Implementation of viscoelastic Hopkinson bars

Knowledge of the properties of soft, viscoelastic materials at high strain rates are important in furthering our understanding of their role during blast or impact events. Testing these low impedance materials using a metallic split Hopkinson pressure bar setup results in poor signal to noise ratios due to impedance mismatching. These difficulties are overcome by using polymeric Hopkinson bars. Conventional Hopkinson bar analysis cannot be used on the polymeric bars due to the viscoelastic nature of the bar material. Implementing polymeric Hopkinson bars requires characterization of the viscoelastic properties of the material used. In this paper, 30 mm diameter Polymethyl Methacrylate bars are used as Hopkinson pressure bars. This testing technique is applied to polymeric foam called Divinycell H80 and H200. Although there is a large body of of literature containing compressive data, this rarely deals with strain rates above 250 s−1 which becomes increasingly important when looking at the design of composite structures where energy absorption during impact events is high on the list of priorities. Testing of polymeric foams at high strain rates allows for the development of better constitutive models.

Published by the University of Cape Town (UCT) in terms of the non-exclusive license granted to UCT by the author.  To start with, the foundation work and principals of the Hopkinson bar are presented.
Thereafter the use of Fourier transforms and the different methods proposed by authors for dealing with the viscoelastic properties of the polymer bars is explored. Finally the different published uses and configurations of the polymer bars are discussed.

The Hopkinson Bar
The split-Hopkinson pressure bar technique is named after Bertram Hopkinson [13] who, in 1914 used induced-wave propagation in a long elastic metallic bar to measure the pressures produced during blast and impact events. Based on this pioneering work, the experimental technique of using elastic stress-wave propagation in long rods to study dynamic material properties has come to be named the Hopkinson pressure bar. Later work by Davies [14,15] and Kolsky [16] used two Hopkinson pressure bars in series, with the sample sandwiched in between, to measure the dynamic stress-strain response of materials. This technique thereafter has been referred to as either the Split Hopkinson pressure bar or Kolsky bar. As shown in Figure 2 direction, called the reflected stress wave σ r , which is again measured by gauge station 1. By comparing these two strain signals (σ i and σ r ) from gauge station 1 it is possible to determine what the stress load applied to the one side of the specimen was together with the displacement of the face.
The stress wave transferred to the specimen propagates through the specimen and reaches the interface of the transmitter bar. As the stress wave reaches the transmitter bar, a portion of the stress wave will be transferred to the transmitter bar, called the transmitted stress σ t , and the remainder will reflect off the interface. The transmitted stress then propagates down the length of the bar to gauge station 2 where it is detected.
By knowing what stress went into the specimen and knowing what comes out, the response of the specimen can be evaluated.

Hopkinson Bar Theory
One Dimentional Wave theory A thorough description of the SHPB and the background theory is provided by Gray [1] and therefore will not be repeated here. However, the salient points will be highlighted.
The geometry of Hopkinson bars allows 1D wave theory to be used, from elementary wave theory the wave equation can be shown to be where C 0 = E ρ is the fundamental wave velocity, u is the displacement and t is the time. The general solution to the wave equation can be written as where f and g give the form of the wave traveling in the positive and negative direction respectively.
By definition the 1D strain is given by Thus Equation 2.2 can be written as and for the transmitter baru Equations 2.5 and 2.6 are valid at all points of the bars including the bar ends or specimen interfaces which are going to be the points of interest. If the instantaneous length of the specimen L s is know then the strain rate in the specimen can be calculated combining equations 2.5, 2.6 and 2.7 it is possible to write the more practical forṁ Finally using the bar area A and the Young's modulus E the forces on the bar ends can be written as After an initial "ring up" 1 period the specimen is in equilibrium and is regarded as deforming uniformly. In which case the forces on the two faces should be in equilibrium 1 The term ring up is described in more detain in the section dealing with specimen equilibrium.
From the forces on the bar ends it is possible to infer stress levels in the specimen with where A s is the area of the specimen.

Impact of Dissimilar Bars
In the preceding explanations, the assumption has been made that the material properties and geometry of both the input and transmitter bars have been the same.
To illustrate the differences that arise, a more general case of the impact of two bars of different material and geometry, adapted from Spotts [17], is presented.
The layout of the two bars can be seen in Figure 2.2. The areas and densities of the two bars are A 1 , A 2 and ρ 1 , ρ 2 respectively.

Figure 2.2: Impact between dissimilar bars
In the generic case, applying the impulse momentum equation to the compressed region seen in Figure 2.2 yields Where C is the wave speed and − → v 2 is the particle velocity in Bar 2 travelling to the right.
Applying a force balance across the interface of the bars yields The velocity ∆ ← − v 1 of the compressive wave moving to the left is sufficient to reduce the initial particle velocity − → v 0 of Bar 1 to the velocity − → v 2 of the interface. Resulting in the following Bar 2 velocity can now be found using v 2 = v 0 − v 1 and substituting in Equation 2.15 The stress in Bar   When this result is interrogated some interesting relations become apparent. One term relates the material properties of both bars, namely which can be further deconstructed into By simply inspecting these terms it becomes possible to gain insight into the expected response of any two bars used in a Split Hopkinson bar setup.
When discussing the impedance it can be defined in two ways namely the acoustic material impedance ρC and the bar impedance ρCA.
The bars are often made of a high strength steel with a high elastic limit σ s > 1GP a and an acoustic material impedance of ρ 0 C 0 of approximately 40 MPa s/m. This is done as the specimen of test material must have a lower strength and lower acoustic wave impedance ρ s C s , allowing it to deform plastically while the bars are still in the elastic state. However if the wave impedance of the specimen ρ s C s A s is much lower than that of the bars, the signal of the transmitted pulse will become too small so that it is unable to be detected accurately. In such a case it makes more sense that a bar material with a lower acoustic impedance, such as polymers, should be used instead of high strength steel. This ultimately results in a lower impedance mismatch between the specimen and the bars meaning that a larger stress wave is transmitted to the output bar.
Hopkinson bars with an impedance of 10MPa s/m have been used by Wang et al [18] in the testing of solid polymers with reported success. However testing of materials with an impedance in the region of 0.1 MPa s/m would prove difficult.
Chen [19] attempted to overcome this problem when testing low impedance materials.
Chen implemented a hollow aluminium output tube in place of conventional solid Hopkinson bars and instrumented a solid input bar with quartz crystals which are significantly more sensitive then conventional strain gauges which are normally used.
Chen et al [20] later used this method on soft rubber specimens reporting good results.

Specimen equlibrium
Assuming that a transmitted signal with high signal-to-noise ratio is obtained and accurately describes the conditions at the specimen interface, dynamic equlibrium of stress in specimen presents another major obstacle in obtaining reliable stress-strain data from a conventional SHPB experiment. Equations 2.1 and 2.2 are based on the assumption of dynamic stress equilibrium in the specimen allowing the use of wave superposition, which is not satisfied automatically when the specimen material has a very low material impedance. Dynamic stress equilibrium can be achieved quickly in a metallic or ceramic specimen due to relatively high wave speeds in those specimens.
However, the stress state in a soft specimen may not be in equilibrium over the entire loading duration in a SHPB experiment as the wave speeds in such materials are often slow. In a SHPB experiment, it takes several stress wave reflections within the specimen for stress wave to "ring up" to a state of dynamic stress quasi equilibrium. The nonequilibrated stresses in specimen during a SHPB experiment may lead to a drastic non-uniform deformation in specimen, which invalidates the experimental results for material property characterization [21].
Song and Chen [22] provide a well motivated argument on this subject showing high speed camera footage of soft materials such as rubber and foams being tested. One such example is shown in Figure 2.3 where a rubber specimen can be seen very clearly deforming from one side as the stress wave propagates through the specimen. The drastic non-uniform deformation in specimen demonstrates that the conventional SHPB experiment may not produce valid dynamic properties of the rubber material.

Dispersion Correction in Metallic Bars
One dimensional wave theory is per definition limited by the assumption that there is no disturbance other than in the direction of wave propagation and that the wave does not change shape in the direction of propagation due to axial or radial dispersion. 3D effects are important as pointed out by Pochammer [23] and Chree [24] who independently  specimen deforming dynamically [22].
The form of the displacement solution is where U (r) and W (r) are time invariant functions which describe the variation in axial or radial displacement respectively as a function of radius.
Bancroft [25] and Davies [14] presented numerical solutions to the Pochammer-Chree equations. These showed a relationship between frequency ω and phase velocity C p , and the variation of axial displacement across the radius of the bar. Davies showed the change in shape of a pulse as it propagates axially (dispersion) by representing a trapezoidal pulse with its Fourier series and phase shifting each term according to the phase velocity. As the deflection can be related to the stress in the bar, it was possible to obtain the stress history of a pulse in the bar. Davies also noted that the energy of a pulse consisting of different wavelengths will be transmitted at a different velocity to the phase velocity of the individual wavelengths. This velocity is termed the "group velocity" C g and is characterised by the following equation as explained by Govender [26].
Dispersion may be described by the various frequency components of the pulse changing their relative phase. This is easily explained by looking at Figure 2.4 which relates the phase velocity C p to the frequency ω.
where f (t) is defined for all real numbers t. For any s ∈ R , integrating f (t) against e −2πist with respect to t produces a complex valued function of s. If t has dimention time then to make st dimentionless, s must have dimention 1/time.
Similarly the inverse Fourier transform can be defined as Assuming that t = time and knowing that s = 1/time we can make the substitution ω = 2π s resulting in a more convenient form of the equation: where θ is know as the complex argument or phase and |z| is known as the modulus or magnitude of the complex number, sometimes written as r = |z| = x 2 + y 2

The Discrete Fourier Transform (DFT)
The Fourier transform deals exclusivly with continuous functions. In order to deal with any function defined at discrete intervals the DFT is used. This is an important step as it is not possible to implement a continuous Fourier transform numerically but a DFT can be implemented numerically. The definition of the DFT is as follows: For a sequence of complex numbers f k the DFT F k is and the Inverse is defined as to artificially increase the amount of data in the time domain by simply "padding" the end of the data with zeros [29]. This has no effect on the data in the time domain and simply increases the number of discrete points in the frequency domain thus increasing the resolution.
An important point to note is that each component of the DFT will split into "positive" and "negative" frequency components, which, depending on the term, can be "odd" or "even" functions through the frequency band. An example of this can be seen in Figure 2.6 where two simple sinusoid functions are combined and overlayed with random noise to hide the obvious periodicity. In Figure 2.6 the origional singnal is shown in the first pane and is generated as follows: The second pane shows the signal with random noise overlayed while the final pane shows the magnitude or modulus of the Fourier transform of y(t). Note how the output is mirrored around the zero frequency.

Viscoelastic Wave Propagation
Characterising viscoelastic wave propagation is important because unlike a metallic bar,

Blanc
Lundberg and Blanc (1998) [5] were among the first to implement an experimental technique which can be used to characterise the material properties of a viscoelastic polymer seen in Figure 2.9. In the paper the authors determine the mechanical properties of a linerly viscoelastic body from the body's response to impact. This is achieved by capturing the one-dimensional transient stress wave propagation at two points in a bar that is impacted on one end with a striker. The specific material properties they considered (characterised in the frequency domain ) are the damping coefficient α(ω) and the wavenumber k(ω) which are the real and imaginary parts respectively of the propagation coefficient γ(ω). From this, the phase velocity C and the complex modulus E = E ′ +E ′′ can be calculated. These functions of angular frequency ω are interrelated in such a way that by the density of the material together with α and k , α and c or E ′ and E ′′ are known then the remaining functions can be determined. Blanc states that two different methodologies are discussed namely one using particle velocities which is dependent on wave superposition and another using particle strains which is independant of wave superposition. They conclude that both methods compliment each other in a frequency range of 20Hz -20kHz.
It is important to note at this point that the complex modulus mentioned here is analagous to the youngs modulus. The difference is that the Youngs modulus is the function that relates stress to strain in the time domain while the complex modulus is the function that relates stress and strain in the frequency domain. 2 In a follow up publication, Blanc (1993) [33] states that any homogeneous viscoelastic medium subjected to one dimensional tension or compression can be characterised in terms of the phase velocity c(ω) and the attenuation co-efficient α(ω) of the longitudinal wave, from which it is possible to deduce the complex modulus. Here the jump from wavenumber k(ω) to phase velocity is made by There are several possible ways of deriving these functions from a captured transient wave.
• One can attempt to integrate it exactly assuming a form of c(ω) and α(ω). This approach is reported by Brodner and Kolsky (1958) [34] and further developed by Blanc and Champomier (1976) [35] for cases where only the wave front can be observed.
• Alternatly, a numerical solution for the wave propagation needs to be obtained, which has since been dealt with by a few authors [36,37] and is discussed in does not take geometric wave dispersion into account as pointed out in a response to a letter to the editor of the journal [38]. The form of the ZWT model is as follows [36]  By looking specifically at high strain rate events this model can be simplified as the first Maxwell element effectively acts elastically leaving us with E a = E 0 + E 1 and To use this model the nine different constants need to be solved for the material being used. One of the main differences between this method and others is that it cannot make use of a Fast Fourier Transform (FFT) because a nonlinear viscoelastic equation is used to describe the material. As a result the equation must be integrated analytically.

Zhao and Gary
Zhao and Gary [6,37,39,40] use a 3D analytical solution for the longitudinal wave propagation which takes into account the geometric effects of the bar by not simply treating the the problem as a one dimensional problem. The difference between the 1D and 3D approaches can be seen in Figure 2.12.
The method is based on the Pochhammer [23] and Chree [24] frequency equation for propagation in an infinite rod studied numerically by Bancroft [25] and Davies [14] and generalised for a cylindrical infinite bar made of a linear viscoelastic material by Coquin [41]. This means that the constitutive law can be written in the frequency domain as follows: where σ * (ω), ε * (ω), λ * (ω), µ * (ω) are respectively the Stress tensor, the Strain tensor and the two material coefficients each of which are frequency dependent. In order to use this method the bar material needs to be characterised and an example is presented briefly to illustrate this point. It is assumed that the functions λ * (ω) and µ * (ω) have a pre-defined form with some parameters to be determined. If the material is assumed to have a form shown in Figure 2.14 then 9 parameters will need to be solved for in order to use this model successfully. It should be noted that this method is calibrated using a two gauge station measurement method for the separation of waves.
14: A rheological model for PMMA [6] The resulting force balance at the bar interface is however very good as can be seen in Figure 2.15 for a test performed on 60mm diameter Nylon bar. It is interesting to note the high frequency oscillations which appear on the plateau of the signal. These oscillations are very similar to those studied by Pochammer and Chree. An explanation for these is the geometric dispersion associated with the large diameter bars which were used in the experimental setup by the author. It was shown by Govender [26], that as the bar diameter increases the influence of dispersion effects increases and Pochammer Chree oscillations will be seen.
whereσ(x, ω) andε(x, ω) denote the Fourier transforms of stress and strain respectively. The angular frequency ω is related to the frequency f by ω = 2πf .
And the viscoelastic behaviour of the material can be described as follows: where E * (ω) is the complex Young's modulus of the material. Note that the form used by Bacon is not the same complex modulus mentioned by previous authors, but should rather be thought of as the function which relates stress and strain in the frequency domain. The propagation coefficient γ = γ(ω) is defined by From the above equations the one dimensional equation of motion for a viscoelastic bar becomes  The definition of a convolution between two functions f and g can be defined by and deconvolution can be thought of as the inversion of a convolution equation, so in the case of Liu where "⊗" stands for the convolution and "/" stands for the deconvolution.
This method is different to others seen as it leaves all measured components in the time domain, negating the need to use a FFT to determine the frequency dependant functions. This method is used successfully by the authors to predict wave behaviour in a bar seen in Figure 2.17, which uses two gauge stations on the same bar. For consistentcy, polymeric materials in this section appear under the original names given by the authors, but can be grouped into the following three categories: • NYLON • Polymethyl methacrylate (PMMA) also known as acrylic or perspex Hopkinson bar setup using NYLON bars and an hydraulic actuator in place of a striker is used, seen in Figure 2.18. The data analysis is carried out using the methods described by Zhao and Gary [6], note the three gauge stations used on both the input and output bars.    polycarbonate, polyurethane and styrofoam. These materials cover a large range of 4 In the paper no explanation is given by the author on how these results were derived. 5 The only reference to this found was a reported private conversation with Fourney. 6 Unfortunatally this paper could not be sourced and as a result no comment on the technique can be offered. Ouellet et al [43] tested polystyrene, high-density polyethylene and polyurethane at low, medium and high strain rates to show the rate dependence of polymer foams.
The high strain rate tests were conducted on a split Hopkinson bar setup using 25.4mm diameter acrylic bars. 1000Ω semiconductor strain gauges were placed on both the incident and transmitted bars which are significantly more sensitive than conventional foil strain gauges and ensure that very small strain signals can be measured. The viscoelastic effects of the bars are dealt with using the same method outlined by Salisbury [42] and the software programmed by Salisbury is used to calculate the final results for the tests.
Subhash tests both low (ρ < 1g/cm 3 ) and high density (ρ > 1g/cm 3 )epoxy based foams [53]. The low density foams were tested on an acrylic Hopkinson bar setup while the more dense foam specimens were tested using metallic bars. Later the authors [54] test only low density epoxy based foams seen in Figure 2 Subhash [54] when not tested in a confined configuration The mechanical response of viscoelastic materials to mechanical excitation has traditionally been modelled in terms of elastic and viscous components such as springs and dashpots. The corresponding theory is analogous to electric circuit theory, which is extensively described in engineering textbooks [55]. In many respects the use of

Determination of Viscoelastic Properties
terms of springs and dashpots does not imply that these elements reflect the molecular mechanisms causing the actual relaxation behaviour of complex materials.
All the methods presented below are easily implemented numerically and can be used to approximate the behaviour of polymers. The greatest difficulty in using these models is finding the correct values to use for the different parameters.

The Maxwell Model
The Maxwell model seen in Figure 3.1 has a spring in series with a dashpot. The important relation between these two is the fact that the stress or force will be the same across each component. When this element is exposed to a step displacement, the strain in the element is given by where ε 1 and ε 2 are the strains of the spring and dashpot respectively. Since the stresses in the spring and dashpot are the same combining these togetherε

Kelvin-Voigt model
The Kelvin-Voigt model seen in When this element is exposed to a step displacement, the stress in the element is given by

Standard Linear Solid model
The Standard Linear Solid model comprises a Maxwell element in parallel with a spring seen in Figure 3.3. In this model the stress is split between the two legs such that where σ 1 = E 1 ε andε

Shim model
The Shim model seen in Figure 3.4 puts a Maxwell element in parallel with a Kelvin-Voigt element. In this instance the strains of each element will be the same and the stresses will be additive.
Now for the Kelvin-Voigt branch we have and for the Maxwell branch we haveε equating these termsσ    It was found however that the model was unable to replicate all three of the materials closely. By changing the parameters it was possible to achieve a response that appears to closely match the initial pulses seen in the tap tests. However as the system is allowed to vibrate the reflection produced by the model deviates from the test data. The material that was most closely matched was the PMMA seen in Figure 3  A round extruded bar, 3m in length, of each of the materials was purchased from Maizey Plastics in Cape Town. Each of the bars was sold nominally as 20mm diameter, however this measurement was not exact due to die swell of the material during the extrusion process as is reflected in Table 4.1.
To start with each of the bars was gauged using KYOWA foil gauges in a diametrically opposed gauge station to cancel any bending effects in the bars. Each of the bars had tap tests performed on them to generally assess the material response.
Following this calibration tests were performed on each of the bars to determine the material properties and finally tests were performed using the each of the bars together with a magnesium bar.

Strain Gauging and Data Capture
Each of the Hopkinson bars used were gauged with diametrically opposed gauges to cancel bending effects in the bars. Dummy gauges were used for both temperature   In addition to the middle gauge station, it was decided that an additional gauge station would be placed 250mm from the front of the Polycarbonate bar. This decision was made in order to investigate the work reported by Martins [48] and Shim [8] who reported that the viscoelastic effects in polycarbonate bars were negligible allowing regular elastic theory to be used to calculate the specimen response.
The gauge stations were amplified through an amplifier designed and built in BISRU labs 4 with a gain of 1000. The signals from these gauge stations were then captured by an ADLINK 9826H data capture card capable of capturing 4 channels of data at 20 MHz.

Tap tests
Tap tests were performed on each of the polymeric bars to initially establish the general behaviour of each bar and to evaluate the response of each bar to different strikers.
The different strikers used were a stainless steel ball bearing, a 10mm diameter 250mm

Polycarbonate
The PC test seen in Figure 4.3b shows very different results to the previous nylon test.
Unlike the nylon test the initial compressive pulse has a very rectangular shape with high frequency oscillations on the plateau similar to those seen in metallic Hopkinson bar tests. A tail seems to develop initially but disappears by the first reflection. The tensile reflection seems to show a small change in magnitude but much smaller than that seen in the nylon test which supports the work presented by Shim [8] stating that polycarbonate can be treated as an elastic material when using it in Hopkinson bars.
As the pulse propagates down the length of the bar a change in the gross shape can be seen which does bring the linear elastic assumption of Shim into question in Section 2.5.2 .

Polymethyl methacrylate
The PMMA test seen in Figure 4.3c again has a different shape to the other two tests.
Instead of the almost rectangular initial PC pulse or the slightly rounded Nylon, the PMMA initial compressive pulse is almost trapezoidal in shape. Keeping in mind that all three tests were performed with the same striker which should nominally produce the same shaped initial pulse, it becomes clear how differently these materials respond.
A small amount of high frequency oscillations can be seem at the top of the initial pulse but die down by the time the reflected pulse returns. The gross shape of the pulse changes faster in the PMMA bar becoming almost saw tooth like by the third reflection.
It would appear that on first inspection the PMMA has the most marked effect on a pressure pulse which has the most visible changes as the pulse propagates through the material. It should be noted that the general shape of the pulse is the most like that

Calibration tests
Two different types of calibration were carried out on each of the bars, namely wave speed characterisation and strain gauge calibration. The wave speed calibration allows for accurate prediction of the response of the bars to a pressure pulse. The strain gauge calibration on the other hand allows for the accurate measurement of the response.
The bar wave speed is measured first as it is not necessary to have an accurate gauge calibration in order to measure this property which is a time dependent measurement.

Wave speed calibration
Other than the static properties such as density, the wave speed of each bar was first property to be calculated. Each bar was impacted with a striker to produce a pressure pulse which propagates down the bar and through the gauge station, which then reflects off the free end of the bar and passes through the gauge station as a tensile wave. The full signal captured from the calibration of the magnesium bar can be seen in Figure   4.5. This signal is then split into two separate components, namely the incident and It is interesting to note that very little dispersion can be seen in this test. This is due to the fact that a small amount of plasticine putty was used to pulse shape the initial compressive pulse. As a result the inital rise time is slightly increased reducing the frequency content and reducing the dispersion effects noted in Section 2.1.1.
When this method is used on polymer bars it is not possible to get a meaningful reading from the material due to the fact that the viscous material effects change the shape of the pressure pulse as it propagates down the bar. This can be clearly seen in

Gauge station calibration
There are three main ways to calibrate a strain gauge station. These are

Theoretical Calibration
The output from a strain gauge based on strain gauge theory [57] is where K gf is the gauge factor, V in is the bridge voltage, ε is the strain and N is the number of active gauges in the wheatstone bridge.
The input to the amplifier V read is multiplied by the gain in the amp G amp so we can And the strain is related to the stress through the Young's Modulus ε = σ E resulting in By making the correct substitutions for the values the calibration factor K can then be multiplied by the voltage from the gauge station V read such that Therefore using the difference in velocity before and after impact and knowing the mass of the striker m striker , the impulse transferred to the bar during impact I becomes The impulse at any cross-section in the bar is defined as

Calibration tests Experimental Methodology
So by numerically integrating the voltage from the gauge station to obtain impulse which must be the same as that of the striker yields:

Maximum stress
From Section 2.1.1 we know that the stress in the bar can be found from This method is dependant on the pulse produced being almost trapazoidal in shape with a defined plateau which defines an average maximum stress. Now by simply assuming that the average maximum stress in the bar σ max b corresponds to the average maximum voltage reading from the gauge station V max read the calibration factor becomes

Bar Calibration Results
The

Multiple gauge station calibration
As mentioned previously the PC bar was gauged with two gauge stations seen in Figure   4.8.

PC Bar
Pulse direction Impact face 0.250m 0.750m Martins [48] and Shim [8] state that the PC bar can be treated the same way as a metallic bar. However that neglects any effect the viscoelastic material might have had on the pulse, and the purpose of this investigation, and so this assertion will not Once the calibration factor achieved from the quasistatic calibration is applied to the signals from the two gauge stations, the magnitudes of the signals become almost the same seen in Figure 4.11. Although these signals appear to be almost the same magnitude a small difference exists when the exact magnitudes are closely interrogated.

Theory
The method chosen to characterise the viscoelastic wave propagation in this dissertation is Bacon's method as discussed in Section 2.4.4. This specific method was chosen due to the fact that the material is characterised experimentally in a very straight forward way which lends itself to routine calibration. It is important because it means that with a few simple calibration tests, before testing specimens, a transfer function for the bars under almost the same conditions as those to be expected during testing can be derived.
In contrast, some of the other methods require at least one analytical component to be used, some requiring significant computational effort to derive a property that is assumed to remain constant. Changes in temperature, for instance, have an effect on these properties which is difficult to characterise in an absolute sense. with a complex modulus E * (ω) which relates the stress and strain at any point.
The central feature of this method is the propagation coefficient γ = γ(ω) which is defined by The general solution to the one dimensional equation of motion becomes: where the functionsP (ω) andÑ (ω) define the strains at x = 0 to waves travelling in directions of increasing and decreasing x respectively. From these it is possible to find the Fourier transform of the particle velocityṽ(x, ω) and the normal forceF (x, ω) at any cross section x.ṽ where k(ω) is the wave number defined by . Where Θ R and Θ I are the unwrapped phase angles of the two complex functions P (ω) andÑ (ω). So by finding α along with either k or c the propagation coefficient can be described.
The following properties of these different functions should be noted: • The Wave number k is an odd function • Phase velocity C is an even function

Deriving the Propagation coefficient
The propagation coefficient γ describes the changes that occur to a stress pulse that travels some distance through a viscoelastic material. When a stress pulse can be measured at two different positions in a Hopkinson bar, then the changes that occur to the pulse between these two points are assumed to be attributable to γ. There are two ways of achieving this; • Having two gauge stations separated by some distance on a Hopkinson bar seen in Figure 4.8 and capturing the data from both. By comparing the two different signals the differences can be calculated.
• Using the reflection of a stress wave off a free end of the bar as discussed in Section 2.1.1. This method requires taking the information from a single gauge station and separating the initial compressive pulse from the tensile reflection so that they can be compared in order to calculate γ.
The differences in the two gauge station setup on the PC bar discussed in Section 4.4.2 make it a more complicated approach to take. The purpose of this investigation is to illustrate the differences in the bar materials and the single gauge station will be sufficient for a comparison to be drawn.
In order to implement Bacon's method a set procedure needs to be followed. This procedure can be seen in Figure 5   The first process is to window or separate the incident and reflected pulses. Each window must contain all the pertinent information from the instant when the pulse starts to deviate from the mean zero to the time when the pulse settles back to the mean zero.
An example is shown in Figure 5

Attenuation Co-efficient
The Attenuation coefficient α is defined as: Where r R and r I are the modulus of the reflected and incident pulses in the frequency domain respectively, x is the distance through which the pulse has propagated. It is important to note that this is an even function in the frequency domain and defines the real part of the transfer function γ. An example of the attenuation coefficient is shown in Figure 5

Phase angle
The phase angle of the complex number that comes from the FFT it may not initially look sensible as seen in Figure 5.6. The angle oscillates between π and −π. To make sense of this an operation called "unwrapping" needs to be performed on the phase angle to account for the jumps from π to −π. This involves adding or subtracting π depending on the jump that occurs.
The unwrapped phase angle Θ can be seen in Figure 5

Wave number
The wave number k is defined as: where (Θ R ) − (Θ I ) is the change in phase angle and x is the distance through which the pulse has propagated. An example of the wave number is shown in Figure 5 It is important to note that the wave number is an odd function in the frequency domain, and comprises the imaginary component of the transfer function γ.

Phase velocity
The Phase velocity can be derived from the wave number k as follows: where ω is the frequency. The phase velocity is an even function in the frequency domain and an example can be seen in Figure 5

Propagation coefficient
From the attenuation coefficient α and the wave number k the transfer function γ can be written as: In order to shift some arbitrary pulse P up or down a bar the transfer function is multiplied by the distance d of the shift and applied in the frequency domain as follows: P shift = P e γx (5.14) Once the shift has been applied then the pulse in the frequency domain is transferred to the time domain using the Inverse Fast Fourier Transform (IFFT). In Figure 5.10 the incident pulse and the tensile reflection have both been shifted to the free end of the bar where they should sum to the zero force boundary condition at the free surface.
When shifting both these pulses in a bar to a common interface, one would expect them coincide. The fact that they do is regarded as confirmation that the method is working.

Shifting pulses in polymer bars
A short description of how the propagation coefficient, derived in the previous section, is applied to shifting pulses is described here. The term "shifting" is used here to describe the process of taking a signal or pulse captured at a gauge station on a Hopkinson bar, and predicting what that pulse would be if it had been captured at some position which is either x m from the gauge station in the same direction as the wave is propagating, or −x m in the opposite direction.
The pulse of interest is firstly selected from the remainder of the data in the same fashion as described in Section 5.2. After the pulse has been separated and the remainder of the data set to zero, the signal is transformed into the frequency domain by means of the FFT function in the same manner as described in Section 5.2 where the use of the FFT is described.
What exists at this point is a frequency domain representation of the strain ε(ω).
At this point the propagation coefficient γ(ω) can be applied to shift the pulse through some linear distance x from the gauge station as shown in: ε shift (ω) = ε(ω)e γx (5.15) At this point the shifted pulse like the unshifted one is frequency dependant and the inverse Fourier transform must be used to transfer the signal back to the time domain where regular Hopkinson bar theory can be applied.
The result of a shift performed on a PC bar signal using the propagation coefficient derived in Section 5.2.6 is used to shift two pulses in the bar. The first pulse that was

PC single gauge station
In Figure 6.9 the data for the attenuation coefficient for the PC bar is presented. This data is tightly grouped up to 10 kHZ. Between 10 -15 kHz the data starts to spread and after 20 kHz it scatters and no distinct trend can be seen. In this instance a larger number of data points can be seen below the zero line indicating a negative attenuation coefficient. These data points together with the asymptotes are a result of the numerical error described previously.
In Figure 6.10 the wave number for the PC bar can be seen. In this instance the data is tightly grouped up till 15 kHz where a portion of the test deviate from the trend in a bilinear fashion as observed in the NYLON bar. However the remainder of the data continues in the same linear trend up to 25 kHz where it starts to disperse.
In Figure 6.11 the phase velocity for the PC bar is presented. In a similar fashion to the wave number the phase velocity data forms a trend up to 15 kHz where a group of data starts to deviate leaving the remainder to continue the trend up to almost 25 kHz.
In order to choose a suitable propagation coefficient for the bar, the attenuation coefficient and wave number functions that most closely match the trends seen in Figure   6.9 and 6.10 were used. These were combined according to Equation 5.13 to describe the propagation coefficient. This propagation coefficient is then used on a pulse from the  PMMA bar that was recorded for an extended duration capturing multiple reflections in the bar. The incident pulse in Figure 6.12 is selected and the propagation coefficient is applied to it as outlined in Section 5.3. The shifting is applied in such a way that the shifted pulses should coincide with the reflected pulses in the bar. The results for the first four reflections in the bar are presented and as can be seen they coincide well with the actual data recorded at the gauge station.
As a final note Pochammer and Chree oscillations were detected on the PC bar. One such example is shown in Figure 6.13 where one of the reflected signals from further down the bar has been shifted back towards the initial pulse on the PC bar with two gauge stations. The Pochammer Chree oscillations can be clearly seen on the plateau of the pulse and the captured data from the two gauge stations can be seen preceding it.

PC double gauge station
The following section presents the results of using a two gauge station method for characterising the bar material of the PC bar. The method is described in Section 5.2 where the decision was made to use a single gauge station for the tests on all the bars. This section is presented here purely as a comparison to the single gauge station method. The key difference between the two methods it that the two gauge method is able to characterise the wave propagation without it reflecting off the free surface.
The data used in this section is the same as that for Section 6.1.3, the only difference is that the signals from both gauge stations are used where as previously only the signal from the middle gauge station was used. By doing this a good contrast between the two methods is achieved.
In Figure 6.14 the attenuation coefficient is calculated using the two gauge method.
Immediately a difference can be seen to Figure 6.9 where the data started to disperse at 10 kHz. Using the two gauge method the data remains much narrower band right up to 25 kHz. This is significant as the data is from the same tests it means the difference is only due to the difference in measuring techniques. The result is that the data between 10 -25 kHz was either affected by the reflection off the free surface, small changes in the bar properties as the pulse propagates through a longer distance, or the effect of the incident pulse passing through the single gauge station while the reflected pulse is being measured. It is interesting to note that there are significantly fewer asymptotes in the double gauge station data indicating fewer numerical errors due to division by very small numbers.
In Figure 6.15 the wave number for the two gauge station method can be seen. With the exception of a small amount of data a very strong linear trend can be seen right up to 25 kHz. This is in contrast to the data seen in Figure 6.10 where a portion of the data appeared to deviate from the trend at 15kHz. This means that the data that deviates was clearly affected by the measurement technique.
Finally looking at the phase velocity seen in Figure 6. 16

Calibration factors for polymer gauge stations
Measuring the stress is important as Hopkinson bar theory uses the stress to calculate the interface force, displacement and velocity. In Section 4.4.2 the conventional approaches to calibrating the gauge stations is discussed. Due to the material effects of the bars it was only possible to get a valid theoretical calibration factor. Having successfully implemented a technique for correcting for these material effects it is now possible to comment on the calibration of the gauge stations by the other techniques. Using the same methods discussed previously but now using the pulses corrected using the derived propagation coefficient the following calibration factors were achieved: For the theoretical value in this section the calculated wavespeed and density were used to attain a Young's Modulus which is drastically different to that used previously in Section 4.4.2. It was found that the maximum stress calculation generally gave the most accurate stress value for comparison with the magnesium bar and this was ultimatly the value that was used.

Configuration 3 -Magnesium ->Polymer
In this section, similar to Section 6.3 the magnesium bar is used to infer the force at the interface of the two bars. In this configuration the magnesium bar is impacted with a 250 mm magnesium striker. The incident and reflected pulses in the magnesium bar are used to infer the force while only the transmitted pulse is used in the polymer bar. As a result of the higher wave speed in the magnesium, the pulses produced in the polymer bar are of a much shorter duration when compared to configuration 2. In this set of tests the difference in the forces at the bar ends was found to be between 1 -2%. One of the most pertinent issues being dealt with in this dissertation is the method by which polymeric bar materials are characterised in Hopkinson bar testing. In the literature many authors make use of analytical parameter driven models, often requiring calibration of many different parameters. One such model is the linear solid model which is very similar to Shim's model. In Section 3 the different models are discussed and one such model is implemented in Section 3.3 . While it was possible to come close to replicating the initial behaviour of different materials to an impulsive loading, when the extended response using these parameters was interrogated, the behaviour deviated from the actual material response. This indicates that the simple models are not able to fully describe the response of these polymeric materials and that ultimately the real material behaviour is more complex.
The alternative to these simple models is to use Bacons' method where the the material response is not assumed to take a parameter driven form. Once the code had been developed, this method is implemented as a matter of routine on any material by simply calibrating the propagation coefficient of the material using experimental data would be a large force mismatch between the two bars. By using the magnesium bars it is also proved that the conversion from strain to stress is correct Zhao and Gary present data using their method for 3D material characterisation clearly showing high frequency Pochammer Chree oscillations due to the geometric effects of the bar. One of the main arguments for the implementation of this method was that it was able to capture these geometric effects and reliably reproduce these effects. It was found however, that Bacons' method was just as capable in reproducing the effects of this 3D behaviour. Other authors found this effect to be significant while using large diameter bars where geometric dispersion is significant. The following conclusions can be drawn from this work: • The polymeric materials used here were not well characterised by a simple linear viscoelastic model.
• Bacons' method for viscoelastic material characterisation was found to be appropriate on PC, PMMA and NYLON Hopkinson bars.
• Bacons' method has been implemented on a PC bar which, to the authors knowledge, has not been presented in the literature. The same method has been successfully implemented on three different bar materials with exactly the same experimental setup which was not found in the literature.
• Both single and double gauge station methods were successfully implemented in characterising the material showing they can be used interchangeably, although a slightly more accurate calibration was achieved using the double gauge station setup.

Conclusions and Recommendations
As a recommendation from this investigation a PC bar will work well as an output bar in a SHPB setup due to the small amount of dispersion and attenuation in the material.
A PMMA bar would however work well as as input bar due to the reliability with which the stress waves can be shifted back to the bar interfaces over longer distances.
This dissertation enables further testing on softer materials such as polymeric foams and biological tissues which can now be conducted using the apparatus and methods implemented.