Wave Dispersion in Cylindrical Tubes: Applications to 

Hopkinson Pressure Bar Experimental Techniques 

by Libo Ren, Mike Larson, Bazle A. Gama, and 

John W. Gillespie, Jr. 

ARL-CR-551 September 2004 

prepared by 

Center for Composite Materials 

University of Delaware 

Newark, DE 19716 

under contract 

DAAD19-01-2-0005 

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

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ARL-CR-551 September 2004 

Wave Dispersion in Cylindrical Tubes: Applications to 

Hopkinson Pressure Bar Experimental Techniques 

Libo Ren, Mike Larson, Bazle A. Gama, and 

John W. Gillespie, Jr. 

Center for Composite Materials 

University of Delaware 

prepared by 

Center for Composite Materials 

University of Delaware 

Newark, DE 19716 

under contract 

DAAD19-01-2-0005 

Approved for public release; distribution is unlimited.
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Wave Dispersion in Cylindrical Tubes: Applications to Hopkinson Pressure Bar 

Experimental Techniques 

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6. AUTHOR(S) 

Libo Ren,* Mike Larson,* Bazle A. Gama,* and John W. Gillespie, Jr.* 

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University of Delaware 

Center for Composite Materials 

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*University of Delaware, Center for Composite Materials, Newark, DE 19716 

14. ABSTRACT 

It is well known that harmonic waves with higher frequencies travel slower than those with lower frequencies and that they are 

known as “wave dispersion” in finite diameter rods. From the frequency equation, the phase speed can be determined if the 

wavelength is known. A dispersion correction methodology uses the frequency equation to disperse a waveform to a specific 

location of interest. Dispersion correction methodologies are generally used in split-Hopkinson pressure bar (SHPB) techniques

to reduce experimental data accurately. This report investigates wave propagation and wave dispersion in cylindrical tubes. 

Based on the work of Mirsky and Herrmann (M-H), the phase speed can be solved for wave motion along a cylindrical tube with 

a specific thickness-to-radius ratio. A numerical algorithm is developed to solve the M-H model and is compared with the 

solutions obtained from a three-dimensional finite element model. It is found that the first mode of the M-H solution gives the 

correct phase speed for wave motion in a tube. A modification to the traditional inverse fast Fourier transform algorithm is 

proposed for better prediction of the dispersed signal. The effects of the tube dimensions and the accuracy of dispersion 

correction in SHPB experiments are also discussed. 

15. SUBJECT TERMS

wave dispersion, split-Hopkinson pressure bar experiment, cylindrical tubes 

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Libo Ren 

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iii 

Contents 

List of Figures iv

Acknowledgments v

1. Introduction 1

2. Solution of M-H Model 3

3. Modification of Traditional Dispersion Algorithm 7

3.1 Solution of Phase Speed ....................................................................................................7 

3.2 Reconstruction of Waveform ............................................................................................7 

4. Finite Element (FE) Analysis 9

5. Results and Discussion 11

6. Conclusions 16

7. References 17

Distribution List 19 
iv 

List of Figures 

Figure 1. Schematic diagram of SHPB setup..................................................................................1 

Figure 2. Solution of the frequency by 3-D theory and shell theory for various m values.............5 

Figure 3. Solution of the frequency equation by shell theory for various m = h/R........................6 

Figure 4. k c c / c when δ = h / Λ approaches infinity, cc = E / ρ . ....................................................6 

Figure 5. Explanation of new dispersion algorithm for SHPB. ......................................................8 

Figure 6. FE model of a cylindrical tube. .....................................................................................10 

Figure 7. Definition of stress wave applied on the end surface of the cylindrical tube................10 

Figure 8. Strain vs. time at the mid-length of the tube. ................................................................11 

Figure 9. Axial strain predicted by M-H model and FE analysis for the tube with 

m = 0.548. ................................................................................................................................12 

Figure 10. Axial strain predicted by M-H model and FE analysis for the tube with 

m = 0.667. ................................................................................................................................13 

Figure 11. Comparison of new and traditional dispersion algorithms..........................................14 

Figure 12. Dispersed wave in rod determined by the new algorithm and traditional 

algorithm. Material Poisson’s ratio = 0.29. ............................................................................14 

Figure 13. Stress history of all elements at z = 1.00 m along X = 0 axis. ....................................16 
v 

Acknowledgments 

This report is prepared through participation in the Composite Materials Technology 

Collaborative Program sponsored by the U.S. Army Research Laboratory under Cooperative 

Agreement DAAD09-01-2-0005. Libo Ren would like to thank Dr. J. R. Xiao and Dr. Y. P. 

Duan at the University of Delaware-Center for Composite Materials for helpful discussions on 

wave propagation theory.
vi 

INTENTIONALLY LEFT BLANK.
1 

1. Introduction 

The split-Hopkinson pressure bar (SHPB) experimental technique has become a standard method 

in determining the dynamic mechanical properties of materials. The Hopkinson pressure bar 

technique was originally developed by B. Hopkinson (1) in 1914 to measure the pressure 

generated by the detonation of explosives. In 1948, R. M. Davies (2) conducted a critical study 

of the Hopkinson pressure bar technique. He developed the axial and radial strain measurement 

system and analyzed the wave dispersion effects. In 1949, H. Kolsky (3) invented the SHPB 

technique. In his work, he used the same system as Davies’ to measure the strain. At the same 

time, he developed a one-dimensional (1-D) model to perform experimental data analysis. The 

work of Hopkinson, Davies, and Kolsky established the foundation for the development of the 

SHPB technique. Figure 1 shows a classic SHPB test setup, which consists of one striker bar 

(SB), one incident bar (IB) and one transmission bar (TB), equal in diameter and usually made 

from the same material. Strain gages are mounted on the surfaces of IB (SG1) and TB (SG2) to 

record the strain. The specimen (SP) is sandwiched between the IB and the TB. The SB, 

powered by pressured gas, will hit the IB. A compressive stress wave will travel along the IB 

and reach the IB-specimen (IB-S) interface. At this interface, one part of it will be reflected back 

into IB, and the rest will be transmitted to the TB through the specimen-transmission bar (S-TB) 

interface. The strain gage SG1 will record two waves: incident wave and reflected wave and the 

strain gage SG2 will record only the transmitted wave. Data reduction on these three waves will 

provide the stress-strain response of the specimen material under dynamic loading. 

Figure 1. Schematic diagram of SHPB setup. 

The classical SHPB data analysis is based on 1-D wave propagation theory in long rods. Several 

assumptions have been made for the 1-D theory. Two important assumptions are as follows: 

(1) stress wave propagation in the bars is 1-D, and (2) the bars are free of dispersion effects. 

Actually, these two assumptions are not satisfied for finite diameter bars. Based on such 

assumptions, the strain data measured at the middle point of the bars are directly used as the 

strain at the interfaces to reduce the data. Strictly speaking, wave propagation in cylindrical bars 

is three-dimensional (3-D) in nature: both axial and radial stresses, and strains are generated 

when the stress wave travel along the bar. Such variables are functions of radial dimension, so 

they are not uniform across the cross section. The wave dispersion effect exists, which means 

Striker Bar 

(SB)

Incident Bar 

(IB)

Transmission Bar 

(TB)

Specimen

(SP)

IB-S S-TB

SG1 SG2

Striker Bar 

(SB)

Incident Bar 

(IB)

Transmission Bar 

(TB)

Specimen

(SP)

IB-S S-TB

SG1 SG2
2 

that the harmonic wave with higher frequencies travels more slowly than the wave with lower 

frequencies. Due to this effect, the wave shape keeps changing as it travels along the axial 

direction. This effect should be considered for accurate data reduction. 

Many authors have investigated the wave dispersion effect in order to modify the classical 1-D 

data analysis algorithm. Their study is based on the work of Pochhammer (4) and Chree (5), 

who independently solved the problem of wave propagation in an infinite cylindrical rod with 

finite diameter and derived the frequency equation. This frequency equation establishes the 

relationship between phase velocities and wavelengths of harmonic waves. Later Davies (2) and 

Bancroft (6) solved the equation numerically for some limiting cases. With the help of modern 

computers, the frequency equation can be easily solved over the complete range of geometric 

and material parameters. Follansbee and Frantz (7) and Gong et al. (8) investigated the effect of 

wave dispersion correction for the SHPB test and found that significant improvement on the 

stress-strain data can be achieved by wave dispersion correction. During dispersion correction, 

Fourier transformation (FT) is used to decompose the measured strain data into its harmonic 

waves. The dispersion correction methodology is then applied for each component frequencies. 

Finally, inverse FT is performed to get the dispersed wave at the IB-S and S-TB interfaces. 

Other dispersion correction methods were also suggested, e.g., in Li and Lambros (9). Currently 

wave dispersion correction has become a standard procedure in SHPB data analysis. 

In this report, the wave dispersion in cylindrical tubes for the SHPB application is investigated. 

This study has become important and indispensable because recently cylindrical tubes have been 

used for SHPB experiments. For example, Dowling et al. (10) performed a dynamic punch shear 

test to study the mechanical properties of metals. This technique uses a transmission tube in 

place of the TB, such that the IB can shear a cylindrical disc through the specimen. This test 

method is still being extensively used today. Another example is determining the mechanical 

properties of low-strength, low-impedance materials under dynamic loading. If solid TBs are 

used for the SHPB test, the noise level is so high that proper interpretation of the transmission 

signal becomes impossible. To solve this problem, some researchers have used visco-elastic bars 

e.g., see Zhao et al. (11). The disadvantage of using visco-elastic bars is that additional 

assumptions have to be made about the visco-elastic material properties, which may not 

represent the true characteristics of the bar materials at high strain rates. Furthermore, the 

material properties are functions of temperature, moisture, and aging conditions, which are 

difficult to control. In addition, the dispersion correction algorithm now becomes very extensive 

and time consuming due to the visco-elastic nature of the bar materials. Other researchers have 

put forward the idea of using a hollow cylindrical tube for the transmission part of the SHPB test. 

Recently, Chen et al. (12) used IB and transmission tube made of high-strength Al alloy to 

measure the dynamic mechanical properties of silicon (Si) rubber materials. In this way, they 

obtained higher amplitude transmission signal that was achievable using a solid bar. The 

problem is that they did not perform any dispersion correction on the incident and transmission 

signals, which were collected from midpoints of the bar/tube. They used pulse shaper to filter 
3 

the high-frequency components in the waveform and then claimed that dispersion correction was 

not necessary. This argument is not correct because the same argument can be applied to the 

traditional SHPB test, which means if pulse shaper is used, dispersion correction is not necessary 

any longer. It is not clear how to define the lower limit of high-frequency wave components 

without a dispersion correction methodology. The use of a pulse shaper can improve the result 

but cannot eliminate the necessity of dispersion correction. It is still necessary to investigate the 

wave propagation in a cylindrical tube and find the appropriate dispersion relations, and is the 

objective of the present study. 

Wave propagation in infinitely long tubes has been discussed by several authors, as early as in 

the 1920s. Ghosh (13) studied the longitudinal motions of a hollow cylinder with 3-D theory. In 

1952, Fay (14) investigated this problem again and derived the frequency equation. But there 

was an error in his equation, which was pointed out by Herrmann and Mirsky (15). The work of 

Mirsky and Herrmann (M-H model) is considered the most comprehensive treatment (15, 16). 

They used both 3-D theory and shell theory to study the wave motion in cylindrical tube. Using 

3-D theory, they derived the exact frequency equation, and their shell theory considered 

membrane effect, bending effect, transverse shear deformation, and rotary inertia. They claimed 

that the results of 3-D theory and shell theory were very close. 

This study is based on the work of Mirsky and Herrmann (16). First, the two frequency 

equations will be solved again to make sure that the shell theory really gives a solution to phase 

speed, which is accurate enough. Then the results of M-H shell theory will be applied to the 

SHPB test. It will be shown that the first mode solution is correct for the SHPB application. A 

modified dispersion correction algorithm will be suggested to make accurate prediction of strain 

at the IB-S and at the specimen-transmission tube (S-TT) interfaces. Finally, the effects of the 

tube geometry on the accuracy of data reduction will be discussed. 

2. Solution of M-H Model 

As mentioned earlier, Herrmann and Mirsky (15, 16) derived the exact frequency equation for 

cylindrical tubes by solving the 3-D problem and one approximate frequency equation through 

shell theory. The 3-D theory gave the exact frequency equation in cylindrical coordinates 

( ) r,θ, z (15): 

F (c,h / ) [K ( )K ( ) K ( )K ( ) (8/( ab)) 2

3D Λ = 10 β 01 γ + 01 β 10 γ + π βγ 

FK ( )K ( ) (1/ F)K ( )K ( )] 11 00 11 00 + γ β + β γ + [(1 B) /(F ab)]K ( )K ( ) 11 11

2 2 + γ β γ 

[(1 B)/( ab)][aK ( )K ( ) bK ( )K ( )] − + γ 11 γ 10 β + 11 γ 01 β 

−[(1+ B)/(Fγab)][aK11 (β)K10 (γ) + bK11 (β)K01 (γ)] = 0 , (1) 
4 

where 

K mn (z) = J m (zb)Yn (za) − J n (za)Ym (zb),cc = (λ + 2μ)/ ρcs = μ / ρ 

and 

βγ

α β = α − γ = α − = − =

2 2

2

s

2

2

s

2

2 2

2

c

2

2 2 B 1), F

2c

c 1),B (

c

c 1), (

c

c

( . (2) 

In equation 2, αis the wave number 2π Λ and m m J ,Y are the Bessel functions, a and b are the 

inner and outer radii of the tube, λ and μare Lame’s constants, ρ is tube material density. 

c is the phase speed to be solved, h = b − a is the thickness of the tube wall and Λ is the wave 

length. Mirsky and Herrmann also derived one approximate frequency equation based on shell 

theory (16). This equation is very lengthy, so it is not listed here. Basically, this equation is in 

following format: 

( ) 6

5

2

4

4

3

6

2

2 2 2 2 2 8

shell 1 F s, δ = A (m)(s − M) (s − k ) δ +[A (m)s + A (m)s + A (m)s + A (m)]δ 

[A (m)s A (m)s A (m)] A (m)[s A (m)] 0 2

10

2

9

4

8

2

7

4 + 6 + + δ − − δ = . (3) 

where s s = c / c , δ = h /Λ , A1 , , , A and M are constants which are 10 functions of tube geometry 

parameter m = h / R (16), ν is the material Poisson’s ratio, k is the shear coefficient in M-H 

shell model which is related to Poisson’s ratio ν (15). Here R = (a + b)/ 2 is the mean tube 

radius. From equation 3, it can be seen that phase speed is a function of both m and δ . 

Equation 1 is transcendental in nature but can be solved with the help of mathematical software 

such as MAPLE or MATHEMATICA. From equation 2, it can be seen that β and/or γ could 

become imaginary. When s c < c and c c < c , both β and γ are imaginary. When s c c < c < c , 

γ is real but β is imaginary. Wherever one variable becomes imaginary, Bessel function 

m K m I , must be used in place of m Ym J , . Such replacement guarantees that the imaginary 

number i will be eliminated from the equation so that the real roots can be obtained. 

Equation 3 can be solved easily using the Newton-Raphson method. There are four real 

solutions. When the wavelength is very long, δ approaches zero and the first and second 

solutions of s approach the shear coefficient k . This coefficient k leads numerically to the 

same phase speed as the limiting solution for wave speed in cylindrical rods when the 

wavelength approaches infinity. So it is interesting to see that M-H shell model explains the 

physical meaning of this limiting solution. 

The solutions of equations 1, 2, and 3 for various tube geometries are shown in figure 2. There 

are infinite solutions for the frequency equations. Here, only the first-mode solutions are shown. 

The following material constants are assumed: E = 206.09 GPa , 3 ρ = 7850 kg /m , and 

Poisson’s ratio ν = 0.30. It can be seen that the two models provide very close solutions. The 

maximum difference occurs at the “valley” of the curves. As an example, for m = 0.250 , 3-D 
5 

Figure 2. Solution of the frequency by 3-D theory and shell theory for 

various m values. 

theory gives c 0.5720, shell theory gives / cs = c 0.5916, and the error is / cs = ~3%. This error is 

0.54%, 0.29%, 0.21% for m = 0.548 , m = 0.667 (2/3) and m =1.000 , respectively. The error 

decreases when m increases. Figure 3 shows the solutions by shell theory for various m values. 

When m = 0.033 (1/30), the phase speed decreases very quickly over the range of low δ . With 

increasing m, the phase speed drops more slowly with δ . There exists one minimum (valley) 

point for all the curves, after which speed starts to increase. It is obvious that all the curves 

approach to one limiting speed when δ goes to infinity. For the material properties used here, 

this limiting phase speed is 0.927 s c . 

One important observation from phase speed solutions is that the general claim “wave 

components with lower frequencies travel faster than wave components with higher 

frequencies,” which is always true for the cylindrical rod, does not always hold true for the 

cylindrical tubes. For small m values, the phase speed first drops then increases. When m 

becomes higher, the claim may hold for the whole range of δ . 

On the other hand, in this study, it will be shown that the phase speed will not fall into the range 

where phase speed increases with δ . When we discuss the results later, for simplicity, we still 

use the claim that the lower frequency wave moves faster. 

Bancroft (6) solved the frequency equation for the cylindrical rod. He gave the value of k c c / c

when δ approaches infinity. Herrmann and Mirsky (15) also gave these values. The 

comparison is shown in figure 4. It is clear that these two sets of solutions are exactly the same, 

which also verifies the correctness of M-H models.

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

M-H Shell theory

3-D theory

m=0.548

m=0.250

m=0.667

m=1.000

δ=h/Λ

c/cs
6 

Figure 3. Solution of the frequency equation by shell theory for various 

m = h/R.

Figure 4. k c c / c when δ = h / Λ approaches infinity, cc = E / ρ . 

Now it is proven that the 3-D theory and the shell theory give solutions that can be considered 

the same. Shell theory solutions will be used for further investigation. The advantage of using 

the shell theory model is that the frequency equation of the shell model is not transcendental, so 

it is straightforward to include the model into a computer code. 

Mirsky and Herrmann discussed the first-mode solution thoroughly. But they were not sure if 

wave motion in tubes always follows the first-mode solution, and they did not mention the 

0

0.4

0.8

1.2

1.6

2.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Cylindrical Bar

k=0.927

2√(1+ν)

m=0.548

m=0.250

m=0.033

m=0.667

m=1.000

δ=h/Λ

c/cs

First Mode

0.54

0.55

0.56

0.57

0.58

0.59

0.60

0.20 0.25 0.30 0.35 0.40 0.45 0.50

M-H Shell Theory

Bancroft Solution

Poisson's Ratio, ν

ck/cc
7 

application of their models to the SHPB experiment either. So it is not clear if the first-mode 

solution can be used to determine the phase speed in the SHPB experiments. Follansbee et al. 

(7) proved that the solid cylindrical bars in the SHPB experiment vibrate in the first mode, so it 

may also be true for the hollow cylindrical tubes. To prove this assumption, 3-D finite element 

analysis is performed to simulate the wave motion in the tubes and is compared with the shell 

theory for the first mode. 

3. Modification of Traditional Dispersion Algorithm 

3.1 Solution of Phase Speed 

In predicting the dispersed waveform by M-H shell model, the first step is to determine the phase 

speed of every harmonic wave component. Another equation is required in addition to the 

c / cs ~ h /Λ relationship shown in figure 3: 

2π(c / cs )⋅(h /Λ) = ( ) 0 s kω h / c , (4) 

which is equivalent to 

c(k) = f(k) Λ = (kω0 / 2π)⋅Λ . (5) 

Here, 2 k = 1,,,K = N / and N are the numbers of strain data collected by the strain gages. Phase 

speed c(k) can be solved by finding the intersection of the two curves represented by equations 

3–5 numerically. Felice (17) performed nonlinear fitting on the solution of frequency equation 

for cylindrical rods to obtain one explicit relationship between s c and / c h /Λ . Then it was easy 

to use this relationship and equation 4–5 together to determine the phase speed. For cylindrical 

tubes, this fitting method does not work because the shape of the c / cs ~ h /Λ curves changes 

significantly with m as shown in figure 3. On the other hand, the nonlinear fitting is not 

necessary. This is because there is one implicit functional relationship between s c and / c h /Λ

in equation 3. Giving one h /Λ value, the first-mode solution s c can be determined / c

numerically from equation 3. With the help of this observation, the phase speed of wave 

components can be determined using equations 3 and 4. 

3.2 Reconstruction of Waveform 

A modification is suggested in this study on dispersion correction. In traditional dispersion 

correction algorithm, the following equation is used to reconstruct the waveform: 

N/2

0

k 0k 0

k 1

A f (t) [A cos(k t ) B sin(k t )] 2 =

= + ω+ ω ∑ ' ' , (6) 

with 
8 

k 0

1 1

tt z

c c

⎛ ⎞ = −∆ − ⎜ ⎟ ⎝ ⎠

' . (7) 

Here, 0 Ak A , and , Bk are the coefficients of harmonic wave components determined by Fourier 

transform (FT). ∆z is the distance between the strain gage and the destination position. k c is 

the phase speed of the wave with frequency kω0 . This modification is explained in figure 5. 

Suppose there are two sinusoidal waves sin(t) and sin(2t) traveling together in a tube. At 

location A, the two waves are in phase. Now let them travel forward to location B. Because the 

wave with higher frequency travels slower than the wave with lower frequency, wave sin(t) will 

arrive at location B earlier (at time T1) than sin(2t) (at time T2). From time T1 to time T2, the 

contribution from wave sin(2t) to stress and strain at location B should be zero because it has 

not arrived yet. In the traditional algorithm, the stress and strain response is taken to be 

sin(2(t- LAB /c2+LAB/c0)), where LAB is the distance between location A and location B, c2 is the 

phase speed for sin(2t), and c0 is the fundamental speed. 

Figure 5. Explanation of new dispersion algorithm for SHPB. 

Based on this observation, the algorithm for inverse fast Fourier transform (FFT) in SHPB 

application should be modified. Suppose the initial time the signal arrives at one certain position 

is t = 0 . At time t = tx , the signal should be reconstructed as in the following pseudocode: 

Signal: =A /2.0; 0

For k =1 to N/2 

 If (tx<( z ∆ / k c - z ∆ / 0 c )) 

-2

-1

0

1

2

0 5 10 15 20

T2 T1

A B

Time

Signal
9 

 Do nothing; 

 Else 

 Signal⇐Signal+ A cos( k kω0 (tx-∆z/ k c +∆z/ 0 c ))+B sin( k 0 kω (tx-∆z/ k c +∆z/ 0 c ));

The higher frequency wave travels slower than the lower frequency wave. The lower frequency 

wave reaches location B at time T1. No response should be generated by the higher frequency 

wave at location B until after time T2. 

After the coefficients A , A , and 0 k Bk are determined, this modified dispersion algorithm is 

applied to each harmonic component in reconstructing the waveform. 

4. Finite Element (FE) Analysis 

FE analysis was performed to check the prediction of the M-H shell model. Stress wave 

propagation in two cylindrical tubes was simulated. These tubes have m = h/R values of 

0.548–0.667. The dimensions of the tube with m = 0.548 are as follows: inner radius 

a = 7.24 mm, outer radius b = 12.7 mm, and the length is L 1.524 m. Tubes with such =

dimensions are commercially available and suitable for Hopkinson bar applications. The 

dimensions of the tube with m = 0.667 are as follows: inner radius a = 5.46 mm, outer radius 

b = 10.92 mm, and the length is L 1.524 m. Both tubes have a wall thickness of 5.46 mm. =

The tube material is modeled as steel with Young’s modulus E = 206.09 GPa, Poisson’s ratio 

ν = 0.3 , and density ρ = 7850 Kg/m3. A trapezoidal stress pulse is applied at one end of the 

tube, and the wave propagation in the tube is monitored. The other end is assumed to be traction 

free, so the stress wave will reflect from that end. Due to the symmetric nature of this problem, 

only one quarter of the tube is analyzed. 

Commercial FE analysis code ABAQUS/Explicit*

 was used to perform the analysis. Element 

type C3D8R, which is 8-node brick element with reduced integration, was used to generate the 

mesh. The numbers of elements along the radial direction, tangential direction, and axial 

direction are 5, 20, and 600, respectively, with a total of 60,000 elements. These elements are 

uniformly distributed along the axial direction. Figure 6 shows the model and the mesh 

generation. The boundary conditions are set as follows: zero x displacement along edge X = 0 ; 

and zero y displacement along edge Y = 0 . Figure 7 defines the trapezoidal stress pulse applied 

on the end face Z = 0 . The stress history is defined with several time and stress parameters as 

described in figure 7. The values used in the present analyses are as follows: t A = 0 μs, 

t B = 25 μs, t 35 C = μs, t D = 105 μs, t E = 115 μs, t F = 140 μs and P 500 MPa. max =

 *

 ABAQUS/Explicit is a registered trademark of Abaqus Inc. 
10 

(a) One Quarter Model (b) Mesh Generation 

Figure 6. FE model of a cylindrical tube. 

Figure 7. Definition of stress wave applied on the end surface of the 

cylindrical tube. 

Figure 8 gives the strain history at the mid-length of tube for a time period up to 700 μs. The 

results of FE analyses are compared with M-H shell theory and are presented in the next section. 

Time, t, μs

Pmax

Pressure

At Bt Ct Dt Et Ft

Time, t, μs

Pmax

Pressure

At Bt Ct Dt Et Ft
11 

Figure 8. Strain vs. time at the mid-length of the tube. 

5. Results and Discussion 

The axial strain predictions by both M-H model and FE analysis for the two cylindrical tubes at 

z / L = 0.50–0.75 are presented in figures 9 and 10. When the M-H model is used, an implicit 

assumption is made that the wave propagation in the tubes is 1-D, and strain is determined from 

stress by Hooke’s law ε ZZ = σ ZZ / E . Plots in the left column show the complete strain history, 

and the closer views in the peak region are listed in the right column. From the four plots in 

figures 9 and 10, it is obvious that the predictions by the M-H model and FEM analysis are very 

close. The rising parts of the waves are almost overlapping. For the peak part, the Pochhammer 

modes match very well. The only difference is that for the M-H model, the beginning part of the 

waves is abnormal, which will be explained later. The results in figures 9 and 10 prove that the 

cylindrical tubes in the SHPB experiment also vibrate in the first mode. 

Figure 11 compares the wave dispersion results by the traditional algorithm, and the new 

algorithm used in this study. Here the strain data are plotted in the opposite sign. The results by 

the traditional algorithm are not smooth, and discontinuities are seen over the peak part of the 

waves. On the contrary, the modified algorithm predicts fairly smooth waveform. From the 

smooth waves, the number of Pochhammer modes can be clearly counted. If the old algorithm is 

compared to FE results, it is nearly impossible to reach the conclusion that wave motion in tubes 

follows the first-mode solution of M-H model. The comparison clearly verifies the correctness 

and necessity of the new algorithm. 

-0.0030

-0.0025

-0.0020

-0.0015

-0.0010

-0.0005

0

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0 100 200 300 400 500 600 700

Reflected Wave

Incident Wave

Original Wave

L = 1.524m

Inner Radius a = 7.24mm

Outer Radius b = 12.7mm

Time, t, μs

Axial Strain, εZZ
12 

-0.0030

-0.0025

-0.0020

-0.0015

-0.0010

-0.0005

0

0.0005

0.0010

100 150 200 250 30

M-H Shell Model 

FE Analysis

Time, t, μs

Axial Strain, εZZ

-0.0030

-0.0025

-0.0020

-0.0015

170 195 220 245 270

M-H Shell Model 

FE Analysis

Time, t, μs

Axial Strain, εZZ

(a) z / L = 0.50 (b) z / L = 0.50 , closer view 

-0.0030

-0.0025

-0.0020

-0.0015

-0.0010

-0.0005

0

0.0005

0.0010

200 250 300 350 400

M-H Shell Model

FE Analysis

Time, t, μs

Axial Strain, εZZ

-0.0030

-0.0025

-0.0020

-0.0015

250 275 300 325 350

M-H Shell Model

FE Analysis

Time, t, μs

Axial Strain, εZZ

(c) z / L = 0.75 (d) z / L = 0.75, closer view 

Figure 9. Axial strain predicted by M-H model and FE analysis for the tube with m = 0.548. 

But why does the traditional algorithm work for rods? Figure 12 gives an example for the use of 

both algorithms on the rod with ν = 0.29. It is seen that the predicted wave by traditional 

algorithm still has some discontinuous response in the later part of the wave peak, although the 

difference between the two algorithms is not as significant as for the tubes. From figure 3, it can 

be seen that the phase speed in rods changes very slowly with δ, compared with phase speed in 

tubes. This leads to more Pochhammer modes for rods than the tubes. Because the irregularities 

occur at the later part of the wave peak, the traditional algorithm still gives wave dispersion 

prediction accurate enough for engineering application, and the shortcoming of the traditional 

algorithm is concealed. In this study of wave dispersion in tubes, the limitation of the traditional 

wave dispersion algorithm is identified first. 

The first part of the wave dispersion prediction by the new algorithm shows abnormal behavior. 

This is because at the beginning, relatively few wave components contribute to the dispersed 

wave, which can be seen from the algorithm described earlier. The M-H model assumes infinite 
13 

-0.0030

-0.0025

-0.0020

-0.0015

-0.0010

-0.0005

0

0.0005

0.0010

100 150 200 250 300

M-H Shell Model

FE Analysis

Time, t, μs

Axial Strain, εZZ

-0.0030

-0.0025

-0.0020

-0.0015

170 195 220 245 270

M-H Shell Model

FE Analysis

Time, t, μs

Axial Strain, εZZ

(a) z / L = 0.50 (b) z / L = 0.50 , closer view 

-0.0030

-0.0025

-0.0020

-0.0015

-0.0010

-0.0005

0

0.0005

0.0010

200 250 300 350 400

M-H Shell Model

FE Analysis

Time,t, μs

Axial Strain, εZZ

-0.0030

-0.0025

-0.0020

-0.0015

250 275 300 325 350

M-H Shell Model

FE Analysis

Time,t, μs

Axial Strain, εZZ

(c) z / L = 0.75 (d) z / L = 0.75, closer view 

Figure 10. Axial strain predicted by M-H model and FE analysis for the tube with m = 0.667. 

long tubes and continuous wave components. Under these assumptions, at any instance, all the 

wave components contribute to the whole waveform, so this abnormal behavior will not be seen. 

This phenomenon cannot be seen in real tests or in an FEM simulation either. When the new 

algorithm is used to predict the dispersed wave, it is assumed that the wave has period of 

1400 s μ during which the FT is performed. This input wave is a pulse wave rather than a 

continuous wave. The new algorithm does not allow any summation of wave components if the 

travel time condition is not satisfied. Thus the initial data are spurious till all the wave 

components have arrived at the particular point of interest. This problem can be resolved by 

taking a wider time window during FT and reducing the window during reconstruction. 

When rods are used in the SHPB test, the only geometric parameter affecting the wave 

propagation in rods is the radius. If tubes are used in the SHPB test, two more parameters are 

involved: the inner radius and the m value. Among the outer radius, inner radius, tube wall 
14 

-0.0010

-0.0005

0

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

-10 30 70 110 150-0.0005

0

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

Traditional Algorithm

New Algorithm

Time, t, μs

Axial Strain, εZZAxial Strain, εZZ

-0.0010

-0.0005

0

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

-10 30 70 110 150-0.0005

0

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

Traditional algorithm

New algorithm

Time, t, μs

Axial Strain, εZZAxial Strain, εZZ

(a) m = 0.548 , z = 0.524 m (b) m = 0.548 , z = 0.762 m 

-0.0010

-0.0005

0

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

-10 30 70 110 150-0.0005

0

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

Traditional algorithm

New algorithm

Time, t, μs

Axial Strain, εZZAxial Strain, εZZ

-0.0010

-0.0005

0

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

-10 30 70 110 150-0.0005

0

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

Traditional algorithm

New algorithm

Time, t, μs

Axial Strain, εZZAxial Strain, εZZ

(c) m = 0.667 , z = 0.524 m (d) m = 0.667 , z 0.762 m =

Figure 11. Comparison of new and traditional dispersion algorithms. 

Figure 12. Dispersed wave in rod determined by the new algorithm and 

traditional algorithm. Material Poisson’s ratio = 0.29. 

-0.0010

-0.0005

0

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0 20 40 60 80 100 120 140-0.0005

0

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

Traditional Algorithm

New Algorithm

Time, t, μs

Axial Strain, εZZAxial Strain, εZZ
15 

thickness h and m, only two are independent. The discussion below will focus on h and m. It 

can be seen from figure 3 that when m takes lower values, the phase speed decreases more 

quickly when δ is small. For a fixed m value, when the tube wall is thinner, the phase speed for 

all the wave components will be closer than when the h value is higher. This can be seen 

graphically from figure 3. So lower m values and higher h values will make the phase speed 

decreases faster as a function of frequency, which will make the wave become wider and wider 

when it propagates in the tube. To catch all the wave components, a wider time window must be 

used to process the strain data. Usually a computer code is used to automatically select the 

processing window. For example, Leber’s method (18) finds the processing window in the 

following way: first find a window between the point where the wave rises and the point where 

the wave drops to zero again. Another window that is ~33% wider than the first window and 

includes the first window in the middle is used as the processing window. If the wave becomes 

too wide, an incorrect window may be chosen which omits some important components. 

Furthermore, the incident wave, and the reflected wave will not contain the same amount of 

information because the reflected wave is wider than the incident wave, but the processing 

windows are still the same width. To obtain an accurate result, it is recommended to use a tube 

with higher m value and lower h value. Such a combination is actually not difficult to realize. 

When tubes are used in the SHPB technique, several configurations of the apparatus are possible. 

If an incident tube and a transmission tube are used as the apparatus, it is recommended that one 

hollow cylindrical specimen be used. This requires that the tubes and the specimen have the 

same inner and outer diameters. The advantage of this configuration is that the 1-D requirement 

of the SHPB technique can be better satisfied than using rods. Figure 13 shows the stress history 

at the integration point of all the five elements at z 1.00 = m along the radial direction. It is seen 

that for all five elements, the axial stress history σzz vs. time is nearly the same. The normal 

stresses in the other two directions are also plotted. Figure 13 also shows that the stressσxx is 

very low (maximum is 13 MPa) compared to σzz (maximum is 580 MPa), and the stress σyy is 

even lower (maximum is 0.7 MPa). So the stress distribution can be considered as 1-D, and the 

axial stress distribution is uniform across the cross-section of the tube. The strains in x and y 

directions can be considered only due to Poisson’s effect. Another option is described by Chen 

(12), where only the TT is used. When this configuration is used, a cap must be inserted 

between specimen and TT to transfer the load. It is not clear what effect this cap might have on 

the stress wave propagation and will be addressed in a separate work. 
16 

Figure 13. Stress history of all elements at z = 1.00 m along X = 0 axis. 

6. Conclusions 

In this report, wave dispersion in cylindrical tubes applied to the SHPB experiments is studied. 

The main conclusions are as follows: 

1. The M-H shell model gives the same result as the 3-D model. 

2. The FE study shows that wave propagation in tubes follows the first-mode solution of the 

M-H shell model. 

3. A new dispersion algorithm is suggested, and it can better predict the dispersion wave. 

4. For purpose of the SHPB application, tubes with higher m and lower h values should be 

used. 

-600

-400

-200

0

200

150 200 250 300 350

σZZ

σXX

σYY

Time, t, μs

Stress, σ, MPa
17 

7. References 

1. Hopkinson, B. A Method of Measuring the Pressure in the Deformation of High Explosives 

by the Impact of Bullets. Philosophical Transactions of the Royal Society of London 1914, 

A213, 437–452. 

2. Davies, R. M. A Critical Study of the Hopkinson Pressure Bar. Philosophical Transactions 

of the Royal Society of London, Series A, Mathematical and Physical Sciences 1948, 240, 

375–457.

3. Kolsky, H. An Investigation of the Mechanical Properties of Materials at Very High Rates 

of Loading. Proceedings of the Royal Society of London 1949, B62, 676–700. 

4. Pochhammer, L. Uber die Fortpflanzungsgeschwindigkeiten kleiner Schwing ungen in 

einem unbegrenzten isotropen Kreiscylinder. Journal fur Mathematick (Crelle) 1876, 81, 

324–336. 

5. Chree, C. The Equations of an Isotropic Elastic Solid in Polar and Cylindrical Co-ords. 

Their Solutions and Applications. Cambridge Philosophical Society: Transactions 1889, 

14, 250–369. 

6. Bancroft, D. The Speed of Longitudinal Waves in Cylindrical Bars. Physical Review 1941, 

59, 588–593. 

7. Follansbee, P. S.; Frantz C. Wave Propagation in the Split Hopkinson Pressure Bar. Journal 

of Engineering Materials and Technology 1983, 105, 61–66. 

8. Gong, J. C.; Malvern, L. E.; Jenkins, D. A. Dispersion Investigation in the Split Hopkinson 

Pressure Bar. Journal of Engineering Materials and Technology 1990, 112, 309–314. 

9. Li, Z. H.; Lambros, J. Determination of the Dynamic Response of Brittle Composites by the 

Use of the Split Hopkinson Pressure Bar. Composite Science and Technology 1999, 59, 

1097–1107. 

10. Dowling, A. R.; Harding, J.; Campell, J. D. The Dynamic Punching of Metals. Journal of 

the Institute of Metals 1970, 98, 215–224. 

11. Zhao, H.; Gary, G.; Klepaczko, J. R. On the Use of a Visco-Elastic Split Hopkinson 

Pressure Bar. International Journal of Impact Engineering 1997, 19, 319–330. 

12. Chen, W.; Zhang, B.; Forrestal, M. J. A Split Hopkinson Bar Technique for Low-Impedance 

Materials. Experimental Mechanics 1999, 39, 81–85. 
18 

13. Ghosh, J. Longitudinal Vibrations of a Hollow Cylinder. Bulletin Calcutta Mathematical 

Society 1923, 14, 31–40. 

14. Fay, R. D. Waves in Liquid-Filled Cylinders. The Journal of Acoustical Society of America

1952, 24, 459–462. 

15. Herrmann, G.; Mirsky, I. Three-Dimensional and Shell-Theory Analysis of Axially 

Symmetric Motions of Cylinders. Journal of Applied Mechanics 1956, 23, 563–568. 

16. Mirsky, I.; Herrmann, G. Axially Symmetric Motions of Thick Cylindrical Shells. Journal 

of Applied Mechanics 1958, 25, 97–102. 

17. Felice, C. W. The Response of Soil to Impulse Loads Using the Split Hopkinson Pressure 

Bar Technique; AFWL-TR-8-92; Air Force Weapons Laboratory: Kirtland Air Force Base, 

NM, May 1986. 

18. Lifshitz, J. M.; Leber, H. Data Processing in the Split Hopkinson Pressure Bar Tests. 

International Journal of Impact Engineering 1994, 15 (6), 723–733.
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 45433-7739 

 5 DIRECTOR 

 LLNL 

 R CHRISTENSEN 

 S DETERESA 

 F MAGNESS 

 M FINGER MS 313 

 M MURPHY L 282 

 PO BOX 808 

 LIVERMORE CA 94550 

 1 AFRL MLS OL 

 L COULTER 

 5851 F AVE 

 BLDG 849 RM AD1A 

 HILL AFB UT 84056-5713 

 1 OSD 

 JOINT CCD TEST FORCE 

 OSD JCCD 

 R WILLIAMS 

 3909 HALLS FERRY RD 

 VICKSBURG MS 29180-6199 
NO. OF NO. OF 

COPIES ORGANIZATION COPIES ORGANIZATION

25

 3 DARPA 

 M VANFOSSEN 

 S WAX 

 L CHRISTODOULOU 

 3701 N FAIRFAX DR 

 ARLINGTON VA 22203-1714 

 2 SERDP PROGRAM OFC 

 PM P2 

 C PELLERIN 

 B SMITH 

 901 N STUART ST STE 303 

 ARLINGTON VA 22203 

 1 OAK RIDGE NATL LAB 

 R M DAVIS 

 PO BOX 2008 

 OAK RIDGE TN 37831-6195 

 1 OAK RIDGE NATL LAB 

 C EBERLE MS 8048 

 PO BOX 2008 

 OAK RIDGE TN 37831 

 3 DIRECTOR 

 SANDIA NATL LABS 

 APPLIED MECHS DEPT 

 MS 9042 

 J HANDROCK 

 Y R KAN 

 J LAUFFER 

 PO BOX 969 

 LIVERMORE CA 94551-0969 

 1 OAK RIDGE NATL LAB 

 C D WARREN MS 8039 

 PO BOX 2008 

 OAK RIDGE TN 37831 

 4 NIST 

 M VANLANDINGHAM MS 8621 

 J CHIN MS 8621 

 J MARTIN MS 8621 

 D DUTHINH MS 8611 

 100 BUREAU DR 

 GAITHERSBURG MD 20899 

 1 HYDROGEOLOGIC INC 

 SERDP ESTCP SPT OFC 

 S WALSH 

 1155 HERNDON PKWY STE 900 

 HERNDON VA 20170 

 3 NASA LANGLEY RESEARCH CTR 

 AMSRD ARL VS 

 W ELBER MS 266 

 F BARTLETT JR MS 266 

 G FARLEY MS 266 

 HAMPTON VA 23681-0001 

 1 NASA LANGLEY RESEARCH CTR 

 T GATES MS 188E 

 HAMPTON VA 23661-3400 

 1 FHWA 

 E MUNLEY 

 6300 GEORGETOWN PIKE 

 MCLEAN VA 22101 

 1 USDOT FEDERAL RAILROAD 

 M FATEH RDV 31 

 WASHINGTON DC 20590 

 3 CYTEC FIBERITE 

 R DUNNE 

 D KOHLI 

 R MAYHEW 

 1300 REVOLUTION ST 

 HAVRE DE GRACE MD 21078 

 1 DIRECTOR 

 NGIC 

 IANG TMT 

 2055 BOULDERS RD 

 CHARLOTTESVILLE VA 

 22911-8318 

 1 SIOUX MFG 

 B KRIEL 

 PO BOX 400 

 FT TOTTEN ND 58335 

 2 3TEX CORP 

 A BOGDANOVICH 

 J SINGLETARY 

 109 MACKENAN DR 

 CARY NC 27511 

 1 3M CORP 

 J SKILDUM 

 3M CENTER BLDG 60 IN 01 

 ST PAUL MN 55144-1000 
NO. OF NO. OF 

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 1 DIRECTOR 

 DEFENSE INTLLGNC AGNCY 

 TA 5 

 K CRELLING 

 WASHINGTON DC 20310 

 1 ADVANCED GLASS FIBER YARNS 

 T COLLINS 

 281 SPRING RUN LANE STE A 

 DOWNINGTON PA 19335 

 1 COMPOSITE MATERIALS INC 

 D SHORTT 

 19105 63 AVE NE 

 PO BOX 25 

 ARLINGTON WA 98223 

 1 JPS GLASS 

 L CARTER 

 PO BOX 260 

 SLATER RD 

 SLATER SC 29683 

 1 COMPOSITE MATERIALS INC 

 R HOLLAND 

 11 JEWEL CT 

 ORINDA CA 94563 

 1 COMPOSITE MATERIALS INC 

 C RILEY 

 14530 S ANSON AVE 

 SANTA FE SPRINGS CA 90670 

 2 SIMULA 

 J COLTMAN 

 R HUYETT 

 10016 S 51ST ST 

 PHOENIX AZ 85044 

 2 PROTECTION MATERIALS INC 

 M MILLER 

 F CRILLEY 

 14000 NW 58 CT 

 MIAMI LAKES FL 33014 

 2 FOSTER MILLER 

 M ROYLANCE 

 W ZUKAS 

 195 BEAR HILL RD 

 WALTHAM MA 02354-1196 

 1 ROM DEVELOPMENT CORP 

 R O MEARA 

 136 SWINEBURNE ROW 

 BRICK MARKET PLACE 

 NEWPORT RI 02840 

 2 TEXTRON SYSTEMS 

 T FOLTZ 

 M TREASURE 

 1449 MIDDLESEX ST 

 LOWELL MA 01851 

 1 O GARA HESS & EISENHARDT 

 M GILLESPIE 

 9113 LESAINT DR 

 FAIRFIELD OH 45014 

 2 MILLIKEN RESEARCH CORP 

 H KUHN 

 M MACLEOD 

 PO BOX 1926 

 SPARTANBURG SC 29303 

 1 CONNEAUGHT INDUSTRIES INC 

 J SANTOS 

 PO BOX 1425 

 COVENTRY RI 02816 

 1 ARMTEC DEFENSE PRODUCTS 

 S DYER 

 85 901 AVE 53 

 PO BOX 848 

 COACHELLA CA 92236 

 1 NATL COMPOSITE CTR 

 T CORDELL 

 2000 COMPOSITE DR 

 KETTERING OH 45420 

 3 PACIFIC NORTHWEST LAB 

 M SMITH 

 G VAN ARSDALE 

 R SHIPPELL 

 PO BOX 999 

 RICHLAND WA 99352 

 1 SAIC 

 M PALMER 

 1410 SPRING HILL RD STE 400 

 MS SH4 5 

 MCLEAN VA 22102 
NO. OF NO. OF 

COPIES ORGANIZATION COPIES ORGANIZATION

27

 1 ALLIANT TECHSYSTEMS INC 

 4700 NATHAN LN N 

 PLYMOUTH MN 55442-2512 

 1 APPLIED COMPOSITES 

 W GRISCH 

 333 NORTH SIXTH ST 

 ST CHARLES IL 60174 

 1 CUSTOM ANALYTICAL 

 ENG SYS INC 

 A ALEXANDER 

 13000 TENSOR LANE NE 

 FLINTSTONE MD 21530 

 1 AAI CORP 

 DR N B MCNELLIS 

 PO BOX 126 

 HUNT VALLEY MD 21030-0126 

 1 OFC DEPUTY UNDER SEC DEFNS 

 J THOMPSON 

 1745 JEFFERSON DAVIS HWY 

 CRYSTAL SQ 4 STE 501 

 ARLINGTON VA 22202 

 3 ALLIANT TECHSYSTEMS INC 

 J CONDON 

 E LYNAM 

 J GERHARD 

 WV01 16 STATE RT 956 

 PO BOX 210 

 ROCKET CENTER WV 

 26726-0210 

 1 PROJECTILE TECHNOLOGY INC 

 515 GILES ST 

 HAVRE DE GRACE MD 21078 

 1 HEXCEL INC 

 R BOE 

 PO BOX 18748 

 SALT LAKE CITY UT 84118 

 1 PRATT & WHITNEY 

 C WATSON 

 400 MAIN ST MS 114 37 

 EAST HARTFORD CT 06108 

 5 NORTHROP GRUMMAN 

 B IRWIN 

 K EVANS 

 D EWART 

 A SHREKENHAMER 

 J MCGLYNN 

 BLDG 160 DEPT 3700 

 1100 WEST HOLLYVALE ST 

 AZUSA CA 91701 

 1 HERCULES INC 

 HERCULES PLAZA 

 WILMINGTON DE 19894 

 1 BRIGS COMPANY 

 J BACKOFEN 

 2668 PETERBOROUGH ST 

 HERNDON VA 22071-2443 

 1 ZERNOW TECHNICAL SERVICES 

 L ZERNOW 

 425 W BONITA AVE STE 208 

 SAN DIMAS CA 91773 

 1 GENERAL DYNAMICS OTS 

 L WHITMORE 

 10101 NINTH ST NORTH 

 ST PETERSBURG FL 33702 

 2 GENERAL DYNAMICS OTS 

 FLINCHBAUGH DIV 

 K LINDE 

 T LYNCH 

 PO BOX 127 

 RED LION PA 17356 

 1 GKN WESTLAND AEROSPACE 

 D OLDS 

 450 MURDOCK AVE 

 MERIDEN CT 06450-8324 

 2 BOEING ROTORCRAFT 

 P MINGURT 

 P HANDEL 

 800 B PUTNAM BLVD 

 WALLINGFORD PA 19086 
NO. OF NO. OF 

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28

 5 SIKORSKY AIRCRAFT 

 G JACARUSO 

 T CARSTENSAN 

 B KAY 

 S GARBO MS S330A 

 J ADELMANN 

 6900 MAIN ST 

 PO BOX 9729 

 STRATFORD CT 06497-9729 

 1 AEROSPACE CORP 

 G HAWKINS M4 945 

 2350 E EL SEGUNDO BLVD 

 EL SEGUNDO CA 90245 

 2 CYTEC FIBERITE 

 M LIN 

 W WEB 

 1440 N KRAEMER BLVD 

 ANAHEIM CA 92806 

 2 UDLP 

 G THOMAS 

 M MACLEAN 

 PO BOX 58123 

 SANTA CLARA CA 95052 

 1 UDLP WARREN OFC 

 A LEE 

 31201 CHICAGO RD SOUTH 

 SUITE B102 

 WARREN MI 48093 

 2 UDLP 

 R BRYNSVOLD 

 P JANKE MS 170 

 4800 EAST RIVER RD 

 MINNEAPOLIS MN 55421-1498 

 1 LOCKHEED MARTIN 

 SKUNK WORKS 

 D FORTNEY 

 1011 LOCKHEED WAY 

 PALMDALE CA 93599-2502 

 1 LOCKHEED MARTIN 

 R FIELDS 

 5537 PGA BLVD 

 SUITE 4516 

 ORLANDO FL 32839 

 1 NORTHRUP GRUMMAN CORP 

 ELECTRONIC SENSORS 

 & SYSTEMS DIV 

 E SCHOCH MS V 16 

 1745A W NURSERY RD 

 LINTHICUM MD 21090 

 1 GDLS DIVISION 

 D BARTLE 

 PO BOX 1901 

 WARREN MI 48090 

 2 GDLS 

 D REES 

 M PASIK 

 PO BOX 2074 

 WARREN MI 48090-2074 

 1 GDLS 

 MUSKEGON OPER 

 M SOIMAR 

 76 GETTY ST 

 MUSKEGON MI 49442 

 1 GENERAL DYNAMICS 

 AMPHIBIOUS SYS 

 SURVIVABILITY LEAD 

 G WALKER 

 991 ANNAPOLIS WAY 

 WOODBRIDGE VA 22191 

 6 INST FOR ADVANCED 

 TECH 

 H FAIR 

 I MCNAB 

 P SULLIVAN 

 S BLESS 

 W REINECKE 

 C PERSAD 

 3925 W BRAKER LN STE 400 

 AUSTIN TX 78759-5316 

 1 ARROW TECH ASSOC 

 1233 SHELBURNE RD STE D8 

 SOUTH BURLINGTON VT 

 05403-7700 

 1 R EICHELBERGER 

 CONSULTANT 

 409 W CATHERINE ST 

 BEL AIR MD 21014-3613 
NO. OF NO. OF 

COPIES ORGANIZATION COPIES ORGANIZATION

29

 1 SAIC 

 G CHRYSSOMALLIS 

 8500 NORMANDALE LAKE BLVD 

 SUITE 1610 

 BLOOMINGTON MN 55437-3828 

 1 UCLA MANE DEPT ENGR IV 

 H T HAHN 

 LOS ANGELES CA 90024-1597 

 2 UNIV OF DAYTON 

 RESEARCH INST 

 R Y KIM 

 A K ROY 

 300 COLLEGE PARK AVE 

 DAYTON OH 45469-0168 

 1 UMASS LOWELL 

 PLASTICS DEPT 

 N SCHOTT 

 1 UNIVERSITY AVE 

 LOWELL MA 01854 

 1 IIT RESEARCH CTR 

 D ROSE 

 201 MILL ST 

 ROME NY 13440-6916 

 1 GA TECH RESEARCH INST 

 GA INST OF TCHNLGY 

 P FRIEDERICH 

 ATLANTA GA 30392 

 1 MICHIGAN ST UNIV 

 MSM DEPT 

 R AVERILL 

 3515 EB 

 EAST LANSING MI 48824-1226 

 1 UNIV OF WYOMING 

 D ADAMS 

 PO BOX 3295 

 LARAMIE WY 82071 

 1 PENN STATE UNIV 

 R S ENGEL 

 245 HAMMOND BLDG 

 UNIVERSITY PARK PA 16801 

 2 PENN STATE UNIV 

 R MCNITT 

 C BAKIS 

 212 EARTH ENGR 

 SCIENCES BLDG 

 UNIVERSITY PARK PA 16802 

 1 PURDUE UNIV 

 SCHOOL OF AERO & ASTRO 

 C T SUN 

 W LAFAYETTE IN 47907-1282 

 1 STANFORD UNIV 

 DEPT OF AERONAUTICS 

 & AEROBALLISTICS 

 S TSAI 

 DURANT BLDG 

 STANFORD CA 94305 

 1 UNIV OF MAINE 

 ADV STR & COMP LAB 

 R LOPEZ ANIDO 

 5793 AEWC BLDG 

 ORONO ME 04469-5793 

 1 JOHNS HOPKINS UNIV 

 APPLIED PHYSICS LAB 

 P WIENHOLD 

 11100 JOHNS HOPKINS RD 

 LAUREL MD 20723-6099 

 1 UNIV OF DAYTON 

 J M WHITNEY 

 COLLEGE PARK AVE 

 DAYTON OH 45469-0240 

 1 NORTH CAROLINA ST UNIV 

 CIVIL ENGINEERING DEPT 

 W RASDORF 

 PO BOX 7908 

 RALEIGH NC 27696-7908 

 5 UNIV OF DELAWARE 

 CTR FOR COMPOSITE MTRLS 

 J GILLESPIE 

 M SANTARE 

 S YARLAGADDA 

 S ADVANI 

 D HEIDER 

 201 SPENCER LAB 

 NEWARK DE 19716 
NO. OF NO. OF 

COPIES ORGANIZATION COPIES ORGANIZATION

30

 1 DEPT OF MTRLS 

 SCIENCE & ENGRG 

 UNIV OF ILLINOIS 

 AT URBANA CHAMPAIGN 

 J ECONOMY 

 1304 WEST GREEN ST 115B 

 URBANA IL 61801 

 1 UNIV OF MARYLAND 

 DEPT OF AEROSPACE ENGRG 

 A J VIZZINI 

 COLLEGE PARK MD 20742 

 1 DREXEL UNIV 

 A S D WANG 

 3141 CHESTNUT ST 

 PHILADELPHIA PA 19104 

 3 UNIV OF TEXAS AT AUSTIN 

 CTR FOR ELECTROMECHANICS 

 J PRICE 

 A WALLS 

 J KITZMILLER 

 10100 BURNET RD 

 AUSTIN TX 78758-4497 

 3 VA POLYTECHNICAL 

 INST & STATE UNIV 

 DEPT OF ESM 

 M W HYER 

 K REIFSNIDER 

 R JONES 

 BLACKSBURG VA 24061-0219 

 1 SOUTHWEST RESEARCH INST 

 ENGR & MATL SCIENCES DIV 

 J RIEGEL 

 6220 CULEBRA RD 

 PO DRAWER 28510 

 SAN ANTONIO TX 78228-0510 

 1 BATELLE NATICK OPERS 

 B HALPIN 

 313 SPEEN ST 

 NATICK MA 01760 

 3 DIRECTOR 

 US ARMY RESEARCH LAB 

 AMSRD ARL WM MB 

 A FRYDMAN 

 2800 POWDER MILL RD 

 ADELPHI MD 20783-1197 

ABERDEEN PROVING GROUND

 1 US ARMY ATC 

 CSTE DTC AT AC I 

 W C FRAZER 

 400 COLLERAN RD 

 APG MD 21005-5059 

 91 DIR USARL 

 AMSRD ARL CI 

 AMSRD ARL O AP EG 

 M ADAMSON 

 AMSRD ARL SL BA 

 AMSRD ARL SL BB 

 D BELY 

 AMSRD ARL WM 

 J SMITH 

 H WALLACE 

 AMSRD ARL WM B 

 A HORST 

 T KOGLER 

 AMSRD ARL WM BA 

 D LYON 

 AMSRD ARL WM BC 

 J NEWILL 

 P PLOSTINS 

 A ZIELINSKI 

 AMSRD ARL WM BD 

 P CONROY 

 B FORCH 

 M LEADORE 

 C LEVERITT 

 R LIEB 

 R PESCE RODRIGUEZ 

 B RICE 

 AMSRD ARL WM BF 

 S WILKERSON 

 AMSRD ARL WM M 

 B FINK 

 J MCCAULEY 

 AMSRD ARL WM MA 

 L GHIORSE 

 S MCKNIGHT 

 E WETZEL 

 AMSRD ARL WM MB 

 J BENDER 

 T BOGETTI 

 L BURTON 

 R CARTER 

 K CHO 

 W DE ROSSET 

 G DEWING 

 R DOWDING 

 W DRYSDALE 
NO. OF NO. OF 

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 R EMERSON 

 D HENRY 

 D HOPKINS 

 R KASTE 

 L KECSKES 

 M MINNICINO 

 B POWERS 

 D SNOHA 

 J SOUTH 

 M STAKER 

 J SWAB 

 J TZENG 

 AMSRD ARL WM MC 

 J BEATTY 

 R BOSSOLI 

 E CHIN 

 S CORNELISON 

 D GRANVILLE 

 B HART 

 J LASALVIA 

 J MONTGOMERY 

 F PIERCE 

 E RIGAS 

 W SPURGEON 

 AMSRD ARL WM MD 

 B CHEESEMAN 

 P DEHMER 

 R DOOLEY 

 G GAZONAS 

 S GHIORSE 

 C HOPPEL 

 M KLUSEWITZ 

 W ROY 

 J SANDS 

 D SPAGNUOLO 

 S WALSH 

 S WOLF 

 AMSRD ARL WM RP 

 J BORNSTEIN 

 C SHOEMAKER 

 AMSRD ARL WM T 

 B BURNS 

 AMSRD ARL WM TA 

 W BRUCHEY 

 M BURKINS 

 W GILLICH 

 B GOOCH 

 T HAVEL 

 E HORWATH 

 M NORMANDIA 

 J RUNYEON 

 M ZOLTOSKI 

 AMSRD ARL WM TB 

 P BAKER 

 AMSRD ARL WM TC 

 R COATES 

 AMSRD ARL WM TD 

 D DANDEKAR 

 T HADUCH 

 T MOYNIHAN 

 M RAFTENBERG 

 S SCHOENFELD 

 T WEERASOORIYA 

 AMSRD ARL WM TE 

 A NIILER 

 J POWELL 
NO. OF NO. OF 

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 1 LTD 

 R MARTIN 

 MERL 

 TAMWORTH RD 

 HERTFORD SG13 7DG 

 UK 

 1 SMC SCOTLAND 

 P W LAY 

 DERA ROSYTH 

 ROSYTH ROYAL DOCKYARD 

 DUNFERMLINE FIFE KY 11 2XR 

 UK 

 1 CIVIL AVIATION 

 ADMINSTRATION 

 T GOTTESMAN 

 PO BOX 8 

 BEN GURION INTRNL AIRPORT 

 LOD 70150 

 ISRAEL 

 1 AEROSPATIALE 

 S ANDRE 

 A BTE CC RTE MD132 

 316 ROUTE DE BAYONNE 

 TOULOUSE 31060 

 FRANCE 

 1 DRA FORT HALSTEAD 

 P N JONES 

 SEVEN OAKS KENT TN 147BP 

 UK 

 1 SWISS FEDERAL ARMAMENTS 

 WKS 

 W LANZ 

 ALLMENDSTRASSE 86 

 3602 THUN 

 SWITZERLAND 

 1 DYNAMEC RESEARCH LAB 

 AKE PERSSON 

 BOX 201 

 SE 151 23 SODERTALJE 

 SWEDEN 

 1 ISRAEL INST OF TECHLGY 

 S BODNER 

 FACULTY OF MECHANICAL 

 ENGR 

 HAIFA 3200 

 ISRAEL 

 1 DSTO 

 WEAPONS SYSTEMS DIVISION 

 N BURMAN RLLWS 

 SALISBURY 

 SOUTH AUSTRALIA 5108 

 AUSTRALIA 

 1 DEF RES ESTABLISHMENT 

 VALCARTIER 

 A DUPUIS 

 2459 BLVD PIE XI NORTH 

 VALCARTIER QUEBEC 

 CANADA 

 PO BOX 8800 COURCELETTE 

 GOA IRO QUEBEC 

 CANADA 

 1 ECOLE POLYTECH 

 J MANSON 

 DMX LTC 

 CH 1015 LAUSANNE 

 SWITZERLAND 

 1 TNO DEFENSE RESEARCH 

 R IJSSELSTEIN 

 ACCOUNT DIRECTOR 

 R&D ARMEE 

 PO BOX 6006 

 2600 JA DELFT 

 THE NETHERLANDS 

 2 FOA NATL DEFENSE RESEARCH 

 ESTAB 

 DIR DEPT OF WEAPONS & 

 PROTECTION 

 B JANZON 

 R HOLMLIN 

 S 172 90 STOCKHOLM 

 SWEDEN
NO. OF 

COPIES ORGANIZATION

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 2 DEFENSE TECH & PROC 

 AGENCY GROUND 

 I CREWTHER 

 GENERAL HERZOG HAUS 

 3602 THUN 

 SWITZERLAND 

 1 MINISTRY OF DEFENCE 

 RAFAEL 

 ARMAMENT DEVELOPMENT 

 AUTH 

 M MAYSELESS 

 PO BOX 2250 

 HAIFA 31021 

 ISRAEL 

 1 TNO DEFENSE RESEARCH 

 I H PASMAN 

 POSTBUS 6006 

 2600 JA DELFT 

 THE NETHERLANDS 

 1 B HIRSCH 

 TACHKEMONY ST 6 

 NETAMUA 42611 

 ISRAEL 

 1 DEUTSCHE AEROSPACE AG 

 DYNAMICS SYSTEMS 

 M HELD 

 PO BOX 1340 

 D 86523 SCHROBENHAUSEN 

 GERMANY 
34

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