Éric Canon
email: eric.canon@univ-st-etienne.fr

Michel Lenczner Femto-St

Keywords: Mathematics Subject Classifications (1991): 35B27 Homogenization of partial differential equations in media with periodic structures, 73B27 nonhomogeneous materials and homogenization linear elasticity, piezoelectricity, homogenization, plate theory, composite materials, prescribed electric potential, local electric circuits, nonlocal electric circuits, transfinite networks, smart materials

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Modeling of thin isotropic elastic plates with small piezoelectric inclusions and distributed electric circuits

Eric Canon, Michel Lenczner

Modelling of thin isotropic elastic plates with small piezoelectric inclusions and distributed electric circuits. Models for inclusions larger or comparable to the thickness of the plate.

1. Introduction 1.1 General. This paper is the third and last part of a systematic work devoted to the derivation of effective models for thin piezoelectric/elastic composite plates including elementary electric circuits connected to the upper and lower faces of piezoelectric transducers. It is motivated by an important development of piezocomposites used for instance for distributed control in vibroacoustics [8, 11 ,12, 25] or as sensors in phased arrays. In [START_REF] Canon | Models of elastic plates with piezoelectric inclusions, part I: models without homogenization[END_REF], we considered three-dimensional elastic plates with a small number of piezoelectric inclusions and derived effective models when the thickness a of the plate tends to zero. In [START_REF] Canon | Modelling of thin elastic plates with small piezoelectric inclusions and distributed electronic circuits. Models for inclusions that are small with respect to the thickness of the plate[END_REF], effective models of thin plates with a large number of ε-periodically distributed piezoelectric inclusions have been obtained in the case ε << a by letting a, ε and ε/a simultaneously tend to zero. The aim of the present paper is to consider the two other possible asymptotic cases, namely a/ε → 0 and ε/a → 1. We note that the models for a/ε → 0 were already presented in the note [START_REF] Canon | Deux modèles de plaques minces avec inclusions piézoélectriques et circuits électroniques distribués[END_REF].

As in [START_REF] Canon | Models of elastic plates with piezoelectric inclusions, part I: models without homogenization[END_REF][START_REF] Canon | Modelling of thin elastic plates with small piezoelectric inclusions and distributed electronic circuits. Models for inclusions that are small with respect to the thickness of the plate[END_REF], different kinds of boundary conditions are considered on the metallized upper faces of the inclusions, corresponding to different possible types of control: prescribed electric potential (or Dirichlet conditions) if the tension is controlled, prescribed electric displacement field (or Neumann conditions) if the current is controlled, local and nonlocal mixed conditions if the inclusions are connected to R-L-C circuits (the nonlocal conditions corresponding to inclusions that are connected to each other via R-L-C circuits).

Following a principle used in [START_REF] Canon | Models of elastic plates with piezoelectric inclusions, part I: models without homogenization[END_REF][START_REF] Canon | Modelling of thin elastic plates with small piezoelectric inclusions and distributed electronic circuits. Models for inclusions that are small with respect to the thickness of the plate[END_REF], the derivation of the models is made in the space of gradients of solutions. This leads to a more synthetic and readable presentation of the results. We combine the two-scale convergence [START_REF] Allaire | Homogenization and two scale-convergence[END_REF][START_REF] Nguentseng | A general convergence result for a functional related to the theory of homogenization[END_REF] for homogenization and classical arguments of thin plates theory [START_REF] Ciarlet | Mathematical Elasticity[END_REF][START_REF] Ciarlet | A justification of the two-dimensional plate model[END_REF][START_REF] Raoult | Contribution à l'étude des modèles d'évolution de plaques et à l'approximation d'équations d'évolution linéaires du second ordre par des méthodes multipas[END_REF]. Let us quote the pioneer work of Caillerie [START_REF] Caillerie | Thin elastic and periodic plates[END_REF], who considered the case of thin static elastic plates with periodic coefficients, using the Tartar's method of oscillating test functions [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF][START_REF] Sanchez-Palencia | Non-Homogeneous Media and Vibration Theory[END_REF]. However, in [START_REF] Caillerie | Thin elastic and periodic plates[END_REF] the parameters a and ε tend successively and independently to zero, except for a ≃ ε.

Despite their relative formal complexity, the effective models have a rather simple structure. In the case a/ε → 0, for Dirichlet, and local mixed conditions (from a mathematical point of view the prescribed electric displacement field conditions happen to be a special case of mixed conditions and is not treated separately), the limit model has the same form as the elastic plate model, the influence of piezoelectric inclusion only appears in the definition of the effective coefficients and as a source term. For nonlocal mixed conditions, a coupling arises between mechanical effects and the transverse component of the electric field, because of a Laplace operator in the in-plane direction induced by the electric circuits. The resulting operator is a special case of those encountered in the homogenization of periodic electric circuits [START_REF] Lenczner | Homogénéisation d'un circuit électrique[END_REF][START_REF] Lenczner | Homogenization of electrical networks including voltage-to-voltage amplifiers[END_REF][START_REF] Lenczner | Homogenization of periodic electrical networks including voltage to current amplifiers[END_REF] or in transfinite networks [START_REF] Zemanian | Infinite Electrical Networks[END_REF][START_REF] Zemanian | Transfinite graphs and electrical networks[END_REF], the later being analyzed from a different point of view. We also quote [START_REF] Ratier | Analog computing of partial differential equations[END_REF] where a wide variety of in-plane operators are generated by a periodic network of resistances. In the case a/ε → 1, for Dirichlet conditions the limit model also has the same form as the purely elastic thin plate model, but a coupling arises even in the case of local mixed conditions.

Concerning thin piezoelectric structures, let us also mention: Ghergu and al [START_REF] Ghergu | Homogenization of thin piezoelectric perforated shells[END_REF] who consider perforated piezoelectric shells with fixed thickness; Licht and Weller [START_REF] Licht | Mathematical Modeling of smart material and structures[END_REF][START_REF] Weller | Analyse asymptotique de plaques minces linéairement piézoélectriques[END_REF]; Sene [START_REF] Sene | Modelling of piezoelectric static thin plates[END_REF] and Figueiredo and Leal [START_REF] Figueiredo | A piezoelectric anisotropic plate model[END_REF] who consider piezoelectric plates without homogenization. Remark however that in [START_REF] Figueiredo | A piezoelectric anisotropic plate model[END_REF] general anisotropic models are considered. Finally, let us remark that the models used in vibro-acoustic applications, as in [START_REF] Casadei | Broadband vibration control through periodic arrays of resonant shunts: experimental investigation on plates[END_REF][START_REF] Collet | Semi-active optimization of 2D wave dispersion into shunted piezo-composite systems for controlling acoustic interaction[END_REF][START_REF] Collet | Structural energy flow optimization through adaptive shunted piezoelectric metacomposites[END_REF][START_REF] Thorp | Attenuation and localization of wave propagation in rods with periodic shunted piezoelectric patches[END_REF], are often based on Bloch wave decompositions which seems a priori not compatible with the homogenization method used in this paper. However, the homogenized model for the wave equation in [START_REF] Brassart | A two-scale model for the periodic homogenization of the wave equation[END_REF] builds a bridge between the two views and constitutes a perspective for further works.

1.2 Detailed contents. Section 2 is devoted to the setting of the initial 3-dimensional equations of static linearized elasticity and piezoelectricity. The piezoelectric inclusions are assumed to be strictly included in an insulating elastic matrix.

As our work is more about introducing piezoelectric plates with elementary electric devices and about mixing homogenization and plates theory, for simplicity and efficiency, following [START_REF] Caillerie | Thin elastic and periodic plates[END_REF], we assume that the material coefficients are constant in the thickness direction. This is usually the case in applications, as the matrix and the piezoelectric ceramics are homogeneous materials. In the same spirit, the upper and lower faces of the piezoelectric inclusions are assumed to be metallized, that is to be covered with a thin conductive film. However, from the mathematical point of view, it might be interesting to obtain more general models by removing these technical assumptions and considering fully nonhomogeneous materials as in [START_REF] Figueiredo | A piezoelectric anisotropic plate model[END_REF], or multi-layered plate by adapting to the present work to formalism proposed in [START_REF] Canon | Models of elastic plates with piezoelectric inclusions, part I: models without homogenization[END_REF].

The mechanical boundary conditions applied to the plate are prescribed surface forces on its lower and upper faces and on part of its lateral boundary, and prescribed mechanical displacement on the remaining part. For the Maxwell-Gauss equations, we consider prescribed electrical potential on the lower faces of the inclusions (in practice these faces are connected to ground and the electric potential is zero). As already mentioned in Section 1.1 various boundary conditions are considered on the upper faces of the inclusions, corresponding to connections to electric or electronic devices. These conditions are detailed in section 2.4. Some of them are, to our knowledge, unusual in plates theory, and thus, constitute one of the interesting point of our work.

The weak formulations of the system are stated in Section 3. For a concise formulation covering all kinds of boundary conditions, we adopt synthetic tensorial notations rather than fully extended formulae. We strongly believe that this allows a better legibility of computations as well as of limit models.

The precise assumptions on the data are presented in Section 4. In particular, we give the correct scalings, or, from a more concrete point of view, how electrical circuits have to be chosen to obtain a significant influence on the effective behaviour of the material.

Resulting a priori estimates and first convergence results are given in Sections 4.2 and 4.3.

Sections 5 and 6 are devoted to the statement of the main results i.e. the effective twodimensional plate model for each type of electrical boundary condition in the case where the plate thickness is much smaller than the inclusions size (Theorem 5.1, Section 5) and in the case where these two parameters are of the same order (Theorem 6.1, Section 6).

Theorem 5.1 is proved in Section 7. The proof is in three steps. The first one, which is mathematically the most difficult consists in characterizing the two-scale limits of the strain and of the electrical field. These results are new, even in the case of pure elasticity.In Caillerie [START_REF] Caillerie | Thin elastic and periodic plates[END_REF] -as two-scale convergence did not exist at the time -only weak limits were considered. The second step consists in eliminating the microscopic variable by computing the microscopic fields in terms of the macroscopic fields. We use here the classical arguments of linear homogenization. The third step consists in eliminating the transverse components, or part of the transverse components of the fields (according to the model), that are computed with respect to the other ones. This elimination slightly departs from the classical plates theory, because of the non standard boundary conditions on the upper and lower faces of the inclusions that are considered in the present paper. Theorem 6.1 is proved in Section 8. As here a ≃ ε is assumed, the proof is only in two steps. First, characterization of the limit, second, simultaneous elimination of the local and of (part of) the transverse components.

We use the same formalism as in [START_REF] Canon | Models of elastic plates with piezoelectric inclusions, part I: models without homogenization[END_REF][START_REF] Canon | Modelling of thin elastic plates with small piezoelectric inclusions and distributed electronic circuits. Models for inclusions that are small with respect to the thickness of the plate[END_REF], based on tensorial notations and products, and on simple algebraic operations such as projections. It allows to deal relatively easily with complex computations. Completely explicit formulae would require a lot of room, to the detriment of legibility.

Step 2 and 3 of our proofs are almost formal computation and are easily adapted from one variant to the other. A coupling with the formalism introduced in [START_REF] Canon | Models of elastic plates with piezoelectric inclusions, part I: models without homogenization[END_REF] for multilayered plates models is easily conceivable.

Equations of 3-dimensional piezoelectricity

2.1 Geometry. The three-dimensional plate with thickness a > 0 is represented by For any inclusion ε(i+Y 1 )×] -a, a[ such that i ∈I ε , the mean value on the upper face ε(i+Y 1 )×{a} is denoted by < . > i . For any function ψ on Ω a , ψ i designates its restriction to the inclusion ε(i+Y 1 )×] -a, a[.

Ω a = ω×] -a,
index i = (i 1 , i 2 ) ∈ Z 2 such that ε(i+Y 1 ) is strictly included in ω; then ω ε 1 = ∪ i∈I ε ε(i+Y 1 ) while ω ε 2 = ω \ ω ε 1 . Let b = (a, ε); the set Ω b 2 = ω ε 2 ×] -a, a[ represents the elastic matrix of the body, while Ω b 1 = ω ε 1 ×] -a, a[ is the set of all piezoelectric inclusions. The boundary of ω is divided into two regular parts γ D and γ N , with |γ D | > 0. The boundary of Ω a is thereby divided into Γ a D = γ D ×] -a, a[ and Γ a N = (γ N ×] -a, a[) ∪ (ω × {-a, a}). The boundary of Ω b 1 is divided into Γ b+ 1 = ω ε 1 × {a}, Γ b- 1 = ω ε 1 × {-a} and Γ b 1 = ∂ω ε 1 ×] -a,

Other notations.

Bold characters are used for vector and matrix valued functions and for the corresponding functional spaces. We constantly use Einstein's convention of summation on repeated indices with the additional convention that latin and greek indices are varying from 1 to 3 and from 1 to 2 respectively. Throughout the paper c and C designate generic positive constants, not depending on a and ε.

Also: inutile (1)
Last, vector and vectorial notations are a priori meant in line. When a given vector V is meant in column, we write: t V. For instance for

V = ( (s αβ (v)) α,β=1,2 , ( a -1 s α3 (v) ) α=1,2 , a -2 s 33 (v)
) ,

with t V, we mean

t V =   (s αβ (v)) α,β=1,2 (a -2 s α3 (v)) α=1,2 a -2 s 33 (v)   .
Note that this convention leads us to writings that slightly differ from the one in [START_REF] Canon | Deux modèles de plaques minces avec inclusions piézoélectriques et circuits électroniques distribués[END_REF][START_REF] Canon | Modelling of thin elastic plates with small piezoelectric inclusions and distributed electronic circuits. Models for inclusions that are small with respect to the thickness of the plate[END_REF], but it seems more coherent to us now.

Equations of 3-dimensional piezoelectricity. The mechanical displacements

u b = (u b i ) i=1,2
,3 and the electric potential φ b are governed by the linearized equations of piezoelectricity in their static version,

{ -∂ j σ b ij = f b i in Ω a , σ b ij n j = g b i on Γ a N , u b i = 0 on Γ a D , -∂ j D b j = 0 in Ω b 1 , D b j .n i = 0 on Γ b 1 , (2)
for i = 1, 2, 3, where

{ σ b ij = R ε ijkl s kl (u b ) + d ε kij ∂ k φ b , D b j = -d ε jkl s kl (u b ) + c ε jk ∂ k φ b , ( 3
)
∀ v ∈ H 1 (Ω a ), s ij (v) = 1 2 (∂ i v j + ∂ j v i ). ( 4
)
The assumptions on the volume and surface forces f b := (f b i ) i=1,2,3 and g b := (g b i ) i=1,2,3 are specified below, in Section 4.1. The stiffness tensor R ε := (R ε ijkl ) i,j,k,l=1,2,3 , the piezoelectricity tensor d ε := (d ε kij ) i,j,k=1,2,3 and the permitivity tensor c ε := (c ε ij ) i,j=1,2,3 are assume to satisfy the symmetry conditions:

∀i, j, k, l ∈ {1, 2, 3}, R ε ijkl = R ε klij = R ε jikl , c ε ij = c ε ji , d ε kij = d ε kji . (5)
Note that, as we assume that the inclusions are electrically insulated from the elastic matrix, no Gauss-Maxwell equation is needed in Ω b 2 . However, we let for convenience

c ε = 0, d ε = 0 in Ω b 2 . ( 6
)
We go now into detail about the different kinds of electric boundary conditions that are considered in this paper.

Electric boundary conditions on

Γ b+ 1 ∪Γ b- 1 .
The four kinds of electric boundary conditions are summarized here. Further explanations and comments can be found in [START_REF] Canon | Modelling of thin elastic plates with small piezoelectric inclusions and distributed electronic circuits. Models for inclusions that are small with respect to the thickness of the plate[END_REF].

(i) Prescribed electric potential (Dirichlet conditions): Let us introduce the shift operators defined from

φ b = φ b m + aφ b c on Γ b+ 1 , φ b = φ b m -aφ b c on Γ b- 1 , ( 7
I ε to Z 2 by { T 1 +1 : i →(i 1 + 1, i 2 ), T 2 +1 : i →(i 1 , i 2 + 1), T 1 -1 : i →(i 1 -1, i 2 ), T 2 -1 : i →(i 1 , i 2 -1).
Let φ b m be a given function on ω ε 1 . For all i ∈ I ε , we pose

φb i = φ b i -φ b m,i .
The boundary conditions for the electrostatic equation ( 2)

2 on Γ b+ 1 ∪ Γ b- 1 are then    ∀i ∈ I ε , < D b .n > i = G 1 aε 2 2 ∑ α=1 ( φb T α -1 (i) -2 φb i + φb T α +1 (i) ) - G a φb i + h b on Γ b+ 1 , φ b = φ b m on Γ b- 1 . (8)
The finite difference operator on φb i is completed by the analog of a discrete Neumann boundary condition to the free ends of the circuit φb

T α -1 (i) -φb i = 0 if T α -1 (i) / ∈ I ε , φb T α +1 (i) -φb i = 0 if T α +1 (i) / ∈ I ε (9)
which also means that the current is vanishing in the corresponding branches.

In [START_REF] Casadei | Broadband vibration control through periodic arrays of resonant shunts: experimental investigation on plates[END_REF], G 1 and G are given nonnegative constants. If G = G 1 = 0 the conditions on Γ b+ Although from a physical point of view it has its own meaning, the case of Neumann conditions does not need to be treated separately; one just have to let G = 0 in the effective models obtained in Theorem 5.1 and Theorem 6.1 for local mixed conditions. In the situation a/ε → 0, local and nonlocal conditions lead to different developments and limit models.

As we assume all the faces to be metallized, the functions φb , φ b m and φb c are constant on each metallized face of inclusions and assuming that the current is provided by a single wire, the same holds true for h b .

Finally, in order to use, as much as possible, common formulations for the different boundary conditions, we define h b , φ b c , G and G 1 in all cases, with the conventions that

h b = 0 and G = G 1 = 0 for Dirichlet conditions, (10)
and φ b c = 0 for mixed conditions. ( 11)

Weak formulations

The aim of the present section is the statement of the weak formulation of the equations of Section 2, on the fixed domain Ω. We use the standard change of variables x a → x = (x 1 , x 2 , x a 3 /a) and the appropriate scaling for volume forces, surface forces and displacements fields, see [START_REF] Ciarlet | A justification of the two-dimensional plate model[END_REF] and also [START_REF] Ciarlet | Mathematical Elasticity[END_REF],

       ∀x ∈ Ω, ûb (x) = (u b 1 (x a ), u b 2 (x a ), au b 3 (x a )), ∀x ∈ Ω, f b (x) = (f b 1 (x a ), f b 2 (x a ), a -1 f b 3 (x a )), ∀x ∈ γ N ×] -1, 1[, ĝb (x) = (g b 1 (x a ), g b 2 (x a ), a -1 g b 3 (x a )), ∀x ∈ ω × {-1, 1}, ĝb (x) = a -1 (g b 1 (x a ), g b 2 (x a ), a -1 g b 3 (x a )).
The current source h b , the electric potential φ b , φ b m and φ b c are left unchanged. As in the sequel, we only work on the reference domain Ω, no confusion might occur, so for simplicity we keep the notation

u b , f b , g b , h b , φ b , φ b m and φ b c , without hats. For any V =(v, ψ) ∈ H 1 (Ω) × H 1 (Ω ε 1 ), we let        K a (v) = ( K a αβ (v), K a α3 (v)), K a 33 (v) ) = (s αβ (v), a -1 s α3 (v), a -2 s 33 (v)) , L a (ψ) = (L a α (ψ), L a 3 (ψ)) = (∂ α ψ, a -1 ∂ 3 ψ) , M a (V) = (K a (v), L a (ψ)) . ( 12
)
We put together the tensors R ε , d ε and c ε in a global stiffness-piezoelectricity-permitivity tensor R ε , which is the 10 × 10 symmetric matrix written in a format compatible with [START_REF] Collet | Structural energy flow optimization through adaptive shunted piezoelectric metacomposites[END_REF]:

R ε =       R ε αβγδ 2R ε αβγ3 R ε αβ33 d ε γαβ d ε 3αβ 2R ε α3γδ 4R ε α3γ3 2R ε α333 2d ε γα3 2d ε 3α3 R ε 33γδ 2R ε 33γ3 R ε 3333 d ε γ33 d ε 333 -d ε αγδ -2d ε αγ3 -d ε α33 c ε αγ c ε α3 -d ε 3γδ -2d ε 3γ3 -d ε 333 c ε 3γ c ε 33       . ( 13
)
The linear forms associated with the mechanical load and the electrical current source are

ℓ b u (v) = ∫ Ω f b i v i dx + ∫ Γ N g b i v i ds, and ℓ b φ ( L3 ) = ∫ Ω ε 1 h b L3 dx.
Note that h b is a priori defined on Γ ε+ 1 (or equivalently on ω ε 1 ) but is trivially extended to a function on Ω ε 1 which does not depend on x 3 . Given the assumption on metallization, the set of admissible electric potential is

H 1 c (Ω ε 1 ) = {ψ ∈ H 1 (Ω ε 1 ); ψ is constant on each connected part of Γ ε+ 1 ∪ Γ ε- 1 }.
The relevant functional space is then specific to each electric boundary condition.

(i) Dirichlet conditions:    W b D = W ε (φ b m , φ b c ) = {(v, φ) ∈ H 1 (Ω) × H 1 c (Ω ε 1 ); v = 0 on Γ D , φ = φ b m + aφ b c on Γ ε+ 1 , φ = φ b m -aφ b c on Γ ε- 1 }, W ε = W ε (0, 0). (ii) Mixed conditions: { W b D = W ε (φ b m ) = {(v, φ) ∈ H 1 (Ω) × H 1 c (Ω ε 1 ); v = 0 on Γ D , φ = φ b m on Γ ε- 1 }, W ε = W ε (0).

The backward difference operator ∇ ε

x is defined inclusion by inclusion by 6), ( 10) and ( 11), the weak formulations on the scaled domain Ω for the coupled problems ( 2)-( 7)-( 8), are summarized by:

∀i ∈ I ε , ∀ψ ∈ H 1 c (Ω ε 1 ), (∇ ε xψ) i = ε -1 (ψ i -ψ T 1 -1 (i) , ψ i -ψ T 2 -1 (i) ). Letting U b = (u b , φ b ), with the conventions (
           U b = (u b , φ b ) ∈ W b D , and for all V = (v, ψ) ∈ W ε , ∫ Ω M a (V) R ε t M a (U b ) dx + 2 ∫ Ω ε 1 GM(L a 3 (φ b ))M(L a 3 (ψ)) dx +2 ∫ Ω ε 1 G 1 ∇ ε xM(L a 3 (φ b )).∇ ε xM(L a 3 (ψ)) dx = ℓ b u (v) + ℓ b φ (L a 3 (ψ)). (14)

Assumptions on the data -A priori estimates -Convergences

4.1 Two-scale convergence. As two-scale convergence is an important tool of the paper, before stating the assumptions on the data and the first convergence results, that are expressed in terms of two-scale convergence, let us recall some basic facts about it.

Let C ∞ ♯ (Y ) designate the space of C ∞ functions on R n that are Y -periodic. Définition 1 (Allaire [1]) Let (u ε ) ε>0 be a family of L 2 (ω) and u ∈ L 2 (ω × Y ). We say that (u ε ) ε>0 two-scale converges to u if for any v ∈ D(ω; C ∞ ♯ (Y )) lim ε→0 ∫ ω u ε (x)v(x, x/ε)dx = ∫ ω×Y u(x, y)v(x, y)dxdy.
The important fact here is that for any bounded family in L 2 (ω), there is a subsequence that two-scale converges to some limit u. Two-scale convergence is a more accurate notion than usual weak convergence, in the sense that the two-scale converging family also weakly converges to ∫ Y u(., y)dy.

Since we need two-scale convergence for functions defined on ω ε 1 , we also use the following practical definition.

Définition 2 A family (u ε ) ε>0 of L 2 (ω ε 1
) is said to two-scale converge to a limit u in L 2 (ω×Y 1 ) if u ∈ L 2 (ω×Y 1 ) and (P ε u ε ) two-scale converges to P u in the sense of definition 1, where P ε and P designate the extension by 0 of functions on ω ε 1 to functions on ω and of functions on ω × Y 1 to functions ω × Y respectively.

In our problems, there are no oscillations in the x 3 -direction. Still, the above definitions evidently apply with x 3 as a dummy variable (as it is the case for the time variables t in other contexts). So in what follows when writing that (u ε ) of L 2 (Ω) two-scale converges to u in L 2 (Ω × Y ), we mean that for any v ∈ D(Ω;

C ∞ ♯ (Y )), lim ε→0 ∫ Ω u ε (x)v(x, x/ε)dx = ∫ Ω×Y u(x, y)v(x, y)dxdy,
and similarly for (u

ε ) in L 2 (Ω × Y 1 ).
Remark that the convergences in Definitions 1 and 2 are weak convergences. In general, the two-scale limit of a product is not the product of the two-scale limits. An additional assumption is needed. Let (u ε ) and (v ε ) be two two-scale converging family in L 2 (Ω) with limits u and v respectively. Then, if in addition

lim ε→0 ∥u ε ∥ L 2 (ω) = ∥u∥ L 2 (ω×Y ) , (15)
the following statement hold true for any regular test functions φ :

lim ε→0 ∫ ω u ε (x)v ε (x)φ(x, x/ε)dx = ∫ ω×Y u(x, y)φ(x, y)φdxdy. ( 16
)
Assumption ( 15) is a kind of strong two-scale convergence notion. See for example Theorem 1.8 and the proof of Theorem 2.3 in Allaire [START_REF] Allaire | Homogenization and two scale-convergence[END_REF] for a proof of (16).

Assumptions on the data and first convergence results.

The tensors R ε , d ε and c ε constituting the stiffness-piezoelectricity-permitivity tensor R ε are assumed to satisfy [START_REF] Canon | Models of elastic plates with piezoelectric inclusions, part I: models without homogenization[END_REF] and

           (R ε ) ∈ L ∞ (Ω) and two-scale converges to some R ∈ L ∞ (Ω × Y ), ||R ε || L ∞ (Ω) ≤ C, R ε does not depend on x 3 , ∀ K = (K ij ) ∈ R 3×3 with K ij = K ji , K R ε t K ≥ c||K|| 2 a.e. in ω, ∀ L ∈ R 3 , t L c ε L ≥ c||L|| 2 a.e. in ω ε 1 , (17)
and

lim ε→0 ||R ε || L 2 (ω) = ||R|| L 2 (ω×Y ) . ( 18
)
Remark: As pointed above, assumption ( 18) is a rather vague, but general assumption that allows to go to the limit in product of two-scale converging functions; namely

( M a (U b )
) and (R ε ). It is of course fulfilled by data satisfying classical periodicity conditions, for instance R ε (x) = R(x, x/ε) where R is a given tensor in L ∞ (ω; C ♯ (Y )), or also, which is more relevant in the present work, in C ∞ (ω; L ∞ ♯ (Y )). See Allaire [START_REF] Allaire | Homogenization and two scale-convergence[END_REF] if necessary. Coercivity for c ε and R ε , together with the symmetry assumptions (5) for d, implies coercivity for R ε . Conversely, two-scale convergence for R ε implies two-scale convergence for R ε , c ε and d ε . The corresponding limits are denoted by R, c and d. The other data are assumed to satisfy

       f b ∈ L 2 (Ω), g b ∈ H 1/2 (Γ N ), (f b ) converges weakly in L 2 (Ω) to some limit f , (g b ) converges weakly in L 2 (Γ N ) to some limit g, (19)
           h b , φ b m and φ b c are constant on each inclusion, (h b ) two scale-converges in L 2 (ω × Y 1 ) to some limit h ∈ L 2 (ω), (φ b m ) two-scale converges in L 2 (ω × Y 1 ) to some limit φ m ∈ H 1 (ω), (φ b c ) two-scale converges in L 2 (ω × Y 1 ) to some limit φ c ∈ L 2 (ω). ( 20
)
We observe that, because φ b c , φ b m and h b are constant on each inclusion, their two-scale limits do not depend on y in Y 1 .

Last, let us introduce the space of Kirchhoff-Love's displacement fields:

V KL = {v ∈ H 1 (Ω); v = 0 on Γ D , (s i3 (v)) i=1,2,3 = 0} = {(v 1 -x 3 ∂ 1 v 3 , v2 -x 3 ∂ 2 v 3 , v 3 ); v := (v 1 , v2 ) ∈ H 1 (ω), v 3 ∈ H 2 (ω), v = ∇ xv 3 = 0 and v 3 = 0 on Γ D }. ( 21
)
The following lemma was proved in [START_REF] Canon | Modelling of thin elastic plates with small piezoelectric inclusions and distributed electronic circuits. Models for inclusions that are small with respect to the thickness of the plate[END_REF], the mean-operator M being defined in (1).

Lemma 4.1. If assumptions [START_REF] Canon | Models of elastic plates with piezoelectric inclusions, part I: models without homogenization[END_REF][START_REF] Lenczner | Homogenization of periodic electrical networks including voltage to current amplifiers[END_REF][START_REF] Nguentseng | A general convergence result for a functional related to the theory of homogenization[END_REF][START_REF] Raoult | Contribution à l'étude des modèles d'évolution de plaques et à l'approximation d'équations d'évolution linéaires du second ordre par des méthodes multipas[END_REF] and conventions [START_REF] Canon | Deux modèles de plaques minces avec inclusions piézoélectriques et circuits électroniques distribués[END_REF][START_REF] Ciarlet | A justification of the two-dimensional plate model[END_REF][START_REF] Collet | Semi-active optimization of 2D wave dispersion into shunted piezo-composite systems for controlling acoustic interaction[END_REF] hold then (i) for each fixed b there is a unique solution to [START_REF] Ghergu | Homogenization of thin piezoelectric perforated shells[END_REF];

(ii) ∥K a (u b )∥ L 2 (Ω) + ∥L a (φ b )∥ L 2 (Ω ε 1 ) + √ G 1 ∥∇ ε xM(L a 3 (φ b ))∥ L 2 (Ω ε 1 ) ≤ C; (iii) there exists M = (K, L) ∈ (L 2 (Ω×Y )) 7 ×(L 2 (Ω×Y 1 )) 3 such that (M a (U b )) two-scale converges to M in L 2 (Ω × Y ) × L 2 (Ω × Y 1 ); (iv) there exists u ∈ V KL and u 1 =(u 1 1 , u 1 2 , 0) with u 1 1 , u 1 2 ∈ L 2 (Ω; H 1 ♯ (Y )/R) such that (u b ) converges weakly to u in H 1 (Ω), (∇ xu b ) two-scale converges to ∇ xu + ∇ y u 1 in L 2 (Ω × Y 1 ), (∂ 3 u b ) two-scale converges to ∂ 3 u in L 2 (Ω × Y 1 ); (v) (φ b ) two-scale converges to φ m in L 2 (Ω × Y 1 ); (vi) there exists φ 1 ∈ L 2 (Ω; H 1 (Y 1 )/R) such that t (L 1 , L 2 ) = ∇ y φ 1 , (vii) M(L 3 ) is independent of y and for Dirichlet conditions M(L 3 ) = φ c ; (viii) In the case of nonlocal mixed conditions, M(L 3 ) ∈ H 1 (ω) and (∇ ε xM(L a 3 (φ b ))) two-scale converges to ∇ xM(L 3 ) in L 2 (Ω × Y 1 ).
5. Limit Models I -Effective composite plate models when the thickness is small with respect to the size of the inclusions. This section is devoted to the statement of the two-dimensional composite plate models when the thickness of the plate is much smaller than the size of the inclusions, that is when ε, a and a/ε tend together to zero. The models are derived by coupling the homogenization method and the asymptotic method for plates. Consequently, they present characteristics of both approaches. Since no other situation occurs when a is small with respect to ε, it would be a posteriori equivalent to derive first the two-dimensional plate model and then to apply the method of homogenization to obtain a homogeneous two dimensional plate model. The results are summarized in Theorem 5.1 below, which is given after introducing a few notations: the projections involved by the plate approach, and the local problems and homogenized tensors involved by the homogenization process. The proof is postponed to Section 7.

Notations related to plate theory

The stiffness -piezoelectric -permittivity coefficients of the two dimensional plate model obtained by eliminating the transverse components are as follows.

           Π and Π 1 are respectively the projections from (L 2 (Ω × Y )) 7 × (L 2 (Ω × Y 1 )) 3 on its subspaces {(0 4 , (K i3 ) i=1..3 , 0 2 , L 3 )} and {(0 9 , L 3 )} , Π 2 = Π -Π 1 , T N = -(ΠRΠ) -1 ΠR, T M = -(Π 2 RΠ 2 ) -1 Π 2 R , R N = (Id + t T N )R(Id + T N ), R M = (Id + t T M )(R + 2GΠ 1 )(Id + T M ). ( 22
)
Remark: with notations like (ΠRΠ) -1 , we mean the inverse application of ΠRΠ as an application onto {(0 4 , (K i3 ) i=1..3 , 0 2 , L 3 )} (for the sake of simplicity in the notation, Π, Π 2 are identified with their transposed applications).

Notations related to homogenization theory

Let R Mαβγδ , and R N αβγδ denote the relevant coefficients of the tensors R M , R N and R M ix M written on the format [START_REF] Figueiredo | A piezoelectric anisotropic plate model[END_REF]. Let

H 2 ♯ (Y ) = {v ∈ H 2 (Y ); v and ∇v are Y -periodic}.
For any v ∈ H 1 (Y ), let

S αβ (v) = 1 2 (∂ yα v β + ∂ y β v α ). ( 23
) Let (u γδ M , u γδ N 3 ) ∈ (H 1 ♯ (Y )) 2 × H 2 ♯ (Y ), for γ, δ ∈ {1, 2}, be the solutions of          ∀v ∈ (H 1 ♯ (Y )) 2 , ∫ Y S αβ (v)R Mαβλµ S λµ (u γδ M ) dy = - ∫ Y S αβ (v)R Mαβγδ dy, ∀v 3 ∈ H 2 ♯ (Y ), ∫ Y ∂ 2 yαy β v 3 R N αβλµ ∂ 2 y λ yµ u γδ N 3 dy = - ∫ Y ∂ 2 yαy β v 3 R N αβγδ dy. ( 24
)
Let

u 3 M ∈ (H 1 ♯ (Y )) 2 be the solution of ∀v ∈ (H 1 ♯ (Y )) 2 , ∫ Y S αβ (v)R Mαβλµ S λµ (u 3 M ) dy = - ∫ Y d M3αβ S αβ (v) dy. ( 25
)
The effective tensors

R H M , R H N , d H M , e H
M and c H 33 are then defined by

R H N γδρξ = ∫ Y (δ αβ,γδ + ∂ 2 yαy β u γδ N )R N αβλµ (δ λµ,ρξ + ∂ 2 y λ yµ u ρξ N ) dy, (26)
         ( R H Mγδρξ d H M3γδ e H M3ρξ c H M33 ) = ∫ Y ( δ αβ,γδ + S αβ (u γδ M ) 0 S λµ (u 3 M ) 1
) ( R Mαβλµ d M3αβ -d M3λµ c M33 ) ( δ λµ,ρξ + S λµ (u ρξ M ) S λµ (u 3 M ) 0 1 ) dy, (27)
where the notations δ λµ,ρξ are Kronecker symbols, that is

δ λµ,ρξ = 1 if λ = ρ and µ = ξ, = 0 if not.
For local mixed conditions we also need

R H,loc Mαβγδ = R H Mαβγδ - ( c H M33 + 2|Y 1 |G ) -1 d H M3αβ e H M3γδ .
Let then

ℓ u (v) = ∫ Ω f .v dx + ∫ Γ N g.v ds, (28)
ℓ(v) =              ℓ u (v) -2 ∫ ω s αβ (v)d H M3αβ φ c dx (Dirichlet cond.), ℓ u (v) -2 ∫ ω s αβ (v)d H M3αβ ( c H M33 + 2|Y 1 |G ) -1 |Y 1 |h dx (local mixed cond.), ℓ u (v) + 2 |Y 1 | ∫ ω
L3 h dx (nonlocal mixed cond.).

Theorem 5.1: effective models

The set V KL of Kirchoff-Love displacement fields have been defined in [START_REF] Ratier | Analog computing of partial differential equations[END_REF].

Theorem 5.1. Assume that the assumptions of Section 2.4 hold and that a, ε and a/ε tend together to zero, (i) in the case of Dirichlet conditions (u b ) converges to

u = (ū 1 -x 3 ∂ 1 u 3 , ū2 -x 3 ∂ 2 u 3 , u 3 ) ∈ V KL which is the unique solution in V KL of ∀ v ∈ V KL , ∫ ω ( 2s αβ (v)R H Mαβγδ s γδ (ū) + 2 3 ∂ 2 αβ v 3 R H N αβγδ ∂ 2 γδ u 3 ) dx = ℓ(v); ( 29
) (ii) in the case of local mixed conditions, (u b ) converges to u which is the unique solution in V KL of ∀ v ∈ V KL , ∫ ω ( 2s αβ (v) R H,loc Mαβγδ s γδ (ū) + 2 3 ∂ 2 αβ v 3 R H N αβγδ ∂ 2 γδ u 3 ) dx = ℓ(v); (iii) for nonlocal mixed conditions, (u b , M(L a 3 (φ b ))) converges to (u,L 0 3 ) = (ū 1 -x 3 ∂ 1 u 3 , ū2 -x 3 ∂ 2 u 3 , u 3 , L 0 3 ) ∈ V KL × H 1 (ω) which is the unique solution of                ∀ (v, L3 ) = (v 1 -x 3 ∂ 1 v 3 , v2 -x 3 ∂ 2 v 3 , v 3 , L3 ) ∈ V KL × H 1 (ω), ∫ ω ( 2(s αβ (v), L3 ) ( R H Mαβγδ d H M3αβ e H M3γδ c H M33 + 2|Y 1 |G ) ( s γδ (ū) L 0 3 )) dx + ∫ ω ( 4|Y 1 |G 1 ∂ α L3 ∂ α L 0 3 + 2 3 ∂ 2 αβ v 3 R H N αβγδ ∂ 2 γδ u 3 ) dx = ℓ(v).
Remarks: (i) Since elasticity coefficients are constant in the thickness, the membrane and the flexion models are discoupled as for a single-layer elastic plate.

(ii) for Dirichlet, and local mixed conditions, the limit model has the same form as the elastic plate model, the influence of piezoelectric inclusion only appears in the definition of the effective coefficients and as a source term.

(iii) The piezoelectric force operates only in the membrane model as for the single-layer piezoelectric plate. A piezoelectric force may be generated in the flexion model by a torque in a multi-layer piezoelectric plate.

(iv) In a thin plate with an imposed voltage φ c on the upper face of a piezoelectric inclusion ω P , the resulting force is concentrated along the boundary of the inclusion and may be expressed as a distribution v → ∑ 2 α,β=1 ⟨d αβ φ c , s αβ (v)⟩ ω P where v is a regular test function, d the piezoelectric coefficient, and ⟨., .⟩ ω P the distribution bracket over the mean surface of the inclusion. For a periodic distribution of piezoelectric inclusions with an imposed voltage φ c on the upper faces, as considered in this paper, the resulting force is distributed over the complete mean surface ω and is still a distribution v → ∑ 2 α,β=1 ⟨d αβ φ c , s αβ (v)⟩ ω because in general, the distributed voltage φ c ∈ L 2 (ω) has no regularity.

(v) For nonlocal conditions, the presence of derivatives in the term

∫ ω ∂ α L3 ∂ α L 3 dx is
due to the electric circuits connecting neighbouring cells, which results in a regularization of the transverse component of the electric field L 3 and in a direct nontrivial coupling between mechanical and electrical effects. Note that one may a priori choose the form of the differential operator on L 3 by choosing the way to connect the upper faces of the inclusions to each others. This last point is interresting in view of controlling the structure.

(vi) For the case of local mixed conditions, the local circuit, i.e. without connections between cells, allows elimination of L 3 from the effective equations. Even if it is actuated by the current source h, the piezoelectric force is similar to the case of an imposed voltage with

φ c = |Y 1 |h/ ( c H M33 + 2|Y 1 |G ) where 1/ ( c H M33 + 2|Y 1 |G
) is the impedance of the local circuit.

(vii) It would also be possible to keep the voltage L 3 in the model with local mixed conditions. The resulting model would be similar to the model with nonlocal condition, i.e. with the current source h, but with G 1 = 0. This formulation is most suited for extension to dynamic problems where the admittance G is generally an integro-differential operator in time.

Limit Models II -Effective composite plate models when a ∼ ε.

This section is devoted to the statement of the two-dimensional limit models when a and ε tend to zero while a/ε tends to a finite positive limit. Then, there is no loss of generality in assuming that a/ε tends to 1. If a/ε tends to any other positive number ℓ ∈ R + * , one just has to normalize the coordinates (x 1 , x 2 ) in an appropriate way. However ℓ must be not too far from 1. If not, one of the two other models, a/ε → 0 if ℓ << 1, ε/a → 0 if ℓ >> 1, would be to consider. The results are summarized in Theorem 6.1 below, which is given after introducing necessary preliminary notations, in particular the local problems to be solved to compute the effective coefficients of the limit models. The proof is postponed to Section 8.

Notations related to the microstructure.

Because a and ε are of the same order of magnitude, we have here a unique three-dimensional microstructure containing both the y and x 3 directions. Let us introduce appropriate notations

Z = Y ×] -1, 1[, Z 1 = Y 1 ×] -1, 1[, z = (y 1 , y 2 , x 3 ), Γ + = Y 1 × {1} , Γ -= Y 1 × {-1} , W 1 = ( L 2 (] -1, 1[; H 1 ♯ (Y )) ∩ H 1 (Z) ) 3 × Ψ 1 where Ψ 1 = {ψ ∈ H 1 (Z 1 ); ψ = 0 on Γ -∪ Γ + }. ( 30
)
∀V 1 = (v 1 , ψ 1 ) ∈ W 1 , M 1 (V) = ((S ij (v 1 )) i,j=1..3 , ∇ z ψ 1 ))
where

S ij (v 1 ) = 1 2 (∂ z i v 1 j + ∂ z j v 1 i ). (31)
Let U λµ M = (u λµ M , φ λµ M ) ∈ W 1 designates for any (λ, µ) ∈ {1, 2} 2 the solutions to

∀V 1 ∈ W 1 , ∫ Z M 1 (V 1 )R t M 1 (U λµ M ) dz = - ∫ Z M 1 (V 1 ) t X λµ dz, (32)
where

X λµ = (R αβλµ , 2R α3λµ , R 33λµ , -d αλµ , -d 3λµ ) is for the first column of R.
Similarly, let U λµ N = (u λµ N , φ λµ N ) ∈ W 1 be solutions to

∀V 1 ∈ W 1 , ∫ Z M 1 (V 1 )R t M 1 (U λµ N ) dz = ∫ Z x 3 M 1 (V 1 ) t X λµ dz, ( 33
)
and U 3 = (u 3 , φ 3 ) ∈ W 1 be solution to

∀V 1 ∈ W 1 , ∫ Z M 1 (V 1 )R t M 1 (U 3 ) dz = ∫ Z M 1 (V 1 ) t X 3 dz , ( 34
)
where X 3 = (d 3αβ , 2d 3α3 , d 333 , c α3 , c 33 ) stands for the last column of R.

The effective coefficients are then given by:

                         R H MMλµρξ = ∫ Z (E λµ + M 1 (U λµ M ))R t (E ρξ + M 1 (U ρξ M )) dz, R H MN λµρξ = ∫ Z (E λµ + M 1 (U λµ M ))R t (-x 3 E ρξ + M 1 (U ρξ N )) dz, R H N Mλµρξ = ∫ Z ( -x 3 E λµ + M 1 (U λµ N ))R t (E ρξ + M 1 (U ρξ M )) dz, R H N N λµρξ = ∫ Z ( -x 3 E λµ + M 1 (U λµ N ))R t (-x 3 E ρξ + M 1 (U ρξ N )) dz,                                    d H MM3λµ = ∫ Z ( E λµ + M 1 (U λµ M ) ) R t ( b + M 1 (U 3 ) ) dz, d H N M3λµ = ∫ Z ( -x 3 E λµ + M 1 (U λµ N ) ) R t ( b + M 1 (U 3 ) ) dz, e H MM3αβ = ∫ Z ( b + M 1 (U 3 ) ) R t (E αβ + M 1 (U αβ M )) dz, e H MN 3αβ = ∫ Z ( b + M 1 (U 3 ) ) R t ( -x 3 E αβ + M 1 (U αβ N ) ) dz, c H MM33 = ∫ Z ( b + M 1 (U 3 ) ) R t ( b + M 1 (U 3 ) ) dz.
where E λµ = ((δ αβ,λµ ) α,β=1,2 , 0 6 ) and b =(0 9 , 1).

Effective models: Theorem 6.1

The set V KL of Kirchoff-Love displacement fields have been defined in [START_REF] Ratier | Analog computing of partial differential equations[END_REF].

Theorem 6.1. Assume that the assumptions of Section 2.4 hold and in addition that a, ε tend to zero and a/ε tends to 1 then (i) in the case of Dirichlet conditions, the limit u of (u b ) b>0 satisfies:

u =(ū 1 -x 3 ∂ 1 u 3 , ū2 -x 3 ∂ 2 u 3 , u 3 ) ∈ V KL and is the unique solution in V KL of          ∀v ∈ V KL , ∫ ω (s αβ (v), ∂ 2 αβ v 3 ) ( R H MMαβγδ R H MN αβγδ R H N Mαβγδ R H N N αβγδ ) ( s γδ (ū) ∂ 2 γδ u 3 ) dx = ℓ u (v) - ∫ ω (s αβ (v), ∂ 2 αβ v 3 )d H MM3αβ φ c dx;
(ii) in the case of mixed conditions, the limit u =(ū 1 -

x 3 ∂ 1 u 3 , ū2 -x 3 ∂ 2 u 3 , u 3 ) of (u b ) b>0 belongs to V KL , the limit L 0 3 of M(L 3 (φ b )) belongs to L 2 (ω) or to H 1 (ω) for nonlocal conditions, and (u, L 0 3 ) is the unique solution in V KL × L 2 (ω) or in V KL × H 1 (ω) for nonlocal conditions of                    ∀(v, L 3 ) ∈ V KL × L 2 (ω)(resp. V KL × H 1 (ω)), ∫ ω (s αβ (v), L 3 , ∂ 2 αβ v 3 )   R H MMαβγδ d H MM3αβ R H MN αβγδ e H MM3αβ c H MM33 d H MN 3αβ R H N Mαβγδ e H MN 3αβ R H N N αβγδ     s γδ (ū) L 0 3 ∂ 2 γδ u 3   dx + ∫ ω ( 4|Y 1 |(G L 3 L 0 3 + G 1 ∂ α L 3 ∂ α L 0 3
)

) dx = ℓ u (v) + 2|Y 1 | ∫ ω L 3 h dx.
Remarks: (i) On the contrary to the model in Section 5, the membrane and flexion models are coupled. This comes from the presence of a microstructure which size is comparable to the plate thickness, and generates a complex displacement field at the scale of the plate thickness.

(ii) The other remarks (iii-vii) also hold for this model.

Proof of Theorem 5.1.

This section is devoted to the derivation of Theorem 5.1. The proof is based on the general results of Lemma 4.1. The proof consists in three steps. The first one is the characterization of the limit M of (M a (V)) and is the aim of Section 7.1. The second one consists in the elimination of the x 3 -variable as usual in plate theory. The last one is the elimination of the local variable as in homogenization theory and obtention of the effective models. The last two steps slightly differ from one boundary condition to the other, and thus are presented separately. In particular, in the case of local mixed conditions, the complete elimination of the transverse component is achieved after the elimination of the local variable. This two steps are presented in Section 7.2. The assumption that R does not depend on x 3 is widely used there to obtain simplifications and compute our limit models.

7.1

Step 1: characterization of the limit M.

In the forthcoming Lemma 7.4, the limit M is related to the limits u, u 1 and φ 1 defined in Lemma 4.1. The limit of equation ( 14) is given in Lemma 7.5. Suitable choices of tests functions V lead here to a list of equations corresponding to the various asymptotic levels of M a (V). But first, we give à few additionnal notations, and technical results, in section 7.1.1.

Further notations, and preliminary lemmas.

Recall that

C ∞ ♯ (Y ) denotes the subspace of all Y -periodic functions of C ∞ (R 2 ). Similarly, let C ∞ ♯ (Y 1
) denote the subspace of all Y -periodic functions of C ∞ (Z 2 +Y 1 ). For functions v on Ω × R 3 which are Y -periodic with respect to the second variable, we designate by v ε the function x -→ v(x, x/ε). We also use test functions in

W 1 ad = { (v 1 , ψ 1 ) ∈ D( Ω, C ∞ ♯ (Y )) × Ψ 1 ad ; v 1 = 0 on Γ D × Y } , ( 35
)
W ad = { (v, ψ) ∈ H 1 (Ω) × Ψ ad ; v = 0 on Γ D } ,
where

Ψ 1 ad = D(ω×] -1, 1[; C ∞ ♯ (Y 1 )) for Dirichlet conditions, = { ψ 1 ∈ D(ω×] -1, 1]; C ∞ ♯ (Y 1 )); ψ 1 is constant for x 3 = 1 } otherwise, Ψ ad = D (ω×] -1, 1[) for Dirichlet conditions, = {ψ ∈ D (ω×] -1, 1]) ; ψ is constant for x 3 = 1} otherwise.
Functions in W ad and in W 1 ad are admissible test functions for [START_REF] Ghergu | Homogenization of thin piezoelectric perforated shells[END_REF]. For V =(v, ψ) ∈ W 1 ad , we have the natural decomposition of M a (V ε ):

M a (V ε ) = (M 00 (V)) ε + 1 ε (M 10 (V)) ε + 1 a (M 01 (V)) ε + 1 εa (M 11 (V)) ε + 1 a 2 (M 02 (V)) ε , ( 36
)
where

                     M 00 (V) = ((s αβ (v)) α,β=1,2 , 0 3 , (∂ α ψ) α=1,2 , 0), M 10 (V) = ((S αβ (v)) α,β=1,2 , 0 3 , (∂ yα ψ) α=1,2 , 0), M 01 (V) = (0 2×2 , (s α3 (v)) α=1,2 , 0 3 , ∂ 3 ψ), M 11 (V) = (0 2×2 , ( 1 2 ∂ yα v 3 ) α=1,2 , 0 4 ), M 02 (V) = (0 2×2 , , 0 2 , s 33 (v), 0 3 ). (37)
Notations s ij (v) and S αβ (v) have been defined in ( 4) and [START_REF] Sene | Modelling of piezoelectric static thin plates[END_REF]. Relevant subspaces M, M 0 , M -1 and M -2 associated to this decomposition are defined by

                 M 0 = {(K αβ , 0 6 ); K αβ = s αβ (v) + S αβ (v 1 ), v ∈ V KL , v 1 ∈ V 1 KL }, M -1 = {(0 2×2 , K α3 , 0 3 , L 3 ); K α3 ∈ L 2 (Ω × Y ), L 3 ∈ L 2 (Ω × Y 1 ), M(L 3 ) is independant of y; and M(L 3 ) = 0 for Dirichlet conditions}, M -2 = {(0 2×2 , 0 2 , K 33 , 0 3 ); K 33 ∈ L 2 (Ω × Y )}, M = M -2 ⊕ M -1 ⊕ M 0 . ( 38
)
where

V 1 KL = {((v 1 α -x 3 ∂ yα v 2 3 ) α=1,2 , 0); v1 α ∈ L 2 (ω; H 1 ♯ (Y )), v 2 3 ∈ L 2 (ω; H 2 ♯ (Y ))}. ( 39
)
Our first lemma contains density results that will lead to a suitable weak formulation for the problem solved by M.

Lemma 7.1. (i) The set {M 02 (V); V ∈ W 1 ad } is dense in M -2 , (ii) The set {M 01 (V); V ∈ W ad , v 3 = 0} is dense in M -1 , (iii) The set {M 0 (V) + M 10 (V 1 ); (V, V 1 ) = ((v,0), (v 1 , 0)) where (V, V 1 ) ∈ W 1 ad , (v, v 1 ) ∈ V KL × V 1 KL } is dense in M 0 . Proof : (i) Let K 33 ∈ C ∞ (Ω × Y ) with compact support. This function K 33 can also be considered as a function of D(Ω, C ∞ ♯ (Y )) by letting ∀x ∈ Ω, ∀y ∈ R 2 , K 33 (x, y) = K 33 (x, y ′ ),
where y ′ is défined by: y ′ ∈ Y and y -y ′ ∈ Z 2 . Then, v 3 defined by:

v 3 (x, y) = ∫ x 3 0 K 33 (x, t, y)dt satisfies: (0 2 , v 3 , 0) ∈ W 1 ad and ∂ 3 v 3 = K 33 . As the set of functions of C ∞ (Ω × Y ) with compact support is dense in L 2 (Ω × Y ), this proves that {M 02 (V); V ∈ W 1 ad } is dense in M -2 .
(ii) The proof is similar for the first two terms K α3 of the elements M -1 . For the third term L 3 : let L 3 be a given function in C ∞ (Ω × Y 1 ) with compact support; we also assume that M(L 3 ) does not depend on y, and that in addition M(L 3 ) = 0 in the case of Dirichlet conditions. As in (i), we now consider L 3 as a function of D(Ω, C ∞ ♯ (Y 1 )); let then ψ be defined by ψ(x, y) =

∫ x 3 -1 L 3 (x, t, y) dt.
With the two conditions above on L 3 , ψ is an admissible test function such that ∂ 3 ψ = L 3 ; with densiry arguments again, this completes the proof of (ii). Part (iii) is just a restatement of the definition of M 0 . 2

We now give two technical lemmas that are used in the next section to caracterize the limit M. Lemma 7.2. Let (u ε ) ε>0 be bounded in H 1 (Ω). Let u ∈ H 1 (Ω) and u 1 ∈ L 2 (Ω; H 1 ♯ (Y )) be functions such that (u ε ) ε>0 converges weakly to u in H 1 (Ω), and (∇u ε ) ε>0 two-scale converges to ∇u + ∇ y u 1 . Then

∀v ∈ D(Ω; C ∞ ♯ (Y )), lim ε→0 ∫ Ω u ε ε ∂ yα v ε dx = ∫ Ω×Y u 1 ∂ yα v dxdy.
Proof : One just has to pass to the limit in

∫ Ω ∂ α u ε (x)v(x, x/ε) dx = - ∫ Ω u ε (x)∂ α v(x, x/ε) dx - 1 ε ∫ Ω u ε (x)∂ yα v(x, x/ε) dx.
An integration by parts of the first term on the right-hand side yields then the result. 2

Lemma 7.3. The family [START_REF] Ghergu | Homogenization of thin piezoelectric perforated shells[END_REF], where ṽα ∈ D(ω; C ∞ ♯ (Y )) and ṽα = 0 on Γ D . Then

( ε 2 a ∫ Ω ( R ε α3kl K a kl (u b ) + d ε α3k L a k (φ b ) ) ṽα dx ) tends to zero for any ṽα ∈ D(ω; C ∞ ♯ (Y )) such that ṽα = 0 on Γ D . Proof : Choose v = ε 2 x 3 (ṽ 1 , ṽ2 , 0) and ψ = 0 in
ε 2 ∫ Ω (R ε αβkl K a kl (u b ) + d ε αβk L a k (φ b ))(s αβ (v) + 1 ε S αβ (v)) dx+ ε 2 a -1 ∫ Ω 2(R ε α3kl K a kl (u b ) + d ε α3k L a k (φ b )) ṽα dx = ε 2 ℓ b u (v).
Passing to the limit yields the result. 2

Characterization of the limit M.

The first lemma of this section relates to the form of the limit M. We prove that M ∈ϕ c +M where ϕ c = (0 9 , φ c ). The second gives a first variational characterization of M.

Lemma 7.4. Let M designates the two-scale limit, up to the extraction of a subsequence, of (M b (U b )) b>0 . Then M ∈ϕ c + M where ϕ c = (0 9 , φ c ), or in other words:

(i) there exist u ∈ V KL and u 1 ∈ V 1 KL such that K aβ = s αβ (u) + S αβ (u 1 ); (ii) L 1 = L 2 = 0; (iii) L 3 ∈ L 2 (Ω × Y 1 ), M(L 3 ) is independant of y; and M(L 3 ) = φ c for Dirichlet conditions.
The third point is just a restatement of Lemma 4.1 (vii).

Proof of (i):

In view of Lemma 4.1, to prove (i), it remains to prove that u 1 ∈ V 1 KL . The choice u 1 3 = 0 has been done in Lemma 4.1 (iv). The computation of u 1 α requires some calculus. First, with a few integration by parts we get

∫ Ω (∂ 1 u b 2 -∂ 2 u b 1 )∂ 3 v ε dx = 2a ∫ Ω K a 23 (u b ) ( ∂ 1 v ε + 1 ε ∂ y 1 v ε ) dx -2a ∫ Ω K a 13 (u b ) ( ∂ 2 v ε + 1 ε ∂ y 2 v ε ) dx (40)
for all v ∈ D(Ω, C ∞ ♯ (Y )). Passing to the limit, as a/ε tends to zero, we get

∀v ∈ D(Ω, C ∞ ♯ (Y )), ∫ Ω×Y (∂ 1 u 2 -∂ 2 u 1 )∂ 3 v dxdy + ∫ Ω×Y (∂ y 1 u 1 2 -∂ y 2 u 1 1 )∂ 3 v dxdy = 0. But because u ∈ V KL , this reduces to ∀v ∈ D(Ω, C ∞ ♯ (Y )), ∫ Ω×Y (∂ y1 u 1 2 -∂ y 2 u 1 1 )∂ 3 v dxdy = 0. (41)
Now, we show that

∀v ∈ D(Ω × Y ), ∫ Ω×Y ∂ y β u 1 α ∂ 2 33 v dxdy = 0. (42) As (K a 33 (u b )) is bounded, ε -1 ∫ Ω ∂ 3 u b 3 (∂ α ṽε + ε -1 ∂ yα ṽε )
∫ Ω u b α ∂ y β ∂ 2
33 v ε dx tends to zero. Then, using Lemma 7.2, we get (42). Now, we are able to conclude the proof. Equation (42) implies that ∇ y u 1 α is affine with respect to x 3 . Thus, since (u

1 1 , u 1 2 ) is defined up to a function of x, u 1 α (x, y) = u 1 α (x, y) + x 3 u 2 α (x, y) where ū1 α , u 2 α ∈ L 2 (ω × Y ). But then ∂ 3 u 1 α = u 1 α ∈ L 2 (ω × Y ).
On the other hand, in view of (41), (∂

3 u 1 1 , ∂ 3 u 1 2
) is curl free, so that there exists c α ∈ L 2 (ω) and

u 2 3 ∈ L 2 (ω; H 1 ♯ (Y )) such that ∂ 3 u 1 α = c α -∂ yα u 2 3
. This finally implies that (u 1 1 , u 1 2 ) may be chosen of the form

u 1 α = ū1 α -x 3 ∂ yα u 2 3 where ū1 ∈ (L 2 (ω × Y )) 2 and u 2 3 ∈ L 2 (ω; H 1 ♯ (Y )). 2
Proof of (ii): Passing to the limit in

a ∫ Ω ε 1 L a 3 (φ b ) ( ∂ α ψ ε + 1 ε ∂ yα ψ ε ) dx = ∫ Ω ε 1 L a α (φ b )∂ 3 ψ ε dx, (43)
because a/ε tends to zero, we get that

∀ψ ∈ D(Ω × Y 1 ), ∫ Ω×Y 1 L α ∂ 3 ψ dxdy = 0.
This proves that L 1 and L 2 are independent of x 3 . On the other hand, a few integrations by parts, with test functions ψ independent of x 3 , yield

a ∫ Ω ε 1 L a 3 (φ b ) (x 3 -1) (∂ α ψ ε + 1 ε ∂ yα ψ ε ) dx = ∫ Ω ε 1 ∂ α φ b ψ ε dx + 2 ∫ ω ε 1 (φ b m -aφ b c )(∂ α ψ ε + 1 ε ∂ yα ψ ε ) dx.
The left-hand side tends to zero because a, ε and a/ε tend to zero. Since φ b m and φ b c are constant on each connected part of ω ε 1 , the last term on the right is zero. Therefore, passing to the limit, we get

0 = 2 ∫ ω×Y 1 L α ψ dxdy.
Since L α does not depend on x 3 , this proves that L α = 0. 2 Lemma 7. [START_REF] Canon | Models of elastic plates with piezoelectric inclusions, part I: models without homogenization[END_REF] The family (M a (U b )) b>0 two-scale converges to M which is the unique

solution in ϕ c + M of        ∀ M = ( K, L) ∈ M, ∫ Ω×Y M R t M dxdy + 2G ∫ Ω×Y 1 M(L 3 )M( L3 ) dxdy +2G 1 ∫ Ω×Y 1 ∂ α M(L 3 )∂ α M( L3 ) dxdy = ℓ u (v) + ℓ φ ( L3 ) ( 44
)
where M is defined in (38), and

ℓ φ ( L3 ) = ∫ Ω×Y 1 h L3 dxdy = 2 |Y 1 | ∫ ω hM( L3 ) dx. ( 45
)
Remark : as usual, the convergence is a priori for a subsequence and the uniqueness of the solution to the limit problem shows a posteriori that the whole family converges.

Proof First, successive multiplications of ( 14) by a 2 , a and 1 and passing to the limit yield for all [START_REF] Ghergu | Homogenization of thin piezoelectric perforated shells[END_REF], so that εM 00 (V ε ) tends to 0, M 02 (V ε ) = 0, and

V ∈ W 1 ad                            ∫ Ω×Y M 02 (V) R t M dxdy = 0, ∫ Ω×Y M 01 (V) R t M dxdy + 2G ∫ Ω×Y 1 M(L 3 )M (∂ 3 ψ) dxdy + 2G 1 ∫ Ω×Y 1 ∂ α M(L 3 )∂ α M(∂ 3 ψ) dxdy = ℓ φ ( L3 ) if M 02 (V) = M 11 (V) = 0, ∫ Ω×Y M 00 (V) R t M dxdy = ℓ u (v) if M 02 (V) = M 11 (V) = M 01 (V) = M 10 (V) = 0. (46) Second, let v = (v α -x 3 ∂ yα v 2 3 , v 2 3 , 0) ∈ V 1 KL ∩ W 1 ad , with v 2 3 = 0 in a neighbourhood of Γ D , and choose V ε = ε (v 1 -x 3 ∂ y 1 v 2 3 , v2 -x 3 ∂ y 2 v 2 3 , εv 2 3 , 0) in
1 a M 01 (V ε ) + 1 aε M 11 (V ε ) = (0 2×2 , ε 2 2a ∂ α v 2 3 , 0 4 ).
Using lemma 7.3, passing to the limit in [START_REF] Ghergu | Homogenization of thin piezoelectric perforated shells[END_REF] yields

∫ Ω×Y M 10 (V) R t M dxdy = 0 ( 47
)
for any V = (v,0) ∈ W 1 ad with v ∈ V 1 KL . Now, summing the equations ( 46)-(47) and using then the density lemma 7.1, we get (44). Last, uniqueness of the solution to this system is a simple application of Lax-Milgram lemma. 2

Proof of Theorem 5.1.

From now on, an important use is made of the assumption that R does not depend on x 3 .

Proof of Theorem 5.1 : Dirichlet conditions.

We recall that here M(L 3 ) = φ c and that M and N have been defined in [START_REF] Allaire | Homogenization and two scale-convergence[END_REF]. The proof is in two steps. The first one consists in eliminating K i3 and N (L 3 ); the second one is for eliminating the y-variable. A key for many simplifications is that R does not depend on x 3 and so M(M) and N (M) bring separate contributions. First step. Remembering that in the present case, G = G 1 = 0 and h = 0, Equation (44) of Lemma 7.5 reduces to

∀ M ∈ M, ∫ Ω×Y M R t M dxdy = ℓ u (v).
Splitting M as M = M(M) + N (M) and taking advantage of the fact that R does not depend on x 3 , this is equivalent to

∀ M ∈ M, ∫ Ω×Y M( M) R t M(M) dxdy + ∫ Ω×Y N ( M) R t N (M) dxdy = ℓ u (v). (48)
Let us chose M : = M( M) as test functions in (48) where M ∈ M -1 ⊕ M -2 . Because the function v associated with such a M is v = 0, and because N • M = 0, we get

∀ M ∈ M -1 ⊕ M -2 , ∫ Ω×Y M( M) R t M(M) dxdy = 0.
Using again that R is independant of x 3 , this may be rewritten as

∀ M ∈ M -1 ⊕ M -2 , ∫ Ω×Y M R t M(M) dxdy = 0.
This holds true in particular for M of the form M =(0 2×2 , Ki3 , 0 3 ) and thus shows that

Π 2 R t M(M) = 0 (49)
On the other hand, Lemma 7.4 shows that one may decompose M as

M = ΠM + M 0 (50)
where ΠM ∈ ϕ c + M -1 ⊕ M -2 , M 0 ∈ M 0 , and that M(ΠM) may be written as

M(ΠM) = (0 2×2 , M(K i3 ), 0 2 , M(L 3 )) = ϕ c + Π 2 M(M).
Hence, applying M to (50) we get

M(M) = ϕ c + M(Π 2 M) + M(M 0 ) = ϕ c + Π 2 M(Π 2 M) + M(M 0 ). (51)
Applying now Π 2 R to this identity, with (49), we obtain then

0 = Π 2 R t ϕ c + (Π 2 RΠ 2 ) t M(Π 2 M) + Π 2 R t M(M 0 ), and therefore t M(Π 2 M) = T M t (ϕ c + M(M 0 )), (52)
where T M has been defined in [START_REF] Sanchez-Palencia | Non-Homogeneous Media and Vibration Theory[END_REF].

Similarly, let us now consider in (48) test functions of the form N ( M) where M = (0 4 , Ki3 , 0 2 , L3 ), Ki3 , L3 being given functions in L 2 (Ω × Y ) and L 2 (Ω × Y 1 ) respectively. As M(N ( L3 )) = 0, N ( M) ∈ M -1 ⊕ M -2 so that this choice is correct. Proceding as for the M-component of M, we get that

∀ M = (0 4 , Ki3 , 0 2 , L3 ), ∫ Ω×Y M R t N (M) dxdy = 0,
or in other words, ΠR t N (M) = 0. Then, from (50),

0 = ΠR t N (ΠM) + ΠR t N (M 0 ), and thus, t N (ΠM) = T N t N (M 0 ) (53)
where T N has been defined in [START_REF] Sanchez-Palencia | Non-Homogeneous Media and Vibration Theory[END_REF]. Thus, with (51), ( 52), ( 50) and (53) we have

M(M) = (Id + T M ) t (M(M 0 ) + ϕ c ), and N (M) = (Id + T N ) t N (M 0 ). (54)
Now, if we choose -in order to maintain a symmetric form to our tensors-in (48) test functions of the special form M =M( M 0 )(Id+ t T M )+N ( M 0 )(Id+ t T N ) with M 0 ∈ M 0 , taking (54) into account, we get that M 0 is the unique solution in

ϕ c + M 0 of        ∀ M ∈ M 0 , ∫ Ω×Y ( M( M) R M t M(M 0 ) + N ( M) R N t N (M 0 ) ) dxdy = ℓ u (v) - ∫ Ω×Y M( M) R M t ϕ c dxdy, ( 55
)
where v is the vector in V KL associated with M, and R M and R N have been defined in [START_REF] Sanchez-Palencia | Non-Homogeneous Media and Vibration Theory[END_REF]. Second step. It remains to eliminate u 1 . For that purpose, we use the usual arguments of linear homogenization. First note that the definitions (21) of V KL and (39

) of V 1 KL imply that    ∀ M ∈ M 0 , M( M) = (s αβ (v) + S αβ (v 1 ), 0 6 ) , ∀ M ∈ M 0 , N ( M) = -x 3 ( ∂ 2 αβ v 3 + ∂ 2 yαy β v 2 3 , 0 6 ) , (56)
where

v ∈ V KL and (v 1 1 , v1 2 , v 2 3 ) ∈ V 1
KL are the fields associated with M. Also, recall that

M 0 = ((s αβ (u) + (S αβ (u 1 )) αβ=1,2 , 0 6 ) with u ∈ V KL and u 1 ∈ V 1 KL .
Considering v = 0 in (55), we thus get that for all

v 1 ∈ (H 1 ♯ (Y )) 2 , ∫ Y S αβ (v 1 )R Mαβλµ S λµ (ū 1 ) dy = - ∫ Y S αβ (v 1 )(R Mαβλµ s λµ (ū) + d M3αβ φ c ) dy, (57)
and for all v

2 3 ∈ H 2 ♯ (Y ), ∫ Y ∂ 2 yαy β v 2 3 R N αβλµ ∂ 2 y λ yµ u 2 3 dy = - ∫ Y ∂ 2 yαy β v 2 3 R N αβλµ ∂ 2 λµ u 3 dy (58)
almost everywhere in ω. Equation (57) implies that ū1 = u ρξ M s ρξ (ū) + u 3 M φ c , where the coefficients functions u ρξ M , u 3 M are defined in [START_REF] Simon | Primitives de distributions et applications[END_REF] and ( 25), and therefore

s λµ (ū) + S λµ (ū 1 ) = (δ λµ,ρξ + S λµ (u ρξ M ))s ρξ (ū) + S λµ (u 3 M )φ c .
This is for the M-part of M 0 . Similarly, Equation (58) implies that and u 2 3 = u ρξ N ∂ 2 ρξ u 3 where the u ρξ N are also defined by [START_REF] Simon | Primitives de distributions et applications[END_REF], and this leads to

∂ 2 λµ u 3 + ∂ 2 y λ yµ u 2 3 = (δ λµ,ρξ + ∂ 2 y λ yµ (u ρξ N ))∂ 2 ρξ u 3 .
Now we take in (55) test functions as in (56), with the special forms v1 = u γδ M s γδ (v) and v 2 3 = u γδ N ∂ 2 γδ v 3 . This leads to the system (29) with

                 d H M3γδ = ∫ Y (δ αβ,γδ + S αβ (u γδ M )) ( R Mαβλµ S λµ (u 3 M ) + d M3αβ ) dy, R H Mγδρξ = ∫ Y (δ αβ,γδ + S αβ (u γδ M ))R Mαβλµ (δ λµ,ρξ + S λµ (u ρξ M )) dy, R H N γδρξ = ∫ Y (δ αβ,γδ + ∂ 2 yαy β u γδ N )R N αβλµ (δ λµ,ρξ + ∂ 2 y λ yµ u ρξ N ) dy.
which are exactly the tensors announced in formulae ( 26)- [START_REF] Zemanian | Infinite Electrical Networks[END_REF]. Note in particular that the coefficient 2 appears on the first term on left hand side of (29) because

∫ 1 -1 dx 3 = 2 while 2 3 = ∫ 1 -1
x 2 3 dx 3 appears on the second term. The proof is very closed to the one for Dirichlet conditions. The main difference is that in the Dirichlet case, M(L 3 ) = φ c is a given data, while here M(L 3 ) is an unknown function which can not be eliminated.

Let us eliminate the other transverse components K α3 , K 33 and N (L 3 ).

The weak formulation (44) of Lemma 7.5, implies now that

       ∀ M ∈ M, ∫ Ω×Y ( M( M) R t M(M) + N ( M) R t N (M) ) dxdy +2 ∫ Ω×Y 1 ( GM(L 3 )M( L3 ) + G 1 ∂ α M(L 3 )∂ α M( L3 ) ) dxdy = ℓ u (v) + ℓ φ ( L3 ). (59)
For M(M), we use test functions such that L3 = 0, as in Section 7.2.1, and thus the computation go on the same way and the result is the same, except that φ c is now replaced by M(L 3 ). For N (M), as M • N and as ℓ φ (N ( L3 )) = 0 (see (45)) the terms in G, G 1 and ℓ φ do not play any part as well, and the computation also go as in 7.2.1. We thus get M(M) = (Id + T M ) t ( M(M 0 ) + Λ 3 ) , and N (M) = (Id + T N ) t N (M 0 ).

Proof of Theorem 6.1.

This section is devoted to the derivation of Theorem 6.1. The proof is based on the general results of Lemma 4.1. As homogenization and plate theory act here on the same level, the proof is in two steps only: first, characterization of the limit M defined in Lemma 4.1; this is the aim of Section 8.1; second, elimination of the local variable (y 1 , y 2 , x 3 ); this is achieved in Section 8.2.

8.1

Step 1: Characterization of the limit M. (ii) there exists φ 1 ∈ L 2 (ω; H 1 (Z 1 )) such that L = ∇ z φ 1 ;

(iii) in the case of Dirichlet conditions, one may chose φ 1 so that φ 1 = x 3 φ c + φ1 with φ1 ∈ L 2 (ω; H 1 (Z 1 )) and φ1 = 0 on Γ + ∪ Γ -;

(iv) in the case of mixed conditions one may chose φ 1 so that φ 1 = (1 + x 3 ) M(L 3 ) + φ1 with φ1 ∈ L 2 (ω; H 1 (Z 1 )) and φ1 = 0 on Γ + ∪ Γ -.

Proof : First, from Lemma 4.1 we know that there is (u, u 1 ) ∈ V KL × L 2 (Ω; H 1 ♯ (Y )/R) 2 such that K αβ = s αβ (u) + S αβ (u 1 ). Then here, passing to the limit in (40) as a/ε → 1, we get, taking u ∈ V KL into account, that for all v ∈ D(Ω; C ∞ ♯ (Y )),

∫ Ω×Y (∂ y 1 u 1 2 -∂ y 2 u 1 1 )∂ 3 v dxdy = 2 ∫ Ω×Y (K 23 ∂ y 1 v -K 13 ∂ y 2 v) dxdy. (61)
Hence, see for instance [START_REF] Simon | Primitives de distributions et applications[END_REF]Theorem 5], there is u 2 3 and c 1 , c 2 not depending on y such that for α = 1, 2, ∂ 3 u 1 α -2K α3 = c α -∂ yα u 2 3 . Also, note that u 2 3 is defined up to the addition of a function of x. Here, as u 1 is also defined up to the addition of a function of x, one may choose u 1 α so that c α = 0. Letting û1 = (u 1 1 , u 2 2 , u 2 3 ) , then K α3 = S α3 ( u 1 ). We now prove that K 33 = ∂ 3 u 2 3 . With a few integrations by parts, one easily gets that for any v ∈ D(Ω × Y ), 3 -K 33 ) = 0. But because u 2 3 is defined up to an additive function of x we may choose u 2 3 such that K 33 = ∂ 3 u 2 3 . Then S ij (û 1 ) ∈ L 2 (Ω × Y ) for each pair (i, j) ∈ {1..3} 2 and therefore û1 ∈ L 2 (ω; H 1 (Z)). Last, using (61), u 2 3 is Y -periodic. This ends the proof of point (i). Now, we prove (ii) and (iii) in the case of Dirichlet conditions. From Lemma 4.1, we already know that there exists φ 1 such that L 1 = ∂ y 1 φ 1 and L 2 = ∂ y 2 φ 1 . Besides, passing to the limit in (43) yields ∫ Hence, since φ 1 is defined up to a function of x, one may choose φ 1 so that L 3 = ∂ 3 φ 1 and we have in addition that φ 1 ∈ L 2 (ω; H 1 (Z 1 )), so (ii) is proven. On the other hand, as we know from Lemma 4.1 that M(L 3 ) = φ c , we have that φ 1 /Γ + -φ 1 /Γ -= 2φ c . Since so far, φ 1 is still defined up to a function of x, we may choose φ 1 so that φ 1 = φ c on Γ + and φ 1 = -φ c on Γ -, or in other words, choose φ 1 as stated in (iii).

∫ Ω ∂ α u b β ∂ 2 33 v ε dx + ∫ Ω ∂ β u b 3 ( ∂ 2 α3 v ε + 1 ε ∂ 2 3yα v ε ) dx = 2a ∫ Ω K a β3 (u b )∂ 2 α3 v ε dx + 2 ∫ Ω K a β3 (u b )∂
The case of mixed conditions is more complicated. Let us reconsider the limit of ( L a (φ b ) )

globally. We remark that ∂ )

. Besides, as

ε -1 ∂ 3 ( φ b -φ b m ) = a ε L a 3 ( φ b -φ b m ) = a ε L a 3 ( φ

  a[, ω being a bounded domain of R 2 , see Figure 1. Using the change of scales and variables introduced in [10], we shall work on the fixed domain Ω = ω×] -1, 1[. The domain ω is divided into two subdomains ω ε 1 and ω ε 2 that are constructed as follows. Let Y =] -1/2, 1/2[ 2 and Y 1 be a strict smooth subdomain of Y , see Figure 2, let I ε be the set of multi-

  a[. The outer unit normals to the boundaries of Ω a and Y are denoted by n and n Y , respectively.

Figure 1 :

 1 Figure 1: Composite plate with piezoelectric inclusions.

Figure 2 :

 2 Figure 2: Elementary cell, with piezoelectric inclusion and metallization.

) φ b m and φ b c being two given functions on ω ε 1 .

 1 (ii) Prescribed electric displacement field (Neumann conditions), local and nonlocal electric circuits (local and nonlocal mixed conditions):

1 are

 1 Neumann condition (prescribed electric displacement field). If G > 0 and G 1 = 0 those are local mixed conditions. If G G 1 > 0 those are nonlocal mixed conditions. If G 1 = 0, the condition (9) is of course not relevant.

1 ∫Ω∂ α u b 3 ∂ 3 1 ∫Ω∂ 3

 1313 dx tends to zero for any ṽ ∈ D(Ω, C ∞ ♯ (Y )); hence, ε -ṽε dx tends to zero. Also, since (K a α3 ) is bounded, ε -1 s α3 (u b ) tends to zero. Consequently, ε -u b α ∂ 3 ṽε dx tends to zero. Choosing now ṽ = ∂ y β v where v ∈D(Ω × Y ), one gets that ε -1

2 7. 2 . 2

 222 Proof of Theorem 5.1 : nonlocal mixed conditions.

8. 1 . 1 Lemma 8 . 1 .

 1181 Characterization of M. Let M = (K, L) be the limit, up to a subsequence, of( M a (U b ) ) , then (i) there exist u ∈ V KL and û1 ∈ (L 2 (Ω; H 1 ♯ (Y )) ∩ L 2 (ω; H 1 (Z)) 3 such that ∀α, β ∈ {1, 2}, K aβ = s αβ (u) + S αβ (û 1 ),∀i ∈ {1, 2, 3}, K i3 = S i3 (û 1 );

Ω×Y 1 L 3 ∂ 1 ∂

 131 yα ψ dxdy = ∫ Ω×Y yα φ 1 ∂ 3 ψ dx dy for all α = 1, 2 and all ψ ∈ D(Ω × Y 1 ). Thus, ∂ 3 φ 1 -L 3 does not depend on y in Y 1 .

3 φ b = ∂ 3 ( φ b -φ b m )) for α = 1 , 2 .

 3m12 , and also, due to the assumption of metallization, that∂ α φ b = ∂ α ( φ b -φ b m The limit L of ( L a (φ b ) ) is therefore also the limit of ( L a (φ b -φ b m )

  bounded in L 2 (Ω ε 1 ) and also two-scale converge to L 3 . Thus, if we poseLb = ( ∂ α (φ b -φ m ), ε -1 ∂ 3 (φ b -φ m ) ) , then ( Lb )two-scale converges to L.

  )tends to K 33 . Thus, integrating by parts again (for the second term on the left hand side) and passing to the limit, we get∫ ∈ D(Ω × Y ). But because u ∈ V KL , this is equivalent to ∫ Choosing v = w α in (62), summing for α = 1 to 2 and using the fact that for eachv in D(Ω × Y ), there is a w in (D(Ω × Y )) 2 such that div y w = v, (62) implies that ∂ 2 33 u 1 β + ∂ y β K 33 -2∂ 3 K β3 = 0 for β = 1, 2. As 2K β3 = ∂ 3 u 1 β + ∂ y β u 2 3 , this is in turn equivalent to ∂ y β (∂ 3 u 2

	Besides, from Lemma 4.1, we know that that ( K a 33 (u b )	(	a 2 K a 33 (u b ) )	and	(	aK a 33 (u b ) )	tend to zero and
					∫			∫
	Ω×Y	(∂ α u β + ∂ yα u 1 β )∂ 2 33 v dxdy +	Ω×Y	K 33 ∂ 2 yαy β v dxdy = 2	Ω×Y	K β3 ∂ 2 yα3 v dxdy
	for all v Ω×Y	∂ yα u 1 β ∂ 2 33 v dxdy +	∫ Ω×Y	K 33 ∂ 2 yαy β v dxdy = 2	∫ Ω×Y	K β3 ∂ 2 3yα v dxdy.	(62)
								2 3yα v ε dx.

where Λ 3 = (0 9 , M(L 3 )). Then, with a suitable choice of test functions, (59) implies that (M 0 , M(L 3 )) is the unique solution in

(60) where Λ 3 = (0 9 , L3 ) and v is the vector of V KL associated with M 0 .

To eliminate the local variable y, we proceed as in the Dirichlet case, with Λ 3 = (0 9 , L3 ) instead of ϕ c . First, letting L 0 3 = M(L 3 ) and considering in (60) test functions such that L3 = 0 and v = 0, we get that ū1 = u ρξ M s ρξ (ū

with a suitable choice of test functions in (60) we obtain the announced model. 2

Note that the above calculations would be more complicated with non metallized inclusions, as M(L 3 ) would then depend on y.

Proof of Theorem 5.1 : local mixed conditions.

Here G 1 = 0 so that the effective equations simplifies into

This simplification allows to eliminate the unknown L 0 3 that can be computed explicitly in terms of s γδ (ū). We choose v = 0 then

e. in ω. Now replacing L 0 3 and restricting ourselves to test functions with L3 = 0, we get the announced model, and thus conclude the proof. 2 Also, by Poincaré inequality, the fact that

)) two-scale converges to φ 1 , and

)) two-scale converges to ∂ 3 φ 1 . Then, passing to the limit in ∫

we get that for any function

This proves that φ 1 actually belongs to L 2 (ω;

Moreover, the continuity of the trace function implies that φ 1 = 0 on Γ -and, with the assumption of metallization, that φ 1 does not depend on y on Γ + . Then passing to the limit in

we get that φ 1 = 2M(L 3 ) on Γ + . Thus, letting φ1 = φ 1 -(1 + x 3 )M(L 3 ), we get the announced result. 2 8.1.2 Intermediate limit model. We state now a first limit model, including both global and local variables, that is directly deduced from the convergence results of Lemma 4.1 and the characterization in Lemma 8.1.

The limit M of ( M a (U b ) ) is completely determined by the macroscopic fields U =(u,M(L 3 )) ∈ (0 3 , φ c ) + W 0 , where we have let

and the microscopic fields

and the operator M 1 has been defined in (31). Now, we state the following intermediate limit model using the definitions (28), (45) of ℓ u , ℓ φ and L 0 3 = M(L 3 ).

Lemma 8.2. The limit M of M a (U b ) takes the form M = M 0 (U) + M 1 (U 1 ) where

Proof: Using the definitions (35) of W 1 ad and the notations (36)-(37), multiplying ( 14) successively by ε 2 , ε and 1, and passing to the limit, one gets

Unlike the preceding models, the first two equations are coupled. Consequently, the computation of K 33 on one hand and of (K 13 , K 23 ) on the other hand cannot be carried out independently. Summing up the three equations above and using usual density results, and restricting ourselves to test functions such that L 3 does not depend on x 3 , we get the announced weak formulation. Uniqueness of the solution follows from Lax-Milgram Lemma, and the convergence of the whole family from the uniqueness of the solution. 2

8.2

Step 2: Derivation of the models. In this section, we complete the proof of Theorem 6.1 by eliminating the local variable z and the corresponding unknown U 1 . We use notations already introduced in Section 6.1.

For W 1 defined in (30), considering (63) with V = 0, we easily get

) , where L 0 3 = φ c for Dirichlet conditions, then (64) may be rewritten as

This proves that

where U λµ M , U λµ N and U 3 have been defined in (32)-( 33)-(34). Moreover,

The choice in (63

in the case of Dirichlet conditions and

in the case of mixed conditions, where (v, L 3 ) belongs to V KL ×{0} for Dirichlet conditions, to V KL ×L 2 (ω) for local mixed conditions, and V KL ×H 1 (ω) for nonlocal mixed conditions, completes the proof. 2


A Coupel-Ledru

Eric Lebon

Angélique Christophe

Agustina Gallo

Pilar Gago

Florent Pantin

Agnes Doligez

Thierry Simonneau

Breeding grapevine for an efficient use of water by lowering night-time transpiration

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Jacques Rigo

Jean-Michel Hascoet

Jean-Charles Picaud

Fabio Mosca

Amandine Rubio

Elie Saliba

Michaël Radkë

Umberto Simeoni

Bernard Guillois

PhD Nicholas P Hays
email: nicholaspaul.hays@nestle.com

J-M Hascoët

J-C Picaud

M Hartweg

C Billeaud

J Spalinger

Comparative study of preterm infants fed new and existing human milk fortifiers showed favourable markers of gastrointestinal status

Keywords: alpha-1 antitrypsin, calprotectin, elastase-1, low birth weight infant, prematurity

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Preterm infants have immature development at birth [START_REF] Abraham | Problems of prematurity[END_REF] and are more likely to need specialised care in the immediate postpartum period than term infants. Immaturity of the gastrointestinal system is a particular concern, as it has been associated with severe morbidities, including necrotising enterocolitis (NEC) [START_REF] Denning | Pathogenesis of NEC: Role of the innate and adaptive immune response[END_REF]. It also influences the ability of the infant to digest, absorb and use the nutrients needed to support further growth and development [START_REF] Neu | Gastrointestinal maturation and implications for infant feeding[END_REF]. Gut immaturity, and the associated increased risk in morbidity and feeding intolerance, make the choice of appropriate enteral nutrition extremely important. Although giving a preterm infant human milk provides many them with benefits [START_REF] Garcia | Bioactive compounds in human milk and intestinal health and maturity in preterm newborn: an overview[END_REF][START_REF] Corpeleijn | Intake of own mother's milk during the first days of life is associated with decreased morbidity and mortality in very low birth weight infants during the first 60 days of life[END_REF], the nutritional profile of unfortified human milk is insufficient to meet the increased demands for growth in this population [START_REF] Carlson | Guidelines for the use of human milk fortifier in the neonatal intensive care unit[END_REF]. As a result, human milk fortification is recommended for all infants born weighing less than 1800g [START_REF] Moro | XII. Human Milk in Feeding Premature Infants: Consensus Statement[END_REF]. While feeding fortified human milk has been shown to have no adverse effect on feeding tolerance in preterm infants [START_REF] Moody | Feeding tolerance in premature infants fed fortified human milk[END_REF], the influence of different human milk fortifiers on markers of gut maturity and gastrointestinal inflammation has not been well studied.

Maturity of gastrointestinal function and inflammation can be assessed using faecal biomarkers, which provide a non-invasive and early method for detecting any increased risk for these conditions [START_REF] Siddiqui | Update on clinical and research application of fecal biomarkers for gastrointestinal diseases[END_REF]. Alpha-1 antitrypsin is a serum trypsin inhibitor that is highly resistant to intestinal proteolysis. The extravasation of alpha-1 antitrypsin into the gut can be measured in the stools and is a classic marker for protein-losing enteropathies or conditions that result in loss of blood proteins into the intestinal tract [START_REF] Kosek | Fecal markers of intestinal inflammation and permeability associated with the subsequent acquisition of linear growth deficits in infants[END_REF]. This process happens in intestinal inflammation, as a result of mucosal ulceration or augmented permeability. The total protein in stools is another marker of protein-losing enteropathy and gut inflammation. Faecal calprotectin reflects various pathological processes that occur in the mucosa of pediatric patients, with elevated concentrations indicating pathogenesis [START_REF] Aydemir | Increased fecal calprotectin in preterm infants with necrotizing enterocolitis[END_REF]. Faecal elastase-1 is a highly specific and sensitive test for determining gut functional maturity, specifically pancreatic exocrine function, and differentiating severe from milder pancreatic insufficiency [START_REF] Kori | Faecal elastase 1 levels in premature and full term infants[END_REF]. In preterm infants, these parameters can be used as noninvasive markers for pancreatic function, intestinal permeability and intestinal complications such as NEC [START_REF] Kori | Faecal elastase 1 levels in premature and full term infants[END_REF][START_REF] Campeotto | Fecal calprotectin: cutoff values for identifying intestinal distress in preterm infants[END_REF][START_REF] Keller | Fecal alpha 1-antitrypsin in newborn infants[END_REF].

The purpose of this study was to assess the effect of two human milk fortifiers with different composition on faecal biomarkers of gastrointestinal inflammation and maturity of gut function in clinically stable preterm infants. Faecal biomarkers were the secondary endpoints of a trial that evaluated weight gain velocity, other growth parameters and nutritional biomarkers, including lipid profiles, in clinically stable preterm infants The new fortifier was PreNAN Human Milk Fortifier (Nestlé, Vevey, Switzerland) and the control fortifier was FM85 Human Milk Fortifier (Nestlé, Vevey, Switzerland).

The primary results of the trial have already been reported [START_REF] Rigo | Growth and nutritional biomarkers of preterm infants fed a new powdered human milk fortifier: a randomized trial[END_REF][START_REF] Billeaud | Effects on fatty acid metabolism of a new powdered human milk fortifier containing medium-chain triacylglycerols and docosahexaenoic acid in preterm infants[END_REF]. Briefly they showed that the unadjusted weight gain rate was 18.3g per kilogram per day for the new fortifier and 16.8 for the control fortifier and that both fortifiers were well tolerated, with a similar incidence of adverse gastrointestinal events. The adjusted mean weight gain was 2.3g per day greater for the new fortifier and the lower limit of the 95% confidence interval of 0.4 g per day exceeded both the noninferiority and superiority margins.

METHODS

This report presents secondary outcomes from a controlled, double-blind, randomised, parallel group study that was conducted in neonatal intensive care units at 11 metropolitan hospitals in France, Belgium, Germany, Switzerland and Italy. Each provided between 25 and 45 beds. We included clinically stable preterm infants born from April 2011 to March 2014 at up to 32 weeks of gestation or with a birthweight of up to 1500g. The mothers agreed to provide expressed or donor breastmilk for the 21-day study duration. The inclusion and exclusion criteria have previously been reported [START_REF] Rigo | Growth and nutritional biomarkers of preterm infants fed a new powdered human milk fortifier: a randomized trial[END_REF][START_REF] Billeaud | Effects on fatty acid metabolism of a new powdered human milk fortifier containing medium-chain triacylglycerols and docosahexaenoic acid in preterm infants[END_REF].

The fortifiers were both based on cows' milk and they provided similar energy, namely 17 kcal per 100 mL of human milk. For every 100 mL of HM, the new human milk formula provided 1.4g partially hydrolyzed whey protein, 0.7g lipids, 1.3g carbohydrate and a blend of micronutrients. The control human milk fortifier provided 1.0g extensively hydrolysed whey protein, no lipids, 3.3g carbohydrate and a blend of micronutrients.

The new fortifier was developed with a higher protein to energy ratio, with protein provided as partially hydrolysed whey and non-protein energy from lipids, as mediumchain triglycerides and docosahexaenoic acid. It also contained higher electrolyte and vitamin levels, while achieving a lower osmolality. The aim was to comply with the recommendation from the European Society for Paediatric Gastroenterology Hepatology and Nutrition [START_REF] Agostoni | Enteral nutrient supply for preterm infants: commentary from the European Society for Paediatric Gastroenterology, Hepatology, and Nutrition Committee on Nutrition[END_REF] and the discussions by an expert group reported by Koletzko et al [START_REF] Koletzko | Recommended nutrient intake levels for stable, fully enterally fed very low birth weight infants[END_REF].

Infants tolerating ≥100 mL/kg/day of human milk for >24 hours were randomised to receive either the new or control fortifier for 21 days after they had achieved full enteral feeding. The fortifiers that were given to the infants started at half-strength and this was called fortification strength day one (FS day one). It was then increased in line with hospital practice, with the full-strength fortification being reached when infants could maintain intakes of 150-180 mL/kg/day, namely full enteral feeds and full fortification. FS day one values were log-transformed and groups were compared using t-tests computed with the Satterthwaite method. Day 21 values were log-transformed and analysed using analysis of covariance (ANCOVA), adjusted for the FS day one value of the relevant parameter, sex, and centre (random effect). Changes from FS day one to day 21 were analysed using ANCOVA, adjusted for postmenstrual age and weight at study day one, when full enteral feeding and full fortification had been achieved, the FS day one value of the relevant parameter, sex, and centre (random effect). As many data were missing for total faecal protein, only summary statistics were calculated for this parameter.

The study was reviewed and approved by the institutional review board or independent ethics committee at each hospital and the parent or legal representative of each participant provided written, informed consent prior to enrollment. The Clinical Trial Registration for the study was NCT01771588.

RESULTS

The study flow diagram, baseline subject characteristics and parental demographics have previously been reported [START_REF] Rigo | Growth and nutritional biomarkers of preterm infants fed a new powdered human milk fortifier: a randomized trial[END_REF]. A total of 153 infants were enrolled and randomised to either the new fortifier (n=77) or the control fortifier (n=76). One infant in the new group was excluded due to a history of systemic disease and two in the control group were small for gestational age, leaving 76 and 74, respectively. The infant characteristics were similar between the new and control groups: 50% and 53% of girls, 32% and 27% of vaginal births and 24% and 22% of twins, respectively. The mean birth weights were 1147 ±258g and 1156 ±289g and the mean gestational ages at birth were 28.8 ±2.1 weeks and 28.7 ±1.18 weeks. As shown in Figure 1, only 56-94% of the population provided data for the stool analyses (depending on time point and parameter), due to the difficulty in obtaining sufficient stool quantities for all subjects.

There were no significant differences in any faecal biomarkers at FS day one (Figure 2).

The geometric mean concentrations of faecal alpha-1 antitrypsin (Figure 2A) and faecal elastase-1 (Figure 2C) were significantly lower in the new fortifier group compared to the control fortifier group at study day 21 (p = 0.010 and 0.016, respectively). In addition, the increase from FS day one to study day 21 for both of these parameters was significantly less in the new fortifier group compared to the control fortifier group (p = 0.007 and 0.004, respectively). Faecal calprotectin concentrations (Figure 2B) tended to increase in both groups during the study, according to the Satterthwaite test.

However, the mean changes from FS day one to study day 21 in the new fortifier and control fortifier groups were not significant (p = 0.051 and p = 0.112, respectively), with no significant difference between groups demonstrated at study day 21 or in the adjusted change over time. The evaluation of the total protein concentration in stools was limited due to inadequate stool material in 89% of cases at FS day one and 85% of cases on study day 21. Nevertheless, no significant difference was observed between the groups on FS day one when we looked at the medians and interquartile ranges: The faecal alpha-1 antitrypsin results observed here may have been due to several factors. First, alpha-1 antitrypsin is present in human milk [START_REF] Thomas | Fecal alpha 1-antitrypsin and hemoglobin excretion in healthy human milk-, formula-, or cow's milk-fed infants[END_REF][START_REF] Oswari | Comparison of stool microbiota compositions, stool alpha1-antitrypsin and calprotectin concentrations, and diarrhoeal morbidity of Indonesian infants fed breast milk or probiotic/prebiotic-supplemented formula[END_REF] and appears to increase with lactation [START_REF] Thomas | Fecal alpha 1-antitrypsin and hemoglobin excretion in healthy human milk-, formula-, or cow's milk-fed infants[END_REF]. Human milk alpha-1 antitrypsin is also resistant to pasteurisation [START_REF] Meredith-Dennis | Composition and variation of macronutrients, immune proteins, and human milk oligosaccharides in human milk from nonprofit and commercial milk banks[END_REF] and proteolysis in young infants [START_REF] Davidson | Fecal alpha 1-antitrypsin in breast-fed infants is derived from human milk and is not indicative of enteric protein loss[END_REF], meaning that faecal concentrations may be high, or change over time, even in the absence of enteropathy. It is likely that this accounted for the higher levels of this protein in the two study groups when they were compared to infants who were predominantly fed preterm formula [START_REF] Thomas | Fecal alpha 1-antitrypsin and hemoglobin excretion in healthy human milk-, formula-, or cow's milk-fed infants[END_REF][START_REF] Oswari | Comparison of stool microbiota compositions, stool alpha1-antitrypsin and calprotectin concentrations, and diarrhoeal morbidity of Indonesian infants fed breast milk or probiotic/prebiotic-supplemented formula[END_REF][START_REF] Sivan | Protein conservation by the immature intestine[END_REF]. For example, Sivan et al reported concentrations of faecal alpha-1 antitrypsin in a sample of preterm infants who were predominantly fed formula. These ranged from 0.34 to 1.05 mg/g stools [START_REF] Sivan | Protein conservation by the immature intestine[END_REF], which was lower than the geometric mean values observed in our study (Figure 2). However, it is unlikely that the alpha-1 antitrypsin content of human milk contributed to the small, but significant, difference between the groups on study day 21, since each group consumed a very similar volume of fortified milk during the study period. This was 153 mL/kg/day in both groups, with 49% of the volume from donor milk in the new fortifier group and 51% in the control fortifier group [START_REF] Rigo | Growth and nutritional biomarkers of preterm infants fed a new powdered human milk fortifier: a randomized trial[END_REF]. Alternatively, it can be speculated that these results indicate a slight, but significant, increase in gastrointestinal protein loss and, or, inflammation among infants in the control than new fortifier groups. However, the geometric mean concentrations in both groups were substantially lower than the cut-off value previously associated with various gastrointestinal diseases in children (2.6 mg/g stools) [START_REF] Thomas | Random fecal alpha-1-antitrypsin concentration in children with gastrointestinal disease[END_REF] and the difference between the groups was quite small (0.30 mg/g stools).

In a 2016 systematic review, Pergialiotis et al concluded that faecal calprotectin was elevated in newborns with NEC [START_REF] Pergialiotis | Calprotectin levels in necrotizing enterocolitis: a systematic review of the literature[END_REF]. However, faecal calprotectin has also been shown to be higher in healthy, exclusively breastfed infants than in infants fed formula or a mixture of breast milk and formula [START_REF] Savino | High faecal calprotectin levels in healthy, exclusively breast-fed infants[END_REF][START_REF] Asgarshirazi | Comparison of fecal calprotectin in exclusively breastfed and formula or mixed fed infants in the first six months of life[END_REF], suggesting that higher levels may not always indicate pathology. Consistent with this hypothesis, Groer et al [START_REF] Groer | Relationships of feeding and mother's own milk with fecal calprotectin levels in preterm infants[END_REF] showed that infants who received their mother's own milk had increasing faecal calprotectin levels over time, while infants who received mixed feeding or pasteurised donor milk did not show this increase over time. Together, these previous findings suggest that faecal calprotectin may not always represent pathological inflammation in preterm infants, but rather feeding pattern and, or, normal maturation of the gut immune axis [START_REF] Groer | Relationships of feeding and mother's own milk with fecal calprotectin levels in preterm infants[END_REF].

Furthermore, no infants in our study experienced NEC, despite the fact that many had calprotectin levels that greatly exceeded those reported to be indicative of this condition [START_REF] Pergialiotis | Calprotectin levels in necrotizing enterocolitis: a systematic review of the literature[END_REF][START_REF] Zhang | Diagnostic value of fecal calprotectin in preterm infants with necrotizing enterocolitis[END_REF]. In our study, the slight increase in faecal calprotectin over time in both the fortifier groups was likely to have reflected both feeding with human milk fortifier plus mixed sources of human milk and normal gastrointestinal maturation and development.

The lack of a significant difference between the groups was consistent with a previous study that showed no differences in this biomarker prior to commencing fortification or 14 days after fortification with either a human milk or bovine milk-based fortifier [START_REF] O'connor | Nutrient enrichment of human milk with human and bovine milk-based fortifiers for infants born weighing <1250 g: a randomized clinical trial[END_REF].

Our results also provide further evidence that more work is needed to identify an appropriate diagnostic cut-off value for NEC. This finding was consistent with the conclusions reached by Pergialiotis et al [START_REF] Pergialiotis | Calprotectin levels in necrotizing enterocolitis: a systematic review of the literature[END_REF].

Reduced elastase-1 secretion in stools is considered to be a marker of pancreatic insufficiency, with values of less than 200 μg/g stools indicating impaired exocrine pancreatic function [START_REF] Nissler | Pancreatic elastase 1 in feces of preterm and term infants[END_REF]. Values have been shown to normalise for most infants, including preterm infants, within the first few weeks after birth [START_REF] Kori | Faecal elastase 1 levels in premature and full term infants[END_REF][START_REF] Campeotto | Low levels of pancreatic elastase 1 in stools of preterm infants[END_REF]. Preterm infants have been shown to exhibit pancreatic immaturity and high sensitivity to environmental stress, including inappropriate nutrient uptake related to defects in the gut barrier function [START_REF] Neu | Gastrointestinal maturation and implications for infant feeding[END_REF]. Given that nutrients within the duodenal lumen are the most important stimulators of the pancreatic exocrine response [START_REF] Keller | Human pancreatic exocrine response to nutrients in health and disease[END_REF], it can be hypothesised that such inappropriate uptake might lead to excessive elastase production and secretion into the gut lumen. Excessive elastase activity in the gut lumen may contribute to maintenance or exacerbation of inflammatory status [START_REF] Antalis | Mechanisms of disease: protease functions in intestinal mucosal pathobiology[END_REF], leading to metabolic stress and an increased risk of feeding intolerance and NEC [START_REF] Claud | Neonatal necrotizing enterocolitis -Inflammation and Intestinal Immaturity[END_REF]. In the present study, preterm infants fed with the new fortifier presented with slightly lower levels of faecal elastase at study day 21 than the group who received the control fortifier. This finding suggests that the new fortifier may have had a modest impact on gut homeostasis.

To our knowledge, this was the first study to assess multiple faecal markers of gastrointestinal inflammation and maturity of gut function in response to different forms of human milk fortification in preterm infants. A limitation was that the number of samples of sufficient quantity for all analyses ranged from 10-85% of the full sample size.

CONCLUSION

These results indicate that different human milk fortifiers can lead to modest differences in markers of gastrointestinal status in clinically stable, preterm infants. The mean values of each marker in both fortifier groups were within the ranges observed in healthy term infants. When they are combined with previously reported anthropometry and clinical biochemistry data, these results further illustrate that the new fortifier was safe and well tolerated by our cohort.

  This was called study day one and occurred approximately three days after fortification began. The measurements then continued for another 21 days. FS day one occurred at a median of 13 (Q1 11, Q3 18) days in the new group and 14 (Q1 10, Q3 20) days in the control group. The mean days of life for study day one were 16 (Q1 13, Q3 20) versus 17 (Q1 13, Q3 23) days, respectively, and 36 (Q1 33, Q3 40) versus 37 (Q1 33, Q3 43) for study day 21.Stool samples were collected on FS day one or the day after and at 21 ±1 days (day 21). They were analysed for the following parameters: gastrointestinal inflammation, namely faecal alpha-1 antitrypsin, calprotectin and total protein, and maturity of gut function, namely faecal elastase-1. Approximately 5-8g was collected from each infant within two hours of a bowel movement. Samples were stored frozen at -20°C and shipped for analysis on dry ice. If the stool sample quantity was below 5.5 g, which was judged to be insufficient, a second collection was made from a later bowel movement on the same or following day. Concentrations of faecal calprotectin and elastase-1 were assessed by enzyme-linked immunosorbent assays, following sample treatment with the Smart Prep faecal extraction device (Roche Diagnostics Mannheim, Germany). These were: EK-Cal (Bühlmann Laboratories, Schönenbuch, Switzerland); ScheBo Pancreatic Elastase 1 (ScheBo Biotech AG, Giessen, Germany) and Euroimmun Analyzer A1 (Euroimmun, Lübeck, Germany). Faecal alpha-1 antitrypsin concentration was assessed by immunonephelometry, using the IMMAGE 800 (Beckman Coulter, Nyon, Switzerland) and faecal total protein by near-infrared spectroscopy with the Fenir 8820 (Perten Instruments, Alpnach, Switzerland). All the analyses were completed in a central laboratory (Rothen Medizinische Laboratorien AG, Basel, Switzerland).

these were 1 .

 1 70 (1.45-1.80) for the new fortifier versus 1.40 (1.30-1.53) for the control fortifier (p = 0.100). The values were also similar in both groups at study day 21: 1.60 (1.50-1.63) for the new fortifier versus 1.60 (1.50-1.70) for the control fortifier. No inferential statistics were performed at study day 21 because 85% of the values were missing. DISCUSSION This report presents faecal biomarkers of gastrointestinal inflammation and gut maturity in clinically stable, preterm infants fed human milk fortified with either a new or control multinutrient, powdered fortifier. Our results indicate that the different human milk fortifiers led to limited differences in faecal biomarkers of gastrointestinal status in this population.

Figure 1 .

 1 Figure 1. Flow of study participants and sample sizes for stool analyses

Figure 2 .

 2 Figure 2. Geometric mean */÷ geometric SD for faecal alpha-1 antitrypsin (panel A),

ACKNOWLEDGMENTS

The authors thank Virginie de Halleux (University of Liége) and Jonathan Jaeger, Laurent Ameye, Philippe Steenhout, Christelle Perdrieu, Samir Dahbane and Jalil Benyacoub for their important contributions to this study. They were all affiliated to Nestlé when the study was completed.

FUNDING

This study was sponsored by Nestlé Nutrition.

CONFLICTS OF INTEREST

NPH and MH are employed by Nestlé SA and JR, J-CP and CB are consultants for Nestlé Nutrition. JR, J-MH, J-CP, FM, AR, ES, MR, US, BG, CB and JS have received research funding from Nestlé Nutrition. US has been a speaker, consultant and expert panel participant for Nestlé, Danone and Bledina for the last three years.

List of abbreviations

ANCOVA, analysis of covariance; FS fortification strength; NEC, necrotising enterocolitis. centre); ‡ p = 0.016 versus control fortifier (by analysis of covariance, adjusting for value at FS day one, sex and centre).