## Abstract

Elastic theory shows that wide spectrum signals in the Hopkinson pressure bar suffer two forms of distortion as they propagate from the loaded bar face. These must be accounted for if accurate determination of the impact load is to be possible. The first form of distortion is the well-known phase velocity dispersion effect. The second form, which can be equally deleterious, is the prediction that at high frequencies, the stress and strain generated in the bar varies with radial position on the cross section, even for a uniformly applied loading. We consider the consequences of these effects on our ability to conduct accurate backward dispersion correction of bar signals, that is, to derive the impact face load from the dispersed signal recorded at some other point on the bar. We conclude that there is an upper limit on the frequency for which the distortion effects can be accurately compensated, and that this can significantly affect the accuracy of experimental results. We propose a combination of experimental studies and detailed numerical modelling of the impact event and wave propagation along the bar to gain better understanding of the frequency content of the impact event, and help assess the accuracy of experimental predictions of impact face load.

## 1. Introduction

Bertram Hopkinson's original development of the pressure bar that was to bear his name [1] was, as is well known, driven by the requirement for a robust and accurate device for recording intense, transient loading pulses. It is clear, however, that the dominant use of the pressure bar in experimental research has been in the split Hopkinson pressure bar (SHPB) format developed by Kolsky [2]. A non-rigorous, but informative, search by the authors of published literature referenced on the Scopus online database revealed 2384 articles published since 1949 which include the term ‘Hopkinson pressure bar’ in the title, abstract or keyword, of which 2167 also include the word ‘split’ in these fields (table 1).

Nevertheless, the original use of the pressure bar is still of value in modern research. The SHPB test, when properly conducted, is a test that characterizes *material* properties at high strain rates, but it is often of interest to determine the response to such loading of specimens whose length scales preclude the use of the SHPB test. In such circumstances, the specimens are actually small *structural elements* and under dynamic loading, longitudinal inertial effects are important and the specimens do not experience the longitudinal stress equilibrium required for the SHPB test to be valid. Experimental testing of these larger specimens generally requires accurate capture of the load–deformation characteristics of the specimens, with the load measured at both the proximal and distal face of the specimen.

While it is possible to use discrete load cells, restrained by stiff end-stops to record the load experienced on the faces of rapidly deformed specimens, in practice the finite rigidity of both the load cells and their restraints can limit the natural frequency, and hence, useful bandwidth of such a recording device. The HPB in its original ‘un-split’ form offers an attractive alternative dynamic force transducer. If the pressure bar is used to record the propagation of an elastic stress pulse generated by an impact of, or onto a specimen, then the distal restraint conditions of the bar are irrelevant so long as the entire loading pulse can be captured at a strain gauge station on the bar before the reflection form the distal face returns to the gauge station. In such conditions, the longitudinal natural frequency of the bar does not impose the bandwidth restriction of a short, restrained load cell.

## 2. Distortion of Hopkinson pressure bar signals

There are limitations of a different kind imposed on the effective bandwidth of the HPB as a dynamic force transducer, due to the fact that signals change in form as they propagate along the HPB. The fundamental work on quantifying these limitations was presented firstly by Bancroft [3] and subsequently, in more detail, by Davies [4]. Both found solutions of the cylindrical equations of motion derived by Pochhammer [5] and Chree [6] and set out in detail by Love [7]. The results of Bancroft and Davies show that there are two mechanisms by which dynamic disturbances in the pressure bar will become distorted.^{1} When using the HPB as a dynamic force transducer, it is of great importance that we understand the nature of this distortion, and are able, as best we can, to correct for it, so that we can obtain good estimates of the actual impact load from records of the bar response some distance from the loaded face.

The first of these distortion mechanisms is the well-known dispersion effect, due to the fact that the phase velocity of oscillations in the bar is a function of the frequency of the oscillation, with the velocity typically decreasing with frequency. Davies's study of this effect led him to conclude that ‘no pretence at accuracy can be made’ when the loading applied to the HPB varied significantly in amplitude at rates of the order of a few microseconds. Consequently, both Davies and Hseih & Kolsky [8] used the approach of assuming an idealized form of the loading pulse in their experiments with the HPB, calculating the Fourier transform of the assumed pulse, predicting how the pulse would disperse as it propagated along the bar to some gauge location, and comparing this prediction with the observed dispersed pulse to assess the accuracy of the assumed form of the loading pulse. The application of the fast Fourier transform method to the analysis of dispersed experimental HPB signals by Yew & Chen [9] led to the development of the frequency domain dispersion correction method by Follansbee & Frantz [10] and Gorham [11] in which corrections to the phase angle of the Fourier components are made to allow for the differences in phase velocity of different harmonics as a pulse travels between two points on the bar. This has become a standard approach for analysing and correcting the dispersion effect in HPB signals.

The second consequence of the Pochhammer–Chree equations, as shown by Bancroft and Davies is that harmonic oscillations will generally produce bar response which is a function of the radial ordinate at any cross section. Davies, in particular, set out this issue in great detail, and presented formulae for calculating the variation of a range of stresses and displacements over the cross section. For example, the axial stress in the bar, which will be required later in this article, is given by 2.1 where 2.2 2.3 2.4 2.5 and 2.6 However, while the phase-shift dispersion effect has been widely recognized, the non-uniform bar response aspect of the theory has received surprisingly little attention in subsequent HPB research. The issue was considered by some users of the SHPB [10,12,13] who concluded that the frequency content of their signals was such that ignoring the non-uniform bar response had a negligible effect on the accuracy of their results. Safford [14] appears to have been the first to propose including a correction for the non-uniform bar response to experimental HPB signals. Tyas & Watson [15] presented experimental evidence that a propagating wave in an HPB produced variations in response over the bar cross section which closely followed Davies's predictions of the response of an infinitely long bar to a sinusoidal forcing function of infinite duration. The same authors consequently presented a modification of the frequency domain dispersion correction method [16] similar to that proposed by Safford, and used explicit dynamic finite-element models of wave propagation in the HPB assess the benefit of the modification. In this work, as in that of Safford, in addition to the standard correction of the phase angle, the amplitude of the Fourier component of stress or strain (recorded by means of strain gauges on the bar perimeter) was also modified to allow for the fact that the perimeter stress and strain were not equal to the average values over the cross section. A similar approach was presented more recently by Merle & Zhao [17]. Both Safford [14] and Tyas & Pope [18] demonstrated that the addition of this amplitude modification could lead to significant improvements in the accuracy of frequency domain dispersion correction when the aim was to recreate the loading on the impact face of a single HPB.

An important consequence of these investigations is to highlight an issue that was very much apparent to Davies, but which has received little subsequent attention. This is the fact that Davies's analysis shows that as the frequency of axial harmonic oscillation in the HPB increases, the ratio of the axial strain at the bar perimeter to the average axial strain over the bar cross section falls, eventually reaching zero at a non-dimensional frequency *x* of around 0.293 for *ν*=0.29 (the precise value of *x* being relatively insensitive to most practical values of *ν*). At this frequency, therefore, the condition may exist where energy is being propagated along the interior of the bar, with no axial response at the bar perimeter. Consequently, at this frequency, the Fourier amplitude modification factor proposed in [16] goes to infinity. In practice, at some lower frequency, the amplification factor magnifies background signal noise to such an extent that the underlying signal becomes badly corrupted. In [16], as a result, it was suggested that the amplitude correction approach would effectively break down at around *x*=0.25, indicating that, for a steel pressure bar, the maximum frequency that could be accurately recorded and corrected using the conventional approach of axially aligned strain gauges was around 1250/*a* Hz.

Aware of this problem, Davies had suggested the use of condenser gauges to record either the radial displacement of the bar surface, or the average displacement of the distal end of the HPB rather than the axial displacement or strain at the bar perimeter, neither value of which has this ‘drop-out’ frequency.

The rest of this paper describes an approach combining experimental HPB studies with numerical modelling in order to attempt both to gain better insights into the mechanics of wave propagation in the HPB, and to investigate the possible bandwidth limitations in the use of the HPB as a dynamic force transducer.

## 3. Experimental methodology

The motive for the study presented in this paper was a series of experimental tests conducted at the University Of Sheffield Blast and Impact Laboratory, designed to investigate the impact behaviour of additively manufactured Ti_{6}Al_{4}V alloy microlattice structures. These materials have the ability to absorb a significant amounts of energy, and in principle, the microstructure can be designed to give controlled collapse under dynamic load. The micro-trusses have the possibility therefore of being used in blast, impact and ballistic damage protection systems as a sacrificial energy absorption layer, say between a front armour face and ahead of a rear protected face.

In this work, unit cells approximately 5 mm square were used to produce multiple-layered lattice structures by replicating unit cells in three dimensions. Changing the size and geometrical arrangement of the lattice bars offers the possibility of producing desired mechanical properties, and the purpose of this initial study was to assess the impact behaviour of different lattices when loaded, and to give guidance on future analysis and design approaches. An example of a unit cell with a re-entrant bar arrangement is shown in figure 1, whereas an example of a 25 mm diameter×25 mm long microlattice built up from this unit cell is shown in figure 2. The bulk density of the microlattice was 625 kg m^{−3}.

The samples were loaded by driving them at speed on to the front face of an HPB. The samples were either initially placed on the front face of the bar and impacted by a projectile, or mounted on the face of a projectile which was then fired at the bar, the two cases giving respectively distal and impact face loading onto the front of the HPB. A 25 mm diameter and 250 mm long EN24T steel bar with a mass of 963 g was fired at velocities in the range 5–21 m s^{−1} for low-velocity impact tests, whereas a 27 mm diameter and 31 mm long nylon 66 projectile with a mass of 19.3 g was fired at velocities in the range 80–250 m s^{−1} for high-velocity impact.

The HPB was made from EN24T steel, 25 mm diameter and 3.4 m. An axial gauge station was located on the bar surface, 250 mm from the impact face, comprising four Kyowa KSP-2-120-E4 semiconductor strain gauges, linked in such a way as to eliminate bending effects in the output strain. The density of the bar was 7850 kg m^{−3}. The bar wave speed (*C*_{0}=5175 m s^{−1}) was determined from measurements of the transit time of repeated reflections of a pulse along the bar and using the bar wave velocity equation, the elastic modulus determined to be 210.2 GPa. Poisson's ratio was 0.29. Full details of the experimental work are given in [19].

## 4. Numerical modelling methodology

All features of the impact events (impactor, microlattice specimen and HPB) were modelled using the multi-purpose explicit, nonlinear FE analysis program LS-DYNA. Modelling the impact event and the bar allowed capture of the stress–time history generated at the bar–specimen interface, as well as the dispersed signal at the location of the experimental strain gauge station, for direct comparison with the experimental records.

Three-dimensional Timoshenko beam elements with appropriate contact properties were used for the modelling of lattices due to the fact that continuum elements for such small bars would be prohibitively computationally expensive. The main failure mechanisms of the lattices such as plasticity, buckling and brittle shear failure were considered in the numerical model. In order to take plasticity into account, a material model with elastoplastic stress–strain behaviour pertaining to Von Mises yield condition with isotropic strain hardening and failure criteria was used. In the material model, Ti_{6}Al_{4}V alloy was assumed to be strain-rate independent, and to have properties determined by static testing, viz. Young's modulus of 114 GPa, Poisson's ratio of 0.3, mass density of 4.43 10^{3} kg m^{−3} and yield stress of 880 MPa.

The interaction forces between parts were transferred using contact algorithms. Two separate single surface contact pairs were used to model the force transfer between impactor and micro-truss sample and the interaction between micro-truss sample and the HPB. Self-contact of the lattices was modelled using a beam-to-beam contact algorithm. The nylon 66 impactor was assumed to exhibit linear elastoplastic behaviour with Young's modulus of 1.7 GPa, Poisson's ratio of 0.4, mass density of 1.088 10^{3} kg m^{−3}, yield stress of 160 MPa and tangent modulus of 1.00 MPa. Following the development of the finite-element model of micro-trusses, a mesh sensitivity analysis was carried out to ensure that the results are not sensitive to the mesh size; it was found that five beam elements per strut provided good convergence. Full details of the numerical modelling methodology are given in [20].

## 5. Results

Figure 3 shows the axial stress–time histories from the experimental and FE model, for the impact face load of a five-layer deep by 25 mm diameter microlattice with an impact velocity of 136 m s^{−1}. Several general features are worthy of comment. First, all signals display five short load spikes, followed by a longer load pulse, associated with the successive collapse of the five layers of the microlattice specimen, followed, finally, by the rapid deceleration of the nylon piston. The load predicted by the model at the impact face of the HPB (figure 3*b*) has a highly transient initial load pulse (rise time less than 1 *μ*s, with a decay back to approximately zero over approx. 10 *μ*s) which is not seen in any of the signals recorded at the gauge station location, 250 mm along the bar; in these latter signals, the first spike has a much smoother rise, over 10 *μ*s or so and a similar decay time. The results show good correlation between the general form of the experimental signal (figure 3*a*) and that of the model at the bar surface (figure 3*c*), albeit that the magnitudes of the transient load spikes vary somewhat (perhaps owing to slightly over-stiff behaviour in the model). Finally, the signals from the FE model at the strain gauge location show significant difference in high-frequency content between the surface measurement and that at the bar axis (figure 3*d*), with the latter appearing to have significantly greater high-frequency content.

These results taken together demonstrate the features discussed in §2. First, classical phase velocity dispersion means that the rapid rise of the initial pulse in figure 3*b* is softened significantly by the time that the pulse has travelled 250 mm along the bar. However, the dispersed high-frequency energy shows up much more markedly on the bar axis signal in the FE model than in either the experimental or FE model signals at the bar surface. Qualitatively, this appears to be related to the reduction in response at the bar surface with increasing frequency.

For quantitative investigation of the extent to which the recorded signals matched the predictions of non-uniform stress distribution from Davies's theory, fast Fourier transform analyses of the signals were conducted, with the frequency spectra shown in figure 4. It is immediately clear that for the experimental signal, the amplitude drops to around that of the background experimental noise around the expected non-dimensional frequency of *x*=0.293. The results from the FE models all follow near-identical paths at low frequencies, *x*<0.05, before diverging, the bar axis spectrum rising above that of the average input stress and the bar surface stress falling below. Around the critical frequency *x*=0.293, the bar surface stress spectrum closely follows that of the experimental signal, its amplitude falling to less than one-twentieth that of the average impact stress. Conversely, in the region of this frequency, the bar axis amplitude is typically around twice that of the average impact stress, compared with a predicted ratio of approximately 2.1–2.2 from equations (2.1)–(2.6).

It seems clear then that, notwithstanding the less than perfect correlation between the experimental results and the FE modelling of this complex impact event, the frequency spectra of axial stress signals from each approach appear generally to follow the predictions set out by Davies. This finding is further strengthened by a comparison of the frequency data for the bar axis and surface signals from the FE model. Figures 5 and 6 show, respectively, the ratio of the Fourier amplitudes of the signals at the bar surface and axis, and the difference in phase angle of these Fourier components. On both graphs, the predictions from Davies's theory (equations (2.1)–(2.6)) are shown, for the assumption of first mode wave propagation.

It should be remembered that, for *x*>0.293, the sign of the stress at the bar surface is opposite to that at the bar axis; i.e. *σ*_{(r=a)}/*σ*_{(r=0)} is negative. In figure 5, we have shown the absolute magnitude of the ratio from Davies's prediction, and in figure 6, the negative ratio manifests itself as a prediction that the bar surface stress will be in anti-phase to that at the bar axis.

Figures 5 and 6 show that, with a few minor, and local deviations, the data from the FE model are in excellent agreement with Davies's predictions for . At higher frequencies, there appears to be a permanent breakdown of the correlation between the FE data and the theoretical values.

## 6. Discussion and conclusion

The results set out in §5 offer strong evidence that, at least for frequencies below *x*=0.35, the aspect of Davies's theory relating to non-uniform response across the bar cross section is both real and measurable. The question that now results is: what does this imply for frequency domain dispersion correction, when we wish to determine the load on the impact face of the bar from a record of the dispersed pulse that we have measured at some remote (and usually, surface mounted) strain gauge station?

The most obvious conclusion was well known to Davies, but bears repeating. If the signal contains energy at frequencies at which the axial response of the bar surface falls towards zero, then this cannot accurately be recorded by a direct measurement of axial stress or strain on the bar surface. While the addition to the frequency domain dispersion correction method presented in [13,14] can be of use at frequencies below that at which the axial surface response falls towards zero, at *x*≈0.28–0.32, it is of questionable value, as it tends simply to apply a large factor to a signal which is little more than background noise, as shown by the data in figure 4. Although not discussed here in detail, this problem can be obviated by measurement of radial stress or strain, and amplitude correction factors based on Davies's theory and similar to those presented in [13,14] may be derived to convert the radial response to axial stress. This approach has the comparative advantage that, at no frequency does the radial response fall to (or close to) zero.

However, the data shown in figures 5 and 6 suggest that there is a much more serious limitation on the effective bandwidth of the HPB. The data in figures 5 and 6 show that the FE model results deviate greatly from the Davies predictions when . This is an important value, for it is the frequency at which, for *ν*=0.29, the second mode of axial disturbance begins to propagate. It appears likely therefore that this is the reasons for the deviation between the FE and theoretical results.

If there were a cut-off at *x*=0.35, with first mode propagation ceasing and second mode commencing, then it would be a relatively simple matter to determine the relevant dispersion relation for conventional phase angle dispersion correction, and calculate suitable Fourier amplitude correction values following the approach in [13,14]. However, it appears that this is not the case. For example, for second mode propagation, it can be shown that the bar surface stress falls to zero at *x*≈0.8. In figure 5, there is no evidence of this. Indeed, if anything, it appears that the bar surface stress may approach zero at *x*≈0.6. This suggests that for at *x*>0.35, energy is being carried in more than one mode. With current knowledge, this appears sets an upper bound on the effective bandwidth of the HPB, because there is as yet no rigorous and accurate way of identifying the energy of a disturbance that is propagated in any given mode at any given frequency and therefore, no way of determining which dispersion relation and amplitude correction factors to use in frequency domain dispersion correction.

This suggests that, for steel HPBs, the practical limit on the frequency that can be recorded and for which dispersion correction can be applied is approximately 1750/*a* Hz, if radial gauges are used on the bar surface and suitable factors applied to the Fourier amplitudes to convert radial response to axial stress. If surface mounted axial gauges are used, the upper limit on frequency is approximately 1250/*a* Hz.

However, this leaves a troubling issue. How are we to determine whether or not there is significant energy in our signal at higher frequencies, if we cannot accurately record the bar response at those frequencies? The results presented in figure 3 indicate the nature of this problem. There is clearly a significant amount of high-frequency energy associated with the rapid rise time of the initial loading spike on the loaded face of the bar (figure 3*b*). By the time the signal has travelled 250 mm along the bar, the phase velocity effect has dispersed this energy along the length of the signal, rounding off the initial spike. This is clear in both figure 3*c*,*d*. However, only on the response at the bar axis, figure 3*c*, can we see these dispersed high-frequency oscillations on the signal. Because of the fall in axial response with frequency, they are hidden on the bar surface signal. So, for intense, transient loading if we have only the bar surface stress data, then it would seem that we have little chance of either being able to reconstruct the actual impact face load, or perhaps of even being aware that there is additional energy that we have not accurately recorded. And as the comparison between the initial load spike magnitudes in figure 3*a*–*c* indicates, this may result in a significant underestimate of both the magnitude and the rise time of important features in the loading.

In the absence of a better understanding of how to separate information being carried at different propagation modes, it seems sensible therefore, to conduct high-fidelity numerical modelling of impact events on HPBs as a standard accompaniment to any experimental testing where there is the possibility that significant energy will be contained at high frequencies. In the ideal outcome, very good correlation between the dispersed HPB surface signals from experimental tests and numerical analysis would indicate that the model was correctly capturing the key features of the loading event, and hence, give confidence that the undispersed load at the impact face of the HPB in the model could be trusted. This presents a challenge for both experimental practitioners of the HPB studies, and numerical modellers to develop methods of combining the insights of both approaches to give us better insights into high-rate impact mechanics.

## Funding statement

The experimental results were obtained from work supported by the MOD's Armour and Protection Science and Technology Centre through project no. DSTLX1000054230.

## Acknowledgements

The authors thank Mr Stephen Fay and Mr Alan Hindle of Blastech Ltd for their assistance in conducting the experimental tests. The authors also thank a reviewer for directing them towards the excellent and little-known publication by Safford.

## Footnotes

One contribution of 11 to a Theme Issue ‘Shock and blast: celebrating the centenary of Bertram Hopkinson's seminal paper of 1914 (Part 2)’.

↵1 The Pochhammer–Chree equations have an infinite number of solution branches, each appertaining to a particular mode of response in the bar. In this paper, we confine our attention to the first mode unless expressly noted otherwise.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.