C Pideri

P Seppecher
email: seppecher@univ-tln.fr

Asymptotics of a non-planar rod in non-linear elasticity

Keywords: beam, rod, non-linear elasticity, 3D-1D, Γ -convergence

published or not. The documents may come

Asymptotics of a non-planar rod in non-linear elasticity

Catherine Pideri, Pierre

Introduction

Thin elastic objects like beams or plates are of crucial importance in structural design for their low weight and cost. Their properties and the link with the properties of the material they are made of, was a constant subject of study for mechanicians. Approximations for the displacements at small scale which are induced by a global displacement of the structure are well known from the pioneer works of Euler, Bernoulli and Navier. The energy induced by such a global displacement can be estimated and then the behavior of the structure is known. The mathematical justifications of these approximations are more recent. A very wide literature is devoted to this subject. Here we are more particularly interested by the so-called 3D-1D reduction, the limit object is one-dimensional (the reader can refer, for instance, to [START_REF] Antman | The theory of rods[END_REF] or [START_REF] Trabucho | Mathematical modelling of rods[END_REF] for a review of one-dimensional elastic models).

Non-planar thin objects like shells or curved rods are also important. Here we restrict our attention to non-planar curved rods: the limit object is a one-dimensional non-planar curve. The mechanical applications are numerous: let us simply mention that helicoidal springs are nothing else but curved elastic rods and that they are widely used in mechanisms.

The mathematical literature devoted to curved rods is much more restricted than to the straight ones. This lack seems essentially due to the difficulty for obtaining fine a priori estimates in this complex geometry. First works based on formal expansions are due to Jamal et al. [START_REF] Jamal | Théorie asymptotique des tiges courbes anisotropes[END_REF] or Sanchez et al. [START_REF] Sanchez-Hubert | Statics of curved rods on account of torsion and flexion[END_REF]. Complete studies have been performed in the last years by Jurak et al. [START_REF] Jurak | Modelling of curved rods[END_REF], by Griso [START_REF] Griso | Asymptotic behaviour of curved rods by the unfolding method[END_REF] and by Pideri et al. [START_REF] Seppecher | Asymptotics of a non-planar beam in linear elasticity[END_REF].

These studies, like besides the major part of the studies devoted to the straight case, are performed in the framework of linear elasticity. The first reason is mathematical: a fundamental tool was missing 34 C. Pideri and P. Seppecher / Asymptotics of a non-planar rod in non-linear elasticity for obtaining a priori estimates in the context of non-linear elasticity where Korn inequality is of no help. This tool has been provided very recently by Friesecke, James and Muller [START_REF] Friesecke | A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity[END_REF]. The second reason comes from mechanics: all theories for rods involves only the parameters (Lamé coefficients or Young modulus) which describe the linear behavior of the material the rod is made of. This is remarkable. Rods are weak structures: submitted to reasonable forces they move far from their initial position. The classical approximation of small displacements is in general not valid, but the parameters which describe the global behavior can be efficiently computed in the linear framework. This may seem a paradox. The point is that the rigidity constant is very small in thin domains: the displacement can be large while its gradient remains close to a rotation. We are in a case of large displacement but with small strain. Starting with a non-linear three-dimensional model of elasticity, we end up with a non-linear model for a bending and torsion line but the material coefficients of this elastic line depend only of the linear approximation of the starting model.

The study of straight rods in the non-linear case has recently been performed by Mora et al. [START_REF] Mora | A nonlinear model for inextensible rods as a low energy Γ -limit of three-dimensional nonlinear elasticity[END_REF] by Mora [START_REF] Mora | Derivation of a rod theory for phase-transforming materials[END_REF] and by Pantz [START_REF] Pantz | Le modèle de poutre inextensionnelle comme limite de l'élasticité non-linéaire tridimensionnelle[END_REF]. The way followed in [START_REF] Seppecher | Asymptotics of a non-planar beam in linear elasticity[END_REF] for estimating the Korn constant of a non-straight rod and the way followed by Mora et al. for estimating the rigidity constant in a straight but non-linear rod are very close: the idea is to use Korn (or Rigidity) inequality on small sections of the rod in order to obtain a step by step estimate for the displacement. The present work is based on this observation. A major part of it is devoted to the estimation of the rigidity constant on the considered domain. The transcription of the method used by Mora and Muller is not straightforward as each small part of the rod is no more similar to a fixed domain. The method used in [START_REF] Seppecher | Asymptotics of a non-planar beam in linear elasticity[END_REF] for estimating the Korn constant cannot either be transcripted. Indeed the homogeneity property of Korn inequality is a crucial tool in [START_REF] Seppecher | Asymptotics of a non-planar beam in linear elasticity[END_REF] and nothing similar can be used in the non-linear case.

We make here an intensive use of the paper by Mora and Muller [START_REF] Mora | A nonlinear model for inextensible rods as a low energy Γ -limit of three-dimensional nonlinear elasticity[END_REF]. We use similar assumptions and try to use close notation. For sake of simplicity, we restrict our attention to homogeneous and isotropic materials but further generalization to anisotropic materials like it is done in [START_REF] Mora | A nonlinear model for inextensible rods as a low energy Γ -limit of three-dimensional nonlinear elasticity[END_REF] does not present great difficulties. Our method for proving the Γ -convergence result is also very close to the one called "refined Γ -convergence and director theories" in this paper. We think that this method is the closest to the method used by mechanicians and the most natural.

The major originality of our work lies in the application of the rigidity lemma in a special geometry: it is a neighborhood of a regular non-planar curve (note that we do not try to write the weakest assumptions for the regularity of the domain and that some of assumptions we make can probably be relaxed). We do not restrict to circular sections: so we deal with a possible rotation of the section along the line.

Rotation relative to what basis?

There is no canonical choice for such a basis. We emphasize that choice of the Frenet basis is neither canonical neither even possible in general. Our choice has the only but fundamental advantage that the basis is always well defined. As we desire to take into account a possible rotation of the section with respect to this basis, there is no further difficulty to take also into account a possible deformation of the section. That is what we do. The resulting geometry looks like Fig. 4.

We describe precisely this geometry together with the initial and resulting energies in Section 2.

The initial energy corresponds to a general three-dimensional hyperelastic material. We assume for sake of simplicity that the material is homogeneous and isotropic but this assumption is not fundamental. The scaling assumed for the energy defines the order of magnitude of the forces which may be applied to the structure. Our choice is such that the structure resists with a finite energy to displacements the order of magnitude of which is the length of the rod. In the case of straight rods the choice of the scaling and its implications on the resulting model has been the subject of long debates but the case of straight rods is quite special: one can consider a scaling which ensures that the structure resists with a finite energy to transverse displacements of order one (this is what is done in [START_REF] Mora | A nonlinear model for inextensible rods as a low energy Γ -limit of three-dimensional nonlinear elasticity[END_REF], leading to a bending-torsion but inextensional model) or a scaling which ensures that the structure resists with a finite energy to longitudinal displacements of order one (this is what is done in [START_REF] Acerbi | A variational definition of the strain energy for an elastic string[END_REF], leading to an extensional string theory). Here we have no real choice: when the rod is curved this decomposition is not so simple and we think it is quite unrealistic to take into account forces (as it is done in [START_REF] Griso | Asymptotic behaviour of curved rods by the unfolding method[END_REF]) which lead to extensional and bending displacements with the same order of magnitude.

The limit energy is similar to the one obtained in the straight case. It takes into account bending and torsion. We write it, like mechanicians do, by invoking the rotation of the section as an internal parameter. This parameter can of course be eliminated. In the simplest case of homogeneous and isotropic straight rods with circular section, this elimination is easy. The resulting energy is then a functional of the displacement only. But in many cases, this elimination leads to a non-local functional. This due to a bending-torsion coupling (which can result for instance from anisotropy [START_REF] Murat | Effets non locaux dans le passage 3d-1d en élasticité linéarisée anisotrope hétérogène[END_REF]). In our case the coupling is, even in the isotropic case, complete and it is due to the non-planar geometry of the rod.

At the end of Section 2 we state our main theorem while Sections 3 and 4 are devoted to the proof. In Section 3 we study the application of the rigidity lemma of Frieseke et al. [START_REF] Friesecke | A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity[END_REF] to the considered domain. We first prove (Lemma 1) that the rigidity constants of two almost similar domains are close. Then we prove that the rigidity constant of the considered domain is of order ε -1 (Theorem 2). The situation is not very different from the case of straight rods. The compactness result follows easily. Section 4 is devoted to the proof of the Γ -convergence result.

The main result

Description of the beam

First let us define the mean line of the considered beam. Let L be a curve (a regular one-dimensional manifold) in the physical space R 3 and let ϕ ∈ C 3 ([0, ], R 3 ) be a curvilinear parametrization of L.

For any x 1 ∈ [0, ], we denote t(x 1 ) := ϕ (x 1 ) the unit vector tangent to the curve. We complete t(x 1 ) in order to get an orthonormal basis (t, n, b)(x 1 ). There are many choices for such a basis. The Frenet basis is a classical choice. However, it is not defined when the curvature of L vanishes. Moreover this choice is arbitrary: even when considering a beam with constant section, there is no reason for this section to be constant in this particular basis. Here we consider the possibility of varying section so the choice of the basis has no importance. Let us choose it in such a way that n ∧t = 0: the basis is then well defined by a first-order differential equation as soon as n(0) is given (arbitrarily in the plane orthogonal to t(0)). We can introduce the two functions τ ,

ξ in C 1 ([0, ], R) such that t (x 1 ) = τ (x 1 )n(x 1 ) + ξ(x 1 )b(x 1 ), n (x 1 ) = -τ (x 1 )t(x 1 ), (1)
b (x 1 ) = -ξ(x 1 )t(x 1 ).

Note that the curvature of the line can be easily recognized as τ 2 + ξ 2 and the torsion of the line as τ ξ -ξτ τ 2 +ξ 2 . Now let us describe the section of the considered beam. Let ω be the "prototype section": a piecewise C 1 domain in R 2 . It is a bounded connected open set. Let C be the cylinder of R 3 : C := [0, ] × ω and let y = (y 2 , y 3 ) be a function in C 2 (C, R 2 ) such that, for any x 1 , the mapping (x 2 , x 3 ) → y(x 1 , x 2 , x 3 ) is a positive diffeomorphism from ω onto its image ω x 1 . The domain ω x 1 describes the rescaled section at abscissa x 1 . We assume that 0 is the inertial center of ω x 1 . For the convenience of notations we consider y as a function from C to R 3 by setting y 1 (x 1 , x 2 , x 3 ) := x 1 . Then the image of y is a domain in R 3 denoted Ω that we call the reference domain. In Fig. 2 we represent a possible prototype section (a rectangle), the corresponding reference cylinder C, while in Fig. 3 we represent the reference domain Ω. For this figure we have assumed a particular choice of y. For any x 1 , the mapping (x 2 , x 3 ) → y(x 1 , x 2 , x 3 ) is a similarity: for sake of simplicity of the drawing, we did not use the possibility of varying the shape of the section but only its scaling and position. Note that the possibility of a rotating section shows that the choice we made for the basis (t, n, b) has no importance.

The regularity assumptions we made for y ensures that ∇y is continuous on the compact C with a positive determinant: there exist positive constants C 1 , C 2 and C 3 such that, for all x ∈ C, (

∇y(x) C 1 (2)
) 3
As we desire to describe a thin non-planar beam, we have now to rescale the sections by introducing a small parameter ε and to plug them along the line L. Throughout this paper ε denotes a sequence tending to zero. Without loss of generality, we assume that ε is such that ε -1 is an integer. Then we denote Ψ ε the function defined on Ω by

Ψ ε (y 1 , y 2 , y 3 ) := ϕ(y 1 ) + ε y 2 n(y 1 ) + y 3 b(y 1 ) (4)
and

Φ ε the composition Φ ε := Ψ ε • y. For ε sufficiently small, Ψ ε is a C 2 -diffeomorphism from Ω onto its image denoted Ω ε (cf. Fig. 4) (and Φ ε is a C 2 -diffeomorphism from C onto Ω ε ).
In the sequel the set Ω ε will be refereed as "the beam" and we will use in it the parameterizations Ψ ε or Φ ε . In order to simplify some expressions appearing in our computations we will use, when no confusion can arise, the notation y for y(x) and x for Ψ ε (y) = Φ ε (x).

The geometry of the beam is then determined by the line L, a unit vector n(0) orthogonal to t(0), the prototype section ω, the function y and the scaling parameter ε.

In Fig. 4 we represent the beam corresponding to the line L represented in Fig. 1, to the prototype section represented in Fig. 2 (a rectangle) and to the particular choice of y represented in Fig. 3. Note that the way the section "turns" around the mean line is a type of "torsion" of the beam which should not be confused with the torsion of the line L nor with the mechanical torsion which can result from the displacement of the beam. We emphasize the fact that, in Fig. 4, the beam is at rest.

The regularity assumptions we made for ϕ lead to the following estimates for ∇Ψ ε : there exist positive constants C 1 , C 2 and C 3 such that, for any y ∈ Ω,

∇Ψ ε (x) -1 C 1 ε (5)
and

(1 -C 2 ε)ε 2 det ∇Ψ ε (x) (1 + C 3 ε)ε 2 . ( 6
)
Indeed

∂ 1 Ψ ε (y) = 1 -ετ (y 1 )y 2 -εξ(y 1 )y 3 t(y 1 ), ∂ 2 Ψ ε (y) = εn(y 1 ), (7)
∂ 3 Ψ ε (y) = εb(y 1 ).
This leads also to estimates of the area of any section ω ε a := Φ ε ({a} × ω) and of the volume of any part of the beam defined by a < x 1 < b. We have (possibly modifying the positive constants C 2 and C 3 )

C 2 |ω|ε 2 φ ε {a} × ω C 3 |ω|ε 2 , ( 8
)
C 2 (b -a)|ω|ε 2 φ ε [a, b] × ω C 3 (b -a)|ω|ε 2 . ( 9
)
Note that, when the context allows no confusion, we denote indifferently | • | the two-dimensional or three-dimensional Hausdorff measure.

Elastic energies 2.2.1. 3-D elastic energy

Our goal is to study the behavior of the beam Ω ε in the framework of non-linear elasticity. A motion of the beam is a function u belonging to the Sobolev space H 1 (Ω ε , R 3 ) which describes the position of the deformed beam. We assume that the beam is fixed on its basis {x 1 = 0}. So any motion u has to satisfy u(x) = x when x 1 = 0. The space of admissible motions u is denoted:

H 1 b (Ω ε ) := u ∈ H 1 Ω ε , R 3 ; u(x) = x when x 1 = 0 .
The vector field u -Id is usually called the displacement field. It belongs to

H 1 0 (Ω ε ) := {v ∈ H 1 (Ω ε , R 3 ); v(x) = 0 when x 1 = 0}.
The elastic energy is a functional

E ε on H 1 b (Ω ε ) of the form E ε (u) := 1 ε 4 Ωε W ∇u(x) dx,
where the energy density W is regular, objective and non-degenerated: it satisfies [8]

W ∈ C 0 R 3×3 , R , ∀F ∈ R 3×3 , ∀R ∈ SO(3), W (RF ) = W (F ), W (Id) = 0, ( 10
)
W is of class C 2 in a neighborhood of Id, ∀F ∈ R 3×3 , W (F ) C d F , SO(3) 2 .
This energy

E ε is defined on H 1 b (Ω ε ). It is naturally extended on L 2 (Ω ε , R 3 ) by setting E ε (u) := +∞ if u does not belong to H 1 b (Ω ε ).
The scaling ε -4 is needed, as we will see later, to obtain a finite energy when passing to the limit. From the mechanical point of view this scaling can be interpreted as a choice for the force unit (and therefore for the energy unit) which is adapted to the weak rigidity of the rod we consider.

We denote

W in (G) := 1 2 ∂ 2 W ∂F 2 (Id)(G, G). ( 11
)
When ∇u is close to Id then W in (∇u -Id) is a good approximation for W (∇u): it is the linearized elastic energy associated to W . It plays an important role in this study. The point is that a thin structure, like the one we consider, allows for large displacements while the strain remains small. This explains why the limit model for the rod is completely non-linear but depends only on the linearized part W in of the elastic energy W of the material the rod is made of. We assume for sake of simplicity that the considered material is homogeneous and isotropic:

∀F ∈ R 3×3 , ∀R ∈ SO(3), W (F R) = W (F ).
Then the linearized energy takes the form

W in ∇(u -Id) = µ e 2 + λ 2 tr(e) 2 ,
where e is linearized strain tensor (the symmetric part of ∇u -Id) and the Lamé coefficients, λ and µ, satisfy µ > 0 and 3λ + 2µ > 0.

The limit one-dimensional model

The limit energy we obtain is the one classically used in mechanics for describing the motion of rods. Let us describe it: it is a one-dimensional model for the line L. The motion of L is described by a vector field u on L but the mechanical description is made easier by the introduction of an extra matrix-valued field r on L. In mechanics, r is interpreted as the rotation of the section of the beam. The space of admissible couples (u, r) is

H ad := (u, r) ∈ H 2 L, R 3 × H 1 L, SO(3) ; u = r • t along L; u ϕ(0) = ϕ(0), r ϕ(0) = Id . ( 12
)
Here L is endowed with the one-dimensional Hausdorff measure and the derivatives are relative to the curvilinear abscissa. The boundary conditions u(ϕ(0)) = ϕ(0), r(ϕ(0)) = Id are known as the "clamping conditions". For any (u, r) ∈ H ad , the function r -1 r belongs to L 2 (L, R 3×3 ) and takes values among skew-symmetric matrices. It is classical to associate to such a skew-symmetric matrix the vector 1 ṙ such that, for any

V ∈ R 3 , ṙ ∧ V = r -1 r • V .
The elastic energy F is characterized by a field of definite positive symmetric matrices A and reads:

F(u, r) := 1 2 L (A • ṙ) • ṙ dH 1 . ( 13
)
The energy is defined on H ad but we extend it on L 2 (L, R 3 × SO(3)) by setting F(u, r) := +∞ when (u, r) does not belong to the admissible space. Note that the dependence upon u is hidden in the constraint u = r • t. Note also that this constraint implies u = 1: the line is non-extensional.

In terms of the motion of the line only, the energy reads

F(u) := min r∈L 2 (L,SO (3)
)

F(u, r). ( 14
)
Computing this infimum is in general very intricate and leads to a non-local functional. Here the coupling between u and r is total. This coupling refereed in mechanics as the "flexion-torsion coupling" is due both to the non-planar geometry of the line L and to the fact that the section is varying.

The matrix A defining F in ( 13) is related to the geometry of the section and to the material properties by

A ϕ(x 1 ) := µJt ⊗ t + Y I 2 n ⊗ n + I 3 b ⊗ b + I 23 (b ⊗ n + n ⊗ b) , ( 15
)
where Y is the Young modulus of the material Y := µ(3λ + 2µ)(λ + µ) -1 , I 2 , I 3 , I 23 are the inertial moments of the section

I 2 (x 1 ) := ωx 1
(y 3 ) 2 dy 2 dy 3 ,

I 3 (x 1 ) := ωx 1 (y 2 ) 2 dy 2 dy 3 , (16)
I 23 (x 1 ) := - ωx 1 y 2 y 3 dy 2 dy 3 , ( 17
)
and J is the solution of the classical problem for torsional rigidity:

J(x 1 ) := min ωx 1 (∂ 3 ψ + y 2 ) 2 + (∂ 2 ψ -y 3 ) 2 dy 2 dy 3 ; ψ ∈ H 1 (ω x 1 , R) . ( 18
)
The parameters J, I 2 , I 3 and I 23 depend only on the geometry of the rescaled section ω x 1 . Finally, on H ad , F reads

F(u, r) = 1 2 L µJ(ṙ • t) 2 + Y I 2 (ṙ • n) 2 + I 3 (ṙ • b) 2 + 2I 23 (ṙ • n)(ṙ • b) dH 1 . ( 19
)

The main result

Let |ω ε x 1 | denote the (two-dimensional) Hausdorff measure of the section ω ε x 1 . For any f ∈ L 2 (Ω ε , R p ), we denote f ∈ L 2 (L, R p ) its mean value on each section:

f ϕ(x 1 ) := 1 |ω ε x 1 | ω ε x 1 f dH 2 .
Our main theorem is the following Theorem 1.

(i) If (u ε ) is a sequence in L 2 (Ω ε , R 3
) with a bounded energy (i.e., E ε (u ε ) < M), then there exists a subsequence still denoted

(u ε ) such that (ū ε , ∇u ε ) converges weakly in L 2 (L, R 3 × R 3×3 ). (ii) For any sequence (u ε ) in L 2 (Ω ε , R 3 ) such that (ū ε , ∇u ε ) converges weakly to (u, r) in L 2 (L, R 3 × R 3×3 ), we have lim inf E ε (u ε ) F(u, r). ( 20
) (iii) For any (u, r) in L 2 (L, R 3 ×R 3×3 ), there exists a sequence in L 2 (Ω ε , R 3 ), (u ε ) such that (ū ε , ∇u ε ) converges to (u, r) in L 2 (L, R 3 × R 3×3 ) and lim sup E ε (u ε ) F(u, r). (21)
One may prefer to reformulate the limit energy in terms of the motion of the line only. A trivial consequence of Theorem 1 reads Corollary 1.

(i) For any sequence (u ε ) in L 2 (Ω ε , R 3 ) such that ūε converges to u in L 2 (L, R 3 ), we have lim inf E ε (u ε ) F(u). (22)
(ii) For any u in L 2 (L, R 3 ), there exists a sequence (u ε ) in L 2 (Ω ε , R 3 ) such that ūε converges to u in L 2 (L, R 3 ) and

lim sup E ε (u ε ) F(u). ( 23
)
Remark 1. We have decided to formulate this theorem in terms of the actual displacement fields, those which arise from the physical problem and are defined on Ω ε and L. One may prefer to refer to a fixed functional space. This is what is usually done in the study of straight beams [] and this is actually what we will do in the proof. Formulating the theorem in a fixed functional space has an important advantage: it can then be written in terms of Γ -convergence. A first disadvantage is that the choice of the fixed functional space is somehow arbitrary and one could then wonder whether the theorem is still valid for a different choice. A second disadvantage is the very intricate expression of the energy in the fixed space.

Remark 2. There is however a canonical way to reformulate the previous theorem in terms of Γ -convergence. Indeed let us associate to any function u ∈ L 2 (Ω ε , R 3 ) the vector valued measure |ω ε x 1 | -1 u(x)1 Ωε (x) dx, where 1 Ωε denotes the characteristic function of Ω ε . In the same way let us associate to any u ∈ L 2 (L, R 3 ) the vector valued measure u dH 1 |L , where H 1 denotes the one-dimensional Hausdorff measure. Let us endow the space of such vector valued measures with the weak* topology. A slightly different version of the previous theorem states the relative compactness of sequences with bounded energy and the Γ -convergence of E ε to F. Indeed it is easy to check that the convergence of u ε to u in the sense of these measures implies, when the energy is bounded, the convergence of ūε to u in L 2 (L, R 3 ). Remark 3. Let f be a continuous field of forces. A property of Γ -convergence (for details about the definition and the properties of Γ -convergence the reader can refer to [START_REF] Maso | An Introduction to Γ -Convergence[END_REF]) shows that Theorem 1 remains valid when adding 2 in E ε (u ε ) and F(u) respectively -ε -2 Ωε f •u ε and -L |ω x 1 |f •u. A second property of Γ -convergence shows that a sequence of equilibrium displacements for the beam (i.e., of minimizers of E ε (u ε )ε -2 Ωε f • u ε ) converges to an equilibrium solution for the line L (i.e., a minimizer of

F(u) -L |ω x 1 |f • u).

Rigidity lemma and compactness

Uniform estimation of the rigidity constant

For any connected bounded Lipschitz domain D ⊂ R 3 we consider the Lebesgue space L 2 (D, R 3×3 ) of matrix valued functions and we denote d D the standard distance induced by the L 2 -norm. We denote S D the subset of those functions which take values among rotations S D := L 2 (D, SO( 3)) and we denote s D ⊂ S D the subset of such functions which are constant.

In a remarkable paper [START_REF] Friesecke | A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity[END_REF] G. Friesecke, D. James and S. Muller proved a rigidity lemma which states the existence of a positive constant K such that

∀v ∈ H 1 D, R 3 , d D (∇v, s D ) Kd D (∇v, S D ). (24)
We denote by K D the rigidity constant: i.e., the smallest constant K which satisfies (24). In order to estimate the asymptotic behavior of the rigidity constant K Ωε when ε tends to zero, we first compare the rigidity constant of two almost identical domains. Lemma 1. Let D be a domain in R N and (D δ ) be a sequence of such domains. Assume that, for any δ > 0, there exists a C 1 -diffeomorphism Ψ δ from D onto D δ satisfying, at every point x ∈ D, ∇Ψ δ (x) -Id δ. Then, for any K K D , we have

K D δ K(1 + δ) 1 -6δ -2Kδ (25)
which clearly implies

lim sup δ→0 K D δ K D . ( 26
) Proof. Let v δ ∈ H 1 (D δ , R 3 ). • Let r δ ∈ s D δ such that d D δ (∇v δ , s D δ ) = ∇v δ -r δ L 2 (D δ
) and let us define

w δ := r -1 δ • v δ and wδ := w δ • Ψ δ + Id -Ψ δ (27)
which belong respectively to H 1 (D δ , R 3 ) and H 1 (D, R 3 ). Then we have

d D δ (∇v δ , s D δ ) = d D δ (∇w δ , s D δ ) = ∇w δ -Id L 2 (D δ ) , ( 28
)
d D δ (∇v δ , S D δ ) = d D δ (∇w δ , S D δ ). ( 29
)
2 Different choices of forces could be considered as in [START_REF] Murat | Comportement asymptotique des solutions du système de l'élasticité linéarisée anisotrope hétérogène dans des cylindres minces[END_REF] to the price of a different formulation of the theorem.

Moreover, defining ∇w δ by ∇w δ (x) = ∇w δ (Ψ δ (x)), we have

(∇ wδ -Id) = (∇w δ -Id)(∇Ψ δ ), = (∇w δ -Id) + (∇w δ -Id)(∇Ψ δ -Id). ( 30
)
Using the assumption ∇Ψ δ -Id δ which implies also that |det(∇Ψ δ ) -1| √ 3δ we get by integration, change of variables and for δ sufficiently small, the following estimates:

∇ wδ -Id L 2 (D) (1 + 3δ) ∇w δ -Id L 2 (D δ ) , ( 31
) ∇ wδ -Id L 2 (D) (1 -3δ) ∇w δ -Id L 2 (D δ ) . ( 32
) • Now, let ρ δ ∈ s D such that d D (∇ wδ , s D ) = ∇ wδ -ρ δ L 2 (D) .
We have

d D (∇ wδ , s D ) = ∇w δ ∇Ψ δ + Id -∇Ψ δ -ρ δ L 2 (D) = (∇w δ -ρ δ ) + (Id -∇Ψ δ )(Id -ρ δ ) + (∇w δ -ρ δ ) L 2 (D) .
By a change of variables, we get the inequality

d D (∇ wδ , s D ) (1 -3δ) ∇w δ -ρ δ L 2 (D δ ) -δ Id -ρ δ L 2 (D) . ( 33
)
The definition of ρ δ implies

∇ wδ -ρ δ L 2 (D) ∇ wδ -Id L 2 (D) .
Using triangular inequality, we get

Id -ρ δ L 2 (D) 2 ∇ wδ -Id L 2 (D) .
From ( 28) and (31) we deduce, for δ sufficiently small,

Id -ρ δ L 2 (D) 3 ∇w δ -Id L 2 (D δ ) 3d D δ (∇w δ , s D δ ).
Then inequality (33) leads to

d D (∇ wδ , s D ) (1 -6δ)d D δ (∇w δ , s D δ ). ( 34
) • Now, let R δ ∈ S D δ such that d D δ (∇w δ , S D δ ) = ∇w δ -R δ L 2 (D δ ) .
We have, for δ sufficiently small,

d D (∇ wδ , S D ) ∇ wδ -R δ • Ψ δ L 2 (D) ∇w δ ∇Ψ δ + Id -∇Ψ δ -R δ • Ψ δ L 2 (D) ∇w δ -R δ • Ψ δ + (Id -∇w δ )(Id -∇Ψ δ ) L 2 (D) (1 + δ) ∇w δ -R δ L 2 (D δ ) + δ(1 + δ) ∇w δ -Id L 2 (D δ ) ,
and, using (28),

d D (∇ wδ , S D ) (1 + δ)d D δ (∇w δ , S D δ ) + 2δd D δ (∇w δ , s D δ ). ( 35
)
Applying the rigidity lemma (24) to D we get

d D (∇ wδ , s D ) K D (1 + δ)d D δ (∇w δ , S D δ ) + 2K D δd D δ (∇w δ , s D δ )
then, using (34),

(1 -6δ)d D δ (∇w δ , s D δ ) K D (1 + δ)d D δ (∇w δ , S D δ ) + 2K D δd D δ (∇w δ , s D δ ), d D δ (∇w δ , s D δ ) K D (1 + δ) 1 -6δ -2K D δ d D δ (∇w δ , S D δ ), ( 36
)
K D δ K D (1 + δ) 1 -6δ -2K D δ . ( 37
)
If K is such that K K δ then

K D δ K(1 + δ) 1 -6δ -2Kδ . ( 38
)
Proof is concluded as the right-hand side of this inequality tends to K as δ tends to zero.

As the rigidity inequality (24) is invariant when rescaling the domain and obviously invariant when rotating it, we easily get the following corollary: Corollary 2. Lemma 1 remains valid if we only assume that each domain D δ is almost similar to D. More precisely if there exist a sequence a δ satisfying lim δ→0 δa -1 δ = 0, a rotation r δ and a diffeomorphism Ψ δ from D onto D δ satisfying, at every point x ∈ D δ , ∇Ψ δ (x)a δ r δ δ, then inequality (25) still holds with δ replaced by δa -1 δ .

Indeed, denoting

D δ := a δ -1 r -1 δ (D δ ), the diffeomorphism Ψ δ := a δ -1 r -1 δ • Ψ δ maps D on D δ and satisfies ∇ Ψ δ -Id δa δ -1
. Now we can state a rigidity theorem for Ω ε . Let us introduce a new notation: we denote S Ωε the subset of those functions r in S Ωε which depend only on the x 1 coordinate. This set can clearly be identify to L 2 (L, SO(3)). In the same way, the set of such functions with bounded variations S Ωε ∩ BV(Ω ε ) can be identified with BV(L, SO(3)). We have Theorem 2. There exists a constant K such that, for ε sufficiently small and for any u in H 1 (Ω ε ),

d Ωε (∇u, S Ωε ) Kd Ωε (∇u, S Ωε ), ( 39
)
d Ωε (∇u, s Ωε ) K ε d Ωε (∇u, S Ωε ). ( 40
)
Moreover, inequality (39) can be precised in the following way: there exists

r ε in S Ωε ∩ BV(Ω ε , SO(3)) such that ∇u -r ε L 2 (Ωε) Kd Ωε (∇u, S Ωε ), (41)
r ε BV(Ωε) Kd Ωε (∇u, S Ωε ). (42)
Proof. Recalling that ω z denotes the (rescaled) section at abscissa z, let us define Ω z := [0, 1] × ω z . Let f z be the diffeomorphism from ω onto ω z defined by f z (x 2 , x 3 ) := y(z, x 2 , x 3 ). Then

g δ := f z+δ •(f z ) -1
is a diffeomorphism from ω z onto ω z+δ which induces a natural diffeomorphism from Ω z onto Ω z+δ . Its satisfies

∇g δ -Id = (∇f z+δ -∇f z )(∇f z ) -1
and so ∇g δ -Id tends to zero as δ tends to zero. Lemma 1 ensures that lim sup δ→0 K Ω z+δ K Ω z : the function z → K Ω z is upper-semi-continuous. Therefore the rigidity constant K Ω z is upper-bounded by some positive real K for z in the compact [-, ]. Now, for any z ∈ [-,ε], let us consider the part Ω z ε of Ω ε defined by

Ω z ε := Φ ε [z, z + ε] × ω . ( 43
)
It is the image of Ω z by the mapping F z ε :

F z ε (u, v, w) := Φ ε z + εu, f -1 z (v, w) ,
which can be rewritten

F z ε (u, v, w) = ϕ(z + εu) + ε(g εu ) 2 (v, w)n(z + εu) + ε(g εu ) 3 (v, w)b(z + εu).
Let us check that the gradient of F z ε is close to a similarity. We denote r z the rotation which transforms the canonical basis (e 1 , e 2 , e 3 ) in (t(z), n(z), b(z)). We have, for some constant C, by using the results [START_REF] Jurak | Modelling of curved rods[END_REF],

∂ 1 F z ε -εr z • e 1 = ε∂ 1 Φ ε z + εu, f -1 z (v, w) -εt(z) Cε 2 .
It is easier to estimate the two other terms together: we have

∂ 2 F z ε -εr z • e 2 + ∂ 3 F z ε -εr z • e 3 ε ∇g εu -Id + n(z + εu) -n(z) + b(z + εu) -b(z) Cε 2 .
Thus ∇F z εεr z 2Cε 2 and we can apply Corollary 2. As, for any z ∈ [-, ], K Ω z < K, then, for ε sufficiently small,

∀z ∈ [-, -ε], K Ω z ε K(1 + 2Cε 2 ) 1 -(6 + 2K)2Cε 2 2K. (44)
A similar reasoning shows, for

Ω z ε := Φ ε ([z, z + 2ε] × ω), the existence of a positive constant K such that ∀z ∈ [-, -2ε], K Ω z ε < 2 K. ( 45
)
Let us now denote n ε := ε (which was assumed to be an integer). For any integer i ∈ {0, . . . , n ε -1}, let us consider the set Ω iε ε and introduce the rotation r i ε such that ∇u -

r i ε L 2 (Ω iε ε ) = d Ω iε ε (∇u, s Ω iε ε )
. Note that for i < 0, we have r i ε = Id. The function

r ε := nε-1 i=0 r i ε 1 Ω iε ε
is a piecewise constant function which belongs to S Ωε . Rigidity lemma applied to Ω iε ε reads

Ω iε ε ∇u -r i ε 2 dx 4K 2 d Ω iε ε (∇u, S Ω iε ε ) 2 . ( 46
)
By summing over i, we get

∇u -r ε L 2 (Ωε) 2Kd Ωε (∇u, S Ωε ). (47)
Inequality (39) is proved. Now let us apply Rigidity lemma to

Ω iε ε = Ω iε ε ∪ Ω (i+1)ε ε
states the existence of a rotation ri ε such that

Ω iε ε ∇u -ri ε 2 dx 4 K 2 d Ω iε ε (∇u, S Ω iε ε ) 2 . ( 48
)
Restricting the integral at the left-hand side of this inequality to Ω iε ε and using (46) we get

Ω iε ε r i ε -ri ε 2 dx 8K 2 d Ω iε ε (∇u, S Ω iε ε ) 2 + 8 K 2 d Ω iε ε (∇u, S Ω iε ε ) 2 . ( 49
)
The measure of Ω iε ε was estimated in [START_REF] Mora | Derivation of a rod theory for phase-transforming materials[END_REF]. Thus

r i ε -ri ε 2 8 C 2 |ω|ε 3 K 2 d Ω iε ε (∇u, S Ω iε ε ) 2 + K 2 d Ω iε ε (∇u, S Ω iε ε ) 2 . ( 50
)
We get a similar estimate for r i+1 ε ri ε 2 by restricting the integral at the left-hand side of (48) to

Ω (i+1)ε ε . Using the fact that d Ω iε ε (∇u, S Ω iε ε ) 2 = d Ω iε ε (∇u, S Ω iε ε ) 2 + d Ω (i+1)ε ε (∇u, S Ω (i+1)ε ε ) 2 we obtain r i+1 ε -r i ε 2 16 C 2 |ω|ε 3 K 2 + 2 K 2 d Ω iε ε (∇u, S Ω iε ε ) 2 . ( 51
)
We get by summation, for any index i,

i j=0 r j ε -r j-1 ε 2 i i j=0 r j ε -r j-1 ε 2 n ε 16(K 2 + 2 K 2 ) C 2 |ω|ε 3 i j=0 d Ω jε ε (∇u, S Ω jε ε ) 2 n ε 32(K 2 + 2 K 2 ) C 2 |ω|ε 3 d Ωε (∇u, S Ωε ) 2 32(K 2 + 2 K 2 ) C 2 |ω|ε 4 d Ωε (∇u, S Ωε ) 2 . ( 52
)
And so,

r i ε -r 0 ε 2 32(K 2 + 2 K 2 ) C 2 |ω|ε 4 d Ωε (∇u, S Ωε ) 2 . ( 53
)
Multiplying by the volume |Ω iε ε | and summing from i = 0 to n ε -1 leads to

r ε -r 0 ε L 2 (Ωε) ε 32C 3 C -1 2 K 2 + 2 K 2 d Ωε (∇u, S Ωε ). (54)
Proof of the second inequality is concluded using (47) and triangular inequality. From (52), we can also deduce an estimate for the variations of r ε . We have

r ε BV(Ωε) = nε j=0 ω ε jε r j ε -r j-1 ε 32 |ω|C 2 3 C -1 2 K 2 + 2 K 2 d Ωε (∇u, S Ωε ). ( 55
)
Hence r ε satisfies the two last inequalities of the theorem.

Corollary 3. There exists a constant K such that, for ε sufficiently small and for any u in

H 1 b (Ω ε ), u -Id H 1 (Ωε) K ε d Ωε (∇u, S Ωε ). ( 56
)
Proof. In order to take easily into account the boundary condition, let us extend (without changing the notations) the domain Ω ε by considering suitable extensions of ϕ on [-, ], of y and Φ ε on [-, ] × ω.

We also extend any u ∈ H 1 b (Ω ε ) by setting u = Id on the new part

Ω ε := Φ ε ([-, 0] × ω]. Theorem 2 states the existence of r ∈ s Ωε satisfying Ωε∪Ω ε ∇u -r 2 dx K 2 ε 2 d Ωε∪Ωε (∇u, S Ωε∪Ωε ) 2 .
Using the fact that ∇u = Id on Ω ε , we have

d Ωε∪Ωε (∇u, S Ωε∪Ωε ) = d Ωε (∇u, S Ωε ).
Restricting the integral on the left-hand side on the previous inequality to Ω ε leads to

| Ω ε | Id -r 2 K 2 ε 2 d Ωε (∇u, S Ωε ) 2 . Thus Id -r 2 L 2 (Ωε) 2C 3 C 2 K 2 ε 2 d Ωε (∇u, S Ωε ) 2
and triangular inequality gives,

∇u -Id L 2 (Ωε) 2C 3 C 2 + 1 K ε d Ωε (∇u, S Ωε ).
The result will be obtained when checked that the Poincaré constant in H 1 0 (Ω ε ) is bounded by a constant independent of ε. Indeed, let us use the change of variables Φ ε . Using estimations ( 5), ( 6) and the fact that is an obvious upper-bound for the Poincaré constant on H 1 0 (C),

u -Id 2 L 2 (Ωε) C 2 ε 2 C u Φ ε (x) -Id 2 ε 2 dx C 2 2 ε 2 C ∇(u • Φ ε )(x) -Id 2 dx 2C 2 2 ε 2 C ∇u Φ ε (x) -Id 2 dx 2C 2 C 3 2 Ωε ∇u(x) -Id 2 dx.

Compactness

We use the notations defined in Theorem 1. Let us begin by a general remark:

Remark 4. If v ε is a sequence in L 2 (Ω ε , R p
) such that, for some constant M , v ε L 2 (Ωε) < εM, then up to a subsequence, the sequences v ε • Ψ ε and vε converges weakly respectively in L 2 (Ω) and L 2 (L).

Indeed, owing to estimation (5) the sequence v ε • Ψ ε is bounded in L 2 (Ω) and, up to a subsequence converges weakly to some v 0 in L 2 (Ω). Owing also to estimation [START_REF] Mora | A nonlinear model for inextensible rods as a low energy Γ -limit of three-dimensional nonlinear elasticity[END_REF], the sequence of functions y → v ε • Ψ ε (y) det(∇Ψ ε (y))|ω ε y 1 | is still bounded in L 2 (Ω). A subsequence converges weakly to some w 0 . It is then easy to check that vε converges weakly to the function x 1 → ωy 1 w 0 (y) dy 2 dy 3 .

and so, owing to Remark 4, that G ε := G ε • Ψ ε converges weakly to some G in L 2 (Ω). We have ∇u ε = r ε (Id + εG ε ) and so

∇u ε = r ε (Id + εG ε )∇Ψ ε .
Applying this equality to a basis vector e i and deriving with respect to the variable y j , we get for any (i, j) ∈ {1, 2, 3} 2 ,

∂ j ∂ i u ε = r ε (Id + εG ε ) • ∂ j ∂ i Ψ ε + ∂ j r ε (Id + εG ε ) • ∂ i Ψ ε . ( 61
)
This equality holds in the sense of distributions on Ω and the quantity has to be symmetric with respect to i and j. For i = 1 and j ∈ {2, 3}, we get

∂ j r ε (Id + εG ε ) • ∂ 1 Ψ ε = ∂ 1 r ε (Id + εG ε ) • ∂ j Ψ ε .
Hence, taking j = 2,

r ε ∂ 2 (G ε ) • ∂ 1 Ψ ε = r ε + ε∂ 1 (r ε G ε ) • n,
which gives, passing to the limit, r

• ∂ 2 (G) • t = r • n. Recalling the definition of ṙ ( ṙ is the vector such that ∀V , ṙ ∧ V = r -1 r • V ), ∂ 2 (G • t) = ṙ ∧ n. (62)
In the same way, taking j = 3 we get

∂ 3 (G • t) = ṙ ∧ b. ( 63
)
From these equalities, which hold in the sense of distributions, we deduce the following structure for G • t:

(G • t)(y) = h(y 1 ) + ṙ(y 1 ) ∧ y 2 n(y 1 ) + y 3 b(y 1 ) . ( 64
)
Now let us define on Ω, wε (y) := ε -1 u ε (y)r ε (y 1 ) • (y 2 n(y 1 ) + y 3 b(y 1 ), on [0, ] wε (y 1 ) := |ω y 1 | -1 ωy 1 wε (y) dy 2 dy 3 , and, on Ω, w ε (y) = wε (y)wε (y 1 ). It is easy to check that

∂ 2 w ε = r ε G ε • n, ∂ 3 w ε = r ε G ε • b. ( 65
)
Hence the gradient of w ε with respect to the variables y 2 and y 3 is bounded in L 2 (Ω) and Poincaré Wirtinger inequality applied to each section ω y 1 ensures that w ε is a bounded sequence in L 2 (Ω). So is r -1 ε w ε , a subsequence of which converges weakly to some w in L 2 (Ω). Passing to the limit in (65) leads to

G • n = ∂ 2 w, G • b = ∂ 3 w. ( 66
)
Note that, for almost all y 1 ∈ [0, ], the function (y 2 , y 3 ) → w(y 1 , y 2 , y 3 ) belongs to H 1 (ω y 1 ).

Let A ε ⊂ Ω the set of points x where G ε (x) > √ ε. Its measure tends to zero with ε and G ε (1-1 Aε ) still converges to G. Let η > 0. From the definition of W in , we know that, for ε sufficiently small,

G ε < √ ε ⇒ W (Id + εG ε ) W in (εG ε )(1 -η).
Recalling that W (Id) = 0, and using the fact that the functional of linear elasticity is quadratic, positive and so lower semi-continuous, we have

lim inf E ε (u ε ) = lim inf 1 ε 4 Ωε W (Id + εG ε ) dx lim inf 1 ε 2 (1 -C 3 ε) Ω W (Id + εG ε ) dy lim inf 1 ε 2 Ω W Id + ε(1 -1 Aε )G ε dy lim inf (1 -η) Ω W in (1 -1 Aε )G ε dy (1 -η) Ω W in (G) dy.
Then, passing to the limit η → 0,

lim inf E ε (u ε ) Ω W in (G) dy. (67)
A simple minimization with respect to the components n

• G • n, n • G • b and b • G • n of G shows that W in (G) Y 2 (t • G • t) 2 + µ 2 (t • G • n + n • G • t) 2 + µ 2 (t • G • b + n • G • b) 2 , ( 68
)
where Y is the combination of µ and λ defined in Section 2.2.2. Let us now integrate this expression over a section ω y 1 . For the first term, using (64), and the assumption that (y 1 , 0, 0) is the inertial center of the section, we get

ωy 1 (t • G • t) 2 dy 2 dy 3 I 2 (y 1 )( ṙ • n) 2 + I 3 (y 1 )( ṙ • b) 2 + I 23 (y 1 )( ṙ • b)( ṙ • n), ( 69
)
where I 2 , I 3 and I 23 are the inertial moments or product defined in Section 2.2.2. For the last two terms, we use (64) and (66) together. We have

ωy 1 (t • G • n + n • G • t) 2 + (t • G • b + n • G • b) 2 dy 2 dy 3 = (t • ṙ) 2 ωy 1 -y 3 + ∂ 2 (w • t) 2 + y 2 + ∂ 3 (w • t) 2 dy 2 dy 3 (t • ṙ) 2 J(y 1 ), (70)
where J(y 1 ) is the geometrical parameter defined by the ad hoc minimization problem in 2.2.2. Collecting (67)-(70), we finally obtain lim inf E ε (u ε ) F(u, r).

Eq. ( 60) already proved that the constraint u = r • t is satisfied. The sequence of inequalities (67)-( 70) shows that all components of ṙ are bounded in L 2 (L, R 3 ). Thus r -1 r belongs to L 2 (L, R 3×3 ) and so does r . Then r belongs to H 1 (L, SO(3)) and Eq. ( 60) shows that u belongs to H 2 (L, R 3 ). The boundary conditions are naturally imposed by the fact that u(ϕ(y 1 )) = ϕ(y 1 ) and r(ϕ(y 1 )) = Id on the extended part (y 1 ∈ [-, 0]). The regularity of u and r imposes u(ϕ(0)) = ϕ(0) and r(ϕ(0)) = Id. Hence (u, r) belongs to the admissible space H ad . Proof of the lower-bound inequality (point (ii) of Theorem 1) is concluded.

Upperbound

As usual in Γ -convergence proofs, we restrict our attention when proving point (iii) of Theorem 1 to a function u such that F(u) is finite. Let r such that F(u) = F(u, r). Using a density argument we also restrict our attention to regular functions u and r. With the same argument we assume that r = Id in a neighborhood of ϕ(0).

Using the curvilinear parametrization of L, we consider u := u • ϕ and r := r • ϕ which are defined on [0, ]. We also consider the vector valued function ṙ defined by ∀V , ṙ(y 1 ) ∧ V = r(y 1 ) -1 r (y 1 ) • V . We introduce the function ψ y 1 , solution of the minimization problem (18) and we set ψ(y 1 , y 2 , y 3 ) := ψ y 1 (y 2 , y 3 ). Then we define u ε on Ω ε by defining u ε = u ε • Ψ ε on Ω: Recalling that F(u, r) < +∞ implies r -1 • u = t, the comparison of the two last equalities leads to

G ε • t = G • t + O(ε).
Then G ε converges uniformly to G. This implies also that (∇u ε ) • Ψ εr converges uniformly to zero. Let η > 0. For ε sufficiently small,

W (∇u ε ) = W r -1 ∇u ε (1 + η)W in r -1 ∇u ε -Id ε 2 (1 + η)W in (G ε ).
Hence

E ε (u ε ) 1 ε 2 (1 + η) Ωε W in G ε (x) dx (1 + η)(1 + C 3 ε) Ω W in G ε (y) dy, lim sup E ε (u ε ) (1 + η) Ω
W in G(y) dy and passing to the limit η → 0,

lim sup E ε (u ε ) Ω W in G(y) dy. ( 72
)
Let us check now that the quantity Ω W in (G(y)) dy coincides with the expression (19) for F(u, r). Indeed, concerning the diagonal terms, we have

λ 2 (t•G•t + n•G•n + b•G•b) 2 + µ (t•G •t) 2 + (n•G•n) 2 + (b•G•b) 2 = µ(3λ + 2µ 2(λ + µ) -( ṙ • b)y 2 + ( ṙ • n)y 3 2 .

Fig. 1 .

 1 Fig. 1. The mean line L of the beam, a non-planar curve.

Fig. 2 .

 2 Fig. 2. The prototype section ω and the reference cylinder C.

Fig. 3 .

 3 Fig. 3. The reference domain Ω.

Fig. 4 .

 4 Fig. 4. The beam Ωε at rest.

2 3 -y 2 2 2 + 2 3 -y 2 2 2 - 2 = r -1 ∇u ε • e 2 == ( 1 -

 2222221 u ε (y) := u(y 1 ) + εr • (y 2 n + y 3 b)+ ε 2 ( ṙ • t)ψ(y)r • t + ε 2 -λ 2(λ + µ) ( ṙ • b) y ( ṙ • n)y 2 y 3 r • n + ε 2 -λ 2(λ + µ) ( ṙ • n) y ( ṙ • b)y 2 y 3 r • b. (71) Let us estimate G ε := ε -1 r -1 (∇u εr) or G ε := G ε • Ψ ε . We have (εG ε + Id)∇Ψ ε = r -1 ∇u ε .We can explicit the partial derivatives of u ε and use the expression[START_REF] Jurak | Modelling of curved rods[END_REF] for the partial derivatives of Ψ ε . Defining G byG := -( ṙ • b)y 2 + ( ṙ • n)y 3 t ⊗ t + -λ 2(λ + µ) (n ⊗ n + b ⊗ b) + ( ṙ • n)y 2 + ( ṙ • b)y 3 -λ 2(λ + µ) (n ⊗ bb ⊗ n) + ( ṙ • t)(-y 3 n ⊗ t + y 2 b ⊗ t + ∂ 2 ψt ⊗ n + ∂ 3 ψt ⊗ b)we have(εG ε + Id) • (εn) = (εG ε + Id)∇Ψ ε • e εn + ε 2 G • n. Hence G ε • n = G • n.In the same way we getG ε • b = G • b.Computations are a bit more tricky in the e 1 direction: We have, using (7)(εG ε + Id)∇Ψ ε • e 1 = (1ετ y 2εξy 3 )(εG ε + Id) • t ετ y 2εξy 3 )t + εG ε • t + O ε 2 ,and on the other hand(εG ε + Id)∇Ψ ε • e 1 =r -1 ∇u ε • e 1 = r -1 u + εr • (y 2 n + y 3 b) + εr • (-y 2 τ ty 3 ξt) + O ε 2 .

This vector is interpreted in rods theories as "the variation of the rotation vector of the sections". Such an interpretation is valid in linear elasticity. In the non-linear case considered here, the vector ṙ does not derive from any "rotation vector" (which has no sense).

Now, let u ε be a sequence with bounded energy (E ε (u ε ) M ). Owing to assumption [START_REF] Murat | Comportement asymptotique des solutions du système de l'élasticité linéarisée anisotrope hétérogène dans des cylindres minces[END_REF], we have

Then Corollary 3 implies

Previous remark states that, up to a subsequence,

and then ūε and ∇u ε converge weakly to some u and r in L 2 (L). Point (i) of Theorem 1 is proved.

Proof of the Γ -convergence result

Lowerbound

First, in order to take easily into account the boundary condition, we extend (without changing the notations) the domain Ω ε by considering suitable extensions of ϕ on [-, ], of y and Φ ε on [-, ] × ω. We also extend any u ∈ H 1 b (Ω ε ) by setting u = Id on the new part

It is clear that it is enough to consider in the proof of point (ii) of Theorem 1 only sequences (u ε ) with bounded energy (E ε (u ε ) M ). Moreover we can restrict our attention to a subsequence (still denoted

. The statements will then be proved, when proved for some subsequence.

In Section 3.2 we did not write all the implications of the rigidity theorem. Under assumption (57), Theorem 2 states also the existence of r ε in S Ωε such that

Inequality (59) implies that r ε is bounded in BV(L) and then converges strongly in L 2 (L) to some r which takes values in SO(3) (in an equivalent way r ε := r ε • Ψ ε converges strongly in L 2 (Ω) to r := r • y). Inequality (58) implies that ∇u ε • Ψ εr ε converges strongly to zero in L 2 (Ω). Therefore ∇u ε • Ψ ε converges strongly in L 2 (Ω) to the function r := r • y which depends only on the first variable.

We have ∇u ε = (∇u ε • Ψ ε )∇Ψ ε , and from expression [START_REF] Jurak | Modelling of curved rods[END_REF] we know that ∇Ψ ε converges uniformly on Ω to t ⊗ e 1 . Passing to the limit, we get, in the sense of distributions on Ω,

Hence u depends only on the first variable, ∂ 1 u = r • t, and finally, all along the line L,

From inequality (58) we deduce also that

The integration of these terms over a section ω y 1 gives the quantity

Concerning the non-diagonal terms, we have n

Taking into account the definition of ψ, the integration of these two last terms over a section ω y 1 gives the quantity J( ṙ • t) 2 . Finally the expression (19) for F(u, r) is recovered.

It is clear that u ε belongs to H 1 b (Ω ε ) and that ūε converges to u. We already noticed that (∇u ε ) • Ψ ε converges uniformly to r. Thus ∇u ε converges to r. This concludes the proof of point (iii) of Theorem 1.


Keywords:

The properties of thermoluminescence (TL) observed from various mixed rare earth (RE) phosphate powders La 1-x-y Ce x Tb y PO 4 are reported. TL peaks appear around -175°C, -135°C, -80°C, -35°C and +25°C and the TL emission spectra of La 1-x-y Ce x Tb y PO 4 are mainly composed of characteristic line transitions of Tb 3+ ions. The results show that the co-doping with Ce 3+ is not required to observe the TL emission of Tb 3+ . A model for the recombination mechanisms of the various TL peaks is discussed.

T T T h

h PHILIBERT IACCONI (1) *, MICHEL JUNKER (1,[START_REF] Bourcet | Proceedings of the 10th Rare Earth Research Conference[END_REF] , BERNARD GUILHOT [START_REF] Bourcet | Proceedings of the 10th Rare Earth Research Conference[END_REF] , DENIS HUGUENIN [START_REF] Denis | [END_REF]

I. Introduction

When they are excited by the 253.7 nm radiation of a low pressure mercury lamp, the La 1-x-y Ce x Tb y PO 4 powders emit a green luminescence [1 and 2] used as the green component in commercial trichromatic lamps. In fact, the spectral emission is composed of various lines of which the main, appearing at 544 nm, is assigned to the 5 D 4 → 7 F 5 transition of the Tb 3+ ion [3]. The Tb 3+ emission results from an energy transfer from Ce 3+ to Tb 3+ ions [1 and 2].

The transfer mechanism can be described in the following manner:

 Excitation of the Ce 3+ ions into 5d levels by an incident photon of wavelength λ exc (220<λ exc <300 nm), then non-radiative deexcitation towards the lowest 5d level followed by an energy transfer by diffusion towards neighbouring Ce 3+ ions.

 Energy transfer from Ce 3+ ion towards a neighbouring Tb 3+ ion. The excitation into the 5 D 2 level is followed by radiationless transitions to the 5 D 3 and 5 D 4 levels, then radiative deexcitation of Tb 3+ ions by relaxing to a 7 F J (0<J<6) level.

The overall efficiency of the Tb 3+ emission can be affected by the presence of lattice defects whi are ch susceptible to:

1. absorb part of the exciting incident radiation,

2.

absorb part of the emitted radiation,

3.

break the diffusion chain between the Ce 3+ ions.

The aim of this work is to study the thermoluminescent properties of La 1-x-y Ce x Tb y PO 4 mixed compounds in order to identify the role of such defects. [4]. After drying and grinding, they are fired at 1250°C, during 1 h, under N 2 atmosphere. At the end of this tre ment, one obtains a powder characterised by a particle size in the 1-3 μm range which at obeys to the La 0.57 Ce 0.29 Tb 0.14 PO 4 formula. This compound is called LaCeT. La 1-x Ce x PO 4 and La 1-y Tb y PO 4 samples were obtained according to the same process: coprecipitation then firing at 1250°C during 1 h under N 2 atmosphere. The values of x and y are the same as that for the LaCeT. So, the obtained powders are characterised by La 0.71 Ce 0.29 PO 4 and La 0.86 Tb 0.14 PO 4 formulas. I II I. .2 2. . T

II. Experimental conditions

Th he e e ex xp pe er ri im me en nt ta al l a ap pp pa ar ra at tu us s The thermoluminescence (TL) measurements were carried out in a secondary vacuum chamber from the liquid nitrogen temperature (LNT) to 350°C. The sample, put down on a sample holder, was first cooled down at LNT and irradiated by X-rays (or by UV-mercury low pressure lamp). Then, it was heated up at the constant rate of 30 Kmin -1 . The TL curves were registered thank to an S13 response photomultiplier (Philips 2018B). The irradiation was provided by a tungsten-target X-ray generator operating at 45 kV, 2 mA with an air kerma rate of 0.035 Gy/min. Usually, the irradiation time was 10 min and the air kerma 0.35 Gy. The spectral analyses were obtained with 10-25 nm bandwidth interference filters and corrected to the spectral characteristics of the photomultiplier, filters and optical windows.

III. Results

I

. Curves of TL

After irradiation, at LNT, by the 253.7 nm of a mercury low pressure lamp, the LaCeT sample does not give any significant TL response. On the other hand, the X-ray irradiation at LNT, gives rise to an intense TL phenomenon. As an example, we report the curve of TL observed for the LaCeT (Figure 1). The glow curve is characterised by an intense peak at -80°C with two shoulders around -135°C and -35°C which indicate the presence of three types of trap defects existing in the LaCeT crystalline matrix. These results must be compared with those of Bourcet [1] who observed that, after irradiation at LNT with a radiation of wavelength λ exc =121 nm, La 0.35 Ce 0.60 Tb 0.05 PO 4 sample presents one TL peak around -145°C and who noted that this peak did not appear for irrad nm. iation with wavelength longer than 121 III. 1

.2. Spectral emissions of TL peaks

The spectral analyses of each TL peak are reported in the Figure 2 (a). Between 450 and 600 nm, one observes the characteristic line transitions of Tb 3+ ions originating from the 5 D 4 level, namely the 5 D 4 → 7 F 6,5,4 transitions which correspond, respectively, to 487, 544 and 584 nm. On the other hand, the emission spectra do not reveal any transitions from 5 D 3 level. This is due to the well-known effect of cross-relaxation which results of a resonant energy transfer between the transitions 5 D 3 → 5 D 4 and 7 F 6 → 7 F 0 of Tb 3+ ions [5]. A broad band, very less intense than the Tb 3+ emission lines, is also observed around 320-340 nm. It is constituted of two components and is probably due to the Ce 3+ ions. Bourcet [1] showed that the emission spectrum of Ce 3+ ions, detected at RT in La 0.35 Ce 0.6 Tb 0.05 PO 4 samples, presents a broad band with two maxima at 315 and 335 nm. co om mp po ou un nd ds s We studied also some samples with composition close to the one of LaCeT and in whom the chain of energy transfer from Ce to Tb does not exist. The TL curves of these products, which obey respectively to La 0.71 Ce 0.29 PO 4 and La 0.86 Tb 0.14 PO 4 formulas, are presented in Figure 1. They were obtained in the same experimental conditions as that for LaCeT TL curve. As expected, the emission of Tb 3+ is found only in the product containing Tb 3+ ions, namely La 0.86 Tb 0.14 PO 4 (Figure 2(b)). Two TL peaks are observed around -75°C and +20°C (Figure 1(b)) with shoulders around -125°C and -35°C and exhibit emission spectra with spectral lines situated at 487, 544 and 584 nm which correspond respectively to the transition 5 D 4 → 7 F 4,5,6 of Tb 3+ . For its part, Bourcet [1] observed two TL peaks at -100°C and -53°C for the emission of Tb 3+ in La 0.95 Tb 0.05 PO 4 irradiated by 121 nm radiation. Moreover, the 320-340 nm emission is detected in the TL emission spectra of the La 0.71 Ce 0.29 PO 4 compound (Figure 3). The TL curve of this sample is constituted of three peaks at -175°C, -135°C and -40°C with a shoulder at -10°C (Figure 1(a)).

IV. Discussion and conclusion

The mechanism of the fluorescence in LaCeT, is very well known. By exposure to λ UV =253.7 nm, Ce 3+ ions are excited and transfer their energy to the Tb 3+ species. As a result, the luminescence spectrum of LaCeT is principally composed by the Tb 3+ emission [1 and 4]. In TL, the problem is quite different. The exciting radiation is initially used to populate the trapping levels with the charge carriers released from the irradiation and the TL emission is then induced by an heating up which causes the detrapping of these charge carriers and their subsequent radiative recombination. So, the mechanism of TL emission is activated by thermal stimulation. The TL curve of LaCeT obtained after X-ray irradiation at -196°C is constituted of two peaks relatively intense at -135°C and -80°C with a weak shoulder around -35°C. The spectral distribution of the two main TL peaks is broken down into two kinds of emission. The one, characteristic of the Tb 3+ ions, is composed of lines occurring from level 5 D 4 towards the levels 7 F 4,5,6 ; the other, very less intense, is observed around 320-340 nm and has been assigned to the transitions between the lowest 5d level (likely 5 D 3 [1]) and the 4f levels 2 F 7/2 and 2 F 5/2 of Ce 3+ ions. Usually, the 5d→4f transitions are very sensitive to the crystal field effects and the Ce 3+ broad band emission might be shifted when the host crystalline matrix is changed. In La 0.71 Ce 0.29 PO 4 and La 0.57 Ce 0.29 Tb 0.14 PO 4 compounds investigated in this work, the two components of Ce 3+ emission appear approximately to the same wavelengths and correspond well to the results published by Bourcet [1] who analysed the emission of the La 1-x-y Ce x Tb y PO 4 , La 1-x Ce x PO 4 and CePO 4 set (Table 1). This could have mean that the variation of the crystal field in La 1-x-y Ce x Tb y PO 4 is weak when x and y vary or that optical properties of Ce 3+ depend very feebly on the crystal field. This interpretation is corroborated by the data reported in Table 1 which gives the spectral characteristics of Ce 3+ observed in various phosphate compounds. Generally speaking, two types of mechanisms can be suggested to explain the trivalent rare earth (RE 3+ ) emission observed in TL. They take into account either an energy transfer or a charge transfer process.

(1) In the first case, the mechanism of TL is entirely determined by the crystalline host matrix. Under ionising radiation, electron-hole pairs are created and trapped into the host lattice. Then, during the thermal stimulation, charge recombinations take place into the matrix and the energy released from the recombination is transferred to the RE 3+ ions which emits their specific emission. The temperature of TL peaks depends solely on the host lattice traps whereas their spectral emission depends on the RE 3+ ions. In this case, an overlap is required between the energy available from the host lattice and the RE electronic states; namely between the host lattice emission (HLE) and the Ce 3+ ( orTb 3+ ) excitation.

(2) In the second case, one considers that, under ionising radiation, the charge state of the RE is changed and that the released charge carriers are trapped in the crystalline matrix. Considering the electronic properties of ions such as Ce 3+ or Tb 3+ , we think that during excitation process, they would have to act as electron donors following scheme (I)

3 4 RE RE    e  (I)
At the end of this excitation mechanism, the electrons e -remain trapped in the lattice. Then, a thermal stimulation causes the release of the trapped electrons and their radiative recombination with RE 4+ ions following scheme (II)

  * 4 3 3 RE RE e RE RE h         (II)
The emission arising from the heating up is characteristic of the involved RE 3+ whereas the temperature of TL peaks depends also on the host lattice structure. Another possibility, which can range in the field of charge transfer mechanisms, can be related to the eventual capacity of Ce 3+ or Tb 3+ ions to trap holes release (under ionising radiations) by the crystalline matrix. Following this assumption, the position of TL peaks observed during the subsequent heating up would be specific to the involved RE ions.

In the actual state of our investigations, we note that:

(i) five TL peaks are observed at around -175°C (peak I), -135°C (peak II), -80°C (peak III), -35°C (peak IV) and +25°C (peak V);

(ii) only TL peaks II and IV appear on the three compounds investigated (LaCeT, La 0.86 Tb 0.14 PO 4 and La 0.71 Ce 0.29 PO 4 ). They exhibit the Tb 3+ (LaCeT and La 0.86 Tb 0.14 PO 4 ) or the Ce 3+ (La 0.71 Ce 0.29 PO 4 ) characteristic emission;

(iii) LaCeT and La 0.86 Tb 0.14 PO 4 have very similar TL properties (same TL curves with presence of peaks II-IV with Tb 3+ emission);

(iv) peak III appears only in the TL curves of LaPO 4 : Tb (namely LaCeT and La 0.86 Tb 0.14 PO 4 ) and is the main peak with Tb 3+ emission (accompanied by a weak Ce 3+ emission for LaCeT);

(v) peak I appears only in La 0.71 Ce 0.29 PO 4 compound (Ce 3+ emission).

If we assume that the three investigated compounds can be considered as LaPO 4 lattice-doped or co-doped by Ce, Tb or Ce+Tb elements and that the emissions of TL peaks of LaCeT are governed by an energy transfer mechanism (mechanism 1) from the host lattice to Ce 3+ then to Tb 3+ , we would have to observe similar TL curves for La 0.71 Ce 0.29 PO 4 and LaCeT products. This is not the case because peak I, present in TL curve of La 0.71 Ce 0.29 PO 4 , is not observed in LaCeT compound (Figure 1) and because TL peak III, which appears in LaCeT compounds, is absent in La 0.71 Ce 0.29 PO 4 sample. Besides, the similarity of TL curves of Tb-doped LaPO 4 (LaCeT and La 0.86 Tb 0.14 PO 4 ) encourages us to think that the presence of Ce 3+ is not required to observe the Tb 3+ emission in TL peaks II-IV of LaCeT. We can equally reject the hypothesis of energy transfer between the host lattice and the Ce 3+ ions to interpret the presence of peak I. If this hypothesis was good, we would have to observe TL peak I in the LaCeT compound. The emission of the main TL peaks of the LaCeT material does not seem to be interpreted by an "HLE→Ce 3+ →Tb 3+ " energy transfer mechanism. So, the energy transfer mechanism between Ce 3+ and Tb 3+ as described by Bourcet [1] does not ensue the detrapping of charge carriers responsible for the various TL peaks of LaCeT. Now, suppose that the hypothesis 2 applies. Under ionising radiation, electrons are released by Ce 3+ as well as Tb 3+ and trapped in different lattice sites. With regard to the TL properties of the compound studied, we can suppose that traps II and IV, common to the three products, are specific to the LaPO 4 lattice. On the other hand, traps I and III are respectively associated to the LaPO 4 :Ce and LaPO 4 :Tb (with or without Ce) lattices.

During the heating, the trapped electrons are released at specific temperature from the LaPO 4 lattice and their recombination with the Ce 4+ (La 0.71 Ce 0.29 PO 4 ) or with the Tb 4+ ions (La 0.86 Tb 0.14 PO 4 ) occurs with characteristic emissions of these trivalent ions (scheme II). In the case of the LaCeT, the two ions Ce and Tb being present, the two characteristic emissions are observed and are probably obtained by disconnected recombinations Ce 4+ +e -and Tb 4+ +e -. In this last case, it is possible that a radiative energy transfer occurs from Ce 3+ to Tb 3+ ions.

As concluding remark, we note that all the TL peaks of LaCeT are situated at temperatures inferior to the working temperature of trichromatic lamps which is superior to room temperature. So, the observed TL phenomenon is not harmful to the application of LaCeT as green luminophors in such lamps because all the traps remain empty. In particular, the LaCeT luminophor induces no delay in the rising time of lighting and, reciprocally, no phosphorescence after the extinction of UV excitation.

  II II I. .1 1. . T TL L o of f L La aC Ce eT T s sa am mp pl le es s III.1.1

Figures 6 F

 6 Figures

  The products studied hereafter are obtained from LaPO 4 , mH 2 O and La 1-x-y Ce x Tb y PO 4 , zH 2 O precursors provided by Rhodia (France) and prepared by co-precipitation

	I II I. .1 1. . P Pr re ep pa ar ra at ti io on n o of f s sa am mp pl le es s


Benoît Piro

Anh Lan

Dang

Minh Chau Pham

Silvia Fabiano

Keywords: Biosensors, Covalent immobilisation, Conducting polymers, Glucose, Electrochemical quartz microbalance

We have constructed and characterised a glucose sensor using glucose oxidase (GOD) covalently attached to carboxylic acid polyethyleneglycol (PEG), called (PEG-GOD). This modified enzyme was entrapped afterwards within poly(3,4-ethylenedioxythiophene) (PEDT) films electrogenerated on glassy carbon (GC) electrodes. The composite (PEG-GOD/PEDT) film is more porous than the film without enzyme (PEDT+PEG). Data from electrochemical quartz microbalance (ECQM) and pH-stat experiments indicate a good relative activity of the modified enzyme, ca. 12-15%. Amperometric measurements, using ferrocenemethanol as the redox mediator, confirms that the modified enzyme is catalytically active. The effect of film thickness was also investigated. The sensitivities were quite similar for modified-GOD electrodes (ca. 3 mA cm -2 M -1 ) and unmodified-GOD electrodes (ca. 2.7 mA cm -2 M -1 ) but a better stability was obtained with modified PEG-GOD electrodes. K

I. Introduction

Amperometric biosensors based on immobilisation of enzymes in an electronically conducting polymer (ECP) matrix are of great interest [1], [2], [3], [4], [5] and [6]. ECPs can be grown selectively on a conductive surface allowing miniaturisation of the biosensor. The properties of the polymerised film can be easily controlled (porosity, thickness) by electrochemical procedures. Moreover, ECPs can be chemically modified to improve the affinity towards enzymes, eventually to form covalent bonds. Two conventional procedures exist for immobilisation of enzymes into ECP-modified electrodes. The first consists of the entrapment of the bulky enzyme within the polymer matrix during its electropolymerisation [7], [8] and [9]. The second consists of two steps: ECP is first grown on the electrode surface and then the enzyme is allowed to react chemically with the polymer surface by covalent bonds or by bioaffinity interactions [9] and [10]. The physical entrapment is the simplest way but the catalytic activity of the enzymes is drastically reduced down to a few percent of the value observed in the bulk [START_REF] Cosnier | Innocent Bioelectrochem[END_REF]. The enzyme attachment on a pre-polymerised film is a more complicated technique, which limits the enzyme quantity to a few monolayers at the film⏐solution interface and the activity is also reduced. Nevertheless, the stability is increased [12] and [13]. In general, the direct electron transfer between the electrode and the active site of the enzyme is difficult, and a redox mediator (cosubstrate) is needed in the catalytic cycle. Glucose oxidase (GOD), the most common oxidase enzyme, needs molecular oxygen as a cosubstrate for its reoxidation after reaction with the glucose substrate, in the following manner:

Glucose + GOD(ox) → Gluconolactone + GOD(red)

GOD(red) + O 2 → GOD(ox) + H 2 O 2
Oxygen is consumed and hydrogen peroxide is produced. Because oxygen cannot be available in all media with the same concentration and because conducting polymers tend to deteriorate in the presence of H 2 O 2 , it would be advantageous to replace oxygen by another mediator (Med): GOD(red) + Med(ox) → GOD(ox) + Med(red) Med(red) → Med(ox) + e - Recently, we have developed a composite film, [poly (3,4-ethylenedioxythiophene) (PEDT)+polyethyleneglycol (PEG)] (PEDT+PEG), by electrochemical oxidation of EDT in the presence of PEG in phosphate buffer. It was shown that the composite is excellent for the entrapment and delivery of very large biomolecules such as oligonucleotides (ODN), in view of in vivo applications [14]. Moreover, the polymer composite seems to be biocompatible. These excellent results motivated us to use the composite (PEDT+PEG) to construct an enzymemodified electrode. The enzyme GOD has already been immobilised in conducting polymers, in particular polypyrrole [15], [16], [17], [18] and [19], but also polythiophene [20]. GOD has also been adsorbed on porous membranes containing polypyrrole, results indicating a direct electron transfer between GOD and polypyrrole [21], [22] and [23]. However, controversial data exist about this direct transfer [18], [24] and [25]. More recently, glucose was detected in a deaerated medium on a platinum electrode coated with polypyrrole [26], but at high potential, and the direct transfer was not obvious. Keeping in mind that the more the polymer is conducting, the more the direct electron transfer can occur, and considering that PEDT has been described in the literature as one of the most conductive ECPs [27], we attempted then to immobilise GOD in a PEDT matrix, with the hope that a direct electron transfer could be achieved. In this work, we describe the elaboration and characterisation of a glucose sensor using GOD. The enzyme was first covalently bonded to a polyethyleneglycol bearing a carboxylic acid group. The modified enzyme (PEG-GOD) was then entrapped in the PEDT matrix during electropolymerisation of EDT in phosphate buffer containing PEG-GOD. PEG plays the role of surfactant for the EDT monomer (it was indeed observed that PEG enhances EDT solubility). The GOD immobilised under these conditions presents a better catalytic activity than that habitually observed in the literature [9], estimated by two methods: pH-stat for the enzyme activity [START_REF] Eisenthal | Enzyme Assays: a Practical Approach[END_REF] and ECQM measurements for the enzyme loading. Because amperometric tests without mediator during glucose monitoring are not satisfactory to demonstrate a direct electron transfer, we have used a ferrocene derivative (ferrocenemethanol, FcOH) as mediator to complete the study of the sensor. were purchased from Sigma. Reagent grade KH 2 PO 4 , K 2 HPO 4 and KCl were purchased from Prolabo. Dicarboxylic acid polyethylene glycol (PEG MM 600) was supplied by Fluka. 3,4-Ethylenedioxythiophene (EDT) was from Bayer. Ferrocenemethanol (FcOH) was from Aldrich. All reagents were used without purification; all solutions were prepared with bi-distilled water. Phosphate buffer (5×10 -2 M, pH 6) contains 5×10 -2 M KCl in all experiments. Glucose stock solution (0.25 M) was allowed to mutarotate for at least 24 h and stored at 4 °C prior to use. I II I. .2 2. . M Me et th ho od ds s II.2.1. Modification of GOD Glucose oxidase bears amine groups on the proteinic envelope. The modification of GOD was achieved by forming peptide bonds between the amine groups of GOD and the carboxylic acid groups of PEG (scheme shown below). This kind of coupling has often been described in the literature [26], [START_REF] Gross | The Peptides: Analysis, Synthesis[END_REF], [30] and [31].

II. Experimental

A 5 ml phosphate buffer (pH 6.0) solution containing GOD (1000 IU ml -1 ), with an excess of PEG MM 600 (10 -4 M), EDC (1.5×10 -4 M) and NHS (3×10 -4 M), was prepared and stirred for 2 h. Experiments were performed by varying the PEG/GOD ratio between 1:1 and 50:1. The 1:1 ratio gave no visible difference with the non-modified enzyme, whereas the 50:1 ratio led to a very deactivated enzyme. This could be explained by a steric hindrance of the PEG chains crosslinked on the enzyme, which impedes diffusion of glucose towards the FAD active centre [26] and [32]. A ratio between 3:1 and 4:1 gave the best result. After the reaction, the solution was dialysed against 0.05 M phosphate buffer for 48 h to eliminate reagents. Dialysis was performed using a 6000 MM cut-off dialysis membrane. The remaining solution, containing the PEG-modified enzyme (PEG-GOD), was stored in a freezer when not in use.

II.2.2. Film elaboration

All the electrochemical studies were carried out using a single-compartment cell with a glassy carbon working electrode, a platinum gauze counter electrode and an AgCl covered Ag wire as the reference electrode (Ag⏐AgCl). Glassy carbon electrodes (area 0.07 cm 2 ) were polished prior to use on a polishing cloth (Struers) sequentially with diamond paste of decreasing particle size (3 and 1 μm), rinsed with ethanol, ultrasonicated for 2 min in ethanol, and then air dried. Two different experiments were used to construct the enzyme electrodes. The first case consists in electropolymerisation of EDT in phosphate buffer containing PEG and GOD. The second consists, as a first step, in the modification of GOD with PEG (described above, leading to the modified enzyme PEG-GOD) and as a second step, electropolymerisation of EDT in phosphate buffer containing PEG-GOD. PEDT electrodes containing PEG and GOD (denoted as GOD/PEDT+PEG in the following) were obtained by electrooxidation at a constant potential of 1.1 V versus Ag⏐AgCl of 10 -2 M EDT in 5 ml phosphate buffer (pH 6)+10 -4 M PEG+1000 IU ml -1 GOD. Electrodes containing modified PEG-GOD (denoted as PEG-GOD/PEDT) were obtained by electrooxidation of 10 -2 M EDT at the same potential, in 5 ml phosphate buffer+1000 IU ml -1 PEG-GOD. Other concentrations of GOD were used, up to 50 000 IU ml -1 . It appeared that in the last case, what we call the specific activity of the enzyme decreased. The electrodes were then constructed using 1000 IU ml -1 GOD.

II.2.3. Amperometric determination

The performances of enzyme electrodes for the detection of glucose were investigated. The amperometric responses of the enzyme electrodes to glucose injections in a 10 ml phosphate Journal of Electroanalytical Chemistry, 2001, 512 (1)(2), 101-109, doi:10.1016/S0022-0728(01)00595- 24 buffer (pH 6.0) solution containing 3×10 -4 M FcOH were examined by measuring the anodic current at a potential of 0.3 V (vs. Ag⏐AgCl) at room temperature under argon. All the solutions used in the amperometric studies were deaerated by bubbling argon for 15 min prior to use. Other experiments were conducted under aerated conditions, in order to evaluate the oxygen sensitivity of our electrode. The background current was allowed to settle to a constant baseline, and successive additions of 0.04 ml stock glucose solution (0.25 M) were injected, with gentle stirring for 5 s after each addition. The current as a function of time was monitored continuously until a steady state value was reached. The spike occurring after each addition of glucose is due to the stirring of the solution, which can be assumed as a transient state where the concentrations of glucose and mediator at the electrode vicinity would be higher than at the equilibrium state. These spikes are usually observed when ferrocenemethanol is used as mediator. When dioxygen is used as mediator, spikes are lower.

II.2.4. Electrochemical quartz microbalance (ECQM) measurements ECQM experiments were conducted on a platinum sputtered AT-cut quartz crystal (0.2 cm 2 , nominal frequency 9 MHz) with an EG&G 273A potentiostat and an EG&G-SEIKO QCA917 frequency analyser. Frequency variations are related to mass change following the Sauerbrey equation.

2 0 2 m f f v A ρ ⎛ ⎞ Δ Δ = -⎜ ⎟ ⎝ ⎠
The sensitivity Δm/Δf=-1.1×10 -9 g Hz -1 is an experimental value determined by the manufacturer.

II.2.5. pH-stat

The pH-stat method was used to determine the apparent enzyme concentration of the modified electrodes. As mentioned in Section I, the oxidation of glucose produces gluconolactone and then gluconic acid, therefore this enzymatic reaction results in a change in pH. The enzyme electrode and pH electrode are immersed in a solution containing the substrate, in this case glucose. Glucose reacts with the immobilised GOD and is converted into gluconic acid, with the production of H + . An alkaline solution (10 -4 M KOH, from a 1 M KOH volumetric solution, Titrinorm, Prolabo) is added sequentially in the solution in order to maintain pH at a constant value of 7.4. Thus, the exact quantity of gluconic acid is monitored versus time, and then the enzyme activity is deduced. The enzyme activity EA (μmol min -1 ) is obtained as follows:

3 10 t EA c = s (1)
where c t is the alkali concentration, i.e. 10 -4 mol l -1 and s is the averaged slope of the curve V alkali =f(t)/ml min -1 .

III. Results and discussion

I II II I. .1 1. . F Fi il lm m c ch ha ar ra ac ct te er ri is sa at ti io on n III.1.1. FTIR spectroscopy Spectra were recorded on a computer-controlled Nicolet 860 FTIR spectrometer. Films, electrosynthesised on GC plates, were analysed by the reflection mode. Figure 1 shows spectra of PEDT (a) and (PEG-GOD/PEDT) (b) films. There are no significant differences between spectra a and b. The main bands [33] are situated at 916 and 830 (C-S stretch), 1046 (-C-OROC-stretch), 1182 cm -1 (P=O stretch of phosphate anions). The bands at 1323, 1393 and 1514 cm -1 can be attributed to C=C in-plane ring vibrations. The band at 1668 cm -1 is due to the dopant anions HPO 4 2-and H 2 PO 4 -. It can be concluded from the above IR data that the polymer matrix is effectively PEDT even when the modified enzyme has been incorporated.

III.1.2. Scanning electron microscopy (SEM)

The films thickness was determined with SEM, by stripping a piece of polymer from the electrode to reveal a cross-section of the film. The incidence angle used with the microscope was 80°. Figure 2a andb shows a (PEDT+PEG) film and a (PEG-GOD/PEDT) film obtained on a GC electrode by the potentiostatic method for 10s. The thicknesses of these films are equivalent at ca. 40 nm (Figure 3), which means that the faradaic yield of electropolymerisation is approximately the same in both cases.

III.1.3. pH-stat The enzymatic activity was evaluated using the pH-stat method for the two kinds of films: with unmodified (GOD/PEDT+PEG) and modified (PEG-GOD/PEDT) films. The surface concentration of the active enzyme Γ active can be estimated by the following relation:

active GOD eu GOD EA x m M x A = Γ (2)
where EA is the enzyme activity of relation (1) /μmol min -1 , m eu is the mass of one enzyme unit, i.e. 2.15×10 -5 g min μmol -1 at pH 7.4 and 22 °C, M GOD is the molar mass of the enzyme, i.e. 186 000 g mol -1 , and A is the electrode surface, i.e. 2.5 cm 2 . The results give:

active GOD Γ =0.9±0.1×10 -12 mol cm -2 for (GOD/PEDT+PEG) and active GOD Γ =1.1±0.1×10 -12 mol cm -2 for (PEG-GOD/PEDT) electrodes. The enzyme activity is slightly higher for the modified enzyme.

III.1.4. Electrochemical quartz microbalance (ECQM)

To determine the quantity of enzyme in each film, EQCM experiments were performed. The masses of each kind of film were compared to evaluate the surface concentrations of unmodified GOD ( app GOD Γ ) and modified PEG-GOD ( app PEG-GOD

Γ

) embedded during electropolymerisation. During these experiments, the quartz admittance was monitored. Without a film, the admittance equals 4.58×10 -3 Ω -1 (in phosphate buffer+10 -2 M EDT+GOD+PEG). This admittance never drops below 95% of the initial value during experiments. Thus, the rigidity constraint can be assumed. At a constant electropolymerisation charge, and assuming that GOD, PEG and PEG-GOD do not impede the polymerisation (see Section III.1.2), app GOD Γ in the case of unmodified enzyme, can be found by the relation given below:

app GOD PEDT PEG GOD m m M x A + - = Γ (3)
where m is the mass of the film/μg, m PEDT+PEG is the mass of a (PEDT+PEG) film obtained with the same charge/μg, M GOD is the molar mass of GOD, i.e. 186 000 g mol -1 , and A is the electrode surface, i.e. 0.2 cm 2 . In the case of the modified enzyme, can be found by the same relation, assuming that the same quantity of PEG is incorporated in both cases. Curves m=f(t) are presented in Figure 4. The results give =7.1±0.2×10

app GOD Γ app GOD Γ -12 mol cm -2 for (GOD/PEDT+PEG) and =7.1±0.2×10 app PEG-GOD Γ -12 mol cm -2 for (PEG-GOD/PEDT) electrodes. It can be noted that equivalent amounts of enzyme are found in both cases. By comparing the values of concentrations of GOD present in the polymer matrix with that of the active GOD concentrations obtained from pHstat experiments, the yield y of active enzyme in the film (also called the specific activity of the enzyme) can be deduced:

active GOD app GOD 100 y x = Γ Γ (4)
The results give y=12±2% for (GOD/PEDT+PEG) and y=15±2% for (PEG-GOD/PEDT) electrodes. These values are not very precise, so it is not possible to discuss the difference between (GOD/PEDT+PEG) and (PEG-GOD/PEDT) electrodes. Nevertheless, these activities are higher than those commonly obtained in the literature for this kind of enzyme entrapment [9]. III.1.5. Amperometry The performances of the enzyme electrodes were investigated by chronoamperometry, under the conditions described in Section II.2.3. Figure 5 shows a typical amperometric response as a function of glucose concentration.

To evaluate the reproducibility, ten electrodes were prepared on different days using the same procedure. The variation is at most 8% across the entire concentration range, which shows a relatively good reproducibility. It should be noted that a (PEDT+PEG) electrode without GOD did not show any response with glucose addition. Thus, the signals observed are effectively due to the enzyme incorporated in the polymer matrix.

The calibration curve for the amperometric response of glucose at the (PEG-GOD/PEDT) electrode is established in Figure 6a, which is linear up to 22 mM glucose. The sensitivities of the biosensors (determined by the slope of the calibration curve) attain a value of 2.7 mA cm -2 M -1 for the (PEG-GOD/PEDT) electrode and 3 mA cm -2 M -1 for (GOD/PEDT+PEG) (result not shown). These values are close to those commonly obtained in the literature for GOD entrapped in conducting polymers [34], often comprised between 2 and 5 mA cm -2 M -1 . To evaluate the oxygen sensitivity of the (PEG-GOD/PEDT) electrode, the amperometric response was determined under the same conditions as previously, but in an aerated medium. The data presented in Figure 6b, show that the sensitivity is very similar to that obtained under deaerated conditions, i.e. 2.6 mA cm -2 M -1 (only the first points are shown). Nevertheless, the background current is larger when oxygen is present. The increase of this background current could be due to the reoxidation of hydrogen peroxide which comes from the GOD/oxygen sidereaction, even if this reoxidation is low on a GC electrode. When amperometry is performed at 0.3 V versus Ag ⏐ AgCl on a GC-modified electrode in an aerated solution without FcOH, a very low response current is observed, corresponding to less than 5% of the current obtained with FcOH. Bartlett and coworkers [35] and [36] have established a relation between the current observed during chronoamperometric experiments and several kinetic parameters as well as the bulk glucose concentration:

( ) active GOD cat s A obs cat s A s M k K s kK a I nFA k K s kK a K s K α ∞ ∞ ∞ ∞ ∞ ⎛ ⎞ = ⎜ + + ⎝ ⎠ Γ ⎟ ( 5
)
where I obs is the measured current/A, n is the number of electrons exchanged per glucose molecule, i.e. two electrons, F is the Faraday constant, i.e. 96 487 C mol -1 , A is the electrode surface, here 0.07 cm 2 , is the surface coverage of the active enzyme/mol cm active GOD Γ -2 , k is the rate constant for the reaction between enzyme and mediator, a ∞ is the bulk concentration of the mediator/mol dm -3 , K S and K A are the partition coefficients for glucose and mediator into the film, respectively, K M and k cat are the kinetic parameters for the reaction between enzyme and substrate, respectively, in mol dm -3 and s -1 , 1/α is the fraction of reduced mediator which reaches the electrode surface and s ∞ is the bulk concentration of glucose/mol dm -3 . The calibration data of (PEG-GOD/PEDT) electrodes were fitted to Equation (5) written in the form of a hyperbolic function:

1 2 obs Ps I nFA P s α ∞ ∞ ⎛ ⎞ = ⎜ ⎟ + ⎝ ⎠
An average value was found:

active GOD 3 2 1 1 6 10 M cat s P K x P nFA k K α - = = Γ ± (6)
The kinetic parameters for the reaction between enzyme and substrate, K M =33×10 -3 mol l -1 , k cat =800 s -1 and αK s ∼1, were taken from Reference [37]. Relation (6) gives =5±1×10

active GOD Γ -13

mol cm -2 . This value is of the same order as that obtained above. Nevertheless, the difference could be explained by αK s <1, i.e. a little loss of reduced mediator into the bulk, and/or a low diffusion of glucose into the film. The α value could be estimated using a rotating ring-disc electrode for example. A method has been proposed to evaluate it in polymerised phenols [32]. The aim of this paper is not to determine these parameters, but we paid particular attention to the film thickness. III.1.6. Effect of film thickness α and K s are the two parameters which are most likely to be influenced by the film thickness. Moreover, it has often been shown that the active enzyme in a polymer film is localised near the polymer⏐solution interface [38] and [39]. Therefore, the critical parameter, as said above, should be the diffusion of the reduced mediator (characterised by α) or that of glucose (characterised by K s ) through the film. The effect of thickness has already been studied with polypyrrole. It was shown that the best amperometric activities were obtained for 0-1 μm thick films [40]. In our case, we have studied sensors with film thicknesses varying between 20 and 1000 nm. The sensitivities are reported in Figure 7 versus thickness. The sensitivity has a maximum value for thicknesses in the range of 20-10 nm and decreases with thicker films. Once again, according to Reference [41], this could be due to a low diffusion of the mediator or the substrate through the film. Nevertheless, when we tried to measure the diffusion coefficient of the mediator through the film (with respect to the thickness), we observed a very surprising result: a fraction of the reduced mediator is reoxidised directly on the PEDT film (the apparent diffusion coefficient increases with film thickness). Therefore, this seems to indicate that the limiting parameter is not the diffusion of the mediator.

III.1.7. Stability The enzyme electrodes were also investigated for their operational stability with two kinds of electrodes containing unmodified and modified enzyme. Each electrode was used in ten experiments over 2 weeks (each point is an average of results from three different electrodes). In the case of the unmodified enzyme, the sensitivity decreases rapidly and is more than 50% lower at the end (Figure 8a). With the modified-enzyme electrode, the loss is only ca. 20% (Figure 8b).

IV. Conclusion

An amperometric glucose sensor was constructed by incorporation of modified-glucose oxidase within a conducting polymer matrix, poly(3,4-ethylenedioxythiophene) (PEDT). The enzyme (GOD) was modified by polyethyleneglycol (PEG) chains to create a hydrophilic environment for GOD. This was achieved by amide bonds formed between amine groups of GOD and carboxylic acid groups of dicarboxylic acid polyethyleneglycol molecules. The modified-enzyme was entrapped during electrochemical oxidation of EDT in phosphate buffer. Data from pHstat experiments, ECQM measurements and amperometry indicate that the resulting glucose biosensors using ferrocene as mediator possess a good sensitivity (2.7 mA M -1 cm -2 ) up to 22

I

  II I. .1 1. . C Ch he em mi ic ca al ls s Glucose oxidase from Aspergillus niger (type VII-S, EC 1.1.3.4), D(+)glucose, 1-ethyl-3-(3dimethylaminopropyl) carbodiimide hydrochloride (EDC) and N-hydroxysuccinimide (NHS) Journal of Electroanalytical Chemistry, 2001, 512(1-2), 101-109, doi:10.1016/S0022-0728(01)00595-2 3

FiguresF 11 F

 11 Figures


Natacha Gondran
email: natacha.gondran@emse.fr

Ecological footprint of an organization: can it really be measured?

Keywords: Environmental Management, Ecological footprint, National park

Introduction

Ecological footprint aims to compare the demand on ecological services to the available supply. Such a metric is needed to make policy makers understand the threat of overshoot of natural resources and to facilitate the emergence of a consensus over the actions that are needed to address the ecological risks. During the 2000's, the Global Footprint Network has endeavored to develop and mature the methodology of the National Footprint Accounts [START_REF] Ewing | The Ecological Footprint Atlas[END_REF]. This metric attends to assess current ecological supply and demand, at a macro-economic scale. The aim of this paper is to question whether applying such a metric at a micro-economic scale is possible and relevant. Which method can be used to estimate the ecological footprint of an organization? Does this estimation make it possible to set goals of improvement, to identify options for action, and to track progress toward these goals? Which are the limits of such an exercise? The method, that will be presented, has been developed and validated for the Vanoise National Park, in the Alps, France. This public organization is in charge of preserving the territory of the Vanoise, getting knowledge about its natural and cultural patrimony and making the public aware about the necessity to protect it.

Methods

Ecological Footprint aims to evaluate the human appropriation of ecosystem products and services in terms of the amount of bioproductive land and sea area needed to supply these services. Its unity is the "global hectare" (gha) defined as a hectare that has the world average productivity of biologically productive land and water in a given year. The area of land or sea that is biologically productive is called biocapacity. It represents the biosphere's ability to meet human demand for biological resources' consumption and CO 2 sequestration. The Ecological Footprint and biocapacity accounts cover six land use types: cropland, grazing land, fishing ground, forest land, built-up land and carbon uptake land. For each component, the ecological footprint is obtained through the consumption of a harvested product (or amount of emission of CO 2 ) divided by the yield for this ecological services. This value is then converted into "global hectares" thanks to yield and equivalence factors [START_REF] Ewing | Calculation Methodology for the National Footprint Accounts[END_REF]. The same principles were considered to estimate the ecological footprint of an organization. The following approach was followed:

-The first step was to define the scope of the activity under study [START_REF]Ecological Footprint Standards 2009[END_REF].

-The second step requires collecting the consumption data of the organization. Five categories of components were considered: infrastructure, mobility, food, manufactured goods that are used for the activities and communication of the public institution (consumables and depreciation), services.

-The third step consists in organizing the information and calculating the conversion factors into global hectares. When possible, information sources must be consistent with the information sources of the Global Footprint Network (for example, FAO for harvested products). Concerning the carbon emission factors, the "Bilan Carbone ® ", which has been developed by the French national energy agency, ADEME [START_REF]Agence de l'Environnement et de la Maîtrise de l'Energie -Mission Interministérielle de l'Effet de Serre[END_REF], was used in order to make the Carbon footprint be consistent with the results of the more official "Bilan Carbone ® ". However, it takes into account 6 greenhouse gases contrary to the GFN method that only considers CO 2 emissions. -The fourth step is to calculate the ecological footprint of the organization and to verify the results by cross-checking. -The fifth step is the synthesis and interpretation of the results in order to identify the components that have the most important contribution to the ecological footprint.

Results and Discussion

The main results obtained by this estimation will be described. For instance, the ecological footprint of the administration of the Vanoise National park was estimated to 189 gha (or 2.52 gha / employee) in 2007 and 148 gha (or 1.70 gha / employee) in 2008. But are such figures easily understandable and interpretable? To what extend can they be compared to national biocapacity results? Which are the conceptual and methodological limits of this exercise ? For example, the Vanoise National Park has committed to reduce its ecological footprint by 3 percent each year. In order to track progress toward this goal, it wants to measure yearly its ecological footprint. This poses methodological difficulty. Indeed, rigorously, when calculating the ecological footprint of a new year, conversion factors should be updated to take into account the annual yields. In this case, the variations of ecological footprint could be explained by changes in the consumptions of the organization and / or changes into conversion factors. The latter are linked to variations of national or even world-wide productivity. With an environmental management point of view, this is not satisfying. Indeed, the aim of the organization is to track only changes that it is responsible for and that are linked with its own consumptions. Then, the choice was made keep the same conversion factors every year.

Conclusion

Although the exercise of calculating ecological footprint for an organization raises several methodological and conceptual questions, it also has some interests as an indicator for an environmental management system. For example, it obliges to implement an information system based on physical data and not only monetary data and it make it possible to identify goals of improvement and to track progress toward these goals.