An Adaptation of Walker-Anderson 

Model Elements Into the Frank-Zook 

Penetration Model for Use in MUVES 

by Steven B. Segletes 

ARL-TR-2336 

.- ._.. -.-_-.-- 

September 2000 
The fmdings in this report are not to be construed as an official 

Department of the Army position unless so designated by other 

authorized documents. 

Citation of manufacturer’s or trade names does not constitute an 

offkial endorsement or approval of the use thereof. 

Destroy this report when it is no longer needed. Do not retmn 

it to the originator. 
Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5066 

ARL-TR-2336 September 2000 

An Adaptation of Walker-Anderson 

Model Elements Into the Frank-Zook 

Penetration Model for Use in MUVES 

Steven B. Segletes 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited 
Abstract 

The performance of the well-respected Frank-Zook (FZ) penetration algorithm is 

examined in light of an anticipated class of target technology involving laminated targets 

whose layers are thin relative to the projectile diameter (in the limiting case, this target 

class incorporates functionally graded materials). Because of the manner in which the FZ 

algorithm anticipates changes in material properties along the shotline of the penetrator, 

the algorithm is susceptible to inaccuracy precisely when the target layers along the 

shotline are thin relative to the projectile diameter. Though the problem can be quite 

severe when the target-layer thickness is a fraction of the projectile diameter, the effect is 

still evident to a much lesser extent, even as the target-element thickness is increased to 

several projectile diameters. A remedy to this type of problem is offered, accomplished 

by way of novel adaptation of elements of a model by Walker and Anderson into the FZ 

framework. In doing so, the target’s material- and nonsteady-kinematic properties are 

dynamically composed via an integration through the plastic zone in the target, ahead of 

the rod/target interface. Additionally, a model to predict the crater-diameter profile 

resulting from a modeled penetration event is optionally offered as a calculation 

enhancement. The modeling remedies and enhancements proposed herein are offered for 

incorporation into the U.S. Army Research Laboratory’s (ARL) modular Unix-based 

vulnerability estimation suite (MUVES) code, to be part of MU’s vulnerability/lethality 

(V/L) calculation methodology. 

ii 
Table of Contents 

1. 

2. 

3. 

4. 

5. 

5.1 

5.2 

5.3 

5.4 

5.5 

5.5.1 

5.5.2 

6. 

7. 

List of Figures ............................................................................................................... 

Introduction .................................................................................................................. 

Motivation ..................................................................................................................... 

Basis for Modifications ................................................................................................ 

Extent of Modifications ................................................................................................ 

Algorithmic Details ...................................................................................................... 

Extent of Plastic Zone ................................................................................................. 

Numerical Integration Approach ................................................................................. 

Crater-Radius Calculation ........................................................................................... 

Nonsteady Terms ......................................................................................................... 

Piecewise Integration of Momentum .......................................................................... 

Obliquity Considerations ......................................................................................... 

Surface Resistance.. .................................................................................................. 

Summary ....................................................................................................................... 

References ..................................................................................................................... 

Distribution List ........................................................................................................... 

Report Documentation Page ....................................................................................... 

V 

1 

1 

6 

8 

10 

10 

12 

14 

19 

19 

25 

27 

29 

33 

35 

47 

. . . 

111 
~NTIONALLY LEFT BLANK. 

iv 
List of Figures 

-. Fipure 

1. 

I 

2. 

m 

3 

4 

5 

6 

10 

11 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

12. 

13. 

14. 

15. 

16. 

Effect of thin backing plate on FZ model .................................................... 

Comparison of Resistance vs. Location for 50 mm target and 48 + 2 mm target ...... 

Effect of thin inter-ply layer on FZ model ................................................... 

Influence of thin inter-ply layer on FZ target resistance estimation ...................... 

Description of the target’s plastic zone in the WA model [8]. ............................ 

Comparison of Walker and Anderson fit to the constant R(a-1) assumption [7] ...... 

Nondimensional Penetration vs. Velocity for original (FZ) and revised (WA) 

models for semi-infinite penetration ......................................................... 

Comparison of proposed crater-diameter modeling with fits from other 

models [7,9] ..................................................................................... 

Example 1: Normalized Penetration vs. Velocity for FZ and 

revised (WA) models ........................................................................... 

Example 1: Target Resistance vs. Location for FZ and 

revised (WA) formulations ..................................................................... 

Example 2: Normalized Penetration vs. Impact Velocity for FZ and revised (WA) 

formulations ...................................................................................... 

Example 2: Target Resistance vs. Location for FZ and revised (WA) 

formulations ...................................................................................... 

Geometrical considerations associated with oblique penetration ......................... 

Comparison of FZ and revised (WA) formulations for Residual Length vs. 

Obliquity ......................................................................................... 

Comparison of FZ and revised (WA) formulations for Residual Velocity vs. 

Obliquity ......................................................................................... 

Comparison of FZ and revised (WA) formulations for Nonperforating Penetration 

vs. Obliquity ..................................................................................... 

15 

18 

23 

23 

24 

24 

25 

27 

27 

28 

V 
~NTIONALLY LEFT BLANK. 

vi 
1. Introduction 

Modifications have been made to the Frank-Zook (FZ) penetration algorithm, for inclusion into 

the modular Unix-based vulnerability estimation suite (MUVES) code [l] of the U. S. Army 

Research Laboratory’s (ARL) Survivability/Lethality Analysis Directorate (SLAD). The 

FZ algorithm is composed of extensions to the Tate-Alekseevskii (TA) [2,3] one-dimensional 

(I-D) model of penetration mechanics, is described in section 9.3 of BRL-MR-3960 [4], and is 

currently a part of SLAD’s terminal-ballistic, vulnerability/lethality (V/L) methodology. The 

current modifications to FZ methodology have been made under the auspices of the Target 

Interaction Lethality/Vulnerability (TILV) Workpackage, a joint program between ARL’s SLAD 

and Weapons and Materials Research Directorate (WMRD). 

2. Motivation 

One of the many enhancements possessed by FZ methodology over the TA equations is the 

, ability of the eroding penetrator to sense a change in target properties (i.e., a target/target interface) 

in advance of the penetrator actually reaching that property interface. This adaptation reflects the 

long-understood principle that a target’s resistance to penetration is composed of an integration of 

properties ‘in a finite volume of target material ahead of the actual penetrator/target interface (e.g., 

Wright and Frank [5]). When this finite volume of target material, contributing to the target 

resistance, crosses the boundary of a property interface, the target properties on both sides of the 

interface will contribute to the net resistance. 

The FZ penetration algorithm can, therefore, accurately sense and respond to the situation 

where, for example, the penetration channel proceeds from a weak target element into a strong 

target element. The effective resistance offered by the target would gradually and smoothly 

. 

transition from the weak value, just reaching the strong value of resistance as the penetrator/target 

interface reaches the strong target element. If a rod of diameter D is penetrating target element i, the 

influence of target element i+l upon the effective target resistanceK is computed by the FZ model 

as 

-2r 

K=Hi+(H,+,-H,)e reAD , (1) 

1 
where & , are the target-element component resistances and T,, is the residual, normal thickness of 

target element i yet to be penetrated. In the absence of this modeling enhancement, the transition in 

target resistance would be unrealistically abrupt. 

However, implementing this realistic enhancement within the FZ algorithm is accomplished 

only a single interface at a time (i.e., target elements i+Z, etc. do not affect the effective resistance). 

Thus, if the fmite target volume contributing to the target resistance were to realistically entrain 

several target/target interfaces, the FZ algorithm would account only for the one closest to the 

penetrator/target interface. Such a situation can realistically arise in several situations. One is 

where there exists a target composed of elements that are thin compared to the penetrator diameter, 

possibly as in the case of targets designed for small- and medium-caliber threats. Another situation 

in which several target/target interfaces could be entrained in the zone of target material 

contributing to target resistance would be the situation where a target element is barely clipped by 

the penetrator shotline. This latter situation can arise even if the individual target elements are 

otherwise thick. And since, from the point of view of MUVES calculation, the process of selecting 

and calculating penetrator shotline geometries is fully automated, the user of MUVES has little or 

no control in preventing very thin target elements from arising along a given shotline geometry, 

even for large-caliber targets. 

Two examples are provided here to demonstrate the nature by which thin target elements can 

inhibit the ability of the FZ algorithm to properly account for interface transition. In the first 

example, a length-to-diameter (L/D) 3 tungsten alloy penetrator of 48 mm length strikes a 50 mm 

plate (a super strong aluminum). In the baseline case, the 50mm target is composed of a single 

plate, whereas, in the test case, it is composed of a 48 mm plate followed by a 2mm plate of 

identical properties. Ballistically, these two cases should respond in a similar fashion. The 

properties and geometry of the simulation are given in Table 1. The results, however, depicted in 

Figures l-2, show otherwise. The normalized penetration vs. striking velocity plot (Figure 

shows discrepancies approaching 100% in penetration at velocities in the 900 m/s range. 

The reason for this discrepancy can be seen by examining the target resistance vs. position 

plot (Figure 2). In the more realistically portrayed 50 mm single-element target, the nominal 

2 
Table 1. Material and Geometry Parameters for Example 1 

PENETRATOR 

Lo D LID PP yo 

(mm) om Wm3) GPa) 

48 16 3 17.3 1.9 

TARGET No. 1: SINGLE PLATE, 50 mm (BASELINE) 

Plate No. Thickness m H HSPAU 

(mm) Wm3) @Pa) Wd 

1 50 2.7 5 1 

TARGET No. 2: 48 mm PLATE + 2 mm PLATE (TEST CASE) 

Plate No. Thickness m H &PALL 

(mm) Wcm3) KW (GW 

1 48 2.7 5 N/A 

2 2 2.7 5 1 

b 

0.5 

kr 

0.5 

kr 

0.5 

0.5 

1.2 

1.0 - H=5GPa 

0.8 - 

4 \ 0.6 - 

n, 

0.4 - 

0.2 - 

0.0 I I 

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1: 

V( m/s) 

Figure 1. Effect of thin backing plate on FZ model. 

, 

target resistance of 5 GPa is seen to immediately, but gradually, decay through the thickness of the 

target, eventually reaching the free-surface (spall) limit strength (for this example, designated as 

1 GPa). This monotonic decay in target resistance is brought on by the fact that the free surface at 

the rear of the target is increasingly unable to provide the support necessary to keep the target 

resistance at its nominal value. By contrast, the 48 + 2 mm target retains its nominal resistance out 

to 48mm of target depth before the FZ algorithm even becomes aware that a free surface exists, 

2 mm hence, to degrade target resistance. In essence, the 48 mm target element never detected the 

3 
1 

0 

4.8 cm + 0.2 cm 

,._..___.__...,._._...,,__..___.,.._______,...._.........,__........................, target 

! plies 

0 10 20 30 40 50 

dmm) 

Figure 2. Comparison of Resistance vs. Location for 50 mm target and 48 + 2 mm target. 

rear surface of the target, but only the subsequent 2 mm ply with identical properties to its own. It 

was up to the 2mm ply to detect the target rear surface. But by then, the error had already been 

introduced, resulting in a vastly overestimated penetration resistance of the target. The discrepancy 

in predicted target penetrations is thus explained. 

In the second example, whose description is provided in Table 2, the baseline target is 

nominally composed of two 25 mm layers, followed by an air gap and a residual rolled 

homogeneous armor @HA) target. While both 25 mm layers have the density of aluminum, the 

first has a 3 GPa target resistance, while the second has a much stronger 8 GPa resistance. In the test 

case of this problem, a weak 0.1 GPa film of 0.1 mm thickness is inserted between the two 25 mm 

plies. Clearly, such a minute modification to the target recipe should have no net effect. 

Yet, the normalized penetration vs. striking velocity plot (Figure 3) shows a vast discrepancy 

(often a factor of 2) between the baseline and test cases. As in the prior example, the source of the 

problem is identical. In the baseline case, without the interply film, penetration through the first ply 

becomes progressively more difficult as the 8 GPa strong target element is detected and approached. 

In the test case, by comparison, the 8 GPa ply is never detected while penetrating the first ply. 

Rather, only the weak 0.1 GPa ply is detected, and so penetration through the first ply becomes 

progressively easier as the rear of the first ply is approached. The target resistance vs. position plot 

4 
Table 2. Material and Geometry Parameters for Example 2 

PENETRATOR 

Lo D LID PP yo 

(mm) bw Wm3) @Pa) 

48 16 3 17.3 1.9 

TARGET No. 1: NO WEAK INTERPLY (BASELINE) 

Plate No. Thickness m H &PALL 

(mm) Wm3) GW (GW 

1 25 2.7 3 N/A 

2 25 2.7 8 2 

3 (air) 80 > -0 0 N/A 

b 

0.5 

kr 

0.5 

0.5 

0.5 

4 m 7.85 5 N/A 0.5 

TARGET No. 2: THIN (0.1 mm) WEAK (0.1 GPa) INTERPLY FILM LAYER (TEST CASE) 

Plate No. 

1 

Interply 

2 

3(air) 

4 

Thickness m H &PALL 

(nJd W-n31 GW @Pa) 

25 2.7 3 N/A 

0.1 1.0 0.1 N/A 

25 2.7 8 2 

80 -0 0 N/A 

m 7.85 5 N/A 

kr 

0.5 

0.5 

0.5 

0.5 

0.5 

1.75 

:. 

1.50 i 

.:. 

,:. 

: 

Residual (following air gap) : : : 

,.25 _ H=5GPa : 

Weak (0.1 GPa) : 

film between ’ 

o.50 ------ 

No film between 

V (m/s) 

Figure 3. Effect of thin interply layer on FZ model. 

5 
in Figure 4 demonstrates this divergence clearly. It also shows how, once the second 25 mm ply is 

entered, the 8 GPa target resistance immediately drops below 6 GPa because of the influence of the 

subsequent air gap. One might infer that even the baseline case, without the interply film, is not 

modeled accurately, while the air gap is not yet detected in the first ply. 

In fairness to the FZ model, it is true that these two examples have been chosen to accentuate 

the discrepancy between baseline and test cases. On the other hand, the intrinsic nature of the 

current FZ algorithm to handle only one interface at a time is a real and valid issue, regardless of the 

magnitude of the discrepancy for more “realistic” scenarios. And as cited previously, in the fully 

automated domain of ray tracing, thin target elements along shotlines will always remain a real 

possibility and should not become the cause for algorithm inaccuracy. 

3. Basis for Modifications 

The zone of influence in the target, ahead of the penetratontarget interface, which contributes to 

the effective target resistance, is a real phenomenon. Wright and Frank [5] examined the process by 

which the target response in this zone contributes to the target resistance. Others [6-81 have also 

studied this problem. This target zone of influence is generally recognized as the finite volume of 

target material undergoing deformation and is, in many models, approximated by the more readily 

Ply 2 Air Gap Backup 

40 60 80 100 120 140 

z(mm) 

Figure 4. Influence of thin interply layer on FZ target resistance estimation. 

6 
identifiable zone of plastic deformation. With this latter physical interpretation, the zone of target 

plasticity ahead of the penetrator/target interface completely governs the effective resistance of the 

target. If this zone of plasticity were to extend across, for example, three target elements, then the 

effective target resistance would be composed of an appropriate combination of properties from 

each of those three layers. But, what combination of properties constitutes an appropriate one? 

To address this question, the Walker-Anderson (WA) model [7] was explored for several 

reasons. First, it approximates the problem of integrating the target properties throughout the 

target’s plastic zone into one of integrating the target properties along the axis of penetration. 

Though such a simplification could lose information about the lateral influence of target properties, 

it simplifies the mathematics considerably, while remaining wholly compatible with the framework 

of a TA or FZ model. Secondly, by assuming a particular profile for the flow field in the target’s 

plastic zone, the WA model can analytically compute the exact stress and strain fields in the target’s 

plastic zone that result from the assumed flow field. It is this resulting stress field that directly 

contributes to the evaluation of the effective target resistance. And, while it may be asserted that the 

assumed flow field of the WA model was selected for homogeneous target penetration, the 

approximation introduced by applying that flow field across material interfaces may be argued as 

reasonable and certainly preferable to ignoring the interface effects altogether. And as shown later, 

for the situation of a single interface within the target’s plastic zone, the WA and FZ results are 

quite comparable. 

Additionally, the WA model includes certain nonsteady inertial terms in the formulation that are 

not part of traditional TA or FZ methodology. In theory, accounting for nonsteady effects offers the 

ability to more accurately model the transient phases of the penetration process. Transient phases of 

penetration always occur at initial impact and during the final moments when the motion of a 

residual penetrator is arrested. For the impact of small L/D penetrators, a significant percentage of 

the total penetration event will occur in these transient stages. Furthermore, large transient gradients 

(e.g., accelerations) can also arise when penetration transitions from one target element to the next. 

For targets comprising many elements, the net effect of these nonsteady terms could significantly 

affect the final result. And, for those cases where nearly steady-state penetration dominates the 

7 
penetration process (e.g., long L/D penetrators into monolithic targets), these nonsteady terms lose 

significance, leaving the original underlying TA or Ez relations to govern. 

4. Extent of Modifications 

Though the WA model is distinct from FZ or TA methodology, there are naturally many 

parallels. Furthermore, the existing FZ methodology is known to accurately model ballistic 

penetration events, when the issues involving multiple material interfaces in the target’s plastic zone 

are not germane. Whereas TA modeling is best suited to model the penetration of long rods into 

semi-infinite targets, the areas of FZ algorithm validity, based upon comparison to data [4], include 

long and short penetrators over a large impact-velocity regime against monolithic targets, as well as 

targets with “occasional” interfaces, accounting for the effects of target obliquity. Thus, given the 

generally exemplary performance of the existing FZ algorithm, there was little desire to “reinvent 

the wheel.” The approach therefore adopted was an adaptation of those elements of the WA model 

into the existing framework of the Fz methodology, so as to enhance the existing capabilities, while 

leaving the l!Z approach intact where possible. So, for example, those features of the WA model 

involving penetrator plasticity zones and wave speeds have not been utilized, since the existing FZ 

treatment of penetrators is excellent, albeit empirical [4]. And, though the FZ obliquity 

methodology could not be directly implemented in the WA framework, a very similar approach in 

the spirit of FZ was adapted. 

Other aspects of the WA methodology, when utilized, were simplified for the sake of efficiency. 

For example, rather than using the rather complicated WA formulation to ascertain the lateral extent 

of plastic zone (from the crater centerline in terms of crater diameters, as a function of impact 

velocity), a simpler formulation was adopted. In particular, a closer examination of the behavior of 

the WA model reveals that, for a given material, the physical thickness of the plastic zone (from the 

crater wall, not the centerline) can be adequately approximated by a constant number of penetrator 

(not crater) diameters independent of impact velocity. 

One aspect of the current modeling that is integral to neither the FZ nor WA models is the 

optional calculation of a time-dependent crater profile. In the case of FZ, crater-dimension 

computations are not part of the model. In the case of WA, because the formulation was developed 

8 
for homogeneous targets, the model formulation enforces a constant crater diameter throughout the 

penetration process for a given impact configuration. For the envisioned ARL application involving 

targets comprising multiple elements, such a limitation was not viable. Thus, two options are 

available in the proposed modeling: (1) crater radius is not explicitly calculated (but is implicitly 

treated as the rod diameter) and (2) a time-dependent crater radius is calculated, based 

approximately upon the notion that crater volume is proportional to the impacting kinetic energy. 

Option 1 provides the most compatibility with the existing FZ methodology since, in both cases, 

effects arising from lateral scale are keyed to the rod (and not the crater) diameter. Option 2, though 

requiring more calculation time, should (in some cases with multi-element targets) provide more 

accurate penetration modeling.* The proportionality between the crater volume and the kinetic 

energy is nominally the target resistance, and will be shown to match well to the data for tungsten 

and steel rods impacting RJM [7,9]. The availability of a crater profile prediction may also, in the 

future, assist SLAD in their V/L assessment of terminal-ballistic events. 

Because the WA model was originally presented for monolithic targets, a novel adaptation was 

required in order to apply it to laminated and/or finite targets. Because the WA model, as 

derived [7], pertains to only semi-infinite targets, the evaluated stress integrals that it presents 

cannot be directly employed in the proposed solution strategy, when the plastic zone ahead of the 

rod/target interface extends into more than one target element or to a free surface. To circumvent 

this limitation, the WA integrals are dynamically re-evaluated at each integration step, using the 

limits of spatial integration corresponding to each successive target element in the entrained plastic 

zone, and are solved for the normal-stress difference across each respective target element. By 

adding the integral contributions for normal-stress difference across all the target elements, the 

stagnation stress at the rod/target interface is reconstituted. The summation of terms constituting 

this stress signify contributions from each affected target element towards the aggregated inertial 

WV*), target resistance (Rr), and nonsteady (dldt) terms. 

*The (radial) size of the crater affects the rate and manner in which the flow field decays from the rod/target 

interface, thus affecting the target-resistance formulation and the (axial) penetration rate. Also, the axial 

penetration equations include dRldt terms not part of the original WA formulation. 

9 
5. Algorithmic Details 

5.1 Extent of Plastic Zone. In WA modeling, target properties are formulated by way of an 

integration process through the plastic zone in the target, located ahead of the rod/target interface. 

The extent of this zone is characterized by a nondimensional, velocity-dependent model parameter, 

CL. Though the crater formed via the WA model is roughly cylindrical, the leading edge of the crater 

(at the rod/target interface) has a hemispherical cap of crater radius R. The length a?? defines the 

distance from the radial center of the hemispherical crater cap to the outer fringes of the target’s 

plastic zone. The actual thickness of the plastic zone, therefore, as measured from the edge of the 

hemispherical crater, is (a-l)R. Figure 5 depicts some of the relevant geometrical parameters. 

In the WA model, a complicated formulation was developed to relate the value of this (r 

parameter to the penetration rate through the target. A simpler alternative was sought and found, 

applicable over a wide range of conditions. Simply put, the thickness of the plastic zone may be 

approximated as a constant for a given target element; namely, R(a - 1) = constant. The constant 

may be a priori specified as a material parameter, “plastic-zone thickness,” for each target element. 

By specifying this plastic-zone thickness as a multiple of rod diameters, as in 50, for example, not 

only is g made nondimensional, but it may be readily related to the low-velocity limiting value for 

a, namely a~,. In this limiting case, the crater radius R approaches the rod radius D /2, and the value 

for C& may thus be related to g, as E= (Q - 1)/2. 

Figure 5. Description of the target’s plastic zone in the WA model [S]. 

10 
To test the reasonableness of the hypothesis that the plastic-zone thickness is independent of 

penetration velocity, examine the case of a tungsten long rod penetrating an RHA target and 

compare it against the a vs. U curve presented by Walker and Anderson [7]. To obtain quantitative 

values for a using this hypothesis, values for R must be known. The empirical fit of Bjerke et 

al. [9] presents the crater to rod diameter ratio as a function of striking velocity, VO (km/s), as 

ZRID =1.1524+0.3388V,, +0.12864,2 . (2) 

Bjerke’s curve had been fit to data over the range of VO from approximately 1 to 4.5 km/s. In order 

to express the result in terms of a vs. U, as did Walker and Anderson [7], the striking velocity VCJ 

must be related to the penetration velocity, U. For purposes of this comparison and for the sake of 

simplicity, relate these quantities through the hydrodynamic relation, VO = (1 + yl’* )U, where y is the 

target to rod density ratio pr/p~ equal to 0.4508 in the case of tungsten on steel. 

The comparison may now be presented in Figure 6 and is very good, except at the lowest of 

velocities, where it is nonetheless adequate. The simplicity of the assumption R(a - 1) = 50 makes 

for a greatly streamlined implementation, with many fewer material parameters and computational 

steps than the original WA formulation. In Figure 6, the case of a~ = 8.9, for example, corresponds 

to a plastic-zone thickness of 3.95 rod diameters. 

8 

7 ‘..,Walker and Anderson (q,=9.4) 

6 

5 

6 4 

3 

2 

1 

1 

0 

R&l) = constant 

(assuming Bjerke fit for R 

hydrodynamic penetration, 

VO-1 .O km/s 

of Validity VO-4.5 km/s 

I 1 

0 1 2 3 4 5 

U (km/s) 

Figure 6. Comparison of Walker and Anderson fit to the constant R(a-1) assumption [7]. 

11 
A rational explanation for the reasonableness of the constant plastic-zone thickness assumption 

is surmised. Namely, for a given target strength, the work required to infinitesimally expand a 

crater from radius R to R + dR should be approximately proportional to the crater-wall area, over 

which the crater-growth pressure is applied, times the applied pressure. At the hemispherical tip of 

the newly forming crater, this area grows with crater size as R*. The required pressure, according to 

WA theory, grows roughly with ln(R/D), so that work required goes as R* ln(R/D). 

Correspondingly, given the assumed constant-thickness plastic zone, the volume of entrained plastic 

material in the hemispherical cap grows roughly as R3 for R-D, while approaching R2 for R>>D. 

Thus, the increment of plastic-strain energy to grow the crater size, being roughly proportional to 

the plastic-zone volume, also grows approximately with R3 to R* functional behavior. Because the 

R* ln(R/D) function also grows as R3 for R-D, while approaching R* for R>>D, it follows that both 

the work required for and the plastic strain energy that results from an infinitesimal expansion of the 

crater are roughly of the same functional form and thus approximately proportional to each other. 

This conclusion follows merely by assuming a plastic zone of constant thickness. 

As a result of adopting the R(a - 1) = constant assumption, not only is the dynamic value of a 

obtainable when the time-dependent value of R is known, but the rates at which R and a vary are 

coupled as well. A simple differentiation of the assumption gives the relation by which the 

derivatives may be interrelated: 

2?(a-I)+drR=O . (3) 

5.2 Numerical Integration Approach. In the FZ methodology, penetration velocity U and rod 

velocity V are algebraically tied through the modified Bernoulli equation, 

k,p,(v-U)2+Y=kTp*U2+H , (4) 

where kp and kr are the penetrator and target “shape factors” of Alekseevskii [3]; pp and pr are the 

rod and target densities; and Y and H are the rod strength and target resistance, respectively. Thus, 

under FZ, the separate integration of one of them (typically V via the rod deceleration equation) was 

sufficient to algebraically determine the other (U). In the current modeling, however, the nonsteady 

Bernoulli equation (representing the force-momentum balance along the centerline) is a function of 

12 
not only U and V, but also their time derivatives dU/dt and dV/dt. Thus, while V is integrated by 

way of the rod deceleration equation 

v=- Y 

PpW-4 

(5) 

(L being the rod length, and s being the rod’s plastic-zone thickness), U must be independently 

obtained from the nonsteady Bernoulli equation [8] by isolating dU/dt and integrating. It is shown 

here, applied to the WA flow field for long-rod penetration: 

p,(V-u)2 +Y - ~(ii+d)=k,PLIZ+H+X,~+X,q+X,~ . (6) 

a 

On the target side of the equation, F, p, Xu, Xor, and XR are time-dependent quantities integrated 

through the plastic zone of the target. If the plastic zone in the target spans several target elements, 

the values taken on by these parameters are summations of properties from all constituent target 

elements, as described in section 5.5. However, if the target’s plastic zone comprises a single target 

element only, then the values taken on by these terms are simplified greatly, expressible as 

P=P?- 7 (7) 

K=H , (8) 

X, = p,UR(a-l)/(a+l) , (9) 

X, = 2p,URal(a + l>’ , and (10) 

X, = p,UR(a -l)/(a + 1) . (11) 

. 

Regardless of the actual evaluation of these parameter values, the nonsteady Bernoulli 

equation (6) shows their generic interrelation. In the actual implementation currently suggested, 

several simplifications have been made to this equation. First, because of the excellent functionality 

provided by the existing FZ treatment of the penetrator [by empirically treating Y = Y(L)], the 

plasticity associated with the rod is currently neglected, by treating both s and dsldt as negligible 

quantities. Future treatments may revisit this simplification. Furthermore, the quantities dcrldt and 

dRldt, independent in the nonsteady Bernoulli equation, are herein treated in a coupled manner [via 

1 equation (3)], an assumption detailed in section 5.1. 

13 
Though an optional capability to model the crater radius is described in section 5.3, the 

equations derived to this point may nonetheless be solved by assuming a constant crater diameter 

( i.e., R =0/2). To retain maximum compatibility with the FZ algorithm, which bases its 

interelement transition dimension upon the rod diameter, the constant-crater-diameter mode of the 

revised algorithm will employ the rod diameter itself as the (constant) crater diameter. When this is 

done, dRldt and thus dcrldt terms [via equation (3)] are identically zero. When employing this mode 

for the case of monolithic penetration, and in conjunction with neglecting s and dsldt, equation (6) 

differs from equation (4) by only the XII IU - dU/dt term. 

Whereas equation (4) is algebraic, equation (6) remains differential, despite the aforementioned 

simplifications, and therefore continues to require numerical integration. However, for monolithic 

penetration, where the penetration velocity U changes slowly except for the onset and final 

moments of penetration, one would expect the dU/dt term to carry only small significance. As such, 

the revised (WA-based) algorithm [employing equation (6)] should, for this impact condition, 

closely match the results of the FZ algorithm [employing equation (4)]. Figure 7 shows this 

comparison for the cases of LID = 3 and 30 tungsten rods penetrating RHA. In both models, a 

5 GPa target resistance was employed. The revised algorithm requires a plastic-zone-thickness 

parameter specification (section 5.1), taken in the present case as 3.75 rod diameters. The only 

input difference between the corresponding FZ- and WA-based simulations was the rod strength, 

which was taken as 1.9 GPa in the FZ formulation and 1.8 GPa in the WA-based formulation. 

5.3 Crater-Radius Calculation. The calculation of a time-dependent crater radius has been 

implemented as an option into the proposed methodology, though it is not part of either the FZ or 

WA models. Crater radius does have a secondary effect on the axial penetration equations. This 

effect is more pronounced at obliquity, where the expanding crater may reach a target/target 

interface prior to the arrival of the rod itself. Additionally, the accurate knowledge of the crater 

dimensions may provide valuable data for subsequent V/L modeling. 

The basic premise of the crater-modeling algorithm is the assumption that the crater volume 

displaced is proportional to the impacting kinetic energy. While this assumption has frequently 

14 
2.0 

Tungsten Rods vs. RHA . ..G 

1.8 -- /+.- - 

1.6 -- 

//’ 

A’ 

FZ/.~ 1.4 -- 

/((. WA 

1.2 -- 

2 1.0 -- 

4 

L/&3 /(’ 

//’ E&./Y 

0.8 -- 

//. 

0.6 -- 

/’ 

/ 

L/D-30 

0.4 -- 

0.2 -- 

0.0 I I I I I I 

0 500 1000 1500 2000 2500 3000 

V(m/.s) 

Figure 7. Nondimensional Penetration vs. Velocity for original (FZ) and revised (WA) 

models for semi-infinite penetration. 

been employed in the crater modeling of small L/D impacts, or even the case of particulated shapedcharge jet penetration, an adaptation was required for the case of continuous long-rod penetration. 

In this case, the rate of energy deposition (per unit depth penetration) by the rod is made 

proportional to the rate of crater-volume creation, the proportionality being nominally proportional 

to the effective radial target resistance HR. Implicit in this analysis is the assumption that both rod 

and target are eroding. The special cases of rigid rod and target are addressed at the end of this 

section. 

Energy is delivered to the target by the rod in the form of both force over distance (work) as 

well as specific KE of eroding rod delivered to the rod/target interface: 

H 4w _ 6 I dw 

R dP dP dP ’ (12) 

By taking the control-volume boundary for energy delivered by the rod to be the onset of the rod’s 

plastic zone, the undeformed dimensions, velocities, and properties of the rod may be used in the 

energy balance. The crater volume per depth of penetration d(VoZ)/dP is the crater’s area n R 2, with 

R being the sought-after variable of this exercise. The rate at which usable kinetic energy is 

delivered to the target per unit depth of penetration is better understood as 

dEK/dP = dEK/dL - (dLldP),f, the kinetic energy of the rod per unit length times the (effective) rate 

15 
at which rod length is consumed in the form of penetration. The specific kinetic energy per unit 

length of rod dEK/dL is simply l/2-ppV 2 (nD 2 /4). Finally, the rate of external work done by the 

rod dW/dP is just the force applied by the rod Yo (nD2/4). Combining these expressions provides 

the following result: 

R=D/2a{[~ppV2(dL/dP),,+Yo]/H,)1’2 ,whenO<UcV. 

In no event is R permitted to be’ less than D /2. 

(13) 

The target resistance offered for radial crater formation HR will generally be identical to that 

offered against axial penetration p. However, near the rear of a target element, the axial resistance 

can and does fall off drastically if that rear surface is a free surface. In contrast, however, the ability 

to resist radial crater growth will, at most, fall off to the level of the material’s yield strength. Thus, 

HR is taken as max(Yr, g ). Based on elements of WA theory, the target element’s yield strength Yr 

may be approximated, in terms of target resistance H and other known parameters, as 

Yr = 3/7-H / In(@), where a is related to the target element’s plastic-zone thickness, as described in 

section 5.1. 

The instantaneous rate at which rod length is consumed per unit of penetration dL/dP is 

kinematically related to velocities V and U as dL/dP = (V - U)IU. For two reasons, however, this 

precise value of dL/dP may not be that taken as the effective value (dL/dP),f used in the crater 

radius calculation. First, there does not in reality exist a one-to-one relation between an 

infinitesimal increment of penetration and the resulting value of crater diameter. Allowances need 

to be made during moments of penetration transient (especially at penetration onset), when the 

instantaneous dL/dP can far exceed its steady-state value. At these moments, it is not reasonable to 

expect a small transient section of the crater to have an excessively large diameter compared to the 

adjacent crater profile. The expedient remedy adopted has been to cap the maximum allowable 

value of (dL/dP)ef (in this crater-radius calculation only) to a value related to the hypothetical 

steady-state (not instantaneous) erosion rate. 

The theoretical value for steady-state dLldP is obtained by solving, in terms of V, for the 

quantity (V - U)IU in the steady-state, modified Bernoulli equation: 

16 
1/2p,(v-u)2+Y0 =1/2pu2 +K . (14) 

However, the allowable cap on (dLldP),f is modified ad hoc, from this steady-state value of dLldP, 

consistent with the experimental record on crater size, and is denoted by max(dLldP),fl The result, 

in terms of V, depends on whether the rod and averaged target densities are equal. In the case that 

theyareequaland r=BIP, =l, 

max(dL/ dP),# = (1 + x)/(1 - x) , (y = l), (15) 

where 

x = 0.5(K - Y,)/(S p,v”> . (16) 

Note that, except for the leading 0.5 multiplier in the definition of X, this result represents the exact 

solution to the modified Bernoulli equation for (V - U)IU. The 0.5 multiplier, however, is an ad 

hoc modification to the theoretical steady-state result, chosen to facilitate a fit to steel-on-steel 

crater-profile data [7]. For the case of unlike densities, where y f 1, the result is 

max (dLl dP>ef = (-y + &)/Cl - &) , (y f l), (17) 

where x is now defined as 

x=~+0.4(1-y)(H-Y,)/(~p,V2) . W3) 

Like the y = 1 case, the only ad hoc modification to the theoretical result is the introduction of a 

multiplier (in this case, 0.4) in the definition of x. The selection of the 0.4 multiplier results in good 

correlation to tungsten-rod on RHA-target crater-profile data fit by Bjerke et al [9]. Figure 8 shows 

these correlations to other fits [7,9] for both steel on steel (y = 1) and tungsten on steel (r f: 1). 

Though there appears to be ample opportunity for max(dL/dP),a to be negative or undefined, the 

mathematical nature of the steady-state penetration equations actually precludes it, when both the 

target and rod are eroding. In those limiting end states of rigid-target and rigid-rod penetration, 

special provisions are taken. In the case of rigid-target rod erosion, the crater size is 

inconsequential, ‘since there is no incremental growth of the crater. In the case of rigid-body 

penetration, there is no kinetic energy flux that would result from a feeding of rod into the 

stagnation zone, since rod material does not feed into the stagnation zone. Consequently, look 

merely to the plastic work done on the target by the actual rod/target interface stress. Since the rod 

is rigidly penetrating, this interface stress is less than the yield strength Y of the rod. However, the 

17 
1 

0 

_ RHA targets 

dE/dVol=H, 

Tungsten: (proposed model) 

I 

0 1000 2000 3000 4000 5000 

V(m/s) 

Figure 8. Comparison of proposed crater-diameter modeling with fits from other 

models [7,9]. 

(steady-state) magnitude of this interface stress may be obtained from the flow kinematics of the 

target. Balancing the work of this applied stress with the implied crater volume (as per the original 

energy/crater-volume proportionality assumption) gives 

R=D/2. [1/2p,U*+K]/H, ,(whenU=V). (19) 

A final aspect of the proposed cratering methodology at variance with that already described is 

the treatment the crater-profile development in the final transient stage, as the ballistic event 

concludes. Empirical experience indicates that an established flow pattern does not cease 

instantaneously and that the bottom of a penetration channel will tend to retain a 

pseudohemispherical shape because of flow-field inertia. To this end, at the moment when the 

crater radius begins decreasing within a target element, current values of R and U are noted as 

“critical” values. Subsequent crater radius is developed according to the following ad hoc 

relationship: 

R = D12 + (R,,i, -D/2) dm , (when dRldt < 0). (20) . 

This form is obeyed for the decay of crater radius until such time that either R as described by the 

In either of these 1 

standard method would indicate that dRldt 2 0 or a new target element is entered. 

cases, the “critical” values of R and U are reset and the standard calculation method prevails. This 

18 
form has the crater radius diminishing from Rcrir to D /2 as the penetration velocity falls from Ucrit to 

zero. 

. 

5.4 Nonsteady Terms. The original WA model does not provide for a varying crater radius 

and, as such, does not incorporate radial variation (dRldt) terms into its formulation. Rather, 

nonsteady terms in WA methodology are limited solely to unsteady penetration rate (dU/dt) and 

extent of plasticity (daldt) terms. Thus, the introduction of dRldt quantities into the governing 

force-momentum balance equation (section 5.2) is an extension beyond the scope of the original 

WA model. This extension will only play a role if the variation of crater radius over time is 

modeled. In this proposed implementation, the choice to dynamically model crater radius is an 

option left to the discretion of the model user. 

5.5 Piecewise Integration of Momentum. Discussion of the momentum equation to this point 

has largely centered upon the effect of including nonsteady penetration terms as an extension to the 

FZ methodology. These nonsteady effects, though brought about naturally to small degree in the 

course of penetrator deceleration, may play a larger role when the unsteadiness of the penetration 

event is purposefully induced via time fluctuations in the penetration (i.e., rod/target interface) 

velocity. In addition, however, to unsteady terms that are introduced as a direct result of interface 

fluctuations (i.e., terms proportional to time rates of change of the kinematic variables U, R, and a), 

the “effective” target material properties themselves (i.e., H and p) change as target/target interfaces 

become entrained within the plastic zone that leads the rod/target interface. 

Though theories like Wright-Frank [5] and WA [7] assert that the plastic zone ahead of the 

rod/target interface is responsible for the makeup of the target-resistance terms, neither analysis 

addresses the situation of a multi-element target. The existing FZ methodology, on the other hand, 

does permit this feature to be modeled in a limited way. While no provisions are made by FZ to 

aggregate density into an “effective” value, the transition in effective target resistance, in advance of 

an interface, is modeled to change exponentially from the original value to the value of the 

succeeding element, reaching the new element’s resistance just as the penetration interface reaches 

the new target element. The growth (or decay) exponent is proportional to the remaining distance 

between the rod/target and target/target interface, normalized by the rod diameter. However, this 

19 
modeling approach, as implemented, accounts for the resistance of the next target element only, 

regardless of whether subsequent elements may also be entrained in the target’s plastic zone. 

While the WA methodology [7] in no way addresses the issue of a plastic zone that extends 

across a target/target interface, a methodology may be inferred based on the WA framework with 

the use of a single additional assumption; namely, that the hemispherical displacement field posited 

by WA methodology is assumed to remain valid, even as that displacement field spans across 

multiple target elements. While such an assumption can never be strictly accurate, it is believed that 

the errors introduced by it are of secondary importance. 

When the displacement field governing the target’s plastic zone is known, the axial centerline 

momentum integral [extended Bernoulli equation (9)] may be evaluated. However, because the 

extended Bernoulli equation has material density within the integrand of the stress integrals [8], the 

evaluation is straightforward only when the limits of integration span a region of constant density. 

A full integration across a plastic zone comprising multiple target elements implies a series of 

piecewise integrals over individual target elements. The piecewise integral (i.e., across a single 

target element of constant density) may be evaluated for the normal-stress difference across the 

element. For the problem of long-rod penetration, this integral across target element i (ranging from 

z = ZI to 22) becomes 

(21) 

where the z coordinate of integration coincides with the position of the target element along the axis 

of symmetry (centerline) of the penetration and V, is the axial component of the material velocity 

field. Both z and V, are measured relative to the moving rod/target interface. When the specific WA 

flow field is assumed, the velocity field is known and the associated stress field may be inferred, 

thus allowing the required integrals to be evaluated.’ For a target element that falls within the 

target’s plastic zone, and whose location along the shotline spans from z1 = (PI - 1)R to 

z2 = (/!& - l)R, the extended Bernoulli integral over that target element gives the following: 

20 
(22) 

pTp2 (28, -1) _ W2 -1) mi 

+ (a2 -1) j3: 822 [ 1 ’ 

If this target element i were to completely span the plastic zone of the target, corresponding to 

pr = 1 and /$ = a, this normal-stress difference would correspond to the stagnation stress and would 

exactly reduce to the target side of equation (6) as presented in section 5.2; namely, yielding 

equations (7)-( 11). 

In the general case, however, where the target’s plastic zone comprises several target elements, 

normal-stress continuity at target-element interfaces dictates that the total normal-stress difference 

across the target may be ascertained by summing the stress difference across each successive target 

element, until the far fringes of the target’s plastic zone have been reached. Because the stress at 

any point beyond the target’s plastic zone is zero in WA methodology, the total stress difference 

thus evaluated represents the magnitude of the stagnation stress at the rod/target interface. 

One may therefore generalize the specification of the target terms, when referring back to 

equation (6). In particular, for a plastic zone that spans across target elements ranging from element 

m to IZ, one obtains the following generalized expressions for the target parameters: 

H=C n WXi+l/ Pi lH. 

i=m In(a) ’ ’ 

@f+a’) <Bi:, +a2) 

Pi - Pi+, ’ I 

(Pi, +l) t/T +l) 

Bi+l - Pi I 

,ad 

(23) 

(24) 

(25) 

(26) 

21 
*, =u~f: PTia2 . 

km [ 

C2Bi -l) _ C2&+ -l) 

(a* -1) 6; r+l I 

(27) 

The limits b = 1 and b+l= a apply in all cases [recall that pi = 1 + (zi/R)], corresponding to the 

geometric limits of the plastic zone [i.e., 0 s z I (a - l)R], with Bi values for target elements 

between m and y1+ 1 falling numerically between values of 1 and a, depending upon their 

relative location within the plastic zone. 

With equations (23)-(27), the extended Bernoulli equation (6), now has the means to more 

accurately characterize the influence of thin target elements upon the effective target resistance 

and density, as well as the nonsteady terms of the momentum balance. As such, the two example 

problems cited as the motivation for pursuing these revisions to the existing FZ methodology 

may be revisited. It was first confirmed that the revised algorithm provided nominally identical 

results when comparing the test case to the corresponding baseline. This confirmation indicates 

that the revised algorithm is insensitive to the mere presence of thin-layer interfaces. 

The next comparison is to the results of the original FZ methodology (in particular to the 

baseline-case FZ results, which do not suffer from the thin-ply deficiencies). For the problem 

earlier cited as Example 1, Figures 9 and 10 provide the comparison. Not only can the revised 

algorithm (WA) match the original FZ result for penetration vs9. velocity (Figure 9) but the 

functional decay of target resistance through the thickness of the target, denoted “WA (Plastic 

Zone = 4.30)” and corresponding to equation (24), correlates excellently with the exponential 

decay form of the FZ model. However, whereas the FZ form is only properly computed if the 

target comprises a single 50 mm plate, the WA result displayed is that obtained for both a 50 mm 

plate, as well as a 48 + 2 mm multiplate configuration. For the revised (WA) simulations, all 

simulation parameters were identical to those employed in the corresponding FZ simulations (see 

Table 1). Additionally, the WA simulations were run in constant-crater-diameter mode, with a 

plastic-zone thickness of 4.3 rod diameters. Figure 10 also includes, for purpose of illustration, 

the revised model’s target resistance when the plastic-zone extent is taken as 3.5, rather than 4.3, 

rod diameters. The difference from the 4.3 curve shows the extent to which the effective target 

resistance is affected by designated changes in the size of the plastic zone. 

22 
1.2 

1.0 - H,,=5 GPa 

(Plastic Zone = 4.30) 

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 

V(mls) 

Figure 9. Example 1: Normalized Penetration vs. Velocity for FZ and revised (WA) models. 

4 

1 

0 

0 10 20 30 40 50 

z(mm) 

Figure 10. Example 1: Target Resistance vs. Location for FZ and revised (WA) 

formulations. 

To compare now the revised model to the problem cited in section 2 as example 2, results are 

presented in Figures 11-12. The FZ results presented correspond to the baseline case, without 

the thin-interply film between the soft and hard plies of the target, whereas the revised (WA) 

results are independent of the presence of the inter-ply film. Though the revised (WA) 

formulation compares more closely to the FZ baseline (in Figure 11) than did the FZ test case 

23 
1.75 i 

1.50 - 

Residual (following air gap) 

,.25- H=5GPa 

z 35 Ply 2 

4 0.75 - H =8GPa 

0.25 - p'y1 H = 3GPa 

no film between 

0 200 400 600 800 1000 1200 1400 

V(m/s) 

Figure 11. Example 2: Normalized Penetration vs. Impact Velocity for FZ and revised 

(WA) formulations. 

r Ply 1 Ply 2 Air Gap Backup 

Figure 12. 

0 20 40 60 80 100 120 140 

2 (mm) 

Example 2: Target Resistance vs. Location for FZ and revised 

formulations. 

WA) 

with thin inter-ply film (in Figure 3), there is still some discrepancy evident in Figure 11. The 

source of this discrepancy may be gleaned from the target-resistance plot in Figure 12. Even 

without the thin-inter-ply film that demonstrated (via the example 2 test case) the inability of FZ 

methodology to perceive more than one target interface in advance, the baseline case itself, to a 

lesser extent, is subject to the same deficiency. In particular, the FZ baseline fails to detect the 

24 
. 

rear surface of the second target ply, until actually in that ply. As such, the target resistance in 

the first ply rises fully to 8 GPa at the ply-l/ply-2 interface before suddenly detecting a loss of 

strength originating from the free surface of ply 2. By contrast, the revised (WA) formulation 

avoids sudden internal changes in target resistance (and other properties) by employing the 

plastic-zone integration process embodied in equations (23)-(27). For a sense of scale, note 

from Table 2 that the thickness of ply 1 and ply 2 are each about 1.5 D. A final note on these WA 

simulations is that they were performed in constant-crater-diameter mode, with a plastic-zone extent 

of 4.0 D. 

55.1 Obliquity Considerations. The presence of target obliquity complicates the problem of 

integrating piecewise across multiple target elements. The reason for this complication arises 

from the fact that the integration technique employed in the current algorithm is that of a line 

integral along the shotline. A consequence of reducing the kinematics of a three-dimensional 

(3-D) penetration event to a line integration is that the presence of obliquity guarantees that 

subsequent target elements will become laterally entrained within the target’s 3-D plastic zone 

prior to becoming entrained along the shotline itself (see Figure 13). If no special provisions are 

made, this lateral entrainment will not be detected by the line-integration algorithm and the effect 

of target obliquity will reduce merely to increased line-of-sight thickness. 

I 2R 

Figure 13. Geometrical considerations associated with oblique penetration. 

25 
The technique adopted, though not identical to, is in the flavor of the FZ approach to the 

obliquity problem. In the FZ algorithm, which allows for a smooth transition in target resistance 

properties over a finite distance, the exponent that governs that transition is proportional to the 

normal distance between the rod/target and target/target interfaces. Also, because of the 

hemispherical nature of the plastic-zone cap, the WA model would likewise indicate that the 

ability of a remote target element to influence the aggregated target properties should be 

determined by the normal distance from the wall of the crater to the target/target interface in 

question. 

Therefore, in order to retain the basic numerical approach of employing a 1-D integration 

along the shotline dimension, the axial location of interfaces and extent of plastic zone are 

transformed as a function of obliquity, by projecting the extent of the plastic zone perpendicular 

to the target normal onto the axial line-of-sight. Using the transformation variable 

M= a-l 

(a/co&)-l ’ (28) 

where 8 is the target obliquity measured from surface normal, the extent of plastic zone (as 

projected onto the shotline) may be seen to span from 0 5 z I (a - l)R/M, with the intermediate 

target/target interfaces being defined by pi = 1 + (ZiMIR). With this transformation variable M 

(which equals unity at zero obliquity), a target element will begin to influence the aggregated 

target properties as soon as any part of the plastic zone overlaps the affected target element. 

For the case of a simple target plate at obliquity, the results of this methodology may be 

compared with the FZ model, which is wholly adequate for this class of problem. Figures 14-16 

depict residual length, residual velocity, and nonperforating penetration, as a function of 

obliquity, for a 1250 m/s impact of an L/D = 3 tungsten penetrator (same as in examples 1 and 2) 

into a hard (H = 5 GPa), 50 mm (t/D = 1.04) aluminum plate. The WA simulations were 

performed in variable crater-diameter mode, with a plastic-zone-extent of 3.5 D. The similarity 

of results indicates that the current methodology captures the essence of the FZ approach, while 

retaining the ability to treat multiple interfaces entrained within the plastic zone. 

26 
0.5 - FZ 

. . . . . . . . . . . . . ..____....._... 

0.4 - 

0.0 I I I 

0 10 20 30 40 50 60 70 

0 (de!8 

Figure 14. Comparison for FZ and revised (WA) formulations for Residual Length vs. 

Obliquity. 

0.0s 

0 10 20 30 40 50 60 70 

8 Wg) 

Figure 15. Comparison for FZ and revised (WA) formulations for Residual Velocity vs. 

Obliquity. 

5.5.2 Surface Resistance. The FZ methodology contains a provision for what is called 

surface resistance. Though this parameter could, in theory, be used to model a target element 

whose strength changes over the distance from the element surface to its core, it appears that the 

more conventional utilization of the parameter is to characterize the phenomenon known as 

ceramic interface-defeat in a simple, ad hoc manner. The parameter, with units of resistance, 

27 
2.00 

1.75 

1.50 

1.25 

s 1 .oo 

a 

0.75 

0.50 

0.25 

40 50 60 70 

0 Wg) 

Figure 16. Comparison for FZ and revised (WA) formulations for Nonperforating 

Penetration vs. Obliquity. 

does not quantify a material characteristic of the ceramic, but is determined so as to best fit a 

known ballistic data set of interest. 

Regardless of the motivation for the utilization of surface resistance in FZ methodology, the 

parameter is designed to behave so that the target resistance of a thick target element varies 

smoothly from the surface resistance to the core resistance. (In the case of partial interfacedefeat, the ceramic surface resistance degrades from a high surface resistance to a lower core 

resistance.) The user of FZ methodology lacks control over the manner of resistance variation, 

only the endpoints (surface and core resistances). The FZ algorithm affects this resistance 

variation using the same exponential decay (or growth) form that is employed when transitioning 

from one target element into the next. When a thin target element possesses a surface resistance, 

there are two competing exponential functions determining the local effective material 

resistance, according to FZ methodology: one alters the target resistance from the surface 

resistance (forward) to the core, while, superimposed, the subsequent target element influences 

the target resistance (backward) prior to actually reaching that target/target interface. 

The approach adopted for the revised methodology is to consider a certain thickness at the 

front of the target element to possess the so-called surface resistance, followed by the remainder 

28 
of the element (if any) at the core-resistance value. The thickness of this surface region is taken 

exactly as that of the constant plastic-zone thickness, 50, associated with that target element.* By 

proceeding in this manner, the effective target resistance, as computed via equation (24), begins at 

the surface resistance and immediately begins growing (or decaying), finally reaching the core 

resistance at a distance equal to the plastic-zone thickness beneath the target surface. While this 

technique has the valid criticism that a change in the plastic-zone thickness has the net effect of 

inadvertently changing the effective surface resistance, it must be recalled that the surface resistance 

was not intended to characterize a material parameter in the first place, as the FZ implementation 

provides no latitude for specifying the manner in which the surface blends into the core. If one 

accepts this parameter only for its ad hoc utility to help characterize target elements that ballistically 

exhibit unique surface phenomena, then the revised implementation provides the same utility, in that 

regard, as the original FZ methodology. 

6. Summary 

With new classes of ballistic targets emerging, anticipated to comprise thin-element layers (or 

even continuously varying property layers), the Frank-Zook (FZ) methodology, by which terminalballistic penetration is currently modeled, was revisited. Though exemplary at modeling penetration 

through thick layered targets, with occasional material interfaces, a problem was identified with the 

I3 methodology when layers of target material were small compared to the penetrator diameter. 

Not only will this situation arise when the target-element layers are intrinsically thin or continuously 

varying in property, but it can also arise for the case of thicker target elements when the terminalballistic shotline, by design or chance, barely clips the target element. In the context of V/L codes 

employed in the ARL, a random shotline selection process is often employed to statistically probe 

the vulnerability of a target vehicle. For such situations, the V/L code user has little control over 

specific shotline selection, therefore making the problem of target-element clipping manifest. The 

penetration algorithms of these V/L codes should be made insensitive to this possibility. 

*If the target-element thickness exceeds 2-cD, the surface resistance will fully blend into the core resistance before the 

next target element exerts any influence. If the target-element thickness is between one and two times ED, there will 

exist an overlap region where surface resistance and the next target element both influence the effective resistance away 

from the value in the core. Finally, if the target thickness is less than ED, the target-element’s core resistance will be 

totally subsumed by the influence of the surface resistance and the next-element influence and thereby plays no role. 

29 
With the problem thus identified, an effort was undertaken to remedy the situation, with a 

minimal of modification to the existing FZ methodology. To this end, elements of the WalkerAnderson (WA) model were extracted and incorporated into the FZ methodology. Key in this 

respect were two notions: (1) that nonsteady terms of the governing equations (whose influence 

increases during fluctuations associated with target-element transition) be retained and (2) that 

the instantaneous effective target properties be made to depend upon the flow-field and material 

properties ahead of the rod/target interface. Both of these revisions in solution strategy are 

believed to be important in achieving an accurate characterization of terminal-ballistic target 

classes that employ thin-element or continuously varying layers. 

The implementation satisfies a primary requirement in that the modifications implemented to 

improve the ability of FZ methodology to address thin-element laminates has not hindered its 

ability to address projectile/target classes for which it had already been well suited. Those 

classes for which FZ was previously well suited include penetration of high- and low-L/D 

penetrators into either monolithic targets or targets with occasional material-interface transitions, 

at normal or oblique angles of incidence over a wide range of velocities of ballistic interest. 

The current revisions to the FZ algorithm specifically strive to prevent an inaccurate 

terminal-ballistic characterization from arising for those targets possessing one or more thin 

target-element layers. To do this, the revised algorithm dynamically formulates the target’s 

kinematic and material properties by way of an integration process through the plastic zone in 

the target, ahead of the rod/target interface. As various target layers become entrained in this 

plastic zone over time, their material properties begin influencing the effective (aggregate) 

properties of the target. Furtherrnore, the target forces that kinematically arise because of 

nonsteady penetration (Le., accelerations of the rod/target interface) are retained in this revised 

formulation. Such terms, previously neglected, are expected to retain a significant magnitude in 

those penetration events that are dominated by transient phenomena. A final extension to the FZ 

algorithm that was not part of WA theory is a model to compute the profile of the penetration 

crater. This optional algorithm is based on the notion that the rate of crater volume growth in a 

target may be related to both the rate at which energy is deposited by the penetrator and the 

30 
penetration resistance of the target. Though such data are not currently employed in V/L code 

methodology, such data might prove useful in future revisions to their methodology. 

31 
INTENTIONALLYLEFTBLANK. 

32 
7. References 

1. Dibelka, R. Personal communication. Aberdeen Proving Ground, Maryland, February 2000. 

2. Tate, A. “A Theory for the Deceleration of Long Rods After Impact.” Journal ofMechanics 

and Physics of Solids, vol. 15, pp. 387-399, 1967. 

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

Explosions and Shock Waves, vol. 2, pp. 63-66, 1966. 

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

the Penetration- Mechanics Branch.” BRL-MR-3960, U. S. Army Ballistic Research 

Laboratory, Aberdeen Proving Ground, Maryland, January 1992. 

5. Wright, T. W., and K. Frank. “Approaches to Penetration Problems.” BRL-TR-2957, U. S. 

Army Ballistic Research Laboratory Aberdeen Proving Ground, Maryland, December 1988. 

6. Tate, A. “Long Rod Penetration Models - Part I. A Flow Field Model for High Speed Long 

Rod Penetration.” International Journal of Mechanical Science., vol. 28, no. 8, pp. 535-547, 

1986. 

7. Walker, J. D., and C. E. Anderson, Jr. “A Time-Dependent Model for Long-Rod Penetration.” 

International Journal of Impact Engineering, vol. 16, no. 1, pp. 19-48, 1995. 

8. Segletes, S. B., and W. P. Walters. “A Note on the Application of the Extended Bernoulli 

Equation.” ARL-TR-1895, U. S. Army Research Laboratory, Aberdeen Proving Ground, 

Maryland, February 1999. 

9. Bjerke, T. W., G. F. Silsby, D. R. Scheffler, and R. M. Mudd. “Yawed Long-Rod Armor 

Penetration.” International Journal of Impact Engineering, vol. 12, no. 2, pp. 28 l-292, 1992. 

33 
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REPORT DOCUMENTATION PAGE f%m Approved 

OMB No. 07060188 

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I. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED 

September 2000 Final, Ott 98 - Jul 00 

1. TITLE AND SUBTITLE 5. FUNDING NUMBERS 

4n Adaptation of Walker-Anderson Model Elements Into the Frank-Zook 

Penetration Model for Use in MUVES 

AH80 

j. AUTHOR(S) 

Steven B. Segletes 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS 8: PERFORMING ORGANIZATION 

REPORT NUMBER 

U.S. Army Research Laboratory 

ATTN: AMSRL-WM-TD 

Aberdeen Proving Ground, MD 21010-5066 

ARL-TR-2336 

3. SPONSORING/MONITORING AGENCY NAMES(S) AND ADDRESS 10.SPONSORING/MONITORlNG 

AGENCY REPORT NUMBER 

11. SUPPLEMENTARY NOTES 

12a. DISTRIBUTION/AVAILABILITY STATEMENT 

Approved for public release; distribution is unlimited. 

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3. ABSTRACT(Maximum 200 words) 

The performance of the well-respected Frank-Zook (FZ) penetration algorithm is examined in light of an anticipated 

:lass of target technology involving laminated targets whose layers are thin relative to the projectile diameter (in the 

imiting case, this target class incorporates functionally graded materials). Because of the manner in which the FZ 

dgorithm anticipates changes in material properties along the shotline of the penetrator, the algorithm is susceptible to 

naccuracy precisely when the target layers along the shotline are thin relative to the projectile diameter. Though the 

lroblem can be quite severe when the target-layer thickness is a fraction of the projectile diameter; the effect is still 

:vident to a much lesser extent, even as the target-element thickness is increased to several projectile diameters. A 

,emedy to this type of problem is offered, accomplished by way of novel adaptation of elements of a model by Walker 

md Anderson into the FZ framework. In doing so, the target’s material- and nonsteady-kinematic properties are 

iynamically composed via an integration through the plastic zone in the target, ahead of the rod/target interface. 

gdditionally, a model to predict the crater-diameter profile resulting from a modeled penetration event is optionally 

offered as a calculation enhancement. The modeling remedies and enhancements proposed herein are offered for 

ncorporation into the U.S. Army Research Laboratory’s (ARL) modular Unix-based vulnerability estimation suite 

:MUVES) code, to be part of ARL’s vulnerability/lethality (V/L) calculation methodology. 

14. SUBJECT TERMS 

lenetration mechanics, plasticity, target resistance, effective density 

15. NUMBER OF PAGES 

49 

16. PRICE CODE 

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT 

OF REPORT OF THIS PAGE OF ABSTRACT 

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48 
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