impact behavior of polymeric foams: a review

21
This article was downloaded by: ["Queen's University Libraries, Kingston"] On: 30 September 2013, At: 13:31 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Polymer-Plastics Technology and Engineering Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/lpte20 Impact Behavior of Polymeric Foams: A Review David M. Schwaber a Monarch Rubber Co., Inc. Published online: 04 Mar 2008. To cite this article: David M. Schwaber (1973) Impact Behavior of Polymeric Foams: A Review, Polymer-Plastics Technology and Engineering, 2:2, 231-249, DOI: 10.1080/03602557308545019 To link to this article: http://dx.doi.org/10.1080/03602557308545019 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.

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This article was downloaded by: ["Queen's University Libraries,Kingston"]On: 30 September 2013, At: 13:31Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number:1072954 Registered office: Mortimer House, 37-41 Mortimer Street,London W1T 3JH, UK

Polymer-Plastics Technologyand EngineeringPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/lpte20

Impact Behavior of PolymericFoams: A ReviewDavid M. Schwabera Monarch Rubber Co., Inc.Published online: 04 Mar 2008.

To cite this article: David M. Schwaber (1973) Impact Behavior of PolymericFoams: A Review, Polymer-Plastics Technology and Engineering, 2:2, 231-249, DOI:10.1080/03602557308545019

To link to this article: http://dx.doi.org/10.1080/03602557308545019

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of allthe information (the “Content”) contained in the publications on ourplatform. However, Taylor & Francis, our agents, and our licensorsmake no representations or warranties whatsoever as to the accuracy,completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views ofthe authors, and are not the views of or endorsed by Taylor & Francis.The accuracy of the Content should not be relied upon and should beindependently verified with primary sources of information. Taylor andFrancis shall not be liable for any losses, actions, claims, proceedings,demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, inrelation to or arising out of the use of the Content.

This article may be used for research, teaching, and private studypurposes. Any substantial or systematic reproduction, redistribution,reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of accessand use can be found at http://www.tandfonline.com/page/terms-and-conditions

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POLYMER-PLAST. TECHNOL. ENG., 2(2), 231-249 (1973)

IMPACT BEHAVIOR OF POLYMERIC FOAMS' A REVIEW

DAVID M. SCHWABER

Monarch Rubber Co., Inc.

I. COMPRESSIVE STRESS-STRAIN BEHAVIOR OF FOAMS . . . 232

II. ENERGY LOSS DURING FOAM DEFORMATION 236

III. ANALYSIS OF IMPACT BEHAVIOR 238

IV. PREDICTION OF IMPACT BEHAVIOR 242

References 248

Polymeric foams designed for impact applications must exhibit specificstress-strain behavior as well as an ability to dissipate energy.

It is important for an impacting object to undergo a constant decelerationfor maximum efficiency of energy absorption. If the decleration is not con-stant, a maximum force will be exerted on the object at some time duringimpact. This maximum force could be destructive to the object. A foamwhich exhibits a plateau region in its stress-strain curve, resembling yielding,will absorb impact energy at a relatively constant deceleration.

The foam should also be able to dissipate most of the energy it absorbedduring impact. If the energy is not dissipated, it will be returned to the im-pacting object. The dangerous implications of this effect, especially in auto-motive safety applications [1], are obvious.

In order to understand the impact behavior of foams, it is essential toconsider their compressive stress-strain behavior as well as their ability todissipate energy.

The mechanics of impact for elastic materials, especially for Hookeansprings and elastic spheres, have been treated extensively [2].

231

Copyright © 1974 by Marcel Dekker, Inc. All Rights Reserved. Neither this work norany part may be reproduced or transmitted in any form or by any means, electronicor mechanical, including photocopying, microfilming, and recording, or by any infor-mation storage and retrieval system, without permission in writing from the publisher.

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232 SCHWABER

Equations for the prediction of the time of impact, the maximum pene-tration, and the velocity profile for systems exhibiting ideal Hookean be-havior have been derived. However, these analyses, assuming linear behaviorand no hysteresis, cannot be used to predict the impact behavior of nonlinearviscoelastic foams.

I. COMPRESSIVE STRESS-STRAINBEHAVIOR OF FOAMS

A cellular material can be represented as an assembly of many smallcolumns connected in a three-dimensional network. In many cases closed cellsare also present. During the deformation of this type of material, many com-plex mechanical phenomena occur.

The stress-strain curve of a compressed open-cell natural rubber is almostlinear at low strains, but at higher strains the modulus tends to decrease [3],as shown in Fig. 1.

o

D

0-8.0

()a buckling columno compressed foam

FIG. 1. Stress-strain behavior for a compressed open-cell foam and a buck-ling column [29].

This phenomenon is similar to the stress-strain behavior of a bucklingcolumn with fixed ends. A large stress is necessary to initiate buckling of the

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IMPACT BEHAVIOR OF POLYMERIC FOAMS

column, but the stress necessary for further deformation is reduced becausethe initially small bending moment increases.

This buckling effect is also apparent in foams (Fig. 1). The load increaseswith a constant modulus at low strains until a critical value is reached atwhich buckling is initiated. The modulus then decreases due to an increase inthe bending moments of the columns [3] within the foam. Buckling of thethin interstices subjects the thicker portions of the structure to torsion, shear,and compression [3]. At larger deformations the modulus increases becausethe struts start to touch and interfere with each other's movement. Themodulus of the foam then approaches the compressive modulus of the solidmaterial.

The most important attempts to predict polymeric foam properties fromthe constitutive equation of the matrix material and the foam structure in-clude the cubic lattice model (compression analysis) and the model of arandom arrangement of foam strands (extension analysis) developed by Gentand Thomas [4,5] and the model by Ko [6].

The cubic lattice model is composed of interconnected threads of length,l0, and thickness D. This model was used for the analysis of the compressionbehavior of foams [4,5]. The compression is directed parallel to a set ofthreads and causes them to buckle. An equation was developed in which thecompressive force is related to the bending moments of the buckling struts inthe model, and takes the form

t = YAK2f(X)l2,(l0 + D)2

Here t is the stress across the foam, Y is the modulus of the solid rubberphase, AK2 is the moment of inertia of the thread cross section, f(X) is an un-known function of the strain ratio X = l/l0, lo the unstrained length of athread, and D is the diameter of the rigid volumes where the threads meet.

Gent and Thomas [4,5] were able in this way to predict load deformationcurves for natural rubber foams using a single empirical function, f(X), todescribe the effect of the foam structure on the shape of the stress-straincurve [4]. However, there has been no success in the prediction of the entirestress-strain behavior of open-cell foams.

For the study of foams in extension, Gent and Thomas [4,5] developed amodel composed of randomly oriented threads. An equation was developed torelate the modulus of the foam, Yp, to the modulus of the matrix material, Y:

YF = YB2/2(1 + B)

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234 SCHWABER

For foams of low density, Young's modulus and the compressive stress of thefoam were found to be proportional to the first and second powers of thedensity respectively [4,5], as shown in Fig. 2.

YflY

FIG. 2. Experimental relation between Young's modulus Yf of the rubberfoam relative to that of the solid rubber Y, and the volume fraction of therubber, Vr [5].

Ko's analysis [6] included tension and shear strain as well as bendingmoments. Ko [6] also considered the effect of the foam's packing geometrywhich affects the thread dimensions; which in turn would affect the Poisson'sratio of the foam.

Gent and Thomas [4,5] and Ko [6] have thus been able to predict boththe Young's modulus (slope of stress-strain curve at zero strain) and Poisson'sratio for small extensions and compressions of low-density foams. However,there has been no success in the prediction of the entire stress-strain behaviorof open-cell foams without the use of an empirical function [4].

In the mechanical analysis of closed cell foam systems, it is necessary toaccount for the internal pressure of the entrapped gases. Based on this, Gentand Thomas [5] developed expressions for the Young's modulus andPoisson's ratio of closed cell foams. Their treatment was based on the iso-thermal compression of the entrapped gases, and on the strain energy for anetwork of random threads. They have shown that if the stiffness of the foamis governed solely by the foam matrix, the Poisson's ratio is 0.25, as shown inthe model applied for open cell foams. On the other hand, if the stiffness isdue predominantly to the compressed gases in the closed cells, Poisson's ratiois 0.50, characteristic of an incompressible solid.

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IMPACT BEHAVIOR OF POLYMERIC FOAMS 235

Skochdopole and Rubens [7] applied the ideal gas law to predict stress-strain behavior for closed-cell foams in compression. The shape of the calcu-lated curve was very similar to the actual stress-strain relation as shown inFig. 3.

-30 -60 -90% compression

FIG. 3. Compressive load vs compression for polyethylene foam [7].

They noticed a dependence of stress on cell size as well as foam density. Atlow compressions, the stress is dominated by the compressive modulus of thecell walls. However, this modulus levels off as the cell walls reach a criticalbuckling stress or strain, and then it rises very slowly as the compression is in-creased. The plateau region does not exist in the stress-strain curves of poly-ethylene foams with very small cells as described by Fig. 4. Since the bendingstress is dependent on the third power of the thread's radius, the bucklingeffect would be more pronounced in a thicker thread (larger cell). Further-more, during the deformation of a small-celled structure, buckling of cellwalls is a small contribution to the compressive stress compared to that of thegas in the closed cells [7].

As the fraction of open cells in a polyethylene foam is increased, both thecompressive modulus and the strength decrease [7]. The stress-strain curvewas similar in shape to the ideal gas-pressure-temperature-volume relation ifthe volume is defined as that of the closed-cell fraction of the sample only [7].

Other effects have to be considered in a complete analysis of the mechanicsof foam deformation. Suminokura [8] studied the effect of chemically re-moving membranes from urethane foam on the foam's modulus. The mem-branes were found to have a major reinforcing effect.

Benning [9-11] studied the effects of orientation in cellular polyethyleneand found the response of these foams to deformation to be anisotropic [10].

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236 SCHWABER

celt walls-

IDDo

% compression

FIG. 4. Mechanical model for polyethylene foam [4].

Benning also noticed an increase in foam compressive strength for materialsfoamed at faster rates [11]. This increase was probably due to orientation inthe cell walls which would produce an increase in modulus.

II. ENERGY LOSS DURING FOAM DEFORMATION

There are four means of energy dissipation (hysteresis) during the defor-mation of cellular polymeric systems:

1. Energy loss in the strained polymer matrix (types of strain which occurin the solid phase of the foam include buckling, compression, tension, andshear).

2. Friction between units of the cellular matrix.3. Loss due to viscous flow of fluids through the open pores of the foam.4. Irreversible compression of gases in closed cells.Energy loss can be measured as the area between the loading and unloading

stress-strain curves of a loading cycle. The work of Evans [12] has shown thatthe energy absorbed in a falling weight impact.test of a solid polymer can bepredicted approximately from the area beneath the loading curve of thatpolymer. He did not consider, however, that the nature of the loading curvewas strongly dependent on the impact velocity [12]. Maxwell and Harrington

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IMPACT BEHAVIOR OF POLYMERIC FOAMS 237

[13] found that although the modulus of the sample increases at higher rates,the ultimate elongation decreases with rate.

Another means of energy dissipation is the physical process of fluids flow-ing through the pores of an open cell foam, or pneumatic damping. Thismechanical damping has been represented by a model developed by Kostenand Zwikker [14]. Their analysis was applied to open-cell materials by Gentand Rusch [15,16] and Hilyard and Kanakkanatt [17,18].

The pressure gradient, dP/dx, necessary to maintain flow through theporous structure of the foam at a constant velocity, v, is given by the sum oftwo terms [16]:

dP 7? v p \2

dx K B

where K and B are coefficients which characterize the structure of the porousmaterial, T? is the fluid viscosity, and p is the fluid density. By means of thisequation, the dynamic mechanical properties of the foam-fluid system wererelated to the geometry and porosity of the sample tested as well as the me-chanical properties of the fluid and polymer matrix composing the system[16,18].

They found this mechanical damping to be small at low frequencies, dueto low flow rates. Flow losses increase with frequency but at higher fre-quencies, the flow resistance forces become large relative to the compressibil-ities of the fluid, resulting in less flow through the structure and therebysmaller losses due to flow as shown in Fig. 5.

Another aspect of pneumatic damping was recently analyzed by Bosscher[19,20]. His model depicted flow through cracked cell walls. He found thatthe damping resulting from fluid flow through broken cell walls relates to thehalf-power of the velocity. This agreed quite well with his model consisting ofcantilever flapper valves with a uniform load across the face. At very highfrequencies, the fluid has no chance to leave the foam in the time of the testand the material behaves as a closed-cell system.

Energy dissipation in a compressed closed-cell system is due both to theviscoelastic behavior and stress softening of the matrix material as well as toirreversible heat transfer between the compressed gases and the solid phase[21,22]. The contribution to impact behavior of irreversible compression ofgases has been investigated by Otis [21,22]. When the foam is compressed,the temperature of the gas in the closed-cells increases. The heat of thetrapped gases is transferred to the cooler solid phase of the foam. In thismanner, the solid phase serves as an energy sink. The heat transfer coefficientis dependent on distance and time of compression.

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238 SCHWABER

a: L/h"3.9 O'-L./h* 22

1.0 1.000.01 0.10

FIG. 5. Tan 6 as a function of f for a foam sample in silicone oil (L = 2.8cm)[15].

The effect of irreversible compression of gas in the closed cells of materialsused for impact absorption is probably negligible compared to hysteresis inthe strained solid phase.

III. ANALYSIS OF IMPACT BEHAVIOR

The impact behavior of foams has been studied by Soper and Dove in orderto define energy absorbing efficiency by use of dimensionless parameters [23].They derived the equation of motion for an impacting mass cushioned by aviscoelastic spring, i.e., a two-parameter Maxwell model.

This model consists of a Hookean spring with modulus E, and Newtoniandashpot with viscosity 77. An energy balance of the strained model yields

Ma + / a dA = 0

with M the mass of the impacting object, a, its acceleration, a, the stress, andA, the area of the sample. The stress-strain relation for a Maxwell model isgiven as

a a— +— = eE T?

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IMPACT BEHAVIOR OF POLYMERIC FOAMS 239

Although a more complex viscoelastic model, for example, a four-parameterliquid, could have been applied, the simpler two-element Maxwell model hasproved to be adequate for describing the cushioning behavior of some cellularmaterials [24].

Soper and Dove [23,24] developed dimensionless groups from the analysisof two equations,

x/h = fj (v/h, Mv2/Ah)

and

K = v2/2ah = f2 (v/h, Mv2/Ah)

where x is the penetration distance, h the sample thickness, v the velocity ofimpact, and Mv2/Ah, a kinetic energy concentration factor which is referredto as I.

In an ideal impact, all the kinetic energy of the impacting object is ab-sorbed at constant deceleration through the total compression range of thefoam. The K factor is a dimensionless quantity describing the efficiency ofthe impact test.

Since the accelerations is a function of the distance x, the material be-havior during impact can be described by the equation

v2/2ah = (Mv2/Ah)

The maximum efficiency of energy dissipation exists at an intermediatevelocity between one too low for penetration and one high enough to causefoam failure [24].

It is desirable in practical applications that the K vs v curve (Fig. 6) be asbroad as possible, that K be close to unity, and that Kmax occur at the mostprobable impact velocity. The shapes of the K versus I curves depend on thefoam structure, while their positions depend on the foam modulus. On theother hand, for an identical matrix geometry, brittle foams were shown to ex-hibit a higher efficiency and a flatter, wider plateau (Fig. 6) in the decelerationvs impact curve than flexible foams [25].

The width of the optimum energy absorption region can be no greater thantwo decades along the I-axis for most foam structures [25]. The practical wayto optimize the absorption region is to increase the modulus with the speedof impact. A strongly rate-dependent material would be desirable in this case.

\

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240 SCHWABER

On the other hand, it is not practical to have a system with a modulus ofappreciable temperature dependence.

icr* icr3 icr2 icr> to(impact-enerpu/volurne) j

modulus

FIG. 6. Energy absorbing efficiency, K versus Impact Energy/(Volume-Modulus) [28].

The use of dimensionless parameters simplified the description of a foamsystem, provides a convenient way to compare systems with widely differentproperties, and also permits a qualitative determination of energy-absorbingcharacteristics of a system.

Rusch [26-28] also analyzed foam behavior during impact with the helpof dimensionless parameters. He used an empirical function rather than visco-elastic model to approximate the sample's response to strain.

This empirical function is the result of factorization of the foam's responseinto a modulus and a strain function. The analytical expression used to de-scribe the stress-strain behavior of foams is

o = E0e ^(e)

where e is the strain, Eo is Young's modulus, a the stress, and i//(e) an empiri-cally determined demensionless function of the strain.

The function ^(e) is dependent only on the density and structural featuresof the foam as shown in Fig. 7, and exhibits no time-temperature dependence.

Rusch [26-28] found that iKe)min is proportional to the 0.28 power ofthe solid fraction, and to the -0.2 power of the cell diameter.

It was expressed by Rusch [25] as

= me"n + res

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IMPACT BEHAVIOR OF POLYMERIC FOAMS 241

Iff®

0.1 —

0.01 0.03 O.I€

1.0

FIG. 7. Dependence of factorized strain-dependent function ('/'(e)) on den-sity 04) of cellular material [27].

The curve-fitting constants (m,n,r,s) are evaluated from a plot of log \!>(e)versus log e (Fig. 8), where —n and s are the slopes and m and r, the interceptsat e = 1.0 of the two straight lines used to approximate the curve. At lowstrains, the term res approaches zero and at high strains the term me"n vanishes.This analytical expression for i//(e), although not exact, is sufficiently accuratefor many purposes [27,28].

experimental methodto evaluate ijl(e)

I 10 100strain (o/o)

FIG. 8. Experimental method to evaluate i^(e) [25].

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IV. PREDICTION OF IMPACT BEHAVIOR

Impact behavior has been predicted [29-32] from the response of foams toother loading histories. This method was chosen because it is mathematicallymore simple than a micro-mechanical analysis. Mechanical factors contributingto the response of compressed foams include distortion of cell walls, resistanceof air flowing through open pores, and the compression of gas in closed cells.Furthermore, most foams have a nonuniform structure consisting of a mixtureof open and closed cells and other irregularities.

The response of complex and simple loading histories of a particular foamare dependent on the same viscoelastic and structural parameters and may becompared directly. Therefore, if one response is used to predict another, theparameters need not be considered independently.

Equations have been developed to correlate impact behavior with stress-strain curves obtained at different rates of strain. This correlation has beenobtained by equating the energy loss of an impacting pendulum, '/imv2, andthe energy of deformation of a compressed foam, /ode. For foams exhibitingrelatively rate-independent behavior [29,30], the stress, a, is a function ofthe strain, e, only.

Energy loss = ̂ mvo2 - &mv£2 = /go(e)de

This energy balance has been used successfully to predict impact behavior,including a velocity profile and maximum penetration, for rate-independentfoams as shown in Figs. 9-12. The velocity profile has been determined forthe unloading process by integrating the unloading stress, h(e), from maximumstrain is

The total energy lost during impact is predicted from the hysteresis, Eh, ofthe compression or the area between the loading and unloading curves as shownin Fig. 11:

Eh = ^m(v02 - vout

2) = /S{a(e) - h(e)}de

The response of foams exhibiting rate-dependent behavior [29-32], asshown in Fig. 13, was approximated by the product of a rate-dependent mod-ulus function, E0(v) and a strain-dependent function, -jr (e). These functions

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IMPACT BEHA VIOR OF POLYMERIC FOAMS 243

v (cm I mm)

100-

0

v 0.5o 20.0

60ppi uref/Jane foam I

0 -40 -80

FIG. 9. Stress-strain behavior of urethane foam (60 ppi) at constant load-ing and unloading rates [29,30].

Q GO ppi 5.5

0 20 40 60TIME (msec)

FIG. 10. Distance vs time for impact test of open-cell urethane foam (60ppi) [29,30].

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-—s

E

"—

Uj

UJ

120-

90-

60-

30-

r\~

v Instron

O Mp=2.06 Kg

&Mp=26\ Kg60 ppi

y1energy to deform £%sample Jdj

h us feres is Ttff

0 -50 -100STRAIN (%)

FIG. 11. Energy to deform and energy loss in compression of 60 ppi ure-thane foam [29,30].

6 0

o

oo

._ .̂_..- 30 ppi, &5°~\ impact

~2 I 60 ppi, 5.5° J Instrov

30-

0 20 40 60 80t(m sec)

FIG. 12. Velocity vs time for impact tests of urethane foams [29,30].

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IMPACTBEHA VIOR OF POLYMERIC FOAMS 245

10001

E

E

600

Uj

0)

SBR dipped 10ppi °uretfiarie foam

Q OS cmfminA 2.0 " "• SO " "v / 0.0" "o

O -35 , . -70STRAIN ('/.)

FIG. 13. Stress-strain behavior of SBR-coated 10 ppi urethane foam atconstant loading and unloading rates [29,31].

were obtained by factorization of the stress-strain relationships at variousrates (Figs. 14 and 15). Although the solution to the impact analysis of rate-dependent foams was more complex than that for the previous case, theequations which were developed accurately predicted impact behavior forfoams from constant-rate data.

The strain-dependent function was expressed as a polynomial series,102 B n e n and was fitted by the least-squares method. The rate-dependentn — o

modulus was expressed as a simple exponential function, Eo = ]The solution to the energy balance

]/imv02 - = /ode

yields an expression for the velocity, v, profile of the impacting object ofmass, m.

Jv = {v02-a-£(2-a)/S(nSoBnen)(

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2O

to

oE

I.G

1.2

1/)

SBR dipped 10 ppiuretfiane foam

O 0.5 cm /mimA 2.0 cm/mina 5.0 cm/mi v

0.4

0 10 30 SOstrain (%)

FIG. 14. Rate-independent strain function (•%- (e)) for SBR-coated foam[29,31].

SBR dipped lOppiurethane foam

i= 10*

Q

io-J to'2 IO1 lO1 IO2

RATE (cm/min)

FIG. 15. Modulus vs rate for SBR-coated foam [29,31].

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5BR dipped 10ppiurethane foam

o impactA calculated

E-4.150 compression• recovery

drop angle-3Cfpendulum mass-.293kg

O 10 20 30STRAIN (%)

FIG. 16. Velocity profile for impact test of SBR-coated foam (pendulummass = 2.39 kg, initial velocity = 228 cm/sec) [29,31 ] .

160

o impactA Instron

unloadingA Instron

MONARCH A-6052

FIG. 17. Velocity profile for impact test of Monarch A-6052 [32].

247

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248 SCHWABER

The application of this equation predicted impact behavior for polymericfoams exhibiting rate dependent moduli (a ¥= 0) [29,31], as shown in Fig. 16,as well as rate-independent moduli (a = 0) [29,30]. This equation has alsobeen used to'accurately predict the impact behavior of commercially avail-able energy absorbing materials [32] as shown in Fig. 17.

ACKNOWLEDGMENT

The author would like to thank Gary Varano, Monarch Rubber Company,for graphic assistance and Dr. Eberhard Meinecke, Department of PolymerScience, University of Akron, for advice in the development of this paper.

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IMPACT BEHA VIOR OF POLYMERIC FOAMS 249

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