Supervisors:

J.G.F. Wismans, MSc.

J.A.W. van Dommelen, Dr. Ir.

Eindhoven University of Technology

Department of Mechanical Engineering

Mechanics of Materials

July 2009

Characterization of

polymeric foams

D.V.W.M. de Vries (0611747)

MT 09.22

1

Table of contents

Introduction ............................................................................................................................................. 2

1 - The mechanics of foams .................................................................................................................... 3

1.1 - Introduction to foams .................................................................................................................. 3

1.2 - Deformation mechanisms in foams ............................................................................................ 3

1.2.1 - Linear elasticity ................................................................................................................... 4

1.2.2 - Elastic collapse and densification ........................................................................................ 5

1.2.3 - Plastic collapse and densification ........................................................................................ 6

1.2.4 - The effect of strain rate ........................................................................................................ 6

1.2.5 - The effect of air ................................................................................................................... 7

2 - Experimental procedure ..................................................................................................................... 9

2.1 - Materials ..................................................................................................................................... 9

2.1.1 - IMPAXX foams ................................................................................................................... 9

2.1.2 - Johnson Controls Foams .................................................................................................... 10

2.2 - experimental set-up ................................................................................................................... 11

2.2.1 - Set-up for IMPAXX foams ................................................................................................ 11

2.2.2 - Set-up for Johnson Controls foams .................................................................................... 12

3 - Results ............................................................................................................................................. 14

3.1 - IMPAXX .................................................................................................................................. 14

3.1.1 - Stress-strain behaviour ....................................................................................................... 14

3.1.2 - Linear elasticity ................................................................................................................. 15

3.1.3 - Plastic collapse................................................................................................................... 18

3.2 - Johnson Controls foams ............................................................................................................ 24

3.2.1 - Stress-strain behaviour ....................................................................................................... 24

3.2.2 - Linear elasticity ................................................................................................................. 24

3.2.3 - Elastic collapse .................................................................................................................. 27

Conclusions and discussion .................................................................................................................. 32

References ............................................................................................................................................. 34

2

Introduction

The macroscopic constitutive behaviour of polymer foams is determined by a subtle interplay of (i)

the intrinsic constitutive behaviour of the polymeric material and (ii) the complex microstructure.

Goal of this project is to mechanically characterize two different types of polymer foams (an open-cell

flexible foam and an elasto-plastic foam with a closed-cell structure) in order to determine the effect

of phenomena, such as flow of air through cells in foams and the influence of intrinsic material

behaviour.

As stated above, the macroscopic constitutive behaviour will partially be determined by the intrinsic

constitutive behaviour of the polymeric material of which the foam is made. There are a lot of models

in literature that relate material properties of the polymer foam to the polymeric material of which the

cell walls of the foam are made [3][4]. Some of these are referenced in chapter 1. The models explained

there will be applied to validate experimental results. The other contribution that partially determines

the material behaviour of foams is the complex microstructure. Besides that, there are a lot of external

conditions that can influence the material behaviour of the foam, like temperature and pressure.

As a result of these contributions (and the interplay between them), the strain rate and flow of air

through cells will affect the macroscopic constitutive behaviour of the foams. This will be further

investigated with experiments. Uni-axial compression tests will be executed at different strain rates

and with specimens of different length scales. For both open- and closed-cell foams, foams with

different densities will be analysed. Because of the great complexity of parameters which influence

the macroscopic constitutive behaviour of polymer foams, a large number of experiments is

performed in order to investigate the effect of these phenomena.

The expectation is that for open-cell foams the influence of air flow will be higher than for closed cell

foams, because the air in foams with an open structure can be forced to flow out of the foam. For

larger length scales the air will pass a longer way to get out of the material and the resistance to it will

grow. Strain rate will influence the mechanical behaviour of both foams, like a viscous response of

the material [6]. A higher strain rate will give also higher resistance due to air flow.

In order to perform an analysis of test results, first an introduction will be given to foams. Both the

mechanics of open- and closed-cell foams will be discussed in here. The most important deformation

mechanisms of foams will be clarified. After this introduction to foams, the materials used for the

experiments will be highlighted and for each material a specific experimental set-up will be clarified.

In the third chapter, the test results will be analysed. The results will also be examined on the

analytical expressions, given in chapter 1. Finally, conclusions will be given about the influence of air

flow and intrinsic material behaviour on the macroscopic constitutive behaviour of polymer foams.

Also some discussion points are reported and some recommendations will be given for future

investigations to the material behaviour of polymer foams.

3

1 - The mechanics of foams

1.1 - Introduction to foams

There are a lot of different applications for foams. Examples of applications are absorbing the energy

during impact events, lightweight structures and thermal insulation. To use foams efficiently a

detailed understanding of their mechanical behaviour is required. The mechanical properties of foams

are related to their complex microstructure and to the properties of the material of which the cell walls

are made, in here a solid polymeric material. Some salient structural features of foams [4]

are:

- the relative density

*

s

Rρ

ρ= , in which the superscript *

refers to the effective properties of

the polymer foam and the subscript s refers to the properties of the solid;

- the degree to which cells are open or closed;

- the geometric anisotropy of the foams.

The most important properties of the solid [4] (which will be used here) are the polymer density ρs ,

Young’s modulus Es and yield stress σys. These material parameters can be found in literature or are

given by companies.

The analytical expressions are based on these parameters and test results will be referenced to the

material properties of the solid polymeric material. Factors such as strain-rate and specimen size will

influence the material behaviour of polymer foams too. The latter two factors form the central topic in

this report. Besides that, some other factors, like temperature, anisotropy and loading conditions all

influence the properties too. These will not be considered in this study. Experiments have to be done

in order to ensure the effect of strain rate and air flow on the macroscopic constitutional behaviour.

With the experimental results, the macroscopic constitutive behaviour of foams can be analysed and

analytical expressions can be validated. In this chapter the mechanics of foams is further explained in

order to analyse the test results later.

1.2 - Deformation mechanisms in foams

Stress-strain responses of foams in compression tests show equivalent properties for different types of

foams. Figure 1.1 and 1.2 show typical schematic compressive stress-strain responses for an

elastomeric foam and for an elasto-plastic foam respectively. Because only uni-axial compression

tests will be executed, only mechanical properties in compression will be of importance, but it should

be noted that the mechanical behaviour of foams in tension is different. For example, a foam can be

plastic in compression but brittle in tension, caused by the stress-concentrating effect of a crack,

which leads to fast fracture in tension [4]

.

For the stress-strain responses in compression tests, a region of linear elasticity (Hookean) at low

stresses is followed by a long collapse plateau in which the stresses do not vary a lot, truncated by a

region of densification in which the stress rises steeply. Each region is determined by some

mechanism of deformation.

Linear elasticity is controlled by cell wall bending and, in case of closed cells, by stretching of the cell

walls. The Young’s modulus E* is the initial slope of the stress-strain response of the polymer foam.

For small strains, the foam will have an elastic response. In this region, the compressive stress can be

determined by * *Eσ ε= (1.1)

4

In compression, the plateau is associated

with collapse of the cells. The plateau

region is different for elastomeric foams

and elasto-plastic foams. For an

elastomeric foam the plateau is determined

by elastic buckling and in elasto-plastic

foams by the formation of plastic hinges [4]

. For a pure elastomeric foam, there is no

plastic deformation, but for an elasto-

plastic material the foam has a plastic

region.

When the cells have almost completely

collapsed opposing cell walls touch and

further strain compresses the solid itself,

giving the final region of rapidly

increasing stress, referred to as

densification. Increasing the relative

density of the foam increases the Young’s

modulus, raises the plateau stress and

reduces the strain at which densification

begins. The influence of different densities

will be validated in the experiments.

Superimposed on the deformation of the

cell edges and cell walls is the effect of the

fluid (air) contained within the cells. When

a closed-cell foam is compressed, the cell

fluid is compressed too. This leads to an

additional force which can be calculated

from Boyle’s law [3]. If the cells are open

and interconnected, deformation forces the

fluid to flow from cell to cell, doing

viscous work, and this generates a force

which must also be overcome [4]

.

As indicated before, in this survey foams

will be tested in compression. In the next

paragraphs, a more detailed analysis will

be given for the mechanical properties of foams for this loading regime and foam properties will be

expressed in terms of properties of the solid polymer. The different regions in the stress-strain

responses will be discussed and also some theorem about strain rate and air flow is given. As

indicated in this paragraph, the mechanical behaviour is different for open-cell and closed-cell foams

and for elastomeric and elasto-plastic foams, so a distinction between them has to be made in the

analysis of foams.

1.2.1 - Linear elasticity

The linear elastic behaviour of a foam is characterized by a set of moduli, that depends on its

(an)isotropy. The determination of the material parameters of foams can be done with different

loading regimes. Only uni-axial compression tests will be performed here.

In order to give some analytical expressions, some simplifications will be made. This analysis is based

on an isotropic foam [4]. Foam properties will be related to the properties of the polymer solid in order

to predict the foam’s Young’s modulus. Distinction is made between open- and closed-cell foams.

Figure 1.2 - Schematic compressive stress-strain response for

elasto-plastic foams [4]

Strain ε 0 1

Str

ess

σ

Elasto-plastic foam

εmax

σpl*

Linear elasticity

Plateau

Densification

E

Strain ε

Str

ess

σ

0 1

σel*

εmax

Elastomeric foam

Densification

Plateau

Linear elasticity

Figure 1.1 - Schematic compressive stress-strain response for

elastomeric foams [4]

5

1.2.1.1 - Open-cell foams

At low relative densities, open-cell foams deform primarily by cell-wall bending. As the relative

density increases (R > 0.1) the contribution of simple extension or compression of the cell walls

becomes more significant.

There are several models in literature to predict the Young’s modulus of the foam which are based on

simple cell shapes in structural arrays, like honeycomb structures [3][4]. In practice, this won’t be the

case, but for understanding the behaviour of foams such analyses are important.

For a cubic array of members simple beam theorem can be applied, and it can be derived that [4]

*

2

1

s

EC R

E=

in which C1 is a constant and C1 ≈ 1. (1.2)

The modulus found here, is the modulus at small strains. As the elastic distortion increases, the axial

load on a cell increases. This exerts an additional moment on the bent edge and in compression the

modulus will decrease. So the part of the stress-strain response which is called linear elastic is

concave downwards. When the axial load on the cell edge reaches a critical load, the edge buckles and

the foam loses stiffness. This will be analysed further in 1.2.2.

1.2.1.2 - Closed-cell foams

In closed-cell foams the cell edges both bend and extend or contract, while the membranes which

form the cell faces stretch, increasing the contribution of the axial cell-wall stiffness to the elastic

moduli. If the membranes do not rupture, the compression of the air in the cells also increases their

stiffness (see 1.2.5.2).

So, for Young’s modulus, there are three contributions to the initial stiffness of foams and therefore

the analysis is more complicated. A model - in which the Poisson’s ratio is assumed to be zero - that

predicts the Young’s modulus for closed-cell foams is given by equation 1.3 [4].

( )( )

*2 2 01

1s s

pER R

E E Rφ φ= + − +

− (1.3)

In which ( )1 φ− indicates the fraction of solid in the cell faces, i.e.φ is the fraction of solid contained

in the cell edges. Reasonable values for φ are 0,6 and 0,8 [4].

1.2.2 - Elastic collapse and densification

For open- and closed-cell foams, the elastic collapse stress and the densification behaviour are

different. This part counts for elastomeric open-cell foams. In compression the stress-strain response

for polymeric foams will show an extensive plateau at a stress level which doesn’t change much. This

stress level is referenced to as the elastic collapse stress and the slope of the plateau in the stress-strain

diagram is called the Plateau modulus in here. The elastic collapse in foams is caused by the elastic

buckling of cell walls. The stress level at which elastic collapse occurs - also referred to as σel*, the

elastic collapse stress of the foam - can also be predicted.

Based on a open-cell structure with cubic cells consisting of interconnected cell edges with length l

and thickness t (square cross section t2), the elastic collapse stress can be estimated with the following

model [4]

. When an elastomeric open-cell foam is compressed the cell walls will bend till a critical

load is applied at which the cell walls buckle. This load can be calculated with the Euler formula: 2 2

2

scrit

n E IF

l

π= (1.4)

With second moment of inertia I and the factor n2, that describes the degree of constraint at the ends

of the column. Elastic collapse will initiate at

*

2 4

crit sel

F E I

l lσ ∝ ∝ (1.5)

6

Using 4

I t∝ and

2t

Rl

∝

which counts for open cells [4]

, the elastic collapse stress can be related to

the relative density and the Young’s modulus of the polymer solid, as stated in equation 1.6. *

2

2el

s

C RE

σ= in which C2 is a constant and C2 ≈ 0.05. (1.6)

At larger compressive strains the opposing walls of the cells crush together. Then the cell wall

material itself will be compressed. This results in a steeply rising stress-strain response, with a slope

approaching the Young’s modulus of the solid polymer at a limiting strain of εmax. This limiting strain

could be given by the porosity ψ - which is given by equation 1.7 - but in practice the cell walls

gather together at a smaller strain. With experimental data it is verified that this limiting strain can be

assessed with equation 1.8.

1 Rψ = − (1.7)

max 1 1.4Rε = − (1.8)

1.2.3 - Plastic collapse and densification

Foams that have a plastic collapse stress, referred to as σpl*, collapse plastically when loaded beyond

the linear-elastic regime. Plastic collapse gives a long, approximately horizontal, plateau to the stress-

strain response. Advantage from the long stress plateau is taken in crash protection, since the energy

absorption per unit of volume is defined as the area under the stress-strain responses (Equation 1.9).

Like elastic buckling, the failure is localized in a band transverse to the loading direction. This band

propagates throughout the foam with increasing strain. max

*

0

dεU

ε

σ= ∫ (1.9)

Since the elasto-plastic foam, that will be tested, has a closed-cell structure, the plastic collapse and

densification will only be further explained for this type of foams.

The plastic collapse stress is affected by stretching as well as bending of cell walls. Besides that, also

the fluid in the cells can give a stress contribution to the plastic collapse stress. This contribution is

further explained in section 1.2.5.2. Plastic collapse causes the cell faces to crumple in the

compression direction. If the cell faces are very thin, they could rupture before full plastic collapse,

and then the closed-cell foams will behave like a foam with an open-cell structure.

Because of complexity, analytical expressions that predict the plastic collapse stress are difficult to

determine. One analytical description is given by [4]

( ) ( )* 3

20.3 0.4 1pl atm

ys ys

p pR R

σφ φ

σ σ

−≈ + − + (1.10)

In which the pressure p represents the pressure of the fluid (air) in the cells and patm is the atmospheric

pressure.

1.2.4 - The effect of strain rate

The material behaviour of foams depends, as one might expect, also on strain rate. There are two

separate contributions to the strain rate-dependence of foam properties. The first one derives from the

polymer solid and will be called inherent strain rate-dependence; the foam inherits the strain rate-

dependence of the solid polymeric material of the cell walls. A relationship for the strain rate

dependency of the yield strength of a polymer is given by Eyring [6]

3

lnys

act

kT

C

εσ

ν

=

ɺ (1.11)

7

In which σys is the yield strength of the solid polymer, k is the Bolzmann constant, T is the current

temperature and νact is a so called activation volume. The strain rate is given by εɺ and temperature

dependent parameter C3 is given by [6]

( ) 03 exp

2

UC T

kT

ε ∆ = −

ɺ (1.12)

In here, 0εɺ is a material property and ∆U is the activation energy.

The second contribution to strain rate dependent behaviour derives from the liquid in the cells of the

foam. When the foam is compressed the liquid (in the foams used this will be air) either deforms or is

forced to flow from cell to cell. In an open-cell foam, the air is expelled out of the foam during

compression. This induces viscous forces that are also dependent on strain rate. This is related to air

flow and will further be explained in section 1.2.5.2.

Data in literature [4] states that the plastic collapse strength of polymer foams linearly increases with

the logarithm of strain rate. This is stated in equation 1.13.

( )0

* * 01 lnpl pl

g

AT

T

εσ σ

ε

= −

ɺ

ɺ (1.13)

In which ( )0

*

plσ is the plastic collapse stress at 0 K, A is material property and Tg is the glass

transition temperature of the polymer.

1.2.5 - The effect of air

1.2.5.1 - Open-cell foams: air-flow

The air flow resistance of foams is one of the main

aspects in this survey. The contribution of it to the

macroscopic constitutive behaviour of polymer

foams is investigated. In contrast to the influence

on mechanical properties of a foam, the cell size

strongly influences air-flow properties. In closed-

cell foams the effect of air-flow can be neglected in

most cases, but if the membranes are very thin, the

cells will burst and the air will flow through the

foam. But in common, the effect of air-flow is of

importance for open-cell foams.

When an open-cell foam is compressed, the air it

contains is squeezed out. Air has a viscosity, so work

is done forcing it through the interconnected porosity

of the foam. The faster the foam is deformed, the more work is done; the air flow phenomenon is

therefore strongly dependent on strain rate. One way to analyse the effect of strain rate is to treat the

foam as a porous medium, characterized by an absolute permeability K; then the fluid through it is

described by Darcy’s law [4]

d

d

K pu

xµ= (1.14)

Where u is the velocity of the fluid, K is the absolute permeability of the foam, µ is the dynamic

viscosity and dp/dx is the pressure gradient. Because of small pore sizes and relative small velocities,

it can be assumed that only laminar flows will occur (Re < 2300) so inertial effects can be neglected.

For a permeable material with pores of average diameter d the permeability is given by [4]

( )3

22

4 1K C d R= − (1.15)

Where C4 is a constant to which the empirical value 0,4 is generally assigned. Foams typically have

permeabilities in the range 10-10 to 10-8 m2. The viscous flow in a block of foam is illustrated in figure

L

H

σ

Air flow

V

σ = 0

ey

ex

Figure 1.3 - Illustration of the cross-sectional area of a

foam specimen during uni-axial compression [4]

8

1.3 .The contribution of viscous flow to foam strength can be calculated with help of this figure. The

air flux through each of the two vertical faces is given by

2 2

VL Lq

H

ε= =

ɺ (1.16)

Where V is the compression speed, H is the height of the block and L is the base length of the foam.

The average flux across any vertical internal surface is one-half of this, because of symmetry.

Inserting a factor of ½ , and substituting the result into equation 1.14, gives

d

4 d

L K p

x

ε

µ= −

ɺ (1.17)

The pressure gradient is in here proportional to L

σ− (see figure 1.3). The pore size d is obviously

proportional to the cell edge-length l at the start of deformation. During compression the pores

become narrower. Gent and Rusch [4]

suggest that

( )1

21d l ε∝ − (1.18)

Substituting these relations into Equation 1.17, with K defined by Equation 1.15, gives the

contribution of the air flow to the strength of open-cell foams like 2

* 5

1g

C L

l

µεσ

ε

=

−

ɺ (1.19)

In this equation, the proportionality constants have been combined in the constant C5 that is of order

unity. The contribution of air flow to the strength σ* of a foam is therefore proportional to the strain

rate and to the viscosity of air and to the reciprocal of the cell size, squared. To drop temperature

influences, the temperature should remain constant, because also viscosity is temperature dependent.

1.2.5.2 - Closed-cell foams: air response

As stated before in the introduction of paragraph 1.2, superimposed on the polymer response for

closed-cell foams to compression must be the effect of fluid (air) contained within the cells.

Skochdopole and Ruben’s (1965) [3]

gave a qualitative model (figure 1.4) that suggests that the cell air

and the polymer microstructure of the closed-cell foam are acting in parallel when they undergo

deformation. The model simply adds the stress due to the polymer structure, σp , to the stress σg

originated from the air in the cells. In here, a simple analysis on the air response will be taken (Rusch,

1970) [3].

Assuming zero lateral expansion, i.e. the Poisson ratio is zero and the volumetric strain is equal to the

compressive strain ε, isothermal gas compression and incompressible polymer cell walls, with figure

1.5 can been proven that

( ) ( )1 1atm

p R p Rε− = − − (1.20)

And therefore the air in the cells give an additional stress equal to

1g atm atmp p p

R

εσ

ε

= − =

− − (1.21)

Air Cell walls

Force

Figure 1.4 - Model (redrawn from Skochdopole and

Rubens) of air response and polymer response

acting in parallel for a closed-cell foam [3]

Polymer Polymer

Air at pressure p0

Air at pressure p

1-R

R

ε

1-R-ε

R

Stress σ

Stress σ

Figure 1.5 - Volumes before and after compression [3]

9

2 - Experimental procedure

In this chapter the experimental set-up and the materials will be clarified. Tests have been performed

on an elasto-plastic polymer foam with closed-cell structure and on an elastomeric polymer foam with

open-cell structure. Both materials will be introduced first and some available technical data will be

given. In here, also some pictures of the foams, made with a Scanning Electron Microscope (SEM),

will be given in order to validate the foam structure. After introducing the materials, a test set-up will

be given which is different for the two polymer foams. Results of the experiments will be given in the

next chapter.

2.1 - Materials

Tests have been performed on two polymer foams with various properties. The first material is called

IMPAXX which is used in automotive industries for crash protection. The second foam will be called

Johnson Controls inc. (JC). This foam is used for interior, like car seats. In this paragraph, both foams

will be introduced.

2.1.1 - IMPAXX foams

For the first tests IMPAXX Energy Absorbing Foams (DOW Automotive) were used. IMPAXX

foams are highly engineered polystyrene-based thermoplastic foams. It is formed by extruding

polystyrene polymer - which contains a halogenated flame-retardant system - that has been formulated

with blowing agents and other additives. The blowing agents expand when pressure is released at the

extrusion die to form the foam. These foams are strong and lightweight and are designed to maximize

efficiency and minimize weight. IMPAXX foams are mainly used for automotive applications. Their

function is to absorb the impact energy in the event of a crash and the foams are for instance installed

within bumpers or doors [9]

.

For the compression tests, three different IMPAXX foams were used: IMPAXX 300, IMPAXX 500

and IMPAXX 700, all with different densities. This foam has a closed-cell structure, which has been

validated with some scans with a SEM. Figures of this scan can be seen in figures 2.1a and 2.1b.

From DOW Automotive, some technical data are available for IMPAXX 300 and 500 [7][8]

. These are

listed in table 2.1. This is just a short guideline to check whether the test results are in the same range.

Figure 2.1a - Side view of closed-cell strucure of IMPAXX

(with SEM, TU/e)

Figure 2.1.b - Top view of closed-cell strucure of

IMPAXX (with SEM, TU/e)

IMPAXX is also said to have a small temperature dependence, so all tests will be executed at room

temperature, which is about 21°C.

Table 2.1 - Technical data IMPAXX

IMPAXX ρ* [kg/m3]

300 35

500 43

2.1.2 - Johnson Controls Foams

This foam is used for the seating i

Controls Inc. with 4 different (relative)

JC100 and JC120. This foam has an open

Figure 2.2 - Microscopic view of open

The material of the solid cell walls of JC foams is p

the JC foams is supposed to be similar to

recovery analysis in 2.2.2.1. A PU is

urethane links. It is formed through

at least two isocyanate functional groups

groups in the presence of a catalyst.

cover a wide range of stiffness, hardness, and densities

said to have a small temperature dependence, so all tests will be executed at room

Technical data IMPAXX foams

Compression

strength (23 °C)

at 10% [MPa]

Compression

strength (23 °C)

at 25% [MPa]

Compression

strength (23 °C)

at 50% [MPa]

0.345 0.375 0.434

0.512 0.544 0.612

Johnson Controls Foams

This foam is used for the seating interior of vehicles. There are specimens delivered by Johnson

(relative) densities. These foams will be indicated with JC80, JC90,

JC100 and JC120. This foam has an open-cell structure which is validated with a SEM,

of open-cell structure of JC80 foam, made with SEM (TU/e)

The material of the solid cell walls of JC foams is polyurethane (PU) and the material

supposed to be similar to elastomeric foams. This will be validated with a material

A PU is any polymer consisting of a chain of organic units joined by

formed through step-growth polymerization by reacting a monomer

functional groups, with another monomer containing at least two

. PU polymers can be built of many different components, and they

of stiffness, hardness, and densities [6][10]

.

10

said to have a small temperature dependence, so all tests will be executed at room

Compression

strength (23 °C)

50% [MPa]

0.434

0.612

specimens delivered by Johnson

foams will be indicated with JC80, JC90,

cell structure which is validated with a SEM, see figure 2.2.

material behaviour of

. This will be validated with a material

units joined by

monomer, that contains

, with another monomer containing at least two hydroxyl

PU polymers can be built of many different components, and they

11

From Johnson Controls Inc., some technical data are available for the different polyurethane foams.

These are listed in table 2.2. In here, σel* represents the elastic collapse stress of the JC foam (is

explained later). Again, this is just a short guideline to check whether the test results are in the same

range.

Table 2.2 - Technical data Johnson Controls foams

JC ρ [kg/m3] σel* [MPa]

80 58 6

90 60 8

100 60 10

120 62 13

2.2 - experimental set-up

2.2.1 - Set-up for IMPAXX foams

In order to determine material parameters of the IMPAXX foams (Young’s modulus, Plateau modulus

and plastic collapse stress) as function of test parameters (diameter of specimens and strain rate), a

great number of compression tests has to be done. The forces and displacements are measured on

cylindrical specimens with a constant height and with three different diameters (25, 50 and 75 mm).

Specimens are cut to the appropriate dimensions using a cavity drilling apparatus with different

diameters. From each dimension, 20 specimens (corresponding to one density) will be tested at 5

different strain rates: 10, 100, 10

-1, 10

-2 and 10

-3 s

-1. The loading is therefore displacement controlled.

At each strain rate, 4 identical set-ups are used to validate the results. The compression tests are

carried out on an MTS 810 Elastomer Test System with a 25 kN load cell. The specimens will be

compressed between two cylindrical platens. All specimens are loaded to an engineering strain of

approximately 80%, well beyond the densification strain. Before starting up a measurement, the force

has to be set to zero (without the specimens on the plates). Then a specimen is placed on the machine

and a small load is set on the specimen (approximately 5N). After this (only) the displacement must

be set zero and then the experiment is ready to start. After each measurement following up after the

first one, only the displacement has to be set zero again. The compressive stress-strain responses are

obtained by dividing the applied load by the original specimen area (engineering stress), and by

dividing the specimen displacement by the original specimen height (engineering strain). In this case,

the engineering stress is approximately identical to true stress, because the Poison’s ratio is almost

zero so no notable changes in cross-sectional

area will occur.

To identify the Young’s modulus, the plastic

collapse stress and the Plateau modulus,

Matlab is used. The Young’s modulus can be

found by fitting a line through the elastic

region of the stress-strain response and

determining the slope of that line. This

elastic region has to meet some conditions to

get an appropriate value without disturbances

due to noise. Afterwards, the Plateau

modulus is found equivalently by a line

through the plateau area. The plastic collapse

stress is approximated by the intersection

point of these two lines. This is illustrated in

figure 2.3. For elastomeric foams, this

intersection point is called the elastic

collapse stress.

E*

Strain ε

Str

ess σ

0

σpl* Epl

Figure 2.3 - Determination of Young’s modulus, Plateau modulus

and collapse stress.

12

Because the experiments will be performed in series consisting of 4 tests with identical circumstances,

the found parameters (Young’s modulus, Plateau modulus and plastic collapse stress of the foam) are

averaged and a standard deviation is assessed. These deviations will also be plotted in the figures.

2.2.2 - Set-up for Johnson Controls foams

Because the Johnson Controls (JC) foams are supposed to be elastomeric, the idea is to measure one

specimen more than once because no (or not much) plastic deformation will occur and not much

specimens are available for testing this material.

In order to set up a proper experiment, first the material will be analysed in order to check the

recovery of the material. Therefore, first a material recovery analysis is done at one specimen.

Afterwards, the test set-up for this material is formulated.

2.2.2.1 - Material recovery analysis

In order to test the recovery of the JC foams (fully elastic deformation), ten compression tests with

JC80 foam have been performed at the MTS 810 Elastomer Test System. For the first three

experiments the stress-strain responses are given in figure 2.4. Between all experiments, the material

was given two minutes to recover. For these tests, the strain rate was 10-2

s-1

but for other strain rates

the same results were found.The legend indicates the the number of the experiment. As can be seen,

there’s a significant difference between the stress-strain response of the first test and that of the

second and third experiment. This implicates some plastic deformation at the first compression test.

After the first test, the quality of the foam remains approximately constant and test data will be

reproducible although the stresses in the successive experiments are lying just below each other. This

aspect should therefore be considered in the test set-up.

As can also been concluded from this stress-strain responses, the macroscopic constitutive behaviour

of JC foams is similar to that of elastomeric foam which is described in 1.2 and visualised in figure

1.1.

Figure 2.4 - Stress-strain responses for 3 successive compression test performed at one specimen of JC80 foam

13

2.2.2.2 - Test set-up

With this information, a test set-up is formulated, equivalent to the test set-up for IMPAXX foams.

The set-up is based on 4 available specimens of each of the 4 different types of Johnson Controls

foams. The forces and displacements are measured on square specimens with a constant height. First,

a set of experiments will be executed on blocks with a length of approximately 50 mm and a height of

approximately 30 mm. After executing tests on these specimens, the blocks will be cut to lengths of

25 mm and height of 30 mm.

To reduce the influence of the plastic deformation in the first tests, first three experiments will be

performed on a specimen to get more corresponding results afterwards. These first tests are done at a

strain rate of 10-2 s-1 and the maximum strain in this test is 60%, well before the densification strain so

less plastic deformation occurs. Between all tests the material is given two minutes to recover.

After this kind of initialisation, from each material, 4 specimens will be tested successively at 4

different strain rates: 10, 100, 10-1 and 10-2 s-1. The order in which this strain rates are applied is

different for each of the 4 specimens so there’s less dependency of the times a specimen is used.

Between all tests, the same recovery time (two minutes) is used. For each material and each strain

rate, 4 data files will be created this way. The loading is again done in displacement control and the

test system is identical to the system for IMPAXX foams (see also 2.2.1). All specimens are -

similarly to the initialisation tests - loaded to an engineering strain of approximately 60%, well before

the densification strain. After testing the specimens with lengths of 50 mm, specimens with lengths of

25 mm will be created. These specimens are tested the same way, but only 1 initialisation test will be

done, because there’s less influence of plastic deformation due to the tests done before.

Initialising forces and displacements is done the same way described before, but in here the

prestressing force is much less and approximately 0.1 to 1.0 N. Identifying the Young’s modulus,

Plateau modulus and the elastic collapse stress of the foam is done similarly to the way described in

section 2.2.1, with help of Matlab.

14

3 - Results

In this chapter the most important results of the experiments will be given. As indicated before, the

results are processed with help of Matlab. First of all, the IMPAXX foams will be analysed and

afterwards the results of the JC foams will be shown.

For each material, first a short view will be taken on the stress-strain response to uni-axial

compression. After that, test data will be analysed and the found material parameters will be checked

in relation with variable sample rates, specimen sizes and relative densities. In addition, it will be

validated whether the analytical expressions to predict specific foam properties, see chapter 1, hold or

not. Besides that, some additional phenomena will be analysed, e.g. the influence of air flow in open-

cell foams on the macroscopic constitutive behaviour of the foam.

3.1 - IMPAXX

3.1.1 - Stress-strain behaviour

For IMPAXX, typical stress-strain responses are shown in figure 3.1. The three lines represents the

three foams with different densities. In here, a strain rate of 10-1

s-1

is applied on specimens with a

diameter of 50 mm. The responses are similar to the schematic compressive stress-strain response for

elasto-plastic foams, see also figure 1.2. The region of linear elasticity is followed (at a strain of

approximately 1,5 to 2,0%) by a long collapse plateau at which the stress only slightly rises. At a

strain of approximately 70% densification starts. The plastic collapse can clearly be seen during

compression because the failure is localized in a band transverse to the loading direction. This band

propagates throughout the foam with increasing strain. There is no expansion of the foam during

compression, so in the analysis, a Poisson’s ratio of zero can be assumed.

Figure 3.1 - Stress-strain responses for IMPAXX with different densities (300, 500 or 700) at a constant strain rate of 10-1 s-1

and a constant specimen diameter of 50 mm.

15

In the stress-strain response, three remarkable points must be concerned. The first point can be found

at the transition between the elastic region and the plateau. In here, sometimes a drop in stress can be

found, especially for IMPAXX 500 at low strain rates (of 10-1, 10-2 and 10-3). An explanation for this

behaviour is that the high pressure in the cells is removed due to plastic collapse of cells in the upper

layer of the foam. As a consequence, the stiffness of the foam decreases and the stress becomes

temporarily lower until the next band of cells is reached. For lower strain rates, the time scale is

larger, resulting in a larger stress drop. The second point that must be concerned is the second

transition area, between the plateau region and the point at which densification starts. At this point,

the stress regularly drops before densification starts. This occurs mainly for IMPAXX 500 and

IMPAXX 700. This indicates that the loss of stiffness at this point may have something to do with the

relative density of the foam. A full explanation for the material behaviour at this point is not gathered

yet and is beyond the scope of this project. The last remarkable point can be noticed at the final point

of deformation, where the stress drops vertical. This is due to stress relaxation (at a constant strain

level).

3.1.2 - Linear elasticity

3.1.2.1 - Prediction of Young’s modulus

In order to validate test results, one may want to relate test data to analytical expressions. Therefore,

measured properties of the IMPAXX foams will be related to material properties of the solid polymer,

which is in here assumed to be pure polystyrene (PS). For IMPAXX foams it’s known that PS has

been formulated with blowing agents and some additive components [9], but because no further

information about solid properties is given by DOW, the properties that will be used for the analytical

expressions are stated in table 3.1 [5]

.

Table 3.1 - Data PS

ρs [kg/m3] Es [MPa]

1051 3300

By measuring the length and the weight of the specimens, the densities (on average) and relative

densities of the different IMPAXX foams are found. These are listed in table 3.2. Also the standard

deviation of the measured densities is given. Besides that, also the densities given by DOW are stated.

Note that these are not the same as the measured densities. This can be due to different measurement

methods.

Table 3.2 - Measured densities for IMPAXX

IMPAXX ρ*measured [kg/m3] R [-] σstd [kg/m

3] ρ*DOW [kg/m

3]

[7][8]

300 38,46 0,0366 0,33 35

500 40,39 0,0384 0,44 43

700 44,70 0,0425 0,58 N.A.

Assuming a Poisson’s ratio of zero, the Young’s modulus of the polymer foam can be estimated with

equation 1.3. Reasonable values for φ are 0,6 and 0,8 [4], and the initial pressure in the cells is

assumed to be atmospheric. The results are listed in table 3.3. In here, also the standard deviation is

given for the measured modulus. It must be noted that the modulus listed in here represents the

average modulus over all measurements, with different specimen diameters and different strain rates.

Table 3.3 - Predicted Young’s modulus for different IMPAXX foams

IMPAXX E*predicted [MPa]

φ = 0,6

E*predicted [MPa]

φ = 0,8

E*measured [MPa] σstd

[MPa]

300 50,00 27,09 21,64 2,58

500 52,59 28,59 30,87 4,42

700 58,39 32,00 42,49 6,34

16

From this, it can be concluded that the predictions of the Young’s modulus are of the same order of

magnitude as the measured moduli, especially for a φ of 0,8, although the prediction is not very

accurate.

As listed in table 3.3, the Young’s modulus is dependent on the density of the foam. Using equation

1.3, the relationship between Young’s modulus and foam density should primarily be linear for

closed-cell structures, since the relative densities are small (< 0,05) for IMPAXX foams and the

contribution of the initial pressure (last term) in the cells to stiffness is small. Therefore, equation 1.3

could be simplified to a linear relation

( )*

1s

ER

Eφ= − (3.1)

With this formula, the Young’s modulus of the different IMPAXX foams can be related to each other,

because *

3006*

300

EC

ρ= with

( )6

1s

s

EC

φ

ρ

−= (3.2)

Therefore, assuming that fraction of solid in the cell faces is equal for the different IMPAXX

densities, the Young’s moduli of the different IMPAXX foams are coupled with * * *

300 500 700

* * *

300 500 700

E E E

ρ ρ ρ= = (3.3)

In table 3.4, this relation has been checked. The relationship doesn’t seem to hold in this case. This

could be attributed to the used model, but it may be more convenient to attribute this difference to

variable values of φ for the different foam densities. For each IMPAXX density, the value of φ is

therefore determined with equation 3.2 and stated in table 3.4. With these new, analytically

determined values of φ , new predictions of the Young’s modulus are made using equation 1.3. Now

it appears that the prediction of the Young’s modulus is more realistic.

However, some simplifications have been made and the determination of φ is complex. The latter

could also be done with an accurate analysis of the microstructure. Besides that - based on the relative

densities - one would expect the solid fraction in the cell edges to increase for larger densities. In here,

the fraction decreases, probably originating from simplifications or microstructure of the foam.

Table 3.4 - Relation between Young’s moduli

IMPAXX *

*

E

ρ

φ E*predicted [MPa]

300 0,563 · 106 0,821 24,70

500 0,764 · 106 0,757 33,69

700 0,951 · 106 0,697 45,50

3.1.2.2 - Analysis of Young’s modulus

Different tests have been executed with IMPAXX foams of three different densities, at five different

strain rates and for three different specimen diameters. Therefore it’s useful to check whether material

parameters, like the Young’s modulus, depend on these variables.

Besides the relative density of the foam and/or it’s complex microstructure, an other phenomenon that

could influence the Young’s modulus of the foam is the specimen diameter. In order to investigate

this, the modulus can be plotted as function of the three different densities (given as 300, 500 and

700) for different specimen diameters at a constant strain rate of 10-1 s-1. At other strain rates,

equivalent plots will show up. This is illustrated in figure 3.2. From this, it can be concluded that the

17

influence of specimen diameter on the Young’s modulus of the foam is small. No clear link exists

between the modulus and the diameter.

The same can be done to investigate the influence of strain rate. This is plotted in figure 3.3. In here,

the Young’s modulus is given at a constant diameter of 50 mm. This will be equivalent for other

diameters. From this, it can be concluded that also the influence of strain rate on the Young’s modulus

of the IMPAXX foam is small.

Therefore, it can be concluded that the Young’s modulus for this closed-cell foam is mainly

determined by the relative density of the foam and by the complex microstructure.

Figure 3.2 - Young’s modulus as function of IMPAXX-density (300, 500 or 700) for different specimen diameters at a

constant strain rate of 10-1 s-1.

Figure 3.3 - Young’s modulus as function of IMPAXX-density (300, 500 or 700) for different strain rates at a constant

specimen diameter of 50 mm.

18

3.1.3 - Plastic collapse

The second region in which some parameters can be found from experimental data, and can be related

to analytic expressions, is the plateau region. In this paragraph, first the Plateau modulus Epl* of the

foam will be determined and it will be validated if this modulus depends on test variables (strain rate,

foam density and specimen diameter). After evaluating the Plateau modulus, the plastic collapse stress

will be determined; analogous to the method described in 2.2.1. This material parameter is also

checked on relations with test variables. Besides that, the found collapse stress is compared with

analytical expressions from paragraph 1.2.

3.1.3.1 - Plateau modulus

The Plateau modulus is small in comparison with the Young’s modulus. Typical values for the

Plateau modulus of IMPAXX foams are between 0,10 and 0,50 MPa. This can be seen in figure 3.1

and table 3.5. As listed, the plateau modulus is the highest for IMPAXX 300 and the lowest for

IMPAXX 500. For IMPAXX 700, the mean value of the Plateau modulus is between the values for

IMPAXX 300 and 500. Thus, not a real dependency on foam density is found; it rather originates

from the foam’s microstructure.

Table 3.5 - Plateau modulus of different IMPAXX foams

IMPAXX Epl*[MPa] σstd [MPa]

300 0,350 0.057

500 0,160 0.034

700 0,289 0.038

Also for the Plateau modulus, the relations with strain rate and specimen diameter have been

determined. As can be seen in figure 3.4, in which the Plateau modulus is given as function of the

IMPAXX density for different specimen diameters and at a constant strain rate of 10-1 s-1, the

dependency of specimen diameter is small. This figure is equivalent for other strain rates. In figure

3.5, the Plateau modulus has been given as function of foam density for different strain rates (with a

constant diameter of 25 mm). In here, for IMPAXX 300 and IMPAXX 500 the Plateau modulus will

raise slightly as function of strain rate, except when the strain rate is raised from 1 to 10 s-1 as the

modulus slightly drops. For the IMPAXX 700 foam, the Plateau modulus doesn’t seem to depend on

strain rate at all. So, in some cases a small dependency on strain rate is found, but no clear link exists.

Thus, it can be concluded that also the Plateau modulus is mainly influenced by the complex

microstructure of the foam. Only for IMPAXX 300 and 500, strain rate slightly influences the

modulus.

19

Figure 3.4 - Plateau modulus as function of IMPAXX-density (300, 500 or 700) for different specimen diameters at a

constant strain rate of 10-1 s-1.

Figure 3.5 - Plateau modulus as function of IMPAXX-density (300, 500 or 700) for different strain rates at a constant

specimen diameter of 50 mm.

3.1.3.2 - Analysis of plastic collapse stress

As found in the experiments and stated in 3.1.2.2 and 3.1.3.1, the Young’s modulus and the Plateau

modulus of the IMPAXX foams both mainly seems to be determined by the relative density of the

foam and/or the complex microstructure. There were no clear relations found between the material

behaviour of the foams and test variables. But as can be gathered from figure 3.6, the plastic collapse

stress is dependent on strain rate. Also the relation between plastic collapse stress and specimen

diameter will be analysed.

20

Figure 3.6 - Stress-strain responses for IMPAXX 300 for different strain rates with constant specimen diameter of 75 mm.

In figure 3.7 the relationship between the plastic collapse stress and strain rate is shown for a constant

specimen diameter of 50 mm. As could also be concluded from figure 3.6, the plastic collapse stress

will increase for larger strain rates. In figure 3.8 the relationship for the plastic collapse stress with the

specimen diameter is shown. In here, the strain rate has a constant value of 10-3

s-1

, but results are

comparable at other strain rates. As can be concluded from this figure, the plastic collapse stress is

hardly influenced by the specimen diameter. For the lowest strain rate, the plastic collapse stress

seems to increase slightly for larger specimen diameters - as can be seen in figure 3.8 - but at larger

strain rates this effect vanishes and no clear relation is found. Overall, the influence of specimen

diameter seems to be small.

Figure 3.7 - Plastic collapse stress as function of strain rate for different IMPAXX densities (300, 500 or 700) at a constant

specimen diameter of 50 mm.

21

Figure 3.8 - Plastic collapse stresss as function of specimen diameter for different IMPAXX densities (300, 500 or 700) at a

constant strain rate of 10-3 s-1.

As stated before in chapter 1.2.3, 1.2.4 and 1.2.5.2, the plastic collapse stress will be influenced by

different phenomena, like pressure build-up in closed-cells and viscous behaviour of cell wall

material. In chapter 1, also some equations were given in order to predict the plastic collapse stress

and to estimate the added stress due to pressure in cells. These are used here in order to determine

whether these analytical expressions can give a proper prediction of the plastic collapse stress.

First of all, the stress addition due to pressure built up in closed cells will be analysed. For this,

equation 1.21 was given. The strain at the point at which plastic collapse starts, εpl*, is different for

each IMPAXX density and each experiment, but the mean values (over all experiments) for each

material are given in table 3.6. In here, also the contribution of stress due to pressure in the cells is

given and compared to the mean plastic collapse stress of each IMPAXX density. From this, it can be

concluded that the contribution of the pressure in the cells of the foam can be neglected at this strain

level.

Table 3.6 - Contribution of inner gas pressure in cells at ε = εpl*

IMPAXX mean σpl* [MPa] mean εpl* [-] σg [kPa] σg / mean σpl* [-]

300 0,393 0,0167 1,793 0,0046

500 0,578 0,0168 1,805 0,0031

700 0,865 0,0185 1,996 0,0023

Nevertheless, if the foam is compressed further, one may expect the contribution of pressure build-up

in the cells may become of significant importance. Therefore, the same analysis is done at a strain of

50%. The stress level at this strain level is computed with

( )* * * *0,5pl pl pl

Eσ σ ε= + − (3.4)

From this, the results are placed in table 3.7.

22

Table 3.7 - Contribution of inner gas pressure in cells at ε = 0,5

IMPAXX σ* [MPa] σg* [MPa] σg* / mean σpl* [-]

300 0,562 0,1093 0,194

500 0,655 0,1098 0,168

700 1,004 0,1107 0,110

From this, it can be concluded that the stress build-up in the closed cells will give a significant stress

addition. The stress increase in the plateau region is mainly determined by the pressure build-up in the

closed cells. However, if cell walls are thin, they could fail at a low pressure levels and the

contribution of the pressure build-up in the closed-cells to the strength of the foam decreases.

3.1.3.3 - Strain-rate dependency of plastic collapse stress

As stated in section 3.1.3.2, the collapse stress of IMPAXX foams depends on strain rate, originating

from the intrinsic material behaviour of the solid polymeric cell walls. Assuming this dependency on

solid material properties, it can be validated if the behaviour of the plastic collapse stress of the foam

is related to the behaviour of the yield stress of PS.

For a double logarithmic scale, the plastic collapse stress as function of strain rate for different

IMPAXX foams is given in figure 3.9. The figure corresponds with a specimen diameter of 50 mm,

but the figure is equivalent for specimens with diameters of 25 and 75 mm. As can be seen, there are

drawn straight parallel lines through the measured data. These fitted lines can be formed analogous to

equation 3.6.

( ) ( )10 * 10log logpl a bσ ε= ⋅ +ɺ with constants a and b (3.6)

Figure 3.9 - Double logarithmic figure with the plastic collapse stress as function of strain rate for different IMPAXX

densities (300, 500 or 700) at a constant specimen diameter of 50 mm.

When these constants are known, the slope of the lines can be compared to that of polystyrene [5]. In

figure 3.10, the yield stress of polystyrene as function of strain rate is given on a double logarithmic

scale.

23

Figure 3.10 - Double logarithmic figure with the yield stress of PS as function of strain rate. [5]

The slope of the fitted line through the data for PS (assuming room temperature) is determined and

compared with that of the IMPAXX foam. These data is listed in table 3.8.

Table 3.8 - Slope of polyfit lines through data of PS and IMPAXX

Material PS IMPAXX 300 IMPAXX 500 IMPAXX 700

Specimen

diameter

[mm]

- 25 50 75 25 50 75 25 50 75

Slope of

line (a)

0.054 0.0348 0.0319 0.0221 0.0338 0.0358 0.0279 0.0383 0.0253 0.0224

It can be concluded from table 3.8 that the slope of the line becomes smaller with increasing specimen

diameter but is approximately the same for the different IMPAXX foams. To illustrate this, the mean

values (for all different densities of IMPAXX foams) of the gradients for the different specimen

diameters are shown in table 3.9. This implies that for larger specimens, the strain rate dependency for

the plastic collapse stress becomes less important.

Table 3.9 - Mean slopes for different specimen diameters

Specimen diameter [mm] 25 50 75

Mean Gradient 0.0356 0.0310 0.0241

Besides that, the gradient of the PS is above all gradients of the plastic collapse stress of IMPAXX. A

reason for this can be found by the choice for the solid cell wall material in this survey. On top of that,

a scale factor, owing to the complex microstructure of the foam, should be used to relate the

properties of the solid material to that of the foam.

24

3.2 - Johnson Controls foams

3.2.1 - Stress-strain behaviour

For the Johnson Controls Foams (JC foams), typical stress-strain responses are shown in figure 3.11.

The four lines represent the four open-cell foams with different densities. In here, a strain rate of 10 s-1

is applied on specimens with an average length scale of 50 mm. The responses are similar to the

schematic compressive stress-strain response for elastomeric foams, see also figure 1.1. The region of

elasticity is initially linear, but after a strain of approximately 4%, it’s concave downwards, till it is

followed (at a strain of approximately 10%) by a long elastic collapse plateau. At a strain of

approximately 60% densification starts. After compression, the foam specimen will (almost) fully

return to its original shape, so the deformation is fully elastic. This was exposed in paragraph 2.2.2.1.

Further on, the stress-strain response clearly shows a material dependent behaviour. Especially the

Young’s modulus and the elastic collapse stress will significantly differ for the various densities of JC

foams, as is obvious from figure 3.11. There is (almost) no expansion of the foam during

compression, so in the analysis, a Poisson’s ratio of zero can be assumed. Again, at the end of

deformation (at a strain of 60%), the stress relaxation can be seen.

Figure 3.11 - Stress-strain responses for JC foams with different densities (80, 90, 100 or 120) at a constant strain rate of 10

s-1 and a constant specimen length of 50 mm.

3.2.2 - Linear elasticity

3.2.2.1 - Prediction of Young’s modulus

In here, again the measured properties of the foam will be related to the solid polymer, which is in

here polyurethane (PU). As stated in 2.1.2, PU polymers can be made in many different ways, so they

cover a wide range of specific material properties. In here, the parameter values for (flexible) PU that

will be used are listed in table 3.10 [4]

Table 3.10 - Data PU

ρs [kg/m3] Es [MPa]

1200 45

25

After measuring specimens of the different JC foams, mean densities (over all specimens) are found

which are listed in table 3.11. Also the relative densities of the foams and the standard deviations of

the measured densities are given. Besides that, also the densities given by Johnson Controls Inc. are

stated. Note that these are approximately the same as the measured densities.

Table 3.11 - Measured densities for JC foams

JC ρ* [kg/m3] R [-] σstd [kg/m

3] ρ*JC Inc. [kg/m

3]

80 58,08 0,0484 0,62 58

90 58,85 0,0490 0,39 58

100 60,39 0,0503 0,86 60

120 62,21 0,0518 1,08 62

The Young’s modulus of this open-cell polymer foam can be estimated with equation 1.2. The results

are listed in table 3.12. Also the Young’s modulus of the foam, determined in the experiments, is

stated and the standard deviation is given. It must be noted that the modulus listed in here represents

the average modulus over all measurements, so with different specimen diameters and different strain

rates.

Table 3.12 - Young’s modulus for different JC foams

JC E*predicted [MPa] E*measured [MPa] σstd

[MPa]

80 0,1054 0,0909 0,0142

90 0,1082 0,0985 0,0163

100 0,1140 0,1553 0,0192

120 0,1210 0,2029 0,0253

Again, it can be concluded that the given analytical expression to determine the foam’s Young’s

modulus does give the right order of magnitude but is not very accurate for all JC foams. For JC 100

and JC 120, the predicted value is not accurate.

3.2.2.2 - Analysis of Young’s modulus

Similar to the experiments with IMPAXX foams a large number of experiments have been executed,

with different test set-ups. Foams of four different densities, four different strain rates and two

different specimen length scales were used. Therefore it’s, again, useful to check whether material

parameters, like the Young’s modulus, depend on these variables. As listed in table 3.12, the Young’s

modulus is clearly dependent on the density of the foam.

A phenomenon that also could influence the Young’s modulus is the specimen length. To investigate

this, the modulus is plotted as function of the four different densities and for different specimen

lengths at a constant strain rate of 10-1

s-1

. At other strain rates, equivalent plots will show up. This is

illustrated in figure 3.12. From this, it can be concluded that there is an influence of specimen length

on the Young’s modulus. The modulus seems to rise for larger specimen sizes. It should be noted that

the decrease of Young’s modulus for smaller length scales of the specimens can partially be due to the

fact that smaller specimens have been cut out of the larger specimens and are tested subsequently. But

the effect could also originate from size effects. One could also relate the specimen size dependency

of the Young’s modulus to the air flow phenomenon, but, as will be evaluated in 3.2.3.2, this is

unlikely because of the small effects of air flow for JC foams.

A similar evaluation can be done to investigate the influence of strain rate. This is plotted in figure

3.13. In here, the Young’s modulus is given at a constant length of 50 mm, but for a specimen length

of 25 mm this will be equivalent. From this, it can be concluded that the strain rate clearly influences

the Young’s modulus for this open-cell foam, originating from the intrinsic material behaviour of the

polymeric cell wall material.

26

Thus, it can be concluded that the Young’s modulus is determined by material parameters of the solid

cell walls and by the complex microstructure of the foam as well as by the specimen size and the

applied strain rate. Therefore, the region of linear elasticity should actually be called the ‘visco-

elastic’ region.

Figure 3.12 - Young’s modulus as function of JC density (80, 90, 100 or 120) for different specimen lengths at a constant

strain rate of 10-1 s-1.

Figure 3.13 - Young’s modulus as function of JC density (80, 90, 100 or 120) for different strain rates at a constant

specimen length of 50 mm.

27

3.2.3 - Elastic collapse

The second region in which some parameters can be found from experimental data, and can be related

to analytic expressions, is the plateau region. In this part, first the Plateau modulus Epl* of the foam

will be determined and it’ll be checked whether this modulus depends on test variables (strain rate,

foam density and specimen diameter). After evaluating the Plateau modulus, the elastic collapse stress

will be determined; analogous to the method described in 2.2.1.This is also checked on relations with

test variables. Besides that, the found collapse stress is compared with analytical expressions from

paragraph 1.2 and also the air flow phenomena will be analysed.

3.2.3.1 - Plateau modulus

The Plateau modulus is small in comparison with the Young’s modulus (< 10%). Typical values for

the Plateau modulus of JC foams are between 5 and 15 kPa. This modulus is dependent on the foam

density. This can be seen in table 3.13. The plateau modulus will slightly increase as function of foam

density.

Table 3.13 - Plateau modulus of different JC foams

JC Epl*[kPa] σstd [kPa]

80 6,4 2,5

90 8,9 3,4

100 8,6 2,3

120 11,6 4,0

In figure 3.14, the Plateau modulus is given as function of the JC foam density for different specimen

lengths at a constant strain rate of 1 s-1. The dependency of the Plateau modulus on specimen size is

small, i.e. differences between the two lines are in the range of the standard deviations. This figure is

equivalent for other strain rates.

Figure 3.14 - Plateau modulus as function of JC density (80, 90, 100 or 120) for different specimen lengths at a constant

strain rate of 1 s-1

.

28

In figure 3.15, the Plateau modulus has been given as function of foam density for different strain

rates (with a constant length of 50 mm). For a specimen length of 25 mm, an equivalent figure can be

shown, but larger standard deviations occur due to larger (relative) measurement errors. As can be

concluded from this figure, the Plateau modulus will rise as function of sample rate.

Thus, it can be concluded that the Plateau modulus of the JC foams is mainly influenced by the

material itself and by the strain rate at which the polymer foam is compressed.

Figure 3.15 - Plateau modulus as function of JC density (80, 90, 100 or 120) for different strain rates at a constant specimen

length of 50 mm.

3.2.3.2 - Elastic collapse stress

Now it is clear that the Young’s modulus is influenced by strain rate, specimen size and the material

itself, i.e. the relative density, the solid polymeric material and the complex microstructure, and that

the Plateau modulus is dependent on both the material and strain rate, it can be expected that also the

elastic collapse stress will be influenced by both the material itself and test parameters too. In figure

3.16 the typical relationship between the elastic collapse stress and the strain rate is shown at a

constant specimen length of 50 mm. The elastic collapse stress increases for larger strain rates. In

figure 3.17 the relationship for the elastic collapse stress with the specimen length is shown. In here,

the strain rate has a constant value of 10-2

s-1

, but the results are comparable with other strain rates. As

can be concluded from this figure, the elastic collapse stress is hardly influenced by the specimen

diameter. In general, the elastic collapse will slightly grow for larger specimen diameters, but for

higher strain rates this effect vanishes. So, no clear link is found between the collapse stress and strain

rate. The latter effect was also seen for the plastic collapse stress of IMPAXX foams.

29

Figure 3.16 - Elastic collapse stress as function of strain rate for different JC densities (80, 90, 100 or 120) at a constant

specimen length of 50 mm.

Figure 3.17 - Elastic collapse stress as function of specimen length for different JC densities (80, 90, 100 or 120) at a

constant strain rate of 10-2 s-1.

Similar to the closed-cell IMPAXX foams, the collapse stress versus the strain rate shows linear

parallel lines when depicted in a double logarithmic scale, as can be seen in figure 3.18. It is assumed

that these curves can be related to the stress-strain response of the solid cell wall material PU, but this

is not analysed in here.

30

Figure 3.18 - Double logarithmic figure with the elastic collapse stress as function of strain rate for different JC densities

(80, 90, 100 or 120) at a constant specimen length of 50 mm.

As stated before in chapter 1.2.2, 1.2.4 and 1.2.5.1, the elastic collapse stress will be influenced by

different phenomena, like air flow and viscous behaviour of cell wall material. With equations given

in chapter 1, the elastic collapse stress can be predicted. Also the added stress due to air flow can be

analysed. These equations are used here in order to determine if these analytical expressions can give

a proper prediction of the elastic collapse stress.

First of all, the elastic collapse stress can be predicted with the analytical expressions given by

equation 1.6. In here, no strain rate dependency is taken into account. The measured elastic collapse

stress is therefore averaged over all tests at different strain rates and different specimen lengths. This

is stated in table 3.14. From this, it can be concluded that this analytical expression only gives an

indication for the order of magnitude of the elastic collapse stress.

Table 3.14 - Predicted elastic collapse stress

JC mean σel, measured* [kPa] σstd [kPa] σel, predicted* [kPa]

80 5,512 0,586 5,271

90 6,450 0,811 5,402

100 9,494 1,209 5,693

120 12,064 1,587 6,037

Secondly, the stress addition due to (the resistance to) airflow in open-cell structures will be analysed.

For this, equation 1.19 was given. In here, again some assumptions have to be made. It can be easily

seen that for the highest strain rate and largest specimen length the effect of air flow will become

more important. Therefore a strain rate of 10 s-1 and a specimen length of 50 mm will be used in this

analysis. For the constant C5 in equation 1.19, a value of 1 is used. The air flow is analysed at a strain

at which elastic collapse starts. This point, εel* is different for each JC foam density and strain rate,

and its average value is stated in table 3.15.

The dynamic viscosity µ for air (at 20 °C) equals 18,27 · 10-6

Ns/m2, and the average edge length is

assumed to be 0.1 mm (estimated with the microscopic view of the JC foams, see figure 2.2). The

stress contribution due to air flow is compared to the mean elastic collapse stress of each JC foam

density.

31

From this, it can be concluded that the contribution of air flow in the cells of the open-cell foam is

small at this strain level. In here, it must be remarked that a number of simplifications is used, e.g. the

edge length will be smaller for foams with a higher density.

Table 3.15 - Contribution of air flow at ε = εel* with L = 50 mm and εɺ = 10 s-1

JC mean σel*

[kPa]

mean εel* [-] mean Eel*

[kPa]

σg* [Pa] σg */ σel* [-]

80 6,376 0,0589 7,838 48,53 0,0076

90 7,839 0,0653 10,884 48,86 0,0062

100 11,416 0,0625 11,468 48,72 0,0043

120 14,623 0,0610 13,234 48,64 0,0033

Nevertheless, if the foam is compressed further, one may expect the contribution air flow may become

of significant importance. Therefore, the same analysis is done at a strain of 50%. The stress level at

this point is computed with

( )* * * *0,5el el elEσ σ ε= + − (3.7)

The stress addition due to air flow equals 91,35 [Pa], using equation 1.19. With this, the results are

stated in table 3.16.

Table 3.16 - Contribution of air flow at ε = 0,5 with L = 50 mm and εɺ = 10 s-1

JC σ* [kPa] σg / mean σel* [-]

80 9,833 0,0093

90 12,570 0,0073

100 16,433 0,0056

120 20,433 0,0045

From this evaluation of the analytical expression (1.19), it can be concluded that the contribution of

air flow in the cells of the open-cell foam is small, even at this high strain level and with relative high

strain rates. In here, it must be remarked that a number of simplifications is used. Nevertheless, under

certain circumstances, e.g. smaller pore sizes, higher strain rates and larger specimen lengths, the air

flow can give a significant stress attribution.

32

Conclusions and discussion

As stated in the introduction, the macroscopic constitutive behaviour of polymer foams is determined

by a subtle interplay of the intrinsic constitutive behaviour of the polymeric material and the complex

microstructure. In this survey, an open-cell elastomeric foam and an elasto-plastic foam with closed-

cell structure have been tested in uni-axial compression tests, in order to determine the effect of

phenomena, such as flow of air through cells in foams and the influence of intrinsic material

behaviour. In order to give a clear overview of the main results and conclusions, the next part is

separated in two parts for the two different foams. Afterwards, a short discussion is reported.

Closed-cell elasto-plastic foam: IMPAXX Energy Absorbing Foams Increasing the relative density raises the foam’s Young’s modulus and plastic collapse stress. The

Young’s modulus is not significantly influenced by the specimen diameter or the applied strain rate,

so this parameter is fully determined by the interplay of the intrinsic behaviour of the polymer and the

microstructure (and thereby relative density). The Plateau modulus can slightly be influenced by

strain rate, but is mainly dependent on the complex microstructure. The effect of air compression in

closed cells attributes to the increasing stress level in the plateau region. It is supposed that the

pressure build-up in closed-cells mainly determines the Plateau modulus. Air flow can be neglected

for closed-cell foams, except when cell walls are thin and fail at low strains. The most interesting

effects show up at the plastic collapse stress. This parameter is influenced by strain rate rather well,

but it is assumed that this strain rate dependency originates from the intrinsic constitutive behaviour of

the polymeric material, because the effect of air compression in closed-cells is small at this point. The

specimen diameter doesn’t influence the plastic collapse stress. The material dependent behaviour of

the plastic collapse stress can be found in the double logarithmic plastic collapse stress-strain rate-

diagram, in which straight parallel lines can be drawn, equivalent to that of the solid cell wall

material, see 3.1.3.3.

Open-cell elastomeric foam: Johnson Controls foams. This foam was expected to exhibit both a strain rate and a specimen size dependent material

behaviour, associated with the intrinsic constitutive behaviour of the polymeric material and the air

flow phenomenon. From the executed tests, it was found that for this open-cell foam, the Young’s

modulus increases for larger strain rates and larger specimen sizes. The influence of specimen size

could be due to size effects, giving lower stress responses at smaller length scales. A larger foam

density raises the Young’s modulus, Plateau modulus and elastic collapse stress. The Plateau modulus

is also increased by raising the strain rate, but specimen size doesn’t seem to affect this parameter. For

the elastic collapse stress again no clear dependency on specimen size was found. This indicates that

for the used specimen lengths, stress additions by air flow remain constant. Evaluating analytical

expressions showed that air flow didn’t contribute significantly to the stress level. This could originate

from test parameters - specimen length and strain rate - or the complex microstructure. More dense

packed open-cell foams (smaller pore sizes) or fluids with higher viscosity (than air) will raise the

stress contribution instigated by fluid (air) flow. Above this, it must be noted that the air flow

phenomenon is complex to analyse; knowledge of the complex microstructure is helpful. Further on,

the elastic collapse stress is influenced by strain rate, originating from the intrinsic behaviour of the

polymer. Again, the material dependent behaviour shows up in the double logarithmic diagram, see

3.2.3.2.

Finally, it must be remarked that the macroscopic constitutive behaviour of polymer foams can be

described with use of a great number of experiments and varying foam density, strain rate and

specimen size. With help of electron microscopy and computer tomography, it could be useful to

analyse the complex microstructure before executing experiments. Analytical expressions in

literature, stated in chapter 1, are useful to predict the material behaviour of the polymer foams - if

properties of the polymeric material of which the solid cell walls are made, are known. With this,

materials can be selected that are useful to describe the phenomenon that has to be studied.

33

Besides that, one should notice that in practice deformation of foams almost never will be fully uni-

axial. For example, in a car crashes different deformation regimes act on the foam. To fully describe

the material behaviour of foams, tests with other loading conditions have to be done. The material

behaviour of foams will be different for other loading conditions. Shear and tensile tests can be done

by gluing the specimen to the specimen fixing.

34

References

[1] P.R. Onck, E.W. Andrews, L.J. Gibson. Size effects in ductile cellular solids. Part I: modeling.

International Journal of Mechanical Sciences 43, (2001) 681-699.

[2] E.W. Andrews, G. Gioux, P. Onck, L.J. Gibson. Size effects in ductile cellular solids. Part II:

experimental results. International Journal of Mechanical Sciences 43, (2001) 700-713.

[3] Nigel Mills, Polymer Foams Handbook – Engineering and Biomechanics Applications and Design

Guide. Butterworth-Heinemann (Elsevier); ISBN 978-0-7506-8069-1 (2007).

[4] Lorna J. Gibson and Michael F. Ashby. Cellular solids - Structure and properties (second edition).

Cambridge University Press, 2001; ISBN 0-521-49911-9.

[5] Harold G.H. van Melick. Deformation and failure of polymer glasses (proefschrift). Technische

Universiteit Eindhoven, 2002; ISBN 90-386-2923-0.

[6] A.K. van der Vegt and L.E. Govaert. Polymeren - van keten tot kunststof, page 135-139. VSSD,

vijfde druk 2003/2005; ISBN 90-71301-48-6

[7] Tech Data Sheet - IMPAXX TM

300 Energy Absorbing Foam, The Dow Chemical Company, can

be found at http://automotive.dow.com/materials/products/impaxx/product.htm

[8] Tech Data Sheet - IMPAXX TM

500 Energy Absorbing Foam, The Dow Chemical Company, can

be found at http://automotive.dow.com/materials/products/impaxx/product.htm

[9] Product Safety Assessment - IMPAXX TM Energy Absorbing Foam, The Dow Chemical

Company, can be found at http://automotive.dow.com

[10] Ian Clemitson, Castable Polyurethane Elastomers, CRC Press, ISBN 978142006576 (2008)