Thin-Walled Beams: the Case of the Rectangular Cross-Section

Thin-walled beams: the case of the rectangular cross-section Lorenzo Freddi ∗ Antonino Morassi† Roberto Paroni‡ Abstract In this paper we present an asymptotic analysis of the three-dimensional problem for a thin linearly elastic cantilever Ωε = ωε × (0, l) with rectangular cross-section ωε of sides ε and ε2 , as ε goes to zero. Under suitable assumptions on the given loads, we show that the threedimensional problem converges in a variational sense to the classical one-dimensional model for extension, flexure and torsion of thin-walled beams. 2001 AMS Mathematics Classification Numbers: 74K20, 74B10, 49J45 Keywords: thin-walled cross-section beams, linear elasticity, Γ-convergence, dimension reduction 1 Introduction It is common practice in structural engineering to consider structures with extension in one or more directions small compared to the remaining. Such a situation arises, for instance, in the study of flat domains with small thickness (plates) or of cylinders with transversal section having small diameter (beams). ∗ Dipartimento di Matematica e Informatica, via delle Scienze 206, 33100 Udine, Italy, email: freddi@dimi.uniud.it † Dipartimento di Georisorse e Territorio, via Cotonificio 114, 33100 Udine, Italy, email: antonino.morassi@uniud.it ‡ Dipartimento di Architettura e Pianificazione, Università degli Studi di Sassari, Palazzo del Pou Salit, Piazza Duomo, 07041 Alghero, Italy, email: paroni@uniss.it 1 Approximate mechanical models for thin structures are thousand of years old and go back to the pioneering works in Mechanics of Euler, D. Bernoulli, Navier and Kirchhoff, see [13]. The classical theories are usually based on some a-priori assumptions, motivated by the smallness of certain dimensions with respect to others, on the deformation of the body or on the induced stress field. In the last two-three decades a considerable amount of work has been done in order to rigorously justify the a-priori assumptions on which classical theories are based. In particular, approaches based on rigorous asymptotic expansion (mainly due to the French school) or inspired by the Γ-convergence of energy functionals (proposed by E. De Giorgi in [8]) have been successfully used, in deriving one or two-dimensional classical mechanical models for thin structures in linear and non-linear elasticity starting from three-dimensional problems. In this paper we shall be concerned with an asymptotic analysis for a class of linearly elastic thin beams when the thickness of the transversal cross-section goes to zero. Dimension reduction problems from three dimensions to one have received a great deal of attention in recent years and numerous and interesting results have been obtained, see [5] for a comprehensive bibliographical and historical survey. In [1] a general framework based on the Γ-convergence on varying domains has been applied to give a justification of the classical one-dimensional mechanical model for extension, flexure and torsion of slender cylinders having circular crosssection. An extension to slender cylinders under more general boundary conditions and with cross-sections having Lipschitz boundary has been presented in [18]. By adapting and refining the ideas introduced by Ciarlet and Destuynder of rescaling domains and field displacements, see [6], Le Dret showed in [12] the convergence of the displacements and stresses for slender cylinders with Lipschitz cross-section and also discussed how to treat some more general cases, involving beam shapes with spikes and holes. All of the above results are based on a common assumption, namely the threedimensional variational problem is formulated on a family of cylinders which are obtained by scaling a reference cylindrical domain Ωε ⊂ R3 by a single factor ε, ε > 0, in its cross-section plane, that is Ωε = ωε × (0, l) ⊂ R3 , where l is the length of the cylinder, ωε = εω is its cross-section and ω ⊂ R2 is a (simply connected) open bounded set with Lipschitz boundary. A variant of these cases has been considered in [4] where the cross-section ω was scaled not simply by ε but by a factor rε (x3 ) (not depending on the coordinates x1 and x2 of the cross-section plane). This way allows very rapid variations of the thickness of the domain and produces a one-dimensional model for thin notched beams. In several areas of civil, aeronautic and mechanical engineering, design and technological requirements force the use of the so-called thin-walled cross-section 2 beams, that is slender cylinders in which the transversal cross-section is the union of several walls, whose thickness is very small compared with the diameter of the cross-section. To give an example, thin-walled tubes are often used in the structures where beams are subjected to high twisting moment or to important transversal forces. Hollow cross-sections are in fact most efficient in resisting torsion and flexure because, as a consequence of the advantageous distribution of stresses, they ensure high rigidity and strength with relatively low weight. From the mathematical point of view, the main novelty in dealing with thinwalled cross-section beams is the presence of two scaling factors: one factor is the ratio between the diameter of the cross-section and the beam length, say ε; the other is the ratio between the wall thickness and the diameter of the cross-section, say εα , with α > 1. As far as the asymptotic behavior of the three-dimensional energy functional when ε goes to zero, known results for thin-walled cross-section beams are based on the De Saint-Venant classical principle and, therefore, they essentially involve an asymptotic study of certain two-dimensional Neumann problems defined on the transversal cross-section when the thickness of the walls goes to zero. The limit behavior of the torsion problem for thin-walled beams has been recently studied in [19], [20], see also [14], [15] for an alternative development based on Γ-convergence arguments applied on varying domains and [16] for an asymptotic analysis of the flexure problem. The acceptation of the De Saint-Venant’s principle has important consequences since, roughly speaking, it implies the loss of one dimension. In fact, all the above investigations involve a reduction from two dimensions to one. This paper represents a first step of a line of research which aims to a rigorous deduction of the one-dimensional theory for thin-walled beams from the threedimensional linear elasticity via Γ-convergence techniques. Here we consider a thinwalled cantilever Ωε = ωε × (0, l), made of homogeneous linear isotropic material, with a rectangular cross-section ωε of sides ε and ε2 . By merging and refining the different techniques of [1], [2] and [18], we prove that the three-dimensional elasticity problem converges in a variational sense to a one-dimensional problem as ε goes to zero. The limit problem is defined by a functional which includes the extension, the flexure and the torsion energies of the classical thin-walled beam model, see Theorem 5.2 for a precise statement. A further step of the analysis, which takes into account the case where the cross-section ωε has a multirectangular shape, is developed in a forthcoming paper [9]. The Γ-convergence of the family of energy functionals defined on Ωε gives not only the convergence of the energy of the three-dimensional problem to the corresponding energy of the limit problem, but permits also to obtain a remarkable amount of information about the structure and the behavior of the minimizers of the three-dimensional problem as ε goes to zero. In particular, the recovering 3 sequence used in the proof of the limsup inequality of Theorem 5.2, allows to obtain a good approximation of the displacement field solution of the three-dimensional problem. The components of this recovering sequence show scaling with different powers of ε, and this reflect the fact, well known in practical applications, that for beams with thin-walled rectangular section some displacements are bigger than others. The plan of the paper is as follows. In the next Section 2 we shall introduce the three-dimensional variational problem and some notations. In Section 3 we rewrite the three-dimensional problem as a variational problem for a rescaled energy defined in a fixed domain Ω1 , which is obtained by making a dilatation of Ωε in the cross-section plane. Section 4 is devoted to the proof of some compactness results for suitable families of functions defined on Ω1 . These compactness results will be obtained through Korn inequalities, stated in two and three dimensions, with a constant independent of ε. In Section 5 we prove the Γ-convergence of the family of three-dimensional energy functionals to the limit energy as ε goes to zero and the variational consequences are discussed in Section 6 and 7. Finally, the strong convergence of minimizers is proved in Section 8. Notation. Throughout this article, and unless otherwise specified, we use the Einstein summation convention. Moreover we use the following convention for indexing vector and tensor components: Greek indices α, β and γ take their values in the set {1, 2} and Latin indices i, j in the set {1, 2, 3}. The symbols L2 (A; B) and H s (A; B) denote the standard Lebesgue and Sobolev spaces of functions defined on the domain A and taking values in B, with the usual norms k · kL2 (A;B) and k · kH s (A;B) , respectively. When B = R or when the right set B is clear from the context, we will simply write L2 (A) or H s (A), sometimes even in the notation used for norms. With a little abuse of notation, and because this is a common practice and does not give rise to any mistake, we use to call “sequences” even those families indicized by a continuous parameter ε ∈ (0, 1]. 2 The 3-dimensional problem We consider a three-dimensional body which is at rest in the placement Ωε := ωε × (0, ℓ) ⊂ R3 , where ωε := {(x1 , x2 ) : |x1 | < aε2 /2, |x2 | < bε/2} ⊂ R2 and ε ∈ (0, 1]. For any x3 ∈ (0, ℓ) we further set Sε (x3 ) := ωε × {x3 }. 4 Henceforth we shall refer to Ωε as the reference configuration of the body and denote by Du(x) + DuT (x) Eu(x) := sym(Du(x)) := , (1) 2 the strain of u : Ωε → R3 . In what follows we consider the situation in which the body is subject only to dead body forces bε , so that the equilibrium equations write as  divT + bε = 0 in Ωε ,    T = CEu in Ωε , (2) Tn = 0 on ∂Ωε \Sε (0),    u=0 on Sε (0). We consider an homogeneous isotropic material, so that CA = 2µA + λ(trA) I for every symmetric matrix A. I denotes the identity matrix of order 3. We assume µ > 0 and λ ≥ 0 so to have, for every symmetric tensor A, CA · A ≥ µ|A|2 , (3) where · denotes the scalar product. Define  1 H# (Ωε ; R3 ) := u ∈ H 1 (Ωε ; R3 ) : u = 0 on Sε (0) . Due to the coercivity condition (3) and the strict convexity of the integrand, the energy functionals Z Z 1 CEu · Eu dx − Jε (u) := bε · u dx, (4) 2 Ωε Ωε admit, for every ε > 0, a unique minimizer among all competing displacements 1 (Ω ; R3 ). As already explained in the introduction our aim is to study u ∈ H# ε the asymptotic behavior of such minimizers as ε goes to 0, through the theory of Γ-convergence, for an account of it we refer to the books of Braides [3] and Dal Maso [7]. 3 The rescaled problem To state our results it is convenient to stretch the domain Ωε along the transverse directions x1 and x2 in a way that the transformed domain does not depend on ε. Let us therefore set ω := ω1 , Ω := Ω1 , S(x3 ) := S1 (x3 ) and let p ε : Ω → Ωε 5 be defined by pε (y) = pε (y1 , y2 , y3 ) = (ε2 y1 , εy2 , y3 ). Let us consider the following 3 × 3 matrix   D1 v D2 v Hε v := , , D v , 3 ε2 ε (5) where Di v denotes the column vector of the partial derivatives of v with respect to xi , i = 1, 2, 3. We will use moreover the following notation Eε v := sym( Hε v), Wε v := skw( Hε v), (6) and also denote by Wv := W1 v, the skew symmetric part of the gradient. Let  1 H# (Ω; R3 ) := v ∈ H 1 (Ω; R3 ) : v = 0 on S(0) ; 1 (Ω; R3 ) → R ∪ {+∞} defined by then we can consider the rescaled energy Iε : H# Iε (v) := ε13 Jε (v ◦ p−1 ε ), i.e., Z Z 1 bε ◦ pε · v dy. Iε (v) = CEε v · Eε v dy − 2 Ω Ω We further suppose the loads to have the following form m(y3 ) y2 , I0 m(y3 ) y1 , bε2 ◦ pε (y) = ε3 b2 (y) + ε2 I0 bε3 ◦ pε (y) = ε2 b3 (y), bε1 ◦ pε (y) = ε4 b1 (y) − ε3 (7) with b = (b1 , b2 , b3 ) ∈ L2 (Ω; R3 ), and m ∈ L2 (0, ℓ). Above I0 denotes the polar moment of inertia of the section ω, Z 1 I0 := y12 + y22 dy1 dy2 = (a3 b + ab3 ). 12 ω We note that while b has the units of a force per unit of volume, m has the units of a force, or, equivalently, of a moment per unit of length. The scalings of the loads are chosen in a way to keep the displacements bounded as ε goes to zero. With the loads given by (7) the energy Iε (v) can be rewritten as Z Z 1 v 2 v3 Iε (v) = CEε v · Eε v dy − ε4 b · (v1 , , 2 ) dy + 2 Ω ε ε Ω (8) Z ℓ 4 ε −ε m ϑ (v) dy3 , 0 6 where we have set 1 ϑ (v)(y3 ) := I0 ε Z ω y1 y2 v2 (y1 , y2 , y3 ) − v1 (y1 , y2 , y3 ) dy1 dy2 . 2 ε ε (9) We note that if v ∈ L2 (Ω; R3 ) then ϑε (v) ∈ L2 (0, ℓ). A similar statement holds if we replace L2 with H 1 . 4 Compactness lemmata On the untransformed domain Ωε , the following Korn-like inequality holds. Theorem 4.1 There exists a constant C > 0 such that Z Z
C 2 2 |u| + |Du| dx ≤ 4 |Eu|2 dx ε Ωε Ωε (10) for every u ∈ H 1 (Ωε ; R3 ) with u = 0 on Sε (0). Proof. Divide the section ωε in squares of size ε2 and apply Korn’s inequality (the one obtained by Anzellotti, Baldo and Percivale in [1]; see also [18] and Kondrat’ev and Oleinik [10], Theorem 2) to each beam of length ℓ and with section a square with side proportional to ε2 . Then sum over all the obtained inequalities. ✷ To prove the compactness of the displacements we need the following scaled Korn inequality. Theorem 4.2 There exists a constant C > 0 such that Z Z
C 2 2 ε 2 |Eε u|2 dy |(u1 , u2 /ε, u3 /ε )| + |H u| dy ≤ 4 ε Ω Ω (11) 1 (Ω; R3 ) and every 0 < ε ≤ 1. for every u ∈ H# R R Proof. The inequality Ω |Hε u|2 dy ≤ εC4 Ω |Eε u|2 dy is simply obtained by rescaling inequality (10). To show that Z Z C 2 2 |Eε u|2 dy, |(u1 , u2 /ε, u3 /ε )| dy ≤ 4 ε Ω Ω it suffices to set v := (u1 , u2 /ε, u3 /ε2 ), notice that |Eε u| ≥ ε2 |Ev| and apply the standard Korn inequality to v on the domain Ω (see for instance [17], Theorem 2.7). ✷ 7 Let  1 HBN (Ω; R3 ) := v ∈ H# (Ω; R3 ) : (Ev)iα = 0 for i = 1, 2, 3 α = 1, 2 , (12) be the space of Bernoulli-Navier displacements on Ω. This space can be characterized also as follows (see Le Dret [11], Section 4.1)  1 (Ω; R3 ) : ∃ ξ ∈ H 2 (0, ℓ), HBN (Ω; R3 ) = v ∈ H# α # 1 (0, ℓ) such that ∃ ξ3 ∈ H# (13) vα (y) = ξα (y3 ), v3 (y) = ξ3 (y3 ) − yα ξα′ (y3 ) . The remaining part of this section will be devoted to prove some compactness lemmata which will be stated under the common assumption that uε be a sequence 1 (Ω; R3 ) such that of functions in H# kEε uε kL2 (Ω;R3×3 ) ≤ Cε2 , (14) for some constant C and every 0 < ε ≤ 1. Lemma 4.3 Let us assume (14). Then, for any sequence of positive numbers εn converging to 0 there exist a subsequence (not relabeled) and a pair of functions v ∈ HBN (Ω; R3 ) and ϑ ∈ L2 (Ω) such that (as n → ∞) uε1n , Wεn uεn uε2n uε3n
→v , εn ε2n  weakly in H 1 (Ω; R3 ),  0 −ϑ D3 v1   ϑ 0 0  weakly in L2 (Ω; R3×3 ). → −D3 v1 0 0 (15) (16) Proof. It is convenient to set vε := (uε1 , uε2 /ε, uε3 /ε2 ). It is easily checked that for ε ≤ 1, |Eε uε | ≥ ε2 |Evε |, hence, by (14), Evε is uniformly bounded in L2 (Ω; R3×3 ), and by Korn’s inequality vε is uniformly bounded in H 1 (Ω; R3 ). It then exists 1 (Ω; R3 ), and a subsequence (not relabeled) of ε such that vεn → v a v ∈ H# n ε ε 1 3 weakly in H (Ω; R ). Again, it is easy to check that |(E u )iα | ≥ ε|(Evε )iα |, thus, using (14), we deduce that Cε ≥ k(Evε )iα kL2 (Ω) and consequently (Ev)iα = 0 for i = 1, 2, 3 and α = 1, 2. Hence v ∈ HBN (Ω; R3 ). Using assumption (14) together with Theorem 4.2 we obtain that the sequence Hεn uεn is bounded in L2 so that, up to subsequences, it weakly converges in L2 (Ω; R3×3 ) to some H ∈ L2 (Ω; R3×3 ). Since, from (14), Eεn uεn → 0 in L2 (Ω; R3×3 ) we have Wεn uεn → H weakly in L2 (Ω; R3×3 ). In particular, 8 H is, almost everywhere, a skew-symmetric matrix. Since, on the other hand, (Hε uε )13 = D3 uε1 = D3 v1ε , and (Hε uε )23 = D3 uε2 = εD3 v2ε we immediately deduce that (H)13 = D3 v1 and (H)23 = 0. Denoting (H)12 = −ϑ we obtain (16). ✷ Let Eαβ denote the Ricci’s symbol, thus E11 = E22 = 0, E12 = 1 and E21 = −1. Define (using the summation convention) ℜ2 = {r ∈ L2 (ω; R2 ) : ∃ ϕ ∈ R, c ∈ R2 s.t. rα (y) = Eβα yβ ϕ + cα }. The elements of ℜ2 are the infinitesimal rigid displacements on ω. It is easy to see that ℜ2 ⊂ H 1 (ω; R2 ), moreover, since ℜ2 is a finite-dimensional vector subspace, it is closed in H 1 (ω; R2 ). Let ℜ⊥ 2 be the Hilbertian orthogonal complement of ℜ2 in L2 (ω; R2 ), i.e., Z ⊥ 2 2 v · r dy1 dy2 = 0 for every r ∈ ℜ2 }. (17) ℜ2 = {v ∈ L (ω; R ) : ω L2 (ω; R2 ) Then = ℜ2 ⊕ Let ℘ be the projection of L2 (ω; R2 ) on ℜ2 . Then if w ∈ L2 (ω; R2 ) and {e1 , e2 } denotes the canonical basis of R2 , it is easily seen, taking as test function r = eα and r = Eβα yβ eα , that  Z  Z 1 1 ℘wα = Eβα yβ Eγδ yγ wδ dy1 dy2 + wα dy1 dy2 . (18) I0 ω |ω| ω ℜ⊥ 2. The two-dimensional Korn’s inequality then writes as kw − ℘wkH 1 (ω;R3 ) ≤ C kEwkL2 (ω;R2×2 ) , (19) for all w ∈ H 1 (ω; R2 ). Lemma 4.4 Under assumption (14) and the notation of Lemma 4.3 and of (9) we have ϑε (uε ) → ϑ weakly in L2 (Ω). Therefore, ϑ does not depend on y1 and y2 . Proof. It is convenient to set wε := (uε1 /ε, uε2 /ε2 , uε3 /ε3 ). Let ℘ be the projection of L2 (ω; R2 ) on ℜ2 . Then for almost every y3 ∈ (0, ℓ) we consider the projection of wε (·, y3 ). From equation (18), and recalling (9), we find Z 1 ε ε ε wε dy1 dy2 . (20) ℘wα = Eβα yβ ϑ (u ) + |ω| ω α Since, furthermore, (Ewε )11 = ε(Eε uε )11 , (Ewε )12 = (Eε uε )12 , and (Ewε )22 = (Eε uε )22 /ε, we have 1 k(Ewε )αβ kL2 (Ω;R2×2 ) ≤ k(Eε uε )αβ kL2 (Ω;R2×2 ) . (21) ε 9 Hence, integrating (19) on (0, ℓ) and taking into account (21) and (14), we deduce that kDα (wε − ℘wε )kL2 (Ω;R) → 0, (22) for α = 1, 2. Since (W℘wε )12 = −ϑε (uε ) and (Wwε )12 = (Wε uε )12 we find from the identity ϑε (uε ) = −(W℘wε )12 = −(Wε uε )12 + (W(wε − ℘wε ))12 (23) the first claim of the Lemma, by letting ε to 0 and recalling (16). From the fact that ϑε (uε ) does not depend on y1 and y2 follows the second claim. ✷ That ϑ does not depend on y1 and y2 , can be also easily proved by using (14) and (16). Indeed, it suffices to take ψ ∈ C0∞ (Ω) and to note that Z Z Z D2 uε1 D1 uε1 ε(Eε uε )11 D2 ψ dy. D1 ψ dy = D2 ψ dy = ε ε Ω Ω Ω Finally, taking the limit as ε goes to zero, we find Z ϑD1 ψ dy = 0, Ω and hence that ϑ is independent of y1 . A similar argument shows also that ϑ does not depend on y2 . Remark 4.5 From (19), (21) and (23) follows that 1 kϑε (uε )kL2 (Ω) ≤ k(Wwε )kL2 (Ω;R3×3 ) + C kEε uε kL2 (Ω;R3×3 ) , ε and hence from Theorem 4.2 we deduce kϑε (uε )kL2 (Ω) ≤ C 1 kEε uε kL2 (Ω;R3×3 ) . ε2 (24) 1 (Ω). We now prove that indeed ϑ ∈ H# Lemma 4.6 Under assumption (14) and with the notation of Lemma 4.3 we have 1 (Ω). ϑ ∈ H# Proof. As before, it is convenient to set wε := (uε1 /ε, uε2 /ε2 , uε3 /ε3 ). Let ξ ∈ C0∞ (ω) be such that Z I0 ξ dy1 dy2 = − . 2 ω 10 Then, taking into account (20), we have Z Z ε ε ε ε ε ε I0 ϑ (u ) = −2ϑ (u ) ξ dy1 dy2 = −ϑ (u ) ξDα yα dy1 dy2 ω Z Z ω = ϑε (uε ) Dα ξ yα dy1 dy2 = ϑε (uε ) Eαγ Eβγ Dα ξ yβ dy1 dy2 ω ω Z Eαγ Dα ξ(Eβγ yβ ϑε (uε )) dy1 dy2 = Zω Z 1 = Eαγ Dα ξ(℘wεγ − wγε dy1 dy2 ) dy1 dy2 |ω| ω Zω ε Eαγ Dα ξ ℘wγ dy1 dy2 = Z Zω ε Eαγ Dα ξ wγ dy1 dy2 − Eαγ Dα ξ(wε − ℘wε )γ dy1 dy2 . = ω ω Hence denoting by ϑ̃ε = 1 I0 Z ω Eαγ Dα ξ wγε dy1 dy2 , and recalling (22), we find ϑε (uε ) − ϑ̃ε → 0 strongly in L2 (Ω). (25) We now show that D3 ϑ̃ε is bounded in L2 . Since Eαγ Dα Dγ ξ = 0 everywhere in ω and Dα ξ = 0 on ∂ω, we have Z ε Eαγ Dα ξ D3 wγε dy1 dy2 I0 D3 ϑ̃ = ω Z Z ε = 2 Eαγ Dα ξ (Ew )γ3 dy1 dy2 − Eαγ Dα ξ Dγ w3ε dy1 dy2 Zω Zω = 2 Eαγ Dα ξ (Ewε )γ3 dy1 dy2 − Dγ (Eαγ Dα ξ w3ε ) dy1 dy2 + ω Zω + Eαγ Dα Dγ ξw3ε dy1 dy2 ω Z ε = 2 Eαγ Dα ξ (Ew )γ3 dy1 dy2 , ω but = (Eε uε )13 /ε and (Ewε )23 = (Eε uε )23 /ε2 , and therefore D3 ϑ̃ε is 2 bounded in L (0, ℓ). Thus, from (25) and Lemma 4.4 we conclude that (Ewε )13 ϑ̃ε → ϑ weakly in H 1 (Ω). 1 (Ω). Therefore, since ϑ̃ε (0) = 0, we conclude that ϑ ∈ H# 11 ✷ Lemma 4.7 Under the same assumptions and with the notation of Lemma 4.3 we have, up to subsequences, (Eε uε )33 → D3 v3 weakly in L2 (Ω), ε2 (Eε uε )23 → y1 D3 ϑ + η weakly in L2 (Ω), ε2 (26) (27) where η ∈ L2 (Ω) is independent of y1 . ε ε uε Proof. To prove (26) it suffices to notice that (E εu2 )33 = D3 ε23 and apply (15). ε ε From (14) we deduce that, up to subsequences, (E εu2 )23 → E23 weakly in L2 (Ω). To characterize E23 ∈ L2 (Ω) note that   D2 uε1 D1 uε2 ε ε 2D3 (W u )12 = D3 − 2 ε ε     ε D2 uε3 D3 uε2 D3 u1 D1 uε3 + 3 − D1 + 2 = D2 ε ε ε3 ε ε ε ε ε (E u )23 (E u )13 = 2D2 − 2D1 , ε ε2 in the sense of distributions. Hence for ψ ∈ C0∞ (Ω) we have Z Z Z (Eε uε )23 (Eε uε )13 ε ε D2 ψ dy − D1 ψ dy. (W u )12 D3 ψ dy = ε ε2 Ω Ω Ω ε ε )13 On the other hand, using (14) we have that (E u → 0 weakly in L2 (Ω). Hence, ε passing to the limit in the previous equality we find Z Z −ϑD3 ψ dy = − E23 D1 ψ dy. Ω Ω Thus D1 E23 = D3 ϑ, in the sense of distributions, and therefore, taking into account that ϑ is independent of y1 we have that E23 = y1 D3 ϑ + η, with η like in the statement of the Lemma. ✷ 5 The limit energy Define f0 (α, β) := min{f (A) : A ∈ Sym, A23 = α, A33 = β} 12 where 1 λ f (A) = CA · A = µ|A|2 + |trA|2 . 2 2 A simple computation shows that (28) 1 f0 (α, β) := 2µα2 + Eβ 2 2 where E is the Young modulus E= µ(2µ + 3λ) . µ+λ 1 (Ω; R3 ). If Lemma 5.1 Let uε be a sequence of functions in H# sup ε 1 Iε (uε ) < +∞, ε4 (29) then (14) holds for some constant C. Proof. It is convenient to set vε := (uε1 , uε2 /ε, uε3 /ε2 ). With this notation and using (3), (8), (11) and (24) we find Z Z ℓ Z 1 Eε uε Eε uε 1 ε ε mϑε (uε ) dy3 C 2 · 2 dy − b · v dy − Iε (u ) = ε4 2 Ω ε ε 0 Ω µ Eε uε 2 ≥ k 2 kL2 (Ω) − kbkL2 (Ω) kvε kL2 (Ω) − kmkL2 (0,ℓ) kϑε (uε )kL2 (0,ℓ) 2 ε µ Eε uε 2 1 C1 ε 2 kbk2L2 (Ω) − kv kL2 (Ω) + ≥ k 2 kL2 (Ω) − 2 ε 2C1 2 1 C2 Eε uε 2 − kmk2L2 (0,ℓ) − k 2 kL2 (Ω) , 2C2 2 ε whenever ε12 kEε uε kL2 (Ω) ≥ 1, and where C1 and C2 are arbitrary positive constants. Choosing C2 = µ/2, we get 1 µ Eε uε 2 1 C1 ε 2 ε I (u ) ≥ k k 2 − kbk2L2 (Ω) − kv kL2 (Ω) + ε ε4 4 ε2 L (Ω) 2C1 2 1 − kmk2L2 (0,ℓ) . µ From Theorem 4.2 we have 1 µ C1
ε 2 1 1 kv kL2 (Ω) − Iε (uε ) ≥ kHε uε k2L2 (Ω) + − kbk2L2 (Ω) 4 ε 4C C 2 2C1 1 − kmk2L2 (0,ℓ) µ 13 (30) where C is the constant of Theorem 4.2. By choosing for instance C1 = 1/C, and using assumption (29), we obtain that there exists a constant M > 0 such that M≥ µ 1 kHε uε k2L2 (Ω) + kvε k2L2 (Ω) 4C 2C from which follows that the sequence vε is bounded in L2 (Ω; R3 ). Using this fact in (30) we finally get the estimate (14). ✷ The above Lemma 5.1 and Lemma 4.3 imply that the family of functionals is coercive with respect to the weak convergence of the sequence   uε uε qε (uε ) := uε1 , 2 , 23 , (Wε uε )12 (31) ε ε 1 I ε4 ε in the space H 1 (Ω; R3 ) × L2 (Ω; R), uniformly with respect to ε. Hence, for any sequence uε which is bounded in energy, that is ε14 Iε (uε ) ≤ C for a suitable constant C > 0, and satisfies the boundary conditions, the corresponding sequence qε (uε ) is weakly relatively compact in H 1 (Ω; R3 ) × L2 (Ω; R), and the following convergence result characterizes the weak limits of such sequences. 1 (Ω; R3 ) × H 1 (Ω; R) → R ∪ {+∞} be defined by Theorem 5.2 Let I : H# # Z ℓ Z Z m ϑ dy3 b · v dy − f0 (y1 D3 ϑ, D3 v3 ) dy − I(v, ϑ) := if v ∈ HBN (Ω; R3 ), and +∞ otherwise. As ε → 0+ , the sequence of functionals in the following sense: 1 I ε4 ε (32) 0 Ω Ω Γ-converges to the functional I, 1. [liminf inequality] for every sequence of positive numbers εk converging to εk εk
1 (Ω; R3 ) such that uεk , u2 , u3 →v 0 and for every sequence {uεk } ⊂ H# 1 εk ε2 weakly in H 1 (Ω; R3 ), and (Wεk uεk )12 → −ϑ weakly in L2 (Ω), lim inf k→+∞ k 1 Iε (uεk ) ≥ I(v, ϑ); ε4k k 2. [recovering sequence] for every sequence of positive numbers εk converging to 1 (Ω; R3 ) × H 1 (Ω; R) there exists a subsequence 0 and for every (v, ϑ) ∈ H# #   un un n 1 3 εkn and a sequence {u } ⊂ H# (Ω; R ) such that un1 , εk2 , ε23 → v weakly n in H 1 (Ω; R3 ), (W εkn un )12 → −ϑ weakly in lim sup n→+∞ L2 (Ω) and 1 Iε (un ) ≤ I(v, ϑ). ε4kn kn 14 kn Proof. We start by proving the liminf inequality. Without loss of generality we may suppose that lim inf k→+∞ 1 1 Iεk (uεk ) = lim 4 Iεk (uεk ) < +∞, 4 k→+∞ εk εk hence the results of Lemma 4.7 hold. Looking at the expression (8) of the functional Iε and observing that, from the definitions of f and f0 given at the beginning of this section, 1 CA · A ≥ f0 (A23 , A33 ), 2 then we have Z (Eεk uεk )23 (Eεk uεk )33 1 εk f0 ( I (u ) ≥ , ) dy + εk 4 εk ε2k ε2k Ω   Z ℓ Z εk uε3k εk u2 , 2 dy − m ϑεk (uεk ) dy3 . − b · u1 , ε ε k 0 Ω k Using the convexity of f0 , Lemma 4.4 and Lemma 4.7 we find Z Z 1 b · v dy + f0 (y1 D3 ϑ + η, D3 v3 ) dy − lim inf 4 Iεk (uεk ) ≥ k→+∞ εk Ω Ω Z ℓ m ϑ dy3 − 0 Z ℓ Z Z m ϑ dy3 + b · v dy − f0 (y1 D3 ϑ, D3 v3 ) dy − = 0 Z Ω Ω Z + 4 µy1 D3 ϑ(y3 )η(y2 , y3 ) dy + 2 µη 2 dy. Ω Ω The first integral on the line above is equal to zero, and hence, taking into account that the second integral in the line above is positive we deduce Z ℓ Z Z 1 εk lim inf 4 Iεk (u ) ≥ m ϑ dy3 . b · v dy − f0 (y1 D3 ϑ, D3 v3 ) dy − k→+∞ εk 0 Ω Ω We now find a recovering sequence. We first note that f0 (α, β) = f (Λ(α, β)), with Λ(α, β) a symmetric matrix with Λ11 (α, β) = Λ22 (α, β) = −νβ, Λ33 (α, β) = β, Λ12 (α, β) = Λ13 (α, β) = 0, Λ23 (α, β) = α, 15 (33) where ν denotes the Poisson’s coefficient, ν= λ . 2(λ + µ) If I(v, ϑ) = +∞ there is nothing to prove. Let I(v, ϑ) < +∞, then v ∈ 1 (Ω; R). HBN (Ω; R3 ) and ϑ ∈ H# We first further assume v and ϑ smooth and equal to zero near y3 = 0. By (13) there exists ξ smooth and equal to zero near y3 = 0 such that vα (y) = ξα (y3 ), and v3 (y) = ξ3 (y3 ) − yα ξα′ (y3 ). Then the function u0,ε , defined by
1 1 = ξ1 − εy2 ϑ − νε4 y1 ξ3′ + (−y12 + 2 y22 )ξ1′′ − y1 y2 ξ2′′ , u0,ε 1 2 ε
1 0,ε 2 3 ′ ′′ = εξ2 + ε y1 ϑ − νε y2 ξ3 − y1 y2 ξ1 + (−y22 + ε2 y12 )ξ2′′ , (34) u2 2
1 u0,ε = ε2 ξ3 − y1 ξ1′ − y2 ξ2′ + ε3 y1 y2 ϑ′ + νε4 y1 y22 ξ1′′′ , 3 2 is equal to zero in y3 = 0 and satisfies the following estimates k Eε u0,ε − Λ(y1 D3 ϑ, D3 v3 )kL2 (Ω) ≤ εC(v, ϑ), ε2 k(Wε u0,ε )12 + ϑkL2 (Ω) ≤ εC(v, ϑ), (35)   u0,ε u0,ε 3 2 , − vkH 1 (Ω) ≤ εC(v, ϑ), , k u0,ε 1 ε ε2 where C(v, ϑ) depends only on v and ϑ. Hence in this case (u0,εk ) is a recovering sequence. 1 (Ω; R), we can find, by In the general case, i.e., v ∈ HBN (Ω; R3 ) and ϑ ∈ H# 1 (Ω; R) which are smooth, convolution, functions vn ∈ HBN (Ω; R3 ) and ϑn ∈ H# equal to zero near y3 = 0 and such that kΛ(y1 D3 ϑn , D3 v3n ) − Λ(y1 D3 ϑ, D3 v3 )kL2 (Ω) ≤ n1 , kϑn − ϑkL2 (Ω) ≤ n1 , kvn − vkH 1 (Ω) ≤ n1 , for every n. Denoting by un,ε the sequence defined as u0,ε in (34) but with (v, ϑ) replaced by (vn , ϑn ), given a sequence εk converging to zero, we can find a diagonal un := un,εkn such that εk n n ≤ 1, k E u − Λ(y D ϑn , D v n )k 2 1 εkn 2 εkn un )12 3 3 3 ϑn kL2 (Ω) L (Ω) 1 n, + ≤ k(W   n n u u k un1 , εk2 , ε23 − vn kH 1 (Ω) ≤ n1 . n kn 16 n Therefore, the sequence un satisfies the recovering sequence condition. ✷ Remark 5.3 As a consequence of the weak metrizability of compact subsets of H 1 (Ω; R3 ) × L2 (Ω; R) and of the Urysohn property of Γ-convergence (see for instance Dal Maso [7], Chapter 8), conditions 1 and 2 of Theorem 5.2 imply that the sequence of functionals ε14 Iε Γ-converges to I with respect to the weak convergence in H 1 (Ω; R3 ) × L2 (Ω; R) of the sequence qε (uε ) (see (31)). 6 Convergence of minima and minimizers For every ε ∈ (0, 1] let us denote by ũε the solution of the following minimization problem min{Iε (u) : u ∈ H 1 (Ωε ; R3 ), u = 0 on Sε (0)}. (36) The existence of the solution can be proved by the direct method of the Calculus of Variations and the uniqueness follows by the strict convexity of the functionals Iε . Corollary 6.1 The following minimization problem for the Γ-limit functional I defined in (32) min{I(v, ϑ) : v ∈ HBN (Ω; R3 ), ϑ ∈ H 1 (0, ℓ), v = 0 on Sε (0), ϑ(0) = 0} admits a unique solution (ṽ, ϑ̃). Moreover, as ε → 0+ ,  ε ε 1. ũε1 , ũε2 , ũε23 → ṽ weakly in H 1 (Ω; R3 ), 2. (Wε ũε )12 → −ϑ̃ weakly in L2 (Ω), 3. 1 I (ũε ) ε4 ε converges to I(ṽ, ϑ̃). Proof. Follows from well known properties of Γ-limits and in particular by putting together Propositions 6.8 and 8.16 (lower semicontinuity of sequential Γlimits), Theorem 7.8 (coercivity of the Γ-limit) and Corollary 7.24 (convergence of minima and minimizers) of Dal Maso [7]. ✷ 17 7 The equations of equilibrium The limit energy functional I(v, ϑ) can be written in a more explicit form by using (13) and the fact that ϑ depends only on y3 . Indeed, the limit strain energy rewrites as Z Z f0 (y1 D3 ϑ, ξ3′ − y1 ξ1′′ − y2 ξ2′′ ) dy f0 (y1 D3 ϑ, D3 v3 ) dy = Ω Ω Z Z
2 1 2 ′ = E ξ3 − y1 ξ1′′ − y2 ξ2′′ dy + 2µ y12 ϑ′ dy 2 Ω Ω Z ℓ 1 1 1 1 2 2 2 2 = EAξ3′ + EJ2 ξ1′′ + EJ1 ξ2′′ + µJϑ′ dy3 2 2 2 0 2 where 1 3 ab , 12 Zω Zω 1 1 y12 dy1 dy2 = a3 b, J := 4 y12 dy1 dy2 = a3 b, J2 := 12 3 ω ω and the work done by the external forces can be written as Z Z ℓZ Z ℓ ′ b · v dy = bα ξα + b3 (ξ3 − yα ξα ) da dy3 = hbi iξi − hyα b3 iξα′ dy3 , A := Ω Z 0 dy1 dy2 = ab, J1 := Z y22 dy1 dy2 = 0 ω R where h·i = ω · da denotes integration over the cross-section ω. Thus hbi i, with i = 1, 2, 3, are forces per unit of length and hyα b3 i, for α = 1, 2, are moments per unit of length. The energy of the beam I(v, ϑ) can therefore be rewritten, with an abuse of notation, as Z ℓ 1 1 1 1 2 2 2 2 I(ξ, ϑ) = EAξ3′ + EJ2 ξ1′′ + EJ1 ξ2′′ + µJϑ′ dy3 + 2 2 2 0 2 Z ℓ hbi iξi − hyα b3 iξα′ − m ϑ dy3 , + 0 2 (0, ℓ), ξ ∈ H 1 (0, ℓ) which has to be minimized over all functions (ξ, ϑ) with ξα ∈ H# 3 # 1 and ϑ ∈ H# (0, ℓ). Clearly the minimization problem can be split in four independent problems, as well as the Euler-Lagrange equations which write as follows  (4)  EJ2 ξ1 + hy1 b3 i′ + hb1 i = 0        EJ1 ξ (4) + hy2 b3 i′ + hb2 i = 0 2 (37)  ′′   EAξ3 − hb3 i = 0      µJϑ′′ + m = 0. 18 From Corollary 6.1 we know that if ũk is a minimizer of Iεk , then  ũε1k , ũε2k ũε3k  , → ṽ weakly in H 1 (Ω; R3 ), εk ε2k (Wεk ũεk )12 → −ϑ̃ weakly in L2 (Ω), where (ṽ, ϑ̃) is the minimizer of I. Conversely, if (ṽ, ϑ̃) is the minimizer of I we can find approximate minimizers ′ of Iεk . Indeed, in this case we have ṽ α (y) = ξ˜α (y3 ) and ṽ 3 (y) = ξ˜3 (y3 ) − yα ξ˜α (y3 ) 3 (0, ℓ), ξ˜ ∈ H 2 (0, ℓ) and ϑ̃ ∈ H 2 (0, ℓ) are the solutions of the equiwhere ξ˜α ∈ H# 3 # # librium equations (37) (the minimizers of the functionals I(ξ, ϑ) defined above). A consequence of this gain of regularity is that the sequence aε defined as the sequence u0,ε in (34) with (v, ϑ) replaced by (ṽ, ϑ̃), i.e., 1 1 ′′ ′′
′ aε1 = ξ˜1 − εy2 ϑ̃ − νε4 y1 ξ˜3 + (−y12 + 2 y22 )ξ˜1 − y1 y2 ξ˜2 , 2 ε 1 ′′
′ ′′ ε 2 3 a2 = εξ˜2 + ε y1 ϑ̃ − νε y2 ξ˜3 − y1 y2 ξ˜1 + (−y22 + ε2 y12 )ξ˜2 , 2
1 ′ ′ ′ ′′′ aε3 = ε2 ξ˜3 − y1 ξ˜1 − y2 ξ˜2 + ε3 y1 y2 ϑ̃ + νε4 y1 y22 ξ˜1 , 2 (38) is well defined. Even if we cannot say that it is a recovering sequence, for it does not satisfy the boundary conditions, the estimates (35) with (v, ϑ) replaced by (ṽ, ϑ̃) holds and therefore lim ε→0+ 1 Iε (aε ) = I(ṽ, ϑ̃). ε4 On the other hand, looking at the minimizers ũε of Iε , by Corollary 6.1 we 1 have lim 4 Iε (ũε ) = I(ṽ, ϑ̃). Thus for ε small enough we have + ε→0 ε 1 1 Iε (aε ) − 4 Iε (ũε ) ≤ 1, 4 ε ε or, said differently, |Iε (aε ) − min Iε | ≤ ε4 , (39) which shows that aε is an approximate minimizer. Of course this sequence can be modified as done in the proof of Theorem 5.2, in order to obtain approximate minimizers which satisfy also the boundary conditions. We notice that the components of aε scale with different powers of ε; this reflects the fact that some displacements are bigger than others. The form of aε3 is also quite interesting: the terms multiplied by ε2 are the classical displacements 19 found by De Saint-Venant, while the term multiplied by ε3 takes into account the axial displacements due to the non-uniform warping of the section. The term which multiplies ϑ′ is, in classical beam theory, called the warping function and is usually denoted by Ψ = y1 y2 . This, as well as the value of J, is in perfect accordance with the results of the approximate theory of thin-walled cross-section beams. 8 Strong convergence of minimizers Hereafter it is noticed that in the proof of Theorem 5.2 we have proved more than what is claimed. Indeed, the recovering sequence un is not only weakly but in fact strongly convergent, in the sense precised in part 2 of the statement of the theorem. The aim of this section is to prove that also the convergence of minimizers stated in Corollary 6.1 is strong.  ε ε Theorem 8.1 With the same notation of Corollary 6.1 we have that ũε1 , ũε2 , ũε23 → ṽ strongly in H 1 (Ω; R3 ) and (Wε ũε )12 → −ϑ̃ strongly in L2 (Ω). Let us start by proving the following lemma. Lemma 8.2 Denoted by ũε the solution of the minimization problem (36) and by aε the approximate minimizers defined in (38), we have that lim ε→0+ Eε (ũε − aε ) ε2 L2 (Ω) = 0. Proof. Let us begin by observing that the quadratic form f (A) defined in (28) satisfies the identity f (Ã) = f (A) + CA · (Ã − A) + f (Ã − A) for every pair of 3 × 3 matrices A and Ã. By (3) we thus obtain the inequality f (Ã) ≥ f (A) + CA · (Ã − A) + µ|Ã − A|2 , which can be used in the expression of the integral functional Iε defined in (8) to 20 obtain that ε ε Iε (ũ ) ≥ Iε (a ) + Z CEε (aε ) · Eε (ũε − aε ) dy + ΩZ |Eε (ũε ) − Eε (aε )|2 dy + +µ + ε4 ZΩ b · ũε1 − aε1 , + ε4 Z m ϑε (ũε − aε ) dy3 . Ω ℓ ũε2 − aε2 ũε3 − aε3
, dy + ε ε2 0 As, by (39), Iε (ũε ) − Iε (aε ) ≤ ε4 for any ε small enough, then for such ε we have Z Z |Eε (ũε ) − Eε (aε )|2 CEε (aε ) · Eε (ũε − aε ) 4 ε ≥ dy + µ dy + ε4 ε4 Ω Ω Z Z ℓ ε ε ε ε
ε ε ũ2 − a2 ũ3 − a3 + b · ũ1 − a1 , m ϑε (ũε − aε ) dy3 . dy + , ε ε2 Ω 0 Let us then prove that lim ε→0+ Z Ω CEε (aε ) · Eε (ũε − aε ) dy = 0 ε4 (40) and the claim of the lemma is obtained by passing to the upper limit as ε → 0+ . In order to prove (40) we observe that for every pair of matrices A and B CA · B = 2µA · B + λ tr(A) tr(B). Then Z Z Eε (aε ) · Eε (ũε − aε ) CEε (aε ) · Eε (ũε − aε ) dy = 2µ dy + ε4 ε4 Ω Ω Z tr(Eε (aε )) tr(Eε (ũε − aε )) +λ dy. ε4 Ω (41) In order to perform the computation, it is convenient to shorten some notation by setting Aεij := (Eε (ũε ))ij (Eε (ũε − aε ))ij (Eε (aε ))ij ε ε , U := , F := , ij ij ε2 ε2 ε2 so that the integrand in (41) has the following expression 3 X   ε . 2µ(Aεij Fijε ) + λAεii Fjj i,j=1 21 By (35) we have Aε → Λ(y1 D3 ϑ̃, D3 ṽ 3 ), strongly in L2 (Ω), (42) where Λ is defined in (33), and by the equation above and Lemma 5.1 it follows that Fε is bounded in L2 (Ω). Thus, from (42) we immediately deduce that Z Z ε ε ε Aε13 F13 dy = 0. A12 F12 dy = lim lim ε→0+ ε→0+ Ω Ω ε → 0 weakly in L2 (Ω), and hence From (42) and Corollary 6.1 follows that F33 Z ε Aεij F33 dy = 0, i, j = 1, 2, 3. lim ε→0+ Ω From Corollary 6.1, Lemma 4.3 and Lemma 4.7 it follows that, up to subsequences, ε weakly converges in L2 (Ω) to y ϑ̃′ (y ) + η̃(y , y ), for some η̃ as specified in U23 1 3 2 3 ′ ε Lemma 4.7. By (42) we have that A23 → y1 ϑ̃ (y3 ) strongly in L2 (Ω) and hence, ε → η̃ weakly in L2 (Ω). Thus up to subsequences, F23 Z Z ′ ε ε y1 ϑ̃ (y3 )η̃(y2 , y3 ) dy = 0. A23 F23 dy = lim ε→0+ Ω Ω ε and F ε , Let F11 and F22 be, up to subsequences, the weak limits in L2 (Ω) of F11 22 + respectively. Summarizing and taking the limit as ε → 0 in (41) we obtain (even for the whole sequence) Z CEε (aε ) · Eε (ũε − aε ) lim dy = ε4 ε→0+ Ω Z ε ε ε 2µ(Aε11 F11 + Aε22 F22 ) + λAεii Fαα dy = lim ε→0+ Ω Z D3 v3 (F11 + F22 )[−2ν(µ + λ) + λ] dy = 0 = Ω because −2ν(µ + λ) + λ = −λ + λ = 0, and the proof is concluded. ✷ Proof of Theorem 8.1 As already remarked in the proof of Theorem 4.2, ũε −aε ũε −aε
setting vε = ũε1 − aε1 , 2 ε 2 , 3ε2 3 we have kEvε kL2 (Ω) ≤ k Eε (ũε − aε ) kL2 (Ω) ε2 and by the application of the standard Korn inequality to vε , and Lemma 8.2, we obtain that kvε kH 1 (Ω) → 0. 22  ε ε From the third equation of (35) applied to the sequence aε follows that ũε1 , ũε2 , ũε23 → ṽ strongly in H 1 (Ω; R3 ). On the other hand by Theorem 4.2 we have that Z |Hε (ũε − aε )|2 dy = 0 lim ε→0+ Ω ε and since the definition of H we deduce from here that 1 1 D1 (ũε2 − aε2 ) → 0, D2 (ũε1 − aε1 ) → 0, ε2 ε strongly in L2 (Ω). Thus, from the second equation of (35) applied to the sequence aε follows that (Wε (ũε ))12 = 1 1 D2 (ũε1 − aε1 ) − 2 D1 (ũε2 − aε2 ) + (Wε (aε ))12 → −ϑ 2ε 2ε strongly in L2 (Ω). ✷ Acknowledgements. The work of L.F. and R.P. has been partially supported by the INDAM intergroup project GNAMPA-GNFM 2004 “Problemi di Γ-convergenza nella meccanica delle strutture sottili” and by Progetto Cofinanziato 2003 “Modellazione e tecniche di approssimazione numerica in problemi avanzati nella meccanica dei continui e delle strutture” while A.M. has been partially supported by MURST, grant no. 2003082352. Also, R.P. acknowledge the support of Progetto Cofinanziato 2002 “Modelli matematici per la scienza dei materiali” and L.F. of Progetto Cofinanziato 2002 “Calcolo delle Variazioni: applicazioni all’ottimizzazione di forma ed a problemi geometrici”. References [1] G. Anzellotti, S. Baldo, and D. Percivale, Dimension reduction in variational problems, asymptotic development in Γ-convergence and thin structures in elasticity, Asymptot. Anal. 9 (1994), no. 1, 61–100. [2] F. Bourquin, P.G. Ciarlet, G. Geymonat, and A. Raoult, Gamma-convergence et analyse asymptotique des plaques minces, C.R. Acad. Sci. Paris, t. 315, Série I, (1992), 1017–1024. [3] A. Braides, Γ-convergence for beginners, Oxford Lecture Series in Mathematics and its Applications, 22. Oxford University Press, Oxford, 2002 [4] E. Cabib, L. Freddi, A. Morassi, D. Percivale, Thin notched beams, J. Elasticity 64 no. 2-3 (2001), 157–178. 23 [5] P.G. Ciarlet, Plates and Junctions in Elastic Multi-Structures: An Asymptotic Analysis, RMA 14, Masson Springer-Verlag, Paris, 1990. [6] P.G. Ciarlet, and P. Destuynder, A justification of the two-dimensional plate model, J. Mécanique 18 (1979), 315–344. [7] G. Dal Maso, An introduction to Γ-convergence, Birkhäuser, Boston, 1993. [8] E. De Giorgi, and T. Franzoni, Su un tipo di convergenza variazionale, Rend. Sem. Mat. Brescia 3 (1979), 63–101. [9] L. Freddi, A. Morassi, and R. Paroni, Thin-walled beams: the case of a multirectangular cross-section, Paper in preparation. [10] V.A. Kondrat’ev and O.A. Oleinik, On the dependence of the constant in Korn’s inequality on parameters characterizing the geometry of the region, Russian Math. Surveys 44 (1989), 187–195. [11] H. Le Dret, Problémes variationnels dans le multi-domaines. Modélisation des jonctions et applications., RMA 19, Masson Springer-Verlag, Paris, 1991. [12] H. Le Dret, Convergence of displacements and stresses in linearly elastic slender rods as the thickness goes to zero, Asymptot. Anal. 10 (1995), 367–402. [13] A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, Dover, New-York, 1944. [14] A. Morassi, Torsion of thin tubes: a justification of some classical results, J. Elasticity 39 (1995), 213–227. [15] A. Morassi, Torsion of thin tubes with multicell cross-section, Meccanica 34 (1999), 115–132. [16] A. Morassi, An asymptotic analysis of the flexure problem for thin tubes, Math. Mech. Solids 4 (1999), 357–390. [17] O.A. Oleinik, A.S. Shamaev, and G.A. Yosifian, Mathematical problems in elasticity and homogenization, North-Holland, Amsterdam, 1992. [18] D. Percivale, Thin elastic beams: the variational approach to St. Venant’s problem, Asymptot. Anal. 20 (1999), 39–59. [19] J.M. Rodrı́guez, and J.M. Viaño, Analyse asimptotique de l’équation de Poisson dans un domain mince. Application à la théorie de torsion des poutres élastiques à profil mince. I. Domains ”sans jonctions”, C.R. Acad. Sci. Paris, t. 317, Série I, (1993), 423–428. 24 [20] J.M. Rodrı́guez, and J.M. Viaño, Analyse asimptotique de l’équation de Poisson dans un domain mince. Application à la théorie de torsion des poutres élastiques à profil mince. I. Domains ”avec jonctions”, C.R. Acad. Sci. Paris, t. 317, Série I, (1993), 637–642. 25