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Dépôt légal à parution 


Katell Derrien

Léo Morin

Pierre Gilormini

Designing isotropic composites reinforced by aligned transversely isotropic particles of spheroidal shape

Azdine Nait-ali

C. R. Mecanique 346 (2018) 1192–1198Contents lists available at ScienceDirect
Comptes Rendus Mecanique
www.sciencedirect.com
Stochastic homogenization of reinforced polymer with very 
small carbon inclusions
Azdine Nait-ali
Institut Pprime, UPR CNRS 3346, Département “Physique et mécanique des matériaux”, ENSMA, Téléport 2, 1, avenue Clément-Ader, BP 40109, 
86961 Futuroscope Chasseneuil cedex, France
a r t i c l e i n f o a b s t r a c t
Article history:
Received 15 June 2018
Accepted 26 September 2018
Available online 25 October 2018
Keywords:
Asymptotic analysis
-convergence
Ergodic theory
In this study, we wish to determine a homogenized model of a material reinforced 
by spherical inclusion that is randomly distributed in space. The method used for the 
transition to the limit is -convergence [1] in the stochastic case. In addition to the 
stochastic framework, the very small size compared to the characteristic size of the 
materials makes the homogenization procedure unconventional. In this study, we want 
to determine a homogenized model of a material reinforced by a spherical inclusion 
distributed randomly in space. The peculiarity here is that these particles are of very 
small size, this generating an energy due to the strong contrast of microstructure. The 
method used for the transition to the limit is -convergence [1] in the stochastic case. The 
random distribution is taken into account during the transition of scales, so as to preserve 
the statistical information, and that in spite of the passage to the limit. In addition to 
the stochastic framework, the very small size compared to the characteristic size of the 
materials makes the homogenization procedure unconventional.
© 2018 Académie des sciences. Published by Elsevier Masson SAS. This is an open access 
article under the CC BY-NC-ND license 
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
We want here to obtain a homogenized model of a material reinforced by very rigid inclusions and distributed randomly. 
The model material considered as an example is a reinforced polymer with very small carbon inclusions; by the manufac-
turing method generating random distributions, as well as the strong contrast of property, this material seems to be a very 
good frame of application for this study. Indeed, the microstructural polymers have a certain characteristic length (denoted 
by ε), and to improve their resistance, it is possible to introduce very small carbon charges of radius r << ε. This charge 
is therefore distributed randomly in the material (see Fig. 1). We make the (realistic) hypothesis of an ergodic distribution 
(see [2–4]). In the case of high densities of inclusion, the ergodic side seems to be appropriate.
2. Notations and basic notions
We consider an open bounded cube O = Ô× (−L/2, L/2) of R3, where Ô is a bounded open interval of R2. We write ε
to denote a sequence of positive real numbers intended to go to 0 and chosen in order to satisfy |Ô \ ⋃z∈Iε ε(Y + z)| = 0, 
where Iε is a finite subset of Z3, and we define Y ∈ R2 the reference surface cell; this domain will be considered here as 
unitary.
E-mail address: azdine.nait-ali@ensma.fr.https://doi.org/10.1016/j.crme.2018.09.002
1631-0721/© 2018 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license 
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
A. Nait-ali / C. R. Mecanique 346 (2018) 1192–1198 1193Fig. 1. Example of carbon distribution.
Given rε << ε, we consider the open ball
Brε (εω0) :=
{
x ∈R : |x − ω0,ε| < rε
}
centered at the center ω0,ε of εY and set, for each ω0,ε + εz, z ∈ Iε , as
Drε :=
⋃
z∈Iε
Brε (ωz,ε)
The boundary of each ball Brε (ωz,ε) is denoted by Srε (ωz,ε). At the same time as the scale transition, we are going to move 
the Y domain so as to cover the entire structure under study. For stochastic homogenization to work, we therefore need 
this domain Y to always include at least one inclusion. This is why we need to use the ergodicty hypothesis. The union of 
the ball is Brε .
For Rε satisfying rε << Rε << ε, we also define in the same way the disks B Rε (ωz,ε), their boundary S Rε (ωz,ε), their 
union B Rε , and the union of the ball B Rε . We finally denote by Crε Rε (ωz,ε) the ring
B Rε (ωz,ε) \ Brε (ωz,ε)
Given aε > 0 satisfying aε|Bε| → a, i.e. kε := aε r
2
ε
ε2
→ k with kπ |Ô| = a, we consider the energy functional defined in 
W 1,p(O) by
Fε(u) =
ˆ
O\Brε
f (∇u) dx + aε
ˆ
Brε
f (∇u) dx
that we sometimes write
Fε(u) =
ˆ
O\Brε
f (∇u) dx + aε|Bε|
 
Brε
f (∇u) dx (1)
where f :R3 → R is a convex function satisfying the standard growth condition of order p > 1
α|ξ |p ≤ f (ξ) ≤ β(1 + |ξ |p) (2)
for two given constants α > 0 and β > 0. Let Rε satisfying rε << Rε << ε. Following [3,5], we split the functional Fε into 
three terms:
Fε(u) =
ˆ
O\B
f (∇u) dx +
ˆ
B \B
f (∇u) dx + aε|Bε|
 
B
f (∇u) dx
Rε Rε rε rε
1194 A. Nait-ali / C. R. Mecanique 346 (2018) 1192–1198We assume the following control between f and its p-recession f ∞,p defined by f ∞,p(ξ) = lim
t→+∞
f (tξ)
|t|p : there exists 
β ′ > 0, 0 < γ < p such that, for all ξ ∈R3
∣∣ f (ξ) − f ∞,p(ξ)∣∣ ≤ β ′(1 + |ξ |p−γ ) (3)
From (2), we infer that f ∞,p satisfies for all ξ ∈ R3
α|ξ |p ≤ f ∞,p(ξ) ≤ β|ξ |p (4)
and, since clearly f ∞,p is a positively p-homogeneous convex function, its satisfies the local Lipschitz condition
| f ∞,p(ξ) − f ∞,p(ξ ′)| ≤ L|ξ − ξ ′|(|ξ |p−1 + |ξ ′|p−1) (5)
For more details see [6].
Definition 2.1. Given two concentric disks Br(ω) ⊂ B R(ω) of R3 with arbitrary center ω, we define the capacity of Br(ω) in 
B R(ω) associated with the function f by
Ener fr,R = inf
⎧⎪⎨
⎪⎩
ˆ
B R (ω)
f ∞,p(∇w) dx : w ∈ W 1,p0 (B R(ω)) : w = 1 in Br(ω)
⎫⎪⎬
⎪⎭
Note that Ener fr,R does not depend on the choice of the center ω of the disks. We assume the following limit behavior 
of Ener frε,Rε
lim
ε→0
Ener frε,Rε
ε2
= γ (6)
with γ ∈ [0, +∞[.
3. Technical lemmas
With each sequence (uε)ε>0 of functions in W 1,p(O) we associate the sequence (vε)ε>0 of the functions in Lp(O)
defined by vε = |O||Brε |1Brε uε . Recall the following compactness property (see [3,5,7,8]).
Lemma 3.1. Let (uε)ε>0 be a sequence of finite energy, i.e. satisfying supε>0 Fε(uε) < +∞. Then there exist (u, v) in W 1,p(O) ×
Lp(Ô, W 1,p(0, π)) and a subsequence not relabeled such that
uε → u W 1,p(O)
vε
∗
⇀ v in Lp(Ô, W 1,p(0,π))
+ boundary conditions
We denote by Dom the subset of W 1,p(O) × Lp(Ô, W 1,p(0, π)) made up of functions satisfying the boundary conditions.
We are going to establish the -convergence of the sequence (Fε)ε>0 in Lp(O) equipped with its strong topology to the 
functional F0 defined in W 1,p(O) by F0(u) = infv∈Lp(O) G(u, v), where the bifunctional G : Lp(O) × Lp(O) →R+ ∪ {+∞} is 
given by
G(u, v) :=
⎧⎪⎨
⎪⎩
ˆ
O
f (∇u) dx + γ
ˆ
O
|u − v|p dx + a
ˆ
O
f ∞,p( ∂v
∂ψ
) dx if (u, v) ∈ Dom
+∞ otherwise
The second term corresponds to a non-local energy (see [6,9]). This induced energy of influencing to long range in-
clusions. Therefore, the constant γ would be a characteristic length. This size depends here on the material, which is not 
always the case in the literature. In fact, in some works, this length corresponds either to a regularization parameter ([10]
and [11]) or it depends on a macroscopic energy [12–14] not related to the microstructure. In our case, it also corresponds 
to an energy (see (6)), but this energy depends on the microstructure. Another important quantity here, v , is a virtual 
field corresponding to the displacement of the inclusions; this can be considered as a memory effect of the inclusions. It is 
thanks to this field that we can maintain information from the micro scale to the macro scale.
A. Nait-ali / C. R. Mecanique 346 (2018) 1192–1198 1195To establish the gamma convergence, we must increase and minimize the preceding energy by the inferior and superior 
limits of (1). For the upper bound, it is enough to follow the same strategy as in [1] and [6] for the first term. For the 
second one, it is obtained by using the following lemma.
Lemma 3.2. Let (u, v) ∈ Dom. There exist (uε, vε = |O||Brε |1Brε uε) in W 1,p(O) × Lp(Ô, W 1,p(0, π)) such that uε → u Lp(O), u ∈
W 1,p(O), vε
∗
⇀ v in Lp(Ô, W 1,p(0, π)) and
lim
ε→0
ˆ
B Rε \Brε
f (∇uε) dx = γ
ˆ
O
|u − v|p dx (7)
Proof. Step 1. We establish (7) when (u, v) ∈ C1(O). Let θε ∈ W 1,p0 (D Rε (ω0,ε)) be the solution to the capacitary problemˆ
B Rε (ω0,ε)
f ∞,p(∇θε) dx̂ = Ener frε,Rε
We extend by εY -periodicity in O = ⋃z∈Iε ε(Y + z), so as to cover the entire domain O.
On the other hand, consider
wε(x̂,ψ) :=
∑
z∈Iε
(  
Brε (ωz,ε)+z
v( ŷ,ψ) d ŷ
)
1ε(Y +z)(x)
and set uε = (1 − θε)u + θε wε . One has
lim
ε→0
ˆ
B Rε \Brε
f (∇uε) dx = lim
ε→0
ˆ
B Rε \Brε
f ∞,p(∇uε) dx
Indeed, from (3) and Hölder’s inequality,
ˆ
B Rε \Brε
∣∣ f (∇uε) dx − f ∞,p(∇uε)∣∣ dx ≤ β ′(|Brε Rε | +
ˆ
B Rε \Brε
|∇uε|p−δ dx
)
≤ β ′
(
|Brε Rε | + |Brε Rε |
δ
p
( ˆ
B Rε \Brε
|∇uε|p dx
) p−δ
p
)
and since clearly sup
ε>0
ˆ
B Rε \Brε
|∇uε|p dx < +∞, the claim follows from the fact that, by the choice Rε << ε, one has 
limε→0 |Brε Rε | = 0.
For each z ∈ Iε , choose ωz,ε the center of sphere. From (5), the regularity assumption on (u, v) and since f ∞,p is 
p-positively homogeneous, we infer
lim
ε→0
ˆ
B Rε \Brε
f (∇uε) dx = lim
ε→0
ˆ
B Rε \Brε
f ∞,p(∇uε) dx
= lim
ε→0
ˆ
B Rε \Brε
f ∞,p((u − v)∇θε) dx
= lim
ε→0
ˆ
B Rε \Brε
|u − v|p f ∞,p(∇θε) dx
= lim
ε→0
ˆ
B Rε
f ∞,p(∇θε) dx̂
∑
z∈Iε
πˆ
0
|(u − v)(ωz,ε,ψ)|p dψ
= lim
ε→0
1
ε2
ˆ
B
f ∞,p(∇̂θε,0) dx̂
∑
z∈Iε
ε2
πˆ
0
|(u − v)(ωz,ε,ψ)|p dψ
Rε
1196 A. Nait-ali / C. R. Mecanique 346 (2018) 1192–1198= lim
ε→0
Ener frε,Rε
ε2
∑
z∈Iε
ε2
πˆ
0
|(u − v)(ωz,ε,ψ)|p dψ
= γ
ˆ
O
|u − v|p dx
where we have used assumption (6) and the fact that 
∑
z∈Iε
ε2
πˆ
0
|(u − v)(ωz,ε, ψ)|p dψ is a Riemann sum in the last equality.
Step 2. We establish (7) in the general case by a standard approximation and diagonalization argument. 
Lemma 3.3. For all (uε, vε), vε = |O||Trε |1Brε uε) such that uε → u in Lp(O) and vε
∗
⇀ v in Lp(Ô, W 1,p(0, π)) one has
lim inf
ε→0
ˆ
B Rε \Brε
f (∇uε) dx = lim inf
ε→0
ˆ
B Rε \Brε
f ∞,p(∇uε) dx ≥ γ
ˆ
O
|u − v|pdx (8)
Proof. Step 1. Equality in (8) is clear. For η > 0 intended to go to 0 we decompose (0, π) into a finite family of intervals 
(K j) j∈ Jη of size η: 
∣∣∣(0,π) \ ⋃ j∈ Jη K j
∣∣∣ = 0. In this step, we modify the function uε into a Sobolev function ũε (depending 
on η) in each ring 
⋃
i∈Iε Srε Rε (ωz,ε) × K j , satisfying:
ũε = 0 on each surface
⋃
i∈Iε
Srε Rε (ωz,ε) × K j
lim inf
ε→0
ˆ
B Rε \Brε
f ∞,p(∇uε) dx ≥ lim inf
ε→0
∑
z∈Iε
∑
j∈ Jη
ˆ
Srε Rε (ωz,ε)×K j
f ∞,p(∇ũε) dx
We make use of a standard truncation method in each slice of a neighborhood of each one of the two bases: we set 
uε,i := ϕiuε where ψ → ϕi(ψ) satisfies ϕi = 1 on the neighborhood (K j)δi and belongs to Cc((K j)δi+1 ), i = 1, . . . , ν where 
δi are small parameters. To perform the slicing method, the important points to note are limε→0
´
B Rε \Brε |uε|p dx = 0 and
lim
ν→+∞
1
ν
sup
ε>0
ˆ
B Rε \Brε
|∇uε|p dx = 0
Step 2. In what follows, we still denote by uε the function ũε . By using Jensen’s inequality, we infer
lim inf
ε→0
∑
z∈Iε
∑
j∈ Jη
ˆ
Srε Rε (ωz,ε)×K j
f ∞,p(∇ũε) dx ≥ lim inf
ε→0
∑
z∈Iε
∑
j∈ Jη
η
ˆ
Srε Rε (ωz,ε)
f ∞,p(
 
K j
∇̂uε(x̂, s) ds,0) dx̂ (9)
In the last step we set ̂uε, j(x̂) :=
 
K j
uε(x̂, s) ds for all j ∈ Jη . Note that ̂uε, j depends on η.
Step 3. (End of the proof) We approximate the functions u and v as follows:
uη(x) =
∑
j∈ Jη
1K j (x3)
 
K j
u(x̂, s) ds
vη(x) =
∑
j∈ Jη
1K j (x3)
 
K j
v(x̂, s) ds
For all fixed j ∈ Jη , our strategy consists first in modifying in each ring Srε Rε (ωz,ε). The function ûε, j is such that the new 
function ũε, j agrees with 
ffl
S Rε (ωz,ε)
ûε, j(x̂) dH1 on C Rε (ωz,ε), with 
ffl
Srε (ωz,ε)
ûε, j(x̂) dH1 on Srε (ωz,ε), and satisfies
lim inf
ε→0
∑
z∈Iε
η
ˆ
Srε Rε (ωz,ε)
f ∞,p(∇ûε, j,0) dx̂ ≥ lim inf
ε→0
∑
z∈Iε
η
ˆ
Srε Rε (ωz,ε)
f ∞,p(∇ũε, j,0) dx̂ (10)
This is done by using a standard De Giorgi slicing argument (see for instance [15], proof of Proposition 11.2.3).
A. Nait-ali / C. R. Mecanique 346 (2018) 1192–1198 1197To shorten notation, we still denote by ûε, j the function ũε, j . For all fixed j ∈ Jη , we define the two step functions in 
Lp(O × K j)
ûε, j(x̂) =
∑
z∈Iε
(  
S Rε (ωz,ε)
ûε, j(x̂) dH1(x̂)
)
1ε(Y +z)
v̂ε, j(x̂) =
∑
z∈Iε
(  
Srε (ωz,ε)
ûε, j(x̂) dH1(x̂)
)
1ε(Y +z)
It is worth noticing that ûε, j − v̂ε, j =
ffl
K j
(
uε, j − vε, j
)
dx3 where uε, j and vε, j are two functions defined like uε and vε
in [5] (replace (0, π) by K j ). Proceeding like in [5,6], it is not difficult to establish that uε, j − vε, j weakly converges in 
Lp(O × K j) to uη − vη .
By using the p-homogeneity of f ∞,p and a l.s.c. argument, we deduce
lim inf
ε→0
∑
z∈Iε
∑
j∈ Jη
η
ˆ
Srε Rε (ωz,ε)
f ∞,p(∇ũε, j,0) dx̂
= lim inf
ε→0
∑
z∈Iε
∑
j∈ Jη
ηε2
∣∣∣∣∣∣∣
 
S Rε (ωz,ε)
ûε, j(x̂) dH1(x̂) −
 
Srε (ωz,ε)
ûε, j(x̂) dH1(x̂)
∣∣∣∣∣∣∣
p
Ener frε Rε
ε2
= lim inf
ε→0
Ener frε Rε
ε2
∑
j∈ Jη
η
ˆ
O
|̂uε, j − v̂ε, j|p dx ≥ γ
ˆ
O
|uη − vη|p dx (11)
The claim follows by combining the first step, (9), (10), (11), and letting η → 0 
4. Summary
In summary, in addition to the stochastic homogenization method, the important result of this work is the form of limit 
energy F0(u) = infv∈Lp(O) G(u, v) with G : Lp(O) × Lp(O) →R+ ∪ {+∞} given by
G(u, v) :=
⎧⎪⎨
⎪⎩
ˆ
O
f (∇u) dx + γ
ˆ
O
|u − v|p dx + a
ˆ
O
f ∞,p( ∂v
∂ψ
) dx if (u, v) ∈ Dom
+∞ otherwise
This energy can be compared to non-local models of the type [14], but it has the advantage of not having a second 
gradient. Indeed, here the non-locality is represented, on the one hand, by the constant γ and, on the other hand, by the 
virtual field v . This model can be extended to a model of damage. In the latter case, the variable v will be the variable 
of damage and γ a term of regularization. This regulation constant depends on the distribution of inclusions, and can be 
computed using covariance methods [16].
5. Conclusion
In this study, we determined a homogeneous deterministic model of a randomly enhanced environment with spherical 
inclusion and very small relative to the size of a REV (Representative Elementary Volume). The method proposed here allows 
us to obtain a homogenized model of statistical data preserved in spite of the scale transition process. The same strategy 
can be proposed for other materials, such as randomly reinforced materials such as long fiber composites [17], but in the 
case of fibers with a very small radius. This result is a good start to understand the damage mechanisms of this type of 
material. Indeed, the second energy obtained at the limit corresponds to a stored energy, which can be at the origin of the 
damage to this type of composite. Following this study, we wish to proceed to a numerical resolution of this model so as to 
compare this result with mechanical experiments, for example.
In addition, from the development of a numerical resolution scheme, we wish to extend this model to a nonlinear 
behavior, more precisely to a damage model [12,14,18]. In fact, the energy γ corresponds to a stored energy that can 
create compensation. This damage is localized at the inclusion-matrix interface. This generates a decohesion very difficult 
to model; it is also by this aspect that this work is original and will help better understand these phenomena.
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Stochastic homogenization of reinforced polymer with very small carbon inclusions

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Impact of local diffusion on macroscopic dispersion in three-dimensional porous media

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Exploring the undrained cyclic behavior of methane-hydrate-bearing sediments using CFD–DEM

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Designing isotropic composites reinforced by aligned transversely isotropic particles of spheroidal shape

Stochastic homogenization of reinforced polymer with very small carbon inclusions

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Limit analysis, rammed earth material and Casagrande test

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