A Solution of the Alekseevski-Tate Penetration Equations 

by William Walters and Cyril Williams 

ARL-TR-3606 September 2005 

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Army Research Laboratory 

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ARL-TR-3606 September 2005 

A Solution of the Alekseevski-Tate Penetration Equations 

William Walters and Cyril Williams 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited. 
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A Solution of the Alekseevski-Tate Penetration Equations 

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William Walters and Cyril Williams 

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14. ABSTRACT 

The Alekseevski-Tate equations have been used for five decades to predict the penetration, penetration velocity, rod velocity, 

and rod length of long-rod penetrators and similar projectiles. These nonlinear equations were originally solved numerically 

and more recently by the exact analytical solution of Walters and Segletes. However, due to the nonlinear nature of the 

equations, penetration was obtained implicitly as a function of time. The current report obtains the velocities, length, and 

penetration as an explicit function of time by employing a perturbation solution of the nondimensional Alekseevski-Tate 

equations. Explicit analytical solutions are advantageous in that they clearly reveal the interplay of the various parameters on 

the solution of the equations. Perturbation solutions of these equations were first undertaken by Forrestal et al., up to the first 

order, and good agreement with the exact solutions was shown for relatively short times. The current study obtains a third-order 

perturbation solution and includes both penetrator and target strength terms. This report compares the exact solution to the 

perturbation solution, and a typical comparison between the exact and approximate solution for a tungsten rod impacting a steel

armor target is shown. Also, alternate ways are investigated to normalize the governing equations in order to obtain an optimum 

perturbation parameter. In most cases, the third-order perturbation solution shows near perfect agreement with the exact 

solutions of the Alekseevski-Tate equations. This report compares the exact solution to the perturbation solution, and comments 

are made regarding the range of validity of the explicit solution. 

15. SUBJECT TERMS

penetration, Tate equations, perturbation theory 

16. SECURITY CLASSIFICATION OF: 

19a. NAME OF RESPONSIBLE PERSON

William Walters 

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Standard Form 298 (Rev. 8/98) 

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iii

Contents 

List of Figures iv

List of Tables vi

Acknowledgments vii

1. Introduction 1

2. Parametric Studies 11

3. Conclusions 29

4. References 30

Appendix. FORTRAN Computer Program 31

Distribution List 36 
iv

List of Figures 

Figure 1. Comparison between rod and penetration velocities for a first-order perturbation 

solution, a third-order perturbation solution, and the exact Tate solution. This case is for 

a tungsten rod impacting a steel target at an impact velocity of 2000 m/s. The initial rod 

length was 0.500 m. ...................................................................................................................6 

Figure 2. Penetration depth for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a tungsten rod impacting a steel 

target at an impact velocity of 2000 m/s. The initial rod length was 0.500 m..........................7 

Figure 3. Rod length for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a tungsten rod impacting a steel 

target at an impact velocity of 2000 m/s. The initial rod length was 0.500 m..........................8 

Figure 4. Comparison between rod and penetration velocities for a first-order perturbation 

solution, a third-order perturbation solution, and the exact Tate solution. This case is for 

a tungsten rod impacting a 25-mm-thick steel target at an impact velocity of 2000 m/s. 

The initial rod length was 0.03912 m. .......................................................................................9 

Figure 5. Penetration depth for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a tungsten rod impacting a 

25-mm-thick steel target at an impact velocity of 2000 m/s. The initial rod length was 

0.03912 m. ...............................................................................................................................10 

Figure 6. Rod length for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a tungsten rod impacting a 

25-mm-thick steel target at an impact velocity of 2000 m/s. The initial rod length was 

0.03912 m. ...............................................................................................................................11 

Figure 7. Comparison between rod and penetration velocities for a first-order perturbation 

solution, a third-order perturbation solution, and the exact Tate solution. This case is for 

a tungsten rod impacting a steel target at an impact velocity of 3000 m/s. The initial rod 

length was 0.500 m. .................................................................................................................12 

Figure 8. Penetration depth for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a tungsten rod impacting a steel 

target at an impact velocity of 3000 m/s. The initial rod length was 0.500 m........................13 

Figure 9. Rod length for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a tungsten rod impacting a steel 

target at an impact velocity of 3000 m/s. The initial rod length was 0.500 m........................14 

Figure 10. Comparison between rod and penetration velocities for a first-order perturbation 

solution, a third-order perturbation solution, and the exact Tate solution. This case is for 

a steel rod impacting a silica-sand target at an impact velocity of 3000 m/s. The initial 

rod length was 0.03912 m........................................................................................................15 

Figure 11. Penetration depth for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a steel rod impacting a silica-sand 

target at an impact velocity of 3000 m/s. The initial rod length was 0.03912 m....................16 
v

Figure 12. Rod length for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a steel rod impacting a silica-sand 

target at an impact velocity of 3000 m/s. The initial rod length was 0.03912 m....................17 

Figure 13. Comparison between rod and penetration velocities for a first-order perturbation 

solution, a third-order perturbation solution, and the exact Tate solution. This case is for 

a 93% tungsten rod impacting a steel target at an impact velocity of 2000 m/s. The initial 

rod length was 0.500 m............................................................................................................18 

Figure 14. Penetration depth for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a 93% tungsten rod impacting a 

steel target at an impact velocity of 2000 m/s. The initial rod length was 0.500 m. ..............19 

Figure 15. Rod length for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a 93% tungsten rod impacting a 

steel target at an impact velocity of 2000 m/s. The initial rod length was 0.500 m. ..............20 

Figure 16. Comparison between rod and penetration velocities for a first-order perturbation 

solution, a third-order perturbation solution, and the exact Tate solution. This case is for 

a 93% tungsten rod impacting a steel target at an impact velocity of 3000 m/s. The initial 

rod length was 0.500 m............................................................................................................21 

Figure 17. Penetration depth for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a 93% tungsten rod impacting a 

steel target at an impact velocity of 3000 m/s. The initial rod length was 0.500 m. ..............22 

Figure 18. Rod length for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a 93% tungsten rod impacting a 

steel target at an impact velocity of 3000 m/s. The initial rod length was 0.500 m. ..............23 

Figure 19. Comparison between rod and penetration velocities for a first-order perturbation 

solution, a third-order perturbation solution, and the exact Tate solution. This case is for 

a depleted uranium rod impacting a steel target at an impact velocity of 2000 m/s. The 

initial rod length was 0.500 m .................................................................................................24 

Figure 20. Penetration depth for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a depleted uranium rod impacting a 

steel target at an impact velocity of 2000 m/s. The initial rod length was 0.500 m. ..............25 

Figure 21. Rod length for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a depleted uranium rod impacting a 

steel target at an impact velocity of 2000 m/s. The initial rod length was 0.500 m. ..............26 

Figure 22. Comparison between rod and penetration velocities for a first-order perturbation 

solution, a third-order perturbation solution, and the exact Tate solution. This case is for 

a depleted uranium rod impacting a steel target at an impact velocity of 3000 m/s. The 

initial rod length was 0.500 m. ................................................................................................27 

Figure 23. Penetration depth for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a depleted uranium rod impacting a 

steel target at an impact velocity of 3000 m/s. The initial rod length was 0.500 m. ..............28 

Figure 24. Rod length for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a depleted uranium rod impacting a 

steel target at an impact velocity of 3000 m/s. The initial rod length was 0.500 m. ..............29 
vi

List of Tables 

Table 1. Material input properties.................................................................................................17 

Table 2. Initial values and perturbation parameters.......................................................................23 
vii

Acknowledgments 

The authors would like to express their thanks to Dr. Steven Segletes of the U.S. Army Research 

Laboratory for his vigilance and thorough review of this report. He provided many helpful 

comments and suggestions during the course of the report. 
viii

INTENTIONALLY LEFT BLANK. 
1

1. Introduction 

The Alekseevski-Tate equations (1) have long been employed to predict the penetration of 

kinetic energy penetrators or rod-like projectiles. Typically, these equations are solved by 

straight-forward numerical integration techniques or by using the exact solution developed by 

Walters and Segletes (2) and refined by Segletes and Walters (3). However, due to the nonlinear 

nature of these equations, the exact solution yields penetration only as an implicit function of 

time. It is desirable to obtain an exact, albeit approximate, solution to these equations which 

would yield the pertinent variables, penetration velocity, penetrator length, penetrator velocity, 

and penetration as an explicit function of time. The analytical nature of the solutions would 

clearly reveal the interplay of the various terms in the governing equations on the solutions. 

Toward this end, as per references (1–2), a perturbation solution was obtained for the following 

equation set: 

( )( ) ( ) P v − u +YP = T u + RT

2 2 ρ 2 ρ 2 (1) 

P

P

l

Y

dt

dv

ρ

− = (2) 

u v

dt

dl = − (3) 

dt

dp u = or p = ∫ udt . (4) 

In these equations, v is the penetrator velocity, u is the penetration velocity, p is the penetration, 

l is the penetrator length, t is the time after impact, RT is the target strength term, YP is the 

penetrator strength term, ρ represents the density, where the subscript P stands for penetrator and 

subscript T represents the target. First, the equations are normalized and the method of 

normalization will depend on the input conditions; namely, the ρ values, the initial velocity, and 

the strength terms. For the usual case of interest to ballisticians studying kinetic energy 

penetrators impacting armor targets, the following normalization parameters are introduced: 

i V = v V , i U = u V , λ = l L , τ = βt , and (5) 

μ = ρ T ρ P

2 , L

i V

⎟

⎠

⎞ ⎜

⎝

⎛

+ = μ

μ β

1 , 

P

T P

Y

R − Y

α = , 2

i V

Y

P

P

ρ

ε = , L

p P = , (6) 

where Vi is the impact velocity, L is the initial penetrator length, and V, U, P, λ, and τ are the 

dimensionless variables, while μ ,α, β, and ε are the dimensionless constants. The parameter β is 

used to normalize the time. The constant ε is the perturbation parameter. This is the same 
2

normalization scheme used by Forrestal et al. (4). Forrestal et al. assumed RT = 0 (i.e., α = −1) 

for a steel projectile impacting a foundry core target (silica sand) at 3.0 km/s. 

Thus, the normalized equations become 

( )

2 2 2 V −U − 2αε = μ U , (7) 

⎟

⎠

⎞ ⎜

⎝

⎛ + = − μ

μ

λ

ε

τ

1

d

dV , (8) 

( ) U V

d

d − + = ⎟

⎠

⎞ ⎜

⎝

⎛

μ

μ

τ

λ 1 , (9) 

and 

⎟∫⎟

⎠

⎞

⎜

⎜

⎝

⎛ + =

τ

τ

μ

μ

0

1 P Ud . (10) 

The perturbation method, as shown by Cole (5), involves letting 

............. 3

3

2

2 V = V0 + εV1 + ε V + ε V + (11) 

......... 3

3

2

2 U = U0 + εU1 + ε U + ε U + (12) 

............. 3

3

2

2 λ = λ0 + ελ1 + ε λ + ε λ + (13) 

.............. 3

3

2

2 P = P0 + εP1 + ε P + ε P + (14)

and the perturbation parameter 1 2 = <<

i V

Y

P

P

ρ

ε is chosen. 

For example, if a third-order perturbation solution is obtained, terms of the order of ε to the 

fourth power are neglected. Hence, the accuracy of the solution depends on the magnitude of ε

and the number of terms in the above equation set. 

One can substitute the expressions for V, U, λ, and P (equations 11–14) into the nondimensional 

equation set (equations 7–10) and obtain, to order zero (considering only terms involving ε

0

) , the 

following set: 

( ) 2

0

2 2 V0 −U0 = μ U , (15) 

0 0 = dτ

dV , (16) 

and 

( ) 0 0

0 1

V U

d

d − + = − ⎟

⎠

⎞ ⎜

⎝

⎛

μ

μ

τ

λ , (17) 

where the initial conditions at τ = 0 are 

0 V (0) 1 = , 0

λ (0) 1 = . 
3

The solution is 

1 0 V = , (18) 

+ μ =1

1

U0 , (19) 

λ = 1 −τ 0 , (20) 

and 

μ

τ

= P0 , (21) 

in agreement with Forrestal et al. (4). 

Proceeding along these same lines (next consider terms of order ε, etc.), one can calculate V0, V1, 

V2, V3, and the corresponding U, P, and λ values. The method is straight forward but tedious. 

The final equations, to the third order, are 

3

3

2

2 V = V0 + εV1 + ε V + ε V , (22) 

3

3

2

2 U = U0 + εU1 + ε U + ε U , (23) 

3

3

2

2 λ = λ0 + ελ1 + ε λ + ε λ , (24) 

and 

3

3

2

2 P = P0 + εP1 + ε P + ε P , (25)

where 

V1 =Θ ln Π , 

[ ] 2 2 2

2 ln

2

1 1 1 ln Θ Π

Π

Θ Ω Π −⎥

⎦

⎤ ⎢

⎣

⎡ V = + − , 

( ) ( ) ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡ − + − + ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = − + + − + 3

ln ln

2

1 2 ln 1

2 2

2 2

2

ln 2 3

2

2 2

2

2

2

3 V3 C

Π

Π Ω Π Π Ω

μ Π

α

μ

α Ω Ω

Π

Ω Θ Ψ Π , 

[ ] Π α

μ − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = ln 1 U1 , 

( ) ( ) 2

3

2

2

2

2 1

2

1 ln

1 μ

μ

α

μ Π

μ

α

μ

+ + + − + = V U , 

( ) ( )

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎝

⎛

+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ⎟ − + − ⎠

⎞ ⎜

⎝

⎛ − + − − + = μ Ω

μ

α

Π

Ω Π Π Ω α

μ

αΘ

μ

2

1

2

1 ln

2

3

2

5

2

3

2

2 3 ln 2

1

3

3

V

U , 

λ Θ[τΩ Π ln Π ] 1 = + , 
4

( ) ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+ + − + ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = − − − +

2

2

2

2 2

2

2 ln

2

1 ln 2 ln

2 2

3 3 λ Π Ω Π ΩΠ Π Π Π

μ

α

μ

α

μ

α λ Θ C , 

( ) ( ) ( )

3

2 3

3 2 2

3 2

ln

2

ln

2

ln ln ln

2

7

λ

Π Π

Π

Ω Π Ω Π Π Σ Π ΦΠ Π ΓΠ Ω λ Θ +C

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡ = − + + + − + + , 

[ ] τ ( ) α Π Π

μ

Θ

P1 = − 1+ + ln , 

( ) 2

2

2 2 2

2 ln

2

1 ln 2 ln

2

2

2

3 P +CP

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ − − + − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = + + − − Π Π Π Π

μ

α Π Ω Π α

μ

α

μ

α α

α

μ

Θ , 

( ) ( ) ( ) 3

2 2 3

3 2 2

3 ln

2

1 ln

2

ln ln

2

ln

2

P +CP

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+ + + + − + ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = − − Π Π Π Ω θ Π δΠ Π

Π

Ω Π ωΠ Π Ω ξ

μ

Θ , 

and 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

= − − +

2 2

2 3

2

2 5 4

μ

α

μ

α

μ

α Ψ , 

Π =1 −τ , 

( )

μ

μ Θ + = 1 ,

μ

α Ω = 1− , 

( ) μ

μ

α Ω Ω − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = − − − 1

2

1 2

2

3

2 2

C3 , 

2

1

2 2

7 2 2

= − +

μ

Ω α Σ , 

μ

α Φ Ω Ω

2

2 2 = 2 + 5 + 4 − , 

( )2

3

2 3

2 1

2 2

5 7 5

2

5 = − − − + + μ − μ

α

μ

α Γ Ω Ω , 

( )2

2

2 2 3

2 1

2 2

3 5

2

9 4 3 7 μ

μ

α α

α

μ

α ξ = − − Ω − Ω + − − − − , 

2

3

2

3 6 3

2

7 2 2 α

α

Ω

ω = + Ω + αΩ + + + , 
5

2

1

2

3

2

5 2 2

= + + − +

μ

α

α

Ω θ Ω , 

δ = − Ω − ( ) 1+α

2

3 2 , 

2

2 2

2 2 2

3 3

μ

α

μ

α

μ

α

λ C = − − + , 

( ) ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = + + − − − 2

3

2 2 3

3

3

1

2 2

5 5

2

13

2

5

μ

μ

α

μ

α Ω Ω Θ λ C , 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = − + + − − μ

α

μ

α α

α

μ

Θ

2

2

2

3

2 2 2

CP2 , 

and 

μ

Θ ξ3

CP3 = . 

The normalization scheme used is deemed appropriate for ballistic applications, namely tungsten 

rods penetrating steel. The third-order solution for U, V, P, and λ, with comparison to the exact 

Alekseevski-Tate equation solution, is shown in figures 1–3. In this case, Vi = 2 km/s, L = 0.5 m, 

ρP = 17,600 kg/m3, YP = 1.0 GPa, ρT = 7,800 kg/m3, and RT = 5.5 GPa. The perturbation 

parameter computes as ε = 0.0142, which is much less than one and terms with a coefficient of ε

4

were neglected. The agreement is excellent up to about 495 μs for the velocities and nearly 

identical for the rod length and penetration. The exact (Tate) solution terminates at about 

518 μs. The appendix lists the FORTRAN program used to calculate the rod velocity, 

penetration velocity, rod length, and penetration all as a function of time. 

A few comments are in order. All equations contain a term like ln(1-τ). Thus, a singularity 

occurs at τ = 1 or ( ) 1

625.5 s

+

= =μ

i

L μ t

μV

 for the case described. Thus, t must be less than this 

value for the perturbation solution to be evaluated. As this singularity is approached, deviation 

from the exact solution occurs. Hence, one can accurately calculate the perforation of finite 

thickness targets if the penetration time is less than this value. As an example, figures 4–6 plot 

the rod velocity, penetration velocity, penetration, and rod length for a tungsten rod impacting a 

finite-thickness (25 mm) steel plate as described, except the initial rod length was 39.12 mm. 

The first-order and third-order perturbation theories are compared with the Tate solution. Again, 

the agreement is excellent. Note that the first-order perturbation theory also agrees very well 

with the exact solution for this case. Thus, approximate formulae may be derived for the 

perforation of thin plates. The first order equations are 

V = 1+ εΘ ln Π , (26) 
6

0

500

1000

1500

2000

2500

0 100 200 300 400 500 600

Time (μs)

Velocity (m/s)

Rod Velocity - 1st Order

Rod Velocity - 3rd Order

Rod Velocity - Tate

Penetration Velocity - 1st Order

Penetration Velocity - 3rd Order

Penetration Velocity - Tate

Figure 1. Comparison between rod and penetration velocities for a first-order perturbation 

solution, a third-order perturbation solution, and the exact Tate solution. This case is 

for a tungsten rod impacting a steel target at an impact velocity of 2000 m/s. The initial 

rod length was 0.500 m. 

[ ] Π α

μ

ε

μ

+ − + = ln

1

1 U , (27) 

λ = Π + εΘ[Ω(1− Π )+ Π ln Π ] , (28) 

and 

[ ] ( )( ) Π α Π Π

μ

εΘ

μ

Π 1 1 ln 1 − − + + − P = . (29) 

When P=H, the finite target plate thickness, equation 29 can be solved for the event duration in 

terms of Π as 

( ) ( )⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+ − −⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ − + − = 1 1 1 1

exp α

Π

Π

μ

Π

Π

Π

L

H

A

 and 

μ

εΘ A = . (30) 
7

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0 100 200 300 400 500 600

Time (μs)

Penetration Depth (m)

1st Order

3rd Order

Tate

Figure 2. Penetration depth for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a tungsten rod impacting a steel 

target at an impact velocity of 2000 m/s. The initial rod length was 0.500 m. 

Further, the exponential term in equation 30 may be approximated by the first two terms of its 

series expansion or exp[x] = 1 + x. In this case, 

2

4 2 B + B + C Π = where B = 2 +α – Aμ

1 and ( ) 1 1 = − + − α +

AL Aμ

H C . (31) 

From equation 31, the time, t, was 25.34 μs and the time calculated from the more exact 

equation 30 via standard iteration techniques was 25.12 μs. The rod velocity was 1948.156 m/s 

according to the approximate theory and 1943.000 m/s from the Tate solution. The penetration 

velocity was 977.524 m/s vs. 976.500 m/s from the Tate solution. The final rod length was 

14.20 mm vs. 14.70 mm from the Tate solution. This close agreement means that the 

perturbation solution is in excellent agreement with the Tate solution for small times, even for a 

first-order perturbation solution. Forrestal et al. (4) also showed good agreement with Tate for 

short times. 
8

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

0.400

0.450

0.500

0 100 200 300 400 500 600

Time (μs)

Length (m)

1st Order

3rd Order

Tate

Figure 3. Rod length for a first-order perturbation solution, a third-order perturbation solution, and 

the exact Tate solution. This case is for a tungsten rod impacting a steel target at an 

impact velocity of 2000 m/s. The initial rod length was 0.500 m. 

As previously mentioned, the perturbation parameter must be much less than one. The value of ε

used in the above equations implies a small penetrator strength or P P i

2 Y << ρ V . If this is not the 

case, for example, for a soft target, i.e., small RT, the equations can be normalized with 

Vi V = v , Vi U = u , λ = l L , L

p P = , (32) 

μ = ρ T ρ P

2 , L

Vi ⎟

⎠

⎞ ⎜

⎝

⎛

+ = μ

μ β

1 , 

T

p T

R

Y − R

α = , 2

P i

T

V

R

ρ

ε = , τ = βt , (33) 

and the equation set becomes 

( )

2 2 2 V −U + 2αε = μ U , (34) 
9

0

500

1000

1500

2000

2500

0 5 10 15 20 25 30

Time (μs)

Velocity (m/s)

Rod Velocity - 1st Order

Rod Velocity - 3rd Order

Rod Velocity - Tate

Penetration Velocity - 1st Order

Penetration Velocity - 3rd Order

Penetration Velocity - Tate

Figure 4. Comparison between rod and penetration velocities for a first-order perturbation 

solution, a third-order perturbation solution, and the exact Tate solution. This case is 

for a tungsten rod impacting a 25-mm-thick steel target at an impact velocity of 

2000 m/s. The initial rod length was 0.03912 m. 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ + = −

T

p

R

Y

d

dV

μ

μ

λ

ε

τ

1 , (35) 

and 

( ) U V

d

d − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ + = μ

μ

τ

λ 1 . (36) 

This is very similar to the previous set of equations and thus follows the same solution scheme. 

Note that the normalization scheme was initiated by dividing the first equation by ρp and defining 

μ

2

 to be ρT/ρP. Alternately, one could divide the equation by ρT and define μ

2

 to be ρP/ρT. 

Other normalization schemes are possible namely if the nondimensional variables are defined as 

λ = l L , μ = ρ T ρ P , P

YP K

ρ = , 

P

T P R Y

ρ

2( − ) Σ = , (37) 
10

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0 5 10 15 20 25 30

Time (μs)

Penetration Depth (m)

1st Order

3rd Order

Tate

Figure 5. Penetration depth for a first-order perturbation solution, a third-order perturbation solution, 

and the exact Tate solution. This case is for a tungsten rod impacting a 25-mm-thick steel 

target at an impact velocity of 2000 m/s. The initial rod length was 0.03912 m. 

V = v Σ , U = u Σ , λ = l Σ , L

t Σ τ = , L

p P = . (38) 

Note that Σ > 0 is required. Based on input, if this is not the case, one may reformulate Σ as 

T

p RT Y

ρ

2( − )

. 

The equation set becomes 

( ) 1 2 2 U −V = μU + , (39) 

U V d

d = − τ

λ , and (40) 

2( ) T P

P

R Y

V K Y

d

d

−

− = − = τ Σ

λ

 or 

λ

ε

τ = d

dV , (41) 
11

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.050

0 5 10 15 20 25 30

Time (μs)

Length (m)

1st Order

3rd Order

Tate

Figure 6. Rod length for a first-order perturbation solution, a third-order perturbation solution, 

and the exact Tate solution. This case is for a tungsten rod impacting a 25-mm-thick 

steel target at an impact velocity of 2000 m/s. The initial rod length was 0.03912 m. 

where 

2( ) T P

P

R Y

K Y

−

− = − = Σ

ε is chosen to be the perturbation parameter. Again, the previous set of 

equations is very similar to the original set of equations and thus follows the same solution 

scheme. The normalization scheme or equation set chosen will depend on the input conditions 

(known initial values) and the requirement to keep ε << 1. Also, it is advantageous to make the 

time where the logarithmic singularity occurs (τ = 1) as large as possible. 

2. Parametric Studies 

Other rod-target configurations were studied using the third-order perturbation theory in order to 

exercise the model. In figure 7, the rod velocity and penetration velocity are plotted vs. time for 

the Tate solution, the first-order, and the third-order perturbation theory. In this case, the initial 

rod velocity was 3 km/s. The agreement between the third-order perturbation model and the 
12

0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200 250 300 350 400 450

Time (μs)

Velocity (m/s)

Rod Velocity - 1st Order

Rod Velocity - 3rd Order

Rod Velocity - Tate

Penetration Velocity - 1st Order

Penetration Velocity - 3rd Order

Penetration Velocity - Tate

Figure 7. Comparison between rod and penetration velocities for a first-order perturbation 

solution, a third-order perturbation solution, and the exact Tate solution. This case is 

for a tungsten rod impacting a steel target at an impact velocity of 3000 m/s. The 

initial rod length was 0.500 m. 

Tate solution is excellent, due to the perturbation parameter ε = 0.0063, being small. The length 

and penetration are also in excellent agreement, as shown in figures 8 and 9. The input, namely 

the strength terms, was obtained from Dr. Steven Segletes of ARL for the tungsten rod vs. the 

RHA target. 

Next, figures 10–12 plot the velocities, penetration, and rod length vs. time for the case studied 

by Forrestal et al. (4), namely a steel rod impacting a geological (silica-sand) target. The firstorder solution, third-order solution, and the Tate solution are compared. The perturbation 

parameter was 0.0192 and the improvement of the third-order solution over the first-order 

solution is shown. The Tate solution completes the penetration process at 56 μs. The first-order 

solution agrees, within 1%, up to 29 μs, which is the Forrestal et al. (4) solution. The third-order 

solution is valid, again within 1%, to 35 μs. 
13

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0 50 100 150 200 250 300 350 400 450

Time (μs)

Penetration Depth (m)

1st Order

3rd Order

Tate

Figure 8. Penetration depth for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a tungsten rod impacting a steel 

target at an impact velocity of 3000 m/s. The initial rod length was 0.500 m. 

As mentioned previously, the strength terms for the steel targets were obtained from Dr. 

Segletes. Input values used by Edward Horwath of ARL were penetrator strengths of 3.6 GPa 

for tungsten, 3.2 GPa for depleted uranium (DU), and the target resistance was 5.1 GPa for RHA. 

All input values used in this study are recorded in table 1. These tungsten values, and the value 

previously given for the RHA target, are designated as 93% tungsten (the only difference in the 

two tungstens is the value assigned to Yp) and the plots are shown in figures 13–15 for an initial 

rod velocity of 2 km/s. Again, the agreement is good. The Tate solution indicates that the 

penetration process is complete at 652 μs and the third-order solution is valid up to 598 μs. 

Figures 16–18 present the same plots for an initial impact velocity of 3 km/s. In this case, again 

because of the lower value of the perturbation parameter, the agreement is excellent. The Tate 

solution terminates at 428 μs and the third-order theory is valid up to 409 μs. The perturbation 

parameter was 0.0511 for the 2 km/s case and 0.0227 for the 3 km/s case. Table 2 lists the 

perturbation parameters for all cases. Finally, figures 19–21 give the plots for an initial velocity 

of 2 km/s for a DU rod impacting RHA. The agreement is good, with the Tate solution 

terminating at 640.5 μs and the third-order solution valid up to 611 μs. The agreement is perfect 

for the same rod/target configuration at an initial impact velocity of 3 km/s, see figures 22–24. 
14

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0 50 100 150 200 250 300 350 400 450

Time (μs)

Length (m)

1st Order

3rd Order

Tate

Figure 9. Rod length for a first-order perturbation solution, a third-order perturbation solution, 

and the exact Tate solution. This case is for a tungsten rod impacting a steel target at 

an impact velocity of 3000 m/s. The initial rod length was 0.500 m. 

Here, the perturbation parameter was 0.0430 for the 2 km/s case and 0.0191 for the 3 km/s case. 

All input and perturbation values are summarized in tables 1 and 2. 
15

0

500

1000

1500

2000

2500

3000

3500

0 10 20 30 40 50 60

Time (μs)

Velocity (m/s)

Rod Velocity - 1st Order

Rod Velocity - 3rd Order

Rod Velocity - Tate

Penetration Velocity - 1st Order

Penetration Velocity - 3rd Order

Penetration Velocity - Tate

Figure 10. Comparison between rod and penetration velocities for a first-order perturbation 

solution, a third-order perturbation solution, and the exact Tate solution. This case is 

for a steel rod impacting a silica-sand target at an impact velocity of 3000 m/s. The 

initial rod length was 0.03912 m. 
16

0.000

0.020

0.040

0.060

0.080

0.100

0.120

0 10 20 30 40 50 60

Time (μs)

Penetration Depth (m)

1st Order

3rd Order

Tate

Figure 11. Penetration depth for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a steel rod impacting a silicasand target at an impact velocity of 3000 m/s. The initial rod length was 0.03912 m. 
17

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0 10 20 30 40 50 60

Time (μs)

Length (m)

1st Order

3rd Order

Tate

Figure 12. Rod length for a first-order perturbation solution, a third-order perturbation solution, and 

the exact Tate solution. This case is for a steel rod impacting a silica-sand target at an 

impact velocity of 3000 m/s. The initial rod length was 0.03912 m. 

Table 1. Material input properties. 

Material ρP

(kg/m3

) 

ρT

(kg/m3

) 

YP

(GPa) 

RT

(GPa) 

Tungsten 17600 — 1.000 — 

93% tungsten 17600 — 3.600 — 

Depleted uranium (DU) 18600 — 3.200 — 

Steel (RHA) — 7800 — 5.500 

Steel (RHA) for 93% tungsten — 7800 — 5.100 

Forrestal penetrator (steel) 8000 — 1.380 — 

Forrestal target (silica-sand) — 1700 — 0.000 
18

0

500

1000

1500

2000

2500

0 100 200 300 400 500 600 700

Time (μs)

Velocity (m/s)

Rod Velocity - 1st Order

Rod Velocity - 3rd Order

Rod Velocity - Tate

Penetration Velocity - 1st Order

Penetration Velocity - 3rd Order

Penetration Velocity - Tate

Figure 13. Comparison between rod and penetration velocities for a first-order perturbation 

solution, a third-order perturbation solution, and the exact Tate solution. This case is 

for a 93% tungsten rod impacting a steel target at an impact velocity of 2000 m/s. The 

initial rod length was 0.500 m. 
19

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0 100 200 300 400 500 600 700 800

Time (μs)

Penetration Depth (m)

1st Order

3rd Order

Tate

Figure 14. Penetration depth for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a 93% tungsten rod impacting a 

steel target at an impact velocity of 2000 m/s. The initial rod length was 0.500 m. 
20

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

0.400

0.450

0.500

0 100 200 300 400 500 600 700

Time (μs)

Length (m)

1st Order

3rd Order

Tate

Figure 15. Rod length for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a 93% tungsten rod 

impacting a steel target at an impact velocity of 2000 m/s. The initial rod length 

was 0.500 m. 
21

0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200 250 300 350 400 450

Time (μs)

Velocity (m/s)

Rod Velocity - 1st Order

Rod Velocity - 3rd Order

Rod Velocity - Tate

Penetration Velocity - 1st Order

Penetration Velocity - 3rd Order

Penetration Velocity - Tate

Figure 16. Comparison between rod and penetration velocities for a first-order perturbation 

solution, a third-order perturbation solution, and the exact Tate solution. This case is 

for a 93% tungsten rod impacting a steel target at an impact velocity of 3000 m/s. The 

initial rod length was 0.500 m. 
22

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0 50 100 150 200 250 300 350 400 450

Time (μs)

Penetration Depth (m)

1st Order

3rd Order

Tate

Figure 17. Penetration depth for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a 93% tungsten rod 

impacting a steel target at an impact velocity of 3000 m/s. The initial rod length 

was 0.500 m. 
23

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0 50 100 150 200 250 300 350 400 450

Time (μs)

Length (m)

1st Order

3rd Order

Tate

Figure 18. Rod length for a first-order perturbation solution, a third-order perturbation solution, 

and the exact Tate solution. This case is for a 93% tungsten rod impacting a steel 

target at an impact velocity of 3000 m/s. The initial rod length was 0.500 m. 

Table 2. Initial values and perturbation parameters. 

Penetrator Target Lo

(m) 

Vi

(m/s) 

ε

Tungsten Steel (RHA) 0.500 2000 0.0140 

Tungsten Steel (RHA) 0.500 3000 0.0063 

93% tungsten Steel (RHA) 0.500 2000 0.0511 

93% tungsten Steel (RHA) 0.500 3000 0.0227 

Depleted uranium (DU) Steel (RHA) 0.500 2000 0.0430 

Depleted uranium (DU) Steel (RHA) 0.500 3000 0.0191 

Forrestal steel Silica-sand 0.039 3000 0.0192 
24

0

500

1000

1500

2000

2500

0 100 200 300 400 500 600 700

Time (μs)

Velocity (m/s)

Rod Velocity - 1st Order

Rod Velocity - 3rd Order

Rod Velocity - Tate

Penetration Velocity - 1st Order

Penetration Velocity - 3rd Order

Penetration Velocity - Tate

Figure 19. Comparison between rod and penetration velocities for a first-order perturbation 

solution, a third-order perturbation solution, and the exact Tate solution. This 

case is for a depleted uranium rod impacting a steel target at an impact velocity of 

2000 m/s. The initial rod length was 0.500 m. 
25

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0 100 200 300 400 500 600 700

Time (μs)

Penetration Depth (m)

1st Order

3rd Order

Tate

Figure 20. Penetration depth for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a depleted uranium rod 

impacting a steel target at an impact velocity of 2000 m/s. The initial rod length 

was 0.500 m. 
26

Figure 21. Rod length for a first-order perturbation solution, a third-order perturbation solution, 

and the exact Tate solution. This case is for a depleted uranium rod impacting a steel 

target at an impact velocity of 2000 m/s. The initial rod length was 0.500 m. 

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

0.400

0.450

0.500

0 100 200 300 400 500 600 700

Time (μs)

Length (m)

1st Order

3rd Order

Tate
27

0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200 250 300 350 400 450

Time (μs)

Velocity (m/s)

Rod Velocity - 1st Order

Rod Velocity - 3rd Order

Rod Velocity - Tate

Penetration Velocity - 1st Order

Penetration Velocity - 3rd Order

Penetration Velocity - Tate

Figure 22. Comparison between rod and penetration velocities for a first-order perturbation 

solution, a third-order perturbation solution, and the exact Tate solution. This case is 

for a depleted uranium rod impacting a steel target at an impact velocity of 3000 m/s. 

The initial rod length was 0.500 m. 
28

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0 50 100 150 200 250 300 350 400 450

Time (μs)

Penetration Depth (m)

1st Order

3rd Order

Tate

Figure 23. Penetration depth for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a depleted uranium rod 

impacting a steel target at an impact velocity of 3000 m/s. The initial rod length was 

0.500 m. 
29

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0 50 100 150 200 250 300 350 400 450

Time (μs)

Length (m)

1st Order

3rd Order

Tate

Figure 24. Rod length for a first-order perturbation solution, a third-order perturbation 

solution, and the exact Tate solution. This case is for a depleted uranium rod 

impacting a steel target at an impact velocity of 3000 m/s. The initial rod 

length was 0.500 m. 

3. Conclusions 

A perturbation solution of the Alekseevski-Tate equations was obtained through the third order. 

Agreement with the exact solution is excellent for a tungsten rod impacting a steel target at a 

velocity of 2.0 km/s, as well as for other cases of interest to ballisticians involved in the 

penetration of steel targets. Also, the velocities and residual length of a tungsten rod perforating 

a finite-thickness steel plate show excellent agreement with the exact equations. The 

approximate (perturbation) solution yields the dependent variables (V, U, λ, and P) as explicit 

functions of time. Also, alternate forms of the normalization of the pertinent equations are 

investigated to obtain a perturbation parameter much less than one for various penetration 

problems depending on the input conditions, namely target and penetrator densities, strengths, 

and initial impact conditions. Also, comments are made regarding the singularity in time. 

Extensions of the method to higher orders (i.e., the fourth order) are possible, but a point of 

diminishing returns has been reached. The current third-order solution is expressible as an 

algebraic equation, amenable to a spread sheet or simple calculator evaluation. 
30

4. References 

1. Tate, A. A Theory for the Deceleration of Long Rods after Impact. J. Mech. Phys. Solids

1967, 15, 387–399. 

2. Walters, W. P.; Segletes, S. B. An Exact Solution of the Long Rod Penetration Equations. 

International Journal of Impact Engineering 1991, 11 (2), 225–231. 

3. Segletes, S. B.; Walters, W. P. Extensions to the Exact Solution of the Long-Rod 

Penetration/Erosion Equations. International Journal of Impact Engineering 2003, 11, 

363–376. 

4. Forrestal, M. J.; Piekutowski, A. J.; Luk, V. K. Long-Rod Penetration into Simulated 

Geological Targets at an Impact Velocity of 3.0 km/s. Presented at the 11th International 

Symposium on Ballistics, Brussels, Belgium, 9–11 May 1989, Vol. 2. 

5. Cole, J. D. Perturbation Methods in Applied Mathematics. Blaisdell Publishing Company: 

Waltham, MA, 1968. 
31

Appendix. FORTRAN Computer Program 

! Pertubation.f90 

! 

! FUNCTIONS: 

! Pertubation - Closed-form equations for Tate's model from a pertubation analysis. 

! 

!*************************************************************************************! 

! PROGRAM: Pertubation 

! 

! PURPOSE: This program calculates the rod velocity, penetration velocity, penetration 

! depth, and instantaneous length using pertubation method up to the third-order 

! correction terms. 

! 

!************************************************************************************* 

 program Pertubation 

 implicit none 

 ! Indentifiers 

 ! RT: Strength resistance of target (Kg/m.s^2) 

 ! RHO_P: Density of penetrator (Kg/m^3) 

 ! RHO_T Density of target (Kg/m^3) 

 ! YP: Flow stress of penetrator (Pa) 

 ! VI: Impact velocity (m/s) 

 ! VR: Rod velocity (m/s) 

 ! UP: Penetration velocity (m/s) 

 ! LO: Original length of penetrator (m) 

 ! LI: Instantaneous length of rod (m) 

 ! P: Penetration Depth (m) 

 ! T: Time (microsecond) 

 ! VN: Normalized rod velocity 

 ! UN: Normalized penetration velocity 

 ! PD: Normalized penetration depth 

 ! LAMDA: Normalized length 

 ! TAU: Normalized time 

 ! MHU: Dimensionless variable 

 ! BETA: Dimensionless variable 

 ! ALPHA: Dimensionless variable 

 ! EPSI: Dimensionless variable 

 ! A,B: Expressions used for equation compactness 

DOUBLEPRECISION RT, RHO_P, RHO_T, YP, VI, VR, UP, LO, LI, P, PD, T, VN, UN, LAMDA, TAU, 

MHU, BETA, ALPHA, EPSI, V0, U0, LAMDA0, PD0, V1, U1, LAMDA1, PD1, V2, U2, LAMDA2, PD2, V, U, 

C0, C1, C2, C3, V3, U3, PD3, LAMDA3, THK, A, B 

 INTEGER:: ORDER 

 

 ! Input values for velocity, length, and material constants 
32

 PRINT*, "Enter impact velocity of penetrator, 'Meters per Second'" 

 READ*, VI 

 

 PRINT*, "Enter original length of penetrator, 'Meters'" 

 READ*, LO 

 

 PRINT*, "Enter density of penetrator material, 'Kilograms per Meter Cubed'" 

 READ*, RHO_P 

 

 PRINT*, "Enter density of target material, 'Kilograms per Meter Cubed'" 

 READ*, RHO_T 

 

 PRINT*, "Enter flow stress of penetrator material, 'Pascals'" 

 READ*, YP 

 PRINT*, "Strength resistance of target, 'Pascals'" 

 READ*, RT 

 ! Check for data inconsistencies 

 ! The "strength resistance of the target (RT)," cannot be equal to the "flow stress of the 

 ! penetrator (YP)," for the pertubation solution to be meaningful. 

 IF(RT.EQ.YP)THEN 

 PRINT*, "*****RT must not equal YP for a meaningful pertubation solution.*****" 

 STOP 

 ENDIF 

 PRINT*, "Thickness of the target, 'Meters'" 

 READ*, THK 

 ! Assume a semi-infinite plate for zero thickness. 

 IF(THK.EQ.0)THEN 

 THK=1 

 ENDIF 

 PRINT*, "What order pertubation solution do you require" 

 READ*, ORDER 

 OPEN(UNIT=12, FILE="Pertubation.dat", STATUS="OLD") 

 

 ! Normalization scheme 

 MHU=SQRT(RHO_T/RHO_P) 

 BETA=MHU/(1.0+MHU)*VI/LO 

 ALPHA=(RT-YP)/YP 

 EPSI=YP/(RHO_P*VI**2) 
33

 A=(1.0+MHU)/MHU 

 B=1.0-ALPHA/MHU 

 T=0.0 

 DO 

 TAU=BETA*T 

! Zero order calculation resulting from pertubation analysis 

 V0=1.0 

 U0=1.0/(1.0+MHU) 

 LAMDA0=1.0-TAU 

 PD0=TAU/MHU 

! First order calculation resulting from pertubation analysis 

 

 V1=((1.0+MHU)/MHU)*LOG(1.0-TAU) 

 U1=(1.0/MHU)*(LOG(1.0-TAU)-ALPHA) 

 LAMDA1=((1.0+MHU)/MHU)*((1.0-ALPHA/MHU)*TAU+(1.0-TAU)*LOG(1.0-TAU)) 

 PD1=-(1.0+MHU)/(MHU**2)*(TAU*(ALPHA+1.0)+(1.0-TAU)*LOG(1.0-TAU)) 

! Second order calculation resulting from pertubation analysis 

 V2=((1.0+MHU)/MHU)**2*(1.0-ALPHA/MHU)*(LOG(1.0-TAU)+1/(1.0-TAU)-1.0)- & 

 0.5*(((1.0+MHU)/MHU)**2)*(LOG(1.0-TAU))**2 

U2=V2/(1.0+MHU)+(ALPHA/MHU**2)*(1.0+MHU)*LOG(1.0-TAU)+(ALPHA**2)*(1.0-& 

MHU**2)/(2.0*MHU**3) 

 C0=3.0-3.0*ALPHA/MHU+ALPHA**2/(2.0*MHU**2)-ALPHA**2/(2.0*MHU) 

 

LAMDA2=((1.0+MHU)/MHU)**2*(-(3.0-3.0*ALPHA/MHU+ALPHA**2/(2.0*MHU**2)-& 

ALPHA**2/(2.0*MHU))*(1-TAU) +(1.0-ALPHA/MHU)*LOG(1.0-TAU)+2.0*(1.0-& 

ALPHA/MHU)*(1.0-TAU)*LOG(1.0-TAU)-0.5*(1.0-TAU)* (LOG(1.0-TAU))**2+C0) 

PD2=(1.0+MHU)**2/MHU**3*(B*(1.0-2.0*TAU-(1.0-TAU)*LOG(1.0-TAU)-LOG(1.0-& TAU))+(1.0-

TAU)/2.0*(LOG(1.0-TAU))**2-(1.0-TAU)*LOG(1.0-TAU)+1.0-& TAU+ALPHA*(1.0-TAU-(1.0-

TAU)*LOG(1.0-TAU))+ ALPHA**2*(1.0-& MHU)*TAU/(2.0*MHU)-(2+ALPHA-ALPHA/MHU)) 

! Third order calculation resulting from pertubation analysis 

 C1= -3.0/2.0*B**2-2.0*B+ALPHA**2/(2.0*MHU)*(1.0-1.0/MHU) 

 

V3=A**3*((4.0-5.0*ALPHA/MHU-ALPHA**2/(2.0*MHU)+3.0*ALPHA**2/(2.0*MHU**2))& 

*LOG(1.0-TAU)-0.5*B**2*1.0/(1.0-TAU)**2+(2.0*B+2.0*B**2-ALPHA**2/(2.0*MHU)+ & 
34

ALPHA**2/(2.0*MHU**2))*1.0/(1.0-TAU)-2.0*B*(LOG(1.0-TAU))**2+0.5*(LOG(1.0-& TAU))**3-

B*(LOG(1.0-TAU))/(1.0-TAU)+C1) 

 U3=(V3*(V0-U0)+(V1-U1)*(V2-U2)-MHU**2*U1*U2)/((V0-U0)+U0*MHU**2) 

C2=(1.0+MHU)**3/MHU**4*(-4.0-7.0/2.0*B**2-7.0*B+9.0/2.0*ALPHA**2/MHU-& 5.0*ALPHA3.0*ALPHA**2/2.0-ALPHA**3/(2.0*MHU**2)*(1.0-MHU)**2+B**2/2.0) 

PD3=-(1.0+MHU)**3/MHU**4*((1.0-TAU)*(-4.0-7.0/2.0*B**2-7.0*B+9.0/2.0*& ALPHA**2/MHU5.0*ALPHA-3.0*ALPHA**2/2.0-ALPHA**3*& 

(1.0-HU)**2/(2.0*MHU**2)) +(1.0-TAU)*LOG(1.0-TAU)*(7.0/2.0+6.0*B+& 

+3.0*ALPHA*B+3.0*B**2/2.0+ALPHA+3.0*ALPHA**2/2.0)+B**2/(2.0*(1.0-TAU)) & 

+(B+5.0/2.0*B**2+ALPHA-3.0*ALPHA**2/(2.0*MHU)+1.0/2.0)*LOG(1.0-TAU)+(1.0-& 

TAU)*(LOG(1.0-TAU))**2*(-2.0*B-3.0/2.0*(1.0+ALPHA))-B/2.0*(LOG(1.0-TAU))**2+(1.0-& 

TAU)/2.0*(LOG(1.0-TAU))**3)+C2 

C3=((1+MHU)/MHU)**3*(5.0/2.0+13.0/2.0*B**2+5.0*B-5.0*ALPHA**2/(2.0*MHU)-& 

ALPHA**3/(2.0*MHU**3)*(MHU-1)**2) 

LAMDA3=((1+MHU)/MHU)**3*((-7.0*B/2.0)*(1.0-TAU)*(LOG(1.0-TAU))**2+(7.0*& B**2/2.0 -

ALPHA**2/(2.0*MHU)+1.0/2.0)*LOG(1.0-TAU) -B/2.0*(LOG(1.0-TAU))**2& +(2.0+5.0*B+4*B**2-

2*ALPHA**2/MHU)*(1.0-TAU)*LOG(1.0-TAU)+(-5.0/2.0-7.0*B**2& 

-5.0*B+5.0*ALPHA**2/(2.0*MHU)+ALPHA**3/(2.0*MHU**3)*(MHU-1)**2)& 

 *(1.0-TAU)+B**2/(2.0*(1.0-TAU))+(1.0-TAU)/2.0*(LOG(1.0-TAU))**3)+C3 

! Summation of series for 1st order, 2nd order, and 3rd order pertubation solution 

 IF(ORDER.EQ.1)THEN 

 

 V=V0+EPSI*V1 

 U=U0+EPSI*U1 

 LAMDA=LAMDA0+EPSI*LAMDA1 

 PD=PD0+EPSI*PD1 

 ELSEIF(ORDER.EQ.2)THEN 

 V=V0+EPSI*V1+(EPSI**2)*V2 

 U=U0+EPSI*U1+(EPSI**2)*U2 

 LAMDA=LAMDA0+EPSI*LAMDA1+(EPSI**2)*LAMDA2 

 PD=PD0+EPSI*PD1+(EPSI**2)*PD2 

 ELSEIF(ORDER.EQ.3)THEN 

 V=V0+EPSI*V1+(EPSI**2)*V2+(EPSI**3)*V3 

 U=U0+EPSI*U1+(EPSI**2)*U2+(EPSI**3)*U3 

 LAMDA=LAMDA0+EPSI*LAMDA1+(EPSI**2)*LAMDA2+(EPSI**3)*LAMDA3
35

 PD=PD0+EPSI*PD1+(EPSI**2)*PD2+(EPSI**3)*PD3 

 ENDIF 

! Converting normalized values into actual values 

 VR=V*VI 

 UP=U*VI 

 P=PD*LO 

 LI=LAMDA*LO 

! Penetration process stop conditions 

! For RT > YP, the minimum rod velocity required to achieve penetration occurs when the penetration 

! velocity is zero (UP = 0). At this point the target begins to behave as a rigid body and the rod 

! bounces off the target. 

 IF(RT.GT.YP.AND.VR.LE.SQRT(2.0*(RT-YP)/RHO_P))EXIT 

 

! For RT < YP, the minimum rod velocity required to achieve penetration occurs when the penetration 

! velocity is equal to the rod velocity (UP = VR). At this point the rod begins to behave as a 

! rigid body. 

 IF(RT.LT.YP.AND.VR.LE.SQRT(2.0*(YP-RT)/RHO_T))EXIT 

! If the penetration is greater than the thickness of the target, perforation is achieved terminate 

! loop 

 

 

 IF(P.GT.THK)EXIT 

! If the instantaneous length is less than zero, then the rod had totally eroded, Terminate Loop 

 IF(LI.LT.0.0)EXIT 

! Output data for graphical processing 

 WRITE(12, 100), T, VR, UP, P, LI 

100 FORMAT(F10.7, 5X, F15.10, 5X, F15.10, 5X, F10.8, 5X, F10.8) 

! Time increment, 1 microsecond step size 

 T=T+0.000001 

 

 END DO 

 PRINT*, EPSI 

 END PROGRAM Pertubation 
36

 1 DEFENSE TECHNICAL 

 (PDF INFORMATION CTR 

 ONLY) DTIC OCA 

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INTENTIONALLY LEFT BLANK.