Split Hopkinson pressure bar (SHPB) testing has traditionally been carried out using metal bars. For testing low stiffness materials such as rubbers or low strength materials such as low density cellular solids considered primarily herein, there are many advantages to replacing the metal bars with polymer bars. An investigation of a number of aspects associated with the accuracy of SHPB testing of these materials is reported. Test data are used to provide qualitative comparisons of accuracy using different bar materials and wave-separation techniques. Sample results from SHPB tests are provided for balsa, Rohacell foam and hydroxyl-terminated polybutadiene. The techniques used are verified by finite-element (FE) analysis. Experimentally, the material properties of the bars are determined from impact tests in the form of a complex elastic modulus without curve fitting to a rheological model. For the simulations, a rheological model is used to define the bar properties by curve fitting to the experimentally derived properties. Wave propagation in a polymer bar owing to axial impact of a steel bearing ball is simulated. The results indicate that the strain histories can be used to determine accurately the viscoelastic properties of polymer bars. An FE model of the full viscoelastic SHPB set-up is then used to simulate tests on hyperelastic materials.
Many factors affect the accuracy of test data that are obtained from split Hopkinson pressure bar (SHPB) tests performed on soft materials to large strains. Here, ‘soft’ refers to materials that deform to large strains (of the order of 50%) at low levels of stress (of the order of 1 MPa), for example low density cellular solids or rubbers. Substantial reviews of both classical SHPB testing and testing of soft materials are available elsewhere [1,2]. Some background on SHPB testing of soft materials is provided in §2. Experimentally, confidence in the test data is achieved by comparing forces at the front face and back face of the specimen, i.e. by checking stress uniformity within the specimen. The differences between viscoelastic polymer bars and elastic metal bars are summarized, and sample results from magnesium alloy and poly(methyl methacrylate) (PMMA) pressure bars are compared. For processing of the PMMA bar strain records, two wave-separation techniques and two bar wave models were used. The material properties of the PMMA bars are derived from impact tests.
Neither the material properties of the viscoelastic PMMA bars nor those of the specimen material are known a priori. The properties of the bar material are derived from impact tests. Experimentally, the properties are determined in the form of a complex elastic modulus, i.e. the properties are not fitted to a rheological model. In this form, the elastic modulus must contain errors and has a limited frequency range. Experimentally, it is then difficult to quantify the errors associated with these derived material properties. The properties are assumed to be the same for the experimental input and output bars. For the simulations, a curve fitted to the experimental elastic properties is used to define the bar properties for the striker, input and output bars. Finite-element (FE) simulations are used to validate the experimental procedures associated with the PMMA Hopkinson bars. As there is little information in the open literature on FE simulations of viscoelastic Hopkinson bars, a methodology for converting the complex elastic modulus obtained through impact tests to a form suitable for FE analysis is outlined. FE simulations of impact tests on viscoelastic bars were carried out in order to investigate the accuracy with which the material properties of the PMMA bars can be derived. The full SHPB arrangement is then simulated in order to compare the stress–strain data from the virtual SHPB tests with the properties of hyperelastic materials that are specified in the simulations.
(a) Soft materials
The use of linear elastic, metal bars in the classical SHPB has certain limitations when testing soft materials. Because the levels of stress in the specimen only correspond to low stresses and strains in solid metal bars, the transmitted signal in the output bar may be low and difficult to measure accurately. Different techniques have been applied to overcome this limitation. For example by using a quartz-embedded crystal rather than strain gauges  and by the use of a tube instead of a solid rod for the output bar .
Experimentally, stress uniformity in the specimen can be assessed by comparing the stress at the front face with that at the back face of the specimen. Generally, it is more difficult to determine accurately the stress at the front face of the specimen [1,2]. The majority of guidance available for selecting optimal specimen dimensions and achieving uniform deformation refers to metals [5,6,7]. The establishment of a uniform stress and strain state within a solid specimen can be affected by a number of factors, including inertia effects; friction; the shape of the incident loading pulse; specimen dimensions; mechanical impedance mismatch between the specimen and the pressure bars; specimen quality. For certain materials, inertia effects in the specimen are negligible when a constant strain rate is achieved. So, a constant strain rate may be used as a check for stress uniformity .
In order to deform a specimen to large strains, large stresses may be required in the incident wave. If the incident wave is rectangular and has much larger stresses than those required to deform the specimen, then the reflected wave will be almost equal in magnitude and opposite in sign to the incident wave. The reconstructed force at the front face of the specimen will contain large oscillations, but the strain rate will be approximately constant. This is not an appropriate check when testing soft materials. The axial and radial inertia terms for cellular materials, for example, will be very different from those for solid materials and neither uniform strains nor strain rates will exist throughout the specimen. Gray & Blumenthal  use examples to highlight the fact that for soft materials, the front and back face forces can be very different, even when the strain rate is constant. For this reason, it is crucial to check the stresses on both bar/specimen interfaces. Additionally, for cellular materials, the specimen length needs to be long enough to contain a representative number of cells. However, increasing the length of a soft specimen generally results in lower agreement between front and back face forces, because soft materials require a longer time to achieve stress equilibrium owing to the lower wave speeds in the specimen .
A common modification to overcome the difficulties associated with determining the front and back face stresses on soft specimens is the use of polymer rods that have much lower mechanical impedance than metal rods. While wave propagation in elastic bars only involves dispersion owing to three-dimensional effects (geometrical dispersion), the frequency-dependent dissipative properties of viscoelastic materials result in both additional dispersion and damping of the propagating waves [9–11]. In the classical SHPB, it is often convenient and sufficiently accurate to assume a single phase velocity for the stress waves. When viscoelastic bars are used, modelling the dispersion and attenuation of stress waves in the bars is usually essential to generate accurate data for the specimen.
(b) Wave-separation techniques
When testing to large strains, it is necessary to use very long bars or to separate the overlapping waves. A large variety of wave-separation techniques is available [10–17], in both the time domain  and the frequency domain . The effect of dispersion is greater in viscoelastic bars, so that the duration of a stress pulse tends to expand more as it travels along a viscoelastic bar than is the case in elastic bars. This makes it more difficult to record a wave without overlap from a reflection. This point is highlighted by Zhao et al. , who note that the ‘tail’ in the wave is affected by both material behaviour and the alignment of the bars. An analytical model for the strain histories resulting from the viscoelastic impact of a cylindrical striker and a long cylindrical bar was reported by Bussac et al. . For one-dimensional wave theory, the duration of the incident wave was limited by using striker bars of smaller cross section than the input bar. Good agreement was shown between the analytical model and test data. However, the experimental strain histories plotted in  illustrate how difficult it is to predict the time at which a stress wave can be considered to have fully passed a location. Geometrical dispersion can also result in overlapping signals and cause difficulties for frequency-domain calculations of the propagation coefficient . The specimen being tested in the SHPB may lead to further increases in the durations of the reflected and transmitted waves.
Zhao & Gary  showed that by using wave separation and accounting for wave dispersion effects, it was possible to achieve large strains in a specimen without requiring extremely long bars. The multi-point wave-separation technique reported by Bussac et al.  is suitable for both elastic and viscoelastic bars and permits measuring times up to 50 times longer than the time available with classical methods. Analytical models were used to quantify the errors associated with, for example, background noise, imprecise measurements and imprecise knowledge of the dispersion relationship. The focus here is on stress uniformity in the specimen and validation of techniques by FE analyses.
(c) Bar wave models
A number of approaches and approximate techniques have been used in order to model wave propagation in viscoelastic Hopkinson bars. The majority of studies on longitudinal stress waves in polymer rods retain the one-dimensional wave theory for the sake of simplicity [20–23]. Zhao & Gary  generalized the Pochhammer–Chree equation for elastic bars to the case of viscoelastic bars and used a nine-parameter rheological model for the bar material. These parameters were determined from experimental strain histories through an inverse analysis. Benatar et al.  simplified the Pochhammer–Chree equation for the case of low and intermediate loss viscoelastic bars in order to correct for both attenuation and dispersion. Wang et al.  incorporated the viscoelastic effect into the one-dimensional wave equation and used a simple three-parameter rheological model. The same rheological model was also used recently to incorporate the effect of lateral inertia in a viscoelastic equivalent to Love's equation .
The propagation coefficient describes how a wave changes shape as it travels along a bar . For elastic bars, the propagation coefficient can be defined in terms of the known material properties and the bar diameter. For viscoelastic bars, accurately determining this propagation coefficient is the crucial first step that is required before the bars can be used in Hopkinson bar tests. The general solution for the case of separated waves was provided by Blanc [28,29] using Fourier transforms of the wave at two different positions. Bacon  described an experimental technique for obtaining the propagation coefficient of polymer rods. A short wavelength incident pulse and its reflection were each measured completely without any overlap. The propagation coefficient was calculated from the ratio of the fast Fourier transforms (FFTs) of the two pulses. Two numerical difficulties arise when analysing the wave propagation using the FFT. However, Bussac et al.  illustrated how these difficulties may be avoided by working in the Laplace instead of the Fourier domain. Ahonsi et al.  investigated different techniques for determining the propagation coefficient. For an analytical bar model, it was shown that the method proposed by Bacon  can be used to reconstruct accurately the known propagation coefficient.
Herein, the propagation coefficient was determined using the technique described by Bacon . However, wave-separation calculations are carried out in the Laplace rather than the Fourier domain. As mechanical properties are plotted as functions of frequency, frequency domain equations are used in this text. A bar wave model is necessary to calculate the complex elastic modulus of the bars and to convert from strain to stress in SHPB tests. Here, the results of one-dimensional theory and the ‘four-mode’ rod theory  are compared. The four-mode rod theory is so named because four deformation modes are considered, i.e. two axial and two radial modes. It should not be confused with higher-order modes of the Pochhammer equation. It is used here because it has been shown to be in excellent agreement with the dominant propagating mode of the Pochhammer equation for frequencies well beyond those encountered in SHPB testing .
(d) Use of finite elements
FE modelling of stress waves in metal bars has been carried out many times, typically to facilitate the design stages of Hopkinson bar testing programmes or to assess the accuracy of test data. The models have been used to examine the effect of specimen dimensions, friction and inertia during testing [32,33]. The use of FE simulations to validate the constitutive models of soft viscoelastic materials is proposed by Gray & Blumenthal  and Cady et al. . Meng & Li  used FE analysis to investigate the accuracy of SHPB tests on annealed 1100 aluminium using steel bars. Tan et al.  used FE analysis to account for the effect of an end cap on the bar. However, the authors are unaware of any published material on FE simulations of viscoelastic Hopkinson bars.
An FE simulation of longitudinal stress waves in solid, circular, polymer bars is presented here. The FE simulations have the advantage that the bar material properties are known. By simulating the axial impact of the bar, the material properties can also be derived from strain histories in the bar. By comparing the known and derived properties, it is possible to quantify errors in a way that is not possible experimentally, because the properties of polymer bars are not known a priori. An FE model of the full viscoelastic SHPB set-up is then used to simulate tests on a hyperelastic material with specified properties. Finally, some advantages and difficulties associated with polymer rods are discussed.
3. Properties of magnesium alloy and poly(methyl methacrylate) bars
SHPB tests were conducted on balsa specimens on both magnesium alloy and PMMA bars. Both experimental configurations are illustrated in figure 1. The input and output bars were 1000 mm long. The input bars had three strain gauge stations. The output bars had two. Unless stated otherwise, the striker bar was made of the same material as the pressure bars, had the same diameter and had a length of 450 mm. A 100 mm radius domed end was machined on the front face of the striker bars in order to reduce misalignment effects and to achieve a finite rise time in the incident wave .
The magnesium alloy AZ31B bars have an elastic modulus of 45 GPa, Poisson's ratio of 0.35 and a density of 1770 kg m−3. All bars had a diameter of 23 mm. The propagation coefficient of the bars was determined analytically from the Young's modulus, Poisson's ratio, density and radius of the bars using the four-mode rod equation using equation (A 1) in appendix A.
The PMMA bars had a density of 1190 kg m−3 and a diameter of 20 mm. The Poisson's ratio cannot be assumed constant for viscoelastic materials  and it has a strong effect on the dispersion relationship [14,27,39]. The Poisson's ratio was obtained by simultaneously taking an axial and lateral strain measurement at the same position along the bar. The imaginary part of the Poisson's ratio was approximately equal to zero, whereas the real part was approximately constant at 0.38 up to a frequency of 20 kHz. This is in agreement with previous studies [22,25].
In the frequency domain, the general solution of the one-dimensional longitudinal wave equation is 3.1 where is the strain at axial location x, ω is circular frequency, and are the strains corresponding to waves propagating in the positive and negative x-directions, respectively, at x=0, and the tilde denotes a complex variable. is the propagation coefficient defined as 3.2 where α(ω) is the attenuation coefficient and κ(ω) is the wavenumber. The attenuation coefficient quantifies the reduction of magnitude of a propagating wave, whereas the wavenumber is related to the phase velocity c(ω) and quantifies the dispersion of waves of different frequencies.
To determine the propagation coefficient of the PMMA bars, they were impacted by a 6 mm diameter steel bearing ball as illustrated in figure 1a. The resulting incident pulse (εI(t)) and its reflection (εR(t)) were measured without overlap at a single strain gauge location at a distance d of 500 mm from the free end of the bar. A transfer function was defined in terms of these two measured waves [28–30], i.e. 3.3 The attenuation coefficient and wavenumber for the bar were then determined according to 3.4 To determine the forces at the bar ends, the Young's modulus must be known. The axial load at any position along the bar is 3.5 where is the Young's modulus and A is the cross-sectional area. Although equations (3.1) and (3.5) are exact only for one-dimensional waves, they are sufficiently accurate for SHPB tests [30,40]. In order to calculate the Young's modulus from the propagation coefficient, a wave model must be used. The Young's modulus was calculated from the four-mode theory using equation (A 2) in appendix A and by one-dimensional wave theory: 3.6 where ρ is density. The experimentally derived complex Young's modulus is plotted against frequency in figure 2a. One-dimensional wave theory agrees well with the four-mode theory for the first 10 kHz.
In order to compare the accuracy of wave separation in the time and frequency domains, both methods were applied. The time-domain wave separation used was that proposed by Bacon . The frequency-domain separation was that outlined by Bussac et al. . An example of the separated waves that were obtained from an impact test on the input bar is provided in figure 3. For clarity, the results obtained are shown together with only the strain measurements from the first strain gauge station. Because the propagation coefficient is frequency dependent, wave separation is carried out for every frequency in the signal. Therefore, frequency-domain calculations are still performed for time-domain wave separation and hence problems with signal truncation still exist. Figure 3 illustrates that the frequency-domain separation gave better results.
4. Split Hopkinson pressure bar tests
(a) Cellular materials tested with both bar materials
Tests were carried out using both the magnesium alloy and PMMA pressure bars in order to compare the stress uniformity that can be achieved using the two bar set-ups.
The stress uniformity that can be achieved with the PMMA bars for an end-grain balsa specimen is shown in figure 4. The specimen had a density of 142.8 kg m−3, a diameter of 16.1 mm and a length of 3.2 mm. The average strain rate was approximately 2600 s−1. Bar strain histories are plotted in figure 4a for one location of each bar. The forces at the two specimen faces are plotted in figure 4b and the nominal stress–strain result is plotted in figure 4c. For cellular materials, results are plotted in terms of nominal stress and nominal strain as this is the convention. For all SHPB test data, the specimen stress was calculated using only the stress at the output bar. Reconstructed force histories at the two specimen faces are calculated to assess the stress uniformity only.
For example, reconstructed force histories for the two specimen faces are plotted in figure 5a,b for tests carried out on the magnesium alloy and PMMA bars, respectively. The force histories in figure 5a,b appear quite different from those in figure 4b. However, there is wide variability in the initial crushing stress and plateau stress for end-grain balsa samples (see ). Additionally, the specimen has been compressed to the densification regime in figure 4b, whereas those in figure 5a,b have not. Both tests in figure 5 were performed on 6 mm long end-grain balsa specimens that compressed at similar force levels of roughly 3 kN. The results are presented solely to illustrate the different levels of stress uniformity that resulted from tests for specimens of the same length that are compressed at approximately the same level of stress using the magnesium alloy and PMMA bars. The difficulty in obtaining accurate data for the magnesium input bar is obvious. Although the dynamic response of end-grain balsa wood has been reported , it is much more difficult to achieve stress uniformity with metal bars.
For specimens that compress at lower stresses and for longer specimens, stress uniformity becomes more difficult to verify, even with PMMA bars. Reconstructed forces histories are plotted in figure 6a for a closed-cell polymethacrylimide (PMI) rigid foam (Rohacell-51WF). This specimen had a diameter of 16 mm, a length of 3.2 mm and a density of 57 kg m−3. The average strain rate was 4040 s−1. The front face force oscillates about the level of the back face force, but the differences are large. Errors with the separation technique are highlighted by the fluctuation in the front face force before the incident wave arrives at about 1 ms. The nominal stress–strain result is plotted in figure 6b, together with that for a quasi-static sample with a density of 58 kg m−3.
(b) Testing of hydroxyl-terminated polybutadiene
Previously published test data for hydroxyl-terminated polybutadiene (HTPB) were obtained using metal SHPB arrangements [34,42]. Cady et al.  estimated the room temperature elastic modulus to be 6.62 MPa at an average strain rate of 2600 s−1. The maximum strain achieved was about 20%. Siviour et al.  tested HTPB specimens in the temperature range of −80°C to +20°C, using bars of Inconel and magnesium AZM alloy. Again, the maximum strain level was about 20%.
A ‘pulse shaper’ is needed to achieve stress equilibrium in soft rubber materials [1,3,4,43–45]. This pulse shaper is generally a thin disc of polymer or copper or sheets of paper. Sheets of paper were used here. The HTPB specimens had a diameter of 10 mm and a length of 3 mm. Figure 7a shows typical bar strain readings from the input and output bars. The result of pulse shaping is evident as there is an increase in the rise time of the incident pulse compared with that in figure 4a. The force histories shown in figure 7b indicate that the input and output forces were in good agreement. The stress–strain result for the HTPB specimen is plotted in terms of true strain and true stress in figure 8 as this is the convention for elastomers. The maximum strain is greater than 1. Use of the pulse shaper promotes uniform stress in the specimen, but makes it more difficult to achieve a constant strain rate. The nominal strain rate was approximately constant at 2000 s−1 beyond a strain of roughly 0.2 in figure 8. Throughout the deformation, the stresses are less than 2 MPa, and the slope of the stress–strain plot is well below previous estimates for the elastic modulus. The accuracy of testing at such low stresses is now explored further using FE models.
5. Finite-element modelling
(a) Material definition from test data
From the test data, the Poisson ratio is known and the complex Young's modulus was calculated using the four-mode rod theory. The complex bulk and shear moduli were then calculated as a function of frequency. The FE code used here is ABAQUS explicit . In common with many FE codes, linear viscoelastic material properties are defined by Prony series expansions of the shear and bulk moduli. The Prony series is equivalent to a generalized Maxwell rheological model, consisting of a spring element in parallel with a series of Maxwell elements. The spring element represents the stiffness at zero frequency and each Maxwell element contributes to the viscoelastic- or frequency-dependent properties. The Fourier transforms of these Prony series expansions were curve-fitted to the test data for the shear and bulk moduli. A total of four Prony series terms were used to generate the material model for the viscoelastic Hopkinson bar simulations. Two Prony series terms (i.e. two Maxwell elements) defined the frequency-dependent shear behaviour of the material model, whereas another two defined the frequency-dependent bulk behaviour. The curve fit to the four-mode theory test data of figure 2a is shown in figure 2b.
(b) Bearing ball impact on viscoelastic rod
As only axial stress waves are considered, four-noded axisymmetric elements were used for the 20 mm diameter bars. Four elements were used along the bar radius, whereas 400 elements were used for the 1 m length.
The axial impact of steel bearing balls on a viscoelastic pressure bar (figure 1a) was simulated first. The resulting strain histories midway along the bar are shown in figure 9 for different diameters of bearing balls. Experimentally, it is necessary to repeat the bearing ball impact test several times to minimize the effect that experimental noise has on the propagation coefficient. For the simulated bar, a single test is sufficient. The 5 mm bearing ball impact was selected to keep the duration of incident and reflected pulses short and thereby to ensure that the propagation coefficient is free from the effect of overlapping waves. A propagation coefficient was then determined as described in §3. The propagation coefficient was combined with the elementary wave theory (equation (3.6)) and the four-mode rod theory (equation (A 2)) to regenerate the Young's modulus for comparison with that defined for the material. The results can be seen in figure 2b. The errors in the reconstructed Young's moduli are small, but increase with frequency. With the four-mode theory, the error is 0.26% at 10 kHz and less than 1% at 20 kHz. Using one-dimensional theory, the error was less than 1% up to 10 kHz and 3.5% at 20 kHz.
(c) Viscoelastic split Hopkinson pressure bar simulation
Full SHPB tests with viscoelastic bars were simulated and the data reduced to determine the material properties of hyperelastic specimens. The geometry was the same as in the experimental set-up (figure 1b). Eight elements were used across the specimen radius, whereas 10 elements were used along its length. In ABAQUS explicit  the behaviour of solid, rubber-like materials that have very little compressibility compared with their shear flexibility is modelled using a strain energy potential, which defines the strain energy stored in the material per unit volume as a function of the strain in the material. The Arruda–Boyce strain energy potential was chosen as it is a function of the first strain invariant and two model parameters only. The hyperelastic properties were defined by adjusting these two model parameters and inspecting the resulting stress–strain plot. The ABAQUS interface permits the user to plot the uniaxial stress–strain curve once the strain energy potential is defined. No attempt was made to fit the hyperelastic model to test data. Rather, a range of material properties were used in the simulations. The grey curves in figure 8 show the true stress–strain curves for three specimen models used for SHPB simulations. Two of the material models are stiffer than the HTPB specimen, and one is less stiff. The specimens had no strain-rate dependency and a density of 1000 kg m−3.
After the simulation, the true stresses and true strains for the specimens were reconstructed using the bar strain histories and the propagation coefficient and the Young's modulus calculated in §5b. The simulation results are the thick black lines in figure 8. Oscillations exist in the reconstructed stress–strain curves for both the simulated and experimental HTPB tests. The reconstructed stress–strain curves contain errors, but match the trend of the material properties that were initially defined.
Polymer bars have clear benefits when compared with metal bars for testing soft materials. It is very unlikely that the agreement between the forces illustrated in figures 5b and 7b can be achieved if metal bars are used. Larger bar displacements (larger specimen strains) can be achieved in a single loading cycle than with metal bars of the same length. This good agreement is achieved because the incident wave and transmitted wave are of similar magnitudes. For PMI foam specimens with a plateau stress of roughly 1 MPa, the level of agreement between forces at the front and back faces gave some confidence to the test data. However, the oscillations in the front face force were large. The difference can be reduced by reducing the magnitude of the incident wave, but this will reduce the strain rate and the final strains reached in the specimen.
One of the main disadvantages of polymer bars is the uncertainty of their material properties. The FE simulations of steel bearing ball impacts on the end of a polymer bar indicate that these properties can be determined accurately.
Another disadvantage is the increased signal processing that is needed to perform an SHPB test. The propagation coefficient determined using Bacon's method  is independent of the wave model that is used and contains all material and geometrical effects on both dispersion and attenuation. The specimen strains that are calculated are therefore also independent of the wave model. The wave model does affect the calculation of the Young's modulus of the pressure bars and the stresses in the specimen tested. However, high-frequency components are heavily damped, so the signal energy of waves at the bar–specimen interfaces is predominantly within 10 kHz. For the 20 mm diameter PMMA bars, the one-dimensional wave theory provided the same answer as the four-mode theory in terms of the stress–strain result for all tests reported here.
There are many advantages to fitting a fully analytical propagation coefficient to bar strain histories. The propagation coefficient would then cover a wider frequency range than can be accurately determined experimentally (see ) and would be free from experimental noise. This may reduce the errors illustrated in figure 6a. The advantage of the experimentally determined propagation coefficient is its simplicity. To date, the two methods have not been compared for accuracy.
Using 20 mm diameter PMMA bars, stress uniformity was verified for SHPB tests carried out on end-grain balsa specimens that were compressed at approximately 10–15 MPa in the plateau region of the stress–strain curve. During this plateau stage of compression, the force at the front face oscillated around the value of the force at the back face but typically remained within 4% of the back face force. For HTPB, a pulse shaper was required to achieve stress uniformity. The true stress remained below 2 MPa even at a true strain of 1 for a test where the nominal strain rate was 2000 s−1.
FE simulations indicate that a polymer bar's material properties can be accurately determined from bearing ball impact tests. The error in the Young's modulus that was derived from strain histories was only 0.26% at 10 kHz and less than 1% at 20 kHz. The results of FE simulations of SHPB tests suggest that the properties of elastomer materials can be captured well using the PMMA Hopkinson bars and signal processing procedures that have been described.
J.J.H. and B.A. gratefully acknowledge the financial support of QinetiQ and EPSRC through the industrial CASE scheme. J.J.H. is grateful for the support provided through the LRF Centre. LRF, a UK registered charity and sole shareholder of Lloyd's Register Group Ltd, invests in science, engineering and technology for public benefit, worldwide.
Appendix A. The four-mode rod theory
For a circular rod, the propagation coefficient γ is defined according to the four-mode theory  by a quartic equation in γ2 defined as A 1 where where a is the radius of the bar, ω is circular frequency and μ and λ are Lame's constants. Note that material properties and coefficients are complex, frequency-dependent variables. The full notation has been omitted in this appendix for brevity. Equation (A 1) can be solved without iteration using the quartic formula . Alternatively, for a known propagation coefficient and Poisson ratio, the Young's modulus can be calculated from A 2 where
One contribution of 11 to a Theme Issue ‘Shock and blast: celebrating the centenary of Bertram Hopkinson's seminal paper of 1914 (Part 2)’.
- © 2014 The Author(s) Published by the Royal Society. All rights reserved.