Wave separation in viscoelastic pressure bars using single-point measurements of strain and velocity

Polymer Testing 22 (2003) 155–164www.elsevier.com/locate/polytest

Test Method

Wave separation in viscoelastic pressure bars using single-point measurements of strain and velocity

D.T. Casema,∗, W. Fourneyb, P. Changc

a US Army Research Laboratory, AMSRL-WM-TD Aberdeen Proving Ground, MD 21005-5069 USAb University of Maryland, College Park, Department of Aerospace Engineering, College Park, MD 20742, USA

c University of Maryland, College Park, Department of Civil Engineering, College Park, MD 20742, USA

Received 21 March 2002; accepted 9 May 2002

Abstract

A wave separation technique that uses the single-point measurements of axial strain and axial particle velocity isinvestigated. The method is capable of treating wave propagation in viscoelastic pressure bars and accounts for disper-sion and attenuation due to both geometric and material effects. An example application using a polycarbonate splitHopkinson pressure bar to test low-density polymeric foams is shown. Emphasis is placed on the ability of the methodto correct for viscoelastic wave propagation and the necessity of this correction during practical testing. 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Viscoelastic wave propagation; Split Hopkinson pressure bar; Kolsky bar; Wave separation; Electromagnetic velocity gage

1. Introduction

This paper investigates a wave separation techniquethat is applicable to viscoelastic bar wave propagation.It is well suited for use with the viscoelastic Split Hop-kinson Pressure Bar (SHPB), which has gained popu-larity as a viable technique for testing low-impedancematerials (e.g., polymeric and metallic foams) at highstrain rates. The method is based on the measurement ofaxial strain and axial particle velocity at a single pointon a viscoelastic bar. It is a natural extension of a well-known one-dimensional linear elastic analog. Accord-ingly, the method is introduced by first presenting thesimpler case as applied to a single pressure bar. It is thengeneralized to include viscoelastic wave propagationusing the dispersion correction described in [1]. Severalseries of experiments are presented to illustrate the stepsinvolved during application of the method. Finally, it isapplied to experimental results obtained from a viscoel-

∗ Corresponding author. Tel.:+1-410-306-0798.E-mail address: [email protected] (D.T. Casem).

0142-9418/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.PII: S0142 -9418(02 )00064-8

astic SHPB, where it is used to extend the durations ofexperiments conducted on samples of low-density poly-meric foam.

1.1. Linear elastic wave separation

The primary function of a pressure bar, or Hopkinsonbar, is to measure stress and velocity at one end of thebar through the measurement of stress waves within thebar itself. Such a bar is depicted in Fig. 1. It is assumedthat the bar is perfectly straight and has a constant cross-sectional area that is sufficiently small that the bar canbe assumed to be in a state of one-dimensional axialstress. The bar is linear elastic with densityρ and elasticmodulus E. The end at x= d is subjected to a dynamicpressure P(t). The resulting stress waves that propagatein the bar are monitored by a set of strain gages at themid-point of the bar.

It is desired to measure the applied pressure, P(t), andthe corresponding particle velocity, ve(t), from thismeasurement of strain. To do this, the bar wave propa-gation must be understood. Since the bar is very thinand made from a linear elastic material, the simple one-

156 D.T. Casem et al. / Polymer Testing 22 (2003) 155–164

Fig. 1. A pressure bar instrumented with a strain gage at its mid-point is used to measure an applied pressure P(t) and the correspond-ing particle velocity, ve(t), of that end.

dimensional theory applies, i.e., the strain measured bythe gage is the superposition of two waves, F and G.

e(x,t) � F(x /c0�t) � G(x /c0 � t) (1)

Here x = 0 is the location of the gage, and compressivestrain is defined as positive. The function F is a wave-train that propagates in the direction of increasing x andthe function G is a wave-train that propagates in thedirection of decreasing x. Both waves propagate withoutchanging form at the bar wave speed c0, given byc0 � √E /r.

In many cases, it is possible to ensure that the timingof the pulses is such that no reflections from the end atx = –d reach the gage during the time period of interest.Therefore, the strain gage measures F(t) directly. In thiscase, the desired quantities P(t) and ve(t) are found bythe following equations.

P(t) � F(d /c0�t)rc20 (2)

ve(t) � F(d /c0�t)c0 (3)

This amounts to shifting the measured strain signals intime by an amount d/c0 to the time at which the pulseacts at the end of interest, and multiplying by the appro-priate acoustic quantities accordingly.

In general, however, both F and G are non-zero, andthere is no way to measure the desired quantities usingonly a single measurement of strain without usingadditional information. There are three common methodsof circumventing this problem. The most commonmethod is that which was mentioned above, i.e., to timethe signals such that when one component is present atthe gage, the other is zero. (In the conventional SHPB,this can be achieved by limiting the length of the incidentpulse to be less than the length of the incident bar, withthe strain gage at the mid-point.) Alternatively, if onehas knowledge of the end condition of the far end of thebar (a free surface, for example), one can use a reflec-tion-based technique to separate the pertinent waves. Arecent example of this technique can be found in [2].More sophisticated techniques attempt to separate thewaveforms using multiple gages. For example, Lundbergand Henchoz, [3] and Zhao and Gary, [4], use two-pointstrain measurements to separate the two waves and arein general quite successful.

This paper explores another multiple-gage techniquethat utilizes a velocity gage to complement the strainmeasurements. Both gages are installed at the mid-point

as shown in Fig. 2. In this case, the measured strain isstill given by Eq. (1), above. The measured axial particlevelocity is given by

v(x,t) � c0[F(x /c0�t)�G(x /c0 + t)] (4)

where, again, x = 0 is the location of the gages. Combin-ing Eqs. (1) and (4), the two wave components can beeasily determined.

F(x /c0�t) �12

[e(x,t) � v(x,t) /c0] (5)

G(x /c0 � t) �12[e(x,t)�v(x,t) /c0] (6)

Once F and G are separated, the desired quantities P(t)and ve(t) can be determined by shifting the wave formsto the time at which they act at x = d, in a fairly straight-forward manner. The resulting equations are as follows.

P(t) � rc20[F(d /c0�t) � G(d /c0 � t)] (7)

ve(t) � c0[F(d /c0�t)�G(d /c0 � t)] (8)

1.2. Viscoelastic wave separation

This paper is primarily concerned with the applicationof the above method to viscoelastic bars of relativelylarge diameter. This introduces two complications overthe linear elastic case discussed above. The first is thatgeometric dispersion is present due to lateralexpansion/contraction of the bar when subjected to axialcompression/tension. This well-known dispersion effect[5–7] causes the phase speed to be frequency dependent.Because of this, transient pulses containing multiple fre-quencies disperse as they travel. The second compli-cation arises from the viscoelastic behavior of polymericbars. Viscoelastic wave propagation is accompanied byboth dispersion and attenuation, i.e., pulses in a viscoel-astic bar attenuate and disperse according to their fre-quency content as they propagate through the material.This is in addition to geometric dispersion effects. Thereare several wave-propagation theories available that canbe used to compensate for viscoelastic wave propagation[1,8–11]. The one used in this paper is adopted fromBacon [1]. This technique has several advantages, themost important of which is its ability to handle disper-sion due to the two effects mentioned above. All wavepropagation parameters are empirically determined from

157D.T. Casem et al. / Polymer Testing 22 (2003) 155–164

Fig. 2. A pressure bar instrumented with both a velocity gage and a strain gage at the mid-point.

the two-point measurement of impulsive transient signalsas they travel in the bar and are characterized by a propa-gation coefficient, γ. This parameter is a complex valuedfunction of frequency, i.e., γ = γ(ω), and is easier to con-ceptualize in terms of its real and imaginary parts.Specifically, the attenuation parameter, α(ω), and thephase speed, c(ω), can be determined from γ(ω) by thefollowing equation

g(w) � a � iwc

(9)

α describes the attenuation of different frequencies asthey propagate in the bar. Similarly, c gives the phasespeed as a function of frequency (including effects dueto both geometry and viscoelasticity). Interested readersare referred to [1] for details of the method.

The viscoelastic wave separation technique is other-wise directly analogous to the elastic case. Consider aviscoelastic bar in the configuration of Fig. 2. The strainmeasured by the strain gage is

e(x,w) � F(w)e�gx � G(w)egx (10)

where the ˆ denotes the Fourier Transform. Notice theexponential terms containing the term γ; through theseterms the attenuation and dispersion are accounted for.The particle velocity measured by the velocity gage is

v(x,w) ��iwg

(F(w)e�gx�G(w)egx) (11)

Provided the propagation coefficient is known, F and Gcan be solved for readily. This provides a very straight-forward wave separation technique that accounts for bothdispersion and attenuation, and requires no assumptionsabout the ends of the bar or the presence/non-presenceof pulses at the measurement location. The resultingequations are

F(w) �12�e(x,w) �

igw

v(x,w)� (12)

G(w) �12�e(x,w)�

igw

v(x,w)� (13)

Once these have been determined, the stress and velocityat any other point in the bar can be found. For example,the pressure and velocity at the end at x = d are

P(w) �rw2

g2(F(w)e�gd � G(w)egd) (14)

ve(w) ��iwg

(F(w)e�gd�G(w)egd) (15)

The similarity between Eqs. (5)–(8) and (12)–(15) shouldbe noted.

2. Experimental verification of the wave separationtechnique

2.1. SHPB test without a specimen

The apparatus shown in Fig. 3 was used to verify thewave separation technique. It is essentially a standardpolymeric SHPB in which electromagnetic velocitygages are installed at the locations of each strain gage.This supplies the necessary data for wave separation inboth bars. In this way, the apparatus is “dual-instru-mented” ; provided the incident pulse is short enough, itis possible to analyze the results from a given test using(i) the wave-separation method outlined above and (ii)the conventional strain gage analysis. This provides acomparison between the separation technique and anaccepted method. Furthermore, it is possible to performeither analysis using the elastic or viscoelastic theories,so that the importance of the viscoelastic correction canalso be evaluated. More information on the operation ofthe conventional SHPB system can be found in Fol-lansbee [12] and Davies [13].

Both bars shown in Fig. 3 are 1260 mm long and have19.1 mm diameters. They are made from polycarbonateand have known propagation coefficients. Note there isno specimen in the figure. Instead, the bars are initiallyseparated by a known distance, δ. The concern is themeasurement of stress and velocity on both “ test-ends”of the bars, i.e., I1 and I2. These are the interfaces thatwould be in contact with a specimen during an actualtest. The incident bar is impacted with a projectile andthe gap closes. Before the gap is fully closed, both I1

and I2 are stress-free, and I2 remains stationary. At themoment of impact, and as long as the interfaces remainin contact, both the stress and velocity are continuousacross the interface. These conditions are summarizedas follows:

Before impact: σ1 = σ2 = 0 and v2 = 0After impact: σ1 = σ2 and v1 = v2

As stated above, two methods are used to analyze this


Fig. 3. Gap test used to verify wave separation technique. The x-coordinate system refers to the incident bar; the y-coordinatesystem refers to the transmitter bar.

test. The first uses only the strain gage data in the stan-dard SHPB analysis (viscoelastic effects included). Theincident and reflected pulses are measured independentlyby the strain gages on the incident bar, SG1 (the incidentpulse is short enough to do so). The transmitted pulse ismeasured by the transmitter bar strain gages, SG2. Thesepulses are then used to determine the stress and velocityat the two interfaces, σ1, σ2, v1, and v2.

The second technique is the separation method andutilizes all four gages (SG1, SG2, VG1, VG2). Ofcourse, the separation technique is unnecessary in thiscase but it is performed so that it can be compared tothe strain gage analysis. The same quantities, σ1, σ2, v1,and v2, are determined. As such, there are numerousways to make this comparison, and stress and velocityfrom both methods are plotted below. First, the stepsinvolved in the separation technique are briefly outlinedfor a test in which δ = 1.5 mm. This is small enoughthat the gap closes due to the first passage of the incidentpulse. Strain and velocity are measured by both sets ofgages. Application of Eqs. (12) and (13) separates thetwo superimposed wave-trains. This is done for each barand plotted (in the time domain) in Figs. 4 and 5.

Using these four waveforms, one can correct for dis-persion and attenuation between the measurement pointsand the I1/I2 interfaces:

1. The velocity of I1 is the sum of the particle velocity

Fig. 4. F and G for the incident bar.

Fig. 5. F and G for the transmitter bar.

generated by both waveforms in the incident bar, Fi

and Gi (the subscript “ i” denotes the incident bar).

v1 ��iwg

[Fiedg�Gie�dg] (16)

2. The velocity of the I2 interface is the sum of the par-ticle velocity generated by both waveforms in thetransmitter bar (the subscript “ t” denotes transmitterbar).

v2 ��iwg

[Fte�dg�Gtedg] (17)

3. The stress on the I1 interface is the sum of the stressesdue to the waves in the incident bar.

s1 �rw2

g2[Fiedg � Gie�dg] (18)

4. The stress on the I2 interface is the sum of the stressesdue to the waves in the transmitter bar.

s2 �rw2

g2[Fte�dg � Gtedg] (19)

The stresses and velocities calculated from these for-mulae are now compared to those calculated from the


Fig. 6. Velocity of the I1 interface — strain gage analysis vs.wave separation (both are full viscoelastic analyses).

standard strain gage technique. Fig. 6 shows the velocityof the I1 interface using both methods. They are in goodagreement, and describe what is expected to happen. Thevelocity is essentially a square pulse up until the timethe incident bar impacts the transmitter bar (about 8.40ms). At this instant, the I1 interface changes from a freesurface to a non-reflecting boundary, as one expectscomplete transmission into the transmitter bar. Thisexplains the drop in velocity from approximately 9 m/sto about 4.5 m/s. As for comparing the two methods, itis believed that what little difference exists is primarilydue to measurement error between the strain and velo-city signals.

The velocity of the I2 interface is measured by bothmethods using the transmitter bar gages. Fig. 7 showsresults from the two techniques. As in Fig. 6, bothaccount for the full viscoelastic solution, and are inexcellent agreement. Physically, they give a reasonabledescription of what happens. The I2 interface is motion-less until impacted by the incident bar. At this instant,the velocity jumps to 4.5 m/s (compare with Fig. 6).

Fig. 7. Velocity of the I2 interface — strain gage analysis vs.wave separation (full viscoelastic analyses).

Fig. 8. Stress on the I1 interface — strain gages vs. wave sep-aration (full viscoelastic analyses).

Fig. 8 shows the stress on the I1 interface. In this case,the agreement between the two techniques is good, butnot as good as the velocity measurements. This has todo mainly with inadequacies in the dispersion correctionfor this type of loading situation. Stress measurementson the incident bar are often poor when the incident baris free (or lightly impeded) because of the presence ofoscillations on the incident and reflected pulses. Eventhough dispersion is accounted for, the oscillationsremain. This is seen clearly in the non-zero portion ofthe stress signal between 0.6 ms and 0.8 ms. The stressat this time must be zero because the two bars are notin contact. This is an unfortunate deficiency of the visco-elastic correction, although the uncorrected data is sig-nificantly worse. For this reason, many researchers forgothe measurement of stress with the incident bar and useonly the transmitter bar to measure stress in a test withan actual specimen.

Fig. 9 shows the stress on the I2 interface calculated byboth methods. Again, the two methods are in excellentagreement. In contrast to Fig. 8, the stress signal is

Fig. 9. Stress on the I2 interface — strain gages vs. wave sep-aration (full viscoelastic analyses).


exactly zero up until impact, as expected (the slight dipin tension at 0.84 ms is another error of the dispersioncorrection, although not very serious; this can also beseen in Fig. 7).

It is important to illustrate the ability of this methodto treat viscoelastic wave propagation, and to show whenand why the viscoelastic analysis is required over thesimpler elastic analysis. There are two places in the waveseparation technique where a viscoelastic correction canbe applied. The first has to do with the use of Eqs. (12)and (13), which determine F and G from measured strainand particle velocity. This, in general, requires the useof the Fast Fourier Transform (FFT) on the gage data.This transformation complicates the analysis because themeasured time window may not be well suited for analy-sis in the frequency domain. However, it was found thatfor all practical cases with the polycarbonate bars used inthis research, the viscoelastic correction is unnecessary atthis point of the analysis. This is because dispersion andattenuation effects are only noticeable when a pulse trav-els over a significant distance along a bar; they are notnoticeable when the measurements and calculations areconfined to a single point (i.e., at the location of thegages). This observation is explained in more detail in[11,14]. The consequence here is that F and G may bedetermined from Eqs. (5) and (6), which are the elasticapproximations of Eqs. (12) and (13). Again, the advan-tage of these equations is that they can be used directlyfrom the gage output without any need for analysis infrequency space.

However, depending on the distance between thegages and the ends of the bars/specimen, a viscoelasticcorrection may be important. This can be demonstratedby the experiment discussed above. Fig. 10 shows thevelocity of the I2 interface using the wave separationmethod. One curve shows the full viscoelastic analysisand the other shows the elastic approximation. The vis-

Fig. 10. Necessity of viscoelastic correction in the wave sep-aration method — comparison of corrected I2 velocity with theuncorrected I2 velocity (“viscoelastic” and “elastic” ,respectively).

coelastic result demonstrates a sharper wave front thanthe elastic result. This is what one would expect fromthe impact of the two bars. The magnitude is slightlyhigher, probably because of slight attenuation that occursbetween the two gages (neglected in the elastic case).Also, the oscillations (which are a product of dispersion)are eliminated when the viscoelastic analysis is used. Itis clear in this case that a viscoelastic correction is ben-eficial. However, it should be noted that because thiswave separation technique allows more freedom in theplacement of gages, it is possible to greatly reduce thedistances between the gage and the specimen. Thisreduces dispersion and attenuation, and can eliminate theneed for a viscoelastic correction entirely. Unfortunately,it is difficult to establish exactly at what distance thisoccurs for the variety of pulses observed during practicaltesting. However, as a general rule, if the distancesbetween the gages and the specimen are reduced, theeffects of dispersion and attenuation are lessened.

2.2. Impact of an aluminum projectile with apolycarbonate pressure bar

As another illustration of the wave-separation tech-nique, consider the impact of an aluminum projectilewith a polycarbonate pressure bar, as shown in Fig. 11.This situation is somewhat more interesting than the gaptest described above because the pulse created by theimpact is long enough that the separation technique isnecessary. As before, an electrical resistance strain gageis installed along with an electromagnetic velocity gageat the mid-point of the bar. Because the impedance ofthe projectile is higher than that of the bar, it does notrebound after the impact (the reflection from thealuminum/polycarbonate interface is tensile). Instead, itgradually transmits its momentum to the pressure barthrough a series of decaying reflections. The result is thatthe pulse transmitted to the bar has a step-like geometricdecay, and is long enough that it superimposes on itsown reflection at the gage location. This is evident in therecorded strain signal in Fig. 12, where it is clear upuntil the arrival of the reflection (approximately 1.515ms). Consistent with the free-end boundary condition ofthe far end of bar, the reflected strain is tensile, and issuperimposed on the continuing incident pulse. A similarthing is observed in the velocity signal (Fig. 13), exceptthat the particle velocity due to the incident pulse andthe reflection both have the same sign (motion awayfrom the projectile). However, through the processdescribed above, the superimposed waves can be separ-ated and the stress at each end of the bar may be determ-ined. This is shown in Fig. 14. Each stress profile is cal-culated using the simple linear elastic theory (labeled“elastic” ) and the full linear viscoelastic theory (labeled“viscoelastic” ). The observed differences are typical ofneglected dispersion effects. For example, the step pro-


Fig. 11. An aluminum projectile impacting a single polycarbonate pressure bar (both have 19.1 mm diameters).

Fig. 12. Axial strain measured at the mid-point of a polycar-bonate pressure bar due to impact with an aluminum projectile.

Fig. 13. Axial particle velocity measured at the mid-point ofa polycarbonate pressure bar due to impact with an aluminumprojectile.

file of the impact event has a steeper rise time when theviscoelastic analysis is used. In addition, the non-zerooscillations observed on the free end of the bar are less-ened in the viscoelastic case. Unfortunately, however,they are not eliminated (a similar observation was madein Fig. 8 for essentially the same reason). Note that theseerrors are associated entirely with the propagation of thepulses between the gages and the ends of the bar; asabove, it was found that the bar was sufficientlydescribed as being linear elastic during the actual step

of separation (i.e., Eqs. (5) and (6) could be used in placeof Eqs. (12) and (13)).

3. Applications of wave separation to the dynamictesting of foams

3.1. Low-rate tests

There are several important practical applications ofwave separation techniques to the SHPB, especiallywhen testing low-impedance materials such as foams.The primary advantage is, in theory, that it extends theduration of the test indefinitely. It does so without anyrestrictions placed on the projectile length or material,or on the end conditions of either bar. With an unlimitedduration available, low-rate tests can be performed with-out sacrificing maximum strain. This is importantbecause it allows a comparison between the SHPB andmore common low-rate methods, e.g., drop-weight tests.One such test is discussed in this section.

Fig. 15 shows the apparatus. It is the same apparatusshown in Fig. 3, except that a foam specimen is placedbetween the bars. Also, notice that the projectile is thesame length as each bar. The projectile strikes the inci-dent bar at a speed of 5.9 m/s and essentially transmitsall of its momentum to it. The incident bar moves as a‘ rigid body’ and compresses the specimen at a very con-stant rate. Note that this test would be impossible to ana-lyze with the conventional strain gage method becausethe incident pulse in the incident bar superimposes withits reflection at the gage location. However, strain andvelocity gages located at the mid-points of each barallow for wave separation.

The analysis that follows uses the separation pro-cedure discussed above in an otherwise standard SHPBanalysis. The resulting stress–strain curve is shown inFig. 16. The strain rate is about 280 s–1, and gooddensification is achieved (83% max strain). To compareto a conventional SHPB without any wave separation,for the same length bars, specimen geometry, and strainrate, the maximum obtainable strain would be only about23%. This is hardly acceptable for such a highly deform-able material. Lower-rate tests are certainly possible; intheory there is no limitation.

It is interesting to note in this case that the viscoelastic


Fig. 14. Stress measured on both ends of the pressure bar. Both analyses (full viscoelastic and the simple one-dimensional linearelastic) are shown.

Fig. 15. Low-rate test performed on a low-density foam sample. Because of the length of the projectile, this test would not bepossible to analyze without a separation technique.

Fig. 16. Stress–strain curve for a foam specimen obtainedfrom a low-rate test using the wave separation technique. Thetwo curves show little difference between the elastic and visco-elastic analyses.

correction makes no significant difference in the finalstress–strain curve. This is also shown in Fig. 16, wherethe viscoelastic result is plotted along with the elasticresult (i.e., the pulses are simply shifted in time accord-ing to c0 without regard for dispersion and attenuation).The two are nearly identical. This is because dispersionand attenuation effects are only important when high-frequency signals are present, as observed in Figs. 10and 14. However, during the low-rate test, high-fre-

quency signals are not present and the elementary elasticanalysis is adequate. This is true even though the gagesare distant from the specimen (about 33 bar diameters).Of course, it is difficult to say with certainty a priori atwhat point viscoelastic effects are negligible and, in gen-eral, the viscoelastic analysis should be performed.

3.2. More complicated loading conditions

Another benefit of the extended test duration offeredby a wave separation technique is the increased varietyof loading regimes that can be applied to a specimen.The stress–strain curve for one such test performed ona foam sample is shown in Fig. 17. The projectile in thistest is short enough that the standard strain gage analysiscan be used. However, the incident pulse is not of suf-ficient magnitude or length to completely compress thespecimen, although further reverberations of this pulsein the incident bar do compress the specimen. Thisadditional compression can be measured using the waveseparation method. The two curves shown in Fig. 17 arefrom a single test using both methods. The strain gageanalysis shows the specimen loaded to about 50% strainand then unloaded, with no additional data. This is dueto the limitation of test duration caused by interferingreflections at the strain gage location. However, usingthe wave separation technique, it is seen that the foamis then loaded to over 85% compression and onceagain unloaded.


Fig. 17. Extended test using wave separation technique. Theconventional strain gage analysis only captures the initial load-ing; the wave separation analysis shows the second loading tofull densification.

Again, it is interesting to note the difference betweenthe viscoelastic and elastic analyses. This is shown inFig. 18. Even though the strain rate for this test is con-siderably higher than in the low-rate test (1200 s�1

instead of 280 s–1) the difference is small. The viscoelas-tic analysis reconstructs a slightly sharper peak at themaximum stress but the remainders of the two curvesappear almost identical.

4. Conclusion

By measuring both strain and particle velocity at thesame point in a viscoelastic bar, a simple and direct waveseparation technique is available that accounts for linearviscoelastic wave propagation. A practical use of this iswith the viscoelastic SHPB, where it can be used toextend test duration. This increased test duration givesmore freedom in the types of loads that can be applied

Fig. 18. Effect of viscoelastic dispersion in a wave separationtest. Notice that the peak stresses are slightly different.

to specimens, e.g., low-rate tests and variable-rate“unload/reload” tests can be performed. The viscoelasticseparation technique has a simpler linear elastic analog,and an important issue with the SHPB is when the errorsassociated with the elastic analysis are large enough towarrant application of the more complicated viscoelasticanalysis. The above examples illustrate this point. Gener-ally speaking, the necessity of the viscoelastic correctiondepends on the nature of the pulses encountered duringtesting. High-frequency, impulsive transient signals exhi-bit more attenuation and dispersion than lower-fre-quency, longer signals; therefore they exhibit more errorwhen viscoelastic effects are neglected. In the gap testpresented above, the square pulses generated by theimpact of the two bars contain enough of a high-fre-quency spectrum that the need for a viscoelastic analysisis evident (Fig. 10). This was also seen in the signalgenerated by the impact of an aluminum projectile (Fig.14). However, during a low-rate test on a low-impedancematerial, the signals are much less likely to suffer fromappreciable dispersion and attenuation. This is why Fig.16 shows no detectable difference between the elasticand viscoelastic analyses. So the low-rate test, whichrequires the use of the wave-separation technique, is notlikely to require the full viscoelastic technique presentedin this paper, i.e., the bars may be treated as linear elas-tic. However, in the fourth example (Fig. 18), a slighterror in the elastic analysis is observed. In general, thereis no way to tell how significant this effect is in advance,so the viscoelastic correction should be applied if for noother reason than to show that it is not needed. It wasobserved, however, at least for the polycarbonate press-ure bar system used by these authors, that viscoelasticeffects could always be neglected during the initial sep-aration step performed at the location of the gages.Noticeable errors due to viscoelastic dispersion onlymanifested during propagation.

As mentioned above, this technique can permit morefreedom in the choice of gage location so that gages canbe moved closer to the specimen. Doing so reduces theamount of dispersion and attenuation that occurs in apulse as it travels between the specimen and themeasurement point. This can mitigate the need for a vis-coelastic analysis and is one advantage the present tech-nique has over two-point techniques. In fact, in anextreme case, the velocity and strain gages could beapplied exactly at the interfaces between the bars and thespecimen so that the velocity and stress1 at each end are

1 In many cases, stress may be obtained from the strain onthe bar using one-dimensional Hooke’s law; see [11,14]. Or onecould use a stress gage, similar to those used by Chen [15].This is a feasible option when testing low-impedance materials,since survivability of transducers is not as much of an issue asit is when testing metals/ceramics in a metallic SHPB.


measured directly, and wave propagation would not needto be considered at all. In any event, because this tech-nique determines the waveforms directly at a single pointwithout any regard for the propagation of these wavesprior to their arrival at the measurement location, thereis never any need to account for propagation over a dis-tance greater than the distance between the gages andthe specimen. This is a distinct advantage over tech-niques that essentially track the progression of a pulseover many transits along the bar because the pulses aresubjected to greater dispersion and are therefore moresusceptible to errors involved in the associated disper-sion corrections.

In terms of limitations of the method, the only disad-vantage is that it requires a measurement of particle velo-city. Electromagnetic velocity gages were used in thesetests and were found to be non-perturbing and relativelyaccurate with good sensitivity and an excellent frequencyresponse. However, they are somewhat cumbersome touse and good results may be obtained with other velocitytransducers (e.g., based on inductance or interferometry).

As a final note, it should be mentioned that this visco-elastic treatment could be applied to the linear elasticcase in which geometric dispersion is non-negligible.The above procedure applies, with the exception that theattenuation coefficient, α(ω), is zero.

References

[1] C. Bacon, An Experimental Method for Considering Dis-persion and Attenuation in a Viscoelastic Hopkinson Bar,Experimental Mechanics 38 (1998) 242.

[2] S.W. Park, M. Zhou, Separation of Elastic Waves in SplitHopkinson Bars Using One-point Strain Measurements,Experimental Mechanics 39 (1999) 287.

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Wave separation in viscoelastic pressure bars using single-point measurements of strain and velocity