistance of the target is set lower than the core resistance. The rationale for this approach is the 

indentation literature that shows the resistance to shallow indentation at a level roughly 1/3 that 

for deep indentation. Coupling this feature with the intrinsic FZ model behavior that precludes 

transition from rigid penetration to eroding penetration within a target element allows the model 

to penetrate rigidly throughout an event if the surface resistance can be set low enough to trigger 

an initially rigid response. Note that the approach of lowering the target resistance by the FZ 

model is the opposite of that proposed in the current model, in which the effective rod strength is 

raised by the presence of lateral confinement from the target. Nonetheless, both approaches have 

the effect of instigating noneroding penetration for rod/target combinations that would normally 

not experience it. 

Allowing for the differences in modeling the rod’s plastic zone cited in subsection 3.4.2, entitled 

“Extended Bernoulli Equation,” the present comparison allows for the target resistances used in 

the FZ model to be elevated over their counterparts in the current model. The elevated values of 

H chosen in the FZ model were selected to produce comparable results for the purely eroding 

and purely rigid configurations. So, for example, where the BHN 97 Al is characterized with a 

target resistance of 1.78 GPa in the current model, a value of 2.10 GPa is employed in the FZ 

model. Other Al alloys in the FZ model are treated as follows: the BHN 112 alloy utilizes 

H = 2.25 GPa, the BHN 143 alloy utilizes H = 2.95 GPa, and the BHN 190 alloy utilizes 

H = 3.55 GPa. The results for the test series, employing purely rigid or purely penetration 

conditions, are given in figures 14–17. 

From figure 15, it is observed that, for the purely eroding configuration, the influence of the thin 

steel backing is overly accentuated, compared with the current model in figure 3. For example, 

the FZ model is seen to predict an excessive 100 m/s difference in limit velocity from the 

addition of the thin target backplate. In the Interply Series results of figure 16, the effect is more 

pronounced, with the FZ model predicting the limit velocity to decrease by 150 m/s, with the 

addition of a thin acrylic element in the middle of the target. The nature of these deficiencies has 

been noted in the past (4, 5) and arises from the fact that the FZ penetration methodology only 

“looks ahead” one target element at a time to anticipate changes in target resistance. But other 

than these noted differences, the FZ model behaves, in most ways, similarly to the revised model 

for both eroding and purely rigid conditions of penetration. 

Noting these differences between FZ and the current model for the purely eroding 

configurations, consider now the more complex problem of noneroding penetration and erosion 

transition. While technical arguments might justify a surface resistance at one third the core 

resistance, such a level in the FZ model produces a noneroding threshold into BHN 97 Al at 

~550 m/s. Recall that the data show noneroding penetration at striking velocities beyond 

1100 m/s. Stretching the justification beyond a prudent level to accentuate the effect, the surface 

resistance will be set as low as possible (zero, to be precise) to give the FZ model its best hope of 
27

Vs (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

FZ Eroding:

Y = 1.1 GPa

FZ Rigid

FZ Rigid

FZ Model

Figure 14. Penetration and residual-velocity data for SemiInfinite Series I tests, with comparison to FZmodel results, for rigid and eroding rods. 

matching the data. Even then, the threshold transition occurs at 900 m/s, as shown in figure 18, 

significantly below the empirically observed levels. In this figure, the nomenclature “-S” is used 

to denote the FZ model with the soft-surface feature employed. Other than the actual velocity of 

the eroding transition, figure 18 appears at first glance consistent with the phenomenology of 

how the eroding transition occurs in deep penetration events. 

When attention is turned to the other test series, however, the FZ soft-surface methodology fares 

even more poorly. Figures 19 and 20 provide the results of the FZ methodology for the FiniteTarget and Interply Series tests. 

The influence of the soft-surface features of the FZ model are primarily limited to the lower 

striking velocities, and do not greatly influence the results at higher striking velocities from the 
28

Vs (m/s)

0 500 1000 1500 2000

L

r /L0

0.0

0.2

0.4

0.6

0.8

1.0

FZ Rigid:

with or w/o

backing

FZ Eroding:

w/o backing

w/ backing

Data:

w/o backing

w/ backing

High-Yaw

Impact

Noneroding Rod:

Measured

Incomplete

 Lr

{

Apparent Rigidto-Eroding

Transition

FZ Model

Vs (m/s)

0 500 1000 1500 2000

Vr (m/s)

0

200

400

600

800

1000

1200

1400

FZ Rigid:

w/o backing

w/backing

FZ Eroding:

w/o backing

w/ backing

Data:

w/o backing

w/ backing

High-Yaw

Impact

FZ Model

 (a) (b) 

Figure 15. Data for Finite-Target Series tests, with comparison to FZ-rigid-model and FZ-erodingmodel calculations, for (a) residual rod length and (b) residual velocity. 

Vs (m/s)

0 500 1000 1500 2000

L

r /L0

0.0

0.2

0.4

0.6

0.8

1.0

FZ Rigid:

with or w/o

backing

FZ Eroding:

w/ and w/o interply

Data:

w/o interply

w/ interply

FZ Model

Vs (m/s)

0 500 1000 1500 2000

Vr (m/s)

0

200

400

600

800

1000

1200

1400

FZ Rigid:

w/ and w/o interply

FZ Eroding:

w/ and w/o interply

Data:

w/o interply

w/ interply

FZ Model

 (a) (b) 

Figure 16. Data for Interply Series tests, with comparison to FZ-rigid-model and FZ-eroding-model 

calculations, for (a) residual rod length and (b) residual velocity. 

basic eroding-only model. Additionally, the exaggerated effect on the FZ model predictions 

from the addition of ballistically negligible target elements is noted for both the Finite Target 

Series and Interply Series tests. 

A comparison of the respective model’s through-target treatment of effective rod strength and 

target resistance is provided in figure 21, for the Finite Target Series simulation at a striking 

velocity of 1250 m/s. The FZ model utilizes the soft-surface feature for target resistance 
29

Vs (m/s)

0 500 1000 1500 2000

P (mm)

0

50

100

150

200

250

300

Eroding Rod

Model:

Y = 1.1 GPa

Rigid Rod

Model

FZ Model

Figure 17. Penetration datum for Semi-Infinite Series II 

test, with comparison to FZ-model results, 

for rigid and eroding rods. 

(figure 21a), whereas the revised model employs a lateral stress (HLAT) augmentation to the axial 

stress state in the rod (figure 21b). In the revised model, when the axial rod stress (sAXIAL) is at 

the maximum allowable level of Y + HLAT, the rod generally will be penetrating in a deforming, 

yet noneroding, fashion. 

The author considered the possibility that the preferred FZ implementation of soft-target surfaces 

was one that was intended not just at the leading surface of the target, but also at subsequent 

interfaces inside the target. But while such an interpretation did, in some cases, improve the 

overall appearance of the residual-length or residual-velocity results arising from the FZ model, 

it would require one to accept strength and resistance variations such as those presented in 

figure 22. In this figure, which reports the FZ model’s strength characterization of the Finite 

Target Series simulation at a striking velocity of 1000 m/s, the nomenclature “-S/I” is used to 

denote the FZ model with the soft-surface and soft-interfaces feature employed. Note that 

figure 22 employs the soft interface only at the material boundary between the two different Al 

alloys, and not at every physical plate boundary in the target. Even so, the purported strength 

variation, as presented, strains the bounds of credulity and so attempts to improve results by 

employing such an interpretation were subsequently discarded. 

The results of modeling the target with a soft surface, for a simple case like that presented in 

figure 18, may give the cursory impression of capturing the penetration dislocation associated 

with the erosion transition of the penetrator. However, a deeper investigation into a wider range 

of targets and engagement configurations reveals the inability of the preexisting FZ-modeling 

framework to capture the more general phenomenon. 
30

Vs (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

FZ Eroding

Rigid Rod

FZ Rigid FZ-S

FZ Model

Figure 18. Penetration and residual-velocity data for 

Semi-Infinite Series I tests, with 

comparison to FZ-model results, for rigid 

and eroding rods and FZ-S model with 

soft target surface. 
31

Vs (m/s)

0 500 1000 1500 2000

L

r /L0

0.0

0.2

0.4

0.6

0.8

1.0

Data:

w/o backing

w/ backing

High-Yaw

Impact

Noneroding Rod:

Measured

Incomplete

 Lr

{

FZ-S:

w/o backing

w/ backing

FZ Model

Vs (m/s)

0 500 1000 1500 2000

Vr (m/s)

0

200

400

600

800

1000

1200

1400

FZ-S:

w/o backing

Data:

w/o backing

w/ backing

High-Yaw

Impact

FZ-S

w/ backing

FZ Model

 (a) (b) 

Figure 19. Data for Finite Target Series tests, with comparison to FZ-S (soft surface) model 

calculations, for (a) residual rod length and (b) residual velocity. 

Vs (m/s)

0 500 1000 1500 2000

L

r /L0

0.0

0.2

0.4

0.6

0.8

1.0

Data:

w/o interply

w/ interply

FZ-S:

w/ and w/o interply

FZ Model

Vs (m/s)

0 500 1000 1500 2000

Vr (m/s)

0

200

400

600

800

1000

1200

1400 Data:

w/o interply

w/ interply

FZ-S

w/ and w/o interply

FZ Model

 (a) (b) 

Figure 20. Data for Interply Series tests, with comparison to FZ-S (soft surface) model 

calculations, for (a) residual rod length and (b) residual velocity. 
32

P (mm)

0 20 40 60 80 100 120 140

H,Y (GPa)

0

1

2

3

4

H (-S)

Y

"Soft"

Target

Surface

Target

Rear

In FZ model, rod

"senses" thin steel ply, 

but not target rear

FZ Model

P (mm)

0 20 40 60 80 100 120 140

H, sAXIAL (GPa)

0

1

2

3

4

Lateral restraint

of rod ceases

H

sAXIAL (Vs = 1400 m/s)

Revised Model

sAXIAL (Vs = 1250 m/s)

Y + Hlat

(a) (b) 

Figure 21. Target resistance and axial rod stress, through the depth of the Finite Target Series target at 

1250 m/s striking velocity, as characterized by (a) the FZ-S (soft surface) model and (b) the 

revised model. 

P (mm)

0 20 40 60 80 100 120 140

H,Y (GPa)

0

1

2

3

4

H (-S/I)

Y

"Soft"

Target

Surface

Once rigid, stays

rigid through

element

"Soft"

Interface

Y follows H

when rigid

Rigid Rod

Onset

Target

Rear

Rod "senses" thin steel plate, but not target rear

FZ Model

Figure 22. Target resistance and axial rod stress, 

through the depth of the Finite Target 

Series target at 1000 m/s striking 

velocity, as characterized by the FZ-S/I 

(soft surface and interfaces) model. 
33

6. Conclusions 

This report analyzes data related to the phenomenon of deforming, yet noneroding, penetration 

and the subsequent erosion transition of hemispherical-nosed WA L/D = 15 rods into laminate 

targets employing various alloys of Al and other materials. While the study of rigid-body 

penetration is well founded, a focus on long-rod penetration in which the penetrator experiences 

significant noneroding deformation has received more limited attention. More vaguely 

understood, still, are the conditions under which noneroding deformation transitions to fully 

eroding penetration. 

The current report documents series of 34 experiments that probe the nature of this phenomenon 

under a variety of target configurations, from thick, monolithic Al to finite laminates composed 

of several materials. The tests were conducted over a striking-velocity range from 906 to 

1722 m/s. Physical and radiographic data were collected, against which to compare any 

modeling, and include penetration, as well as residual length and residual velocity of the 

penetrator. Additionally, crater profiles were available for examination and occasional residual 

penetrator fragments were recovered. A model was formulated and tested against the data in an 

attempt to capture this wide range of data collected. 

Aside from the traditional treatments of fully eroding or rigid-body penetration, the model has 

several key phenomenological elements that have been found necessary to address the issues at 

hand. They include the following: 

• The presence, during the noneroding phase of penetration, of lateral confinement on the rod 

by the target wall, applying a lateral resistance ~1.2 × the target element’s ballistic (axial) 

resistance; 

• An augmentation, during the noneroding phase of penetration, of the rod’s axial stress, in 

accordance with Tresca plasticity, and resulting from the lateral confinement of the rod; 

• A trigger for erosion that is governed by a level of noneroding rod deformation associated 

with the target element’s resistance (rod properties and especially rod-nose geometry will 

also affect the erosion trigger, but these are kept essentially invariant throughout the test 

series); and 

• An accounting of the large transient impact pressure, insofar as its influence on rod 

deceleration. 

In addition to these elements of model phenomenology, the solution methodology employs 

variations from the traditional method. In particular, for the noneroding phase of penetration, the 

rod deceleration equation is solved using a control volume that includes the complete rod, not 

just the elastic portion. Further, the rod’s plastic zone size, during this noneroding phase, is 
34

assumed to follow a generic functional form selected to minimize the net influence on the 

rod/target stress balance. These changes in solution methodology alleviate the need to quantify 

the difficult-to-know size of the plastic zone during the noneroding phase of penetration. 

With this revised methodology, the experimental results for residual velocity and rod length have 

been nicely captured, for a variety of target configurations and over a range of impact velocities. 

Without the revised methodology, or using alternate methodologies (e.g., velocity-dependent 

erosion trigger and/or lateral resistance) that were explored, the match to data was drastically 

different and unacceptably poor. Further, attempts to employ the preexisting FZ methodology to 

the problem of noneroding penetration and erosion transition proved largely unsuccessful. The 

phenomenology proposed by the FZ model for noneroding penetration was one based upon the 

surface of a target having a lower resistance than the target’s core. The implementation of the 

FZ methodology has the net effect of making the erosion transition a striking-velocity-dependent 

phenomenon, which was found to be a poor trigger for erosion onset. 

One is led to the conclusion that a phenomenology like the one proposed, or very close to it, 

prevails during the special circumstances when deforming, yet noneroding, penetration occurs, 

leading up to the point where the transition to a traditional eroding event ensues. Perhaps the 

phenomenon is limited to the penetration of ductile low-density target materials, in which the 

crater radius is naturally small due to the high-penetration efficiencies. Nonetheless, unexplored 

areas warranting further study would include the influence of rod-nose geometry, rod material 

properties, and target materials other than Al. 
35

7. References 

1. Segletes, S. Analysis of the Noneroding Penetration of Tungsten Alloy Long Rods Into 

Aluminum Targets; ARL-TR-3075; U.S. Army Research Laboratory: Aberdeen Proving 

Ground, MD, 2003. 

2. 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. 

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

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

4. 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: 

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5. 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 

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

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

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

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

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

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

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

11. 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. 

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

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

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

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

U.K., May 1986. 
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14. 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. 

15. 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: Aberdeen Proving Ground, MD, 1998. 

16. 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 the 2nd 

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

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

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

Israel, 21−24 May 1995. 

18. Warren, T. L. Simulations of the Penetration of Limestone Targets by Ogive-Nosed 4340 

Steel Projectiles. International Journal of Impact Engineering 2002, 27, 475−496. 

19. 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. 

20. 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. 

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

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

22. 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. 

23. 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. 

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

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

28. Tabor, D. The Hardness of Metals, Chapter IV; Oxford University Press: New York, 1951.

29. Hopkins, H. G. Dynamic Expansion of Spherical Cavities in Metals. Progress in Solid 

Mechanics 1960, 1 (3), Sneddon, I. N., Hill, R., Eds.; North Holland Publishing: 

Amsterdam. 

30. Goodier, J. N. On the Mechanics of Indentation and Cratering in Solid Targets of StrainHardening Metal by Impact of Hard and Soft Spheres. Proceedings of the 7th HyperVelocity Impact Symposium, Volume III—Theory, Orlando, FL, 1965, 215−259. 

31. Hanagud, S.; Ross, B. Large Deformation, Deep Penetration Theory for a Compressible 

Strain-Hardening Target Material. AIAA Journal 1971, 9 (5). 

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

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ABERDEEN PROVING GROUND (CONT’D)

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