Efficient Solution of the 

Long-Rod Penetration Equations 

of Alekseevskii-Tate 

by Steven B. Segletes and William P. Walters 

ARL-TR-2855 September 2002

Approved for public release; distribution is unlimited.
Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5066 

ARL-TR-2855 September 2002

Efficient Solution of the 

Long-Rod Penetration Equations 

of Alekseevskii-Tate 

Steven B. Segletes and William P. Walters 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited.
Contents 

1. Background 1

2. Closed Form Solution for L(V) 2

3. Choice of Model Variable 4

4. Model-Variable Transformation 5

5. Penetration 6

5.1 R = Y................................................................................................................................6 

5.2 γ = 1 .................................................................................................................................7 

5.3 General Case....................................................................................................................8 

6. Implicit Time 12

7. Terminal Rod Length, etc. 13

8. Residual Erosion/Penetration Behaviors 14

8.1 Residual Rod Erosion....................................................................................................15 

8.2 Residual Rigid-Body Penetration..................................................................................16 

9. Conclusions 17

10. References 18

Report Documentation Page 19

i
1. Background 

The penetration equations that describe the behavior of a long rod that erodes while it penetrates 

at high velocity were formulated independently by Alekseevskii [1] and Tate [2] in the mid1960s and are given by 

 (rod deceleration), (1) LV Y ρ R = − / &

R ( ) V −U + Y = TU + R 2 2

2

1

2

1

ρ ρ (interface stress balance), (2) 

V U L (erosion kinematics), and (3) = − &

P = U (penetration definition). (4) &

In these equations, V is the rod velocity, U is the penetration rate, P is the rod penetration, and L

is the rod length, all functions of time. The constant parameters include the rod strength Y, the 

target resistance R, and the target-to-rod density ratio ρ T ρ R γ = / . The dots signify time

differentiation. These equations have typically been integrated numerically to achieve a solution. 

A decade ago, Walters and Segletes [3] obtained an exact solution to these equations. However, 

the solution was not expressed in terms of the primitive variables that appear in the original 

equations, but rather in terms of an oblique transformation variable that was presented without 

explanation. Furthermore, little attempt was made to collate variables into an orderly fashion, 

thus leaving an incomplete sense for the term groupings that actually drive the solution. While 

mathematically rigorous, the solution was somewhat cumbersome to use. 

This equation set has been re-examined, in search of improvements and extensions to the 

solution method. Several improvements to the solution approach are offered herein to improve 

the solution efficiency. A primary hindrance of the original solution was in the evaluation of the 

rod velocity as a function of time. While this hindrance remains with the current approach, it 

may be circumvented by choosing an independent variable other than time, in the evaluation of 

rod erosion. Indeed, it is often more useful to express the solution in terms of, for example, rod 

velocity, rather than the canonical function-of-time solution. And while this alternative was 

available to the original solution [3], the presentation of the original solution perhaps left the 

false impression with the reader that the numerical evaluation of V(t) was a necessary 

intermediate step in the solution of L(V) and P(V).

Though the governing equations (1)−(4) pertain only to the time during which penetration and 

erosion simultaneously occur, extensions to the original solution [3] are herein provided for the 

subsequent stage of rigid-body penetration or rigid-target rod erosion. In addition to the generalcase solution to the penetration problem being addressed, several special-case conditions, 

including the cases for which R = Y and ρR = ρT, respectively, will also be solved. Not addressed 

1
herein, however, because of their simplicity, are three special cases for which R = 0, Y = 0, and 

R = Y = 0, respectively. The present method, described subsequently, can be used to describe the 

R = 0 solution up until the moment that rigid-body penetration commences. Subsequent 

behavior, however, will be governed by Poncelet flow. In the case of both Y = 0 and R = Y = 0, 

the solution becomes trivial in that the rod velocity remains constant until the rod is totally 

consumed, at which point the event ceases. The penetration velocity and rod erosion rate also 

remain constant for these cases, in accordance with equations (2) and (3). For the case of 

R = Y = 0, the steady-state erosion rates are governed by the Bernoulli equation. 

2. Closed Form Solution for L(V) 

Without delay, we present the solution to the rod erosion equations, which is valid for all cases 

(special cases [R = Y, ρR = ρT] and the general case): 





 



 −

















−

− =





 



 −

R

Y

R

Y

V L V L

U L

U L

L

L

γ ρ

γ γ

2 / exp 0 0

1 1

0 0 0

& &

&

& , (5)

where the “0” subscripts signify conditions at the onset of the penetration event. It is worthy to 

note that while −L& is the rate of rod erosion, the term γU would be the rate of rod erosion 

were the case hydrodynamic (i.e., where R=Y=0). 

The presentation of the solution, given by equation (5), is obtained by solving equation (1) for L

and differentiating with respect to time; then, using equations (2) and (3) to obtain dL/dt in terms 

of rod velocity V; and finally eliminating dL/dt from the two resulting equations, which gives 

d 2

V/dt 2 in terms of dV/dt and V. The particular expression for dL/dt varies depending on 

whether the special or general problem cases are considered, and this will affect the form of the 

governing equation, as will be shown. The resulting equation is integrated to provide dV/dt in 

terms of V. While the traditional technique is to separate variables and attempt to integrate again 

to obtain V(t), as was done by Walters and Segletes [3], this step is not necessary to obtain L(V). 

Equation (1) provides a direct algebraic link between L and dV/dt, and thereby allows dV/dt to be 

eliminated in favor of L, immediately following the first integration. The result is L(V), which is 

a desirable way to express the result, as an alternative to L(t). 

When special- and general-case problems are considered, the solutions, at first glance, take on 

different appearances. However, equation (5) was discerned from those solutions (given here, 

expressed in terms of a single independent variable, V, the rod velocity) by realizing that the 

various grouping of V terms in the various L(V) solutions all satisfy the elegant form of 

equation (5): 

2
R = Y : 

Governing Equation: V Y V = − 2 V , (6)

R

& &&

γ ρ

γ

1+

( 













 − +

− = 2 2

0

0 2 (1 ) exp V V

L Y

L R

γ

ρ γ ) , (7)

γ = 1: 

Governing Equation: V

V

V Y

R R

& &&

ρ  ρ





 −

2

Y V R = − 

 

 + 2 , (8) 

( 



 



 − −















 =





 



 −

2 2

0

1

0 0 4 exp V V

V Y

V

L

L R

Y

R

ρ ) , (9)

General case: 

 Governing Equation: ( ) V Y V V R Y V

R

R & &&

ρ

γ γ γ ρ

γ − + + − − = − −

2 2 2( )(1 )/

1

1 , (10) 

. 1

1 2( )(1 )/( )

1

1 2( )(1 )/( )

2 (1 ) exp

1 1 2( )(1 )/( )

1 1 2( )(1 )/( )

2

2

2

2 0

0

1 1

2

0

2

0 0

























−

+ − − − − ⋅ −

+ − − − ⋅

×









+

− ⋅ 















+ + − −

+ + − − = ⋅





 



 −

γ

γ γ ρ γ

γ

γ γ ρ γ

γ

ρ γ

γ γ ρ

γ γ ρ γ

R Y V

V

R Y V

V

R Y V Y

R Y V

V

V

L

L

R R

R

Y

R

R

R

 (11) 

Equations (7), (9), and (11) have been organized and presented in a manner to demonstrate the 

functional linkage between the special- and general-case solutions. For example, when either 

R = Y or γ = 1, the extended square-root terms of equation (11) become unity, leading to the 

simpler (V/V0) monomial and (V0

2

 − V2) exponential terms of equations (9) and (7). When γ = 1, 

the leading multiplier on the exponential term in equation (11) matches that of equation (9). And 

when R = Y, the exponent on the monomial becomes zero, leading to the form of equation (7). 

While the forms for U(V) and , obtainable from equations (2) and/or (3), are vastly 

different in appearance for the special and general cases, the solutions for L(V) nonetheless all 

share a common structured form described by equation (5). 

L(V ) &

3
3. Choice of Model Variable 

While equations (6), (8) and (10) of the previous section choose to cast the problem in terms of 

rod velocity and its derivatives, this is by no means the only option. Through equations (2) and 

(3), V may be algebraically expressed in terms of U or . Therefore, instead of expressing rod 

length as L = L(V) in equation (11), alternate expressions of the result, given as L = L(U) or 

 may be obtained as: 

L&

L L(L) = &

























+

+ − + − ⋅ +

+ − + ⋅

×







− + ⋅ 















+ + −

+ + − = ⋅





 



 −

γ

γ ρ γ

γ

γ ρ γ

ρ γ γ

γ ρ

γ ρ γ

1

1 2( )/( )

1

1 2( )/( )

2

(1 )

exp

1 1 2( )/( )

1 1 2( )/( )

2

2

2

2 0

0

1 1

2

0

2

0 0

R Y U

U

R Y U

U

R Y U Y

R Y U

U

U

L

L

R R

R

Y

R

R

R

 (12) 

 , 

. 1

1 2( )/( )

1

1 2( )/( )

2

(1 )

exp

1 1 2( )/( )

1 1 2( )/( )

2

2

2

2 0

0

1 1

2

0

2

0 0

























+

− − + − ⋅ +

− − + ⋅

×







− + ⋅ 















+ − −

+ − − = ⋅





 



 −

γ

ρ γ

γ

ρ γ

γ

ρ γ

ρ

ρ γ

R Y L

L

R Y L

L

R Y L Y

R Y L

L

L

L

L

R R

R

Y

R

R

R

& & & &

&

&

&

&

 (13) 

In the derivation of these and subsequent relations, there are several closely related, algebraic 

expressions that can facilitate expression and/or transformation of results. These include: 

γ − = [ ] γ + γ + 2(1−γ )( − )/ ρ /(1+ γ ) 2 U L V V R Y R & ; (14) 

γ + = [ ] γ − γ + 2(1−γ )( − )/ ρ /(1− γ ) 2 U L V V R Y R & ; (15) 

= [ ] γ − γ + 2(1− γ )( − )/ ρ /( ) 1− γ 2 L V V R Y R & ; (16) 

= [ ] − γ + 2(1− γ )( − )/ ρ /( ) 1−γ 2 U V V R Y R . (17) 

4
4. Model-Variable Transformation 

The complications of having the model variable V, U, or under the square root for the general 

case of equation (11), (12), or (13), respectively, may be circumvented with the selection of a 

mathematically more “natural” variable than the velocity V, U, or . Looking to equation (5) 

for guidance, success has been found in 

L&

L&

| Σ |

− Φ = U L& γ , (18) 

where the constant Σ is defined as 2(R − Y)/ρR. The variable Φ, proportional to the expression of 

equation (14), is always nonnegative and follows somewhat the trend of rod velocity V (it 

actually equals V/ | Σ | when γ = 1). Not surprisingly, Φ is also proportional to z , which was 

the key transformation variable employed in the original derivation [3]. The key benefit to using 

the Φ transformation is that and U, rather than requiring square root terms as did 

equations (16) and (17) when expressed in V, may be expressed in more simply in terms of Φ as 

L&





 





Φ

Φ + Σ

Σ = − 1 sgn( ) 2

| | L& , (19) 

and 





 





Φ

Φ − Σ

Σ = 1 sgn( ) 2| |

γ

U , (20) 

where the signum function, sgn(x), denotes the sign of the argument [sgn(x) = x/|x| for x ≠ 0, and 

sgn(x) = 0 for x = 0], in this case the sign of Σ. The rod velocity, V, may also be obtained 

directly, by substituting these expressions into equation (3): 

















Φ

− + Φ + Σ

Σ = ( 1) ( 1) sgn( ) 2| | γ

γ

γ

V . (21) 

When Φ is used in preference to rod velocity V as the independent variable, the governing 

equation (5) leads to the following expression: 

















































Φ

− − + Φ + 















Φ

− − − + Φ + 















Φ

Φ =





 



 −

2

2

2

0

2

0

1 1

0 0

( 1) ( 1) ( 1) 1 ( 1) 4

1

exp

γ

γ

γ

γ

γ

γ

Y

R

L

L Y

R

. (22) 

With minimal rearrangement, the variable Φ can be made to appear always in squared form. It is 

for this reason that Walters and Segletes [3] selected their transformation variable, z, 

5
proportional to Φ2

. We will do the same here, though with a different proportionality constant, 

so that 

1

1 2

−

+ = Φ ⋅ γγ

z . (23) 

By so doing, the expression for residual rod length, equation (22), becomes 

















− ± − ± − = − 



 



 −

1 [( 1/ ) ( 1/ )] 4

1

( / ) exp 0 0

1

2

1

0

0

z z z z

Y

R

z z

L

L Y

R

γ

γ γ , (24) 

where the conditional operators in equation (24) are chosen as “+” for γ > 1 and “−” for γ < 1, 

and 

( ) [ ]1/ 4 3 2

2

1/ 4

2

1( 1)

(1 )

1

1

− + Σ

+ + − Σ ⋅ − = 



















− Σ

+ =

γ γ

γ γ γ

γ

γ

γ V V

z U L& . (25) 

Like equation (11), the result given by equation (24) expresses rod length in terms of a single 

independent variable, in this case z. The advantage of equation (24) over equation (11) is in 

removing the model variable from under a radical. The choice of a proportionality constant 

different than that used in the prior work [3], when defining z, provides a result that reduces the 

number of constant parameters in the exponent. More importantly, however, the appearance in 

the exponential of the model variable in the specific form of (z ± 1/z) will greatly expedite the 

evaluation of rod penetration, as will be subsequently shown. 

5. Penetration 

The evaluation of penetration by way of integrating equation (4) may be transformed with 

equation (1) to give 

∫ ∫ = = −

0

0 0 0

1 V

V

t

U dV

L

L

V

P U dt & . (26)

The particular functional forms for L and U will govern the form of the solution. 

5.1 R = Y

Penetration may be directly evaluated in closed form for the simple case of R=Y, wherein L is 

given by equation (7), and U is proportional to V throughout the penetration event. In this case, 

6
the final penetration (given by Pf as U and V approach zero) is 

 R = Y: 



























+

− − 2

0

0

2 (1 )

1 V

Y

R

γ

ρ γ

γ 

 = exp

L

Pf . (27) 

5.2 γ = 1

For the γ = 1 special case, where the penetration velocity U is given algebraically by 

U = (V − Σ/V)/2, the penetration may be calculated, per equation (26), in closed form if the value 

of the V exponent in equation (9), given as(R − Y)/Y , is an even integer (i.e., R/Y is an odd 

integer). For these limited cases, integration by parts permits the problem reduction according to

one of the following two recursion rules: 

2 2 2 2 1 2 2 2 2

( ) ( ) 1 ( ) V e dV

b

a V e

b

V e dV a bV a bV a bV ∫ − = − ∫ , (a > 0) ; (28a) 

2 2 2 1 2 1 2 2 2 2

( ) ( 1) ( ) ( 1)

1 ( ) V e dV

a

b V e

a

V e dV a bV a bV a bV ∫ ∫ + +

+ − + = , (a < 0) ; (28b) 

which is repeated until the reduced exponent on V becomes zero, whereupon the process 

terminates with the rule 

2

. (29) ∫ e dV = e b bV bV / 2 2 2

For other cases without the appropriate integral exponents, a recursion-type solution, with 

tabulation is still plausible, in theory. The recursion rule would be applied j times until the 

exponent a j falls between zero and unity. At that point, a tabulated solution for values of a

between zero and unity (along the lines of the Gamma-function solution) is used to close out the 

integration. With clever use of velocity-normalization, the integral from V to V can be broken 

into two integrals, each evaluated between 0 and 1. Unfortunately, the integral remains a 

function of two parameters, a and b. And while tabulating a function of one parameter can be an 

efficient solution technique, tabulating solutions for functions of two real parameters quickly 

become more cumbersome than a series expansion or numerically integrated solution. 

±

2

0

2

x

As an alternative then, the penetration for the γ = 1special case may be evaluated by way of 

series solution in terms of velocity. One way to achieve this is to express the penetration as 

γ = 1: 





























 −

























 ∑ ∑

∞

=

∞

= 0

2

0 0

2

0

0 4 j 4

j

R

j

j

j

Y

V

a

L

L

Y

V

L

ρ ρ   =  R a j

P , (30) 

and match the derivative of P to the terms of U, given by U = (V − Σ/V)/2. With this approach, 

one obtains a0 = −1, a1 = 2/(1 + ρRΣ/4Y), and for the remaining terms, aj = − aj−1/( j + ρRΣ/4Y). 

Note that ρRΣ/4Y equals (R/Y − 1)/2. While the series terms alternate in sign, the fact that j is in 

7
the denominator of the recursion formula indicates that the rate of convergence for this solution 

approach should be similar to that for the exponential series. To confirm that this expression 

approaches the proper form for the special case when R=Y (when Σ equals zero), the recursion 

relation is observed to then approach the series definition for the exponential, 

1 2exp[ /4 ]. This series takes on the value 1 − 2(L 2 − −V K f /L0) when evaluated at V0, and –1 

when evaluated at V = 0. Here, Lf is the terminal length of the rod. The final penetration, 

therefore, becomes L0 − Lf, as expected for R=Y and γ = 1. 

Perhaps a more forthright approach for the evaluation of penetration for the γ = 1 special case 

(and less prone to the precision problems of evaluating an alternating series) is to directly 

integrate LU·dV, per equation (26), by initially expanding the exponential term of L into a series, 

( ) ( ) 















+ + + 





























 Σ − 













 − ⋅ =

Σ

... 2!

/4

1!

/4 1

4 exp2

1

2 2 2 2

0

2

0

0

V Y V Y

V

V

V

V

Y

V U

L

L R R

Y

R

R

ρ ρ ρ

ρ

, (31) 

and integrating term by term to the desired level of precision. As before, R Y ρ R Σ = 2( − )/ . By 

integrating this expression with respect to V, per equation (26), one may obtain 

γ = 1: 









































⋅ Σ

+

Σ −

⋅ 















− 















⋅ Σ

+

Σ −

⋅ 



 



− = ∑ ∑

∞

=

Σ

∞

=

j

R

j R

R

Y j

R

j R

R

R

Y

V

Y j

Y j

V j

V

Y

V

Y j

Y j

Y j

V

L

P R

4

4

4

!

1

4

4

4

!

1

4 exp

2

0

4

2

0

2 2

0

0

2

0

0

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

. (32) 

5.3 General Case 

In evaluating the penetration for the general case, the solution becomes more complicated but 

can nonetheless be made more efficient compared to the method presented in the original 

solution [3]. Efficiencies are achieved in several ways. The use of rod length L in the form of

equation (24) retains integer-powered polynomials in the exponential term. As such, the series 

expansion of the exponential, by which the integrals are evaluated, does not require the 

evaluation of fractionally powered polynomial expansions, as did the original method [3]. But

more importantly, by having transformed L into a form where the exponential argument is of the 

explicit form c(z ± 1/z), a method may be used to expand the exponential in an efficient way, 

reducing the expansion of the exponential to power n from a cost of (n + 1)(n + 2)/2 monomial 

evaluations in z, to one of 2n + 1 evaluations in z. 

The equation describing the penetration, equation (26), may be reorganized to obtain an 

expression in terms of the transformation variable, z, 

dz

dz

dV U

L

L

V

dz

dz

dV

V

U P U dt

z

z

z

z

t

∫ ∫ ∫ = = = −

0

0 0 0 0

1

& & . (33) 

8
Using equations (21) and (23), the rod velocity is expressible in terms of z as 

( )



















 − + + − Σ − Σ = z

V z

1/ 2

1/ 2

1/ 4 2 1

( 1) sgn[( 1) ]

2

1 γ

γ γ

γ

γ , (34) 

so that dV/dz may be computed as 

( )



















 − − − Σ − Σ + = 3 / 2

1/ 2

1/ 2

1/ 2 1/ 4 2 1

sgn[( 1) ] ( 1)

4

1

dz z z

dV γ

γ

γ

γ

γ . (35) 

In a similar vein, from equations (20) and (23), U may be expressed in terms of z as 









































−

+ − Σ 



















+

Σ − = z

U z

1

1

1

sgn( ) 1

1

2

| |

1/ 4 1/ 4

γ

γ

γ

γ

γ

. (36) 

The product, U·dV/dz, may therefore be computed as 























 − + −













































+

− + − 















−

+ − − Σ

× − Σ =

2

1/ 4

1/ 4 1/ 4 3

3

1/ 4

1/ 4

1

sgn( 1) 1

( 1)

1

sgn( 1)

1

( 1) 1 sgn( )

8

1

z z

dz

dV U

γ

γ

γ

γ

γ

γ

γ

γ

γ

γ

, (37) 

which is of the form

( ) 2

0 1 2 U ⋅ dV/dz = A a + a /z + a /z . (38)

Substituting this result and the transformed expression for L, given by equation (24), into 

equation (33) allows the integral for penetration to take the form

, (39) ∫ = + + ±

0

( / / ) exp[ ( 1/ )] 2

0 1 2

z

z

b

P P B a a z a z z c z z dz

where the conditional minus sign in the exponential is taken when γ < 1, and ai, b, c, and BP are 

all constants, expressible as 

1/ 4

a0 = γ −1 , (40)

9












































+

− + − 















−

+ = − −

1/ 4 1/ 4 3

3

1

( 1)

1

sgn( 1)

1

( 1) sgn( )

γ

γ

γ

γ

γ

a R Y , (41) 

1/ 4

a2 = sgn(γ −1) γ −1 , (42) 





 



 = − 1

2

1

Y

R b

γ , (43) 

1

1

4

1 − − = YR

c

γ

γ , (44) 

and 

exp[ ( sgn( 1)/ )] 4

1

1 0 0

0

1/ 4

0 c z z

Y z

R B L P b − + − ⋅

− = − γ

γ

γ . (45) 

The form of equation (39) is basically identical to an intermediate step of the original 

solution [3], though with differently defined constants. The prior work [3] opted to transform the 

equation again to eliminate the leading polynomials, but did so at the expense of introducing 

noninteger powers into the exponential term. Then, the penetration equation was solved by 

expanding the exponential into a power series of (A1·zg + A2·z−g

)j terms and expanding each 

(A1·zg + A2·z−g

)j term into j + 1 monomials, using a binomial expansion. The net result of the 

total expansion was that, to include terms out to a power of j = n, a total of (n + 1)(n + 2)/2 

monomials was generated, and then integrated term by term. With n routinely exceeding 20 to 

obtain the desired precision, and approaching 100 for certain initial conditions, the computational 

burden was substantial, though still more efficient than a numerical integration of 

equations (1)−(4). 

While the currently proposed method still relies on a series expansion of the exponential to 

perform the integration, a technique permits a streamlined method for achieving the expansion. 

In particular, a method exists to expand the subject exponential series with the form

∑ , (46) ∞

=−∞

± ± = j

j

j exp[c(z 1/z)] C z

where the Cj

+

 or Cj

−

 coefficients are a function only of the parameter c. In particular, the Cj

−

constants are given by evaluations of Bessel functions of the first kind, such that Cj

−

 = Jj(2c). 

The Cj

+

 constants, by contrast, are given by modified Bessel functions of the first kind, such that

Cj

+

 = Ij(2c). The expansion using the form of equation (46), to include terms of power z±n, 

10
requires the evaluation of only 2n + 1 monomials in z, and therefore represents a significant 

improvement over the method previously employed [3], which required the evaluation of 

(n + 1)(n + 2)/2 monomials in z for identical precision. 

While there is an overhead associated with the evaluation of the Cj

±

 parameters, given by the 

converging series that defines the Bessel functions for integer order, 















± <

≥

+

±

=

±

−

∞

=

+

±

∑

( 1) , 0

, 0

!( )!

( 1)

0

2

C j

j i i j

c

C

j

j

i

i i j

j , (47) 

the parameter c is fixed by the initial conditions (material properties) of the penetration problem. 

As such, the Cj

+

 or Cj

−

 terms may be calculated once at the onset of the analysis, regardless of

how many z values (i.e., velocities) for which the solution needs evaluation. Furthermore, there 

exists a recursive technique for evaluating the Cj

±

 parameters of equation (47), based on the 

recursions 

+

+

+

+

−

+

+

=

j

j j

j

C

C

c

C j

C

1 1

1 , (48a) 

and 

−

−

+

−

−

−

−

=

j

j j

j

C

C

c

C j

C

1 1

1 , (48b) 

which thereby offers further computational savings. 

The integration for penetration is, thus, finally achieved by employing this optimized expansion 

and integrating term by term and evaluating at the desired limits. When b is not an integer, 

which is the typical case, the result may be expressed as 

General case: ( ) 0

0 1 1 2 1

z

z j

j b

P j j j j b

z P B ∑ a C a C a C ∞

=−∞

+

±

+

± ±

− + = + + , (49) 

where the C+

 terms are used when γ > 1 and the C− terms are used when γ < 1. For the case when 

b is an integer, the single term of equation (49) that would otherwise produce a zero in the 

denominator (i.e., the term for which j = −b) originated from a 1/z integration and would actually 

have produced, upon integration, the logarithmic term ln(z), instead of zj+b

/(j+b). 

11
6. Implicit Time 

Though these solutions for L(V) and P(V) bypass the intermediate evaluation of V(t), the 

penetration variables may, if needed, be implicitly expressed in terms of time, by integration of 

L(V), 

∫ ∫ = = −

0

0 0 0

1 V

V

V

V

dV

L

L

V V

dV

t & & , (50)

in order to obtain t(V). As in the case of penetration, a closed-form solution to equation (50) will 

be possible only for the special case of γ = 1 and then only when (R − Y)/Y is an odd integer 

(i.e., R/Y is even). In all other cases, the integration of equation (50) will take the form of a 

series solution. Of the several ways to obtain a series integration of the special case solutions, a 

power-series expansion is preferable to a repeated integration-by-parts solution because it avoids 

an alternating series, for the case when the “c” constant associated with the exp[−c(V0

2

 − V2)] 

term is positive. Such is always the case for penetration problems. Both the R = Y and γ = 1 

special cases can be reduced to an integral of the form

∫ ∑

∞

=

+

+ + =

a

i

i

b b

i i b

ca V cV dV a

0 0

2

2 1

!(2 1)

( ) exp( ) . (51) 

Thus, the special-case solutions for t(V) may be evaluated as 

R = Y: 

































+ + − 















 + +















+

− = ∑ ∑

∞

=

∞

= 0

2

0 0

2

0

2

0 0 0

!(2 1) 2 (1 )

1

!(2 1) 2 (1 )

1

2 (1 ) exp

i

i

R

i

i

R R R

Y

V

V i i

V

Y

V

Y i i

V

Y

L V

t

γ

ρ γ

γ

ρ γ

γ

ρ ρ γ , (52) 

γ = 1: 

































+ 













 − 















+ 



 



− = ∑ ∑

∞

=

∞

= 0

2 /

0 0

2

0

2

0 0 0

!(2 / ) 4

1

!(2 / ) 4

1

4 exp i

i

R

R Y

i

i

R R R

Y

V

V i i R Y

V

Y

V

Y i i R Y

V

Y

L V

t ρ ρ ρ ρ . (53) 

For the general case, a solution is most profitably obtained in a manner analogous to the 

penetration evaluation, in which a transformation to z facilitates a streamlined series solution: 

∫ ∫ = = −

0

0 0 0

1 z

z

V

V

dz

dz

dV

L

L

V V

dV

t & & . (54)

12
This integration may be staged through the substitution of equations (24) and (35), to give the 

following form: 

, (55) ∫ = + ±

0

( / / ) exp[ ( 1/ )] 3 / 2

1

1/ 2

0

z

z

b

t t B d z d z z c z z dz

where the conditional minus sign is taken when γ < 1. Here, b and c are defined as before, by 

equations (43) and (44), while 

1/ 2

0 d = ( γ +1) , (56)

1/ 2

d1 = −sgn[(γ −1)(R −Y)] γ −1 , (57)

and 

exp[ ( sgn( 1)/ )]

1

1

8

1

0 0

0

1/ 2 1/ 4

0 c z z

Y z

R

Y

B L b

R

t − + − −















 = ⋅ − γ

γ

γ

ρ . (58) 

By using a method analogous to that in equations (46)–(49) and with the same definitions for Cj

±

[given by equation (47), where the “+” solution applies for γ > 1, and the “−” solution for γ < 1], 

the expression for t given by equation (55) may be expanded in a series as 

General case: ( ) 0

1 2

1 2

0 1 1

z

z j

j b

t j j j b

z B ∑ d C d C ∞

=−∞

+ −

± ±

− + − t = + . (59) 

Like equation (49), there is one exception to the general validity of this result, specifically for the 

case when b is precisely a half-integer. If and only if this is the case, a single term of

equation (59) will require modification: namely, the term for which j + b − 1/2 exactly equals 

zero, originating from a 1/z integration. This integration would, for this one term only, rightfully 

have produced a ln(z) term, instead of z j+b−1/2

/( j+b−1/2). As with the evaluation of penetration, 

the summation of equation (59) is carried out for j over some finite range from −n to +n so as to 

achieve the desired level of precision. 

7. Terminal Rod Length, etc. 

The “terminal” rod length may be ascertained for the various solution cases [from equations (7), 

(9) or (11)], by setting V to its terminal value, Vx = Σ for the case of R > Y and = − Σ / γ x

= 0

V

for R < Y, with the parameter Σ given by Σ = 2(R − Y)/ρR. When R > Y, this termination 

corresponds to the point where U = 0, when the penetration ceases (though the rod may continue 

to erode thereafter). For R < Y, the termination corresponds to the point where L , when the &

13
rod erosion ceases (though the rod may continue to penetrate as a rigid body thereafter). This 

terminal state, denoted with the subscript “x,” corresponds not to the end of the ballistic event, 

but rather to the time at which the governing equations (1)−(4) cease to apply. In those 

governing equations, developed for the case of a simultaneously eroding rod and target, the 

subscript “x” condition corresponds to the moment at which either the rod or the target stops 

eroding. In general, these two conditions do not occur simultaneously. The rod length 

(normalized) at the terminal state “x” for the various cases is expressible as: 

 R=Y: 

















+

− = 2

0

0 2 (1 ) exp V

L Y

Lx R

γ

ρ γ , (60)

γ = 1: 































 − Σ  − −



 





Σ =





 



 − −

1

| |

1

2

1

exp | |

2

0

1

2

1

2

0

0

V

Y

V R

L

L Y

R

x , (61) 

General case: 

. 2

1 sgn( )

1

( / ) ( / ) sgn( )(1 ) ( / )

exp 1

1

( / ) ( / ) sgn( )(1 )

2

0

2

0 0

1 1

2

0 0

0









































+ Σ − −

Σ Σ + Σ − − Σ − −

×





















+

Σ + Σ + Σ − =





 



 − −

γ

γ γ γ

γ

γ γ γ

γ

V V V

Y

R

V V

L

L

Y

R

x

 (62) 

For cases where R > Y, this terminal length corresponds to that length of rod as of the moment 

that penetration ceases. For R < Y, this is the rod length at the onset of rigid-body penetration. 

Terminal values (at state “x”) for penetration and time may likewise be obtained by evaluating 

the respective relations [equations (30), (32), or (49) for penetration and equations (52), (53), or

(59) for time] with the substitution of V = Vx [and L = Lx in the case of equation (30)]. Their 

presentation is omitted, however, because these relations are summations and not in closed form

like those for residual length previously given. As such, there is little clarity of reduction gained 

in restating these earlier equations with the V = Vx substitution in place. 

8. Residual Erosion/Penetration Behaviors 

Equations (1) and (2) are valid only while there is simultaneous target penetration and rod 

erosion. Except for the special case of R = Y, and U will not simultaneously approach zero. 

In the general case then, the physical event will continue with either residual rod erosion 

L&

14
following the cessation of penetration (when R > Y) or residual rigid body penetration following 

the cessation of rod erosion (when R < Y). These afterflow events are amenable to closed-form

analytical solution. Continuing to denote the state at this transition point (the moment of

transition to either rigid target or rigid rod) with the use of the subscript “x,” the absolute final 

state, when the rod velocity itself finally reaches zero, will be denoted with the subscript “f.” 

Recall that Vx = Σ when R > Y, while = − Σ / γ x

= −L&

VV Y R & ( / 2 = − ρ

V when R < Y, where Σ = 2(R − Y)/ρR. 





− ( 2 exp 2 V

Y x

R

x

ρ

8.1 Residual Rod Erosion 

For the case of R > Y, the target becomes rigid while rod erosion continues. To deal with this, 

equation (2) is discarded and is substituted with the constraint U = 0. The kinematic constraint 

of equation (3) becomes, as a result, V . Equation (1) remains valid for the eroding-rod 

case. Solving equation (1) for L, differentiating, and substituting the revised kinematic 

constraint to eliminate L , one obtains as the governing equation &

V . (63) && )

This may be integrated to obtain V in terms of V, whereupon equation (1) may be used to 

eliminate V in favor of L. The result (as a function of V) is that 

&

&





 = − )

2 L L V . (64)

Evaluating the penetration and rod length at the final state (where V = 0), one obtains Pf = Px

and 





 



− = Y

V L L R x

f x 2 exp

2 ρ . (65)

Because of the similarity between the governing equation here, equation (63), and the special 

case R = Y governing equation, equation (6), the duration of this residual-erosion phase of the rod 

may likewise be calculated with the same series-solution form used to calculate event duration 

for the special cases, described by equation (51). Use of this form leads to 

































+ − 















+ 













 − − = ∑ ∑

∞

=

∞

= 0

2

0

2 2

!(2 1) 2

1

!(2 1) 2

1

2 exp

i

i

R

i x

i

R x x R x R x

x Y

V

V i i

V

Y

V

Y i i

V

Y

L V

t t ρ ρ ρ ρ , (66) 

which, as V approaches zero, becomes the following result: 

∑

∞

= 















+ 













 − − = 0

2 2

!(2 1) 2

1

2 exp i

i

R x x R x R x

f x Y

V

Y i i

V

Y

L V

t t ρ ρ ρ . (67) 

15
8.2 Residual Rigid-Body Penetration 

For the alternate case of R < Y, a state of rigid-body penetration is reached after the rod erosion 

ceases. As before, equation (2) is discarded and is substituted with the constraint . The 

kinematic constraint (3) becomes, as a result, V = U. However, there is one additional 

modification required to the governing equations. In particular, the force causing the rod 

deceleration in equation (1) is no longer Y, since the rod is no longer in a plastic state. Rather, it 

is a diminished stress state applied by the pressure head and resistance of the target, 

. But since, kinematically, V = U and L remains fixed at , the rod deceleration 

equation becomes 

L = 0 &

TU + R 2 1/2ρ Lx

. (68) LxV ρ TV R ρ R (1/2 )/ 2 & = − +

There is no algebraic relation between P and V analogous to that which equation (1) affords 

between L and V . Therefore, this equation will be solved by separating the variables V and t, as 

follows: 

&

&

dt

R V

L dV

T

x = − + ⋅ 2 2 /

2

γ ρ . (69)

This may be solved as 































 = = − + −

R

t t V R

L

R V U T

x x

T x T 2 ( ) tan

2

tan

2 1 ρ

ρ

γ

ρ

. (70) 

The final time, at which the velocity drops to zero, is found to be 















 = + −

R

V

R

L

t t T

x

x T

f x 2

tan

2ρ 1 ρ

γ . (71)

The expression for U, which is equation (70), may be integrated one more time to obtain the 

differential penetration that occurs during the afterflow phase. One obtains 































  −



 



 − = − −

R

t t V R

L

L

P P T

f x

x T

x

x 2 ( ) log costan

2 log cos

2 1 ρ

ρ

γ

γ

. (72) 

When evaluated at t = t , and employing some trigonometric substitutions, the final result is that

L

f

f = Lx and the afterflow penetration is 















 − = +

R

L V P P x T x

f x 2 log 1

2 ρ

γ . (73)

16
9. Conclusions 

This report presents updated results related to the exact solution of the long-rod penetration 

equations, formulated by Alekseevskii [1] and Tate [2], and first solved by Walters and 

Segletes [3]. While the original solution [3] is accurate and comprehensive, there have been a 

number of improvements or enhancements, both to the presentation and the solution approach. 

Equation (5) is a concise analytical presentation of rod length as a function of rod velocity, valid 

for both special and general cases, providing an enhanced sense for the terms that drive the 

analytical solution. Equations (6)−(11) compare and contrast the special- and general-case 

analytical solutions, while equations (12) and (13) present the result in terms of an alternate 

model variable. The key independent variable transformation (to z), unexplained but 

indispensable to the original solution, is herein developed more fully and much of its mystery is 

thereby uncloaked. Further, its expression is slightly altered from the original solution, resulting, 

by comparison, in a form amenable to a highly streamlined series solution for penetration P(z), 

as equation (49), or implicit time, t(z), as equation (59). 

Not only are results derived to the point where the penetration equations cease validity, but 

extensions to the original solution are presented, which account for the period of rigid-body 

penetration or rigid-target rod erosion that follows the period of eroding-body penetration 

addressed by the original penetration equations. 

While not taking anything from the original solution of Walters and Segletes [3], the current

work offers enhanced appreciation and understanding of the original effort, as well as extensions 

to the original work. Finally, the streamlined techniques presented herein make any 

implementation of the solution significantly more efficient than the originally offered solution 

technique. 

17
10. References 

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

Explosion and Shock Waves, vol. 2, pp. 63−66, 1966. 

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

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4 INSTITUTE OF CHEMICAL PHYSICS 1 RESEARCH INSTITUTE OF MECHANICS 

RUSSIAN ACADEMY OF SCIENCES NIZHNIY NOVGOROD STATE 

G I KANEL UNIVERSITY 

A M MOLODETS A SADYRIN

S V RAZORENOV P R GAYARINA 23 KORP 6 

A V UTKIN NIZHNIY NOVGOROD 603600

 142432 CHERNOGOLOVKA RUSSIAN REPUBLIC 

 MOSCOW REGION

 RUSSIAN REPUBLIC 2 RUSSIAN FEDERAL NUCLEAR 

 CENTER VNIIEF 

L F GUDARENKO 

R F TRUNIN 

MIRA AVE 37 

 SAROV 607190

 RUSSIAN REPUBLIC 

12 
NO. OF NO. OF 

COPIES ORGANIZATION COPIES ORGANIZATION

1 SAMARA STATE AEROSPACE UNIV 2 K&W THUN 

L G LUKASHEV W LANZ 

 SAMARA W ODERMATT 

 RUSSIAN REPUBLIC ALLMENDSSTRASSE 86 

 CH-3602 THUN 

1 TOMSK BRANCH OF THE INSTITUTE SWITZERLAND 

FOR STRUCTURAL MACROKINETICS 

 V GORELSKI 2 AWE 

8 LENIN SQ GSP 18 M GERMAN 

 TOMSK 634050 W HARRISON 

 RUSSIAN REPUBLIC FOULNESS ESSEX SS3 9XE

 UNITED KINGDOM 

1 UNIVERSIDAD DE CANTABRIA 

FACULTAD DE CIENCIAS 1 CENTURY DYNAMICS LTD 

 DEPARTMENTO DE FISICA APLICADA N FRANCIS 

 J AMOROS DYNAMICS HOUSE 

AVDA DE LOS CASTROS SN HURST RD 

 39005 SANTANDER HORSHAM 

 SPAIN WEST SUSSEX RH12 2DT

 UNITED KINGDOM 

1 CARLOS III UNIV OF MADRID 

 C NAVARRO 5 DERA 

 ESCUELA POLITEENICA I CULLIS

 SUPERIOR J P CURTIS Q13 

C BUTARQUE 15 A HART Q13 

28911 LEGANES MADRID K COWAN Q13 

 SPAIN M FIRTH R31 

 FORT HALSTEAD 

1 DYNAMEC RESEARCH AB SEVENOAKS KENT TN14 7BP 

 A PERSSON UNITED KINGDOM 

PO BOX 201 

S 151 23 SODERTALJE 6 DEFENCE RESEARCH AGENCY 

 SWEDEN W A J CARSON 

 I CROUCH 

 7 FOI C FREW 

SWEDISH DEFENCE RESEARCH T HAWKINS 

 AGENCY B JAMES 

GRINDSJON RESEARCH CENTRE B SHRUBSALL

L GUNNAR OLSSON CHOBHAM LANE CHERTSEY

 B JANZON SURREY KT16 0EE 

 G WIJK UNITED KINGDOM 

 R HOLMLIN 

 C LAMNEVIK 1 UK MINISTRY OF DEFENCE 

 L FAST G J CAMBRAY 

 M JACOB CBDE PORTON DOWN SALISBURY 

SE 147 25 TUMBA WITTSHIRE SPR 0JQ 

 SWEDEN UNITED KINGDOM 

2 SWEDISH DEFENCE RSCH ESTAB 

DIVISION OF MATERIALS 

S J SAVAGE 

 J ERIKSON 

 STOCKHOLM S17290 

 SWEDEN 

13 
NO. OF 

COPIES ORGANIZATION 

7 INSTITUTE FOR PROBLEMS IN 

 MATERIALS SCIENCE 

 S FIRSTOV 

 B GALANOV 

 O GRIGORIEV

 V KARTUZOV

 V KOVTUN 

 Y MILMAN 

 V TREFILOV 

3 KRHYZHANOVSKY STR 

252142 KIEV 142 

 UKRAINE 

1 INSTITUTE FOR PROBLEMS OF 

 STRENGTH 

 G STEPANOV 

TIMIRYAZEVSKAYU STR 2 

 252014 KIEV 

 UKRAINE 

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