We investigate the shear elastic modulus of soft polymer foams loaded with
hard spherical particles and we show that, for constant bubble size and gas
volume fraction, strengthening is strongly dependent on the size of those
inclusions. Through an accurate control of the ratio $\lambda$ that compares
the particle size to the thickness of the struts in the foam structure, we
evidence a transition in the mechanical behavior at $\lambda \approx 1$. For
$\lambda < 1$, every particle loading leads to a strengthening effect whose
magnitude depends only on the particle volume fraction. On the contrary, for
$\lambda > 1$, the strengthening effect weakens abruptly as a function of
$\lambda$ and a softening effect is even observed for $\lambda \gtrsim 10$.
This transition in the mechanical behavior is reminiscent of the so-called
"particle exclusion transition" that has been recently reported within the
framework of drainage of foamy granular suspensions [Haffner B, Khidas Y,
Pitois O. The drainage of foamy granular suspensions. J Colloid Interface Sci
2015. In Press.]. It involves the evolution for the geometrical configuration
of the particles with respect to the foam network, and it appears to control
the mechanics of such foamy systems

Haffner, Benjamin

Khidas, Yacine

Pitois, Olivier

English

arXiv

Cette étude porte sur le renforcement de matériaux moussés par des inclusions rigides. Pour cela nous avons mesuré le module de cisaillement d’une mousse de polymère chargée avec des sphères rigides. Nous montrons que le renforcement est fortement dépendant de la taille de ces inclusions. En contrôlant précisément le paramètre géométrique λ qui compare la taille des particules à celle des constrictions dans le réseau de la mousse, nous avons mis en évidence une transition des propriétés mécaniques autour de λ ≈ 1. Pour λ 1, ce renforcement diminue rapidement avec λ et on observe même un ramollissement lorsque λ ≳10. En partant d’une approche de milieu effectif pour décrire l’élasticité d’une matrice élastique avec inclusions rigides, nous proposons un modèle phénoménologique pour les comportements asymptotiques λ > 1 et nous obtenons un bon accord avec nos données expérimentales

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22
ème
 Congrès Français de Mécanique                                               Lyon, 24 au 28 Août 2015 
 
 
Strengthening of foamed composite materials 
 
O. Pitois, Y. Khidas and B. Haffner 
Université Paris Est, Laboratoire Navier, UMR 8205 CNRS – École des Ponts ParisTech – 
IFSTTAR, cité Descartes, 2 allée Kepler, 77420 Champs-sur-Marne, France 
 
Résumé :  
Cette étude porte sur le renforcement de matériaux moussés par des inclusions rigides. Pour cela nous 
avons mesuré le module de cisaillement d’une mousse de polymère chargée avec des sphères rigides. 
Nous montrons que le renforcement est fortement dépendant de la taille de ces inclusions. En 
contrôlant précisément le paramètre géométrique λ qui compare la taille des particules à celle des 
constrictions dans le réseau de la mousse, nous avons mis en évidence une transition des propriétés 
mécaniques autour de λ ≈ 1. Pour λ <1, toute charge particulaire renforce la mousse et l’ampleur de 
ce renforcement dépend directement de la concentration en particules. A l’inverse, pour λ >1, ce 
renforcement diminue rapidement avec λ et on observe même un ramollissement lorsque λ ≳10. En 
partant d’une approche de milieu effectif pour décrire l’élasticité d’une matrice élastique avec 
inclusions rigides, nous proposons un modèle phénoménologique pour les comportements 
asymptotiques λ <<1 et λ >> 1 et nous obtenons un bon accord avec nos données expérimentales. 
 
Abstract:  
We consider the strengthening of foamy materials by hard inclusions. We investigate the shear elastic 
modulus of soft polymer foams loaded with hard spherical particles and we show that strengthening is 
strongly dependent on the size of those inclusions. Through an accurate control of the ratio λ that 
compares the particle size to the thickness of the struts in the foam structure, we evidence a transition 
in the mechanical behavior at λ≈1. For λ<1, every particle loading leads to a strengthening effect 
whose magnitude depends only on the particle volume fraction. On the contrary, for λ>1, the 
strengthening effect weakens abruptly as a function of λ and a softening effect is even observed for 
λ≳10. From an effective approach which describes the elastic moduli of an elastic matrix loaded with 
rigid inclusions, a phenomenological model is proposed to describe each asymptotic behavior λ <<1 
and λ >> 1 and which is in good agreement with our experimental data 
 
Mots clefs : mousse ; composite ; particule ; module élastique ; polymère  
 
1 Introduction  
 
The incorporation of particles into materials is a well-known strategy to improve their mechanical 
behavior. When dealing with foamy materials, one has to consider the typical size of the struts that 
form the solid skeleton compared to the filler size. Several studies have been devoted to the 
reinforcement of such foams. For example, polyurethane foams or aerated cementitious materials were 
loaded with different kind of fillers [1,2]. In the current climate of sustainable development, those 
foam-based materials are destined to expand and in certain cases to replace advantageously 
conventional building materials. Numerous works were focused on those materials in order to improve 
22
ème
 Congrès Français de Mécanique                                               Lyon, 24 au 28 Août 2015 
 
their mechanics. One of the essential features of foamy composite materials is that the filler size has to 
be fine enough for strengthening to be observed, and it has been shown that reinforcement is not 
efficient if the added particles are bigger than the bubbles [1]. Surprisingly, whereas this filler size 
effect is expected to be based on geometry only, it has never been studied from the purely geometrical 
point of view. Indeed, the bubble size and the gas volume fraction cannot be kept constant when 
studying the effect of the loading, which increases the difficulty to disentangle all types of effect. 
In this study, we propose a generic approach to better understand the subtle interplay between the 
complex geometry of particulate foams and their mechanics. We investigate the shear elastic modulus 
of systems which consist of soft polymer foams loaded with hard spherical particles. The experimental 
study is conducted in such a way that monodisperse precursor foams are unaltered by the loading 
process, allowing us to assess the strengthening effect due solely to the presence of the particles. 
 
2 Materials and methods  
 
Our model system is a soft polymer foam loaded with hard spherical inclusions. The soft polymer is a 
gelatin. Above Tgel≈ 29°C gelatin is a liquid solution and below it becomes a soft elastic solid. The 
inclusions are polystyrene beads characterized by a week polydispersity.  
Inside a box above Tgel, the liquid gelatin solution is mixed with nitrogen in order to produce a 
precursor monodisperse foam. The gas fraction of this foam is set by controlling the flow rates of the 
liquid and the gas. In a second step, a suspension of particles in liquid gelatin is pushed and mixed to 
the precursor foam. The resulting particulate foam is pushed in a transparent shear cell and we rapidly 
quench the system at a temperature T = 5°C. 
In the present study, the gas volume fraction of the particle-laden foam and the bubble diameter are 
kept constant with respective values: ϕ = 0.8 and Db = 400 µm. The particle size dp varies from 6 to 
500 µm. Their volume fraction in the whole foam ϕp varies from 0 to 6%, which gives with respect to 
the foam skeleton: φp = ϕp/(1- ϕ) = 0 - 30%. 
Mechanical properties of our samples are characterized by the shear elastic modulus measured in a 
plane shear test. All the results presented in this paper are performed at the constant shear rate = 
1.3·10
-3
 s
-1
. Thanks to the transparent cell, we measure the surface area of each sample, allowing for 
the stress to be plotted as a function of the strain. We remain in the linear elastic regime and the shear 
modulus G(ϕp,dp) can be simply deduced from the slope of the linear fit over a strain of 2.6 %. 
 
3 Results and discussions  
 
The influence of the particle loading is characterized by normalizing the shear modulus of particle-
laden foams by the value for the corresponding particle-free foam, i.e. G*=G(ϕ,φp )/G(ϕ,0). Fig 1 
shows the evolution of the normalized shear modulus as a function of the particle volume fraction for 
several particle sizes, where two behaviors can be distinguished. The shear moduli of foams loaded 
with particles of diameter below 40 µm follow the same increasing function of ϕp. When the particle 
size increases, the stiffening can become negligible, as observed for the 250 µm particles. We can 
even notice a weak softening effect for the 500 µm particles. For that case, note that the particle size 
becomes comparable to the bubble size. 
22
ème
 Congrès Français de Mécanique                                               Lyon, 24 au 28 Août 2015 
 
 
Fig. 1: The shear modulus of particle-laden foams normalized by the modulus of the corresponding 
particle-free foam as a function of the particle volume fraction φp in the interstitial volume or the 
corresponding fraction ϕp in the whole sample volume. 
 
To understand this stiffening-softening transition, it is necessary to consider the foam at the scale of 
the solid skeleton, or foam network. The size of interest is dc, the size of passage through constrictions 
within the foam pore space. In order to compare the particle size with dc, we will use the so-called 
confinement parameter λ=dp/dc. From [3] and for ϕ=0.8, λ ≃ 8.5 dp / Db. When λ < 1, particles can 
explore freely the whole interstitial space whereas when λ ≳ 1, they have to be forced to enter the 
constrictions in deforming the bubbles surface. One can also imagine that for λ ≫ 1 the particles 
cannot be included anymore in the space defined by the foam network but instead the network is 
restructured around them. Thus, for a given particle volume fraction, the resulting particle 
configuration in the foam network depends crucially on λ, which is expected to have a significant 
influence on the mechanical behavior. 
To quantify this influence, let’s first consider the asymptotic case of small particles, i.e. λ≪1. The 
corresponding shear modulus will be called G*max. In one hand, the elastic modulus of solid foams 
normalized by the modulus of the solid skeleton is a decreasing function of the gas volume fraction: 
G(ϕ)/G(0) = f(ϕ). On the other hand, the modulus of an elastic matrix with rigid inclusions normalized 
by the modulus of the matrix is an increasing function of the particle fraction g(φp). Thus, when the 
particles are small enough, the stiffness of the elastic skeleton is set by the particle volume fraction it 
contains and the reduced shear modulus of particulate foams can be written as a combination of those 
two functions and one finally obtains: G*max= g(φp). We make use of the Krieger-Dougherty equation 
to fit our data as suggested by previous studies on the elastic modulus of yield stress fluids loaded with 
particles [4]: 
 
     
*5.2**
max 1
p
pppp gG



      (1) 
 
where φp*≈ 0.6. This phenomenological function can be used to estimate G*max(φp ), which is found to 
be in good agreement with experimental data for particles smaller than 40 µm, i.e. λ ≤ 0.84, as shown 
in Fig 1.  
Let’s now consider the case of G*min, which corresponds to the case of very large particles compared 
to the size of constrictions, i.e. λ≫1. That situation should be understood as particles large enough to 
be fully excluded from the space defined by the solid network. In such a case, the system is composed 
of particle-free foam that embeds large inclusions. As the particles no more contribute to the volume 
22
ème
 Congrès Français de Mécanique                                               Lyon, 24 au 28 Août 2015 
 
of the network, the gas fraction of the embedding foam, ϕ', is therefore increased with respect to the 
gas fraction in the whole system, ϕ. The relation between ϕ' and ϕ is: ϕ'=ϕ/(1- ϕp). In combining the 
effect of ϕp on ϕ', with the effect of gas fraction on the shear modulus of foams, i.e. f(ϕ), one can 
deduce that the shear modulus of the embedding foam decreases as ϕp increases. Consequently, two 
opposing effects interact for particle-laden foams characterized by λ≫1: the particles stiffen the 
embedding foam whose shear modulus is decreased with respect to the corresponding unloaded foam. 
The global behavior of G*min depends on the magnitude of those two effects. Similarly to what has be 
done for G*max, one can write:                0,0,00,0,0,0,0,,  GGGGGGGG pppp  . 
Therefore, the corresponding global shear modulus is: 
 
   
 
 



f
f
gG
p
pp


1*
min      (2). 
 
The function f has been determined from measurements on particle-free gelatin foams. Fig1 illustrates 
that eqn 2 predicts the global softening of the foamy material, in spite of the presence of solid 
inclusions. 
 
4 Conclusion  
 
We have addressed the issue of the strengthening of foamy materials by hard inclusions. The shear 
elastic modulus of particle-laden polymer foams has been measured as a function of the particle 
loading, in such a way that the ratio λ of particle size to the foam network typical size was controlled. 
This geometric parameter was found to have a crucial influence on the mechanical behavior. For λ<1, 
every particle loading leads to a strengthening effect whose magnitude depends only on the particle 
volume fraction. On the contrary, for λ>1, the strengthening effect weakens abruptly as a function of 
λ, and a softening effect was even observed for λ≳10. This transition in the mechanical behavior of 
foamy composite materials has been interpreted in terms of the evolution for the particles 
configuration in the foam network.  
 
Références 
[1] S.H. Goods, C.L. Neuschwanger, L.L. Whinnery, W.D. Nix, Mechanical properties of a particle-
strengthened polyurethane foam, J. Appl. Polym. Sci. 74 (1999) 2724; F. Saint-Michel, L. Chazeau, J.-
Y. Cavaille, Mechanical properties of high density polyurethane foams: II Effect of the filler size, 
Comp. Sci. Technol. 66 (2006) 2709 
[2] T.D. Tonyan and L.J. Gibson, Structure and mechanics of cement foams, J. Mater. Sci. 27 (1992) 
6371 ; M.R. Jones and A. McCarthy, Preliminary views on the potential of foamed concrete as a 
structural material  Mag. Concr. Res. 57 (2005) 21 
[3] N. Louvet, R. Höhler and O. Pitois, Capture of particles in soft porous media, Phys. Rev. E 82, 
041405 (2010) 
[4] F. Mahaut, X. Château, P. Coussot and G.Ovarlez, Yield stress and elastic modulus of suspensions 
of noncolloidal particles in yield stress fluids, J. Rheol. 52 (2008), 28 
 
 


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