ASM Metals HandBook Vol. 8 - Mechanical Testing and Evaluation
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incident bar may be considerably greater than the actual sample displacement. For two tests, if the experimental
conditions and the area of contact between the sample and the inserts and that between the inserts and the bars
are kept the same, the error produced by the presence of the inserts will also be essentially the same in both
tests.
To apply the DSM technique, two high strain rate tests are conducted on the Hopkinson bar, with two samples
of different lengths. All other experimental conditions such as the striker bar length, the striker velocity, the
pulse-shaper dimensions, and the sample temperature are maintained the same in both tests. After the two tests
are conducted for the same stress, the difference in the strain can be attributed to the difference in the sample
lengths. The displacement of the shorter sample is subtracted from that of the longer sample at several stress
levels, and the differential displacement is obtained as a function of the stress. The displacement data thus
obtained correspond to the displacement data for a sample with a length equaling the difference in the lengths
of the two test samples. Thus, the stress-displacement data obtained now are the actual data that would
correspond to a sample with a length equal to the difference in the lengths of the test samples.
Using a well-designed ramped loading, it is possible to measure the elastic modulus of ceramics at high
temperatures and high strain rates by this procedure with considerable accuracy. Figure 25 shows the strain as a
function of time for two samples of silicon nitride tested at 730 °C (1350 °F). The samples are of different
lengths and are both tested in the elastic regime. It may be noted that the strain rate is nearly the same for the
two tests.
Fig. 25 Strain-time data for two samples of different lengths tested at 730 °C (1350 °F)
References cited in this section
1. S. Nemat-Nasser and J.B. Isaacs, Direct Measurement of Isothermal Flow Stress of Metals at Elevated
Temperatures and High Strain Rates with Application to Ta and Ta-W Alloys, Acta Metall., Vol 45,
1997, p 907–919
16. U.F. Kocks, A.S. Argon, and M.F. Ashby, Thermodynamics and Kinetics of Slip, Prog. Mater. Sci.,
Pergamon Press, New York, Vol 19, p 68–170
Recovery Hopkinson Bar Techniques
Sia Nemat-Nasser, University of California, San Diego
Acknowledgments
This work has been supported by an ARO/MURI DAAH04-96-1-0376 grant to the University of California,
San Diego.
Recovery Hopkinson Bar Techniques
Sia Nemat-Nasser, University of California, San Diego
References
1. S. Nemat-Nasser and J.B. Isaacs, Direct Measurement of Isothermal Flow Stress of Metals at Elevated
Temperatures and High Strain Rates with Application to Ta and Ta-W Alloys, Acta Metall., Vol 45,
1997, p 907–919
2. S. Nemat-Nasser and Y. Li, Flow Stress of FCC Polycrystals with Application to OFHC Cu, Acta
Mater., Vol 46, 1998, p 565–577
3. S. Nemat-Nasser, W. Guo, and M. Liu, Experimentally-Based Micromechanical Modeling of Dynamic
Response of Molybdenum, Scr. Mater., Vol 40, 1999, p 859–872
4. K.H. Hartman, H.D. Kunze, and L.W. Meyer, Metallurgical Effects in Impact Loaded Materials, Shock
Waves and High-Strain-Rate Phenomena in Metals, M.A. Meyers and L.E. Murr, Ed., Plenum, London,
1986, p 325
5. S. Nemat-Nasser, J.B. Isaacs, G. Ravichandran, and J.E. Starrett, High Strain Rate Testing in the U.S.,
Proc. TTCP TTP-1 Workshop on Techniques of Small Scale High Strain Rate Studies, 26 April 1988,
Materials Science Laboratory, Melbourne, Australia, 1988
6. S. Nemat-Nasser and S.N. Chang, Compression-Induced High Strain Rate Void Collapse, Tensile
Cracking, and Recrystallization in Ductile Single and Polycrystals, Mech. Mater., Vol 10, 1990, p 1–17
7. S. Nemat-Nasser, J.B. Isaacs, and J.E. Starrett, Hopkinson Techniques for Dynamic Recovery
Experiments, Proc. R. Soc. (London) A, Vol 435, 1991, p 371–391
8. A. Thakur, S. Nemat-Nasser, and K.S. Vecchio, Bauschinger Effect in Haynes 230 Alloy: Influences of
Strain Rate and Temperature, Metall. Mater. Trans., Vol 27A, 1996, p 1739–1748
9. A. Thakur, S. Nemat-Nasser, and K.S. Vecchio, Dynamic Bauschinger Effect, Acta Mater., Vol 44,
1996, p 2797–2807
10. W.P. Rogers, J.B. Isaacs, and S. Nemat-Nasser, Effect of Microstructural Damage on Ultrasonic
Velocity and Elastic Moduli of Partially Stabilized Zirconia, Review of Progress in Quantitative
Nondestructive Evaluation, D.B. Thomson and D.E. Chimenti, Ed., Plenum, New York, 1989, p 1827–
1832
11. W.P. Rogers and S. Nemat-Nasser, Transformation Plasticity at High Strain Rate in Magnesia-Partially-
Stabilized Zirconia, J. Am. Ceram. Soc., Vol 73, 1990, p 136–139
12. V. Sharma, S. Nemat-Nasser, and K.S. Vecchio, Dynamic-Compression Fatigue of Hot-Pressed Silicon-
Nitride, Exp. Mech., Dec 1994, p 315–323
13. G. Subhash and S. Nemat-Nasser, Dynamic Stress-Induced Transformation and Texture Formation in
Uniaxial Compression of Zirconia Ceramics, J. Am. Ceram. Soc., Vol 76, 1993, p 153–165
14. S. Nemat-Nasser, Y Li, and J.B. Isaacs, Experimental/Computational Evaluation of Flow Stress at High
Strain Rates with Application to Adiabatic Shearbanding, Mech. Mater., Vol 17, 1994, p 111–134
15. R. Kapoor and S. Nemat-Nasser, Determination of Temperature Rise during High Strain Rate
Deformation, Mech. Mater., Vol 27, 1998, p 1–12
16. U.F. Kocks, A.S. Argon, and M.F. Ashby, Thermodynamics and Kinetics of Slip, Prog. Mater. Sci.,
Pergamon Press, New York, Vol 19, p 68–170
Split-Hopkinson Pressure Bar Testing of Soft
Materials
George T. (Rusty) Gray III and William R. Blumenthal, Los Alamos National Laboratory
Introduction
THE HIGH-STRAIN-RATE, STRESS-STRAIN RESPONSE of polymers and polymeric-composite materials
has received increased scientific and industrial attention in recent years. This interest is driven by the need for
predictive constitutive model descriptions to support large-scale finite-element simulations of polymers
subjected to high-strain-rate (impact) loading. Applications for which high-strain-rate loading represents an
important service environment for polymers include:
• Automotive crashworthiness, where polymers, polymer foams, and polymeric composites have seen
significantly increased structural usage
• Aerospace damage tolerance, including foreign-object damage such as during bird ingestion in jet
engines equipped with polymeric composite blades and vanes
• Sporting goods applications where dynamic behavior and impact resistance of polymer composites is
critical, such as tennis rackets, golf shafts, mountain bike frames, and so on
• Polymeric binder constituents used in polymer-bonded explosives (PBXs) and rocket propellants
The establishment of more physically based constitutive models to describe complex loading paths and to
understand the effect of manufacturing processes on polymers and polymeric composites subjected to dynamic
loading environments requires a detailed knowledge of the separate and synergistic effects of temperature and
strain rate on their mechanical response. Split-Hopkinson pressure bar (SHPB), or Kolsky-bar, testing remains
the main experimental method to characterize the high-strain-rate mechanical behavior of these materials. The
specialized aspects required to accurately quantify the behavior of these “soft” materials using the SHPB is the
subject of this article.
Split-Hopkinson Pressure Bar Testing of Soft Materials
George T. (Rusty) Gray III and William R. Blumenthal, Los Alamos National Laboratory
Background
A significant number of previous studies have probed the high-rate constitutive response of a wide variety of
polymers and polymer composites, starting with Kolsky in 1949 through the present (Ref 1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11). Several in-depth reviews by Walley and others summarize the broad spectrum of polymers (Ref 12, 13,
14). Beginning with the high-rate pressure-bar studies of Kolsky (Ref 1) on polythene (polymerized ethylene)
and rubber, it has been shown that:
• The effective elastic moduli of polymeric materials are strongly influenced by strain rate.
• The flow stress of a ductile polymer reaches a maximum strength at some moderate strain level under
constant, high-strain-rate loading.
• Some polymers exhibit no permanent plastic flow after substantial high rate straining, the strain
recovery effect being time dependent.
• Stress-wave propagation through some polymers, such as polythene, is so dispersive that over a certain
sample length no transmitted Hopkinson bar signal is measured.
Experimental results for a large number of polymers over a wide range of strain rates have exhibited a linear
relationship between the compressive yield stress of the polymer and log (strain rate) (Ref 13, 14).
The low inherent flow strength and low elastic impedance of polymers and polymeric composites pose
additional challenges to the accurate measurement of their high-strain-rate uniaxial stress mechanical response
using an SHPB (Ref 3). Resolution of the low flow strengths in polymers has prompted the adoption of
alternate pressure-bar materials in deference to the high-strength steels classically used. Because lower modulus
bar materials increase the signal-to-noise level of measurement strain gages, their selection is desirable to
facilitate high-resolution dynamic testing of low-strength materials such as polymers, as long as the yield
strength of the sample material remains well below that of the bar materials. Researchers have documented the
use of bar materials possessing a range of elastic stiffnesses from maraging steel to titanium to aluminum to
magnesium (Ref 15, 16) and, finally, to polymer bars (Ref 3, 17, 18, 19). Table 1 summarizes the elastic
modulus and impedance values for a range of typical pressure-bar materials.
Table 1 Comparison of the impedance properties of bar materials for split-Hopkinson pressure bar
testing
Elastic modulus
(E)
Bar material
GPa 10
6
ksi
Wave impedance
(ρ/C
0
),
km · g/s · cm
3
Impedance relative to
steel, (ρ/C
0
)
x
/(ρ/C
0
)
steel
,
%
Steel 212 31 40.8 100
Ti-6Al-4V 115 17 22.8 56
Aluminum 90 13 13.5 33
Magnesium 45 7 8.6 21
Polymethyl methacrylate
(PMMA)
4 1 2.2 5
The ideal materials properties desired for pressure bars used in an SHPB are multifaceted and reflect a range of
effects to control physical parameters that either improve measurement fidelity and robustness or are the chief
sources of potential errors that can complicate data analysis and resolution and, thereby, test accuracy. The
selection of an appropriate pressure-bar material represents a balance of the tradeoffs pursuant to a given
material. The traditional use of metallic pressure bars reflects the fact that many metals and alloys exhibit a
number of positive attributes in regard to their use as “load cells” in parallel with the sample and as wave
guides for high-fidelity wave propagation. Ideal pressure bars should exhibit all of the following properties, and
any deviations in practice constitute a compromise:
Good wave guide properties:
• Very low attenuation. Signal loss in perfectly elastic materials is essentially zeroed.
• Little or no time-dependent response, including frequency dependence, anelasticity, and dispersion
• Little radial wave attenuation. A low Poisson ratio is desirable.
• Large thermal range, both above and below ambient, over which the elastic properties are constant
• Low interference potential—low susceptibility to picking up stray signals during testing (high
impedance mismatch with air, little coupling between bar and bar stanchions due to large impedance
mismatch)
• Large input range before bars exhibit distortion—high strength/high elastic limit
Good load cell properties:
• High sensitivity for high-gage signal-to-noise, low-modulus bars
• Large input range—high-strength and high-elastic-limit bars
• High-wave-speed bar material resulting in good dynamic range (high bandwidth)
• Little signal interference from strain gages and glue joints—high modulus ratio between the pressure bar
material and the glue and strain gages
• Minimal coupling of pressure bars with stanchions, strain-gage cables, and connections
• Low Poisson ratio of pressure bars to maximize one-dimensional strain conditions in pressure bars since
longitudinal strain is measured (i.e., minimize two-dimensional strain effects on gage measurements)
The adoption of increasingly lower impedance pressure bar materials for Hopkinson bar testing over the past
ten years is a direct result of the desire to increase the signal-to-noise gain to adequately resolve the low elastic
strains produced in the pressure bars during testing of low-strength materials. Even with a high-gain, strain-
gage signal conditioner using a 500× gain, it can be difficult or impossible to resolve the transmitted pressure
pulse of a soft polymer using traditional steel pressure bars. In essence, if the wave impedance (ρC
0
) of the
sample material under investigation is much lower than that of the pressure bars, the magnitude of the
transmitted pulse can become very weak. In addition, as described in the article “Classic Split-Hopkinson
Pressure Bar Testing,” several different wave analyses should be used to calculate the sample stress from the
incident, reflected, and transmitted bar time-resolved measured strains to validate stress-state equilibrium.
Issues of stress-state equilibrium are addressed subsequently as pertains to polymer testing.
The additional complications inherent to testing viscoelastic materials and the care that must be exercised to
accurately measure their mechanical behavior in an SHPB were introduced in the seminal work of Kolsky (Ref
1) on plastics and rubber. Kolsky (Ref 1, 20) discussed the difficulty in resolving the low-magnitude
transmitted pulses for rubber or polythene and, thereafter, the importance of using thin sample thicknesses to
reduce the time a sample takes to “ring up” to a state of uniform uniaxial stress. His studies demonstrated that
thin samples were critical when studying the high-rate constitutive response of plastics and rubber or the
“assumption that the pressure on both sides of the specimen are sensibly the same is no longer valid” (Ref 1). In
this article, some of the specialized SHPB developments within the last ten years to facilitate testing soft
materials are detailed. These techniques include the data-reduction techniques and assumptions required to use
polymer pressure bars, the importance of sample-size considerations to polymer testing, and temperature-
control methodologies germane to measuring the high-strain-rate uniaxial stress response of polymers and other
soft materials.
References cited in this section
1. H. Kolsky, An Investigation of the Mechanical Properties of Materials at Very High Rates of Loading,
Proc. Phys. Soc. (London) B, Vol 62, 1949, p 676–700
2. S.H. Ahmad, Flow Stress of High Density Polyethylene and Nylon 66 at High Rates of Strain, Polym.
Int., Vol 28, 1992, p 291–294
3. G.T. Gray III, W.R. Blumenthal, C.P. Trujillo, and R.W. Carpenter II, Influence of Temperature and
Strain Rate on the Mechanical Behavior of Adiprene L-100, J. Phys. (France) IV Colloq., C3
(EURODYMAT 97), Vol 7, 1997, p 523–528
4. N.N. Dioh, A. Ivankovic, P.S. Leevers, and J.G. Williams, The High Strain Rate Behaviour of
Polymers, J. Phys. (France) IV Colloq., C8 (DYMAT 94), Vol 4, 1994, p 119–124
5. A. Trojanowski, C. Ruiz, and J. Harding, Thermomechanical Properties of Polymers at High Rates of
Strain, J. Phys. (France) IV Colloq., C3 (EURODYMAT 97), Vol 7, 1997, p 447–452
6. A.M. Bragov and A.K. Lomunov, Mechanical Properties of Some Polymers and Composites at Strain
Rate of 1000/s, J. Phys. (France) IV Colloq., C8 (DYMAT 94), Vol 4, 1994, p 337–342
7. S.M. Walley, D. Xing, and J.E. Field, Mechanical Properties of Three Transparent Polymers in
Compression at a Very High Rate of Strain, Impact and Dynamic Fracture of Polymers and
Composites, J.G. Williams and A. Pavan, Ed., Mechanical Engineering Publications Ltd., 1995, p 289–
303
8. S.M. Walley, J.E. Field, G.M. Swallowe, and S.N. Mentha, The Response of Various Polymers to
Uniaxial Compressive Loading at Very High Rates of Strain, J. Phys. (France) Colloq., C5 (DYMAT
85), Vol 46, 1985, p 607–616
9. S.M. Walley, J.E. Field, P.H. Pope, and N.A. Safford, The Rapid Deformation Behaviour of Various
Polymers, J. Phys. (France) III, Vol 1, 1991, p 1889–1925
10. S.M. Walley, J.E. Field, and N.A. Safford, A Comparison of the High Strain Rate Behaviour in
Compression of Polymers at 300K and 100K, Proc. 8th Int. Conf. on Deformation, Yield, and Fracture
of Polymers, edited by p. paper 16, Plastics and Rubber Institute, London, 1991
11. S. Hamdan and G.M. Swallowe, The Strain-Rate and Temperature Dependence of the Mechanical
Properties of Polyetherketone and Polyetheretherketone, J. Mater. Sci., Vol 31, 1996, p 1415–1423
12. S.M. Walley, J.E. Field, P.H. Pope, and N.A. Safford, A Study of the Rapid Deformation Behaviour of a
Range of Polymers, Philos. Trans. R. Soc. (London) A, Vol 328, 1989, p 1–33
13. S.M. Walley and J.E. Field, Strain Rate Sensitivity of Polymers in Compression from Low to High
Strain Rates, DYMAT J., Vol 1, 1994, p 211–228
14. S.N. Kukureka and I.M. Hutchings, Yielding of Engineering Polymers at Strain Rates of up to 500 s
-1
,
Int. J. Mech. Sci., Vol 26, 1984, p 617–623
15. G.T. Gray III, D.J. Idar, W.R. Blumenthal, C.M. Cady, and P.D. Peterson, High- and Low-Strain Rate
Compression Properties of Several Energetic Material Composites as a Function of Strain Rate and
Temperature, 11th Detonation Symposium, (Snow Mass, CO), J. Short, Ed., Amperstand Publishing,
2000
16. J.E. Field, S.M. Walley, N.K. Bourne, and J.M. Huntley, Review of Experimental Techniques for High
Rate Deformation Studies, Proc. Acoustics and Vibration Asia '98, 1998 (Singapore), p 9–38
17. G. Gary, J.R. Klepaczko, and H. Zhao, Generalization of Split Hopkinson Bar Technique to Use
Viscoelastic Materials, Int. J. Impact Eng., Vol 16, 1995, p 529–530
18. G. Gary, L. Rota, and H. Zhao, Testing Viscous Soft Materials at Medium and High Strain Rates,
Constitutive Relation in High/Very High Strain Rates, K. Kawata and J. Shioiri, Ed., Springer-Verlag,
Tokyo, 1996, p 25–32
19. H. Zhao and G. Gary, On the Use of SHPB Techniques to Determine the Dynamic Behavior of
Materials in the Range of Small Strains, Int. J. Solids Struct., Vol 33, 1996, p 3363–3375
20. H. Kolsky, The Propagation of Stress Pulses in Viscoelastic Solids, Philos. Mag., 8th series, Vol 1,
1956, p 693–710
Split-Hopkinson Pressure Bar Testing of Soft Materials
George T. (Rusty) Gray III and William R. Blumenthal, Los Alamos National Laboratory
Split-Hopkinson Bar Testing of Polymers
The quantification of the uniaxial stress-strain compressive response of low mechanical impedance soft
materials in the SHPB has been the subject of significant innovation since 1992. Higher-resolution
measurement of the stress-strain responses of increasingly lower-flow-strength materials has led to the adoption
of lower-impedance pressure bars, specifically titanium-alloy, aluminum-alloy, magnesium or magnesium-
alloy, and, finally, polymer pressure bars. The technical advantages and disadvantages of each are discussed in
this section.
Polymer Pressure Bars
Based on the low impedance of many structural plastics, such as polymethyl methacrylate (PMMA) and
polycarbonate (PC), a number of researchers have explored the benefits of using polymers for the pressure bars
in an SHPB to test low-impedance polymers and foams (Ref 17, 18, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32, and 33). These researchers have pursued the use of polymer bars to access the significant signal-to-noise
gains that polymeric bars promise, as summarized in Table 1. While the lower inherent impedance of polymers
offers positive attributes to SHPB testing of soft materials, their use requires additional analysis for data
reduction, temperature complications, and additional restrictions compared with traditional metallic pressure
bars.
Polymeric materials, unlike metals and alloys loaded in their elastic regime, are not ideally linear elastic, nor do
they exhibit undistorted wave propagation. Polymeric materials, on the contrary, exhibit significant stress-wave
attenuation, wave dispersion due to time and frequency dependency, a strong temperature dependency between
±50 °C (±90 °F), and are susceptible to gradual degradation in their “elastic” properties during prolonged usage
due to the effects of oxidation and ultraviolet radiation. Notwithstanding these shortcomings, numerous
researchers have explored the opportunities that the lower impedance and low modulus of polymers offers to
the field of SHPB testing.
Researchers exploring the high-strain-rate, stress-strain response of lower-impedance soft materials have
worked to quantify how the nonlinear response of polymeric bars alters the classic SHPB data reduction. In
metallic bars, the linear elastic response of the bars defines that the measured strain pulse describes not only the
local strain wave but also the associated stress wave and the particle velocity at different positions along the
pressure bar. On the contrary, a viscoelastic material, such as PMMA, exhibits both wave attenuation and wave
dispersion. As a wave travels down a polymer pressure bar, the wave amplitude decreases due to attenuation,
and the shape of the waveform becomes distorted. Accordingly, strains measured at a given position on a
viscoelastic bar do not represent the strain pulse at another position along the pressure bar without complex
manipulation and corrections. Therefore, the “classic” SHPB analysis based on the undistorted character of
wave propagation must be modified when viscoelastic bars are used instead of linear-elastic metallic bars. The
measured strain pulse on a viscoelastic bar represents the strain wave only at the measured point, but neither the
stress nor the particle velocity can be directly deduced from a measured strain signal in a viscoelastic bar by
simply using a proportional coefficient as in the case of metals where the elastic modulus is used. This fact has
been demonstrated by a number of researchers (Ref 17, 21, 23, 24, and 25). Quantification of the stresses,
strains, and strain rates in a sample to be tested in an SHPB equipped with polymeric pressure bars, therefore,
requires a viscoelastic wave analysis of the pressure-bar responses. Use of a simple linear elastic data reduction
for polymeric bars is inappropriate and will lead to errors as large as 52% as quantified by Wang and others
(Ref 23).
Use of polymeric bars in an SHPB, therefore, requires the derivation of an accurate predictive material
constitutive model description of the dispersion and dissipation in the viscoelastic bars to deduce the stress-
strain behavior of the sample being tested. Zhao and Gary (Ref 24) proposed using a stress-strain constitutive
model derived by using the wave recorded at one point on the incident bar as input data. The model parameters
are thereafter determined through comparison with the predicted wave and the wave recorded at another point
on the incident bar. A similar model reduction can be conducted for the transmitted pressure bar. Once the
model parameters have been identified, this data can also be used for the wave dispersion correction (Ref 34).
Quantification and comparison of the forces in the transmitted bar are conducted using a bars-together
configuration, where no sample is present. Here, the stress in the incident and transmitted bars are ideally equal.
Using this technique, Zhao and Gary (Ref 24) characterized the frequency and phase velocity of their
viscoelastic bars. The complex modulus of the polymer pressure bars is then calculated from the phase velocity
and attenuation. Both of these quantities are bar-diameter dependent, and the frequency characteristics become
less and less accurate as the wave frequency and the bar diameter increase. The work of Zhao and Gary (Ref
24) showed that in the case of PMMA pressure bars, selection of bar with diameters less than 10 mm (0.4 in.)
minimized errors.
Sawas, Brar, and Brockman (Ref 33) similarly proposed measurement of the strain or particle velocity at two
different locations along a polymer pressure bar. The strain measurements for the two positions were then
reduced analytically to derive the model parameters for a viscoelastic description of the pressure bars. Figure 1
presents an example of the similitude of stress-strain mechanical property for 1100 aluminum alloy using a
conventional SHPB and what can be obtained using cast acrylic pressure bars. In the experimental setup
described by Sawas, Brar, and Brockman (Ref 33), high-strength titanium anvils were placed between the
aluminum alloy sample and the two polymeric pressure bars to prevent damage to the bars during testing and to
minimize elastic deformation at the specimen bar interface.
Fig. 1 Comparison of the stress-strain data measured for 1100 aluminum alloy using a conventional
metallic pressure bar and an acrylic pressure bar for split-Hopkinson pressure bar testing. Source: Ref
33
Zhao, Gary, and Klepaczko (Ref 29) have additionally analyzed the use and modeling of polymeric pressure
bars in the SHPB. In this model, a Fourier stationary wave approach is presented. This modeling approach was
proposed because the numerical efficiency and precision are independent of the distance between the
bar/specimen interface and the gage location. Additionally, this approach addresses the need to deal with the
three-dimensional effects, such as bar-diameter effects, without introducing additional computational penalties.
Their analysis of the solution of the propagation of the harmonic waves in an elastic bar reduces to the
Pochhammer-Chree frequency equation, which gives the relation between the wave number and the frequency
of the wave. Further analysis by Bacon (Ref 35, 36) supports the accuracy of harmonic wave modeling and its
applicability to any solid cylindrical cross-sectional linear viscoelastic-infinite pressure bar.
The constitutive behavior of the polymeric bars, independent of whether a transient wave analysis (Ref 23) or a
Fourier stationary harmonic wave analysis (Ref 29) is employed, is modeled using a Maxwell element-based
approach to describe linear viscoelastic behavior. Zhao, Gary, and Klepaczko (Ref 29) proposed using a 4
Voigt element and 1 spring in a series Maxwell element rather than a more classic 3-element Maxwell model (a
mechanical analog composed of a spring in parallel with a dashpot in series with a spring). Due to the almost
constant damping and phase velocity, they selected the more complex form rather than a 3-element Maxwell
model to model the high frequencies. Upon implementation of this model, Zhao, Gary, and Klepaczko (Ref 29)
demonstrated that analyses of an SHPB test cannot be performed by simply shifting the effects of elastic
geometry dispersion and the effects of the one-dimensional viscoelastic wave propagation. This is because of
the strong effect of bar geometry on wave attenuation in the polymeric pressure bars.
Researchers interested in implementing one of these constitutive modeling approaches should examine in detail
the appropriate papers. The main problem with using polymeric pressure bars to quantify the constitutive
response of soft materials is the dependency on a complex geometry-dependent viscoelastic model analysis of
the pressure bar signals to deduce sample behavior. This is in contrast to the relatively simple “classic” linear
elastic data reduction where metallic pressure bars are used. These papers have demonstrated that a detailed
solution to the dissipation and dispersion behavior of polymeric pressure bars can be accurately obtained for
constant ambient temperature testing. Accordingly, sufficiently accurate dynamic stress-strain behavior of soft
materials can be measured using polymeric pressure bars.
However, modeling to date of the polymer pressure-bar constitutive response has only been successfully
demonstrated for ambient-temperature testing. In the case of metallic bars, the elastic properties are essentially
constant well above and below ambient temperature. This allows testing without modification of the analysis
and without the extremely difficult task of controlling the entire bar system and sample temperature to within a
few degrees. In the case of polymer bars, testing at temperatures below and above ambient temperature also
requires a strain-rate- and temperature-dependent constitutive model and a thermal profile of each test if the
temperature of the entire bar cannot be maintained at the same level for accurate modeling of the pressure-bar
response and thereafter the data reduction of the sample mechanical behavior.
Lower-Impedance Metallic Pressure Bars
The use of increasingly lower impedance titanium (Ti) (Ref 3, 16), aluminum (Al) (Ref 13, 37, 38), and
magnesium (Mg) (Ref 15, 16, 39) metallic pressure bars to achieve higher-resolution transmitted bar signals
requires no additional data reduction assumptions or caveats. Analysis of the stresses and strains in test samples
using any metallic pressure bar material maintained in their elastic regimes can be readily conducted per the
linear-elastic bar analysis detailed in the article “Classic Split-Hopkinson Pressure Bar Testing.” Unlike the
case of using polymeric pressure bars, the data reduction to determine the stress-strain response of a sample
tested using metallic pressure bars benefits from not only the validity of a simple linear-elastic bar response but
also the relatively undistorted character of wave propagation in metallic bars (wave dispersion must still be
corrected for maximum fidelity, as discussed in the article “Classic Split-Hopkinson Pressure Bar Testing”).
The absolute resolution of metallic pressure bars can be additionally increased through the concurrent use of
high-gain, wide bandwidth, strain-gage signal amplifiers. Low-noise signal conditioners capable of gains up to
1000× and 3 MHz bandwidths are now commercially available (Ref 15). The use of pressure bars of 6061-T6
aluminum alloy or AZ31B magnesium alloy have proved capable of accurately resolving the mechanical
response of a number of polymeric materials. Figure 2 illustrates the measurement of the dynamic stress-strain
behavior of a soft Estane (B.F. Goodrich, Richfield, OH)-based binder material using magnesium pressure bars
(Ref 15). The resolution of the 1- and 2-wave stress-strain responses for this binder material at stress levels
below 1 MPa (145 psi) demonstrates that a classic metallic SHPB using magnesium bars can accurately resolve
low-amplitude transmitted strain pulses. The delay in equilibrium, which is evident in Fig. 2 by the difference
between the 1- and 2-wave signals, is consistent with the pronounced dispersive damping of stress-wave
propagation through low-strength viscoelastic specimens. The ramifications of this pronounced lag in stress-
state stability is discussed in the next section of this article.
Fig. 2 Stress-strain response of a viscoelastic estane-based polymeric energetic binder tested at 0 °C (30
°F) measured with magnesium pressure bars and using an incident bar strain-gage gain of 200× and
transmitted bar strain-gage gain of 500×. W1, one-wave signal; W2, two-wave signal. Source: Ref 15
An additional recent approach to increase the sensitivity of an SHPB to resolve the stress-strain behavior of
polymers is the use of a hollow aluminum transmitted pressure bar. Chen and others (Ref 38) have
demonstrated that using a hollow high-strength aluminum alloy transmitted pressure bar can significantly
increase the signal to noise measured during the SHPB testing of RTV630 silicon rubber. A pulse shaper was
used with the hollow aluminum alloy transmitted bar to simultaneously increase the rise time of the incident
pulse as well as to filter out the high-frequency components of the incident waveform, thereby reducing the
effect of the end cap of the hollow transmitted bar (Ref 38). While an end cap can be used to suppress damage
to the ends of the bar and not interfere with the transmitted stress signal, the presence of an additional interface
at the end cap/incident bar intersection will significantly complicate the reflected strain signal, which is needed
to calculate the sample strain and strain rate. The influence of an end cap on a hollow pressure bar on stress-
state equilibrium in a polymer sample should be evaluated by comparing the 1- and 2-wave analyses of the state
of stress in the sample, as described subsequently.
While lower-impedance metallic bars can significantly increase the signal to noise to resolve the stress-strain
response of low-strength solids, there remain classes of soft polymeric materials that cannot be analyzed using
conventional split-Hopkinson bar data analysis. For all types of Hopkinson bar testing, the assumption of one-
dimensional wave propagation in the bar and sample requires that true stress in a sample in the Hopkinson bar
cannot be extracted for materials whose volumes are not conserved. The instantaneous sample area used in Eq 6
in the article “Classic Split-Hopkinson Pressure Bar Testing” is deduced from the reflected strain signal in the
incident bar, assuming that the constancy of volume assumption is valid in the sample (i.e., there is a fixed
relationship between sample cross-sectional area and its length). In addition, samples not possessing constant
volumes by their very nature of compacting or exhibiting nonhomogeneous deformation processes prior to
achieving a “bulk” response during testing will exhibit significant time delays before the sample attains a
uniform state of uniaxial stress-state equilibrium (if at all) during testing. These materials include metallic and
polymeric foams, honeycomb structures, and porous compacts, including types of wood, for which mechanical
loading produces densification or porosity during testing.
Valid Hopkinson bar testing of materials for which the constant volume criterion is not valid, therefore, requires
additional sample diagnostics during the duration of the test to calculate true stress; simultaneous use of high-