Impact response of high density flexible polyurethane foam(3)

The reshock R of the amplitude s 2travelling towards the sample rear surface meets the unloading wave U1,generated at the rear surface of the 5.5-mm sample or U2generated at the rear surface of the 8.9-mm sample.As a result,the partially unloaded shocks R1and R2(for 5.5-mm and 8.9-mm samples,respectively)with amplitude s 3arrive at the sample rear surface,accelerating it from w 1¼196m/s (the free surface velocity corresponding to the P1wave amplitude)to w 2¼354m/s (the free surface velocity corre-sponding to the P2wave amplitude).The later arrival of the P2wave (Fig.4b)at the rear surface of the 5.5-mm sample,47.1m s with respect to 44.1m s in the case of the 8.9-mm sample,is the result of certain relations between the speeds of unloading (U 1,U 2)and reloading (R ,R 1,R 2)waves.It is apparent from Fig.4,that for accu-rate description of the second,P2,wave knowledge of the s 1Às 2,s 2Às 3,and s 3À0paths is required.Since reliable data are avail-able only for the description of the s 1Às 2path,the analysis in the next section will be limited to the shock states s 1behind the P1wave and to those produced by unloading from the s 1states.

The waveform recorded after a single spall-oriented impact is shown in Fig.5.Since the foam impactor used in this experiment was very thin,a part of the waveform amplitude is eliminated by hydrodynamic decay.(The 355-m/s velocity of the primary PMMA impactor corresponds to some 600-m/s impact of the secondary foam impactor on the foam sample.)Although the spall signature (the velocity pull-back D w )at the waveform of Fig.4is very small,of about 3m/s,the post-spall oscillations of the free surface suggest that the foam sample was fractured by the tensile pulse caused by collision of the rarefaction waves generated at the free surfaces of the sample and the secondary impactor.4.Impact response of the foam 4.1.Hugoniot of the foam

In order to obtain the Hugoniot states at the top of the P1wave from the free surface velocity histories shown in Fig.3,the velocity

U S of the propagation of the wave front should be determined.Since the wave fronts are not step-like,we assume U S for the velocity of propagation of the wave half-height.The wave leading edge trav-elling with velocity U LE higher than U S gives rise to the multiple wave re flections,as it is shown in Fig.6a.These re flections distort the timing of the arrival of the wave half-height at the sample surface.Accounting in that the U LE is the velocity of propagation of low-stress perturbations,we assume that all the multiple re flec-tions shown in Fig.6a propagate with the same velocity equal to U LE .In such case,the Lagrangian velocity U S is determined by the times t 1and t 2of the arrival of,respectively,the wave leading edge and the re flection from the wave half-height at the free surface of the sample of thickness d .

U S ¼2

d Àðt 2Àt 1ÞU LE =2

t 2þt 1

(1)

The principal Hugoniot of the foam,namely the stress s 1and the speci fic volume V 1at the top of the P1wave,may be obtained by applying the mass and momentum conservation laws to the wave front propagating with the velocity U S :

V 1V 0

¼ðU S Àu 1Þ

U S ;

s 1¼

U S u 1

V 0

(2)

where V 0¼1=r 0¼2:44Â10À3m 3=kg is the initial speci fic volume of the ing conservation of mass and momentum across the unloading wave (path u 1w 1in Fig.4a),yields the esti-mates of the average velocity of the unloading wave U UL and of the speci fic volume V UL of the foam unloaded after loading up to the stress s 1.The corresponding estimates are given in Table 2.

The velocities U LE ,U S ,and U UL and the difference U LE ÀU S are shown in Fig.6b as a functions of the particle velocity u 1¼v 0/2at the top of P1wave.

The linear expression U S ¼14.8þ1.318u (dashed line in Fig.6)fits the data on the velocity of the propagation of the wave half-

Table 1

Parameters of the planar impact experiments performed with dense flexible polyurethane foam.Test impactor a sample a imp.velocity,m/s Test impactor a sample a imp.velocity,m/s PFA foam foam 43.5PFE1b

foam foam 314PFB foam foam 72.5PFF foam foam 384PFC foam foam 141PFG foam foam 499PFD foam foam 235PFH foam foam 605PFE

foam

foam

311

PFAS c

foam

foam

355

a Thickness of both foam impactors and samples was 8.9Æ0.1mm.The impactor diameter is 55mm,the samples were 60mm Â60mm squares.

b The foam sample was a 60mm Â60mm square of 5.5(Æ0.1)-mm thick.

c

The thickness of the primary PMMA impactor and of the secondary foam impactor were 11.80Æ0.01mm and 1Æ0.1mm,respectively.

impactor

b

the sample holder

a

Fig.2.Schematics of the symmetric planar impact tests with primary foam impactor (a),and of the spall-oriented experiment with the secondary foam impactor (b).In both cases the velocity of the free rear surface of the foam sample was monitored by VISAR.

E.Zaretsky et al./International Journal of Impact Engineering 39(2012)1e 7

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