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

Georges Griso
email: griso@ann.jussieu.fr

Decomposition of the deformations of a thin shell. Nonlinear elastic models

Keywords: shells, Korn's inequality, nonlinear elasticity, Γ-convergence. 2000 MSC: 74B20, 74K20, 74K25, 49J45

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 justify a nonlinear bending model for elastic shells for an elastic energy of order δ 3 .

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 type inequality for shells. In a second part of the paper, we use such a decomposition to derive a nonlinear elastic shell model.

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] by Friesecke, James and Müller 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.

In the above expression, the fields V and R are defined on S, s 3 is the variable in the direction n which is a unit vector field normal to S and 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 "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 | Junction of a periodic family of elastic rods with a thin plate[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. Using Poincaré inequality as done in [START_REF] Friesecke | Derivation of nonlinear bending theory for shells from three-dimensionnal nonlinear elasticity by Gamma convergence[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[)
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

∇ x v+(∇ x v) T -2I 3 (L 2 (S×]-δ,δ[)) 9 ≤ C||dist(∇ x v, SO(3))|| L 2 (S×]-δ,δ[) 1+ 1 δ 5/2 ||dist(∇ x v, SO(3))|| L 2 (S×]-δ,δ[)
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].

In the present paper we focus on the case where ||dist(∇ x v δ , SO(3))|| L 2 (S×]-δ,δ[) ∼ δ 3/2 which is the highest level of energy which can be analyzed through our technique. The lower level of energies will be studied in a forthcoming paper.

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 the second part of the paper, we strongly use the results of the first part in order to derive limit 2D shells models. As a general reference on elasticity theory we refer to [START_REF] Ciarlet | Mathematical Elasticity[END_REF] and we start from a total energy W (∇v)f • v where W is the local elastic energy. We assume that

W (∇v) = W (∇v) T ∇v -I 3 if det(∇v) > 0, +∞ if det(∇v) ≤ 0.
The assumptions on the function W are similar to those adopted in [START_REF] Friesecke | A theorem on geometric rigidity and the derivation of nonlinear plate theory from the three-dimensional elasticity[END_REF]. We assume that W is continuous from S 3 (the space of symmetric 3 × 3 matrices) into R and that

∃c > 0 such that ∀E ∈ S 3 W (E) ≥ c|||E||| 2 , ∀ε > 0, ∃θ > 0, such that ∀E ∈ S 3 |||E||| ≤ θ =⇒ |W (E) -Q(E)| ≤ ε|||E||| 2 ,
where Q is a positive quadratic form. Recall that W (∇v) ≥ c(dist(∇v, SO(3)) 2 .

Using the nonlinear Korn's inequality for shells mentioned above, we are in a position to scale the applied forces f in order to obtain a given order of total energy (with respect to δ) . Then we deduce the order of the quantity ||dist(∇ x v δ , SO(3))|| L 2 (S×]-δ,δ[) with respect to the scaling of the forces in the general case. In the following, we choose the applied forces so that ||dist(∇ x v δ , SO(3))|| L 2 (S×]-δ,δ[) ∼ δ 3/2 . At last we derive the "limit" energy as δ goes to 0 using a Γ-limit procedure on sequences of fields V δ , R δ and v δ . Through a possible elimination of v in the Γ-limit energy, we finally obtain a minimization problem for the mean deformation V and the rotation R under a constraint between ∇V et R.

As general references on the theory of nonlinear elasticity, we refer to [START_REF] Ball | Convexity conditions and existence theorems in nonlinear elasticity[END_REF], [START_REF] Ciarlet | Mathematical Elasticity[END_REF] and [START_REF] Marsden | Mathematical Foundations of Elasticity[END_REF] and to the extensive bibliographies of these works. For the justification of plates or shell models in nonlinear elasticity 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]. A general introduction of Γ-convergence can be found in [START_REF] Maso | An Introduction to Γ-convergence[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 plate[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 strain tensor of a sequence of deformations such that ||dist(∇ x v δ , SO(3))|| L 2 (S×]-δ,δ[) ∼ δ 3/2 in Section 6. In Section 7 we consider nonlinear elastic shells and we use the results of the proceeding sections to scale the applied forces in order to obtain a priori estimates on the deformation. In Section 8 is devoted to derive the Γ-limit for energies of order δ 3/2 . At last two appendices contain a few technical results on the interpolation of rotations and an algebraic elimination for quadratic forms.

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 that 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 e.g. 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 first 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 )

3
which remains injective and such that the 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 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

′ 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 made of a finite number of connected components (whose closure are disjoint) and v be a deformation such that

v = I d on Γ 0,δ = Φ(γ 0 ×] -δ, 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 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 it 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 | Junction of a periodic family of elastic rods with a thin plate[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 of rotations R a . 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 it leads

(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

(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( 3)

and v belongs to H 1 (Q δ ) 3 .
The first term V is always 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 and 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.

Indeed these conditions depend on the boundary condition on the field v. We discuss essentially the usual case of a clamped condition on the part of the lateral boundary of Q δ . Let γ 0 as in Subsection 3.2, and recall that

Γ 0,δ = Φ(γ 0 ×] -δ, δ[).
We assume that

v(x) = x on Γ 0,δ .
Due to the definition (3.3) of V, we first 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 decompositions 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 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: Lemma 4.1. In Theorem 3.3 (respectively in Theorem 3.4), we can choose R a (resp. R) such that R a = I 3 on γ 0 , (resp. R = I 3 on γ 0 ), without modifications in the estimates of these theorems.

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 .
which gives with (3.2)

(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. 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.

       ||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 Korn's 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 [START_REF] Ciarlet | Un modèle bi-dimentionnel non linéaire de coques analogue à celui de W.T. Koiter[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)

(4.11) ||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 the previous estimate 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 ) (which will be a bound from below of the elastic energy)

||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 3/2 ).

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 (4.8) 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 ).

Limit behavior of the deformation for

||dist(∇ x v, SO(3))|| L 2 (Q δ ) ∼ δ 3/2
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.

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])

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)) 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 (R a,δ -R a,δ ) ⇀ 1 2 ∂R ∂s 1 + ∂R ∂s 2 weakly in (L 2 (ω) 3×3 .
Now 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 strain tensor of the sequence v δ .

Corollary 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 A α ∈ L 2 (ω; R 3×3 ) such that ∂R ∂s α = RA α . Since there exists also 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)
.

With the same notation as in the proof of Corollary 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
.

Nonlinear elastic shells

In this section we consider a shell made of an elastic material. Its thickness 2δ is fixed and belongs to ]0, 2δ 0 ]. The local energy W : S 3 -→ R + is a continuous function of symmetric matrices which satisfies the following assumptions which are similar to those adopted 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] and [START_REF] Friesecke | Derivation of nonlinear bending theory for shells from three-dimensionnal nonlinear elasticity by Gamma convergence[END_REF] (the reader is also referred to [START_REF] Ciarlet | Mathematical Elasticity[END_REF] for general introduction to elasticity)

∃c > 0 such that ∀E ∈ S 3 W (E) ≥ c|||E||| 2 , (7.1) ∀ε > 0, ∃θ > 0, such that ∀E ∈ S 3 |||E||| ≤ θ =⇒ |W (E) -Q(E)| ≤ ε|||E||| 2 , (7.2)
where Q is a positive quadratic form defined on the set of 3 × 3 symmetric matrices. Remark that Q satisfies (7.1) with the same constant c. From (7.2) and the continuity of W we deduce that

(7.3) ∀C 0 > 0, ∃C 1 > 0, such that ∀E ∈ S 3 |||E||| ≤ C 0 =⇒ |W (E)| ≤ C 1 |||E|||.
Still following [START_REF] Ciarlet | Mathematical Elasticity[END_REF], for any 3 × 3 matrix F , we set

(7.4) W (F ) =    W 1 2 (F T F -I 3 ) if det(F ) > 0 + ∞ if det(F ) ≤ 0.
Remark that due to (7.1), (7.4) and to the inequality

|||F T F -I 3 ||| ≥ dist(F, SO(3)) if det(F ) > 0, we have (7.5) W (F ) ≥ c 4 dist(F, SO(3)) 2
for any matrix F .

Remark 7.1. As a classical example of a local elastic energy satisfying the above assumptions, we mention the following St Venant-Kirchhoff's law (see [START_REF] Ciarlet | Mathematical Elasticity[END_REF], [START_REF] Ciarlet | A justification of a nonlinear model in plate theory[END_REF]) for which

W (F ) =    λ 8 tr(F T F -I 3 ) 2 + µ 4 tr (F T F -I 3 ) 2 if det(F ) > 0 + ∞ if det(F ) ≤ 0.
In order to take into account the boundary condition on the admissible deformations we introduce the space (7.6)

U δ = v ∈ (H 1 (Q δ )) 3 | v = I d on Γ 0,δ .
Now we assume that the shell is submitted to applied volume forces f δ ∈ (L 2 (Ω δ )) 3 and we define the total energy J(v) over U δ by (7.7)

J(v) = Q δ W (∇ x v)(x)dx - Q δ f δ (x) • v(x)dx.
To introduce the scaling on f δ , let us consider f in (L 2 (ω)) 3 and g in (L 2 (Ω)) 3 such that (7.8)

1 -1 g(s 1 , s 2 , S 3 )dS 3 = 0 for a.e. (s 1 , s 2 ) ∈ ω.
Let κ ≥ 1 and assume that the force f δ is given for x = Φ(s) by (7.9)

f δ (x) = δ κ f (s 1 , s 2 ) + δ κ-1 g s 1 , s 2 , s 3 δ for a.e. x ∈ Q δ .
The fact remains that to find a minimizer or to find a deformation that approaches the minimizer of J(v) or of J(v) -J(I d is the same. Let v be in U δ , thanks to (2.3), (4.6), (4.7), (7.8) and (7.9), we obtain (7.10)

Q δ f δ (x) • (v -I d )(x)dx ≤ Cδ κ-1/2 (||f (L 2 (ω)) 3 + ||g|| (L 2 (Ω)) 3 )||dist(∇ x v, SO(3))|| L 2 (Q δ ) .
In general, a minimizer of J does not exist on U δ . In what follows, we will investigate the behavior of the functional 1 δ 2κ-1 J(v) -J(I d ) using Γ-convergence (see Remark 7.2 for a few arguments which justify the study of this quantity). Hence, we consider a deformation v of U δ such that

(7.11) 1 δ 2κ-1 J(v) -J(I d ) ≤ C 1
where C 1 does not depend on δ and v. Using (7.5) and (7.10) we obtain

C||dist(∇ x v, SO(3))|| 2 L 2 (Q δ ) -Cδ κ-1/2 (||f (L 2 (ω)) 3 + ||g|| (L 2 (Ω)) 3 )||dist(∇ x v, SO(3))|| L 2 (Q δ ) ≤ C 1 δ 2κ-1 .
Hence, we have

(7.12) ||dist(∇ x v, SO(3))|| L 2 (Q δ ) ≤ Cδ κ-1/2
where the constant C depends on the sum ||f || (L 2 (ω)) 3 + ||g|| (L 2 (Ω)) 3 and of C 1 .

In the spirit of what follows the proof of Theorem 4.1, let us notice that if one uses (4.3) to estimate the contribution of the forces in the energy, one obtains

||dist(∇ x v, SO(3))|| L 2 (Q δ ) ≤ Cδ (1+κ)/2 .
Indeed, the above estimate is the same as (7.12) for forces of order δ 2 , which is, as far as we know, a critical case above which no asymptotic studies were achieved. For smaller forces, estimate (7.12) is much better and this shows that for smaller energies our technique of decomposition which lead to the second Korn's inequality given in Theorem 4.1 is more relevant in order to justify the limit models for κ > 2.

From the assumption (7.9) on the applied forces and the estimate (7.12), the results of Sections 3 and 4 permit to obtain estimates of V, R a , v a and ∇ x v -R a with respect to δ. Still for the deformation v ∈ U δ satisfying (7.11), we have using (7.4) and (7.5)

c 4 ||(∇ x v) T ∇ x v -I 3 || 2 (L 2 (Q δ )) 3×3 ≤ J(v) -J(I d ) + Q δ f δ • (v -I d ) ≤ C 1 δ 2κ-1 + Cδ κ-1/2 (||f || (L 2 (ω)) 3 + ||g|| (L 2 (Ω)) 3 )||dist(∇ x v, SO(3))|| L 2 (Q δ ) .
Due to (7.12) we obtain the following estimate of the Green-St Venant's tensor:

(7.13) 1 2 (∇ x v) T ∇ x v -I 3 (L 2 (Q δ )) 3×3 ≤ Cδ κ-1/2 .
We first deduce from the above inequality that v ∈ (W 1,4 (Q δ )) 3 and moreover since κ ≥ 1

(7.14) ||∇ x v|| (L 4 (Q δ )) 3×3 ≤ Cδ 1 4 .
Furthermore, there exists two strictly positive constants c and C which does not depend on δ such that for any v ∈ U δ satisfying (7.11) we have

(7.15) -cδ 2κ-1 ≤ J(v) -J(I d ) ≤ Cδ 2κ-1 .
We set (7.16)

m δ,κ = inf v∈U δ J(v) -J(I d ) .
a consequence of the inequality in (7.15) we have

(7.17) -c ≤ m δ,κ δ 2κ-1 ≤ 0.
We denote (7.18)

m κ = lim δ→0 m δ,κ δ 2κ-1 .
Remark 7.2. Under assumption (7.8) and (7.9) on the forces, the quantity J(I d ) is of order δ κ+1 . Hence if κ > 2 then due to (7.15) the infimum of J(v) is of order δ κ+1 . As a consequence for κ > 2, the relevant quantity which describes the limit behavior is

J(v δ ) -J(I d ) δ 2κ-1
. For κ = 2, J(I d ) and the infimum of J(v) have the same order δ 3 and we derive the Γ-limit of the energy J(v δ ) δ 3 .

Limit model in the case κ = 2

In this section we derive the Γ-limit of the functional J(v δ ) δ 3 which corresponds to the case κ = 2. We begin with the lim-inf condition.

Let v δ 0<δ≤δ0 be a sequence of deformations belonging to U δ and such that

(8.1) lim δ→0 J(v δ ) δ 3 < +∞.
Upon extracting a subsequence (still indexed by δ) we can assume that the sequence (v δ ) satisfies the condition (7.11) (recall that J(I d ) is of order δ 3 ). From the estimates of the previous section we obtain (8.2)

         ||dist(∇ x v δ , SO(3))|| L 2 (Q δ ) ≤ Cδ 3/2 , 1 2 ∇ x v T δ ∇ x v δ -I 3 (L 2 (Q δ )) 3×3 ≤ Cδ 3/2 , ||∇ x v δ || (L 4 (Q δ )) 3×3 ≤ Cδ 1/4
For any fixed δ ∈ (0, δ 0 ], the deformation v δ is decomposed following (3.6) such a way that Theorem 3.3 is satisfied. There exists a subsequence still indexed by δ such that (see Section 7)

(8.3)                  V δ -→ V strongly in H 1 (ω) 3 R a,δ ⇀ R weakly in H 1 (ω) 3×3 , 1 δ ∂V δ ∂s α -R a,δ t α ⇀ Z α weakly in L 2 (ω) 3 , 1 δ 2 Π δ v a,δ ⇀ v weakly in L 2 (ω; H 1 (-1, 1)) 3 where R(s 1 , s 2 ) belongs to SO(3) for a.e. (s 1 , s 2 ) ∈ ω, V ∈ H 2 (ω) 3 together with (8.4) V = φ, R = I 3 , on γ 0 , and
∂V ∂s α = Rt α .
Furthermore, we also have (see (6.5), (6.9) and estimate (8.2)) (8.5)

         Π δ v δ -→ V strongly in H 1 (Ω) 3 , Π δ (∇ x v δ ) -→ R strongly in L 2 (Ω) 3×3 . 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 2 (Ω)) 9 , where

E =        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       
Remark that, due to the decomposition (3.6), the convergences (8.3) and (8.5) imply that

(8.6) Π δ (v δ -V δ ) δ -→ S 3 (R -I 3 )n strongly in L 2 (Ω) 3 .
Now, recall that (8.7)

       J(v δ ) δ 3 = Ω 1 δ 2 W 1 2 Π δ (∇ x v δ ) T ∇ x v δ -I 3 |Π δ det(∇Φ)| - Ω f • Π δ v δ |Π δ det(∇Φ)| - Ω g • Π δ v δ δ |Π δ det(∇Φ)|
In order to pass to the lim-inf in (8.7) we first recall that that det(∇Φ) = det(t We now consider the first term of the right hand side. Let ε > 0 be fixed. Due to (7.2), there exists θ > 0 such that

(8.8) ∀E ∈ 3 , |||E||| ≤ θ, W (E) ≥ Q(E) -ε|||E||| 2 .
We now use a similar argument given in [START_REF] Blanchard | Decomposition of deformations of thin rods. Application to nonlinear elasticity[END_REF]. Let us denote by χ θ δ the characteristic function of the set

A θ δ = {s ∈ Ω; |||Π δ (∇ x v δ ) T ∇ x v δ -I 3 (s)||| ≥ θ}. Due to (8.
2), we have

(8.9) meas(A θ δ ) ≤ C δ 2 θ 2 .
Using the positive character of W , (8.2) and (8.8) give

Ω 1 δ 2 W Π δ (∇ x v δ ) |Π δ det(∇Φ)| ≥ Ω 1 δ 2 W 1 2 Π δ (∇ x v δ ) T ∇ x v δ -I 3 (1 -χ θ δ )Π δ det(∇Φ) ≥ Ω Q 1 2δ Π δ (∇ x v δ ) T ∇ x v δ -I 3 (1 -χ θ δ ) Π δ det(∇Φ) -Cε
In view of (8.9), the function χ θ δ converges a.e. to 0 as δ tends to 0 while the weak limit of

1 2δ Π δ (∇ x v δ ) T ∇ x v δ - I 3 (1-χ θ δ
) is given by (8.5). As a consequence and also using the convergence of Π δ det(∇Φ) obtained above, we have

lim δ→0 Ω 1 δ 2 W Π δ (∇ x v δ ) Π δ det(∇Φ) ≥ Ω Q (t 1 | t 2 | n) -T E(t 1 | t 2 | n) -1 det(t 1 |t 2 |n) -Cε.
As ε is arbitrary, this gives (8.10) lim

δ→0 Ω 1 δ 2 W Π δ (∇ x v δ ) Π δ det(∇Φ) ≥ Ω Q (t 1 | t 2 | n) -T E(t 1 | t 2 | n) -1 det(t 1 |t 2 |n).
Using the convergences (8.5) and (8.6), it follows that

lim δ→0 Ω f • Π δ (v δ )Π δ det(∇Φ) + Ω g • Π δ (v δ ) δ Π δ det(∇Φ) = L(V, R)
where L is defined by

(8.11)          L(V, R) = ω 2f det(t 1 |t 2 |n) + 1 -1 g(., ., S 3 )S 3 det ∂n ∂s 1 |t 2 | n + det t 1 | ∂n ∂s 2 | n dS 3 • V + ω 1 -1 g(., ., S 3 )S 3 dS 3 • R n det(t 1 |t 2 |n).
From (8.7), (8.10) and the above limit, we conclude that (8.12) lim δ→0

J(v δ ) δ 3 ≥ Ω Q (t 1 | t 2 | n) -T E(t 1 | t 2 | n) -1 det(t 1 |t 2 |n) -L(V, R).
In order to bound from below the right hand side of (8.12), we first write

E =        S 3 ∂R ∂s 1 n • Rt 1 + Z 11 S 3 ∂R ∂s 1 n • Rt 2 + Z 12 1 2 ∂v ∂S 3 • Rt 1 * S 3 ∂R ∂s 2 n • Rt 2 + Z 22 1 2 ∂v ∂S 3 • Rt 2 * * ∂v ∂S 3 • Rn       
where v is defined by

v • Rt α = v • Rt α + S 3 Z α • Rn v • Rn = v • Rn
and where

Z αβ = 1 2 Z α • Rt β + Z β • Rt α .
Now for almost any fixed (s 1 , s 2 ) in ω, we apply Lemma B.1 of the appendix to the quadratic form

1 -1 Q (t 1 | t 2 | n) -T E(t 1 | t 2 | n) -1 det(t 1 |t 2 |n)dS 3 = 1 -1 A        S 3 a 1 + b 1 S 3 a 2 + b 2 S 3 a 3 + b 3 c 1 (S 3 ) c 2 (S 3 ) c 3 (S 3 )        •        S 3 a 1 + b 1 S 3 a 2 + b 2 S 3 a 3 + b 3 c 1 (S 3 ) c 2 (S 3 ) c 3 (S 3 )        dS 3
where A is a symmetric positive definite constant 6 × 6 matrix

A =    A 1 . . . A 2 • • • • • • A T 2 . . . A 3   
and where a = ∂R ∂s

1 n • Rt 1 , ∂R ∂s 1 n • Rt 2 , ∂R ∂s 2 n • Rt 2 , b = Z 11 , Z 12 , Z 22 and c = 1 2 ∂v ∂S 3 • Rt 1 , 1 2 ∂v ∂S 3 • Rt 2 , ∂v ∂S 3 • Rn . Then, if we define the function w ∈ (L 2 (ω; H 1 (-1, 1))) 3 by (8.13) w = v + S 3 A -1 3 A T 2 (Z 11 e 1 + Z 12 e 2 + Z 22 e 3 )
we have

Ω Q (t 1 | t 2 | n) -T E(t 1 | t 2 | n) -1 det(t 1 |t 2 |n) ≥ Ω Q (t 1 | t 2 | n) -T E(t 1 | t 2 | n) -1 det(t 1 |t 2 |n)
where

(8.14) E =        S 3 ∂R ∂s 1 n • Rt 1 S 3 ∂R ∂s 1 n • Rt 2 1 2 ∂w ∂S 3 • Rt 1 * S 3 ∂R ∂s 2 n • Rt 2 1 2 ∂w ∂S 3 • Rt 2 * * ∂w ∂S 3 • Rn       
Moreover since w is defined up to a function of (s 1 , s 2 ), we can assume that

1 -1 w(s 1 , s 2 , S 3 )dS 3 = 0 for almost any (s 1 , s 2 ) in ω.
Let U nlin be the set

U nlin = V ′ , R ′ , w ′ ∈ (H 2 (ω)) 3 × (H 1 (ω)) 3×3 × (L 2 (ω; H 1 ((-1, 1)))) 3 | V ′ = φ, R ′ = I 3 , on γ 0 , 1 -1 w ′ (s 1 , s 2 , S 3 )dS 3 = 0 for a.e. (s 1 , s 2 ) ∈ ω, R ′ (s 1 , s 2 ) ∈ SO(3) for a.e. (s 1 , s 2 ) ∈ ω, ∂V ′ ∂s α = R ′ t α .
The set U nlin is closed in the product space. For any V ′ , R ′ , v ′ ∈ U nlin , we denote by J N L the following limit energy (8.15)

J N L V ′ , R ′ , w ′ = Ω Q (t 1 | t 2 | n) -T E ′ (t 1 | t 2 | n) -1 det(t 1 |t 2 |n) -L(V ′ , R ′ )
where (8.16)

E ′ =         S 3 ∂R ′ ∂s 1 n • R ′ t 1 S 3 ∂R ′ ∂s 1 n • R ′ t 2 1 2 ∂w ′ ∂S 3 • R ′ t 1 * S 3 ∂R ′ ∂s 2 n • R ′ t 2 1 2 ∂w ′ ∂S 3 • R ′ t 2 * * ∂w ′ ∂S 3 • R ′ n        
Indeed notice that in U nlin and J N L , the matrix R ′ could be eliminated using the relation

∂V ′ ∂s α = R ′ t α
and the fact that R ′ ∈ SO(3). Doing such a elimination would lead to an intricate expression of the strain tensor E ′ and this is why we prefer to work with the two fields V ′ and R ′ and to keep the constraint in the definition of U nlin . (8.12)-(8.15) we deduce that (8.17)

J N L V, R, w ≤ lim δ→0 J(v δ ) δ 3 ,
which gives the lim-inf condition in the definition of the Γ-convergence.

In order to obtain the lim-sup condition for the Γ-limit, we first prove the following Lemma.

Lemma 8.1. Let V, R, w be in U nlin , there exists a sequence

V δ , R δ , w δ δ>0 of (W 2,∞ (ω)) 3 × (W 1,∞ (ω)) 3×3 × (W 1,∞ (Ω)) 3 such that (8.18) V δ = φ, R δ = I 3 on γ 0 , w δ = 0, on γ 0 ×] -1, 1[, with (8.19)
                               V δ -→ V strongly in (H 2 (ω)) 3 R δ -→ R strongly in (H 1 (ω)) 3×3 1 δ (R δ -R) -→ 0 strongly in (L 2 (ω)) 3×3 1 δ ∂V δ ∂s α -R δ t α -→ 0 strongly in (L 2 (ω)) 3 w δ -→ w strongly in (L 2 (ω; H 1 ((-1, 1))) 3 , δ ∂w δ ∂s α -→ 0 strongly in (L 2 (Ω)) 3 ,
and moreover

(8.20)        ||dist R δ , SO(3) || L ∞ (ω) ≤ 1 8 , ∂V δ ∂s α -R δ t α (L ∞ (ω)) 3 ≤ 1 8 , ||R δ || 2 (W 1,∞ (ω)) 3×3 + ||w δ || 2 ((W 1,∞ (Ω)) 3 ≤ 1 (4c ′ 1 δ) 2 . The constant c ′ 1 is given by (2.3). Proof. For h > 0 small enough, consider a C ∞ 0 (R 2 )-function ψ h such that 0 ≤ ψ h ≤ 1 ψ h (s 1 , s 2 ) = 1 if dist (s 1 , s 2 ), γ 0 ≤ h ψ h (s 1 , s 2 ) = 0 if dist (s 1 , s 2 ), γ 0 ≥ 2h.
Indeed we can assume that (8.21)

||ψ h || W 1,∞ (R 2 ) ≤ C h , ||ψ h || W 2,∞ (R 2 ) ≤ C h 2 .
Since ω is bounded with a Lipschitz boundary, we first extend the fields V and R n = Rn into two fields of (H 2 (R 2 )) 3 and (H 1 (R 2 )) 3 (and we use the same notations for these extentions). We define the 3

× 3 matrix field R ′ ∈ (H 1 (R 2 )) 3×3 by the formula (8.22) R ′ = ∂V ∂s | ∂V ∂s 2 |R n t 1 |t 2 |n -1
.

By construction we have

∂V ∂s α = R ′ t α in R 2 and R ′ = R in ω.
At least, we introduce below the approximations V h and R h of V and R as restrictions to ω of the following fields defined into R 2 :

(8.23)

       V ′ h (s 1 , s 2 ) = 1 9πh 2 B(0,3h) V(s 1 + t 1 , s 2 + t 2 )dt 1 dt 2 , R ′ h (s 1 , s 2 ) = 1 9πh 2 B(0,3h) R ′ (s 1 + t 1 , s 2 + t 2 )dt 1 dt 2 , a.e. (s 1 , s 2 ) ∈ R 2 and (8.24) V h = φψ h + V ′ h (1 -ψ h ), R h = I 3 ψ h + R ′ h (1 -ψ h ), in ω.
Notice that we have

(8.25) V ′ h ∈ (W 2,∞ (R 2 )) 3 , R ′ h ∈ (W 1,∞ (R 2 )) 3×3 , V h ∈ (W 2,∞ (ω)) 3 , R h ∈ (W 1,∞ (ω)) 3×3 , V h = φ, R h = I 3 on γ 0 .
Due to the definition (8.22) of R ′ and in view of (8.23) we have (8.26)

V ′ h -→ V strongly in (H 2 (R 2 )) 3 , R ′ h -→ R ′ strongly in (H 1 (R 2 )) 3×3
and thus using estimates (8.21)

(8.27) V h -→ V strongly in (H 2 (ω)) 3 , R h -→ R strongly in (H 1 (ω)) 3×3
Moreover using again (8.23) and the fact that R ′ -R h strongly converges to 0 in (H 1 (R 2 )) 3×3 we deduce

that 1 h R ′ h -R ′ ) -→ 0 strongly in (L 2 (R 2 )) 3×3
and then together with (8.21), (8.22), (8.24) and (8.27) we get

1 h R h -R -→ 0 strongly in (L 2 (ω)) 3×3 , 1 h ∂V h ∂s α -R h t α -→ 0 strongly in (L 2 (ω)) 3 .
We now turn to the estimate of the distance between R h (s 1 , s 2 ) and SO(3) for a.e. (s 1 , s 2 ) ∈ ω. We apply the Poincaré-Wirtinger's inequality to the function (u 1 , u 2 ) -→ R ′ (u 1 , u 2 ) in the ball B((s 1 , s 2 ), 3h). We obtain B((s1,s2),3h)

|||R ′ (u 1 , u 2 ) -R ′ h (s 1 , s 2 )||| 2 du 1 du 2 ≤ Ch 2 ||∇R ′ || 2 (L 2 (B((s1,s2),3h))) 3
where C is the Poincaré-Wirtinger's constant for a ball. Since the open set ω is boundy with a Lipschitz boundary, there exists a positive constant c(ω), which depends only on ω, such that

| B((s 1 , s 2 ), 3h) \ B((s 1 , s 2 ), 2h) ∩ ω| ≥ c(ω)h 2 .
Setting m h (s 1 , s 2 ) the essential infimum of the function (u 1 , u 2 ) → |||R(u 1 , u 2 ) -R ′ h (s 1 , s 2 )||| into the set B((s 1 , s 2 ), 3h) \ B((s 1 , s 2 ), 2h) ∩ ω, we then obtain

c(ω)h 2 m h (s 1 , s 2 ) 2 ≤ Ch 2 ||∇R ′ || 2 (L 2 (B((s1,s2),3h))) 3
Hence, thanks to the strong convergence of R ′ h given by (8.26), the above inequality shows that there exists h ′ 0 which does not depend on (s 1 , s 2 ) ∈ ω such that for any h ≤ h

′ 0 dist(R ′ h (s 1 , s 2 ), SO(3)) ≤ 1/8 for any (s 1 , s 2 ) ∈ ω.

Now,

• in the case dist (s 1 , s 2 ), γ 0 > 2h, (s 1 , s 2 ) ∈ ω, by definition of R h and thanks to the above inequality

we have dist(R h (s 1 , s 2 ), SO(3)) ≤ 1/8, • in the case dist (s 1 , s 2 ), γ 0 < h, (s 1 , s 2 ) ∈ ω, by definition of R h we have R h (s 1 , s 2 ) = I 3 and then dist(R h (s 1 , s 2 ), SO(3)) = 0,
• in the case h ≤ dist (s 1 , s 2 ), γ 0 ≤ 2h, (s 1 , s 2 ) ∈ ω, due to the fact that R ′ = I 3 onto γ 0 , firstly we have

||R ′ -I 3 || 2 (L 2 (ω 6h,γ 0 )) 3×3 ≤ Ch 2 ||∇R ′ || 2 (L 2 (ω 6h,γ 0 )) 3×3
where

ω kh,γ0 = {(s 1 , s 2 ) ∈ R 2 | dist (s 1 , s 2 ), γ 0 ≤ kh}, k ∈ N * . Hence ||R ′ h -I 3 || 2 (L 2 (ω 3h,γ 0 )) 3×3 ≤ Ch 2 ||∇R ′ || 2 (L 2 (ω 6h,γ 0 )) 3×3 .
The constants depend only on ∂ω.

Secondly, we set M h the maximum of the function (

u 1 , u 2 ) → |||I 3 -R ′ h (u 1 , u 2 )||| into the closed set (u 1 , u 2 ) ∈ ω | h ≤ dist (u 1 , u 2 )
, γ 0 ≤ 2h , and let (s 1 , s 2 ) be in this closed subset of ω such that

M h = |||I 3 -R ′ h (s 1 , s 2 )|||.
Applying the Poincaré-Wirtinger's inequality in the ball B (s 1 , s 2 ), 4h we deduce that

∀(s ′ 1 , s ′ 2 ) ∈ B (s 1 , s 2 ), h , |||R ′ h (s ′ 1 , s ′ 2 ) -R ′ h (s 1 , s 2 )||| ≤ C||∇R ′ || (L 2 (B((s1,s2),4h))) 3 .
The constant depends only on the Poincaré-Wirtinger's constant for a ball.

If M h is larger than C||∇R ′ || (L 2 (B((s1,s2),4h))) 3 we have

πh 2 M h -C||∇R ′ || (L 2 (B((s1,s2),4h))) 3 2 ≤ ||R ′ h -I 3 ||| 2 (L 2 (B((s1,s2),h))) 3 ≤||R ′ h -I 3 || 2 (L 2 (ω 3h,γ 0 )) 3×3 ≤ Ch 2 ||∇R ′ || 2 (L 2 (ω 6h,γ 0 )) 3×3
then, in all the cases we obtain

M h ≤ C||∇R ′ || (L 2 (ω 6h,γ 0 )) 3×3 .
The constant does not depend on h and R ′ . The above inequalities show that there exists h ′′ 0 such that for any h ≤ h

′′ 0 |||R ′ h (s 1 , s 2 )-I 3 ||| ≤ C||∇R ′ || (L 2 (ω 6h,γ 0 )) 3×3 ≤
∀(s 1 , s 2 ) ∈ ω, ∂V ′ h ∂s α (s 1 , s 2 )-R ′ h (s 1 , s 2 )t α (s 1 , s 2 ) 2 ≤ Ch||φ|| (W 2,∞ (ω)) 3 +C ||V|| (H 2 (ω 3h )) 3 +||R ′ || (H 1 (ω 3h )) 3×3
where

ω 3h = {(s 1 , s 2 ) ∈ R 2 | dist (s 1 , s 2 ), ∂ω ≤ 3h}.
We have

∂V h ∂s α -R h t α = (1 -ψ h ) ∂V ′ h ∂s α -R ′ h t α + ∂ψ h ∂s α φ -V ′ h .
Thanks to the above inequality, (8.21) and again the estimate of

|||R ′ h -I 3 ||| in the edge strip h ≤ dist (s 1 , s 2 ), γ 0 ≤ 2h we obtain for all (s 1 , s 2 ) ∈ ω ∂V h ∂s α (s 1 , s 2 ) -R h (s 1 , s 2 )t α (s 1 , s 2 ) 2 ≤C h||φ|| (W 2,∞ (ω)) 3 + ||V|| (H 2 (ω 3h )) 3 + ||R ′ || (H 1 (ω 3h )) 3×3 + ||φ -V|| (H 2 (ω 5h,γ 0 )) 3×3 .
The same argument as above imply that there exists h 0 ≤ max(h ′ 0 , h ′′ 0 ) such that for any 0 < h ≤ h 0 and for any (s 1 , s 2 ) ∈ ω we have (8.28)

∂V h ∂s α (s 1 , s 2 ) -R h (s 1 , s 2 )t α (s 1 , s 2 ) 2 ≤ 1 8 .
From (8.21), (8.22), (8.23) and (8.24) there exists a positive constant C which does not depend on h such that (8.29)

||R h || (W 1,∞ (ω)) 3×3 ≤ C h ||V|| (H 2 (ω)) 3 + ||R|| (H 1 (ω)) 3×3 .
Now we can choose h in term of δ. We set

h = κδ, δ ∈ (0, δ 0 ]
and we fixed κ in order to have h ≤ h 0 and to obtain the right hand side in (8.29) less than

1 4 √ 2c ′ 1 δ ( c ′ 1 is
given by (2.3)). It is well-known that there exists a sequence w δ δ∈(0,δ0] satisfying (8.20), the convergences in (8.18) and the estimate

||w δ || (W 1,∞ (Ω)) 3 ≤ 1 4 √ 2c ′ 1 δ
.

Let us now consider an arbitrary element V, R, w of U nlin and the corresponding sequences V δ , R δ , w δ δ>0

given by Lemma 8.1. The deformation v δ is then defined by

(8.30) v δ (s) = V δ (s 1 , s 2 ) + s 3 R δ (s 1 , s 2 )n(s 1 , s 2 ) + δ 2 w δ s 1 , s 2 , s 3 δ , for s ∈ Ω δ .
Step 1. Estimate on

||Π δ ∇ x v δ -R δ || L ∞ (Ω)) 3×3 and ||dist ∇ x v δ , SO(3) || L ∞ (ω) .
Using (3.11) and (8.30), we first have (8.31)

       (∇ x v δ -R δ )t α = ∂V δ ∂s α -R δ t α + s 3 ∂R δ ∂s α n + δ 2 ∂w δ ∂s α -(∇ x v δ -R δ )s 3 ∂n ∂s α (∇ x v δ -R δ )n = δ ∂w δ ∂S 3 . We first estimate the L ∞ -norm of Π δ (∇ x v δ -R δ ). We have (8.32) Π δ (∇ x v δ -R δ ) • Π δ (∇ s Φ) = ∂V δ ∂s 1 -R δ t 1 + S 3 δ ∂R δ ∂s α n + δ 2 ∂w δ ∂s α | ∂V δ ∂s 2 -R δ t 2 + S 3 δ ∂R δ ∂s α n + δ 2 ∂w δ ∂s α | δ ∂w δ ∂S 3 .
Thanks to (2.3), (8.20) and (8.28) we obtain (8.33)

||Π δ ∇ x v δ -R δ || (L ∞ (Ω)) 3×3 ≤ 1 4 .
From (8.20) and (8.33) we deduce that there exists a positive constant C 0 such that

(8.34) ||Π δ (∇ x v δ ) T ∇ x v δ -I 3 || (L ∞ (Ω)) 3×3 ≤ C 0 .
In view of (8.18) and (8.33) we deduce that

||dist ∇ x v δ , SO(3) || L ∞ (ω) ≤ 1 2
and then we obtain (8.35) for a.e. s ∈ Ω δ det ∇ x v δ (s) > 0.

Step

2. Strong limit of 1 2δ Π δ (∇ x v δ ) T ∇ x v δ -I 3 .
Tanks to the convergences of Lemma 8.1, (8.20) and (8.32) we have

(8.36) ||Π δ ∇ x v δ -R δ || (L 2 (Ω)) 3×3 ≤ Cδ.
We write the identity (∇

x v δ ) T ∇ x v δ -I 3 = (∇ x v δ -R δ ) T R δ + R T δ (∇ x v δ -R δ ) + (∇ x v δ -R δ ) T (∇ x v δ - R δ ) + (R δ -R) T R δ + R T (R δ -R).
(∇ x v δ ) T ∇ x v δ -I 3 || (L 2 (Ω)) 3×3 ≤ Cδ.
In view of (8.18), (8.20), (8.31) we deduce that (8.38)

       1 δ Π δ (∇ x v δ -R)t α -→ S 3 ∂R ∂s α n strongly in (L 2 (Ω)) 3 1 δ Π δ (∇ x v δ -R)n -→ ∂w ∂S 3 n strongly in (L 2 (Ω)) 3
Now thanks (8.33) and the strong convergences (8.38) we obtain

1 √ δ Π δ (∇ x v δ -R) -→ 0 strongly in (L 4 (Ω)) 3
then using the above identity, we get

(8.39) 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 strongly in (L 2 (Ω)) 3×3 ,
where E is given by (8.14).

Step 3. The lim

δ→0 J(v δ ) δ 3 .
Let ε be a fixed positive constant and let θ given by (7.2). We denote χ θ δ the characteristic function of the set

A θ δ = {s ∈ Ω; |||Π δ (∇ x v δ ) T ∇ x v δ -I 3 (
1 δ 2 (1 -χ θ δ ) W Π δ (∇ x v δ ) Π δ det(∇Φ) ≤ Ω Q (t 1 | t 2 | n) -T E(t 1 | t 2 | n) -1 det(t 1 |t 2 |n) + ε Ω |||(t 1 | t 2 | n) -T E(t 1 | t 2 | n) -1 ||| 2 det(t 1 |t 2 |n)
where E is given by (8.14). Thanks to (7.3), (7.4), (8.37), the strong convergence (8.39) and the weak convergence

1 δ χ θ δ ⇀ 0 in L 2 (Ω) we obtain lim δ→0 Ω 1 δ 2 χ θ δ W Π δ (∇ x v δ ) Π δ det(∇Φ) ≤ C 1 lim δ→0 Ω 1 δ χ θ δ ||| 1 2δ Π δ (∇ x v δ ) T ∇ x v δ -I 3 ||| Π δ det(∇Φ) = 0
Hence for any ε > 0 we get

lim δ→0 Ω 1 δ 2 W Π δ (∇ x v δ ) Π δ det(∇Φ) ≤ Ω Q (t 1 | t 2 | n) -T E(t 1 | t 2 | n) -1 det(t 1 |t 2 |n) + ε Ω |||(t 1 | t 2 | n) -T E(t 1 | t 2 | n) -1 ||| 2 det(t 1 |t 2 |n) Finally (8.41) lim δ→0 Ω 1 δ 2 W Π δ (∇ x v δ ) Π δ det(∇Φ) ≤ Ω Q (t 1 | t 2 | n) -T E(t 1 | t 2 | n) -1 det(t 1 |t 2 |n).
As far as the contribution of the applied forces is concerned, proceeding as in the proof of the lim-inf condition and using the convergences (8.18) gives

(8.42) lim δ→0 Ω f • Π δ (v δ )Π δ det(∇Φ) + Ω g • Π δ (v δ ) δ Π δ det(∇Φ) = L(V, R).
From (8.41) and (8.42), we conclude that

lim δ→0 J(v δ ) δ 3 ≤ Ω Q (t 1 | t 2 | n) -T E(t 1 | t 2 | n) -1 det(t 1 |t 2 |n) -L(V, R) = J N L V, R, w .
The following theorem summarizes the above results.

Theorem 8.2. The functional J N L is the Γ-limit of J(.) δ 3 in the following sense: • consider any sequence of deformations v δ 0<δ≤δ0 belonging to U δ and satisfying lim δ→0 J(v δ ) δ 3 < +∞ and let V δ , R a,δ , v a,δ be the terms of the decomposition of v δ given by Theorem 3.3. Then there exists V, R, w ∈ U nlin such that (up to a subsequence )

V δ -→ V strongly in H 1 (ω) 3 , R a,δ ⇀ R weakly in H 1 (ω) 3×3 , 1 δ ∂V δ ∂s α -R a,δ t α ⇀ Z α weakly in L 2 (ω) 3 , 1 δ 2 Π δ v a,δ ⇀ v weakly in L 2 (ω; H 1 (-1, 1)) 3
where w is defined by (8.13) as a function depending of R, Z α and v. We have

J N L V, R, w ≤ lim δ→0 J(v δ ) δ 3
• for any V, R, w ∈ U nlin there exists a sequence v δ 0<δ≤δ0 belonging to U δ such that

lim δ→0 J(v δ ) δ 3 ≤ J N L V, R, w .
Moreover, there exists V 0 , R 0 , w 0 ∈ U nlin such that

(8.43) m ′ 2 = lim δ→0 1 δ 3 inf v∈U δ J(v) = J N L V 0 , R 0 , w 0 = min (V,R,w)∈U nlin J N L V, R, w .
The next theorem shows that the variable w can be eliminated in the minimization problem (8.43).

Theorem 8.3. Let (V 0 , R 0 ) be given by Theorem 8.2. The minimum m 2 of the functional J N L over U nlin satisfies the following minimization problem:

(8.44) m ′ 2 = F N L V 0 , R 0 = min (V,R)∈V nlin F N L V, R ,
where

V nlin = V, R ∈ (H 2 (ω)) × (H 1 (ω)) 3×3 | V = φ, R = I 3 on γ 0 , R(s 1 , s 2 ) ∈ SO(3) for a.e. (s 1 , s 2 ) ∈ ω, ∂V ∂s α = Rt α , and
F N L V, R = ω a αβα ′ β ′ ∂R ∂s α n • Rt β ∂R ∂s ′ α n • Rt β ′ det(t 1 |t 2 |n) -L(V, R)
where a αβα ′ β ′ are constants which depend only of the quadratic form Q and the vectors (t 1 , t 2 , n).

Proof of Theorem 8.3. In order to eliminate w, we fix V, R ∈ V nlin and we minimize the functional

J N L V, R, • over the space W = w ∈ (L 2 (ω; H 1 (-1, 1))) 3 | 1 -1
w(s 1 , s 2 , S 3 )dS 3 = 0 for a.e. (s 1 , s 2 ) ∈ ω .

We first write

Ω Q (t 1 | t 2 | n) -T E(t 1 | t 2 | n) -1 | det(t 1 |t 2 |n)| = Ω    Q 1 . . . Q 2 . . . . . . Q T 2 . . . Q 3            E 11 E 12 E 22 E 13 E 23 E 33         •         E 11 E 12 E 22 E 13 E 23 E 33         det(t 1 |t 2 |n)
where Q 1 , Q 2 and Q 3 are matrices belonging to (L ∞ (ω)) 3×3 moreover Q 1 and Q 3 are symmetric. They depend of the coefficients of the quadratic form Q and the vectors (t 1 , t 2 , n) and they satisfy where c 0 is a positive constant.

   Q 1 (s 1 , s 2 ) . . . Q 2 (s 1 , s 2 ) . . . . . . Q T 2 (s 1 , s 2 ) . . . Q 3 (s 1 , s 2 )          
Through solving simple variational problems (see [START_REF] Griso | Asymptotic behavior of curved rods by the unfolding method[END_REF]), we find that the minimum of the functional J N L V, R, • over the space W is obtained with

1 3 Q -1 3 Q T 2       
J N L V, R, w = F N L V, R = 2 3 ω Q 1 -Q 2 Q -1 3 Q T 2        ∂R ∂s 1 n • Rt 1 ∂R ∂s 1 n • Rt 2 ∂R ∂s 2 n • Rt 2        •        ∂R ∂s 1 n • Rt 1 ∂R ∂s 1 n • Rt 2 ∂R ∂s 2 n • Rt 2        det(t 1 |t 2 |n) -L(V, R).
The symmetric matrix 3×3 and moreover it satisfies 9. A few remarks.

Q 1 -Q 2 Q -1 3 Q T 2 belongs to (L ∞ (ω))
Q 1 -Q 2 Q -1 3 Q T
1. In the case of a St-Venant-Kirchhoff material, the above analysis and a classical energy argument show that if v δ 0<δ≤δ0 is a sequence such that

′ 2 = lim δ→0 J(v δ ) δ 3 ,
then there exists a subsequence and (V 0 , R 0 ) ∈ V nlin , which is a solution of Problem (8.44), such that the sequence of the Green-St Venant's deformation tensors satisfies

1 2δ Π δ (∇ x v δ ) T ∇ x v δ -I 3 -→ (t 1 | t 2 | n) -T E 0 (t 1 | t 2 | n) -T strongly in (L 2 (Ω)) 3×3 ,
where E 0 is defined in (8.14) with w 0 given by (8.45) (replacing R by R 0 ).

2.

Let us give the explicit expression of the limit energy F N L in the case where S is a developable surface such that the parametrization φ is locally isometric

∀(s 1 , s 2 ) ∈ ω ||t α (s 1 , s 2 )|| 2 = 1 t 1 (s 1 , s 2 ) • t 2 (s 1 , s 2 ) = 0.
We consider a St Venant-Kirchhoff's law for which we have

W (F ) =    λ 8 tr(F T F -I 3 ) 2 + µ 4 tr (F T F -I 3 ) 2 if det(F ) > 0 + ∞ if det(F ) ≤ 0,
so that Q = W . For any V, R, w ∈ U nlin , the expression (8.15) gives

J N L V, R, w = Ω λ 2 tr( E) 2 + µ tr ( E) 2 -L(V, R)
where E is defined by (8.16). It follows that the elimination of w in Theorem 8.2 is identical to that of standard linear elasticity (see [START_REF] Griso | Asymptotic behavior of structures made of plates[END_REF]) hence we have w(., ., S 3 ) = -λ λ + 2µ S 2 3 -

1 3 ∂R ∂s 1 n • Rt 1 + ∂R ∂s 2 n • Rt 2 Rn
and then -L(V, R).

F N L V, R = 2E 3(1 -ν 2 ) ω (1 -ν)

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).

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

(A.3)    f (0) = R 0 , f (1) = R 1 , df dt (L ∞ (0,1)) 9 ≤ π 2 |||R 1 -R 0 ||||, |||R 0 -f (t)||| ≤ |||R 0 -R 1 |||.
Proof One has df dt (L ∞ (0,1)) Proof. We write

9 = √ 2θ 2 ≤ π 2 |||R 2 -I 3 ||| = π 2 |||R 1 -R 0 |||. Moreover |||R 0 -f (t)||| = |||I 3 -R -1 0 f (t)||| = 2 √ 2 sin θ 2 t 2 ≤ 2 √ 2 sin θ 2 2 = |||I 3 -R 2 ||| = |||R 0 -R 1 ||||.
A =    A 1 . . . A 2 • • • • • • A T 2 . . . A 3   
where A 1 and A 3 are symmetric positive definite 3 × 3 matrices. We set

d = -A -1 3 A T 2 b.
By a straightforward calculation and using the fact that The positivity of Q then gives the result.

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

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

22  ≥ c 0 |ξ| 2 for any ξ =   ξ 11 ξ 12 ξ 22  ∈ R 3

 22223 and for a.e. (s 1 , s 2 ) ∈ ω.

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(A.2) ∀x ∈ R 3 , R a,θ (x) = cos(θ)x + 1 1 + cos(θ) < x, b > b + b ∧ x.

Lemma A. 2 . 2 , 2 , 2 , 2 . 3 where

 222223 Let R 00 , R 01 , R 10 and R 11 be four matrices belonging to SO(3) and satisfying(A.4) |||R 10 -R 00 ||| ≤ 1 |||R 01 -R 00 ||| ≤ 1 |||R 11 -R 01 ||| ≤ 1 |||R 11 -R 10 ||| ≤ 1 There exists a function R ∈ 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 |||| . A is a symmetric positive definite constant 6 × 6 matrix. For any (a, b, c) ∈ R 3 × R 3 × L 2 (-1, 1) 3 ,there exists d in R 3 which depends only on b and A such thatQ m (a, b, c) ≥ Q m (a, 0, cd).

1 - 1 S 3

 113 dS 3 = 0, we obtainQ m (a, b, c) = Q m (a, 0, cd) + Q m (0, b, d).

  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)

  The convergences (6.2) are direct consequences of Theorem 3.3 and estimate (4.8) excepted for what concerns the last convergence which will be established below. The compact imbedding of H 1 (ω)

3×3

in L 4 (ω)

  1 |t 2 |n)+s 3 det | n so that indeed Π δ det(∇Φ) strongly converges to det(t 1 |t 2 |n) in L ∞ (Ω) as δ tends to 0.

	s 3 det t 1 |	∂n ∂s 2	| n + s 2 3 det	∂n ∂s 1	|	∂n ∂s 2	∂n ∂s 1	|t 2 | n +

  1/8 for any (s 1 , s 2 ) ∈ ω such that h ≤ dist (s 1 , s 2 ), γ 0 ≤ 2h. By definition of R h , that gives |||R h (s 1 , s 2 ) -I 3 ||| ≤ 1/8.

	Finally, for any h ≤ max(h	′ 0 , h	′′ 0 ) and for any (s 1 , s 2 ) ∈ ω we have
			dist(R h (s 1 , s 2 ), SO(3)) ≤ 1/8.
	Using (8.22) and (8.23) we obtain (recall that • 2 is the euclidian norm in R 3 )

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

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,

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

and where the functions

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

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 -1 00 f 01,11 (t) we associate the vector b 01,11 (t). Let b be the vectors field defined by

This function belongs to W 1,∞ (Y ) 3 and satisfies

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

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