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

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1 Decomposition of the deformations of a thin shell Asymptotic behavior of the Green-St Venant s strain tensor Dominique Blanchard Georges Griso To cite this version: Dominique Blanchard Georges Griso Decomposition of the deformations of a thin shell Asymptotic behavior of the Green-St Venant s strain tensor 009 <hal > HAL Id: hal Submitted on 3 Jul 009 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents whether they are published or not The documents may come from teaching and research institutions in France or abroad or from public or private research centers L archive ouverte pluridisciplinaire HAL est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche publiés ou non émanant des établissements d enseignement et de recherche français ou étrangers des laboratoires publics ou privés

2 Decomposition of the deformations of a thin shell Asymptotic behavior of the Green-St Venant s strain tensor Dominique Blanchard a and Georges Griso b a Université de Rouen UMR Saint Etienne du Rouvray Cedex France dominiqueblanchard@univ-rouenfr blanchar@annjussieufr b Laboratoire d Analyse Numérique Université P et M Curie Case Courrier Paris Cédex 05 - France griso@annjussieufr Keywords: shells Korn s inequality large deformations 000 MSC: 74B0 74K0 74K5 Abstract 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 -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/ 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 (ie which is H with respect to the variables and is fixed on a part of the lateral boundary which take into account the fact that the thickness δ of such a domain is small This decomposition technique has been developed in the framework of linearized elasticity for thin structures in [4] [5] [6] and for thin curved rods in nonlinear elasticity in [4] As far as large deformations are concerned these decompositions are obtained through using the Rigidity Theorem proved in [] together with the geometrical precision of this result given in [4] Let us consider a shell with mid-surface S and thickness δ 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 vso(3 L (S ] δδ[ and the thickness of the shell

3 In the first decomposition the field R satisfies dist(rso(3 L (S C δ / dist( xvso(3 L (S ] δδ[ 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 vso(3 L C(Sδ 3/ where C(S is a geometrical constant the field R is valued in SO(3 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 [4] ( see also [8] and Subsection 4 of the present paper leads in the case of a shell to the following inequality v I d (L (S ] δδ[ 3 + xv I 3 (L (S ] δδ[ 9 C(δ/ + dist( x vso(3 L (S ] δδ[ 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 (S ] δδ[ 3 + xv I 3 (L (S ] δδ[ 9 C δ dist( xvso(3 L (S ] δδ[ 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/ 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 I (L 3 C dist( (S ] δδ[ 9 x vso(3 L (S ] δδ[ + } δ dist( xvso(3 5/ L (S ] δδ[ which shows that dist( x v δ SO(3 L (S ] δδ[ δ 5/ 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 [4] 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 (S ] δδ[ is of order δ κ / (κ The order δ 3/ is the highest level of energy which can be analyzed through our technique For dist( x v δ SO(3 L (S ] δδ[ δ 3/ 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 (S ] δδ[ δ κ / with κ > 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 < 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

4 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 [5] [6] [7] [9] [0] [] [6] [8] [9] [0] The rigidity theorem and its applications to thin structures using Γ-convergence arguments are developed in [] [] [7] [8] The decomposition of the deformations in thin structures is introduced in [4] [5] and a few applications to the junctions of multi-structures and homogenization are given in [] [] [3] The paper is organized as follows Section 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 (S ] δδ[ δ κ / in Section 6 for κ = and in Section 7 for κ > 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 [4] for a detailed presentation Let ω be a bounded domain in R with lipschitzian boundary and let φ be an injective mapping from ω into R 3 of class C We denote S the surface φ(ω We assume that the two vectors φ s (s s and φ s (s s are linearly independent at each point (s s ω We set ( t = φ s t = φ n = t t s t t The vectors t and t 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 ] δδ[ We set Ω δ = ω ] δδ[ Now we consider the mapping Φ : ω R R 3 defined by ( Φ : (s s s 3 x = φ(s s + s 3 n(s s There exists δ 0 (0] depending only on S such that the restriction of Φ to the compact set Ω δ0 = ω [ δ 0 δ 0 ] is a C diffeomorphism of that set onto its range (see eg [6] Hence there exist two constants c 0 > 0 and c c 0 which depend only on φ such that δ (0δ 0 ] s Ω δ0 c 0 s Φ(s c and for x = Φ(s c 0 x Φ (x c Definition For δ (0δ 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 ] δδ[ The fibers of the shell are the segments Φ ( {(s s } ] δδ[ (s s ω We respectively denote by x and s the running 3

5 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 ae 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 s R dist ( (s s ω < η } We assume that there exist η 0 > 0 and an extension of the mapping φ (still denoted φ belonging to ( C (ω η0 3 which remains injective and such that the vectors φ (s s and φ (s s are linearly independent at s s each point (s s 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 diffeomorphism of that set onto its range Then there exist four constants c 0 c c and C such that (3 { s ωη0 [ δ 0 δ 0 ] c 0 s Φ(s c and for x = Φ(s c 0 x Φ (x c c x v(x s v(s C x v(x for ae x = Φ(s At the end we denote by I d the identity map of R 3 3 Decompositions of a deformation In this Section we recall the theorem of rigidity established in [] (Theorem 3 of Section 3 In Subsection 3 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 to a covering of the shell In Subsections 34 and 35 we introduce the two decompositions of a deformation and we established estimates on these decompositions in term of dist( x vso(3 L 3 Theorem of rigidity We equip the vector space R n n of n n matrices with the Frobenius norm defined by A = ( a ij ij n A = n n a ij i= j= We just recall the following theorem established in [] in the version given in [4] Theorem 3 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 ( O;R (0 < R R For any v ( H (Ω n there exist R SO(n and a R n such that (3 { x v R (L (Ω n n C dist( xv;so(n L (Ω v a Rx (L (Ω n CR dist( x v;so(n L (Ω where the constant C depends only on n and R R 3 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 34 or a field of rotations (see Subsection 35 4

6 and a residual one we must extend any deformation belonging to ( H (Q δ 3 in a neighborhood of the lateral boundary Γ δ of the shell To this end we will use Lemma 3 below The proof of this lemma is identical to the one of Lemma 3 of [4] 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 Let δ be fixed in (0δ 0 ] such that 3δ η 0 and set Q δ = Φ(ω 3δ ] δδ[ There exists an extension operator P δ from ( H (Q δ 3 into ( H (Q δ 3 such that v ( H (Q δ 3 Pδ (v ( H (Q δ 3 Pδ (v Qδ = v dist ( x P δ (vso(3 L (Q δ c dist( x vso(3 L (Q δ with a constant c which only depends on ω and on the constants appearing in inequalities (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 0δ = Φ(γ 0 ] δδ[ Consider now a deformation v such that v = I d on Γ 0δ Let γ 0δ be the domain γ 0δ = { (s s γ 0 dist((s s E 0 > 3δ} where E 0 denotes the extremities of γ 0 We set Q δ = Φ ( {(s s (ω 3δ \ ω dist((s s γ 0δ < 3δ} ] δδ[ Q δ = Φ ( {(s s ω 3δ dist((s s γ 0 < 6δ} ] δδ[ Indeed up to choosing δ 0 small enough we can assume that Q δ has the same number of connected components as γ 0 The open set Q δ is included into Q δ \ Q δ According to the construction of P δ given in [4] we can extend the deformation v by choosing P δ (v = I d in Q δ together with the following estimates (3 { x P δ (v I 3 (L (Q δ 9 C dist( xvso(3 L (Q δ P δ (v I d (L (Q δ 3 Cδ dist( xvso(3 L (Q δ From now on we assume that 3δ η 0 and then any deformation v belonging to ( H (Q δ 3 is extended to a deformation belonging to ( H (Q δ 3 which we still denote by v Now we are in a position to reproduce the technique developed in [4] in order to obtain a covering of the shell (the reader is referred to Section 33 of this paper for further details Let N δ be the set of every (k l Z such that the open set ω δ(kl =]kδ(k + δ[ ]lδ (l+δ[ 5

7 is included in ω 3δ and let N δ be the set of every (k l N δ such that ((k + δlδ (kδ(l + δ (k + δ (l + δ are in N δ We set Ω δ(kl = ω δ(kl ] δδ[ By construction of the above covering we have ω (kl N δ ω δ(kl According to [4] there exist two constants R and R which depend on ω and on the constants c 0 c c and C (see(3 such that for any δ (0η 0 /3] the open set Q δ(kl = Φ(Ω δ(kl has a diameter less than Rδ and is star-shaped with respect to a ball of radius R 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 c and C depends on the mid-surface S and we write C(S Since the ratio Rδ R δ of each part Q δ(kl does not depend on δ Theorem 3 gives a constant C(S Let v be a deformation in (H (Q δ 3 extended to a deformation belonging to (H (Q δ 3 Applying Theorem 3 upon each part Q δ(kl for (k l N δ there exist R δ(kl SO(3 and a δ(kl R 3 such that (33 { x v R δ(kl (L (Q δ(kl 3 3 C(S dist( xv;so(3 L (Q δ(kl v a δ(kl R δ(kl ( x φ(kδlδ (L (Q δ(kl 3 C(SR(Sδ dist( x v;so(3 L (Q δ(kl For any (kl N δ such that (k + l N δ the open set Q δ(kl = Φ(](k + /δ (k + 3/δ[ ]lδ (l + δ[ ] δδ[ also have a diameter less than R(Sδ and it is also star-shaped with respect to a ball of radius R (Sδ (see Section 33 in [4] We apply again Theorem in the domain Q δ(kl This gives a rotation R δ(kl In the domain Q δ(kl Q δ(kl we eliminate x v in order to evaluate R δ(kl R δ(kl Then we evaluate R δ(k+l R δ(kl Finally it leads to (34 R δ(k+l R δ(kl C(S { } dist( x v;so(3 δ 3/ L (Q δ(kl + dist( x v;so(3 L (Q δ(k+l In the same way we prove that for any (kl N δ such that (k l + N δ we have (35 R δ(kl+ R δ(kl C(S { } dist( x v;so(3 δ 3/ L (Q δ(kl + dist( x v;so(3 L (Q δ(kl+ 33 First decomposition of a deformation In this section any deformation v ( H (Q δ 3 of the shell Qδ is decomposed as (36 v(s = V(s s + s 3 R a (s s n(s s + v a (s s Ω δ where V belongs to ( H (ω 3 Ra belongs to ( H (ω 3 3 and va belongs to ( H (Q δ 3 The map V is the mean value of v over the fibers while the second term s 3 R a (s s n(s s is an approximation of the rotation of the fiber (of the shell which contains the point φ(s s The sum of the two first terms V(s s + s 3 R a (s s n(s s is called the elementary deformation of first type of the shell 6

8 The matrix R a is defined as the Q interpolate at the vertices of the cell ω δ(kl =]kδ(k+δ[ ]lδ (l+δ[ of the four elements R δ(kl R δ(k+l R δ(kl+ and R δ(k+l+ belonging to SO(3 (see the previous subsection We can always define paths in SO(3 from R δ(kl to R δ(k+l R δ(kl to R δ(kl+ R δ(k+l to R δ(k+l+ and R δ(kl+ to R δ(k+l+ That gives continuous maps from the edges of the domain ω δ(kl into SO(3 If it is possible to extend these maps in order to obtain a continuous function from ω δ(kl into SO(3 then it means that the loop passing trough R δ(kl R δ(k+l R δ(k+l+ and R δ(kl+ is homotopic ( to the constant loop equal to R δ(kl But the fundamental group π SO(3Rδ(kl is isomorphic to Z (the group of odd and even integers hence the extension does not always exist That is the reason why we use here a Q 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+l R δ(k+l+ and R δ(kl+ are in a neighborhood of R δ(kl then this extension exists and we give in Theorem 34 a simple condition to do so Theorem 33 Let v ( H (Q δ 3 there exist an elementary deformation (of first type V + s3 R a n and a deformation v a satisfying (36 and such that v a (L (Ω δ 3 Cδ dist( xvso(3 L (Q δ s v a (L (Ω δ 9 C dist( xvso(3 L (Q δ R a (L C s α (ω 9 δ dist( xvso(3 3/ L (Q δ (37 V (L R a t α C s α (ω 3 δ dist( xvso(3 / L (Q δ x v R (L a C dist( (Ω δ 9 x vso(3 L (Q δ dist(r a SO(3 L (ω C δ dist( xvso(3 / L (Q δ where the constant C does not depend on δ Proof The field V is defined by (38 V(s s = δ Then we define the field R a as following δ δ v(s s s 3 ds 3 ae in ω (k l N δ R a (kδlδ = R δ(kl and for any (s s ω δ(kl R a (s s is the Q interpolate of the values of R a at the vertices of the cell ω δ(kl Finally we define the field v a by v a (s = v(s V(s s s 3 R a (s s n(s s ae in Ω δ From (34 and (35 we get the third estimate in (37 By definition of R a we obtain (39 (kl N δ R a R δ(kl (L (ω δ(kl 9 C δ dist( xv;so(3 L (Q δ Taking the mean value of v on the fibers and using definition (38 of V lead to ( (30 V a δ(kl R δ(kl φ φ(kδlδ (L (ω δ(kl Cδ dist( xvso(3 3 L (Q δ (kl N δ 7

9 From (33 (39 (30 and the definition of v a we get the first estimate in (37 We compute the derivatives of the deformation v to get (3 v s = x v ( t + s 3 n s v = x v ( n t + s 3 s s v s 3 = x v n We consider the restrictions of these derivatives to Ω δ(kl Then from (33 and (39 we have (3 v ( n R a tα + s 3 s α s + v R a n α (L (Ω δ 3 s C dist( xvso(3 3 (L (Ω δ 3 L (Q δ By taking the mean value of Observe now that v ( n R a tα + s 3 on the fibers we obtain the fourth inequality in (37 s α s α (33 v a = v V n R a s 3 R a s 3 n s α s α s α s α s α v a = v R a n s 3 s 3 Then from (3 and the third and fourth inequalities in (37 we obtain the second estimate in (37 The fifth inequality in (37 is an immediate consequence of (33 and (39 The last estimate of (37 is due to (34 (35 and to the very definition of the field R a Since the matrices R δ(kl 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 s ω δ(kl we obtain R a (s s R T a (s s I 3 C { R δ(kl R δ(k+l + R δ(kl R δ(kl+ + R δ(kl+ R δ(k+l+ + R δ(k+l R δ(k+l+ } det ( R a (s s C { R δ(kl R δ(k+l + R δ(kl R δ(kl+ + R δ(kl+ R δ(k+l+ + R δ(k+l R δ(k+l+ } where C is an absolute constant Hence from (34 and (35 we deduce that (34 R a R T a I 3 (L (ω 9 C δ dist( xvso(3 / L (Q δ det(r a L (ω C δ dist( xvso(3 / L (Q δ Notice that the function R a R T a belongs to ( H (ω 3 3 and satisfies (35 R ar T a s α (L (ω 9 C δ 3/ dist( xvso(3 L (Q δ 34 Second decomposition of a deformation In this section any deformation v ( H (Q δ 3 of the shell Qδ is decomposed as (36 v(s = V(s s + s 3 R(s s n(s s + v(s s Ω δ 8

10 where V belongs to ( H (ω 3 ( R belongs to H (ω 3 3 and satisfies for ae (s s ω: R(s s SO(3 and v belongs to ( H (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 s n(s s describes the rotation of the fiber (of the shell which contains the point φ(s s The sum of the two first terms V(s s +s 3 R(s s n(s s is called the elementary deformation of second type of the shell Theorem 34 There exists a constant C(S (which depends only on the mid-surface S such that for any v ( H (Q δ 3 verifying (37 dist( x vso(3 L (Q δ C(Sδ 3/ then there exist an elementary deformation of second type V + s 3 Rn and a deformation v satisfying (36 and such that v (L (Ω δ 3 Cδ dist( xvso(3 L (Q δ (38 s v (L (Ω δ 9 C dist( xvso(3 L (Q δ R (L s C α (ω 9 δ dist( xvso(3 3/ L (Q δ V (L Rt α C s α (ω 3 δ dist( xvso(3 / L (Q δ x v R (L C dist( (Ω δ 9 x vso(3 L (Q δ where the constant C does not depend on δ Proof In this proof let us denote by C (S the constant appearing in estimates (34 and (35 If we assume that (39 then for each (kl N δ C (S δ 3/ dist( x v;so(3 L (Q δ we have using (34 and (35 R δ(k+l R δ(kl R δ(kl+ R δ(kl Thanks to Lemma A in Appendix A there exists a function R ( W (ω 3 3 such that for any (s s ω the matrix R(s s belongs to SO(3 and such that (k l N δ R(kδlδ = R δ(kl From (34 (35 and Lemma A we obtain the estimates (38 of the derivatives of R Due to the corollary of Lemma A we have (30 R R a (L (ω 9 C δ / dist( x v;so(3 L (Q δ All remainder estimates in (38 are consequences of (37 and (30 9

11 4 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 0δ (see Subsection 3 In Subsection 4 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 is the Korn s inequality for shells given Let v be in (H (Q δ 3 such that v(x = x on Γ 0δ Due to the definition (33 of V we have (4 V = φ on γ 0 4 First H - Estimates Using the boundary condition (4 estimates (37 or (38 and the fact that R a (L (ω and R (L (ω it leads to (4 V ( (H C + (ω 3 δ dist( xvso(3 / L (Q δ With the help of the decompositions (36 or (36 estimates (37 or (38 and (4 we deduce that v (L (Q δ 3 + δ v V (L (Q δ 3 + xv (L (Q δ 9 C (δ / + dist( x vso(3 L (Q δ The above inequality leads to the following first nonlinear Korn s inequality for shells : (43 v I d (L (Q δ 3 + xv I 3 (L (Q δ 9 C (δ / + dist( x vso(3 L (Q δ together with (v I d (V φ (L (Q δ 3 Cδ (δ / + dist( x vso(3 L (Q δ Let us notice that inequality (43 can be obtained without using the decomposition of the deformation Indeed we first have so that by integration v(x dist( v(xso(3 + 3 for ae x x v (L (Q δ 9 C (δ / + dist( x vso(3 L (Q δ Poincaré s inequality then leads to (43 This is the technique used to derive estimates in [3] 4 Further H - Estimates In this subsection we derive a boundary condition on R a and R on γ 0 using the extension given in Subsection 3 We prove the following lemma: Lemma 4 In Theorem 33 (respectively in Theorem 34 we can choose R a (resp R such that R a = I 3 on γ 0 (resp R = I 3 on γ 0 0

12 without modifications in the estimates of these theorems Proof Recall that γ 0δ Q δ and Q δ are defined in subsection 3 We also set Let us consider the following function Q 3 δ = Φ ( {(s s ω 3δ dist((s s γ 0 < 3δ} ] δδ[ ρ δ (s s = inf { sup ( 0 3δ dist((s s γ 0 } (s s R This function belongs to W (R and it is equal to if dist((s s γ 0 > 6δ and to 0 if dist((s s γ 0 < 3δ Let v δ be the deformation defined by v δ (s = φ(s s + s 3 n(s s + ρ δ (s s ( v(s φ(s s s 3 n(s s for ae s ω 3δ ] δδ[ By definition of v δ we have v δ = v in Q δ \ Q δ v δ = I d in Q 3 δ Recall that v = I d on Q δ Since the L -norm of ρ δ is of order /δ and the two estimates in (3 lead to (44 { x v x v δ (L (Q δ 9 C dist( x vso(3 L (Q δ v v δ (L (Q δ 3 Cδ dist( x vso(3 L (Q δ Hence (45 { dist( x v δ SO(3 L (Q δ xv x v δ (L (Q δ 9 + dist( x vso(3 L (Q δ C dist( x vso(3 L (Q δ where the constant does not depend on δ Since v δ = I d in Q δ the R a s and the R s given by application of Theorem 33 or 34 to the deformation v δ are both equal to I 3 over γ 0 Estimate(37 and (38 of these theorem together with (44-(45 show that Theorems 33 and 34 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 (43 for energies of order smaller than δ 3/ and an estimate on v V which permit to precise the scaling of the applied forces in Section 7 Theorem 4 (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 (Q δ 3 such that v = Id on Γ 0δ (46 v I d (L (Q δ 3 + xv I 3 (L (Q δ 9 C δ dist( xvso(3 L (Q δ and (47 (v I d (V φ (L (Q δ 3 C dist( xvso(3 L (Q δ where V is given by (38

13 Proof From the decomposition (36 Theorem 33 and the boundary condition on R a given by Lemma 4 the use of Poincaré s inequality gives (48 R a I 3 (H (ω 9 C δ dist( xvso(3 3/ L (Q δ V (L t α C s α (ω 3 δ dist( xvso(3 3/ L (Q δ Using the fact that t α = φ s α and the boundary condition (4 on V it leads to V φ (L (ω 3 C δ 3/ dist( xvso(3 L (Q δ Using again the decomposition (36 and Theorem 33 the above estimate implies that v I d satisfies the nonlinear Korn s inequality (46 At last the decomposition (36 which implies that (v I d (V φ = (R a I 3 s 3 n + v a the first estimate in (37 and (48 permit to obtain (47 Let us compare the two inequalities (43 and (46 Indeed they are equivalent for energies of order δ 3/ For energies order smaller than δ 3/ (46 is better (43 which is then more relevant in general for thin structures 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 43 There exists a constant C which does not depend upon δ such that for all v ( H (Q δ 3 such that v = I d on Γ 0δ (49 x v + ( x v T I 3 (L (Q δ 9 C dist( x vso(3 L (Q δ { + } δ dist( xvso(3 5/ L (Q δ Proof In view of the decomposition (36 and Theorem 33 we have (40 x v + ( x v T I 3 (L (Ω δ 9 C dist( x vso(3 L (Q δ + Cδ / R a + R T a I 3 (L (ω 9 Due to the equalities R a + R T a I 3 = R a R a R T a + R T a R a R T a + R a (I 3 R a R T a + (R a R T a I 3 and to the first estimate in (34 it follows that = (R a I 3 R T a + R a (I 3 R a R T a + (R a R T a I 3 (4 R a + R T a I 3 (L (ω 9 C (R a I 3 (L (ω 9 + C δ / dist( xvso(3 L (Q δ Since (R a I 3 (L (ω 9 C R a I 3 (L 4 (ω 9 and the fact that the space ( H (ω 3 3 is continuously imbedded in ( L 4 (ω 3 3 we deduce that (4 (R a I 3 (L (ω 9 C δ 3 dist( xvso(3 L (Q δ From (40 (4 and (4 we finally get (49

14 Remark 44 In view of (37 and since the field R a belongs to (L (ω 3 3 the function (R a I 3 belongs to ( H (ω 3 3 with Hence with Lemma 4 (R a I 3 (L (ω 9 (40-(4 (R a I 3 s α (L (ω 9 C δ 3/ dist( xvso(3 L (Q δ C δ 3/ dist( xvso(3 L (Q δ which gives together with x v + ( x v T I 3 (L (Q δ 9 C δ dist( xvso(3 L (Q δ Notice that the above estimate is worse than (49 at least as soon as the energy is smaller than δ / Let us emphasize that in view of estimates (37-(38 (43 and (49 one can distinguish two critical cases for the behavior of the quantity dist( x vso(3 L (Q δ { O(δ 3/ dist( x vso(3 L (Q δ = O(δ 5/ Estimates (4-(43 show that the behavior dist( x vso(3 L (Q δ O(δ / also corresponds to an interesting case but the estimates (37 and (48 show that the decompositions given in Theorems 33 and 34 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 (Q δ O(δ κ / κ 5 Rescaling Ω δ As usual when dealing with a thin shell we rescale Ω δ using the operator (Π δ w(s s S 3 = w(s s s 3 for any s Ω δ defined for eg w L (Ω δ for which (Π δ w L (Ω The estimates (37 on v a transposed over Ω lead to (5 Π δ v a (L (Ω 3 Cδ/ dist( x vso(3 L (Q δ Πv a (L s (Ω 3 C δ dist( xvso(3 / L (Q δ Πv a (L (Ω s 3 C δ dist( xvso(3 / L (Q δ and estimates (46 on v I d give Πv a S 3 (L (Ω 3 Cδ/ dist( x vso(3 L (Q δ (5 Π δ (v I d (L (Ω 3 C δ dist( xvso(3 3/ L (Q δ Π δ(v I d s (L (Ω 3 C δ 3/ dist( xvso(3 L (Q δ Π δ(v I d s (L (Ω 3 C δ 3/ dist( xvso(3 L (Q δ Π δ(v I d S 3 (L (Ω 3 C δ / dist( xvso(3 L (Q δ 3

15 6 Asymptotic behavior of the Green-St Venant s strain tensor in the case κ = Let us consider a sequence of deformations v δ of ( H (Q δ 3 such that (6 dist( x v δ SO(3 L (Q δ Cδ 3/ For fixed δ > 0 the deformation v δ is decomposed as in Theorem 33 and the terms of this decomposition are denoted by V δ R aδ and v aδ If moreover the hypothesis (37 holds true for the sequence v δ then v δ can be alternatively decomposed through (36 in terms of V δ R δ and v δ so that the estimates (38 of Theorem 35 are satisfied uniformly in δ In what follows we investigate the behavior of the sequences V δ R aδ and v aδ Indeed due to (30 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 (37 (5 and (5 lead to the following lemma Lemma 6 There exists a subsequence still indexed by δ such that (6 V δ V strongly in ( H (ω 3 R aδ R weakly in ( H (ω 3 3 and strongly in ( L (ω 3 3 δ Π δv aδ v weakly in ( L (ω;h ( 3 ( Vδ R aδ t α Z α weakly in ( L (ω 3 δ s α (R T δ aδr aδ I 3 0 weakly in ( L (ω 3 3 where R belongs SO(3 for ae (s s ω We also have V ( H (ω 3 and (63 V s α = Rt α The boundaries conditions (64 V = φ R = I 3 on γ 0 hold true Moreover we have (65 { Π δ v δ V strongly in ( H (Ω 3 Π δ ( x v δ R strongly in ( L (Ω 9 Proof The convergences (6 are direct consequences of Theorem 33 and estimate (48 excepted for what concerns the last convergence which will be established below The compact imbedding of ( H (ω 3 3 in ( L 4 (ω 3 3 and the first convergence in (6 permit to obtain (66 { R aδ R strongly in ( L 4 (ω 3 3 det(r aδ det(r strongly in L 4/3 (ω 4

16 These convergences and estimates (34 prove that for ae (s s ω: R(s s SO(3 The relation (63 and (64 and the convergences (65 are immediate consequences of Theorem 33 and of the above results We now turn to the proof of the last convergence in (6 We first set (δ[ s ] R aδ (s s = R aδ δ δ [ s δ ] ae in ω where [t] denote the integer part of the real t From (34 (35 and (6 we have (67 R aδ R aδ (L (ω 3 3 Cδ From (66 and the above estimate we deduce that (68 Raδ R strongly in (L (ω 3 3 Now we derive the weak limit of the sequence δ (R aδ R aδ Let Φ be in C0 (Ω 3 3 and set M δ (Φ(s s = ( [ s ] [ s ] δ + z δδ + z δ dz dz for ae (s s in ω We recall that (see [] δ δ ]0[ Φ ( Φ Mδ (Φ 0 weakly in (L (ω 3 3 δ M δ (Φ Φ strongly in (L (ω 3 3 We write ω δ (R aδ R aδ Φ = = ω ω ( R aδ Φ Mδ (Φ + δ ( R aδ Φ Mδ (Φ + δ δ (R aδ R aδ M δ (Φ ω ω ( Raδ s + R aδ M δ (Φ + K δ s where K δ Cδ R aδ (L (ω 3 3 Φ (L (ω 3 3 In view of the properties of M δ(φ recalled above of (6 and (6;4 we deduce from the above equality that δ (R aδ R aδ ( R + R s s In order to prove the last convergence of (6 we write weakly in (L (ω 3 3 (R T δ aδr aδ I 3 = ( RT δ aδ (R aδ R aδ + (R aδ R aδ T Raδ + (R aδ R aδ T (R aδ R aδ and we use estimates (34 and (67 the strong convergence (68 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 For the same subsequence as in Lemma 6 we have (69 (60 δ Π ( δ ( x v δ T x v δ I 3 (t t n T E(t t n weakly in (L (Ω 9 where the symmetric matrix E is equal to R S 3 n Rt + Z Rt s R S 3 n Rt + { } Z Rt + Z Rt s R S 3 n Rt + Z Rt s 5 v Rt + S 3 Z Rn v Rt + S 3 Z Rn v Rn S 3

17 and where (t t n denotes the 3 3 matrix with first column t second column t and third column n and where (t t n T = ( (t t n T Proof First from estimate (37 equalities (33 and the convergences in Lemma 6 we obtain Then thanks to the identity ( R Πδ ( x v δ R aδ tα S 3 n + Z α weakly in ( L (Ω 3 δ s α ( v Πδ ( x v δ R aδ n weakly in ( L (Ω 3 δ S 3 δ Π ( δ ( x v δ T x v δ I 3 = δ Π ( δ ( x v δ R aδ T ( x v δ R aδ + δ RT aδπ δ ( x v δ R aδ + δ Π δ( x v δ R aδ T R aδ + ( R T δ aδ R aδ I 3 and again to estimate (37 and Lemma 6 we deduce that (6 δ Π ( δ ( x v δ T x v δ I 3 (t t n T( R R S 3 n + Z S 3 n + Z s s +R T( S 3 R s n + Z S 3 R s n + Z weakly in (L (Ω 9 v TR S 3 v (t t n S 3 Now remark that (6 R n Rt = R n Rt s s Indeed deriving the relation R T R = I 3 with respect to s α shows that R T R s α + RT s α R = 0 Hence there exists an antisymmetric matrix field A α L (ω; R 3 3 such that R s α = RA α Moreover there exists a field a α belonging to ( L (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 V = R t α + R t α = RA β t α + R φ s α s β s β s β s α s β It implies that A t = A t from which (6 follows Taking into account the definition of the matrix E convergence (6 and the equality (6 show that (69 holds true Remark 63 There exists a constant C such that R (L + R ( (L R L C n Rt + R L n Rt + R L n Rt s (ω 9 s α (ω 9 s (ω s (ω s (ω With the same notation as in the proof of Theorem 6 we have R s = Aα = a α (L (ω 9 (L (ω 9 α (L (ω 3 6

18 Recalling that a t = a t we obtain a α n = 0 and then R n = a α n s α (L (ω 3 (L (ω = a α 3 (L (ω = R 3 s α (L (ω 9 Remark It is well known that the constraint V = Rt and V = Rt together the boundary s s conditions are strong limitations on the possible deformation for the limit d 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 > Let us consider a sequence of deformations v δ of ( H (Q δ 3 such that (7 dist( x v δ SO(3 L (Q δ Cδ κ / with κ > We use the decomposition (36 of a deformation and the estimates (38 of Theorem 34 These estimates and the boundary condition (Lemma 4 lead to the following convergences: R δ I 3 strongly in ( H (ω 9 (7 Π δ v δ φ strongly in ( H (Ω 3 Π δ ( x v δ I 3 strongly in ( L (Ω 9 In view of these convergences we now study the asymptotic behavior of the sequence of displacements belonging to ( H (Q δ 3 Due to the decomposition (36 we write u δ (x = v δ (x x (73 u δ (s = U δ (s s + s 3 (R δ I 3 (s s n(s s + v δ (s s Ω δ where U δ (s s = V δ (s s φ(s s Thanks to the estimates (38 we obtain the following lemma: Lemma 7 There exists a subsequence still indexed by δ such that (74 ( ( Rδ δ κ I 3 A weakly in H (ω 9 and strongly in ( L (ω 9 δ κ U δ U strongly in ( H (ω 3 δ κ Π δv δ v weakly in ( L (ω;h ( 3 ( Uδ δ κ (R δ I 3 t α Z α weakly in ( L (ω 3 s α and (75 δ κ Π δu δ U strongly in (H (Ω 3 δ κ Π δ( x u δ A strongly in (L (Ω 9 7

19 where A (H (ω 9 U (H (ω 3 v ( L (ω;h ( 3 and Zα (L (ω 3 Moreover we have (76 U = 0 A = 0 on γ 0 and U s α = At α and U ( H (ω 3 We now show that the matrix A is actually an antisymmetric matrix Using the first convergence in (7 and the first convergence in (74 we get ( ( δ κ RT δ Rδ I 3 A weakly in H (ω 3 3 ( The matrix R δ belongs to SO(3 hence R T δ Rδ I 3 = I3 R T δ It follows that the matrix A is an antisymmetric matrix There exists a field R ( H (ω 3 (with R = 0 on γ0 due to (76 such that for all x R 3 we have (77 Ax = R x From (76 and the above equality we obtain (78 U s α = R t α 7 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 γ 0 = We equip ( H γ 0 (ω 3 with the following inner product: {v H (ω v = 0 on γ 0 } (UV ( H γ 0 (ω 3 ( H γ0 (ω 3 < U V >= ω [ U s V s + U s V s ] For any U (H (ω 3 we set e (U = U t e (U = { U t + U } t e (U = U t s s s s The spaces of inextentional and extensional displacements are respectively defined by D In = { U ( Hγ 0 (ω } 3 e (U = e (U = e (U = 0 D Ex = ( D In where ( D In is the orthogonal of DIn in the space ( H γ 0 (ω 3 For all U DIn there exists a unique field R ( L (ω 3 such that U s = R t and { R U D In } is a closed subspace of ( L (ω 3 8 U s = R t

20 We equip D Ex with the norm (79 U D Ex U Ex = e(u L (ω + e (U L (ω + e (U L (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 { Un } n N converges to U D Ex the sequences { e (U n } n N { e (U n } n N and { e (U n } n N strongly converge in L (ω and their limits depend only on U That is the reason why we will denote these limits e (U e (U and e (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 s = (s s hence t = e t = e 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 γ 0 (ω and due to Korn s inequality in ( H γ0 (ω the norm Ex is equivalent to the H norm on D Ex 7 Limit of the Green-St Venant s strain tensor for κ > We consider the sequence v δ introduced in Section 7 which satisfies dist( x v δ SO(3 L (Q δ Cδ κ / and the associated displacement u δ = v δ I d We write the displacement U δ of the mid-surface as the sum of an inextensional displacement and an extensional one as in Section 6 (70 U δ = U Iδ + U Eδ U Iδ D In U Eδ D Ex We first give the estimates on U Iδ and U Eδ in the following lemma Lemma 7 We have (7 UIδ (H (ω 3 Cδ κ UEδ (H (ω 3 Cδ κ U Eδ Ex Cδ κ ( + δ κ 3 The constants do not depend on δ Then we can choose the subsequence in Lemma 7 such that (7 δ κ U Iδ U weakly in (H (ω 3 δ κ U Eδ 0 weakly in (H (ω 3 and moreover (73 if < κ < 3 if κ 3 δ κ 4 U Eδ U E weakly in D Ex δ κ U Eδ U E weakly in D Ex The convergences in (73 are equivalent to the weak convergences in L (ω of e (U Eδ e (U Eδ and e (U Eδ Proof The two first estimates of (7 follow from (74 and from the orthogonality of U Iδ and U Eδ Now notice that (74 e αα (U δ = e αα (U Eδ and e αβ (U δ = e (U Eδ 9

21 We denote by A δ the antisymmetric part of R δ Notice that R δ + R T δ I 3 (L (ω 3 3 R δ I 3 (L 4 (ω 3 3 C R δ (L (ω 8 Then from estimates (38 and (7 we deduce that U δ (L A δ t α Cδ κ + Cδ κ 4 s α (ω 3 Then by definition of the norm Ex we get the third estimate in (7 and then the convergences in (73 The following theorem gives the expression of the limit of the Green-St Venant s tensor Theorem 73 Let us set (75 Z αβ = { Zα t β + Z β t α } u = v + S 3 where ( t t is the contravariant basis of ( t t For a subsequence we have ( Z n t + S 3 ( Z n t (76 δ κ Π ( δ ( x v δ T x v δ I 3 (t t n T E(t t n weakly in (L (Ω 9 where the symmetric matrix E is defined by (77 E = Moreover if < κ < 3 then we have [ R ] [ R S 3 n t + Z S 3 s (78 e αβ (U E + U U = 0 s α s β and if κ 3 then we have ] u n t + Z t s S 3 [ R ] u S 3 n t + Z t s S 3 u n S 3 e αβ (U E + U U if κ = 3 (79 Z αβ = s α s β e αβ (U E if κ > 3 Proof First we have v s = x v ( t + s 3 n s v s = x v ( t + s 3 n s v s 3 = x v n As a consequence of the above formulaes of (33 and of the convergences in Lemma 5 we have (70 ( R Πδ δ κ ( x v δ R δ tα S 3 n + Z α weakly in ( L (Ω 3 s α ( v Πδ δ κ ( x v δ R δ n weakly in ( L (Ω 3 S 3 0

22 Then thanks to the identity δ κ Π ( δ ( x v δ T x v δ I 3 = δ κ Π ( δ ( x v δ R δ T x v δ + δ κ RT δ Π δ ( x v δ R δ and to convergences (7 and (70 we deduce that δ κ Π ( δ ( x v δ T x v δ I 3 (t t n T E(t t n weakly in (L (Ω 9 where the symmetric matrix E is equal to E = ( R R v T(t S 3 n + Z S 3 n + Z t n s s S 3 + (t t n T( S 3 R s n + Z S 3 R s n + Z Deriving the equality (78 with respect to s and s gives hence U s s = R s t + R φ = R t + R s s s [ R ] [ R ] n t = n t s s Introducing Z αβ and u we obtain the expression (77 for E Below we show (78 and (79 Due to (74 we first have (7 v S 3 φ s s [e δ κ αβ (U δ ( Rδ + R T ] δ I 3 tα t β Z αβ weakly in L (ω Recalling the identity ( R δ + R T δ I 3 tα t β = (R δ I 3 t α (R δ I 3 t β and using the first convergence in (74 we deduce that (7 δ κ 4 ( Rδ + R T δ I 3 tα t β At α At β strongly in L (ω In the case < κ < 3 we have [e δ κ 4 αβ (U δ ( Rδ + R T ] δ I 3 tα t β 0 strongly in L (ω δ κ 4 e αβ(u δ e αβ (U E weakly in L (ω Thanks to (76 and (7 we obtain (78 In the case κ 3 then convergences (73 (7 and (7 permit to obtain the expression of Z αβ in terms of e αβ (U E and U 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 Y =]0[ B 3 = { } { } x R 3 ; x S 3 = x R 3 ; x =

23 We denote R aθ the rotation with axis directed by the vector a S 3 and with angle of rotation about this axis θ R (A x R 3 R aθ (x = cos(θx + ( cos(θ < xa > a + sin(θa x Let R 0 and R be two matrices in SO(3 Matrix R 0 represent the rotation R a0θ 0 and matrix R represent the rotation R aθ The linear transformation in R 3 x ( sin(θ a sin(θ 0 a 0 x has for matrix R R 0 (R R 0 T and we have sin(θ a sin(θ 0 a 0 = R R 0 (R R 0 T R R 0 To any matrix R in SO(3 we associate the vector b = sin(θa where R is the matrix of the rotation R aθ This map is continuous from SO(3 into B 3 and from the above inequality we obtain b R I 3 If cos(θ using the vector b we can write the rotation R aθ as (A x R 3 R aθ (x = cos(θx + Let R 0 and R be two matrices in SO(3 such that < xb > b + b x + cos(θ R 0 R < Now we define a path f from R 0 to R : if R = R 0 we choose the constant function f(t = R 0 t [0] if R R 0 we set R = R 0 R there exists a unique pair (a θ S 3 ]0π[ such that the matrix R represent the rotation with axis directed by the vector a and with the angle θ We consider the rotations field R a(tθ(t given by formula (A where a(t = a θ(t = tθ t [0] and we define f(t as the matrix of the rotation R 0 R a(tθ(t where R 0 is the rotation with matrix R 0 Lemma A The path f belongs to W (0;SO(3 and satisfies (A3 f(0 = R 0 f( = R R 0 f(t R 0 R df dt π (L (0 9 R R 0 Proof One has df dt = θ π (L (0 9 R I 3 = π R R 0

24 Moreover R 0 f(t = I 3 R 0 f(t = sin ( θ t sin ( θ = I 3 R = R 0 R Lemma A Let R 00 R 0 R 0 and R be four matrices belonging to SO(3 and satisfying (A4 R 0 R 00 R 0 R 00 R R 0 R R 0 There exists a function R W (Y ;SO(3 such that (A5 { R(00 = R00 R(0 = R 0 R(0 = R 0 R( = R R (L (Y 9 C{ R 0 R 00 + R 0 R 00 + R R 0 + R R 0 } and where the functions x R(x 0 x R(x x R(0x and x R(x are paths given by Lemma A Proof We denote f 000 the path from R 00 to R 0 f 0 the path from R 0 to R f 000 the path from R 00 to R 0 and f 0 the path from R 0 to R given by Lemma A From Lemma A we have t [0] { f000 (t R 00 f 0 (t R 00 f 000 (t R 00 f 0 (t R 00 For any t [0] to matrix R 00 f 000(t we associate the vector b 000 (t to matrix R 00 f 0(t we associate the vector b 0 (t to matrix R 00 f 000(t we associate the vector b 000 (t and to matrix R 00 f 0(t we associate the vector b 0 (t Let b be the vectors field defined by b 000 (0 ( = b 000 (0 if (x x = (00 x b 000 (x + x b 000 (x if 0 < x + x x + x x + x b(x x = x b 0 (x + x b 0 (x if x + x < x x x x b 0 ( ( = b 0 ( if (x x = ( This function belongs to ( W (Y 3 and satisfies (x x Y b(x x Now we introduce the rotations field R(x x given by formula (A where b(x x is defined above and where θ(x x = arccos < b(x x b(x x > (x x Y 3

25 Let R(x x be the matrix of the rotation R 00 R(x x where R 00 is the rotation with matrix R 00 It is easy to check that R satisfies the conditions (A5 Corollary of Lemma A Let R a be the Q interpolate of the matrices R 00 R 0 R 0 and R There exists a strictly positive constant C such that R R a (L (Y 9 C{ R 0 R 00 + R R 0 + R 0 R 00 + R R 0 } References [] D Blanchard A Gaudiello G Griso Junction of a periodic family of elastic rods with a 3d plate I J Math Pures Appl (9 88 (007 no [] D Blanchard A Gaudiello G Griso Junction of a periodic family of elastic rods with a thin plate II J Math Pures Appl (9 88 (007 no -33 [3] D Blanchard G Griso Microscopic effects in the homogenization of the junction of rods and a thin plate Asympt Anal 56 (008 no -36 [4] D Blanchard G Griso Decomposition of deformations of thin rods Application to nonlinear elasticity Analysis and Applications Vol 7 Nr (009-7 [5] PG Ciarlet Mathematical Elasticity Vol II Theory of plates North-Holland Amsterdam (997 [6] PG Ciarlet Mathematical Elasticity Vol III Theory of shells North-Holland Amsterdam (000 [7] PG Ciarlet Un modèle bi-dimentionnel non linéaire de coques analogue à celui de WT Koiter C R Acad Sci Paris Sér I 33 ( [8] PG Ciarlet and C Mardare Continuity of a deformation in H as a function of its Cauchy-Green tensor in L J Nonlinear Sci 4 (004 no (005 [9] PG Ciarlet and C Mardare An introduction to shell theory Preprint Université PM Curie (008 [0] PG Ciarlet and P Destuynder A justification of a nonlinear model in plate theory Comput Methods Appl Mech Eng 7/8 ( [] G Friesecke R D James and S Müller A theorem on geometric rigidity and the derivation of nonlinear plate theory from the three-dimensional elasticity Communications on Pure and Applied Mathematics Vol LV (00 [] G Friesecke R D James and S Müller A hierarchy of plate models derived from nonlinear elasticity by Γ-convergence (005 [3] G Friesecke R D James MG Mora and S Müller Derivation of nonlinear bending theory for shells from three-dimensionnal nonlinear elasticity by Gamma convergence C R Acad Sci Paris Ser I 336 (003 [4] G Griso Decomposition of displacements of thin structures J Math Pures Appl 89 ( [5] G Griso Asymptotic behavior of curved rods by the unfolding method Math Meth Appl Sci 004; 7: 08-0 [6] G Griso Asymptotic behavior of structures made of plates Analysis and Applications 3 ( [7] H Le Dret and A Raoult The nonlinear membrane model as variational limit of nonlinear threedimensional elasticity J Math Pures Appl 75 ( [8] H Le Dret and A Raoult The quasiconvex envelope of the Saint Venant-Kirchhoff stored energy function Proc R Soc Edin A 5 (

26 [9] O Pantz On the justification of the nonlinear inextensional plate model C R Acad Sci Paris Sér I Math 33 (00 no [0] O Pantz On the justification of the nonlinear inextensional plate model Arch Ration Mech Anal 67 (003 no