The Compressive Deformation of Polymeric Foams JAMES T. TSAI

Ra ychem Corporation, 300 Constitution Drive, Menlo Park, California 94025

An alternative approach has been developed to evaluate compressive stress on polymeric foams. Compressive stress depends primarily on three factors: foam density, deformation strains, and deformation rates. The density dependency de- rived from this approach agrees closely with the empirical models reported in the literature. Correlations between the dy- namic compression and shear deformation are also derived. Experimental data are presented which show that G‘ andG”in- crease with increasing strain rates, while the damping factor reaches a maximum at low strain rates. Note that the proposed model for the prediction ofcell-wall rupture would not apply to foams with high open-cell contents.

INTRODUCTION

e major characteristics of polymeric foams are their Th very low density, relatively low cost, and attractive compression-to-weight ratio. An important commercial use of these properties is in the design of energy-ab- sorbing structures. In many instances, foam materials are selected by trial and error rather than by analytical techniques, because information on the deformation characteristics of the foam structural variables is not available.

The mechanism of deformation in cellular plastics has been studied for compression and impact absorption (1-8). There are two components of deformation when polymeric foams are under compression:

deformation of cell structure, a n d build up gas pressure within the cells.

Rusch (1) has proposed that the compressive stress u can be expressed by the following relation:

c = E E ~ + (E) (1)

where E is the compressive strain, Efis the Young’s mod- ulus of the foam, and + ( E ) is a dimensionless function that reflects the buckling of the foam matrix. However, + (E) cannot be predicted by physical parameters for low-density foams, and the hoop stress due to gas pres- sure build-up has not been considered.

Nagy, et al (3), have shown that the dynamic stress- strain behavior of polymeric foams is strongly sensitive to the rate ofdeformation. They suggested that the rate- dependent response may be due to a combination of vis- coelastic behavior of polymeric cellular structure and to the resistance of air flow through the cell structure.

The aim of this paper is to investigate the compressive and shear deformations of polymeric foams using a gen- eral class of constitutive equations that account for volu- metric changes during the deformation process. Corre- lations between compressive and shear deformations

are given because instrumentation to measure the dy- namic shear moduli is readily available.

THE PHENOMENON OF CELL WALL RUPTURE

Figure 1 shows the schematic diagram of a cellular structure under compression. Consider the cell as a symmetric membrane specified by the meridian curve R(z) and a thickness h. The two principal radii of surface curvature are R, and R, in the meridian and circumfer- ential directions. Neglecting gravity and acceleration forces, and for membranes of small thickness gradients, the force balances (9) give:

AP = - h ( F (+I1 + -) (+33

z Rc

where AP is the pressure difference exerting a net nor- mal force per unit area. For the polymeric foams, which typically have low Poisson’s ratios in compression (6),

F

Fig. 1 . Schemutic diagram of u cellular structure.

POLYMER ENGINEERING AND SCIENCE, JUNE, 1982, Vol. 22, No. 9 545

Janies T. Tsai

the overall force balances of the foam under uniaxial compression can be described by the following approxi- mation:

F , A, hP -h A, (2 + 3) (3) R,.

where F , is the external uniaxial compression force, Af is the foam surface area under compression, and A, is the cross-sectional area of the polymeric material. To solve the problem at hand, a constitutive equation is needed to relate stresses to deformations. Nagy, KO, and Lindholm (3) have shown that the compressive de- formation of foams can best be fit into a power law ex- pression using a single modulus.

The strain rate can be expressed by:

where 9 is the overall compression speed, cmls, H is the overall sample thickness, and D is the compressive ratio:

Using the power law model, the stress of the cell wall can be expressed by:

0- = m(T) q 1nD (5)

where m(N * sn/m2), and dimensionless n are parame- ters called the consistency and power law index, respec- tively.

For a first approximation, one can estimate the stress exerted on the cell wall for given foam structural param- eters. Assuming spherical cells, the effective loading cross-section ratio can be represented by:

APIAf = rrRh/rrR2 = h /R (6)

Further derivation in the Appendix shows h/ R equal to one-third of the ratio of foam density d f and polymer density d,. Combining Eqs 3 , 5 , and 6 , the compressive force corresponding to the rupture stress of the cell be- comes:

F = -m(d,/d,)* (8,)' Ad9 (7)

This result suggests that the restoring force increases with increasing foam density, compression strains, as well as strain rates. The density dependence agrees closely with the empirical model'surnmarized by Progel- hof and Throne (7) for a static mode.

THE BULK LONGITUDINAL MODULUS

The deformations of low-density foams are character- ized hy a change in volume or a combined change in volume and shape. To describe the mechanical behavior under such conditions, it is necessary to use a more com- plicated constitutive equation, including all compo- nents of the stress tensor with two time-dependent functions (10):

546

where 6 is the Kronecker delta, G(t - t ' ) is the stress relaxation modulus, and K ( t - t ' ) has the physical sig- nificance of a volumetric relaxation modulus. When perceptible changes in volume occur, both G(t - t ' ) and K ( t - t ' ) are found to depend heavily on the density or degree of compression.

Bulk longitudinal compression occurs when the di- mension changes in one direction while constrained in the two mutually perpendicular directions. Under these

conditions ys2 = y33 = 0. If a sudden strain E = - y, ,

occurs and the stress relaxation is followed as a function of time, E q 8 becomes:

1 2

0-11 = + 4G(E)/31 + (4 (9) The quantity in brackets is denoted as M ( t ) , and may be called the bulk longitudinal relaxation modulus. For the polymeric foams, which typically have very low Pois- son's ratios in compression (6), the following approxi- mation can be derived:

M * (w) = 2 [G'(w) + iG"(w)] (10)

where o is the frequency. This suggests that the bulk dynamic moduli of foams can be obtained from dynamic shear devices, which are readily available.

EXPERIMENTAL

Low-density polyethylene and polystyrene foams were prepared by extrusion. Rectangular specimens of square cross-section were tested in uniaxial compres- sion on an Instron machine. These tests involved defor- mation at a constant strain rate (.02 in./min). The open cell content was measured by a Pycnometer using the air displacement technique (11). The cell volume was mea- sured using a light reflection method proposed by Co- lumbo and Tsai (12).

The dynamic viscoelastic behavior was measured with an eccentric rotating disc (ERD) rheometer commer- cialized by the Rheometrics Company. The principle of the ERD rheometer is to submit the sample to eccentric shearing of amplitude A, at frequency w. The forces F , and F , applied to the lower disc give a measure of the real and imaginary part of complex modulus G ' and G".

F , rrR2 al l C ' (0 ) =

F , C"(w) = ~ rrR2 all

where 1 is the gap between the two discs, 2R is the di- ameter of the sample, and the deformation a is applied along the Y-axis.

DISCUSSION Figure 2 shows the compressive stress-strain curves

measured by uniaxial compression with an Instron ma- chine for three polyethylene foams. Curve B, repre- senting a sample with 35 percent open cells, shows an extremely low modulus. This suggests that the gas phase contribution to the elasticity is critical. For closed cell foams at small strains, the gas pressure increases with

POLYMER ENGINEERING AND SCIENCE, JUNE, 1982, Vol. 22, No. 9

The Conipressiue Deformution of Polymeric Foums

9 - - z)

0 c - - n 0

6 - n n‘ DENSITY YI Y c n

increasing compressive strains to a point where the pressure equilibration among cells occurs either by flow through the porous structure or by gas pressure build-up with the cells. At medium compressive strains, the buckling of the cell structures is intensified; the compressive stress accelerates the gas pressure build-up, which then causes the cell walls to collapse. At the last stage of compression, the stress rises rapidly, since all cells have collapsed, leaving a bulk solid rather than a foam.

Figure 3 compares the initial compressive moduli for closed-cell foams with various densities. A log-log plot of initial compressive moduli vs. foam density gives a fairly linear correlation.

Table 1 summarizes the effects of cell size, cell-wall thickness, and foam density on the viscoelastic proper- ties of polystyrene foams. Both the cell size and the av- erage cell-wall thickness correlate highly with the stor- age modulus of foams, while the damping factors increase with decreasing density and cell size, then level off at low-density regions.

Compressive stress has been found to be directly de- pendent on the rate of deformation. Figure 4 shows that both G’ and G” increase slightly with increasing strain rates, while complex viscosity decreases with strain rates following a power law. This property is very impor-

S L O P E = 1 .9 I m I

N 0 - 5 -

4 -

Y

E

2 3 - d 3 n 0 I 2 -

‘. G’

\ * \. \

\ T a n 6

1

‘c C A

l C

0.1 1 10 100

FREQ. R/SEC

tant because it is related to creep, which limits a materi- al’s usefulness for packaging applications.

Figure 5 shows that G‘ and G” decrease with increas- ing temperature. The negative slopes of the stress-tem- perature curves suggest that enthalpy changes in the system are greater than its restoring force (13). Figure 5 also shows that the damping factor reaches a maximum at about 50°C. It has been known that the modulus of the base polymer decreases linearly with temperatures to the glass transition point, while the air pressure of the closed-cell foam will depend on the balance of permea- tion and PVT relationship. Harding (14) has shown that the gas pressure tends to increase with increasing tem-

Table 1. Effect of Cell Size on Dynamic Properties of Foams*

Cell Wall Density Cell Size (XK) Damping Modulus G’

Sample gmlcc cm3 R . d, Tan S (lo7) DYNE/cm2

0.109 0.084 0.079 0.071 0.068 0.066 0.066

7.57 5.23 3.62 1.75 1.23

.99

.64

5.45 3.71 3.05 2.1 2 1 .82 1.70 1.46

0.115 0.131 0.153 0.176 0.183 0.183 0.185

3.25 2.69 2.20 1 .a0 1.51 1.50 1.45

‘Closed-cell polystyrene foams measured at a frequency 10 Hz and 20%.

POLYMER ENGINEERING AND SCIENCE, JUNE, 1982, Vol. 2 2 , No. 9 547

James T. Tsai

9 ~ _.C. - ’ . -.-.

I

1

’ 7 I..

01 40 60 80 2 0

T E M P . C o

F i g . 5 . G ‘ und G” us o fuiiction of teniperuture f o r u closed-cell poZyeth!ylene foum.

perature and then level off. Therefore, the maximum damping factors in Fig. 5 appear to correspond to the en- tropy effects contributed by both solid and gas phases.

CONCLUSION An alternative approach has been developed to evalu-

ate compressive stress on polymeric foams. Compres- sive stress depends primarily on three factors: foam density, deformation strains, and deformation rates. The density dependency derived from this approach agrees closely with the empirical models reported in the literature. Correlations between the dynamic compres- sion and shear deformation are also derived. Experi- mental data are presented that show that G’ and G” in- crease with increasing strain rates, while the damping factor reaches a maximum at low strain rates. For foams with a given density, closed-cell structure provides the best compressive strength and damping characteristics. Note that the proposed model for the prediction of cell wall rupture would not apply to foams with high open- cell contents. This information is useful for designers and engineers in order to optimize cell structures for ap- plications in extrusion and forming processes.

APPENDIX Disregarding the weight ofair in the foam, the weight

of the polymer in each cell, W,, is:

W, = d f . V, (8) where df = density of the foam and V, = average volume of the cell. Since the cell wall is made of the polymeric phase:

W, = A, . h . d, (9)

where A, = surface area of a cell h = cell window thickness d, = density of polystyrene

Therefore, the thickness of cell window h, assuming spherical cells, becomes:

where R is the average cell radius.

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