Dominique Blanchard
email: dominique.blanchard@univ-rouen.fr

Georges Griso
email: griso@ann.jussieu.fr

Decomposition of the deformations of a thin shell. Asymptotic behavior of the Green-St Venant's strain tensor

Keywords: shells, Korn's inequality, large deformations. 2000 MSC: 74B20, 74K20, 74K25

We investigate the behavior of the deformations of a thin shell, whose thickness δ tends to zero, through a decomposition technique of these deformations. The terms of the decomposition of a deformation v are estimated in terms of the L 2 -norm of the distance from ∇v to SO(3). This permits in particular to derive accurate nonlinear Korn's inequalities for shells (or plates). Then we use this decomposition technique and estimates to give the asymptotic behavior of the Green-St Venant's strain tensor when the "strain energy" is of order less than δ 3/2 .

Introduction

The concern of this paper is twofold. We first give a decomposition technique for the deformation of a shell which allows to established a nonlinear Korn's type inequality for shells. In a second part of the paper, we use such a decomposition to derive the asymptotic behavior of the Green-St Venant's strain tensor.

In the first part, we introduce two decompositions of an admissible deformation of a shell (i.e. which is H 1 with respect to the variables and is fixed on a part of the lateral boundary) which take into account the fact that the thickness 2δ of such a domain is small. This decomposition technique has been developed in the framework of linearized elasticity for thin structures in [START_REF] Griso | Decomposition of displacements of thin structures[END_REF], [START_REF] Griso | Asymptotic behavior of curved rods by the unfolding method[END_REF], [START_REF] Griso | Asymptotic behavior of structures made of plates[END_REF] and for thin curved rods in nonlinear elasticity in [START_REF] Blanchard | Decomposition of deformations of thin rods. Application to nonlinear elasticity[END_REF]. As far as large deformations are concerned these decompositions are obtained through using the "Rigidity Theorem" proved in [START_REF] Friesecke | A theorem on geometric rigidity and the derivation of nonlinear plate theory from the three-dimensional elasticity[END_REF] together with the geometrical precision of this result given in [START_REF] Blanchard | Decomposition of deformations of thin rods. Application to nonlinear elasticity[END_REF]. Let us consider a shell with mid-surface S and thickness 2δ. The two decompositions of a deformation v defined on this shell are of the type

v = V + s 3 Rn + v,
where s 3 is the variable in the direction n which is a unit vector field normal to S. In the above expression, the fields V and R are defined on S while v is a field still defined on the 3D shell. Let us emphasize that the terms of the decompositions V, R and v have at least the same regularity than v and satisfy the corresponding boundary conditions. Loosely speaking, the two first terms of the decompositions reflect the mean of the deformation over the thickness and the rotations of the fibers of the shell in the direction n. For the above decomposition, it worth noting that the fields V, R and v are estimated in terms of the "strain energy" ||dist(∇ x v, SO(3))|| L 2 (S×]-δ,δ[) and the thickness of the shell.

In the first decomposition, the field R satisfies ||dist(R, SO(3))|| L 2 (S) ≤ C δ 1/2 ||dist(∇ x v, SO(3))|| L 2 (S×]-δ,δ[) which shows that the field R is close to a rotation field for small energies.

In the second decomposition, for which we assume from the beginning that ||dist(∇ x v, SO(3))|| L 2 ≤ C(S)δ 3/2 where C(S) is a geometrical constant, the field R is valued in SO [START_REF] Blanchard | Microscopic effects in the homogenization of the junction of rods and a thin Asympt[END_REF].

For thin structures, the usual technique in order to rescale the applied forces to obtain a certain level of energy is to established nonlinear Korn's type inequalities. In order to simplify the analysis, we consider here that the deformation v is equal to the identity on a part of the lateral boundary of the shell (clamped condition). Using Poincaré's inequality as done in [START_REF] Blanchard | Decomposition of deformations of thin rods. Application to nonlinear elasticity[END_REF] ( see also [START_REF] Ciarlet | Continuity of a deformation in H 1 as a function of its Cauchy-Green tensor in L 1[END_REF] and Subsection 4.1 of the present paper) leads in the case of a shell to the following inequality

||v -I d || (L 2 (S×]-δ,δ[)) 3 + ||∇ x v -I 3 || (L 2 (S×]-δ,δ[)) 9 ≤ C(δ 1/2 + ||dist(∇ x v, SO(3))|| L 2 (S×]-δ,δ[) ).
The first important consequence of the decomposition technique together with its estimates is the following nonlinear Korn's inequality for shells

||v -I d || (L 2 (S×]-δ,δ[)) 3 + ||∇ x v -I 3 || (L 2 (S×]-δ,δ[)) 9 ≤ C δ ||dist(∇ x v, SO(3))|| L 2 (S×]-δ,δ[) .
Another important technical argument involved in the proof of the above inequalities is the possible extension of a deformation in a neighborhood of the lateral boundary without increasing the order (with respect to δ) of the strain energy. Indeed the two inequalities identify for energies of order δ 3/2 which is the first interesting critical case. For smaller levels of energy, the second estimate is more relevant. We also establish the following estimate for the linear part of the strain tensor which shows that ||dist(∇ x v δ , SO(3))|| L 2 (S×]-δ,δ[) ∼ δ 5/2 is another critical case. For such level of energy, our Korn's inequality for shells turns out to appear as an important tool. We have established and used the analogue of these inequalities for rods in [START_REF] Blanchard | Decomposition of deformations of thin rods. Application to nonlinear elasticity[END_REF].

∇ x v+(∇ x v) T -2I 3 (L 2 (S×]-δ,δ[)) 9 ≤ C||dist(∇ x v, SO (3)
In the second part of the paper, we strongly use the results of the first part in order to derive the asymptotic behavior of the Green-St Venant's strain tensor. We focus on the case where the "strain energy" ||dist(∇ x v δ , SO(3))|| L 2 (S×]-δ,δ[) is of order δ κ-1/2 (κ ≥ 2). The order δ 3/2 is the highest level of energy which can be analyzed through our technique.

For

||dist(∇ x v δ , SO(3))|| L 2 (S×]-δ,δ[) ∼ δ 3/2
, we deduce the expression of the limit of the Green-St Venant's strain tensor from the decompositions, the associated estimates and a standard rescaling and the result is the same using the two decompositions. In this case the limit deformation is pure bending but the limit Green-St Venant's strain tensor contains a field which measures the defect between the mean deformation and a pure bending deformation.

For

||dist(∇ x v δ , SO(3))|| L 2 (S×]-δ,δ[) ∼ δ κ-1/2
with κ > 2, the displacements of the fibers of the shell are rigid displacements. To describe the limit behavior, we introduce the inextentional and extentional displacements which correspond respectively to the bending and to a generalization of membrane displacements for a plate. The value κ = 3 is a critical case. For 2 < κ < 3, the inextentional and extentional displacements are coupled. If κ ≥ 3, the defect field mentioned above can be expressed in terms of the extentional displacement (κ > 3) and also of the inextentional displacement (κ = 3).

A byproduct of the decomposition technique and the derivation of the limit of the Green-St Venant's strain tensor introduced in this paper is a simplification of the obtention of limit elastic shell models through Γ-convergence (that we will present in a forthcoming paper).

As general references on the theory of shells, we refer to [START_REF] Ciarlet | Mathematical Elasticity[END_REF], [START_REF] Ciarlet | Mathematical Elasticity[END_REF], [START_REF] Ciarlet | Un modèle bi-dimentionnel non linéaire de coques analogue à celui de W.T. Koiter[END_REF], [START_REF] Ciarlet | An introduction to shell theory[END_REF], [START_REF] Ciarlet | A justification of a nonlinear model in plate theory[END_REF], [START_REF] Friesecke | A hierarchy of plate models derived from nonlinear elasticity by Γ-convergence[END_REF], [START_REF] Griso | Asymptotic behavior of structures made of plates[END_REF], [START_REF] Dret | The quasiconvex envelope of the Saint Venant-Kirchhoff stored energy function[END_REF], [START_REF] Pantz | On the justification of the nonlinear inextensional plate model[END_REF], [START_REF] Pantz | On the justification of the nonlinear inextensional plate model[END_REF]. The rigidity theorem and its applications to thin structures using Γ-convergence arguments are developed in [START_REF] Friesecke | A theorem on geometric rigidity and the derivation of nonlinear plate theory from the three-dimensional elasticity[END_REF], [START_REF] Friesecke | A hierarchy of plate models derived from nonlinear elasticity by Γ-convergence[END_REF], [START_REF] Dret | The nonlinear membrane model as variational limit of nonlinear threedimensional elasticity[END_REF], [START_REF] Dret | The quasiconvex envelope of the Saint Venant-Kirchhoff stored energy function[END_REF]. The decomposition of the deformations in thin structures is introduced in [START_REF] Griso | Decomposition of displacements of thin structures[END_REF], [START_REF] Griso | Asymptotic behavior of curved rods by the unfolding method[END_REF] and a few applications to the junctions of multi-structures and homogenization are given in [START_REF] Blanchard | Junction of a periodic family of elastic rods with a 3d plate[END_REF], [START_REF] Blanchard | Junction of a periodic family of elastic rods with a thin plate[END_REF], [START_REF] Blanchard | Microscopic effects in the homogenization of the junction of rods and a thin Asympt[END_REF].

The paper is organized as follows. Section 2 is devoted to describe the geometry of the shell and to give a few notations. In Section 3 we introduce the two decompositions of the deformations of a thin shell and we derive the estimates on the terms of these decompositions. We precise the boundary conditions on the deformation and we establish a nonlinear Korn's inequality for shells in Section 4. Section 5 is concerned with a standard rescaling. We derive the limit of the Green-St Venant's strain tensor of a sequence of deformations such that ||dist(∇ x v δ , SO(3))|| L 2 (S×]-δ,δ[) ∼ δ κ-1/2 in Section 6 for κ = 2 and in Section 7 for κ > 2. At last the appendix contains a few technical results on the interpolation of rotations.

The geometry and notations.

Let us introduce a few notations and definitions concerning the geometry of the shell (see [START_REF] Griso | Decomposition of displacements of thin structures[END_REF] for a detailed presentation).

Let ω be a bounded domain in R 2 with lipschitzian boundary and let φ be an injective mapping from

ω into R 3 of class C 2 .
We denote S the surface φ(ω). We assume that the two vectors ∂φ ∂s 1 (s 1 , s 2 ) and ∂φ ∂s 2 (s 1 , s 2 ) are linearly independent at each point (s 1 , s 2 ) ∈ ω.

We set (2.1)

t 1 = ∂φ ∂s 1 , t 2 = ∂φ ∂s 2 , n = t 1 ∧ t 2 t 1 ∧ t 2 2 .
The vectors t 1 and t 2 are tangential vectors to the surface S and the vector n is a unit normal vector to this surface. The reference fiber of the shell is the segment ]δ, δ[. We set

Ω δ = ω×] -δ, δ[.

Now we consider the mapping

Φ : ω × R -→ R 3 defined by (2.2) Φ : (s 1 , s 2 , s 3 ) -→ x = φ(s 1 , s 2 ) + s 3 n(s 1 , s 2 ).
There exists δ 0 ∈ (0, 1] depending only on S, such that the restriction of Φ to the compact set Ω δ0 = ω ×[-δ 0 , δ 0 ] is a C 1diffeomorphism of that set onto its range (see e.g. [START_REF] Ciarlet | Mathematical Elasticity[END_REF]). Hence, there exist two constants c 0 > 0 and c 1 ≥ c 0 , which depend only on φ, such that

∀δ ∈ (0, δ 0 ], ∀s ∈ Ω δ0 , c 0 ≤ |||∇ s Φ(s))||| ≤ c 1 , and for x = Φ(s) c 0 ≤ |||∇ x Φ -1 (x))||| ≤ c 1 .
Definition 2.1. For δ ∈ (0, δ 0 ], the shell Q δ is defined as follows:

Q δ = Φ(Ω δ ).
The mid-surface of the shell is S. The lateral boundary of the shell is Γ δ = Φ(∂ω×]δ, δ[). The fibers of the shell are the segments Φ {(s 1 , s 2 )}×]δ, δ[ , (s 1 , s 2 ) ∈ ω. We respectively denote by x and s the running points of Q δ and of Ω δ . A function v defined on Q δ can be also considered as a function defined on Ω δ which we will also denote by v. As far as the gradients of v are concerned we have ∇ x v and ∇ s v = ∇ x v.∇Φ and for a.e. x = Φ(s)

c|||∇ x v(x)||| ≤ |||∇ s v(s)||| ≤ C|||∇ x v(x)|||,
where the constants are strictly positive and do not depend on δ.

Since we will need to extend a deformation defined over the shell Q δ , we also assume the following.

For any η > 0, let us denote the open set

ω η = (s 1 , s 2 ) ∈ R 2 | dist (s 1 , s 2 ), ω < η .
We assume that there exist η 0 > 0 and an extension of the mapping φ (still denoted φ) belonging to C 2 (ω η0 )

0 , c ′ 1 , c ′ and C ′ such that (2.3) ∀s ∈ ω η0 × [-δ 0 , δ 0 ], c ′ 0 ≤ |||∇ s Φ(s)||| ≤ c ′ 1 , and for x = Φ(s) c ′ 0 ≤ |||∇ x Φ -1 (x)||| ≤ c ′ 1 c ′ |||∇ x v(x)||| ≤ |||∇ s v(s)||| ≤ C ′ |||∇ x v(x)|||, for a.e. x = Φ(s).
At the end we denote by I d the identity map of R 3 .

Decompositions of a deformation.

In this Section, we recall the theorem of rigidity established in [START_REF] Friesecke | A theorem on geometric rigidity and the derivation of nonlinear plate theory from the three-dimensional elasticity[END_REF] (Theorem 3.1 of Section 3.1). In Subsection 3.2 we recall that any deformation can be extended in a neighborhood of the lateral boundary of the shell with the same level of "energy". Then we apply Theorem 3.1 to a covering of the shell. In Subsections 3.4 and 3.5, we introduce the two decompositions of a deformation and we established estimates on these decompositions in term of ||dist(∇ x v, SO(3))|| L 2 .

Theorem of rigidity.

We equip the vector space R n×n of n × n matrices with the Frobenius norm defined by

A = a ij 1≤i,j≤n , |||A||| = n i=1 n j=1 |a ij | 2 .
We just recall the following theorem established in [START_REF] Friesecke | A theorem on geometric rigidity and the derivation of nonlinear plate theory from the three-dimensional elasticity[END_REF] in the version given in [START_REF] Blanchard | Decomposition of deformations of thin rods. Application to nonlinear elasticity[END_REF].

O; R 1 (0 < R 1 ≤ R). For any v ∈ H 1 (Ω) n , there exist R ∈ SO(n) and a ∈ R n such that (3.1) ||∇ x v -R|| (L 2 (Ω)) n×n ≤ C||dist(∇ x v; SO(n))|| L 2 (Ω) , ||v -a -R x|| (L 2 (Ω)) n ≤ CR||dist ∇ x v; SO(n) || L 2 (Ω) ,
where the constant C depends only on n and R R 1 .

Extension of a deformation and splitting of the shell.

In order to make easier the decomposition of a deformation as the sum of an elementary deformation given via an approximate field of rotations (see Subsection 3.4) or a field of rotations (see Subsection 3.5) and a residual one, we must extend any deformation belonging to H 1 (Q δ )

3 in a neighborhood of the lateral boundary Γ δ of the shell. To this end we will use Lemma 3.2 below. The proof of this lemma is identical to the one of Lemma 3.2 of [START_REF] Griso | Decomposition of displacements of thin structures[END_REF] upon replacing the strain semi-norm of a displacement field by the norm of the distance between the gradient of a deformation v and S0(3). Lemma 3.2. Let δ be fixed in (0, δ 0 ] such that 3δ ≤ η 0 and set

Q ′ δ = Φ(ω 3δ ×] -δ, δ[).
There exists an extension operator

P δ from H 1 (Q δ ) 3 into H 1 (Q ′ δ ) 3 such that ∀v ∈ H 1 (Q δ ) 3 , P δ (v) ∈ H 1 (Q ′ δ ) 3 , P δ (v) | Q δ = v, ||dist ∇ x P δ (v), SO(3) || L 2 (Q ′ δ ) ≤ c||dist ∇ x v, SO(3) || L 2 (Q δ )
, with a constant c which only depends on ∂ω and on the constants appearing in inequalities (2.3).

Let us now precise the extension operator P δ near a part of the boundary where v = I d . Let γ 0 be an open subset of ∂ω which is made of a finite number of connected components (whose closure are disjoint). Let us denote the lateral part of the boundary by

Γ 0,δ = Φ(γ 0 ×] -δ, δ[).
Consider now a deformation v such that v = I d on Γ 0,δ . Let γ ′ 0,δ be the domain

γ ′ 0,δ = (s 1 , s 2 ) ∈ γ 0 | dist((s 1 , s 2 ), E 0 ) > 3δ}
where E 0 denotes the extremities of γ 0 . We set

Q 1 δ = Φ {(s 1 , s 2 ) ∈ (ω 3δ \ ω) | dist((s 1 , s 2 ), γ ′ 0,δ ) < 3δ}×] -δ, δ[ , Q 2 δ = Φ {(s 1 , s 2 ) ∈ ω 3δ | dist((s 1 , s 2 ), γ 0 ) < 6δ}×] -δ, δ[ .
Indeed, up to choosing δ 0 small enough, we can assume that Q 2 δ has the same number of connected components as

γ 0 . The open set Q 1 δ is included into Q ′ δ \ Q δ .
According to the construction of P δ given in [START_REF] Griso | Decomposition of displacements of thin structures[END_REF], we can extend the deformation v by choosing P δ (v) = I d in Q 1 δ together with the following estimates

(3.2) ||∇ x P δ (v) -I 3 || (L 2 (Q 2 δ )) 9 ≤ C||dist(∇ x v, SO(3))|| L 2 (Q δ ) , ||P δ (v) -I d || (L 2 (Q 2 δ )) 3 ≤ Cδ||dist(∇ x v, SO(3))|| L 2 (Q δ ) .
From now on we assume that 3δ ≤ η 0 and then any deformation v belonging to

H 1 (Q δ ) 3 is extended to a deformation belonging to H 1 (Q ′ δ )
3 which we still denote by v.

Now we are in a position to reproduce the technique developed in [START_REF] Griso | Decomposition of displacements of thin structures[END_REF] in order to obtain a covering of the shell (the reader is referred to Section 3.3 of this paper for further details). Let N δ be the set of every (k, l) ∈ Z 2 such that the open set

ω δ,(k,l) =]kδ, (k + 1)δ[×]lδ, (l+)δ[ is included in ω 3δ and let N ′ δ be the set of every (k, l) ∈ N δ such that ((k + 1)δ, lδ), (kδ, (l + 1)δ), (k + 1)δ, (l + 1)δ) are in N δ . We set Ω δ,(k,l) = ω δ,(k,l) ×] -δ, δ[.
By construction of the above covering, we have

ω ⊂ (k,l)∈N ′ δ ω δ,(k,l) .
According to [START_REF] Griso | Decomposition of displacements of thin structures[END_REF], there exist two constants R and R 1 , which depend on ω and on the constants

c ′ 0 , c ′ 1 , c ′ and C ′ (see(2.3)), such that for any δ ≤ (0, η 0 /3] the open set Q δ,(k,l) = Φ(Ω δ,(k,l) )

has a diameter less than

Rδ and is star-shaped with respect to a ball of radius R 1 δ.

As a convention and from now on, we will say that a constant C which depends only upon ∂ω and on the constants

c ′ 0 , c ′ 1 , c
′ and C ′ depends on the mid-surface S and we write C(S).

Since the ratio

Rδ R 1 δ of each part Q δ,(k,l) does not depend on δ, Theorem 3.1 gives a constant C(S). Let v be a deformation in (H 1 (Q δ )) 3 extended to a deformation belonging to (H 1 (Q ′ δ )) 3 . Applying Theorem 3.1 upon each part Q δ,(k,l) for (k, l) ∈ N δ , there exist R δ,(k,l) ∈ SO(3) and a δ,(k,l) ∈ R 3 such that (3.3) ||∇ x v -R δ,(k,l) || (L 2 (Q δ,(k,l) )) 3×3 ≤ C(S)||dist(∇ x v; SO(3))|| L 2 (Q δ,(k,l) ) ||v -a δ,(k,l) -R δ,(k,l) x -φ(kδ, lδ) || (L 2 (Q δ,(k,l) )) 3 ≤ C(S)R(S)δ||dist ∇ x v; SO(3) || L 2 (Q δ,(k,l) ) .
For any (k, l)

∈ N δ such that (k + 1, l) ∈ N δ , the open set Q ′ δ,(k,l) = Φ(](k + 1/2)δ, (k + 3/2)δ[×]lδ, (l + 1)δ[×] -δ, δ[)
also have a diameter less than R(S)δ and it is also star-shaped with respect to a ball of radius R 1 (S)δ (see Section 3.3 in [START_REF] Griso | Decomposition of displacements of thin structures[END_REF]). We apply again Theorem 2.1 in the domain Q ′ δ,(k,l) . This gives a rotation R ′ δ,(k,l) . In the domain

Q ′ δ,(k,l) ∩ Q δ,(k,l) we eliminate ∇ x v in order to evaluate |||R δ,(k,l) -R ′ δ,(k,l) |||. Then we evaluate |||R δ,(k+1,l) -R ′ δ,(k,l) |||. Finally it leads to (3.4) |||R δ,(k+1,l) -R δ,(k,l) ||| ≤ C(S) δ 3/2 dist(∇ x v; SO(3)) L 2 (Q δ,(k,l) ) + ||dist(∇ x v; SO(3))|| L 2 (Q δ,(k+1,l) ) .
In the same way, we prove that for any (k, l) ∈ N δ such that (k, l + 1) ∈ N δ we have

(3.5) |||R δ,(k,l+1) -R δ,(k,l) ||| ≤ C(S) δ 3/2 dist(∇ x v; SO(3)) L 2 (Q δ,(k,l) ) + ||dist(∇ x v; SO(3))|| L 2 (Q δ,(k,l+1 )

First decomposition of a deformation

In this section any

deformation v ∈ H 1 (Q δ ) 3 of the shell Q δ is decomposed as (3.6) v(s) = V(s 1 , s 2 ) + s 3 R a (s 1 , s 2 )n(s 1 , s 2 ) + v a (s), s ∈ Ω δ ,
where

V belongs to H 1 (ω) 3 , R a belongs to H 1 (ω) 3×3 and v a belongs to H 1 (Q δ ) 3 .
The map V is the mean value of v over the fibers while the second term s 3 R a (s 1 , s 2 )n(s 1 , s 2 ) is an approximation of the rotation of the fiber (of the shell) which contains the point φ(s 1 , s 2 ). The sum of the two first terms

V(s 1 , s 2 ) + s 3 R a (s 1 , s 2 )n(s 1 , s 2
) is called the elementary deformation of first type of the shell.

The matrix R a is defined as the Q 1 interpolate at the vertices of the cell ω δ,(k,l) =]kδ, (k+1)δ[×]lδ, (l+1)δ[ of the four elements R δ,(k,l) , R δ,(k+1,l) , R δ,(k,l+1) and R δ,(k+1,l+1) belonging to SO(3) (see the previous subsection). We can always define paths in SO(3) from R δ,(k,l) to R δ,(k+1,l) , R δ,(k,l) to R δ,(k,l+1) , R δ,(k+1,l) to R δ,(k+1,l+1) and R δ,(k,l+1) to R δ,(k+1,l+1) . That gives continuous maps from the edges of the domain ω δ,(k,l) into SO [START_REF] Blanchard | Microscopic effects in the homogenization of the junction of rods and a thin Asympt[END_REF]. If it is possible to extend these maps in order to obtain a continuous function from ω δ,(k,l) into SO(3), then it means that the loop passing trough R δ,(k,l) , R δ,(k+1,l) , R δ,(k+1,l+1) and R δ,(k,l+1) is homotopic to the constant loop equal to R δ,(k,l) . But the fundamental group π 1 SO(3), R δ,(k,l) is isomorphic to Z 2 (the group of odd and even integers), hence the extension does not always exist. That is the reason why we use here a Q 1 interpolate in order to define an approximate field R a of rotations. In the next subsection we show that if the matrices R δ,(k+1,l) , R δ,(k+1,l+1) and R δ,(k,l+1) are in a neighborhood of R δ,(k,l) then this extension exists and we give in Theorem 3.4 a simple condition to do so.

Theorem 3.3. Let v ∈ H 1 (Q δ )
3 , there exist an elementary deformation (of first type) V + s 3 R a n and a deformation v a satisfying (3.6) and such that

(3.7)                                  ||v a || (L 2 (Ω δ )) 3 ≤ Cδ||dist(∇ x v, SO(3))|| L 2 (Q δ ) ||∇ s v a || (L 2 (Ω δ )) 9 ≤ C||dist(∇ x v, SO(3))|| L 2 (Q δ ) ∂R a ∂s α (L 2 (ω)) 9 ≤ C δ 3/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ ) ∂V ∂s α -R a t α (L 2 (ω)) 3 ≤ C δ 1/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ ) ∇ x v -R a (L 2 (Ω δ )) 9 ≤ C||dist(∇ x v, SO(3))|| L 2 (Q δ ) ||dist(R a , SO(3))|| L 2 (ω) ≤ C δ 1/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ )
where the constant C does not depend on δ.

Proof. The field V is defined by

(3.8) V(s 1 , s 2 ) = 1 2δ δ -δ v(s 1 , s 2 , s 3 )ds 3 , a.e. in ω.
Then we define the field R a as following

∀(k, l) ∈ N δ , R a (kδ, lδ) = R δ,(k,l)
and for any (

s 1 , s 2 ) ∈ ω δ,(k,l) , R a (s 1 , s 2 ) is the Q 1 interpolate of the values of R a at the vertices of the cell ω δ,(k,l) .
Finally we define the field v a by

v a (s) = v(s) -V(s 1 , s 2 ) -s 3 R a (s 1 , s 2 )n(s 1 , s 2 ) a.e. in Ω δ .
From (3.4) and (3.5) we get the third estimate in (3.7). By definition of R a we obtain (3.9)

(k,l)∈N ′ δ R a -R δ,(k,l) 2 (L 2 (ω δ,(k,l) )) 9 ≤ C δ ||dist(∇ x v; SO(3))|| 2 L 2 (Q δ ) .
Taking the mean value of v on the fibers and using definition (3.8) of V lead to (3.10)

(k,l)∈N ′ δ ||V -a δ,(k,l) -R δ,(k,l) φ -φ(kδ, lδ) || 2 (L 2 (ω δ,(k,l) )) 3 ≤ Cδ||dist(∇ x v, SO(3))|| 2 L 2 (Q δ ) .
From (3.3), (3.9), (3.10) and the definition of v a we get the first estimate in (3.7).

We compute the derivatives of the deformation v to get

(3.11) ∂v ∂s 1 = ∇ x v t 1 + s 3 ∂n ∂s 1 , ∂v ∂s 2 = ∇ x v t 2 + s 3 ∂n ∂s 2 , ∂v ∂s 3 = ∇ x v n.
We consider the restrictions of these derivatives to Ω δ,(k,l) . Then, from (3.3) and (3.9) we have

(3.12) ∂v ∂s α -R a t α + s 3 ∂n ∂s α 2 (L 2 (Ω δ )) 3 + ∂v ∂s 3 -R a n 2 (L 2 (Ω δ )) 3 ≤ C ||dist(∇ x v, SO(3))|| 2 L 2 (Q δ ) .
By taking the mean value of ∂v ∂s α -R a t α + s 3 ∂n ∂s α on the fibers we obtain the fourth inequality in (3.7).

Observe now that

(3.13) ∂v a ∂s α = ∂v ∂s α - ∂V ∂s α -s 3 R a ∂n ∂s α -s 3 ∂R a ∂s α n, ∂v a ∂s 3 = ∂v ∂s 3 -R a n.
Then, from (3.12) and the third and fourth inequalities in (3.7) we obtain the second estimate in (3.7). The fifth inequality in (3.7) is an immediate consequence of (3.3) and (3.9). The last estimate of (3.7) is due to (3.4), (3.5) and to the very definition of the field R a .

Since the matrices R δ,(k,l) belong to SO(3), the function R a is uniformly bounded and satisfies

||R a || (L ∞ (ω)) 9 ≤ √ 3.
Let (k, l) be in N δ . By a straightforward computation, for any (s 1 , s 2 ) ∈ ω δ,(k,l) we obtain

|||R a (s 1 , s 2 )R T a (s 1 , s 2 ) -I 3 ||| ≤ C |||R δ,(k,l) -R δ,(k+1,l) ||| + |||R δ,(k,l) -R δ,(k,l+1) ||| +|||R δ,(k,l+1) -R δ,(k+1,l+1) ||| + |||R δ,(k+1,l) -R δ,(k+1,l+1) ||| | det R a (s 1 , s 2 ) -1| ≤ C |||R δ,(k,l) -R δ,(k+1,l) ||| + |||R δ,(k,l) -R δ,(k,l+1) ||| +|||R δ,(k,l+1) -R δ,(k+1,l+1) ||| + |||R δ,(k+1,l) -R δ,(k+1,l+1) |||,
where C is an absolute constant. Hence, from (3.4) and (3.5) we deduce that

(3.14)      ||R a R T a -I 3 || (L 2 (ω)) 9 ≤ C δ 1/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ ) , || det(R a ) -1|| L 2 (ω) ≤ C δ 1/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ ) .
Notice that the function R a R T a belongs to H 1 (ω) 3×3 and satisfies

(3.15) ∂R a R T a ∂s α (L 2 (ω)) 9 ≤ C δ 3/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ ) .

Second decomposition of a deformation.

In this section any deformation

v ∈ H 1 (Q δ ) 3 of the shell Q δ is decomposed as (3.16) v(s) = V(s 1 , s 2 ) + s 3 R(s 1 , s 2 )n(s 1 , s 2 ) + v(s), s ∈ Ω δ ,
where V belongs to H 1 (ω) 3 , R belongs to H 1 (ω) 3×3 and satisfies for a.e. (s 1 , s 2 ) ∈ ω: R(s 1 , s 2 ) ∈ SO [START_REF] Blanchard | Microscopic effects in the homogenization of the junction of rods and a thin Asympt[END_REF] and v belongs to H 1 (Q δ ) 3 . The first term V is still the mean value of v over the fibers. Now, the second one s 3 R(s 1 , s 2 )n(s 1 , s 2 ) describes the rotation of the fiber (of the shell) which contains the point φ(s 1 , s 2 ).

The sum of the two first terms V(s 1 , s 2 ) + s 3 R(s 1 , s 2 )n(s 1 , s 2 ) is called the elementary deformation of second type of the shell.

Theorem 3.4. There exists a constant C(S) (which depends only on the mid-surface S) such that for any

v ∈ H 1 (Q δ ) 3 verifying (3.17) ||dist(∇ x v, SO(3))|| L 2 (Q δ ) ≤ C(S)δ 3/2
then there exist an elementary deformation of second type V + s 3 Rn and a deformation v satisfying (3.16) and such that

(3.18)                          ||v|| (L 2 (Ω δ )) 3 ≤ Cδ||dist(∇ x v, SO(3))|| L 2 (Q δ ) ||∇ s v|| (L 2 (Ω δ )) 9 ≤ C||dist(∇ x v, SO(3))|| L 2 (Q δ ) ∂R ∂s α (L 2 (ω)) 9 ≤ C δ 3/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ ) ∂V ∂s α -Rt α (L 2 (ω)) 3 ≤ C δ 1/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ ) ∇ x v -R (L 2 (Ω δ )) 9 ≤ C||dist(∇ x v, SO(3))|| L 2 (Q δ )
where the constant C does not depend on δ.

Proof. In this proof let us denote by C 1 (S) the constant appearing in estimates (3.4) and (3.5). If we assume that

(3.19) √ 2C 1 (S) δ 3/2 dist(∇ x v; SO(3)) L 2 (Q δ ) ≤ 1 2 .
then, for each (k, l) ∈ N ′ δ we have using (3.4) and (3.5)

|||R δ,(k+1,l) -R δ,(k,l) ||| ≤ 1 2 , |||R δ,(k,l+1) -R δ,(k,l) ||| ≤ 1 2 .
Thanks to Lemma A.2 in Appendix A there exists a function R ∈ W 1,∞ (ω) 3×3 such that for any (s 1 , s 2 ) ∈ ω the matrix R(s 1 , s 2 ) belongs to SO(3) and such that

∀(k, l) ∈ N δ , R(kδ, lδ) = R δ,(k,l) .
From (3.4), (3.5) and Lemma A.2 we obtain the estimates (3.18) of the derivatives of R. Due to the corollary of Lemma A.2 we have

(3.20) ||R -R a || (L 2 (ω)) 9 ≤ C δ 1/2 ||dist ∇ x v; SO(3) || L 2 (Q δ ) .
All remainder estimates in (3.18) are consequences of (3.7) and (3.20).

Two nonlinear Korn's inequalities for shells

In this section, we first precise the boundary conditions on the deformations. We discuss essentially the usual case of the clamped condition on Γ 0,δ (see Subsection 3.2). In Subsection 4.1, we deduce the first estimates on v and ∇v. Then we show that the elementary deformations of the decompositions can be imposed on the same boundary than v. The main result of Subsection 4.2 is the Korn's inequality for shells given.

Let v be in (H

1 (Q δ )) 3 such that v(x) = x on Γ 0,δ .
Due to the definition (3.3) of V, we have

(4.1) V = φ on γ 0 .

First H 1 -Estimates

Using the boundary condition (4.1), estimates (3.7) or (3.18) and the fact that

||R a || (L ∞ (ω)) 3×3 ≤ √ 3 and ||R|| (L ∞ (ω)) 3×3 ≤ √ 3, it leads to (4.2) V (H 1 (ω)) 3 ≤ C 1 + 1 δ 1/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ ) .
With the help of the decompositions (3.6) or (3.16), estimates (3.7) or (3.18) and (4.2) we deduce that

||v|| (L 2 (Q δ )) 3 + 1 δ ||v -V|| (L 2 (Q δ )) 3 + ||∇ x v|| (L 2 (Q δ )) 9 ≤ C δ 1/2 + ||dist(∇ x v, SO(3))|| L 2 (Q δ ) .
The above inequality leads to the following first "nonlinear Korn's inequality for shells":

(4.3) ||v -I d || (L 2 (Q δ )) 3 + ||∇ x v -I 3 || (L 2 (Q δ )) 9 ≤ C δ 1/2 + ||dist(∇ x v, SO(3))|| L 2 (Q δ )
together with

||(v -I d ) -(V -φ)|| (L 2 (Q δ )) 3 ≤ Cδ δ 1/2 + ||dist(∇ x v, SO(3))|| L 2 (Q δ ) .
Let us notice that inequality (4.3) can be obtained without using the decomposition of the deformation. Indeed, we first have

|||∇v(x)||| ≤ dist(∇v(x), SO(3)) + √ 3, for a.e.
x so that by integration

||∇ x v|| (L 2 (Q δ )) 9 ≤ C δ 1/2 + ||dist(∇ x v, SO(3))|| L 2 (Q δ ) .
Poincaré's inequality then leads to (4.3). This is the technique used to derive estimates in [START_REF] Friesecke | Derivation of nonlinear bending theory for shells from three-dimensionnal nonlinear elasticity by Gamma convergence[END_REF].

Further H 1 -Estimates

In this subsection, we derive a boundary condition on R a and R on γ 0 using the extension given in Subsection 3.2. We prove the following lemma: Proof. Recall that γ ′ 0,δ , Q 1 δ and Q 2 δ are defined in subsection 3.2. We also set

Q 3 δ = Φ {(s 1 , s 2 ) ∈ ω 3δ | dist((s 1 , s 2 ), γ 0 ) < 3δ}×] -δ, δ[
Let us consider the following function

ρ δ (s 1 , s 2 ) = inf 1, sup 0, 1 3δ dist((s 1 , s 2 ), γ 0 ) -1 , (s 1 , s 2 ) ∈ R 2 .
This function belongs to W 1,∞ (R 2 ) and it is equal to 1 if dist((s 1 , s 2 ), γ 0 ) > 6δ and to 0 if dist((s 1 , s 2 ), γ 0 ) < 3δ. Let v δ be the deformation defined by

v δ (s) = φ(s 1 , s 2 ) + s 3 n(s 1 , s 2 ) + ρ δ (s 1 , s 2 ) v(s) -φ(s 1 , s 2 ) -s 3 n(s 1 , s 2 ) for a.e. s ∈ ω 3δ ×] -δ, δ[.
By definition of v δ , we have

v δ = v in Q ′ δ \ Q 2 δ , v δ = I d in Q 3 δ . Recall that v = I d on Q 1 δ .
Since the L ∞ -norm of ρ δ is of order 1/δ and the two estimates in (3.2) lead to

(4.4) ||∇ x v -∇ x v δ || (L 2 (Q ′ δ )) 9 ≤ C||dist(∇ x v, SO(3))|| L 2 (Q δ ) , ||v -v δ || (L 2 (Q ′ δ )) 3 ≤ Cδ||dist(∇ x v, SO(3))|| L 2 (Q δ ) . Hence (4.5) ||dist(∇ x v δ , SO(3))|| L 2 (Q ′ δ ) ≤ ||∇ x v -∇ x v δ || (L 2 (Q ′ δ )) 9 + ||dist(∇ x v, SO(3))|| L 2 (Q ′ δ ) ≤ C||dist(∇ x v, SO(3))|| L 2 (Q δ )
where the constant does not depend on δ.

Since v δ = I d in Q 2 δ , the R a 's and the R's given by application of Theorem 3.3 or 3.4 to the deformation v δ are both equal to I 3 over γ 0 . Estimate(3.7) and (3.18) of these theorem together with (4.4)-(4.5) show that Theorems 3.3 and 3.4 hold true for v with R a = I 3 and R = I 3 on γ 0 .

The next theorem gives a second nonlinear Korn's inequalities, which is an improvement of (4.3) for energies of order smaller than δ 3/2 and an estimate on v -V which permit to precise the scaling of the applied forces in Section 7.

Theorem 4.2. (A second nonlinear Korn's inequality for shells) There exists a constant C which does not depend upon δ such that for all

v ∈ H 1 (Q δ ) 3 such that v = I d on Γ 0,δ (4.6) ||v -I d || (L 2 (Q δ )) 3 + ||∇ x v -I 3 || (L 2 (Q δ )) 9 ≤ C δ ||dist(∇ x v, SO(3))|| L 2 (Q δ ) , and
(4.7) ||(v -I d ) -(V -φ)|| (L 2 (Q δ )) 3 ≤ C||dist(∇ x v, SO(3))|| L 2 (Q δ ) ,
where V is given by (3.8).

Proof. From the decomposition (3.6), Theorem 3.3 and the boundary condition on R a given by Lemma 4.1, the use of Poincaré's inequality gives (4.8)

       ||R a -I 3 || (H 1 (ω)) 9 ≤ C δ 3/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ ) ∂V ∂s α -t α (L 2 (ω)) 3 ≤ C δ 3/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ )
Using the fact that t α = ∂φ ∂s α and the boundary condition (4.1) on V, it leads to

||V -φ|| (L 2 (ω)) 3 ≤ C δ 3/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ .
Using again the decomposition (3.6) and Theorem 3.3, the above estimate implies that v -I d satisfies the nonlinear Korn's inequality (4.6). At last the decomposition (3.6), which implies that (v - Let us compare the two inequalities (4.3) and (4.6). Indeed they are equivalent for energies of order δ 3/2 . For energies order smaller than δ 3/2 , (4.6) is better (4.3) which is then more relevant in general for thin structures.

I d ) -(V -φ) = (R a -I 3 )s 3 n + v a ,
The decomposition technique given in Section 3 also allows to estimate the linearized strain tensor of an admissible deformation. This is the object of the lemma below.

Lemma 4.3 There exists a constant C which does not depend upon δ such that for all

v ∈ H 1 (Q δ ) 3 such that v = I d on Γ 0,δ (4.9) ∇ x v + (∇ x v) T -2I 3 (L 2 (Q δ )) 9 ≤ C||dist(∇ x v, SO(3))|| L 2 (Q δ ) 1 + 1 δ 5/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ ) .
Proof. In view of the decomposition (3.6) and Theorem 3.3 we have (4.10)

∇ x v + (∇ x v) T -2I 3 (L 2 (Ω δ )) 9 ≤ C||dist(∇ x v, SO(3))|| L 2 (Q δ ) + Cδ 1/2 R a + R T a -2I 3 (L 2 (ω)) 9 .
Due to the equalities

R a + R T a -2I 3 = R a R a R T a + R T a -2R a R T a + R a (I 3 -R a R T a ) + 2(R a R T a -I 3 ) = (R a -I 3 ) 2 R T a + R a (I 3 -R a R T a ) + 2(R a R T a -I 3 )
and to the first estimate in (3.14), it follows that (4.11) [START_REF] Ciarlet | An introduction to shell theory[END_REF] and the fact that the space H 1 (ω) 3×3 is continuously imbedded in L 4 (ω) 3×3 , we deduce that (4.12)

||R a + R T a -2I 3 || (L 2 (ω)) 9 ≤ C||(R a -I 3 ) 2 || (L 2 (ω)) 9 + C δ 1/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ ) . Since ||(R a -I 3 ) 2 || (L 2 (ω)) 9 ≤ C||R a -I 3 || 2 (L 4 (ω))
||(R a -I 3 ) 2 || (L 2 (ω)) 9 ≤ C δ 3 ||dist(∇ x v, SO(3))|| 2 L 2 (Q δ ) .
From (4.10), (4.11) and (4.12) we finally get (4.9).

Remark 4.4. In view of (3.7) and since the field R a belongs to (L ∞ (ω)) 3×3 , the function (R a -I 3 )

2 belongs to H 1 (ω) 3×3 with ∂(R a -I 3 ) 2 ∂s α (L 2 (ω)) 9 ≤ C δ 3/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ ) . Hence, with Lemma 4.1, ||(R a -I 3 ) 2 || (L 2 (ω)) 9 ≤ C δ 3/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ )
which gives together with (4.10)-(4.11)

∇ x v + (∇ x v) T -2I 3 (L 2 (Q δ )) 9 ≤ C δ ||dist(∇ x v, SO(3))|| L 2 (Q δ ) .
Notice that the above estimate is worse than (4.9) at least as soon as the energy is smaller than δ 1/2 .

Let us emphasize that in view of estimates (3.7)-(3.18), (4.3) and (4.9) one can distinguish two critical cases for the behavior of the quantity

||dist(∇ x v, SO(3))|| L 2 (Q δ ) ||dist(∇ x v, SO(3))|| L 2 (Q δ ) = O(δ 3/2 ), O(δ 5/2 ). Estimates (4.2)-(4.3) show that the behavior ||dist(∇ x v, SO(3))|| L 2 (Q δ ) ∼ O(δ 1/2
) also corresponds to an interesting case, but the estimates (3.7) and (4.8) show that the decompositions given in Theorems 3.3 and 3.4 are not relevant in this case which, as a consequence, must be analyzed by a different approach.

In the following we will describe the asymptotic behavior of a sequence of deformations v δ which satisfies

||dist(∇ x v δ , SO(3))|| L 2 (Q δ ) ∼ O(δ κ-1/2 ), κ ≥ 2.

Rescaling Ω δ

As usual when dealing with a thin shell, we rescale Ω δ using the operator (Π δ w)(s 1 , s 2 , S 3 ) = w(s 1 , s 2 , s 3 ) for any s ∈ Ω δ defined for e.g. w ∈ L 2 (Ω δ ) for which (Π δ w) ∈ L 2 (Ω). The estimates (3.7) on v a transposed over Ω lead to (5.1)

                     ||Π δ v a || (L 2 (Ω)) 3 ≤ Cδ 1/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ ) || ∂Πv a ∂s 1 || (L 2 (Ω)) 3 ≤ C δ 1/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ ) || ∂Πv a ∂s 2 || (L 2 (Ω)) 3 ≤ C δ 1/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ ) || ∂Πv a ∂S 3 || (L 2 (Ω)) 3 ≤ Cδ 1/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ ) ,
and estimates (4.6) on v -I d give (5.2)

                       ||Π δ (v -I d )|| (L 2 (Ω)) 3 ≤ C δ 3/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ ) || ∂Π δ (v -I d ) ∂s 1 || (L 2 (Ω)) 3 ≤ C δ 3/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ ) || ∂Π δ (v -I d ) ∂s 2 || (L 2 (Ω)) 3 ≤ C δ 3/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ ) || ∂Π δ (v -I d ) ∂S 3 || (L 2 (Ω)) 3 ≤ C δ 1/2 ||dist(∇ x v, SO(3))|| L 2 (Q δ ).
6. Asymptotic behavior of the Green-St Venant's strain tensor in the case κ = 2.

Let us consider a sequence of deformations

v δ of H 1 (Q δ ) 3 such that (6.1) ||dist(∇ x v δ , SO(3))|| L 2 (Q δ ) ≤ Cδ 3/2 .
For fixed δ > 0, the deformation v δ is decomposed as in Theorem 3.3 and the terms of this decomposition are denoted by V δ , R a,δ and v a,δ . If moreover the hypothesis (3.17) holds true for the sequence v δ , then v δ can be alternatively decomposed through (3.16) in terms of V δ , R δ and v δ so that the estimates (3.18) of Theorem 3.5 are satisfied uniformly in δ.

In what follows we investigate the behavior of the sequences V δ , R a,δ and v a,δ . Indeed due to (3.20) all the result of this section can be easily transposed in terms of the sequence R δ and the details are left to the reader.

The estimates (3.7), (5.1) and (5.2) lead to the following lemma. Lemma 6.1. There exists a subsequence still indexed by δ such that (6.2)

                         V δ -→ V strongly in H 1 (ω) 3 , R a,δ ⇀ R weakly in H 1 (ω) 3×3 and strongly in L 2 (ω) 3×3 , 1 δ 2 Π δ v a,δ ⇀ v weakly in L 2 (ω; H 1 (-1, 1)) 3 , 1 δ ∂V δ ∂s α -R a,δ t α ⇀ Z α weakly in L 2 (ω) 3 , 1 δ R T a,δ R a,δ -I 3 ⇀ 0 weakly in L 2 (ω) 3×3 ,
where R belongs SO(3) for a.e. (s 1 , s 2 ) ∈ ω. We also have V ∈ H 2 (ω) 3 and

(6.3) ∂V ∂s α = Rt α .
The boundaries conditions

(6.4) V = φ, R = I 3 on γ 0 ,
hold true. Moreover, we have

(6.5) Π δ v δ -→ V strongly in H 1 (Ω) 3 , Π δ (∇ x v δ ) -→ R strongly in L 2 (Ω) 9 .
Proof. 3×3 and the first convergence in (6.2) permit to obtain

(6.6) R a,δ -→ R strongly in L 4 (ω) 3×3 , det(R a,δ ) -→ det(R) strongly in L 4/3 (ω).
These convergences and estimates (3.14) prove that for a.e. (s 1 , s 2 ) ∈ ω: R(s 1 , s 2 ) ∈ SO(3). The relation (6.3) and (6.4) and the convergences (6.5) are immediate consequences of Theorem 3.3 and of the above results. We now turn to the proof of the last convergence in (6.2). We first set

R a,δ (s 1 , s 2 ) = R a,δ δ s 1 δ , δ s 2 δ a.e. in ω
where [t] denote the integer part of the real t. From (3.4), (3.5) and ( 6.1) we have

(6.7) ||R a,δ -R a,δ || (L 2 (ω)) 3×3 ≤ Cδ.
From (6.6) and the above estimate, we deduce that

(6.8) R a,δ -→ R strongly in (L 2 (ω) 3×3 .
Now we derive the weak limit of the sequence

1 δ (R a,δ -R a,δ ). Let Φ be in C ∞ 0 (Ω) 3×3 and set M δ (Φ)(s 1 , s 2 ) = ]0,1[ 2 Φ δ s 1 δ + z 1 δ, δ s 2 δ + z 2 δ dz 1 dz 2 for a.e. (s 1 , s 2 ) in ω.
We recall that (see [START_REF] Blanchard | Junction of a periodic family of elastic rods with a thin plate[END_REF]) 3×3 . In view of the properties of M δ (Φ) recalled above, of (6.2) and (6;4), we deduce from the above equality that

1 δ Φ -M δ (Φ) ⇀ 0 weakly in (L 2 (ω) 3×3 M δ (Φ) -→ Φ strongly in (L 2 (ω) 3×3 We write ω 1 δ (R a,δ -R a,δ )Φ = ω R a,δ 1 δ Φ -M δ (Φ) + ω 1 δ (R a,δ -R a,δ )M δ (Φ) = ω R a,δ 1 δ Φ -M δ (Φ) + 1 2 ω ∂R a,δ ∂s 1 + ∂R a,δ ∂s 2 M δ (Φ) + K δ where |K δ | ≤ Cδ||∇R a,δ || (L 2 (ω)) 3×3 ||∇Φ|| (L 2 (ω))
1 δ (R a,δ -R a,δ ) ⇀ 1 2 ∂R ∂s 1 + ∂R ∂s 2 weakly in (L 2 (ω) 3×3 .
In order to prove the last convergence of (6.2), we write

1 δ R T a,δ R a,δ -I 3 = 1 δ R T a,δ (R a,δ -R a,δ ) + (R a,δ -R a,δ ) T R a,δ + (R a,δ -R a,δ ) T (R a,δ -R a,δ )
and we use estimates (3.14) and (6.7), the strong convergence (6.8) and the above weak convergence.

The following Corollary gives the limit of the Green-St Venant's strain tensor of the sequence v δ .

Theorem 6.2. For the same subsequence as in Lemma 6.1 we have

(6.9) 1 2δ Π δ (∇ x v δ ) T ∇ x v δ -I 3 ⇀ (t 1 | t 2 | n) -T E(t 1 | t 2 | n) -1 weakly in (L 1 (Ω)) 9 ,
where the symmetric matrix E is equal to

(6.10)        S 3 ∂R ∂s 1 n • Rt 1 + Z 1 • Rt 1 S 3 ∂R ∂s 1 n • Rt 2 + 1 2 Z 2 • Rt 1 + Z 1 • Rt 2 1 2 ∂v ∂S 3 • Rt 1 + 1 2 Z 1 • Rn * S 3 ∂R ∂s 2 n • Rt 2 + Z 2 • Rt 2 1 2 ∂v ∂S 3 • Rt 2 + 1 2 Z 2 • Rn * * ∂v ∂S 3 • Rn       
and where (t 1 | t 2 | n) denotes the 3 × 3 matrix with first column t 1 , second column t 2 and third column n and where (t

1 | t 2 | n) -T = (t 1 | t 2 | n) -1 T .
Proof. First from estimate (3.7), equalities (3.13) and the convergences in Lemma 6.1, we obtain

1 δ Π δ (∇ x v δ ) -R a,δ t α ⇀ S 3 ∂R ∂s α n + Z α weakly in L 2 (Ω) 3 , 1 δ Π δ (∇ x v δ ) -R a,δ n ⇀ ∂v ∂S 3 weakly in L 2 (Ω) 3 .
Then thanks to the identity

1 2δ Π δ (∇ x v δ ) T ∇ x v δ -I 3 = 1 2δ Π δ (∇ x v δ -R a,δ ) T (∇ x v δ -R a,δ ) + 1 2δ R T a,δ Π δ (∇ x v δ -R a,δ ) + 1 2δ Π δ (∇ x v δ -R a,δ ) T R a,δ + 1 2δ R T a,δ R a,δ -I 3 ,
and again to estimate (3.7) and Lemma 6.1 we deduce that (6.11)

           1 δ Π δ (∇ x v δ ) T ∇ x v δ -I 3 ⇀(t 1 | t 2 | n) -T S 3 ∂R ∂s 1 n + Z 1 | S 3 ∂R ∂s 2 n + Z 2 | ∂v ∂S 3 T R +R T S 3 ∂R ∂s 1 n + Z 1 | S 3 ∂R ∂s 2 n + Z 2 | ∂v ∂S 3 (t 1 | t 2 | n) -1
weakly in (L 1 (Ω)) 9 .

Now remark that (6.12)

∂R ∂s 1 n • Rt 2 = ∂R ∂s 2 n • Rt 1 .
Indeed, deriving the relation R T R = I 3 with respect to s α shows that R T ∂R ∂s α + ∂R T ∂s α R = 0. Hence, there exists an antisymmetric matrix field A α ∈ L 2 (ω; R 3×3 ) such that ∂R ∂s α = RA α . Moreover there exists a field

a α belonging to L 2 (ω) 3 such that ∀x ∈ R 3 , A α x = a α ∧ x.
Now we derive the equality ∂V ∂s α = Rt α with respect to s β and we obtain

∂ 2 V ∂s α ∂s β = ∂R ∂s β t α + R ∂t α ∂s β = RA β t α + R ∂ 2 φ ∂s α ∂s β .
It implies that A 1 t 2 = A 2 t 1 from which (6.12) follows. Taking into account the definition of the matrix E, convergence (6.11) and the equality (6.12) show that (6.9) holds true.

Remark 6.3.

There exists a constant C such that

∂R ∂s 1 (L 2 (ω)) 9 + ∂R ∂s α (L 2 (ω)) 9 ≤ C ∂R ∂s 1 n • Rt 1 L 2 (ω) + ∂R ∂s 2 n • Rt 2 L 2 (ω) + ∂R ∂s 1 n • Rt 2 L 2 (ω)
.

the same notation as in the proof of Theorem 6.2, we have

∂R ∂s α 2 (L 2 (ω)) 9 = A α 2 (L 2 (ω)) 9 = 2||a α || 2 (L 2 (ω)) 3 .
Recalling that a 1 ∧ t 2 = a 2 ∧ t 1 , we obtain a α • n = 0 and then

∂R ∂s α n 2 (L 2 (ω)) 3 = ||a α ∧ n|| 2 (L 2 (ω)) 3 = ||a α || 2 (L 2 (ω)) 3 = 1 2 ∂R ∂s α 2 (L 2 (ω)) 9
.

Remark.

It is well known that the constraint ∂V ∂s 1 = Rt 1 and ∂V ∂s 2 = Rt 2 together the boundary conditions are strong limitations on the possible deformation for the limit 2d shell. Actually for a plate or as soon as S is a developable surface, the configuration after deformation must also be a developable surface.

In the general case, it is an open problem to know if the set V nlin contains other deformations than identity mapping or very special isometries (as for example symetries).

7. Asymptotic behavior of the Green-St Venant's strain tensor in the case κ > 2.

Let us consider a sequence of deformations

v δ of H 1 (Q δ ) 3 such that (7.1) ||dist(∇ x v δ , SO(3))|| L 2 (Q δ ) ≤ Cδ κ-1/2 , with κ > 2.
We use the decomposition (3.16) of deformation and the estimates (3.18) of Theorem 3.4. These estimates and the boundary condition (Lemma 4.1) lead to the following convergences:

(7.2)        R δ -→ I 3 strongly in H 1 (ω)) 9 , Π δ v δ -→ φ strongly in H 1 (Ω) 3 , Π δ (∇ x v δ ) -→ I 3 strongly in L 2 (Ω) 9 .
In view of these convergences, we now study the asymptotic behavior of the sequence of displacements belonging to

H 1 (Q δ ) 3 u δ (x) = v δ (x) -x.
Due to the decomposition (3.16) we write

(7.3) u δ (s) = U δ (s 1 , s 2 ) + s 3 (R δ -I 3 )(s 1 , s 2 )n(s 1 , s 2 ) + v δ (s), s ∈ Ω δ , where U δ (s 1 , s 2 ) = V δ (s 1 , s 2 ) -φ(s 1 , s 2 ).
Thanks to the estimates (3.18) we obtain the following lemma:

Lemma 7.1. There exists a subsequence still indexed by δ such that

(7.4)                      1 δ κ-2 R δ -I 3 ⇀ A weakly in H 1 (ω) 9
and strongly in L 2 (ω) 9 ,

1 δ κ-2 U δ -→ U strongly in H 1 (ω) 3 , 1 δ κ Π δ v δ ⇀ v weakly in L 2 (ω; H 1 (-1, 1)) 3 , 1 δ κ-1 ∂U δ ∂s α -(R δ -I 3 )t α ⇀ Z α weakly in L 2 (ω) 3 , and
(7.5)      1 δ κ-2 Π δ u δ -→ U strongly in (H 1 (Ω)) 3 , 1 δ κ-2 Π δ (∇ x u δ ) -→ A strongly in (L 2 (Ω)) 9
where A ∈ (H 1 (ω)) 9 , U ∈ (H 1 (ω)) 3 , v ∈ L 2 (ω; H 1 (-1, 1)) 3 and Z α ∈ (L 2 (ω)) 3 . Moreover, we have (7.6) U = 0, A = 0 on γ 0 , and ∂U ∂s α = At α , and U ∈ H 2 (ω) 3 .

We now show that the matrix A is actually an antisymmetric matrix. Using the first convergence in (7.2) and the first convergence in (7.4) we get

1 δ κ-2 R T δ R δ -I 3 ⇀ A weakly in H 1 (ω) 3×3 .
The matrix R δ belongs to SO(3), hence R T δ R δ -I 3 = I 3 -R T δ . It follows that the matrix A is an antisymmetric matrix. There exists a field R ∈ H 1 (ω) 3 (with R = 0 on γ 0 due to (7.6)) such that for all

x ∈ R 3 we have

(7.7) Ax = R ∧ x.
From (7.6) and the above equality we obtain

(7.8) ∂U ∂s α = R ∧ t α .
7.1. Inextensional and extensional displacements of the shell.

Now we define the inextensional displacements and extensional displacements sets of the mid-surface of the shell. We set

H 1 γ0 = v ∈ H 1 (ω) | v = 0 on γ 0 .
We equip H 1 γ0 (ω) 3 with the following inner product:

∀(U, V ) ∈ H 1 γ0 (ω) 3 × H 1 γ0 (ω) 3 , < U, V >= ω ∂U ∂s 1 • ∂V ∂s 1 + ∂U ∂s 2 • ∂V ∂s 2 .
For any U ∈ (H 1 (ω)) 3 we set

e 11 (U ) = ∂U ∂s 1 • t 1 , e 12 (U ) = 1 2 ∂U ∂s 1 • t 2 + ∂U ∂s 2 • t 1 , e 22 (U ) = ∂U ∂s 2 • t 2 .
The spaces of inextentional and extensional displacements are respectively defined by

D In = U ∈ H 1 γ0 (ω) 3 | e 11 (U ) = e 12 (U ) = e 22 (U ) = 0 D Ex = D In ⊥ ,
where D In ⊥ is the orthogonal of D In in the space H 1 γ0 (ω) 3 . For all U ∈ D In there exists a unique field

R ∈ L 2 (ω) 3 such that ∂U ∂s 1 = R ∧ t 1 , ∂U ∂s 2 = R ∧ t 2 ,
and R | U ∈ D In is a closed subspace of L 2 (ω) 3 .

We equip D Ex with the norm (7.9)

∀U ∈ D Ex , ||U || Ex = e 11 (U ) 2 L 2 (ω) + e 12 (U ) 2 L 2 (ω) + e 22 (U ) 2 L 2 (ω) .
Generally, D Ex is not a Hilbert space. We denote by D Ex a Hilbert space in which D Ex is a dense subspace.

In the general case an element belonging to D Ex is neither a function nor a distribution. If a sequence U n n∈N converges to U ∈ D Ex , the sequences e 11 (U n ) n∈N , e 12 (U n ) n∈N and e 22 (U n ) n∈N strongly converge in L 2 (ω) and their limits depend only on U . That is the reason why we will denote these limits e 11 (U ), e 12 (U ) and e 22 (U ). But notice that we use here improper notations because the element U has not always derivatives in the distribution sense.

If the shell is a plate, we have φ(s 1 , s 2 ) = (s 1 , s 2 ) hence t 1 = e 1 , t 2 = e 2 and n = e 3 . In this case the extensional displacements are the membrane displacements and the inextensional displacements have the form U 3 e 3 where U 3 is the bending. We have

D Ex = D Ex = H 1 γ0 (ω)
2 and due to Korn's inequality in

H 1 γ0 (ω) 2 , the norm || • || Ex is equivalent to the H 1 norm on D Ex .
7.2. Limit of the Green-St Venant's strain tensor for κ > 2.

We consider the sequence v δ introduced in Section 7 which satisfies ||dist(∇

x v δ , SO(3))|| L 2 (Q δ ) ≤ Cδ κ-1/2 and the associated displacement u δ = v δ -I d .
write the displacement U δ of the mid-surface as the sum of an inextensional displacement and an extensional one as in Section 6 (7.10)

U δ = U I,δ + U E,δ U I,δ ∈ D In , U E,δ ∈ D Ex .
We first give the estimates on U I,δ and U E,δ in the following lemma.

Lemma 7.2. We have

(7.11) U I,δ (H 1 (ω)) 3 ≤ Cδ κ-2 , U E,δ (H 1 (ω)) 3 ≤ Cδ κ-2 , U E,δ Ex ≤ Cδ κ-1 (1 + δ κ-3 ).
The constants do not depend on δ. Then, we can choose the subsequence in Lemma 7.1 such that (7.12) 3 and moreover

     1 δ κ-2 U I,δ ⇀ U weakly in (H 1 (ω)) 3 1 δ κ-2 U E,δ ⇀ 0 weakly in (H 1 (ω))
(7.13)      if 2 < κ < 3, 1 δ 2κ-4 U E,δ ⇀ U E weakly in D Ex if κ ≥ 3, 1 δ κ-1 U E,δ ⇀ U E weakly in D Ex
The convergences in (7.13) are equivalent to the weak convergences in L 2 (ω) of e 11 (U E,δ ), e 12 (U E,δ ) and e 22 (U E,δ ).

Proof. The two first estimates of (7.11) follow from (7.4) and from the orthogonality of U I,δ and U E,δ . Now notice that (7.14) e αα (U δ ) = e αα (U E,δ ) and e αβ (U δ ) = e 12 (U E,δ ).

We denote by A δ the antisymmetric part of R δ . Notice that

||R δ + R T δ -2I 3 || (L 2 (ω)) 3×3 ≤ ||R δ - I 3 || 2 (L 4 (ω)) 3×3 ≤ C||∇R δ || 2 (L 2 (ω)) 18 .
Then, from estimates (3.18) and (7.1) we deduce that

∂U δ ∂s α -A δ t α (L 2 (ω)) 3 ≤ Cδ κ-1 + Cδ 2κ-4 .
Then by definition of the norm || • || Ex we get the third estimate in (7.11) and then the convergences in (7.13).

The following theorem gives the expression of the limit of the Green-St Venant's tensor.

Theorem 7.3. Let us set

(7.15) Z αβ = 1 2 Z α • t β + Z β • t α , u = v + S 3 2 Z 1 • n t ′ 1 + S 3 2 Z 2 • n t ′ 2 where t ′ 1 , t ′ 2
is the contravariant basis of t 1 , t 2 . For a subsequence we have

(7.16) 1 2δ κ-1 Π δ (∇ x v δ ) T ∇ x v δ -I 3 ⇀ (t 1 | t 2 | n) -T E(t 1 | t 2 | n) -1
weakly in (L 1 (Ω)) 9 ,

where the symmetric matrix E is defined by

(7.17) E =        S 3 ∂R ∂s 1 ∧ n • t 1 + Z 11 S 3 ∂R ∂s 1 ∧ n • t 2 + Z 12 1 2 ∂u ∂S 3 • t 1 * S 3 ∂R ∂s 2 ∧ n • t 2 + Z 22 1 2 ∂u ∂S 3 • t 2 * * ∂u ∂S 3 • n       
Moreover, if 2 < κ < 3 then we have

(7.18) e αβ (U E ) + 1 2 ∂U ∂s α • ∂U ∂s β = 0
and if κ ≥ 3 then we have

(7.19) Z αβ =    e αβ (U E ) + 1 2 ∂U ∂s α • ∂U ∂s β if κ = 3 e αβ (U E ) if κ > 3
Proof. First we have

∂v ∂s 1 = ∇ x v t 1 + s 3 ∂n ∂s 1 , ∂v ∂s 2 = ∇ x v t 2 + s 3 ∂n ∂s 2 , ∂v ∂s 3 = ∇ x v n.
As a consequence of the above formulaes, of (3.3) and of the convergences in Lemma 5.1, we have (7.20)

       1 δ κ-1 Π δ (∇ x v δ ) -R δ t α ⇀ S 3 ∂R ∂s α ∧ n + Z α weakly in L 2 (Ω) 3 , 1 δ κ-1 Π δ (∇ x v δ ) -R δ n ⇀ ∂v ∂S 3
weakly in L 2 (Ω) 3 .

Then thanks to the identity

1 2δ κ-1 Π δ (∇ x v δ ) T ∇ x v δ -I 3 = 1 2δ κ-1 Π δ (∇ x v δ -R δ ) T ∇ x v δ + 1 2δ κ-1 R T δ Π δ (∇ x v δ -R δ )
and to convergences (7.2) and (7.20), we deduce that

1 2δ κ-1 Π δ (∇ x v δ ) T ∇ x v δ -I 3 ⇀ (t 1 | t 2 | n) -T E(t 1 | t 2 | n) -1
weakly in (L 1 (Ω)) 9 , where the symmetric matrix E is equal to

E = 1 2 S 3 ∂R ∂s 1 ∧ n + Z 1 | S 3 ∂R ∂s 2 ∧ n + Z 2 | ∂v ∂S 3 T (t 1 | t 2 | n) + 1 2 (t 1 | t 2 | n) T S 3 ∂R ∂s 1 ∧ n + Z 1 | S 3 ∂R ∂s 2 ∧ n + Z 2 | ∂v ∂S 3 .
Deriving the equality (7.8) with respect to s 1 and s 2 gives

∂ 2 U ∂s 1 ∂s 2 = ∂R ∂s 1 ∧ t 2 + R ∧ ∂ 2 φ ∂s 1 ∂s 2 = ∂R ∂s 2 ∧ t 1 + R ∧ ∂ 2 φ ∂s 1 ∂s 2 , hence ∂R ∂s 1 t 2 = ∂R ∂s 2 ∧ n t 1 .
Introducing Z αβ and u we obtain the expression (7.17) for E.

Below we show (7.18) and (7.19). Due to (7.4), we first have

(7.21) 1 δ κ-1 e αβ (U δ ) - 1 2 R δ + R T δ -2I 3 t α • t β ⇀ Z αβ weakly in L 2 (ω). the identity R δ + R T δ -2I 3 t α • t β = -(R δ -I 3 )t α • (R δ -I 3
)t β and using the first convergence in (7.4), we deduce that

(7.22) 1 2δ 2κ-4 R δ + R T δ -2I 3 t α • t β -→ - 1 2 At α • At β strongly in L 2 (ω).
In the case 2 < κ < 3 we have

1 δ 2κ-4 e αβ (U δ ) - 1 2 R δ + R T δ -2I 3 t α • t β -→ 0 strongly in L 2 (ω) 1 δ 2κ-4 e αβ (U δ ) ⇀ e αβ (U E ) weakly in L 2 (ω).
Thanks to (7.6) and (7.22) we obtain (7.18). In the case κ ≥ 3 then convergences (7.13), (7.21) and (7.22) permit to obtain the expression of Z αβ in terms of e αβ (U E ) and U.

We denote R a,θ the rotation with axis directed by the vector a ∈ S 3 and with angle of rotation about this axis θ ∈ R, (A.1) ∀x ∈ R 3 , R a,θ (x) = cos(θ)x + (1cos(θ)) < x, a > a + sin(θ) a ∧ x.

Let R 0 and R 1 be two matrices in SO(3). Matrix R 0 represent the rotation R a0,θ0 and matrix R 1 represent the rotation R a1,θ1 . The linear transformation in R 3

x -→ 2 sin(θ 1 )a 1sin(θ 0 )a 0 ∧ x has for matrix R 1 -R 0 -(R 1 -R 0 ) T and we have sin(θ 1 )a 1sin(θ 0 )a

0 2 = 1 2 √ 2 |||R 1 -R 0 -(R 1 -R 0 ) T ||| ≤ 1 √ 2 |||R 1 -R 0 |||.
To any matrix R in SO(3) we associate the vector b = sin(θ)a where R is the matrix of the rotation R a,θ . This map is continuous from SO(3) into B 3 and from the above inequality, we obtain Let R 0 and R 1 be two matrices in SO(3) such that

|||R 0 -R 1 ||| < 2 √ 2.
Now we define a path f from R 0 to R 1 :

• if R 1 = R 0 we choose the constant function f (t) = R 0 , t ∈ [0, 1],

• if R 1 = R 0 , we set R 2 = R -1 0 R 1 , there exists a unique pair (a 2 , θ 2 ) ∈ S 3 ×]0, π[ such that the matrix R 2 represent the rotation with axis directed by the vector a 2 and with the angle θ 2 . We consider the rotations field R a(t),θ(t) given by formula (A.1) where

a(t) = a 2 , θ(t) = tθ 2 , t ∈ [0, 1]
and we define f (t) as the matrix of the rotation R 0 • R a(t),θ(t) where R 0 is the rotation with matrix R 0 .

Lemma A.1. The path f belongs to W 1,∞ (0, 1; SO(3)) and satisfies For any t ∈ [0, 1], to matrix R -1 00 f 00,01 (t) we associate the vector b 00,01 (t), to matrix R -1 00 f 01,11 (t) we associate the vector b 01,11 (t), to matrix R -1 00 f 00,10 (t) we associate the vector b 00,10 (t) and to matrix R This function belongs to W 1,∞ (Y ) 3 and satisfies

(A.3)    f (0) = R 0 , f (1)
∀(x 1 , x 2 ) ∈ Y , b(x 1 , x 2 ) 2 ≤ 1 √ 2 .
Now we introduce the rotations field R(x 1 , x 2 ) given by formula (A.2) where b(x 1 , x 2 ) is defined above and where

θ(x 1 , x 2 ) = arccos 1-< b(x 1 , x 2 ), b(x 1 , x 2 ) >, (x 1 , x 2 ) ∈ Y .
Let R(x 1 , x 2 ) be the matrix of the rotation R 00 • R(x 1 , x 2 ) where R 00 is the rotation with matrix R 00 . It is easy to check that R satisfies the conditions (A.5).

Corollary of Lemma

  )|| L 2 (S×]-δ,δ[) 1+ 1 δ 5/2 ||dist(∇ x v, SO(3))|| L 2 (S×]-δ,δ[)

Theorem 3 . 1 .

 31 Let Ω be an open set of R n contained in the ball B O; R and star-shaped with respect to the ball B

Lemma 4 . 1 .

 41 InTheorem 3.3 (respectively in Theorem 3.4), we can choose R a (resp. R) such thatR a = I 3 on γ 0 , (resp. R = I 3 on γ 0 ),without modifications in the estimates of these theorems.

  the first estimate in (3.7) and (4.8) permit to obtain (4.7).

b 2 1 2 |||R -I 3

 23 |||.If cos(θ) = -1, using the vector b we can write the rotation R a,θ as∀x ∈ R 3 , R a,θ (x) = cos(θ)x + 1 1 + cos(θ) < x, b > b + b ∧ x.

1 1 -x 1 2 - 2 b

 122 -1 00 f 01,11 (t) we associate the vector b 01,11 (t). Let b be the vectors field defined byb(x 1 , x 2 ) =                b 00,10 (0) = b 00,01 (0) if (x 1 , x 2 ) = (0, 0), x 1 x 1 + x 2 b 00,10 (x 2 ) + x 2 x 1 + x 2 b 00,01 (x 1 ) if 0 < x 1 + x 2 ≤ x 1x 2 b 10,11 (x 2 ) + 1x 2 2x 1x 2 b 01,11 (x 1 ) if 1 ≤ x 1 + x 2 < 01,11(1)= b 10,11 (1) if (x 1 , x 2 ) = (1, 1).

  , s 2 ) ∈ ω η0 . The function Φ (introduced above) is now defined on ω η0 × [-δ 0 , δ 0 ] and we still assume that it is a C 1diffeomorphism of that set onto its range. Then there exist four constants c

	3
	which remains injective and such that the vectors each point (s 1 ′ ∂φ ∂s 1 (s 1 , s 2 ) and ∂φ (s 1 , s 2 ) are linearly independent at ∂s 2

  |||I 3 -R 2 ||| = |||R 0 -R 1 ||||.Lemma A.2. Let R 00 , R 01 , R 10 and R 11 be four matrices belonging to SO(3) and satisfying There exists a function ∈ W 1,∞ (Y ; SO(3)) such that(A.5) R(0, 0) = R 00 , R(0, 1) = R 01 , R(1, 0) = R 10 , R(1, 1) = R 11 , ||∇R|| (L ∞ (Y )) 9 ≤ C |||R 10 -R 00 |||| + |||R 01 -R 00 |||| + |||R 11 -R 01 |||| + |||R 11 -R 10 |||| .and where the functionsx 1 -→ R(x 1 , 0), x 1 -→ R(x 1 , 1), x 2 -→ R(0, x 2 ) and x 2 -→ R(1,x 2 ) are paths given by Lemma A.1. Proof. We denote f 00,01 the path from R 00 to R 01 , f 01,11 the path from R 01 to R 11 , f 00,10 the path from R 00 to R 10 and f 01,11 the path from R 01 to R 11 given by Lemma A. From Lemma A.1, we have ∀t ∈ [0, 1], |||f 00,01 (t) -R 00 ||| ≤ 1, |||f 01,11 (t) -R 00 ||| ≤ 1, |||f 00,10 (t) -R 00 ||| ≤ 1, |||f 01,11 (t) -R 00 ||| ≤ 1.

	Moreover
	|||R 0 -f (t)||| = |||I 3 -R -1 0 f (t)||| = 2 √ = (A.4) 2 sin θ 2 t 2 ≤ 2 √ 2 sin θ 2 2 |||R 10 -R 00 ||| ≤ 1 2 , |||R 01 -R 00 ||| ≤ 1 2 , |||R 11 -R 01 ||| ≤	1 2	,	|||R 11 -R 10 ||| ≤	1 2	.

= R 1 , df dt (L ∞ (0,1)) 9 ≤ π 2 |||R 1 -R 0 ||||, |||R 0f (t)||| ≤ |||R 0 -R 1 |||. Proof One has df dt (L ∞ (0,1)) 9 = √ 2θ 2 ≤ π 2 |||R 2 -I 3 ||| = π 2 |||R 1 -R 0 |||.

  A.2. Let R a be the Q 1 interpolate of the matrices R 00 , R 01 , R 10 and R 11 . There exists a strictly positive constant C such that||R -R a || (L 2 (Y ) 9 ≤ C |||R 10 -R 00 |||| + |||R 11 -R 01 |||| + |||R 01 -R 00 |||| + 11 -R 10 |||| .

Appendix A

In this section the vector space R n×n of all matrices with n rows and n is equipped with the Frobenius norm. We set


Göpfert, Mirco: Policing the Frontier: An Ethnography of Two Worlds in Niger,

'Police/Worlds: Studies in Security, Crime, and Governance' series. 192 pp., Ithaca and London: Cornell University Press 2020, ISBN: 9781501747229.

Mirco Göpfert belongs to a generation of young scholars who, from the mid-2000s, greatly contributed to documenting the routines of state police work in Africa through an in-depth ethnographic methodology. Partly drawing on the 'states at work' research program initiated in the late 1990s by Thomas Bierschenk, Jean-Pierre Olivier de Sardan and Giorgio Blundo (2004;2006;2014), they rediscovered the day-to-day functioning of public policing institutions that had long remained outside the interests of the social sciences. Policing the frontier must be read and appreciated as part of this emerging field of studies in relation to the author's past collaborations with some of these scholars.

Mirco Göpfert's book focuses on the day-to-day interactions between the gendarmes and the general public in Niger and places them within their social, historical and institutional contexts. The discussion is structured around the notion of the frontier, which designates the tension, inherent in police work, between 'bureaucratic form' and social 'life ' (p. 141). Inspired by Veena Daas and Deborah Poole's concept of the 'margins' of the state, the frontier refers to both a space 'between life governed by public institutions and life outside of the state's grasp' (p. 9) and the project to extend this grasp. At the frontier, different moral and legal registers emanating from these two worlds 'collide and merge at the same time' (p. 140). As such, this is a space where uncertainty prevails, where excessive arbitrariness and rigidity can easily arise, but which also favours local arrangements, innovation and creativity. It is on this second aspect of the frontier that the book largely focuses. In that sense, as the author shows, what is at stake in policing the frontier is largely about the application, 'repairing' or suspension of the law. Its reality lies somewhere between the 'state of exception', when cases are reduced to 'bare text' in order to defend the law from the influence of social norms, and the 'exception of the state', when people and situations are reduced to 'bare life' to 'protect them from the law ' (p. 123).

The author proposes to tackle this set of questions by studying the routine work of a fictional gendarmerie brigade located in Godiya, the fictional capital of a rural depart-ment not far from the Nigerien border. The brigade is a paragon of the dozen brigades and administrative units across Niger that Göpfert visited during his eighteen months of fieldwork between 2009 and 2014. Nonetheless, the handful of gendarmes in this brigade and all the very detailed situations in which the demonstration is grounded are all real and come from his different research sites. Even though it helps in guarantee-ing the gendarmes' anonymity, reminding us of the difficulties involved in conducting and publishing research within such an uncertain institutional environment, one can nonetheless regret that such a 'narrative strategy' (p. 14) may also result in erasing the specifics of each local context and their influence on police work.

The book comprises three parts and nine chapters, each discussing a specific dimension of policing (at) the frontier. Chapters two and three complete the introductory part by exploring the historicity of the Nigerien gendarmerie and describing its concrete presence in rural areas. Chapter two combines second-hand sources with archives and first-hand testimonies. It provides important thoughts on the centrality of the 'frontier project' across the gendarmerie's history. Gendarmes and their predecessors first served as a military force to push forward and maintain both the territorial frontiers and, later on, the 'bureaucratic frontier' of the state (p. 35), thereby contributing to the discipline and legibility of the internal population. Purely narrative, chapter three describes some of the central aspects of the gendarmes' uneven presence in rural areas, which appear more salient in the difficult handling of an exceptional murder.

Chapters four, five and six form the second part of the book, which concentrates on the 'bureaucratic form' of the gendarmes' work. Focusing on the gendarmes' 'senses', these chapters reveal crucial aspects of police work in Africa that have so far remained understudied. Chapter four highlights the orality in police work. The gendarmes' 'bureaucratically attuned ear' (p. 60) reorders the stories people tell them into 'criminal' and 'civil' categories and then decide what to do about them. But the process reveals the uncertainty of these legal dichotomies, as it is based on the gendarmes' personal and moral appreciation of the gravity of an offence. Chapter five addresses the problem of gendarmes' access to the knowledge of citizens and crime in what is a vast and largely unknown terrain. Despite recent expenditure on 'high-tech surveillance' to fight terrorism, the state's capacity 'to make society legible' (p. 62) is not only poor but also inaccessible to local gendarmes. Göpfert then suggests that the actual capacity of the state to maintain social order lies greatly in the gendarmes' ability to maintain reciprocal relations with 'knowledge brokers'. The chapter discusses the strategic and moral implications of their relations with personal informants, and the chiefs as influential power-holders, friends and acquaintances. Chapter six focuses on the aesthetical dimension of the gendarmes' writing of 'procès-verbaux', which vertically circulate through the state apparatus and constitute one of the major ways though which their work is evaluated by both their peers and their superiors. He argues that the definition of a positive professional identity, articulated around both bureaucratic and intellectual values, is at stake in the dedication and pride they put into the fabrication of 'beautiful' documents, combining bureaucratic rigour with creativity.

The last three chapters explore the gendarmes' adaptative practices to the expectations of the different audiences they interact with and whom they all try to please: the public, the prosecutors, and their superiors, who answer to the authorities. Chapters seven and eight go more deeply into their day-to-day arrangements with civilians who seek their help to settle their disputes in the form of restorative justice, while avoiding the uncertainties and complexities of the judicial system. Agreeing to handle civil cases (Chapter seven) or to settle criminal charges through informal arrangements (Chapter eight), the gendarmes circumvent the law and adapt to popular conceptions of offences and justice. Apart from their financial interest in 'assisting' the people, their motivation lies in the restoration of 'peace' between the disputants, between themselves and the civilians, and within the larger community (p. 108). Therefore, rather than simply compensating the victim and punishing the offenders, the gendarmes also use their bureaucratic authority and the threat of legal procedure to frame the disputants' mediation and help them establish a consensus within which every party can save face. However, these arrangements, being outside legal procedure, expose the gendarmes to severe backlash if, after closing a case, one dissatisfied party complains to their superiors. Chapter nine therefore addresses the gendarmes' frustration with and adaptation to the uncertainties of their lives and careers, which are at the mercy of both powerful civilians' and their superiors' unpredictable desires and actions. Caught in this 'bureaucratic limbo' (p. 126), they have to maintain patron-client relationships with all of them, in which the distribution of money plays a central part.

In a nutshell, the book provides a very detailed, sensitive and well-written ethnographic account of the day-to-day negotiation of policing between the population, state agents and their superiors in a context in which the state has only a limited capacity to enforce the law. However, its observations are not limited to the Nigerien or African contexts alone. As the author convincingly argues, the frontier runs across the state apparatus as a whole: everywhere in the world, 'bureaucrats work in that frontier space' and have to juggle between the 'abstract norms that guide and sanction their actions' and 'the concreteness of practical reality' (p. 141). For that matter, it makes an important contribution to the study of the state in Africa and beyond.

Through extensive interaction with classical studies of police work in other countries, Mirco Göpfert proves 'what an "anthropological criminology" may have to offer to broader criminological debates about crime, punishment, and the application of the law' (p. 122). To that extent, the promises are fully satisfied. But it is also where the perspective adopted in the book encounters its limits. Admittedly, the author argues that 'this book does not focus on policing as a mode of governance from the outset and certainly not as a means of enforcing unequal social order in the name of public security' but instead 'examines policing primarily as work ' (p. 5). Yet, as it aims at discussing the ambiguous and uneven tensions between the bureaucratic sphere and the social world, I suggest that some of its reflections could have gained from being rooted in a perspective that takes better account of the influence of social stratification and the political dimensions of policing.

First, the public policies that frame the gendarmes' mandate and those who promote them are largely missing. Certainly, considering policing primarily as 'work' greatly helps us depart from the normative vision of African police institutions that had long prevailed in the academic debate: as corrupt, absent from people's lives, politically oriented and mainly repressive. But it also seems to set aside central questions about relations between policing and the political sphere, which can conversely help us understand how the gendarmes perform their work. For example, the analysis of day-to-day policing would have gained from being better connected to the new democratic political context and the rising influence of human rights' groups, briefly evoked in chapter two. Taking into account the new forms of pressure they have exerted on the gendarmes' work and careers could have brought new insights into the handling of criminals, the diversion of criminal cases away from the judicial system, and the gendarmes' relations with their superiors.

Second, while the bureaucratic world is thoroughly analysed, the social world sometimes seems to be treated as a black box: for example, the gendarmes' and their interlocutors' social backgrounds and trajectories remain largely unexplored. Nevertheless, as the sociology of state action has shown, 'popular ideas' of justice (p. 115), as well as the resources people can mobilise in their interactions with the gendarmes, may differ greatly from one social milieu to another. Moreover, as the author suggests, the gendarmes presented in the book form a group of 'idealists', less brutal and 'attuned to the sound of money' than others (p. 143). An analysis of their social trajectories, beyond a brief mention of their shared 'poor rural origins' (p. 112) with some of their interlocutors, could have helped explain their specificity and what pushes them to 'break [and repair] the law to be just' (p. 143).

These, I argue, constitute two avenues for future research that would draw on Mirco Göpfert's and others' critical rediscovery of the daily routine of African state-policing institutions.

Paul Grassin University of Paris 1 Panthéon/Sorbonne Centre Européen de Sociologie et de Science Politique


We are observing the groth of multiple superimposed urban times (fast, slow, instant, settled, sporadic etc.) and their ampli fication due to the reinforcement of the networks : transportation and energy, ones added on older ones, plots of land, water ones. But, without no doubt, the most determining one is the information network. It is becoming as the one that orders the others (emission, reception, feedback), from now one increasing the coexistence of different uses in the same time, in a same place, a sort of use-time-space, that modify the de finition (classical, modern) of the city.

The information is becoming strategical to consider our relationships with the urban reality.

Our hypothesis :

The information is now integrative : this supposes that it orders, manages and regulates all the others things. The main consequence is that it transforms the state and nature of the technical things, and their uses in time and space. To develop this hypothesis, I will based it, mainly here, on the philosopher Gilbert Simondon's studies about technics and mécanologie (Gilbert Simondon 1958, 2005, 2010.). This neologism and the field it supposes, needs to take few minutes to explain. In Gilbert Simondon's thinking, the technical things are not taken off just one scienti fic principle, or separated ones. To think about technical things, it needs to consider the correlations between the scienti fic principals : he names this "mécanologie", as science of correlations and transformations (Gilbert Simondon 1958.) Gilbert Simondon calls "allagmatique générale" (Gilbert Simondon 1958, 2005.) the theory of exchanges and changes of states in technical things. The cybernetics ( Norbert Wiener 1965.) is, may be, an equivalent of "allagmatique générale" as an inter-technics science (Gilbert Simondon 1958.) It's interesting to know that the word "mécanologie" comes from an architect, Jacques La fitte, who de fines it as science of machines. (Jacques La fitte, 1932.) Gilbert Simondon refering Jacques La fitte's studies, makes out 3 steps of mécanologie : the first one is the level of passive machines, traying to maintain a permanent balance. These machines improve adaptations and a re-organisation of the MILIEU (Umwelt, Uekhüll 1956.), to keep stability. For Jacques La fitte, the architecture and the buildings are part of this group. Traditionally the aim of construction is to maintain the stability of the edi fice. The adaptations are, for example, the footings, the flying buttresses, the voult, etc. They are some tertium quid holding a dialogue between the structure and the MILIEU. On the second level are the active machines : these ones are assemblies of tools. These machines, through the sciences progress, develop auto-correlation, control of energy, an ideal of reversibility of transformation to maximize output of engines (Gilbert Simondon 2005). This supposes the negative feedback as capability of control (Wiener 1965.) and in the same time, a disregard to the question of balance. So, this second generation of machines is indifferent to the question of MILIEU, out of extraction of matter and energy (in environment). The indistinction of energy canal and information canal characterises it. The main aim is power. The engines based on Thermodynamics are main part of the category.

The third category is opening a new perspective : the one of re flex machines, with the importance of reliability of signal, transmission of ampli fied signals, so positive feedback, for a continious research of metastable state. These machines are technical sets of small sizes for more reliability, for a best informational pro fitability. Their ability to memorize offers a strategical opportunity in the relationship humanbeeing-machine. More than a set of machines, it is a network of machines, able to communicate each others (producing, receiving, treating, storing informations) but also with the MILIEU. So that, this network is co-extendible to the world : the MILIEU is "technicaled "-integrated in this technical (or technological) informational network. Here, the communication is not only exchanged and relayed, treated and used device, it is the origin, the basis of the organization. From now, it is constituting the apparatus, as beginning and end of the process, element and all. It's why we qualify the today information as integrative one. It is the main apparatus.

We situate our today research and presentation in this context. This observation has obviously a great importance on time and space -as insubstantial, information brakes the traditional-modern relationship between space and time based ont the supremacy of the matter and energy. As we said (E.Mortamais 2005) the time is becoming first, braking the modern space hegemony. The today question is : do architecture, landscape design and urban design may really be transformed to belong to the third phase of this genealogy ? Gilbert Simondon, himself foresees an open when he says, precisely about architecture, that the postulate of immobility of its products implies the irreversibility of its genesis and of its damaging. For him, it is the cultural barrier (the one between property and mobile elements) that stops the evolution from the first to the third phase. In fact, he thinks, the passive machine tends towards active ones, and the active one to the third one (complex interactive one). He proposes one clue : "Si les constructions devenaient déplaçables, et démontables, elles auraient les principaux caractères des machines actives qui (…) admettent des réparations, des échanges de pièces." (Gilbert Simondon 2005) ["If the contructions could be movable, demantled, they would get the main characteristics of the active machines (…) able to be repaired, and accepting new exchanging pieces"] This could have been a de finition for some representative projects : for example Buckminster Fuller's (Dymaxion Houses, 4D House etc.) as soon as 1928, or Cedric Price's (Fun Palace, Generator, etc.) as soon as 1960.

We suggest here an other open to jump directly toward the third phase, de fining urban reality as a network of machines, able to communicate each other, and also with the MILIEU (producing, receiving, treating, storing and exchanging informations). It is rather easy making this experience of thinking : regarding the today urban phenomenon and its concrete situation as a complex apparatus, made of stable matters and objects, mobile ones, natural and arti ficial elements, animals, people, users in constant interacting and co-acting, deliberatly or not. In this context, the urban reality itself can be considered as a network of interactive machines (including leaving creatures), able to communicate each other.

The following project is a practical application of this proposition. It is named sharing street.

Short description :

We base it on the fact that the today main occupations of the street are due to the cars and trucks and buses, all vehicules of transportation. In this way, the vocation of the street as public space is neutralized. However, the today citizens expect and try to revive numerous uses of it, for example second-hand trades, street-festivals, kids' activities, societies' meetings, neighbours' lunches etc. Different public or private amenities could take place, providing liveliness, attractiveness to ordinary streets. All these activities could be coordinated according to needs, opportunities, suitabilities, offers and demands, con fines etc. They can be planned for few hours in a day, regularly, or exceptionally, on several days, on a week, few days per month or year etc. Providing an intense use of the public space ; the concept of intensity considered more interesting than just density : implying quality and not only quantity.

Imagine the street on a Monday of September : it is open for cars, buses, vans, small trucks during a part of the morning for example from 5.00 am to 9.30 am. Then a second hand book-show takes place. At 12.00 one way of the street is opened and the show is closing. At 4.00 pm the pavement is given to the kids going out from school : to celebrate autumn start of school, the teachers' staff organize a snack for kids, and the parents are coming with cakes and soft drinks to meat and discuss. At 6.00 pm the street gets back to its classical and functional vocation. Imagine it on a Sunday of April : Spring is just coming. Societies of gardening are organising a festival to promote their skills and share plants and seeds1 . Shopkeepers are participating and exceptionally open their stores. All the year long, every Friday afternoon, the city government opens a street-of fice with computers to help people in their administrative procedures. In the same time, several societies are present to support illiterate foreigners in these procedures and popularise their actions. Buses and cars are allowed to move very slowely in one direction, on one way. We can combine a lot of other setups... The best required needs are co-actors, sharing and creating datas. The first ones, of course, are the city-government and residents : their agreement and participation are vital. So is the associative web to lead the place also. Right now, we are discussing with some of these different groups to build the experience . Other co-actors are drivers, shopkeepers, transportation compagnies, and all the ones liable to use the street as public space. It's why we call them co-actors. The project does not need heavy and big devices, as we could imagine only seven or eight years ago. Actually, since few years, smart-phones modify the situation and technological context. Speci fic applications are able to produce, give and share informations on time, instantaneously. The users can participate in the management, just connecting with the application to get information. When they get it, as positive feedback, they also give some more. About the traf fic : we are conscious that the sharing street (as any other street and any infrastructure more generally) is a part of a complexe web, and the different uses need to be correlated with the traf fic all around the place. The drivers' smart-phones supply the information about the quantity of cars, their ways, and may inform about their destinations (these datas are a real stakes today in this field). Here also, the devoted application we talk about above, can provide the information about the position of the sharing street, on real time. So that the drivers could be guided, according to the actual position and occupation of the street. One ask for a car park ? Via the application, he gets the good information to reserve, and pay his place. Doing that, he participates in the street-administration. A society is organizing a second-hand trade ? So, motivated citizens book a place ; rent it via their smart-phone and the information is immediately transfered in the data basis, to be available for everybody : customers, society, transportation company, drivers. The street as space, times and uses, becomes in the center of free datas growing and modifying on time, according with the participation of people. It's why users and informational machines become co-actors, getting, delivering, creating, and sharing informations. The technical project is also currently in progress : technical studies with private companies' engineers and developers. Now let us try to link this description with the first part of this communication.

In the present project, the "object" of design, of management, use, and observation, is not any more a set of fixed elements (buildings, streets, squares and gardens) as in neither the first step, nor the second one, of the Gilbert Simondon's categories of mécanologie. In the first phase, all the urban elements try to keep their stability, independently from the others. The main goal is to secure adaptation to the conditions : at the city scale, the different parts (buildings, streets, squares and gardens) are set to be stable and perennial, according to the context : gravity, topography, orientation, wind, rain, natural contingencies in general, and people actions and damages (for example against wars : the forti fications as construction to secure the urban activities and humanbeeings' life). In the second phase, the urban concept is changing to become a frame, a support for industry and trade, mainly depending on energy and movements. It became a sort of big machine, promoted and developed as it, by the main modern urban designers. The real strategical items were not the buildings but the infrastructures, the vehicules, the engines, the machines, they were supporting. The lifts order the size of buildings, the cars order the width of streets, the escalators and the cars order the shopping malls etc. Electricity, gas, petrol transformed the urban set in a huge thermodynamic set of pipes, supplying the engines of production. The concept of tabula rasa is completely coherent with this state of mind. The negative feed-back implies if necessary, to come back to the starting point, to reboot the machine.

As described above, the principles developed with the sharing street take place clearly in the third phase context. How ? The project supposes to build an informative network, connecting very speci fic elements : movements of vehicules and people, daily urban needs, sporadic events, common goods and amenities, private requirements and groups' wishes, uses and places, people and matters. This apparent heterogeneity falls down with the today common langage of information and communication. More than that, if we consider each of these components as a strategical provider of signals. Every one is a generator, or a part of a big generator, constantly evolving, muting according to needs and situations ; a member of a complete organism receiving, treating, producing, sharing, recording information. The calculation of relationships between these elements and the Milieu, and the diffusion of the results, in real time to everyone, allow to answer to the different states to keep the all system in good metastable condition. "...chaque objet ou chaque société d'objets comporte des éléments qui s'envoient les uns aux autres des informations, ou plus exactement qui sont dans une relation telle que leurs structures et leurs fonctionnements deviennent compossibles les uns par les autres... " [...every objets or sets of objets contain elements that send informations to the others, or more than that which are in relationship allowing their structures and workings able to be com-possible each others..." ] (Gilbert Simondon, 2010) Today, the internet of things is literally the tool of translation of this : all the territory becomes a big data shelf, and a big data itself. We are able to use it for more urbanity, to build new links and new uses.

We are realizing that the positive feedback implies some added thing : an ampli fication, a possible evolution, a continual adaptation, cause the process supposes creating informations, so new conditions, new signals, new answers etc. Following Gilbert Simondon's theory about the third generation of technical objects, we have planned the sharing street as a complex network which makes itself possible : the instantaneous, and constant coordination between the different elements becomes the condition of the existence of every one, a complete auto-correlation to keep the set in meta-stability. An amplification is a really good perspective. In this optimistic context, the urban reality is not condemned just to maintain concrete structures face to entropy as in the first generation, or to fall down, worn out, against thermodynamic yoke, the future of the second generation. The dependence to modernist codes is overrun to think and design urban concepts. The separation of uses, the separation of time and space in the pure rational modernist vision is obsolete and can not answer to the challenges of sustainability and intensity. The integrative information as our today technicological context, allows to answer to ecological aims in urban field, considering ecology as a complex system of interacting elements always in metastability. This project is a sample of very interesting perspective : the coordinated actions of the sharing street create and maintain it, as an ecosystem lives, according to its members and their communication. The sustainability of the all system (including all the factors we described above) is due to its faculties to stay and evoluate, through sharing and communication. Contemporaneous concept of ecology is wholly associated with the question of communication.

  information and communication (France) Habilitation à diriger des recherches HDR (France) member of EVCAU laboratory -PARIS VAL DE SEINE associated member ICT laboratory -(PARIS 7 -DENIS DIDEROT) Sharing street Composing multiple uses times and spaces in the today communication phase of urban evolution.

  Short selected bibliography : HUEXKÜLL Jakob Von, Streifzüge durch die Umwelten von Tieren und Menschen, 1956. LAFITTE Jacques, Ré flexions sur la science des machines 1911/1932, Librairie Bloud & Gay, 1932. RIFKIN Jeremy, The age of Access, Tacher /Putman, 2000. SIMONDON Gilbert, Communication et information, cours et conférences, La Transparence, 2010. SIMONDON Gilbert, L'invention dans les techniques, cours et conférences, Seuil, 2005. SIMONDON Gilbert, Du monde d'existence des objets techniques, Aubier, 1958. WIENER Norbert, Cybernetics or Control and Communication in the Animal and the Machine, MIT Press, 1965. Le temps des hypermedias dans l'architecture, prospective. Doctorate thesis. 2005.

	Periodical :
	Volume, n°28, The internet of things.
	Thesis :
	MORTAMAIS Elizabeth,

Elizabeth Mortamais, PHYTOBUL', for Autolib', awarded project,

ELIZABETH MORTAMAIS -HYPERURBAIN5 -Florence Mai 2015 -Sharing street. -4