Mechanical Response of an Al-PTFE Composite to Uniaxial 

Compression Over a Range of Strain Rates and 

Temperatures 

by Daniel T. Casem 

ARL-TR-4560 September 2008 

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Army Research Laboratory 

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ARL-TR-4560 September 2008 

Mechanical Response of an Al-PTFE Composite to Uniaxial 

Compression Over a Range of Strain Rates and 

Temperatures 

Daniel T. Casem 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited. 
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Mechanical Response of an Al-PTFE Composite to Uniaxial Compression Over a 

Range of Strain Rates and Temperatures 

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13. SUPPLEMENTARY NOTES 

14. ABSTRACT 

Compressive stress-strain curves were generated for a pressed and sintered mixture of polytetrafluoroethylene and aluminum 

powders. Experiments were performed at strain rates ranging from 0.001 to 8000/s using a servo-hydraulic load frame and a 

split Hopkinson pressure bar. High-rate experiments were also performed over a temperature range of 22–78 °C. The data is fit

to the Johnson-Cook and Modified Johnson-Cook constitutive equations and also to a Zerilli-Armstrong equation adapted for 

use with polymers. Although the material is known to be reactive, only the inert behavior is described in this report. 

15. SUBJECT TERMS

reactive material, split Hopkinson pressure bar, Zerilli-Armstrong polymer model 

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ii
Contents 

List of Figures iv

List of Tables v

1. Introduction 1

2. Material 1

3. Experimental Procedure 2

3.1 Low-Rate Tests................................................................................................................2 

3.2 High-Rate Tests...............................................................................................................2 

3.3 Temperature Tests...........................................................................................................2 

4. Results 3

5. Constitutive Modeling 8

5.1 Johnson-Cook and Modified Johnson Cook ...................................................................8 

5.2 Zerilli-Armstrong Model for Polymers.........................................................................11 

6. Conclusion 12

7. References 15

Appendix. Strain Measures Defined 17

Distribution List 19 

iii
List of Figures 

Figure 1. Engineering stress vs. engineering strain for Al/PTFE (compression). All 

specimens were initially at room temperature (22 °C). .............................................................3 

Figure 2. Lateral strain as measured by a laser extensometer in the SHPB tests plotted in 

figure 1 (thin black curves). The heavy red curve shows that which would be expected 

from incompressible behavior....................................................................................................4 

Figure 3. True stress-strain curves for Al/PTFE. All samples were initially at room 

temperature (22 °C). ..................................................................................................................5 

Figure 4. The data of figure 3 repeated on a finer scale to illustrate the low-strain behavior. .......5 

Figure 5. The variation of true strain rate for the SHPB tests.........................................................6 

Figure 6. Strain-rate hardening observed in the experiments compared to the MJC, JC, and 

ZA fits. Stress is measured at 7% total true strain, all at room temperature (295 K). ..............6 

Figure 7. Stress-strain curves obtained at a rate of 4000/s over a range of initial 

temperatures...............................................................................................................................7

Figure 8. Three fits to the high-rate temperature data (10% total strain, 4000/s). The JC 

equation is fit over two temperature ranges: the lower temperature range of 295 K ≤ T 

≤ 329 K and the full temperature range. A fit is also shown for the ZA model (full 

range). ........................................................................................................................................7 

Figure 9. MJC model predictions, assuming adiabatic conditions, compared to the high-rate 

(SHPB) data. The JC model predictions are essentially identical. .........................................10 

Figure 10. MJC model predictions, assuming isothermal deformations, compared to the lowrate experiments. The JC model predictions are essentially identical. ...................................10 

Figure 11. Comparison of the ZA model to the SHPB experiments assuming adiabatic 

conditions.................................................................................................................................13 

Figure 12. Comparison of the ZA model to the low-rate experiments assuming isothermal 

conditions. Note the great improvement over the JC fits to the same data, figure 10. ...........14 

iv
List of Tables 

Table 1. JC and MJC constants for Al/PTFE. The fits are performed over two temperature 

ranges. ........................................................................................................................................9 

Table 2. The ZA constants for Al/PTFE. The values determined by Cai et al. (13) and Zerilli 

and Armstrong (11) for PTFE-Al-W and PTFE are included for comparison. .......................13 

v
INTENTIONALLY LEFT BLANK. 

vi
1. Introduction 

Reactive materials can be loosely categorized as composites of inert solid materials which, when 

subjected to a violent mechanical stimulus such as an impact, react exothermally with a rapid 

release of energy. This reaction, while aptly described as “explosive,” differs from a true 

detonation or deflagration in that it requires a mechanical stimulus to not only initiate the 

reaction but also to sustain it. Such materials can also be fairly robust mechanically and can 

serve as substantial structural components. Because of these properties, reactive materials have a 

number of potential ordnance applications. Various compositions have been investigated to 

tailor properties of reactivity, strength, and density, to suit particular needs. 

This report addresses a specific aluminum/polytetrafluoroethylene (Al/PTFE) formulation that 

serves as an important benchmark for current reactive material development. In addition to 

material development, a good deal of work is underway to develop physics-based modeling 

capabilities for reactive materials. To aid in this latter effort, this report provides basic 

constitutive models for the inert behavior of this material. Compression tests have been 

performed over a range of strain rates and temperatures relevant to the conditions present during 

low-speed impact. This data, presented in the following sections, is used to generate parameters 

for both the Johnson-Cook (JC) and Modified Johnson-Cook (MJC) constitutive models. 

Although not ideally suited to represent this material, these parameters are given primarily due to 

the wide-spread use of these models and their availability in the current suite of hydrocodes. An 

additional, and more appropriate, fit is given for the Zerilli-Armstrong (ZA) model for polymers. 

The remainder of this document is organized as follows: a basic material description is given in 

section 2; section 3 describes the testing techniques used in the experiments, and the data is 

presented in section 4; finally, the model parameters and descriptions of the fitting processes are 

given in section 5. Unless otherwise noted, the use of the words “stress” and “strain” denote 

positive values in compression. 

2. Material 

The samples tested during this program were supplied by General Sciences, Inc. (GSI). They 

were made from a material designated by GSI as GSI-0017. It is a pressed and sintered mixture 

of aluminum and PTFE powders, 26.5% and 73.5% by weight, respectively. The initial powder 

sizes are 44 and 31 μm, respectively. The density of the compacted material, as measured by a 

buoyancy method based on Archimedes principle, is 2.29 g/cm3

. The longitudinal wave speed, 

measured by ultrasound at 10 MHz, is 1570 m/s.

1
3. Experimental Procedure 

3.1 Low-Rate Tests 

Low-rate tests were performed with an Instron, Inc., model 1331 servo-hydraulic load frame. 

The load applied to the specimen was measured with a load cell, and the specimen deformation 

was measured using a linear variable differential transformer (LVDT) measurement of the crosshead displacement and includes a correction for machine compliance. The specimens were 

cylindrical, nominally 6.35 mm in both diameter and length. Contact surfaces were lubricated 

with a heat-stable silicone lubricant. All of the low-rate tests were performed at room 

temperature (22 °C). 

3.2 High-Rate Tests 

A 6.35-mm-diameter 7075-T6 aluminum Split Hopkinson Pressure Bar (SHPB) was used for the 

high-rate tests. A series of six tests were performed at rates from 600 to 8000/s, all initially at 

room temperature (22 °C). The specimens were cylindrical, 3.18 mm in diameter and length, 

and contact surfaces were lubricated with the same silicone lubricant used in the low-rate tests. 

All data analysis accounts for bar-wave dispersion using a method based on Follansbee and 

Franz (1) and Gong et al. (2). 

As is typical with the SHPB, axial deformation of the specimen is determined by the bar analysis 

(see, for example, Follansbee [3]). However, the radial deformation is unknown, and is usually 

determined under the assumption that the deformation (both elastic and inelastic) in the specimen 

occurs at constant volume. To verify this assumption, a laser extensometer (also known as a 

laser occlusive radius detector, or LORD [4]) was used to measure the radial deformation of the 

specimens during dynamic compression. The frequency response of this device is very high 

(~10 MHz), and in our experience it is able to resolve deformations as small as 0.01 mm. The 

results of these experiments are discussed in section 4. 

3.3 Temperature Tests 

A final set of experiments was performed at elevated temperatures to quantify the thermal 

softening behavior of the material. These were performed at a consistent strain rate of 4000/s 

using the SHPB. Heating was accomplished by circulating heated air into a chamber which 

enclosed the specimen and the adjacent ~65 mm sections of the bars. Ideally, specimen 

temperature would have been monitored directly with a thermocouple glued to each specimen. 

However, this proved impractical because of the small sample size and also because of the 

difficulty in adhering gages to the specimen.* Instead, specimen temperature was measured with 

a thermocouple probe placed within 1 cm of the specimen, i.e., the probe measures ambient air 

 *

PTFE, more commonly known as Teflon, is widely known for its “non-stick” properties. 

2
temperature and not the specimen temperature directly. The temperature in the chamber was 

allowed to equilibrate over a 20 min period prior to each test to ensure that the specimen and 

relevant sections of the bars were allowed to reach the ambient temperature. The temperature 

gradient in the bars is believed to have negligible effects on the bar wave propagation and the 

strain gage measurements. Temperature measurements made in this way are estimated to be 

accurate to within ±2 °C. 

4. Results 

Figure 1 shows the engineering stress-strain curves obtained from the room temperature tests. 

Because of the brief duration of the deformation, the high rate curves (600/s and beyond) are 

considered adiabatic. In contrast, the lowest rate curves, 0.001 and 0.01/s, are considered 

isothermal. The intermediate rate, at 0.1/s, is probably somewhere between the two limits. 

0

50

100

150

200

250

300

350

0.00 0.10 0.20 0.30 0.40 0.50 0.60

engineering strain

engineering stress (MPa)

0.001/s

0.01/s

0.1/s

600/s

600/s

4000/s

4000/s

9000/s

9000/s

Figure 1. Engineering stress vs. engineering strain for Al/PTFE (compression). All specimens were 

initially at room temperature (22 °C). 

Figure 2 shows the tensile radial engineering strain as measured by the laser extensometer as a 

function of compressive axial engineering strain for the six SHPB experiments shown in 

figure 1. These are plotted as thin black curves. The heavy red curve is the relationship that 

3
0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

compressive axial engineering strain

tensile radial engineering strain

black - experiments (6 total)

red - incompressible behavior

Figure 2. Lateral strain as measured by a laser extensometer in the SHPB tests plotted in figure 1 (thin 

black curves). The heavy red curve shows that which would be expected from incompressible 

behavior. 

would result from incompressible deformation. The agreement is good, and what deviation can 

be measured could easily be explained by barreling. Therefore, the remainder of this report will 

speak in terms of true stress and strain as calculated from the engineering values under the 

assumption of incompressibility. These terms and the assumption of incompressibility are 

reviewed in the appendix. 

Figures 3 and 4 show the true stress-strain curves for the tests of figure 1. The data are plotted 

with two scales to show more detail in the low-strain range. Figure 5 shows the rate of true 

strain for the SHPB experiments. Note that in all cases the strain rate is reasonably constant. 

The low-rate tests are controlled in the servo-hydraulic load frame to give constant rates of true 

strain. Figure 6 shows stress at 7% strain as a function of strain rate, and shows the trend of ratehardening of this material. 

Figure 7 shows the stress-strain curves from the experiments at 4000/s, which were conducted at 

initial temperatures of 22, 43, 56, and 78 °C. A substantial softening is observed over this range 

of temperatures. Stresses at a strain of 10% are plotted with temperature in figure 8. From this 

graph, it is apparent that the softening is approximately linear with stress over the range of 

22–56 °C. As seen in figure 7, there is a significant change in the form of the stress-strain curve 

for the 78 °C test. It is suggested that this may be due to a phase change in the PTFE that occurs 

at some temperature between 56 and 78 °C, although this is merely speculative.

4
0

20

40

60

80

100

120

140

160

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80

true strain

true stress (MPa)

0.001/s

0.01/s

0.1/s

600/s

600/s

4000/s

4000/s

9000/s

9000/s

Figure 3. True stress-strain curves for Al/PTFE. All samples were initially at room temperature (22 °C). 

0

10

20

30

40

50

60

0.00 0.05 0.10 0.15 0.20 0.25 0.30

true strain

true stress (MPa)

0.001/s

0.01/s

0.1/s

600/s

600/s

4000/s

4000/s

9000/s

9000/s

Figure 4. The data of figure 3 repeated on a finer scale to illustrate the low-strain behavior.

5
-4000

-2000

0

2000

4000

6000

8000

10000

12000

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80

true strain

true strain-rate (1/s)

Figure 5. The variation of true strain rate for the SHPB tests. 

0

5

10

15

20

25

30

35

0.0001 0.001 0.01 0.1 1 10 100 1000 10000 100000

strain-rate (1/s)

true stress (MPa)

experiments

JC model

MJC model

ZA model

Figure 6. Strain-rate hardening observed in the experiments compared to the MJC, JC, and ZA fits. 

Stress is measured at 7% total true strain, all at room temperature (295 K).

6
0

20

40

60

80

100

120

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70

true strain

true stress (MPa)

295 K

295 K

316 K

329 K

351 K

All tests at ~ 4000/s

Figure 7. Stress-strain curves obtained at a rate of 4000/s over a range of initial temperatures. 

10

15

20

25

30

35

290 300 310 320 330 340 350 360 370

temperature (K)

true stress (MPa)

exp. data at 4000/s

JC - low temperature range

JC - full temperature range

ZA model

Figure 8. Three fits to the high-rate temperature data (10% total strain, 4000/s). The JC equation is 

fit over two temperature ranges: the lower temperature range of 295 K ≤ T ≤ 329 K and 

the full temperature range. A fit is also shown for the ZA model (full range).

7
5. Constitutive Modeling 

5.1 Johnson-Cook and Modified Johnson Cook 

The data of the preceding sections have been fitted to the JC and MJC constitutive equations. 

These models were chosen primarily because of their widespread use, and not necessarily because 

they are particularly well-suited to this material. Details of the models are discussed in Johnson 

and Cook (5) and Holmquist and Johnson (6), and are reviewed briefly in the following. 

The JC and MJC equations are written as 

( ) ( n

p

0

A + B 1+ Cln 1– T*

⎛ ⎞ ⎛ ⎞ ε

σ= ε ⎜ ⎟ ⎜ ⎟ ε ⎝ ⎠ ⎝ ⎠



 ) m , (1) 

and 

( ) ( n

p

0

A+B 1–T*

λ

⎛ ⎞ ε

σ= ε ⎜ ⎟ ε ⎝ ⎠



 ) m , (2) 

respectively, where T* is the homologous temperature, 

r

m r

T–T T* =

T –T . (3) 

The term εp denotes plastic strain, i.e., the total strain less the elastic strain, assumed in the 

following fits to be 3.5%. A, B, n, C, λ, and m are all material constants. Tm, which is generally 

taken as the melting temperature, is here treated as a free-parameter to improve the quality of the 

fits. Tr is a reference temperature, 295 K, the temperature of the lowest temperature tests. Note 

that the form of the thermal softening term is in general not defined for T < Tr, so extrapolation 

below Tr is not possible. The reference strain-rate, , is taken as 1/s. 0 ε

The only difference in the two equations is the form of the strain-rate hardening; the original 

version uses a logarithmic form and the modified version uses a power-law form. The form of 

the equations, , is very convenient for fitting the available data. By 

assuming ε ~ 0 and T = 295 K in equation 1, estimates of A and C can be determined from a 

simple least squares fit to the rate data of figure 6. This can then be repeated with equation 2, 

giving estimates for A and λ for the MJC equation. 

σ= ε ⋅ ε ⋅ f g hT () () ( ) 

Next, the temperature data of figure 8 can be fit to each equation under the assumptions that 

ε ~ 0 and ε = 4000/s to give estimates of A, m, and Tm. Two fits are made, one over the lower 

three temperatures, a range of K, and the other the full range T K. T

former provides the better fit, and it is suggested that it be used if temperatures above 329 K are 

not encountered.



T [295,329] ∈ ∈[295,351] he 

8
The strain-hardening terms, A, B, and n, can be fit to the stress-strain curves of figure 3, under 

the assumption that the high-rate curves are adiabatic and the low-rate curves are isothermal. 

This is done in an ad hoc manner to roughly match the data. To fit the adiabatic curves, it is 

assumed that cV, the specific heat capacity, is 1160 J/kg-K, based on an estimate used by 

Raftenberg et al. (7). The conversion of plastic work to heat, β, is assumed to be 0.9.* Finally, 

the parameters are arbitrarily adjusted such that the overall fit of the model adequately matches 

the experimental data of figures 3, 4, 6–8, simultaneously, using the actual conditions at which 

those graphs present the data (e.g., actual plastic strain values, adiabatic/isothermal conditions). 

More sophisticated fitting procedures can be developed but are hardly justifiable as they do not 

offer any real improvement to the quality of the fit obtained. 

Four sets of final parameters are given in table 1, for each model over the two temperature ranges 

mentioned above. Comparisons of selected model fits against the experimental data are given in 

figures 6, 8, 9, and 10. In general, the difference between the quality of the fits between the JC 

and MJC is small, with the MJC providing a slightly superior fit to the data (see figure 6) in a 

least-squares-residuals sense. Because there is no rate dependence of the strain-hardening terms, 

neither model is able to capture the strain-hardening effect at both high (>600/s, figure 9) and 

low (<0.1/s, figure 10) rates. This is obvious in the figures. Because it is anticipated that these 

models will be applied to high-rate scenarios, the data is fit to the high-rate data essentially 

ignoring the strain hardening observed at the low rates. Users are cautioned against using these 

parameter sets if their problems involve significant deformation at low rates. 

Table 1. JC and MJC constants for Al/PTFE. The fits are performed over two temperature ranges. 

295 K ≤ T ≤ 329 K 295 K ≤ T ≤ 351 K 

Parameter JC MJC JC MJC 

Unit of 

Measurement 

A 22.71 22.71 22.71 22.71 MPa 

B 160 160.1 160 160 MPa 

n 1.8 1.8 1.8 1.8 — 

C 0.0339 Not used 0.0339 Not used — 

λ Not used 0.0324 Not used 0.0324 — 

(dε/dt)0 1 1 1 1 s

–1

m 1 1 0.707 0.707 — 

Tm 417 417 541 541 K 

Tr 295 295 295 295 K 

ρ 2290 2290 2290 2290 kg/m3

cv 1161 1161 1161 1161 J/kg-K 

Note: Tm as used here is simply a free-parameter in the fitting process and is not the melting temperature of the material. 

 *

Rae and Dattelbaum (8) measured the beta factor for pure PTFE and found it to be in the range from 0.8 to 0.6 at room 

temperature and at a strain rate of 10/s. 

9
0

20

40

60

80

100

120

140

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80

true strain

true stress (MPa)

model, 600/s

model, 4000/s

model, 9000/s

exp. 4000/s

exp. 4000/s

exp. 600/s

exp. 600/s

exp. 9000/s

exp. 9000/s

Figure 9. MJC model predictions, assuming adiabatic conditions, compared to the high-rate 

(SHPB) data. The JC model predictions are essentially identical. 

0

10

20

30

40

50

60

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80

true strain

true stress (MPa)

0.001

0.01

0.1

exp. 0.001/s

exp. 0.01/s

exp. 0.1/s

Figure 10. MJC model predictions, assuming isothermal deformations, compared to the low-rate 

experiments. The JC model predictions are essentially identical.

10
5.2 Zerilli-Armstrong Model for Polymers 

Zerilli and Armstrong have developed constitutive models for metals based on thermally 

activated dislocation motion (9, 10). Recently, they have adapted the basic framework to apply 

to viscoplastic deformation of polymers (11, 12). They have used it to model PTFE (11), and 

Cai et al. (13) have used it to model a composite mixture of PTFE-Al-W. Based on the success 

of these applications, it was decided to use this model to represent the present data. Although the 

model is physically based, the approach taken here is to treat it as a curve fit, i.e., the values 

presented below may not necessarily be realistic beyond the observation that they match the 

existing data. 

The Zerilli-Armstrong equation for polymers relates effective stress, σ, to effective plastic strain, 

εp, temperature, T, and pressure, p, according to 

p –

T

0

1–e Be B e

ωε

−β −α σ= +

ω

T , (4) 

where 

p β β β ln ε −= 10 , (5) 

p α α α ln ε −= 10 , (6) 

ω ω ω ln ε ++= ω ppba p , (7) 

( )

Bpn

B = B 1+ B p pa pb , (8) 

and 

( ) . (9) 

B0pn B = B 1+ B p 0 0pa 0pb

There are 13 parameters that need to be determined. Due to the relatively limited set of data and 

the complexity of the model, a number of parameters are set to be zero. Foremost, all parameters 

leading to a pressure dependent yield stress are set to zero. This is not to imply that the material 

has no pressure dependence, the data is simply not available.* Thus, Bpb = ωp = B0pn = Bpn = B0pb

= 0, and the model simplifies to 

( ) ( )

( )

( T

pba

pa

T

pa

p

ppba

p e

e eB B εαα

εεωω

εββ

εωω

σ 







ln

ln

0

10 ln 10

ln

1 −−

+−

−−

+

− = + )

 

. (10) 

Assuming εp = 0 and T = 295 K, the right-hand term vanishes and a power-law least-squares fit 

to the rate data (figure 6) can be made. 

( ) ⇒ . (11) T

p

T

paeB β 0 β1 σ ε − = Tp k β1 = εσ 

*

PTFE has been shown to have a pressure dependent flow stress (10). 

11
The constant k and β1 are determined by the curve fit. Through k, a relationship is established 

between β0 and Bpa. 

0

pa

1 k β = – ln

T B

⎛ ⎞ ⎜

⎜

⎝ ⎠

⎟

⎟

)

. (12) 

Using this relationship and assuming ε = 4000/s and εp = 0, Bpa can be determined by that value 

which gives an optimum fit to the temperature data of figure 8. Equation 12 then gives β0, and 

the first term on the right-hand side of equation 10 is determined. 



The remaining term deals with strain hardening. Aside from the behavior seen in figure 7 at 

351 K, which may be due to a phase change, there is no convincing data that the strain hardening 

changes with temperature, so set α0 = α1 = 0.* This leaves three terms to be determined: ωa, ωb, 

and B0pa. A least absolute value best fit† is made to the stress-strain data shown in figure 3, 

assuming the high-rate curves are isothermal, i.e., the data are fit to 

( ) ( )

( pba

pa

T

pa

ppba

p e eB B

εωω

σ

εεωω

εββ







ln

1 ln

0

10 ln

+

− = +

+−

−− , (13) 

to obtain estimates for ωa, ωb, and B0pa. This is technically incorrect, but fortunately the 

differences between adiabatic and isothermal behaviors for these experiments are not drastically 

different, and thus the fitting process is simplified. 

As with the Johnson-Cook fits, the resulting parameter set is adjusted in an ad hoc manner to 

obtain the “best” simultaneous fits to the isothermal and adiabatic stress-strain curves, the rate 

hardening, and the thermal softening. The resulting optimum parameters are listed in table 2, 

and the quality of the fit is shown against the experimental data in figures 6, 8, 11, and 12. To 

facilitate comparisons, the parameter sets of Cai et al. (13) for PTFE-Al-W, and Zerilli and 

Armstrong (11) for PTFE and also listed in the table. 

6. Conclusion 

The experiments conducted in this program have been used to populate the JC, MJC, and ZA for 

polymers constitutive models. These models address the inert behavior of the material only. It is 

worth noting that in none of these experiments was any evidence of a reaction, either during the 

experiment or post-mortem, observed.

 *

It was hoped that adjustment of α0 and α1 would help produce a better match between the model and the temperature tests at 

larger deformations. However, this did not appear to be the case so these two parameters were left at zero values. 

†

A numerical technique is used. 

12
Table 2. The ZA constants for Al/PTFE. The values determined by Cai et al. (13) and Zerilli and 

Armstrong (11) for PTFE-Al-W and PTFE are included for comparison. 

Parameter RM-4 PTFE-Al-W ZA PTFE 

Unit of 

Measurement 

β0 0.011672 0.020100 0.020100 1/K 

β1 0.000139 0.000264 0.000264 1/K 

α0 0.000000 0.004780 0.004780 1/K 

α1 0.000000 0.000050 0.000050 1/K 

ωa –3.000 –2000 –3.600 — 

ωb –0.500 –0.625 –0.625 — 

ωp 0.000 –0.031 –0.040 1/MPa 

Bpa 550 4016 4016 MPa 

Bpb 0.000 0.020 0.020 1/MPa 

Bpn 0.000 0.714 0.714 — 

B0pa 25.0 72.4 72.4 MPa 

B0pb 0.000 0.022 0.022 1/MPa 

B0pn 0.000 0.500 0.500 — 

0

20

40

60

80

100

120

140

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80

true strain

true stress (MPa)

ZA 600/s

ZA 4000/s

ZA 8000/s

exp. 600/s

exp. 600/s

exp. 4000/s

exp. 4000/s

exp. 8000/s

exp. 8000/s

Figure 11. Comparison of the ZA model to the SHPB experiments assuming adiabatic conditions. 

13
0

10

20

30

40

50

60

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80

true strain

true stress (MPa)

ZA 0.001/s

ZA 0.01/s

ZA 0.1/s

exp. 0.001/s

exp. 0.01/s

exp. 0.1/s

Figure 12. Comparison of the ZA model to the low-rate experiments assuming isothermal conditions. 

Note the great improvement over the JC fits to the same data, figure 10. 

It is believed that these models are suitable for simulations of low-speed impacts (e.g., the 

cylinder impacts of Casem and Raftenberg [14] and Mock and Holt [15]). However, readers 

shown remain aware of the comments and caveats made above, especially the inability of the JC 

models to capture the strain hardening effects over the full range of strain-rates investigated. It is 

suggested that the JC model could be made to capture this effect by making the strain hardening 

parameters suitable functions of strain-rate. However, it is recommended that the ZA model, 

which is more firmly rooted by a physical foundation, be used in place of the JC fits whenever 

possible. As with any model, readers should exercise caution when using these models outside 

of their calibrated ranges. Note that the elastic behavior of the material has not been characterized. 

Any reader who implements this model will need to address this. One should also take care when 

using these models to represent Al-PTFE mixtures that are not manufactured using this same 

process. For example, there is a considerable difference between this material behavior and that 

measured in Raftenberg (7), although superficially both are basically the same mixture. 

Finally, failure is not addressed in this work. When failure was observed in these tests, it 

appeared to be primarily due to ductile fracture. It is not believed that this material exhibits 

shear banding, and the mechanical properties listed in this work support this (in terms of the 

relative effects of strain and strain-rate hardening and thermal softening). Some fracture work 

has been preformed on Al/PTFE composites; refer to Joyce and Joyce (16).

14
7. References 

1. Follansbee, P. S.; Franz, C. Wave Propagation in the Split-Hopkinson Pressure Bar. J. Eng. 

Mat. Tech. 1983, 105, 61. 

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

Pressure Bar. J. Eng. Mat. Tech. 1990, 112, 309–314. 

3. Follansbee, P. S. Metals Handbook, 8; American Society for Metals: Metals Park, OH, 

1985; p 198. 

4. Ramesh, K. T.; Narasimhan, S. Finite Deformations and the Dynamic Measurement of 

Radial Strains in Compression Kolsky Bar Experiments. Int. J. Solids Struct. 1996, 33 (25), 

3723–38. 

5. Johnson, G. R.; Cook, W. H. A Constitutive Model and Data for Metals Subjected to Large 

Strains, High Strain Rates, and High Temperatures. In: Proceedings from the 7th 

International Symposium on Ballistics, 1983, pp 541–547. 

6. Holmquist, T. J.; Johnson, G. R. Determination of Constants and Comparison of Results for 

Various Constitutive Models. J. de Physique IV 1991. 

7. Raftenberg, M.; Mock, W.; Kirby, G. Modeling the Impact Deformation of Rods of a 

Pressed PTFE/Al Composite Mixture. Accepted for publication in Int. J. Impact Eng. 

8. Rae, P. J.; Dattelbaum, D. M. The Properties of poly(tetrafluoroethylene) (PTFE) in 

Compression. Polymer 2004, 45, 7615–7625. 

9. Zerilli, F. J.; Armstrong, R. W. Dislocation-Mechanics Based Constitutive Relations for 

Material Dynamics Calculations. J. Appl. Phys. 1987, 61 (5), 1816–1825. 

10. Zerilli, F.; Armstrong, R. J. Description of Tantalum Deformation Behavior by Dislocation 

Mechanics Based Constitutive Relations. Appl. Phys. 1990, 68, 1580–1591. 

11. Zerilli, F.; Armstrong, R. A Constitutive Equation for the Dynamic Deformation Behavior 

of Polymers. J. Material Sci. 2007, 42 (12), 4562–4574. 

12. Zerilli, F.; Armstrong, R. Thermal Activation Based Constitutive Equations for Polymers. 

J. Phys. IV France 2000, 10, Pr9.3–8. 

13. Cai, J.; Walley, S.; Hunt, R.; Pround, W.; Nesterenko, V.; Meyers, M. High-Strain, High 

Strain-Rate Flow and Failure in PTFE/Al/W Gransular Composites. Mater. Sci. Eng. A

2007, doi:10.1016/j.msea.2007.03.068.

15
14. Casem, D.; Raftenberg, M. Initiation Experiments on Aluminum/PTFE. Proceedings of the 

2006 JANNAF Conference, San Diego, CA, December 2006. 

15. Mock, W.; Holt, W. Impact Initiation of Rods of Pressed Polytetraflouroethylene (PTFE) 

and Aluminum Powders. In: Shock Compression of Condensed Matter-2005, 2006; 1097. 

16. Joyce, J.; Joyce, P. Toughness Characterization of a Metal Filled PolyTetraFlouroEthylene 

using the J-integral. Eng. Fracture. Mech. 2004, 71, 2512–2531. 

16
Appendix. Strain Measures Defined 

This appendix clarifies the definitions of engineering stress and strain and true stress and strain 

as used in this report. The unqualified use of any of these terms implies positive values in 

compression. 

Engineering stress, s, is defined as 

A0

P

s = , (A-1) 

where P is the compressive load applied to the specimen and A0 is its original cross-sectional 

area. 

True stress, σ, is defined as 

A

P

=σ , (A-2) 

where A is the instantaneous cross-sectional area of the specimen, i.e., the area at any instant in 

time as the specimen deforms. 

Engineering strain, e, is defined by 

L0

e

δ = , (A-3) 

where δ is the amount the specimen is compressed and L0 is the initial length of the specimen. 

Note that L, the instantaneous length, can then be found by L . L0 δ−=

True strain, ε, is defined by 

L

dL dε −= . (A-4) 

Here, dL is an increment in specimen length, i.e., the negative sign is needed to make this 

positive in compression. This can be integrated for the conditions of the compression test to give 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−=ε

L0

L ln . (A-5) 

By equation (A-3), true strain can be related to engineering strain by 

−−=ε )e1ln( . (A-6)

17
If the deformation of a specimen is incompressible, and if the initially cylindrical specimen 

remains cylindrical during the deformation process,* the instantaneous volume is equal to the 

initial volume at any instant, i.e., 

, (A-7) 0

2

0

2 = LdLd

where d0 and d are the initial and instantaneous diameters of the specimen. Using this 

relationship, true stress can be determined from the engineering quantities by 

( −=σ e1s ). (A-8) 

Furthermore, the tensile radial engineering strain of the specimen, defined as 

0

0

r d

dd

e − = , (A-9) 

can be related to the axial engineering strain of the specimen by 

1

e1

1

er − − = . (A-10) 

This applies only to incompressible behavior. It is this equation that was plotted in figure 2 of 

this report. 

 *

This is only approximately true in compression testing because of “barreling” that occurs due to friction at the platens. 

18
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