Impact response of high density flexible polyurethane foam(5)

s ¼ð6:053u þ0:539u 2ÞÂ10À3MPa,where the particle velocity units are m/s.The foam Hugoniot in stress s -relative volume V /V 0

coordinates is shown in Fig.7along with the static compression curve of Fig.1b.The departure of the Hugoniot data from the isothermal compression curve,Fig.7,seems to be caused by large irreversible heating of the shocked foam.An accurate estimate of the temperature caused by this heating is,however,hardly possible.Within assump-tion of constant speci fic heat C V and constant ratio of Gruneisen parameter to speci fic volume G /V ,the temperature T H behind the shock front is related to the temperature T 0ahead of the shock as [15]

T H

¼T 0exp G ε 1þ1þs 6T 0a

ε2

þ.

(3)

Here a is the coef ficient of the volumetric thermal expansion,s is the slope of the material Hugoniot,and ε¼ðV 0ÀV Þ=V 0.The thermal expansion coef ficient a of polyurethane is of about 2Â10À4K À1[16],For the polyurethane foam of close,300kg/m 3,density Maw et al.[17]suggest the value G z ing such values of G and a yields even for modest,V =V 0z 0:6,compression T H z 1000K.Another approach based on the direct thermodynamic

de finition G ¼a C 2b =C p

with characteristic for polyurethane speci fic heat C p z 500J =kg [18]seems to underestimate the T H ;for the strongest,V =V 0z 0:27,foam compression in the 605-m/s test it yields T H z 500K with G ¼0.064.4.2.Foam ’s compressive strength

As mentioned above the material strength mechanisms are responsible of the shock front rise time t rt .The latter may be quanti fied through the assessment of the average strain rate

during

a Fig.4.Stress-particle velocity (a)and time-distance (b)diagrams of the tests PFE and PFE1corresponding to the waveforms shown in Fig.3c.The impact velocity is assumed to be equal to 312m/s in both tests.Waves ’timing (b)corresponds to the arrival of the leading wave edge at the sample rear

surface.

Fig.5.The waveform recorded after impact of 1-mm foam impactor on the 8.9-mm foam sample (shot PFAS of Table 1).The insert shows vertical magni fication of the spall-related signal with the velocity pull-back D w pb z 3m =s.

100200300

400

L a g r a n g i a n v e l o c i t i e s U S ,U L E a n d U U L , m /s

particle velocity u , m/s

V e l o c i t y d i f f e r e n c e U L E - U S , m /s

U S

U LE

U LE

U LE t 2

t 18.9

time

h,mm

a

b

Fig.6.Time t -Lgrangian distance h diagram of the propagation of the spread shock front through the foam sample of 8.9-mm thickness (a)and velocities of propagation of the leading edge U LE (filled circles),of the wave half-height U S (open circles),and of the release wave U UL (open triangles)as a function of the particle velocity at the top of P1wave (b).The dashed line is the linear approximation U S ¼U S 0þsu .The crosses show the difference U LE ÀU S (right ordinate).

Table 2

Foam states behind compressive and unloading waves.Test

u1,m/s U LE ,m/s US,m/s s 1,

MPa V 1/V 0w1,m/s U UL ,m/s V UL /V 0

00 1.0PFA 21.75104.740.70.360.46543.540.7 1.0PFB 36.25129.061.20.910.4085370.70.534PFC 70.50157.8109.5 3.160.35699124.90.461PFD 117.5192.6174.08.360.325152226.90.383PFE 155.5236.1222.714.20.302198288.30.354PFE1157.0236.1219.414.20.295199275.40.348PFF 192.0270.5266.420.90.279240345.80.325PFG 249.5345.0342.334.90.271315419.00.321PFH 302.5

415.9

413.0

51.0

0.266

384

488.0

0.319

E.Zaretsky et al./International Journal of Impact Engineering 39(2012)1e 75