COMPOSITES

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Composites Science and Technology 66 (2006) 2709–2718

SCIENCE ANDTECHNOLOGY

Mechanical properties of high density polyurethane foams: II Effectof the filler size

Fabrice Saint-Michel, Laurent Chazeau *, Jean-Yves Cavaille

Groupe d’Etude de Metallurgie Physique et Physique des Materiaux, UMR CNRS 5510, Bat. Blaise Pascal, 69621 Villeurbanne Cedex, France

Received 9 August 2005; received in revised form 4 March 2006; accepted 14 March 2006Available online 5 May 2006

Abstract

This article presents the mechanical behaviour of rigid polyurethane reinforced by mineral fillers, for a density range (qf/qs) >0.27. The influence of the particle size on the foam microstructure and its mechanical behaviour has been studied using calciumcarbonate with two different sizes (with average diameter of 1 and 30 lm, respectively) and crystallised silica particles with anintermediate size. The microstructural characterisation has been performed by electron microscopy. It has shown closed sphericalcells whose size is decreased when the fillers are added. The results of mechanical characterisation by large and small deformation(mechanical spectrometry) tests have been compared to different modelling approaches. It appeared that a correct description ofthe viscoelastic properties and of the yield stress needs to take into account the filler size compared to the wall size. Two mod-elling hypothesis which consider the composite foam as either a filler dispersion in a foam or a void dispersion in a filled polymerhave been made. The first one is adapted to the description of PU filled with the big carbonate fillers, when the second one isbetter to describe the PU filled with the small carbonate fillers. The use of the crystallized silica particles gives composite foamswith properties in between that of the two calcium carbonate composites, showing that the filler size in the range studied is a keyparameter to take into account in the reinforcement level.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Polyurethane foam; A. Particle reinforced composites; B. Mechanical properties; B. Modelling

1. Introduction

Due to the presence of a skeleton made of cells (open orclosed) more or less regular [1,2] foams present very inter-esting mechanical properties such as a high capacity ofenergy absorption, particularly useful for shock damping.The mechanical response of these cellular materials dependon the architecture and the intrinsic properties of the poly-mer constitutive of the cell walls [1]. The parameters of thefoam architecture are the wall width, the size distributionand the shape of the cell.

For some years, their study have been extended to com-posite foams reinforced by the introduction of fibres [3] or

0266-3538/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compscitech.2006.03.008

* Corresponding author.E-mail address: laurent.chazeau@insa-lyon.fr (L. Chazeau).

metallic or mineral particles [4–7]. This modification of thepolymer matrix can lead to a better adsorption of the heatreleased during the polymerisation reaction, an increase ofthe matrix rigidity, with still a large capacity of energyabsorption, and an acceptable density. The granulometry,the surface state, and the filler type are important parame-ters governing the mechanical properties. The foam archi-tecture obviously depends on the filler dispersion, and thewall width compared to the particle size. Works of Goodset al. [5], have shown a more important increase of the elas-tic modulus when the foam density is higher. Moreover, themechanical properties of the foam being also governed bythe mechanical properties of the constitutive material, thereinforcement level of the foam depends on the interactionbetween the filler and the polymer matrix [4]. Eventually,filler addition can lead to an embrittlement of the cell walls

200

300

400

500

600

700

-10 10 30 50 70 90 110 130

0.63

0.27

0.01

0.1

1

Temperature (˚C)

ρ f /ρs

Tan

φ

(mW

/g)

Fig. 1. Tests by differential scanning calorimetry (a) and by DMA (b) forfoams reinforced by calcium carbonate. (a) T = �10 at 130 �C,DT = 10 �C/min. (b) f = 0.1 Hz, T = �10 at 130 �C, DT = 1 �C/min.

2710 F. Saint-Michel et al. / Composites Science and Technology 66 (2006) 2709–2718

and therefore favour their rupture instead of theirdeformation.

Theoretical works on foam have mainly been devoted tolow density foams. The foam structure is then modelled asa compact assembly of cells constituted of walls and struts.The most famous modelling to predict the mechanicalbehaviour of cellular materials is that of Gibson and Ashby[1]. For higher density foams, these materials are made ofclosed and isolated spherical cells, which lead to considerthem as porous materials. Then, the modelling can be per-formed using modelling developed for composite descrip-tion. For instance, Siegmann et al. [7], have correctlypredicted elastic properties of polyurethane foam usingthe Kerner equations [8] and the present authors [9] havemodelled polyurethane foams in the density range (0.3–0.85) using the Christensen and Lo [10] or Herve and Zaouiequations [11].

The accounting for filler presence is a supplementarydifficulty for the modelling of composite foam behaviour.For this purpose, with filled foam in the density range(0.3–0.6), Goods et al. [5] have combined the Gibson andAshby [1] modelling with the Kerner equation [8]. How-ever, their determination of the PU modulus from the Gib-son and Ashby modelling is questionable since the latterinvolves a parameter which is not justified by the micro-structural characterisation. Siegmann et al. [7] have alsostudied foam in the same density range, filled with varioustype of filler with size below 10 lm. They correctly mod-elled the linear behaviour of their materials using the Ker-ner equations [8] in two steps. However, they did notprovided microstructural characterisation of the foamand no discussion was done about the possible influenceof the filler size.

The aim of this article is to address this question by astudy of the mechanical behaviour of rigid polyurethanefoams reinforced by mineral particles with different sizes.The fillers mainly studied are calcium carbonate withtwo different granulometries. Another filler type, crystal-lised silica, with an intermediate size is also used. Themorphology of the obtained foams are characterised byscanning Electron Microscopy. Their mechanical charac-terisation associates large and small deformation tests.The experimental data are then compared to differentmodelling such as the Gibson and Ashby [1] and the2 + 1 phases of Christensen and Lo [10] (or Herve andZaoui [11]) modelling which is modified to account forthe filler size effect. Like in a previous article [9], asimulation of the non linear behaviour is alsoproposed.

Table 1Physico-chemical properties, elastic modulus and Poisson coefficient of minera

Mineral fillers Density (g/cm3) Specificsurfaces (m2/g)

d10 (lm) d50

Calcium carbonate C1 2.7 0.4 8.2 44Calcium carbonate C2 2.7 3.3 1.8 4.4Crystallised silica 2.35 1 0.3 10.4

2. Materials and experiments

2.1. Materials

2.1.1. Composite PU foam formulation

The composite foams are formulated from a polyolmatrix made of polyols, catalysts, a surfactant and mineralfillers. This matrix is mainly made of two polypropylenetriols (polyols) supplied by Shell Chemicals. The siliconesurfactant is supplied by Air Products and is a PDMS/POE copolymer. The two catalysts used are salts suppliedby Air Products. The catalyst A is a dibutyltin dilaurate(DBTDL). The catalyst B is a 2,2 0-[(dimethylstannyl-ene)bis(thio)]diacetate. Their role is to accelerate the reac-tion of chain polymerisation and regulate the reactionkinetic during the foam expansion.

The calcium carbonate fillers C1 and C2 are supplied byProvencale SA. They are made of crushed angular parti-cles. Their size have been characterised by laser granulom-etry with a Malvern apparatus. The size distribution of theC1 calcium carbonate shows a large polydispersity. Thedistribution spreads from 1 up to 200 lm. The size distribu-tion of the calcium carbonate C2 goes from 0.7 up to 20 lmand presents a peak at 4 lm. One can describe these twodistributions with the diameters d10, d50 and d90, which rep-resent the diameters for which x% in volume of particleshave a diameter below dx. These values, as well as the den-sities, the specific surfaces and the average diameter arereported in Table 1. The crystallised silica, supplied byLa Provencale SA, has also been used in the frame of thisstudy. Its size distribution is very large (from 0.1 up to

l fillers used

(lm) d90 (lm) Averagediameter (lm)

Modulus (GPa) Poisson coefficient

108.1 52 14 0.279.9 5.2 14 0.27

37.2 15 30.8 0.17

F. Saint-Michel et al. / Composites Science and Technology 66 (2006) 2709–2718 2711

100 lm) and bimodal with two peaks at 0.3 and 30 lm. Itcan be considered as an intermediate between the calciumcarbonate C1 and the calcium carbonate C2.

The foams are obtained by mixing the polyol matrixwith a polyisocyanate, diphenylmethanediisocyanate-4-4 0

(so-called MDI). Its reaction with the polyols products ure-thane bonds [12]. The water introduced with the polyolreacts with the MDI to produce carbon dioxide at the ori-gin of the foam expansion.

2.1.2. Processing

The initial water concentration being the main parame-ter regulating the final foam density, its estimation and itscontrol are extremely important. The components of thepolyol matrix (polyols, surfactant, catalysts) are mixed at600 rpm during 2 min with a mixer (Rayneri) with rotatingblades. Then this mix is dried on a 4 A molecular sieve for48 h to eliminate the residual water. The mineral fillers arekept in a drying oven during 24 h to eliminate wateradsorbed on the filler surface. For calcium carbonate thetemperature of drying is 200 �C. This temperature is only100 �C for the silica to avoid any modification of its surfacechemistry by a degradation of the silanol groups. A volumefraction /cc of calcium carbonate or of crystallised silica /s

is incorporated in the polyol matrix. This suspension ismixed at 1500 rpm for 10 min. All these mixings are per-formed under a nitrogen gas flux to avoid the introductionof water via the air humidity. The samples are thendegassed under vacuum during 24 h to eliminate air bub-bles introduced in the resin during the mixing.

Distilled water and the polyisocyanate are then added tothe polyol matrix. The mix (suspension/polyisocyanate) isthen stirred at 1500 rpm for 20 s, and is poured into a cylin-drical mould. The foam is obtained after free expansion inthis open mould. The expansion phase lasts about 3 min,then the gelation phase begins (polymerisation and cross-linking). The foam is then unmoulded after 30 min.

The block obtained is then cut in three pieces, two with45 · 20 · 30 mm dimensions and one with 80 · 30 · 40 mmdimensions. The measurement of the apparent density (Dr)is performed from the measurement of the volume and themass of this three blocks. Dr is defined as the ratio of thefoam density qf to that of the solid qs. This relative densityexpresses the volume fraction of matter in the foam. It wasimpossible to elaborate a material without bubbles andtherefore to measure qs. Indeed, the mixing of the suspen-sion with polyisocyanate introduces bubbles which areimprisoned during the polymerisation. qs has been esti-mated by calculation, considering the density of the differ-ent ingredients of the polyurethane foam, their quantity,and a mixing law. Taking a volume fraction of calcium car-bonate /�cc equal to 21.6% (in the polymer) the calculateddensity qs of the filled polyurethane is 1467 g l�1. For thepolyurethane filled with 21.6 vol% of crystallised silica /s,it is found qs = 1393 g l�1. Then, the relative densities ofthe different filled foams obtained can be calculated. Theyvary from 0.27 up to 0.75.

2.2. Experiments

2.2.1. Differential scanning calorimetry (DSC)

The apparatus to characterise the glass transition tem-perature of the samples is a Pyris Diamond from Perkin–Elmer. To erase the thermal history of the samples, a firsttemperature ramp is applied with a heating rate of 10 �C/min from �10 up to 130 �C. After cooling, a second rampis performed in the same conditions.

2.2.2. Electron microscopy

The characterisation of the cellular microstructure andof the cell size of the polyurethane foam is performed witha scanning electron microscope (SEM) JEOL 840 ALGS,with a filament tension of 10 kV. First, the samples are bro-ken in liquid nitrogen. Then the fracture surfaces are goldcoated to make them conducting. The micrographsobtained are treated by image analysis with the softwareScion Image to quantify the size distribution of the voidsin the foam. The apparent pore diameters are then deter-mined with the assumption of their sphericity. The Saltikovmethod [13] enables the determination of the distributionof the real diameter of the voids.

2.2.3. Mechanical tests

Preliminary tests on unfilled foams have shown an effectof physical ageing on their mechanical properties [9]. Thatis why a thermal treatment has been performed to be surethat the thermal history of each sample is the same. Beforeany mechanical test, the samples are kept 10 min at 80 �C(rejuvenation above Tg) then 24 h at 25 �C.

The elastic (G 0), and viscous (G00) moduli, and tan/have been measured by dynamic mechanical analysis onan apparatus Mecanalyser (Metravib S.A.) developed inthe laboratory [14]. It is an inverted torsion pendulumworking in forced harmonic regime, at low frequency,from 10�5 up to 5 Hz, in a temperature range from �170up to 400 �C. The samples tested are parallelepipeds of15 mm length, 5 mm width and 1.5 mm thickness. Theyare submitted to sinusoidal torsion, with controlled defor-mation at a frequency f = 0.1 Hz. A temperature ramp isperformed during the test from �6 up to 130 �C with aheating rate of 1 �C/min. Preliminary tests have shownthat humidity does not affect the mechanical propertiesof the foams.

Uniaxial compression tests have been performed on atensile test machine Instron 8510. The compression pla-teaus are maintained parallel by a guidance device with ballbearings. This machine possesses a thermo-regulated cham-ber to perform tests at temperature in between �90 and230 �C. The tested samples are cylinders with 10 mm diam-eter and 20 mm height (a compromise to avoid bucklingand barrel deformation) [15]. The samples are carefullyprepared so that their contact surfaces be strictly parallel.To decrease friction, they are polished and a lubricant isused (molybdenum bisulphur). Tests have been performedat 30 �C with a constant crosshead speed corresponding to

2712 F. Saint-Michel et al. / Composites Science and Technology 66 (2006) 2709–2718

an initial strain rate of 10�3 s�1. The results are presentedin nominal stress and strain.

3. Results and discussions

3.1. Physical characterisation

The foams have been analysed by DSC. For the unfilledfoams, we had measured a glass transition temperature (Tg)equal to 74 ± 2 �C [9]. As illustrated by the Fig. 1 for foamscontaining 21.6 vol% calcium carbonate C1, Tg is still equalto 74 ± 2 �C and is therefore not modified by the filleraddition.

Moreover, measurements performed by DynamicMechanical Analysis (Fig. 1) show a similar temperatureof the main relaxation, measured at the maximum of thetan/, equal to 83 �C whatever the density. In the case ofunfilled foam with low density, (qf/qs = 0.33) a supplemen-tary very wide relaxation had been detected at around10 �C [9]. The occurrence of this secondary relaxation hasbeen attributed to the formation of domains made of reac-tion sub-products. Indeed, urea, formed during the reac-

Fig. 2. SEM micrographs of foams reinforced by (a) calcium carbonate C1 (qf

carbonate C2 (qf/qs = 0.51, 1000·), (d) crystallized silica. (qf/qs = 0.65, 300·).

tion between the water and the isocyanate can then reactwith the isocyanate to form ureas. However, though theexistence of this supplementary relaxation, the very goodmodelling of the viscoelastic behaviour of the unfilled foamwith the assumption of no modification of the matrix indi-cated that this slight modification is negligible. For filledfoams, this secondary relaxation is not observed. The intro-duction of fillers enables a better absorption of the heatreleased during the polymerisation reaction. This couldinduce a decrease of the kinetic of formation of these by-products which normally occur at temperature above100 �C. From these results we can conclude that the differ-ent foams processed have a similar polymer matrix.

3.2. Morphological characterisation

Fig. 2 presents an example of SEM micrographsobtained for foams reinforced by both calcium carbonatesC1 and C2 and by the crystallised silica respectively. In thedensity range studied (0.27 < q*/qs < 0.75), the cells basi-cally appear spherical and closed. Whatever the filler typeand the foam densities, particles are correctly dispersed.

/qs = 0.31, 70·), (b) calcium carbonate C1 (qf/qs = 0.51, 300·), (c) calcium

106

107

108

109

1010

-10 10 30 50 70 90 110 130

0.620.27

103

104

105

106

107

108

Assumption 1

G' (

MP

a)

106

107

108

109

1010

G' (

MP

a)

G''

(MP

a)

103

104

105

106

107

108

G''

(MP

a)

Temperature (˚C)

-10 10 30 50 70 90 110 130Temperature (˚C)

G'

G''

ρf / ρ

s

0.750.35

Assumption 2

G'

G' '

ρf / ρ

s

(a) Foams with C1

(b) Foams with C2

Fig. 4. Evolution of elastic modulus (G 0) and viscous modulus (G 0) as afunction of temperature obtained by DMA for foams reinforced by:(a) calcium carbonate and (b) calcium carbonate C2. The values on thecurves represents the relative density. The lines represents the predictedvalues with the 2 + 1 phases modelling [10]. f = 0.1 Hz, T = �10 at130 �C, DT = 1 �C/min.

F. Saint-Michel et al. / Composites Science and Technology 66 (2006) 2709–2718 2713

The apparent cell size distribution as a function of the foamdensity was evaluated from image analysis. A calculationby the Saltikov method [13] allowed to deduce the real sizedistributions (Fig. 3). These results have been comparedwith those obtained with unfilled foams [9]. The filler addi-tion does not induce any modification of the cell morphol-ogy but a decrease of its size. In the density range studied,the cell size L, expressed in microns can be roughly fitted bythe linear equation:

L ¼ A� Dþ B; ð1Þwhere D is the foam relative density and A and B are twoadjustable parameters expressed in microns. By this linearleast square regression method, A is found equal to �76,�100, and �52 and B is found equal to 83, 88 and 59 forthe C2 filled PU, the C1 filled PU and the silica filledPU, respectively. Barma et al. [4] have also observed sucha decrease in the case of silica fillers. Conversely, Goodset al. [5] and Siegmann et al. [7] did not observe it. Goodset al. [5] have studied alumina filled foams. Three types offiller have been studied by Siegmann et al. [7]: graphite par-ticles, glass fibres and glass beads. Note that in all theseworks, the filler concentration, expressed in mass fraction,is below 30% , that means volume fraction below that ofour study. Moreover, their processing conditions are differ-ent, such as the mixing time of the filled suspension withthe isocyanate, the mould size, the ingredients, on whichthe cellular structure is very dependent. In our case, the de-crease of the cell size is probably due to a nucleating effectof the fillers. Moreover, this effect can be enhanced by theresidual water at the particle surface which induces a sup-plementary CO2 release. However, conversely to our expec-tation, the particle size does not influence the cell size. Thissuggests that the nucleating effect, if it exists, is astonish-ingly not proportional to the particle surface. The authorshave presently no explanation for this.

3.3. Mechanical properties

The evolution of the G 0 modulus as a function of tem-perature has been measured by DMA on the different sam-ples (Fig. 4). For a better clarity of the figures, only two

0

20

40

60

80

100

120

140with C2with C1with silicaunfilled

0.3 0 .4 0.5 0 .6 0.7 0 .8 0.9

Ave

rage

rea

l dia

met

er (

µm)

ρρf / ρ/ ρs

Fig. 3. Average real diameter of the cells foams as a function of relativedensity.

densities are presented on each one. The curves with differ-ent densities seem parallel and only shifted by a factor (inlogarithmic scale). The G 0 modulus measured at 30 �C asa function of the relative density is reported on theFig. 5. To evaluate the influence of the filler addition,experimental data obtained with the unfilled foams [9] arealso reported. Whatever the filler type, one can observe adecrease of the G 0 modulus with the decrease of the density.For C1 filled foams with relative density below 0.5, theaddition of fillers does not influence the modulus value.The increase of the modulus with the filler addition is onlyvisible for higher density, but this increase is relatively low.Conversely, the C2 filled foams present G 0 values abovethose of the unfilled foams and the C1 filled foams, what-ever the density. The silica filled foams with relative densitybelow 0.5 presents G 0 values in between those of the C1filled foams and those of the C2 filled foams. For higherdensity, their G 0 values tends toward those of the C2 filledfoams. These experimental results clearly show an effect ofthe filler size on the elastic behaviour of the filled foams:the modulus values increase when the average size of thefillers decreases.

Figs. 6a,b and 7 present the compression curves of PUfoams of different density filled with calcium carbonate

0

200

400

600

800

1000with C2with C1unfilledwith silica

0.3 0.4 0.5 0.6 0.7 0.8 0.9

G' (

MP

a)

0

200

400

600

800

1000

G' (

MP

a)

with φ = 0.89

with φ = 1.18

withφ = 1.03

unfilledwith silica

with C2with C1

Assumption (2)

Assumption (1)

ρρf / ρ/ ρs

0.3 0.4 0.5 0.6 0.7 0.8 0.9

ρρf / ρ/ ρs

(a)

(b)

Fig. 5. Evolution of elastic modulus (G 0) as a function of relative densityobtained by DMA. Unfilled and reinforced foams. (a) The lines representthe predicted values with the Gibson and Ashby modelling [1]. (b) Thelines represents the predicted values with the 2 + 1 phase modelling [10].f = 0.1 Hz, T = 30 �C.

0

5

10

15

20

25

30

35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Stre

ss (

MP

a)

0

5

10

15

20

25

30

35

Stre

ss (

MP

a)

Nominal strain

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Nominal strain

0.62

0.42

0.31

0.27

0.5

0.35

0.520.75 0.42

(a)

(b)

Fig. 6. Compression tests for different relative density filled foams:(a) foams reinforced by calcium carbonate C1 (b) foams reinforced bycalcium carbonate C2. The values on the curves represents the relativedensity. The dash lines represents the predicted values with the 2 + 1phases modelling [10]. Initial strain rate = 10�3 s�1, T = 30 �C.

2714 F. Saint-Michel et al. / Composites Science and Technology 66 (2006) 2709–2718

C1, calcium carbonate C2 and crystallised silica respec-tively. They are typical of that of a PU foam, with an elas-tic domain, followed by a densification plateau, ahardening domain and a final rupture. It is noticeable thatthe plateau occurs at constant strain level around 0.4. TheC2 filled foam appeared to be more brittle, as shown byinstabilities on the curve (Fig. 6b). All the curves show,at a given strain level, a decrease of the stress value withthe decrease of density. Compression elastic modulusof the different samples can be deduced from the slope atthe origin of their compression curve. However, due tothe foam nature of the material, the error made to estimateit is large. In fact this leads to an underestimation of thereal value. The important result is that they show the sametrend as the results with G 0 and lead to the same commenton the influence of the filler size. The yield stress (rf) of thedifferent filled foams can be more precisely estimated. Theirvalues are plotted as a function of the density for the differ-ent fillers on Fig. 8. They increase with the density. Fordensity below 0.6, the addition of C1 filler does not induceany significant change of the yield stress value. However,

one can observe that the addition of C2 calcium carbonateor of crystallised silica induces an increase of the rf valuefor foam density above 0.5.

3.4. Modelling of the mechanical behaviour

The most common modelling to predict the mechanicalbehaviour of cellular materials is the Gibson and Ashbyapproach [1]. In this modelling, the foam is described asan arrangement of cubic cells made of struts and walls. Itleads to a simple expression of the elastic modulus Gf ofthe foam as a function of its density (qf/qs), the solid frac-tion in the struts / and the mechanical properties of theconstitutive material (Gs), (in the present case the modulusof the filled polyurethane):

Gf

Gs

� /2 qf

qs

� �2

þ ð1� /Þ qf

qs

. ð2Þ

To obtain this equation, Gibson and Ashby [1] made theassumption that the Poisson coefficient is constant what-ever the density (mf = 1/3). For closed cell foam, / is in

0.65 0.43 0.360.55

0.65 0.43 0.360.55

0

5

10

15

20

25

30

35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Stre

ss (

MP

a)

Nominal strain

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Nominal strain

(a)

0

5

10

15

20

25

30

35

Stre

ss (

MP

a)

(b)

Fig. 7. Compression tests for different relative density filled foamsreinforced by crystallized silica. The values on the curves represents therelative density. The dash lines represents the predicted values with the2 + 1 phases modelling [10]: (a) assumption 1; (b) assumption 2. Initialstrain rate = 10�3 s�1, T = 30 �C.

F. Saint-Michel et al. / Composites Science and Technology 66 (2006) 2709–2718 2715

between 0.6 and 0.9. For open cell foam, / is equal to 1.This modelling is normally adapted to foam with relativedensity below 0.4.

0

5

10

15

20

25

30

35with C2

with C1

with silica

unfilled

0.3 0.4 0.5 0.6 0.7 0.8 0.9

σ f (M

Pa)

Assumption (1)

Assumption (2)

ρρf / ρ/ ρs

Fig. 8. Evolution of the stress value at the beginning of the plateau (rf) asa function of relative density obtained by compression tests. Unfilled andreinforced foams. The line represents the predicted values with the 2 + 1phases modelling [10]. Initial strain rate = 10�3 s�1, T = 30 �C.

An alternative approach considers the polyurethanefoam as a composite made of void inclusions in a polymermatrix. Using the 2 + 1 phases from Herve and Zaoui [11],it is possible to predict the elastic modulus and the yieldstress of unfilled foam as a function of the density [9]. Thisapproach models the material from a representative ele-mentary volume made of a void inclusion embedded in ashell of the polymer matrix, itself embedded in theunknown effective medium. In the case of calcium carbon-ate filled foams, the modelling must account for the fillerpresence. Moreover, the effect of the experimentallyobserved filler size must also be considered. Two differentassumptions, depicted on Fig. 9, have been tested toaccount for this effect. The first assumption (1) depictsthe filled foam as a filler dispersion in a foam then consid-ered as an homogenised medium. The filler volume fractionis related to the whole material, and has to be recalculatedas a function of the foam density. This leads to filler frac-tion values which are lower and lower when the densitydecreases. In this modelling, the first step is the calculationof the void fraction (the inclusions) in the material made ofthe unfilled foam (the matrix), and of its modulus. The sec-ond step is then the calculation of the modulus of this foam(the matrix) reinforced by the fillers (the inclusions). Thesecond assumption (2) considers the composite foam as avoid dispersion in a filled polymer. In this case, the fillerfraction is that in the polymer. It is constant and equalto that initially calculated, 21.4%. The first step calculatesthe filled PU modulus (the fillers are the inclusions andthe matrix is the PU), then the second step consists in cal-culating the modulus of this material (the matrix) filled witvoids (the inclusions).

The application of the equivalence principle of Dickie[16], used later by Hashin for the self-consistent scheme[17] , enables the extension of this modelling to the descrip-tion of the viscoelastic behaviour. It can be simply done byreplacing the elastic terms in the equations by their com-plex counterpart.

Filler

PU foam

Homogeneous medium

Void

PU matrix

Homogeneous medium

Assumption(1)

Void dispersion ina filled polymer

Filler dispersion in afoam

Step 1

Step 2 Void

Filled PU

Homogeneous medium

Filler

PU matrix

Homogeneous medium

Assumption(2)

Fig. 9. Schematic representation of homogenisation method used for the2 + 1 phases modelling [10]. Assumptions and steps used for the modelingof the filled foams.

0.3 0.4 0.5 0.6 0.7 0. 8 0.90

2

4

6

8

10

12

14

Cal

cula

ted

aver

age

cell

wid

th (

µm)

ρρf / ρ/ ρs

Fig. 10. Calculated width of the cell walls as a function of the density, forthe C1 filled foams (- - -), the C2 filled foams (– -– -), and the silica filledfoams (—).

2716 F. Saint-Michel et al. / Composites Science and Technology 66 (2006) 2709–2718

3.4.1. Linear domain

It was impossible to process polyurethane without bub-bles and therefore to measure the mechanical property ofthe constitutive polymer. The shear modulus of the poly-urethane polymer has been estimated from the dilutesphere modelling, which is equivalent to the 2 + 1 phasemodelling at low inclusion fraction and easier to invert[9]. The deduced value from the modulus of the 0.85 den-sity foam was found equal to Gs = 920 MPa. The value ofthe elastic shear modulus of the calcium carbonate filledpolymer has then been calculated from the 2 + 1 phasesmodelling, taking for the filler shear modulus and Poissoncoefficient, 14 Gpa and 0.27, respectively. It was foundGs = 1400 MPa and ms = 0.3. Taking 30.8 GPa and 0.17for the crystallised silica shear modulus and Poisson coef-ficient, the silica filled polymer modulus has been foundequal to 1455 Mpa, and its Poisson coefficient equal to0.3.

These values have been introduced in the Gibson andAshby equation [1] to calculate the composite foam modu-lus. The elastic shear modulus of the C1 and C2 filledfoams as a function of the relative density is correctly pre-dicted by the Gibson and Ashby [1] equation when / isequal to 1.03 and 1.18, respectively (Fig. 5a). For a closedcell foam, the / value is generally between 0.6 and 0.9 [1].When / = 1, the foam is made of open cells. However,electron microscope observations show closed cell foamfor all the relative density range studied (0.27 < qf/qs < 0.75). Thus, as for the study of unfilled foam [9]),the / parameter has no clear significance and must be con-sidered as an adjustable parameter. For instance, it isfound / = 1.07 for the foam containing the calcium car-bonate C2 when the density range considered is reducedbetween 0.35 and 0.42. Therefore, it can be concluded thatthis modelling is not adapted in the case of our material.

The modulus values of the composite foam calculatedfrom the 2 + 1 phase modelling applied in two steps arereported on Fig. 5b. For the C1 filled foams, the moduliare more correctly evaluated when the assumption (1) ischosen. Conversely, the moduli of the C2 filled foams aremore properly evaluated when the assumption (2) is made.However, in the latter case, one can see that the experimen-tal data tends to diverge from the theoretical fit and tobecome closer to the theoretical calculation with theassumption (1). The same figure presents theoretical datawith assumption (1) and assumption (2), and experimentalmodulus of the silica filled foam (the dash lines). In thedensity range studied, the experimental moduli are inthe transition zone between the assumption (1) fit and theassumption (2) fit.

It is possible to estimate the filler fraction value at whichthe particle size is equal to the width of the wall. With theapproximation of centred cubic arrangement of the voidsinto the foam, using the Eq. (1) giving the particle size asfunction of the relative density, the width of the cell walle (taken as the smallest distance between particles andexpressed in microns) can be written as:

e ¼ p3ð1� DÞ

� �13ffiffiffi3p

2� 1

" #½ADþ B�. ð3Þ

The calculated cell wall as a function of the density is pre-sented in Fig. 10 for the different filler types. This can becompared to the experimental average particle size of thedifferent fillers. As expected, in the density range studied,the C1 fillers are in average much bigger then the cell wallthickness of the C1 filled foams. Conversely, the cell wallthickness of the C2 filled foams are in the same range thanthe calculated cell wall thickness. This explains why onecan observe for the modulus value of these foams thebeginning of the transition from a value predicted withassumption 2 to one predicted with assumption 1. Con-cerning the crystallised silica, the calculation is less con-vincing since the average particle size is above that of thecalculated wall thickness. However, one must notice thatthe silica size distribution present two peaks, whose oneis below 1 lm, i.e. well below the calculated wall thickness.This probably explains the modulus evolution of the silicafilled foam as function of the density, which is always in be-tween the predictions with the two different assumptions.

The modelling has been extended to the viscoelasticbehaviour. Fig. 4 compares experimental data and theoret-ical curves and shows that the previous modelling, suitablein the glassy domain does not correctly predict the G 0

(Fig. 4a) and G00 (Fig. 4b) values at temperature aboveTg. It largely underestimates the moduli in the rubberydomain. However, it was previously shown [9] that thismodelling correctly predicts the G 0 modulus of the unfilledfoams in the whole density range and in the whole temper-ature range studied (from �10 up to 130 �C). One assump-tion to explain the underestimation observed in the case offilled foams is that the introduction of filler leads to anincrease of the resin crosslink density: an influence of thefiller in the foam processing is already evidenced by thechange of size distribution of the bubbles, as discussed pre-viously. Moreover, CaCO3 filler surface is highly hydro-philic and probably absorbs a large part of the wateradded in the formulation. This might modify the complex

F. Saint-Michel et al. / Composites Science and Technology 66 (2006) 2709–2718 2717

kinetic balance between the chemical reactions involved inthe CO2 release (and therefore the bubble formation) andthose involved in the resin polymerisation. Of course, thiscrosslink density modification is only detectable abovethe glass transition temperature (when the crosslink densityis the key parameter of the modulus level).

3.4.2. Non linear domain

Gibson et Ashby [1] have linked the foam yield stress(rf) to the yield stress of the constitutive material (rs) withthe assumption that the deformation is the result of thebuckling of the struts:

rf

rs

� 0:3 /qf

qs

� �32

þ 0:4ð1� /Þ qf

qs

. ð4Þ

This assumption of buckling is justified in the present studybecause of experimental results of tests of deformationrecovery. After heating treatment above the glass transitiontemperature of the foam, a total recovery of compressed C1filled polyurethane foam has been observed for deforma-tion level below 0.5. However, like in the linear domain,the use of such modelling with our material has been disap-pointing. In particular, the / parameter has to be adjustedwithout physical justification, and the yield stress rs de-duced from the same fit is much higher than the experimen-tal one.

In the case of composite materials, the mechanicalmodelling in the nonlinear domain, are often the extensionof the modelling used in the linear domain [18,19]. Besidesthe choice of the most appropriate modelling to describethe linear domain, the choice of a linearization method isneeded. This passage between linearity and non linearitycan be done using the tangent modulus, the secant modu-lus, or the affine method along the compression curves[20]. In this manner, the problem to solve is replaced bya symbolic problem of heterogeneous elasticity with initialdeformation. In Ref. [10], the authors tested the secantmethod, which did not provide a correct modelling of thenon-linear behaviour. However, in the same article, itwas found that a phenomenological approach based onthe 2 + 1 phase modelling correctly simulates the behav-iour of the unfilled foams. The more general question ofthe modelling of PU foam in the non-linear domain isnot in the frame of the present article and will be addressedin future publication. However, one can use this phenome-nological approach as a discussion basis of the experimen-tal results.

In this simulation, the assumption is made that thesecant modulus of the foam at a given strain is linked tothe secant modulus of the polyurethane matrix at the samestrain via the 2 + 1 phase modelling. One writes:

rf

ef ¼ em

¼ Frm

em

;qf

qs

� �ð5Þ

with F the 2 + 1 phase modelling applied to the matrixmodulus at the strain level em for a foam with a relativedensity qf/qs. The compression curve of the unfilled poly-

urethane is that calculated from the study of the unfilledpolyurethane foam in Ref. [10].

The first assumption (1) considering the filled foams as afiller dispersion in a foam, the first step consists in calculat-ing for each composite materials the compression curve ofthe foamed polymer matrix, with its void fraction related tothe material without the fillers. This uses Eq. (5). The sec-ond step calculates the compression curve of the filled foamfrom this calculated curve, with the filler fraction related tothe whole material using the following equation:

rfc

efc ¼ efð1� cÞ ¼ F 0rf

ef

; c� �

ð6Þ

with F 0 the 2 + 1 phase modelling applied to the foam mod-ulus at the strain level ef for a composite foam with a fillervolume fraction c, and a filler modulus and Poisson coeffi-cient taken from Table 1. Note that the non deformabilityof the filler is assumed by writing that the composite foamdeformation is (1 � c) times the deformation of the foam.For easier calculation, the Poisson coefficient has been ta-ken constant whatever the deformation since it has beenchecked that its value m, when between 0.35 and 0.4999,has no significant influence on the results.

The second assumption (2) considers the compositefoam as a bubble dispersion in a filled polyurethane. Inthe first step, the compression curve of the filled matrix iscalculated from the unfilled polyurethane compressioncurve and the 2 + 1 phases model. The inclusion is the fillerand the assumption is again used that the fillers do not par-ticipate to the composite deformation. The inclusion frac-tion is the filler fraction in the polymer and is equal to21.6%. From the compression curve of this filled polyure-thane and the 2 + 1 phase model, the stress-strain curvesof the filled foam are then deduced with the voids consid-ered as the inclusion (Eq. (5)).

The results of these calculations with the assumption(1) for all the densities studied are compared to the C1filled foam stress–strain curve. As shown on Fig. 6a,for strain level above 0.4, the modelling overestimatesthe stress level. Such an overestimation was alsoobserved when modelling the unfilled foam stress–straincurves [9]. However, there is a good agreement betweenexperimental yield stress and calculated ones (Fig. 8).For C2 filled foams, conversely to the results in the lin-ear domain, the modelling overestimates the stress levelwhatever the density range (Figs. 6b and 8). The yieldstress is in fact more properly described with the assump-tion (1). These results might be explained by the impor-tant brittleness of the C2 filled foam that is not takeninto account in the simulation approach. Indeed, thisbrittleness has been observed during the compressiontests. In the C1 materials, some struts in the foam archi-tecture do not contain filler (as shown on Fig. 2). There-fore, some plastic strain localisations are more likely tooccur. This explains their better toughness compared tothe C2 filled foams, in which all the struts can be embrit-tled by the fillers they contain.

2718 F. Saint-Michel et al. / Composites Science and Technology 66 (2006) 2709–2718

As expected, the modelling of the silica filled foams leadsto curves, which, either overestimate or underestimate theexperimental stress-strain curves, depending on the modelassumptions.

4. Conclusion

In the density range studied, the viscoelastic propertiesof polyurethane foam reinforced by a 21.6% volume frac-tion of mineral fillers such as calcium carbonate and crys-tallised silica can be correctly simulated as far as themodelling takes into account the filler granulometry.Indeed, depending on the particle dimension compared tothe average wall thickness, these composite foams can bemodelled with the following assumptions:

(i) when the fillers have a large size, these foams are con-sidered as a filler dispersion in an homogeneised med-ium with the foam characteristics.

(ii) When the fillers have small sizes compared to the wallthickness, the composite foams are considered as abubble dispersion embedded in a filled polyurethane.

From a practical point of view, the results shows thatthe reinforcement of foam (in the non linear domain) isnot efficient if the filler added are bigger than the bubblesize. Moreover the comparison with the model above Tg

suggests that the fillers lead to an increase of the crosslinkdensity of the matrix.

In the non-linear domain, in spite of its roughness, themethod proposed leads to a correct simulation of the curveof the C1 filled foams for deformation below 0.5. More-over, it enables to differentiate the silica and the C2 filledfoams from the C1 filled materials. However a deeperexperimental and theoretical work is needed to correctlysimulate the large deformation behaviour. For instance,mechanisms of deformation localisation, as well as thedevelopment of the microstructural anisotropy during thedeformation and the evolution of the void fraction haveto be addressed and taken into account in the mechanicaldescription. These points are currently studied in the labo-

ratory, in the frame of a GDR ‘‘solid foam’’ [21] and will bethe topic of future publications [22].

Acknowledgements

The authors acknowledge S. Youssef for his contribu-tion to the experimental work and G. Vigier for fruitfulldiscussions.

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