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-dimen-sional 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 three-dimensional problem converges in a variational sense to the classicalone-dimensional model for extension, flexure and torsion of thin-walledbeams.

2001 AMS Mathematics Classification Numbers: 74K20, 74B10, 49J45

Keywords: thin-walled cross-section beams, linear elasticity, Γ-convergence, di-mension reduction

1 Introduction

It is common practice in structural engineering to consider structures with exten-sion in one or more directions small compared to the remaining. Such a situationarises, for instance, in the study of flat domains with small thickness (plates) orof 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, Universita 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 oldand go back to the pioneering works in Mechanics of Euler, D. Bernoulli, Navierand Kirchhoff, see [13]. The classical theories are usually based on some a-prioriassumptions, motivated by the smallness of certain dimensions with respect toothers, 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 inorder to rigorously justify the a-priori assumptions on which classical theories arebased. In particular, approaches based on rigorous asymptotic expansion (mainlydue 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 oneor two-dimensional classical mechanical models for thin structures in linear andnon-linear elasticity starting from three-dimensional problems.

In this paper we shall be concerned with an asymptotic analysis for a class oflinearly elastic thin beams when the thickness of the transversal cross-section goesto zero. Dimension reduction problems from three dimensions to one have receiveda great deal of attention in recent years and numerous and interesting results havebeen obtained, see [5] for a comprehensive bibliographical and historical survey.In [1] a general framework based on the Γ-convergence on varying domains hasbeen applied to give a justification of the classical one-dimensional mechanicalmodel for extension, flexure and torsion of slender cylinders having circular cross-section. An extension to slender cylinders under more general boundary conditionsand with cross-sections having Lipschitz boundary has been presented in [18]. Byadapting and refining the ideas introduced by Ciarlet and Destuynder of rescalingdomains and field displacements, see [6], Le Dret showed in [12] the convergenceof the displacements and stresses for slender cylinders with Lipschitz cross-sectionand also discussed how to treat some more general cases, involving beam shapeswith spikes and holes.

All of the above results are based on a common assumption, namely the three-dimensional variational problem is formulated on a family of cylinders which areobtained by scaling a reference cylindrical domain Ωε ⊂ R

3 by a single factor ε,ε > 0, in its cross-section plane, that is Ωε = ωε × (0, l) ⊂ R

3, where l is the lengthof the cylinder, ωε = εω is its cross-section and ω ⊂ R

2 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 wasscaled not simply by ε but by a factor rε(x3) (not depending on the coordinatesx1 and x2 of the cross-section plane). This way allows very rapid variations of thethickness of the domain and produces a one-dimensional model for thin notchedbeams.

In several areas of civil, aeronautic and mechanical engineering, design andtechnological 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 unionof several walls, whose thickness is very small compared with the diameter ofthe cross-section. To give an example, thin-walled tubes are often used in thestructures where beams are subjected to high twisting moment or to importanttransversal forces. Hollow cross-sections are in fact most efficient in resisting tor-sion and flexure because, as a consequence of the advantageous distribution ofstresses, they ensure high rigidity and strength with relatively low weight.

From the mathematical point of view, the main novelty in dealing with thin-walled cross-section beams is the presence of two scaling factors: one factor is theratio between the diameter of the cross-section and the beam length, say ε; theother 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-dimensionalenergy functional when ε goes to zero, known results for thin-walled cross-sectionbeams are based on the De Saint-Venant classical principle and, therefore, theyessentially involve an asymptotic study of certain two-dimensional Neumann prob-lems defined on the transversal cross-section when the thickness of the walls goesto zero. The limit behavior of the torsion problem for thin-walled beams has beenrecently studied in [19], [20], see also [14], [15] for an alternative development basedon Γ-convergence arguments applied on varying domains and [16] for an asymp-totic analysis of the flexure problem. The acceptation of the De Saint-Venant’sprinciple has important consequences since, roughly speaking, it implies the lossof one dimension. In fact, all the above investigations involve a reduction fromtwo dimensions to one.

This paper represents a first step of a line of research which aims to a rigorousdeduction of the one-dimensional theory for thin-walled beams from the three-dimensional linear elasticity via Γ-convergence techniques. Here we consider a thin-walled 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 refiningthe different techniques of [1], [2] and [18], we prove that the three-dimensionalelasticity problem converges in a variational sense to a one-dimensional problemas ε goes to zero. The limit problem is defined by a functional which includesthe extension, the flexure and the torsion energies of the classical thin-walledbeam model, see Theorem 5.2 for a precise statement. A further step of theanalysis, which takes into account the case where the cross-section ωε has a multi-rectangular shape, is developed in a forthcoming paper [9].

The Γ-convergence of the family of energy functionals defined on Ωε givesnot only the convergence of the energy of the three-dimensional problem to thecorresponding energy of the limit problem, but permits also to obtain a remarkableamount of information about the structure and the behavior of the minimizersof 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 obtaina good approximation of the displacement field solution of the three-dimensionalproblem. The components of this recovering sequence show scaling with differentpowers of ε, and this reflect the fact, well known in practical applications, that forbeams with thin-walled rectangular section some displacements are bigger thanothers.

The plan of the paper is as follows. In the next Section 2 we shall introducethe three-dimensional variational problem and some notations. In Section 3 werewrite the three-dimensional problem as a variational problem for a rescaled en-ergy 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 compactnessresults for suitable families of functions defined on Ω1. These compactness resultswill 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 thefamily of three-dimensional energy functionals to the limit energy as ε goes to zeroand the variational consequences are discussed in Section 6 and 7. Finally, thestrong convergence of minimizers is proved in Section 8.

Notation. Throughout this article, and unless otherwise specified, we use theEinstein summation convention. Moreover we use the following convention forindexing vector and tensor components: Greek indices α, β and γ take their valuesin the set 1, 2 and Latin indices i, j in the set 1, 2, 3. The symbols L2(A;B) andHs(A;B) denote the standard Lebesgue and Sobolev spaces of functions definedon the domain A and taking values in B, with the usual norms ‖ · ‖L2(A;B) and‖ · ‖Hs(A;B), respectively. When B = R or when the right set B is clear from thecontext, we will simply write L2(A) or Hs(A), sometimes even in the notationused for norms. With a little abuse of notation, and because this is a commonpractice and does not give rise to any mistake, we use to call “sequences” eventhose 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,

whereωε := (x1, x2) : |x1| < aε2/2, |x2| < bε/2 ⊂ R

2

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 anddenote by

Eu(x) := sym(Du(x)) :=Du(x) +DuT (x)

2, (1)

the strain of u : Ωε → R3.

In what follows we consider the situation in which the body is subject only todead body forces bε, so that the equilibrium equations write as

divT + bε = 0 in Ωε,T = CEu in Ωε,Tn = 0 on ∂Ωε\Sε(0),u = 0 on Sε(0).

(2)

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. Weassume 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

H1#(Ωε; R

3) :=

u ∈ H1(Ωε; R3) : u = 0 on Sε(0)

.

Due to the coercivity condition (3) and the strict convexity of the integrand,the energy functionals

Jε(u) :=1

2

∫

Ωε

CEu · Eu dx−

∫

Ωε

bε · u dx, (4)

admit, for every ε > 0, a unique minimizer among all competing displacementsu ∈ H1

#(Ωε; R3). As already explained in the introduction our aim is to study

the asymptotic behavior of such minimizers as ε goes to 0, through the theory ofΓ-convergence, for an account of it we refer to the books of Braides [3] and DalMaso [7].

3 The rescaled problem

To state our results it is convenient to stretch the domain Ωε along the transversedirections 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

pε : Ω → Ωε

5

be defined bypε(y) = pε(y1, y2, y3) = (ε2y1, εy2, y3).

Let us consider the following 3 × 3 matrix

Hεv :=

(

D1v

ε2,D2v

ε,D3v

)

, (5)

where Div denotes the column vector of the partial derivatives of v with respectto xi, i = 1, 2, 3. We will use moreover the following notation

Eεv := sym(Hεv), Wεv := skw(Hεv), (6)

and also denote by Wv := W1v, the skew symmetric part of the gradient.Let

H1#(Ω; R3) :=

v ∈ H1(Ω; R3) : v = 0 on S(0)

;

then we can consider the rescaled energy Iε : H1#(Ω; R3) → R ∪ +∞ defined by

Iε(v) := 1ε3Jε(v p−1

ε ), i.e.,

Iε(v) =1

2

∫

ΩCEεv · Eεv dy −

∫

Ωbε pε · v dy.

We further suppose the loads to have the following form

bε1 pε(y) = ε4b1(y) − ε3m(y3)

I0y2,

bε2 pε(y) = ε3b2(y) + ε2m(y3)

I0y1,

bε3 pε(y) = ε2b3(y),

(7)

with b = (b1, b2, b3) ∈ L2(Ω; R3), and m ∈ L2(0, ℓ). Above I0 denotes the polarmoment of inertia of the section ω,

I0 :=

∫

ω

y21 + y2

2 dy1 dy2 =1

12(a3b+ ab3).

We note that while b has the units of a force per unit of volume, m has the unitsof a force, or, equivalently, of a moment per unit of length. The scalings of theloads 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

Iε(v) =1

2

∫

ΩCEεv · Eεv dy − ε4

∫

Ωb · (v1,

v2ε,v3ε2

) dy+

− ε4∫ ℓ

0mϑε(v) dy3,

(8)

6

where we have set

ϑε(v)(y3) :=1

I0

∫

ω

y1

ε2v2(y1, y2, y3) −

y2

εv1(y1, y2, y3) dy1 dy2. (9)

We note that if v ∈ L2(Ω; R3) then ϑε(v) ∈ L2(0, ℓ). A similar statement holds ifwe replace L2 with H1.

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∫

Ωε

(

|u|2 + |Du|2)

dx ≤C

ε4

∫

Ωε

|Eu|2 dx (10)

for every u ∈ H1(Ωε; R3) with u = 0 on Sε(0).

Proof. Divide the section ωε in squares of size ε2 and apply Korn’s inequality (theone obtained by Anzellotti, Baldo and Percivale in [1]; see also [18] and Kondrat’evand Oleinik [10], Theorem 2) to each beam of length ℓ and with section a squarewith side proportional to ε2. Then sum over all the obtained inequalities. 2

To prove the compactness of the displacements we need the following scaledKorn inequality.

Theorem 4.2 There exists a constant C > 0 such that∫

Ω

(

|(u1, u2/ε, u3/ε2)|2 + |Hεu|2

)

dy ≤C

ε4

∫

Ω|Eεu|2 dy (11)

for every u ∈ H1#(Ω; R3) and every 0 < ε ≤ 1.

Proof. The inequality∫

Ω |Hεu|2 dy ≤ Cε4

∫

Ω |Eεu|2 dy is simply obtained byrescaling inequality (10). To show that

∫

Ω|(u1, u2/ε, u3/ε

2)|2 dy ≤C

ε4

∫

Ω|Eεu|2 dy,

it suffices to set v := (u1, u2/ε, u3/ε2), notice that |Eεu| ≥ ε2|Ev| and apply the

standard Korn inequality to v on the domain Ω (see for instance [17], Theorem2.7). 2

7

Let

HBN (Ω; R3) :=

v ∈ H1#(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 charac-terized also as follows (see Le Dret [11], Section 4.1)

HBN (Ω; R3) =

v ∈ H1#(Ω; R3) : ∃ ξα ∈ H2

#(0, ℓ),

∃ ξ3 ∈ H1#(0, ℓ) such that

vα(y) = ξα(y3),v3(y) = ξ3(y3) − yαξ

′α(y3)

.

(13)

The remaining part of this section will be devoted to prove some compactnesslemmata which will be stated under the common assumption that uε be a sequenceof functions in H1

#(Ω; R3) such that

‖Eεuε‖L2(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 εnconverging to 0 there exist a subsequence (not relabeled) and a pair of functionsv ∈ HBN (Ω; R3) and ϑ ∈ L2(Ω) such that (as n→ ∞)

(

uεn

1 ,uεn

2

εn,uεn

3

ε2n

)

→ v weakly in H1(Ω; R3), (15)

Wεnuεn →

0 −ϑ D3v1

ϑ 0 0

−D3v1 0 0

weakly in L2(Ω; R3×3). (16)

Proof. It is convenient to set vε := (uε1, u

ε2/ε, u

ε3/ε

2). It is easily checked that forε ≤ 1, |Eεuε| ≥ ε2|Evε|, hence, by (14), Evε is uniformly bounded in L2(Ω; R3×3),and by Korn’s inequality vε is uniformly bounded in H1(Ω; R3). It then existsa v ∈ H1

#(Ω; R3), and a subsequence (not relabeled) of εn such that vεn → v

weakly in H1(Ω; R3). Again, it is easy to check that |(Eεuε)iα| ≥ ε|(Evε)iα|, thus,using (14), we deduce that Cε ≥ ‖(Evε)iα‖L2(Ω) and consequently (Ev)iα = 0 fori = 1, 2, 3 and α = 1, 2. Hence v ∈ HBN (Ω; R3).

Using assumption (14) together with Theorem 4.2 we obtain that the se-quence Hεnuεn is bounded in L2 so that, up to subsequences, it weakly con-verges in L2(Ω; R3×3) to some H ∈ L2(Ω; R3×3). Since, from (14), Eεnuεn → 0

in L2(Ω; R3×3) we have Wεnuεn → H weakly in L2(Ω; R3×3). In particular,

8

H is, almost everywhere, a skew-symmetric matrix. Since, on the other hand,(Hεuε)13 = D3u

ε1 = D3v

ε1, and (Hεuε)23 = D3u

ε2 = εD3v

ε2 we immediately deduce

that (H)13 = D3v1 and (H)23 = 0. Denoting (H)12 = −ϑ we obtain (16). 2

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 seethat ℜ2 ⊂ H1(ω; R2), moreover, since ℜ2 is a finite-dimensional vector subspace,it is closed in H1(ω; R2). Let ℜ⊥

2 be the Hilbertian orthogonal complement of ℜ2

in L2(ω; R2), i.e.,

ℜ⊥

2 = v ∈ L2(ω; R2) :

∫

ω

v · r dy1 dy2 = 0 for every r ∈ ℜ2. (17)

Then L2(ω; R2) = ℜ2 ⊕ ℜ⊥2 . 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 yβeα, that

℘wα = Eβα yβ

(

1

I0

∫

ω

Eγδ yγwδ dy1 dy2

)

+1

|ω|

∫

ω

wα dy1 dy2. (18)

The two-dimensional Korn’s inequality then writes as

‖w − ℘w‖H1(ω;R3) ≤ C ‖Ew‖L2(ω;R2×2), (19)

for all w ∈ H1(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, u

ε2/ε

2, uε3/ε

3). Let ℘ be the projectionof L2(ω; R2) on ℜ2. Then for almost every y3 ∈ (0, ℓ) we consider the projectionof wε(·, y3). From equation (18), and recalling (9), we find

℘wεα = Eβα yβ ϑ

ε(uε) +1

|ω|

∫

ω

wεα dy1 dy2. (20)

Since, furthermore, (Ewε)11 = ε(Eεuε)11, (Ewε)12 = (Eεuε)12, and (Ewε)22 = (Eεuε)22/ε,we have

‖(Ewε)αβ‖L2(Ω;R2×2) ≤1

ε‖(Eεuε)αβ‖L2(Ω;R2×2). (21)

9

Hence, integrating (19) on (0, ℓ) and taking into account (21) and (14), we deducethat

‖Dα(wε − ℘wε)‖L2(Ω;R) → 0, (22)

for α = 1, 2. Since (W℘wε)12 = −ϑε(uε) and (Wwε)12 = (Wεuε)12 we find fromthe identity

ϑε(uε) = −(W℘wε)12 = −(Wεuε)12 + (W(wε − ℘wε))12 (23)

the first claim of the Lemma, by letting ε to 0 and recalling (16). From the factthat ϑε(uε) does not depend on y1 and y2 follows the second claim. 2

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 ∈ C∞

0 (Ω) and to note that

∫

Ω

D2uε1

εD1ψ dy =

∫

Ω

D1uε1

εD2ψ dy =

∫

Ωε(Eεuε)11D2ψ dy.

Finally, taking the limit as ε goes to zero, we find

∫

ΩϑD1ψ dy = 0,

and hence that ϑ is independent of y1. A similar argument shows also that ϑ doesnot depend on y2.

Remark 4.5 From (19), (21) and (23) follows that

‖ϑε(uε)‖L2(Ω) ≤ ‖(Wwε)‖L2(Ω;R3×3) + C1

ε‖Eεuε‖L2(Ω;R3×3),

and hence from Theorem 4.2 we deduce

‖ϑε(uε)‖L2(Ω) ≤ C1

ε2‖Eεuε‖L2(Ω;R3×3). (24)

We now prove that indeed ϑ ∈ H1#(Ω).

Lemma 4.6 Under assumption (14) and with the notation of Lemma 4.3 we haveϑ ∈ H1

#(Ω).

Proof. As before, it is convenient to set wε := (uε1/ε, u

ε2/ε

2, uε3/ε

3). Let ξ ∈C∞

0 (ω) be such that∫

ω

ξ dy1 dy2 = −I02.

10

Then, taking into account (20), we have

I0ϑε(uε) = −2ϑε(uε)

∫

ω

ξ dy1 dy2 = −ϑε(uε)

∫

ω

ξDαyα dy1 dy2

= ϑε(uε)

∫

ω

Dαξ yα dy1 dy2 = ϑε(uε)

∫

ω

EαγEβγDαξ yβ dy1 dy2

=

∫

ω

EαγDαξ(Eβγyβϑε(uε)) dy1 dy2

=

∫

ω

EαγDαξ(℘wεγ −

1

|ω|

∫

ω

wεγ dy1 dy2) dy1 dy2

=

∫

ω

EαγDαξ ℘wεγ dy1 dy2

=

∫

ω

EαγDαξ wεγ dy1 dy2 −

∫

ω

EαγDαξ(wε − ℘wε)γ dy1 dy2.

Hence denoting by

ϑε =1

I0

∫

ω

EαγDαξ wεγ dy1 dy2,

and recalling (22), we find

ϑε(uε) − ϑε → 0 strongly in L2(Ω). (25)

We now show that D3ϑε is bounded in L2. Since EαγDαDγξ = 0 everywhere in ω

and Dαξ = 0 on ∂ω, we have

I0D3ϑε =

∫

ω

EαγDαξ D3wεγ dy1 dy2

= 2

∫

ω

EαγDαξ (Ewε)γ3 dy1 dy2 −

∫

ω

EαγDαξ Dγwε3 dy1 dy2

= 2

∫

ω

EαγDαξ (Ewε)γ3 dy1 dy2 −

∫

ω

Dγ(EαγDαξ wε3) dy1 dy2 +

+

∫

ω

EαγDαDγξwε3 dy1 dy2

= 2

∫

ω

EαγDαξ (Ewε)γ3 dy1 dy2,

but (Ewε)13 = (Eεuε)13/ε and (Ewε)23 = (Eεuε)23/ε2, and therefore D3ϑ

ε isbounded in L2(0, ℓ). Thus, from (25) and Lemma 4.4 we conclude that

ϑε → ϑ weakly in H1(Ω).

Therefore, since ϑε(0) = 0, we conclude that ϑ ∈ H1#(Ω). 2

11

Lemma 4.7 Under the same assumptions and with the notation of Lemma 4.3we have, up to subsequences,

(Eεuε)33ε2

→ D3v3 weakly in L2(Ω), (26)

(Eεuε)23ε2

→ y1D3ϑ+ η weakly in L2(Ω), (27)

where η ∈ L2(Ω) is independent of y1.

Proof. To prove (26) it suffices to notice that (Eεuε)33ε2 = D3

uε3

ε2 and apply (15).

From (14) we deduce that, up to subsequences, (Eεuε)23ε2 → E23 weakly in

L2(Ω). To characterize E23 ∈ L2(Ω) note that

2D3(Wεuε)12 = D3

(

D2uε1

ε−D1u

ε2

ε2

)

= D2

(

D3uε1

ε+D1u

ε3

ε3

)

−D1

(

D2uε3

ε3+D3u

ε2

ε2

)

= 2D2(Eεuε)13

ε− 2D1

(Eεuε)23ε2

,

in the sense of distributions. Hence for ψ ∈ C∞0 (Ω) we have

∫

Ω(Wεuε)12D3ψ dy =

∫

Ω

(Eεuε)13ε

D2ψ dy −

∫

Ω

(Eεuε)23ε2

D1ψ dy.

On the other hand, using (14) we have that (Eεuε)13ε

→ 0 weakly in L2(Ω). Hence,passing to the limit in the previous equality we find

∫

Ω−ϑD3ψ dy = −

∫

ΩE23D1ψ dy.

ThusD1E23 = D3ϑ, in the sense of distributions, and therefore, taking into accountthat ϑ is independent of y1 we have that E23 = y1D3ϑ+ η, with η like in thestatement of the Lemma. 2

5 The limit energy

Definef0(α, β) := minf(A) : A ∈ Sym, A23 = α, A33 = β

12

where

f(A) =1

2CA · A = µ|A|2 +

λ

2|trA|2. (28)

A simple computation shows that

f0(α, β) := 2µα2 +1

2Eβ2

where E is the Young modulus

E =µ(2µ+ 3λ)

µ+ λ.

Lemma 5.1 Let uε be a sequence of functions in H1#(Ω; R3). If

supε

1

ε4Iε(u

ε) < +∞, (29)

then (14) holds for some constant C.

Proof. It is convenient to set vε := (uε1, u

ε2/ε, u

ε3/ε

2). With this notation andusing (3), (8), (11) and (24) we find

1

ε4Iε(u

ε) =1

2

∫

ΩC

Eεuε

ε2·Eεuε

ε2dy −

∫

Ωb · vε dy −

∫ ℓ

0mϑε(uε) dy3

≥µ

2‖Eεuε

ε2‖2

L2(Ω) − ‖b‖L2(Ω)‖vε‖L2(Ω) − ‖m‖L2(0,ℓ)‖ϑ

ε(uε)‖L2(0,ℓ)

≥µ

2‖Eεuε

ε2‖2

L2(Ω) −1

2C1‖b‖2

L2(Ω) −C1

2‖vε‖2

L2(Ω)+

−1

2C2‖m‖2

L2(0,ℓ) −C2

2‖Eεuε

ε2‖2

L2(Ω),

whenever 1ε2 ‖E

εuε‖L2(Ω) ≥ 1, and where C1 and C2 are arbitrary positive con-stants. Choosing C2 = µ/2, we get

1

ε4Iε(u

ε) ≥µ

4‖Eεuε

ε2‖2

L2(Ω) −1

2C1‖b‖2

L2(Ω) −C1

2‖vε‖2

L2(Ω) +

−1

µ‖m‖2

L2(0,ℓ).

(30)

From Theorem 4.2 we have

1

ε4Iε(u

ε) ≥µ

4C‖Hεuε‖2

L2(Ω) +( 1

C−C1

2

)

‖vε‖2L2(Ω) −

1

2C1‖b‖2

L2(Ω)

−1

µ‖m‖2

L2(0,ℓ)

13

where C is the constant of Theorem 4.2. By choosing for instance C1 = 1/C, andusing assumption (29), we obtain that there exists a constant M > 0 such that

M ≥µ

4C‖Hεuε‖2

L2(Ω) +1

2C‖vε‖2

L2(Ω)

from which follows that the sequence vε is bounded in L2(Ω; R3). Using this factin (30) we finally get the estimate (14). 2

The above Lemma 5.1 and Lemma 4.3 imply that the family of functionals1ε4 Iε is coercive with respect to the weak convergence of the sequence

qε(uε) :=

(

uε1,uε

2

ε,uε

3

ε2, (Wεuε)12

)

(31)

in the space H1(Ω; R3) × L2(Ω; R), uniformly with respect to ε. Hence, for anysequence uε which is bounded in energy, that is 1

ε4 Iε(uε) ≤ C for a suitableconstant C > 0, and satisfies the boundary conditions, the corresponding sequenceqε(u

ε) is weakly relatively compact in H1(Ω; R3) × L2(Ω; R), and the followingconvergence result characterizes the weak limits of such sequences.

Theorem 5.2 Let I : H1#(Ω; R3) ×H1

#(Ω; R) → R ∪ +∞ be defined by

I(v, ϑ) :=

∫

Ωf0(y1D3ϑ,D3v3) dy −

∫

Ωb · v dy −

∫ ℓ

0mϑdy3 (32)

if v ∈ HBN (Ω; R3), and +∞ otherwise.As ε → 0+, the sequence of functionals 1

ε4 Iε Γ-converges to the functional I,in the following sense:

1. [liminf inequality] for every sequence of positive numbers εk converging to

0 and for every sequence uεk ⊂ H1#(Ω; R3) such that

(

uεk

1 ,u

εk2

εk,

uεk3

ε2k

)

→ v

weakly in H1(Ω; R3), and (Wεkuεk)12 → −ϑ weakly in L2(Ω),

lim infk→+∞

1

ε4kIεk

(uεk) ≥ I(v, ϑ);

2. [recovering sequence] for every sequence of positive numbers εk converging to0 and for every (v, ϑ) ∈ H1

#(Ω; R3) × H1#(Ω; R) there exists a subsequence

εknand a sequence un ⊂ H1

#(Ω; R3) such that

(

un1 ,

un2

εkn

,un3

ε2kn

)

→ v weakly

in H1(Ω; R3), (Wεknun)12 → −ϑ weakly in L2(Ω) and

lim supn→+∞

1

ε4kn

Iεkn(un) ≤ I(v, ϑ).

14

Proof. We start by proving the liminf inequality. Without loss of generality wemay suppose that

lim infk→+∞

1

ε4kIεk

(uεk) = limk→+∞

1

ε4kIεk

(uεk) < +∞,

hence the results of Lemma 4.7 hold. Looking at the expression (8) of the functionalIε and observing that, from the definitions of f and f0 given at the beginning ofthis section,

1

2CA · A ≥ f0(A23, A33),

then we have

1

ε4kIεk

(uεk) ≥

∫

Ωf0(

(Eεkuεk)23ε2k

,(Eεkuεk)33

ε2k) dy+

−

∫

Ωb ·

(

uεk

1 ,uεk

2

εk,uεk

3

ε2k

)

dy −

∫ ℓ

0mϑεk(uεk) dy3.

Using the convexity of f0, Lemma 4.4 and Lemma 4.7 we find

lim infk→+∞

1

ε4kIεk

(uεk) ≥

∫

Ωf0(y1D3ϑ+ η,D3v3) dy −

∫

Ωb · v dy+

−

∫ ℓ

0mϑdy3

=

∫

Ωf0(y1D3ϑ,D3v3) dy −

∫

Ωb · v dy −

∫ ℓ

0mϑdy3 +

+ 4

∫

Ωµy1D3ϑ(y3)η(y2, y3) dy + 2

∫

Ωµη2 dy.

The first integral on the line above is equal to zero, and hence, taking into accountthat the second integral in the line above is positive we deduce

lim infk→+∞

1

ε4kIεk

(uεk) ≥

∫

Ωf0(y1D3ϑ,D3v3) dy −

∫

Ωb · v dy −

∫ ℓ

0mϑdy3.

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,(33)

15

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 ∈HBN (Ω; R3) and ϑ ∈ H1

#(Ω; R).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) − yαξ

′α(y3). Then the function u0,ε, defined by

u0,ε1 = ξ1 − εy2ϑ− νε4

(

y1ξ′

3 +1

2(−y2

1 +1

ε2y22)ξ

′′

1 − y1y2ξ′′

2

)

,

u0,ε2 = εξ2 + ε2y1ϑ− νε3

(

y2ξ′

3 − y1y2ξ′′

1 +1

2(−y2

2 + ε2y21)ξ

′′

2

)

, (34)

u0,ε3 = ε2

(

ξ3 − y1ξ′

1 − y2ξ′

2

)

+ ε3y1y2ϑ′ +

1

2νε4y1y

22ξ

′′′

1 ,

is equal to zero in y3 = 0 and satisfies the following estimates

‖Eεu0,ε

ε2− Λ(y1D3ϑ,D3v3)‖L2(Ω) ≤ εC(v, ϑ),

‖(Wεu0,ε)12 + ϑ‖L2(Ω) ≤ εC(v, ϑ),

‖(

u0,ε1 ,

u0,ε2

ε,u0,ε

3

ε2

)

− v‖H1(Ω) ≤ εC(v, ϑ),

(35)

where C(v, ϑ) depends only on v and ϑ. Hence in this case (u0,εk) is a recoveringsequence.

In the general case, i.e., v ∈ HBN (Ω; R3) and ϑ ∈ H1#(Ω; R), we can find, by

convolution, functions vn ∈ HBN (Ω; R3) and ϑn ∈ H1#(Ω; R) which are smooth,

equal to zero near y3 = 0 and such that

‖Λ(y1D3ϑn, D3v

n3 ) − Λ(y1D3ϑ,D3v3)‖L2(Ω) ≤

1n,

‖ϑn − ϑ‖L2(Ω) ≤1n,

‖vn − v‖H1(Ω) ≤1n,

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 diagonalun := un,εkn such that

‖Eεknun

εkn2 − Λ(y1D3ϑ

n, D3vn3 )‖L2(Ω) ≤

1n,

‖(Wεknun)12 + ϑn‖L2(Ω) ≤1n,

‖(

un1 ,

un2

εkn

,un3

ε2kn

)

− vn‖H1(Ω) ≤1n.

16

Therefore, the sequence un satisfies the recovering sequence condition.2

Remark 5.3 As a consequence of the weak metrizability of compact subsets ofH1(Ω; R3) × L2(Ω; R) and of the Urysohn property of Γ-convergence (see for in-stance Dal Maso [7], Chapter 8), conditions 1 and 2 of Theorem 5.2 imply that thesequence of functionals 1

ε4 Iε Γ-converges to I with respect to the weak convergencein H1(Ω; R3) × L2(Ω; R) of the sequence qε(u

ε) (see (31)).

6 Convergence of minima and minimizers

For every ε ∈ (0, 1] let us denote by uε the solution of the following minimizationproblem

minIε(u) : u ∈ H1(Ωε; R3), u = 0 on Sε(0). (36)

The existence of the solution can be proved by the direct method of the Calculusof Variations and the uniqueness follows by the strict convexity of the functionalsIε.

Corollary 6.1 The following minimization problem for the Γ-limit functional Idefined in (32)

minI(v, ϑ) : v ∈ HBN (Ω; R3), ϑ ∈ H1(0, ℓ), v = 0 onSε(0), ϑ(0) = 0

admits a unique solution (v, ϑ). Moreover, as ε→ 0+,

1.(

uε1,uε

2

ε, u

ε

3

ε2

)

→ v weakly in H1(Ω; R3),

2. (Wεuε)12 → −ϑ weakly in L2(Ω),

3. 1ε4 Iε(u

ε) converges to I(v, ϑ).

Proof. Follows from well known properties of Γ-limits and in particular byputting together Propositions 6.8 and 8.16 (lower semicontinuity of sequential Γ-limits), Theorem 7.8 (coercivity of the Γ-limit) and Corollary 7.24 (convergence ofminima and minimizers) of Dal Maso [7].

2

17

7 The equations of equilibrium

The limit energy functional I(v, ϑ) can be written in a more explicit form byusing (13) and the fact that ϑ depends only on y3. Indeed, the limit strain energyrewrites as

∫

Ωf0(y1D3ϑ,D3v3) dy =

∫

Ωf0(y1D3ϑ, ξ

′

3 − y1ξ′′

1 − y2ξ′′

2 ) dy

=1

2E

∫

Ω

(

ξ′3 − y1ξ′′

1 − y2ξ′′

2

)2dy + 2µ

∫

Ωy21ϑ

′2

dy

=

∫ ℓ

0

1

2EAξ′

2

3 +1

2EJ2ξ

′′2

1 +1

2EJ1ξ

′′2

2 +1

2µJϑ′

2

dy3

where

A :=

∫

ω

dy1dy2 = ab, J1 :=

∫

ω

y22 dy1dy2 =

1

12ab3,

J2 :=

∫

ω

y21 dy1dy2 =

1

12a3b, J := 4

∫

ω

y21 dy1dy2 =

1

3a3b,

and the work done by the external forces can be written as

∫

Ωb · v dy =

∫ ℓ

0

∫

ω

bαξα + b3(ξ3 − yαξ′

α) da dy3 =

∫ ℓ

0〈bi〉ξi − 〈yαb3〉ξ

′

α dy3,

where 〈·〉 =∫

ω· da denotes integration over the cross-section ω. Thus 〈bi〉, with

i = 1, 2, 3, are forces per unit of length and 〈yαb3〉, for α = 1, 2, are moments perunit of length. The energy of the beam I(v, ϑ) can therefore be rewritten, with anabuse of notation, as

I(ξ, ϑ) =

∫ ℓ

0

1

2EAξ′

2

3 +1

2EJ2ξ

′′2

1 +1

2EJ1ξ

′′2

2 +1

2µJϑ′

2

dy3+

+

∫ ℓ

0〈bi〉ξi − 〈yαb3〉ξ

′

α −mϑdy3,

which has to be minimized over all functions (ξ, ϑ) with ξα ∈ H2#(0, ℓ), ξ3 ∈ H1

#(0, ℓ)

and ϑ ∈ H1#(0, ℓ). Clearly the minimization problem can be split in four indepen-

dent problems, as well as the Euler-Lagrange equations which write as follows

EJ2ξ(4)1 + 〈y1b3〉

′ + 〈b1〉 = 0

EJ1ξ(4)2 + 〈y2b3〉

′ + 〈b2〉 = 0

EAξ′′3 − 〈b3〉 = 0

µJϑ′′ +m = 0.

(37)

18

From Corollary 6.1 we know that if uk is a minimizer of Iεk, then

(

uεk

1 ,uεk

2

εk,uεk

3

ε2k

)

→ v weakly in H1(Ω; R3),

(Wεk uεk)12 → −ϑ weakly in L2(Ω),

where (v, ϑ) is the minimizer of I.Conversely, if (v, ϑ) is the minimizer of I we can find approximate minimizers

of Iεk. Indeed, in this case we have vα(y) = ξα(y3) and v3(y) = ξ3(y3) − yαξ

′

α(y3)where ξα ∈ H3

#(0, ℓ), ξ3 ∈ H2#(0, ℓ) and ϑ ∈ H2

#(0, ℓ) are the solutions of the equi-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 thesequence u0,ε in (34) with (v, ϑ) replaced by (v, ϑ), i.e.,

aε1 = ξ1 − εy2ϑ− νε4

(

y1ξ′

3 +1

2(−y2

1 +1

ε2y22)ξ

′′

1 − y1y2ξ′′

2

)

,

aε2 = εξ2 + ε2y1ϑ− νε3

(

y2ξ′

3 − y1y2ξ′′

1 +1

2(−y2

2 + ε2y21)ξ

′′

2

)

, (38)

aε3 = ε2

(

ξ3 − y1ξ′

1 − y2ξ′

2

)

+ ε3y1y2ϑ′+

1

2νε4y1y

22 ξ

′′′

1 ,

is well defined. Even if we cannot say that it is a recovering sequence, for it doesnot satisfy the boundary conditions, the estimates (35) with (v, ϑ) replaced by(v, ϑ) holds and therefore

limε→0+

1

ε4Iε(a

ε) = I(v, ϑ).

On the other hand, looking at the minimizers uε of Iε, by Corollary 6.1 we

have limε→0+

1

ε4Iε(u

ε) = I(v, ϑ). Thus for ε small enough we have

∣

∣

∣

1

ε4Iε(a

ε) −1

ε4Iε(u

ε)∣

∣

∣≤ 1,

or, said differently,|Iε(a

ε) − min Iε| ≤ ε4, (39)

which shows that aε is an approximate minimizer. Of course this sequence canbe modified as done in the proof of Theorem 5.2, in order to obtain approximateminimizers which satisfy also the boundary conditions.

We notice that the components of aε scale with different powers of ε; thisreflects the fact that some displacements are bigger than others. The form of aε

3

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 accountthe axial displacements due to the non-uniform warping of the section. The termwhich multiplies ϑ′ is, in classical beam theory, called the warping function andis usually denoted by Ψ = y1y2. This, as well as the value of J , is in perfectaccordance with the results of the approximate theory of thin-walled cross-sectionbeams.

8 Strong convergence of minimizers

Hereafter it is noticed that in the proof of Theorem 5.2 we have proved more thanwhat is claimed. Indeed, the recovering sequence un is not only weakly but infact strongly convergent, in the sense precised in part 2 of the statement of thetheorem.

The aim of this section is to prove that also the convergence of minimizersstated in Corollary 6.1 is strong.

Theorem 8.1 With the same notation of Corollary 6.1 we have that(

uε1,uε

2

ε, u

ε

3

ε2

)

→

v strongly in H1(Ω; R3) and (Wεuε)12 → −ϑ strongly in L2(Ω).

Let us start by proving the following lemma.

Lemma 8.2 Denoted by uε the solution of the minimization problem (36) and byaε the approximate minimizers defined in (38), we have that

limε→0+

∥

∥

∥

Eε(uε − aε)

ε2

∥

∥

∥

L2(Ω)= 0.

Proof. Let us begin by observing that the quadratic form f(A) defined in (28)satisfies the identity

f(A) = f(A) + CA · (A − A) + f(A − A)

for every pair of 3 × 3 matrices A and A. By (3) we thus obtain the inequality

f(A) ≥ f(A) + CA · (A − A) + µ|A − A|2,

which can be used in the expression of the integral functional Iε defined in (8) to

20

obtain that

Iε(uε) ≥ Iε(a

ε) +

∫

ΩCEε(aε) · Eε(uε − aε) dy+

+µ

∫

Ω|Eε(uε) − Eε(aε)|2 dy+

+ ε4∫

Ωb ·

(

uε1 − aε

1,uε

2 − aε2

ε,uε

3 − aε3

ε2)

dy+

+ ε4∫ ℓ

0mϑε(uε − aε) dy3.

As, by (39), Iε(uε) − Iε(a

ε) ≤ ε4 for any ε small enough, then for such ε we have

ε4 ≥

∫

Ω

CEε(aε) · Eε(uε − aε)

ε4dy + µ

∫

Ω

|Eε(uε) − Eε(aε)|2

ε4dy+

+

∫

Ωb ·

(

uε1 − aε

1,uε

2 − aε2

ε,uε

3 − aε3

ε2)

dy +

∫ ℓ

0mϑε(uε − aε) dy3.

Let us then prove that

limε→0+

∫

Ω

CEε(aε) · Eε(uε − aε)

ε4dy = 0 (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∫

Ω

CEε(aε) · Eε(uε − aε)

ε4dy = 2µ

∫

Ω

Eε(aε) · Eε(uε − aε)

ε4dy+

+λ

∫

Ω

tr(Eε(aε)) tr(Eε(uε − aε))

ε4dy.

(41)

In order to perform the computation, it is convenient to shorten some notation bysetting

Aεij :=

(Eε(aε))ij

ε2, U ε

ij :=(Eε(uε))ij

ε2, F ε

ij :=(Eε(uε − aε))ij

ε2,

so that the integrand in (41) has the following expression

3∑

i,j=1

[

2µ(AεijF

εij) + λAε

iiFεjj

]

.

21

By (35) we have

Aε → Λ(y1D3ϑ,D3v3), strongly in L2(Ω), (42)

where Λ is defined in (33), and by the equation above and Lemma 5.1 it followsthat Fε is bounded in L2(Ω). Thus, from (42) we immediately deduce that

limε→0+

∫

ΩAε

12Fε12 dy = lim

ε→0+

∫

ΩAε

13Fε13 dy = 0.

From (42) and Corollary 6.1 follows that F ε33 → 0 weakly in L2(Ω), and hence

limε→0+

∫

ΩAε

ijFε33 dy = 0, i, j = 1, 2, 3.

From Corollary 6.1, Lemma 4.3 and Lemma 4.7 it follows that, up to subsequences,

U ε23 weakly converges in L2(Ω) to y1ϑ

′(y3) + η(y2, y3), for some η as specified in

Lemma 4.7. By (42) we have that Aε23 → y1ϑ

′(y3) strongly in L2(Ω) and hence,

up to subsequences, F ε23 → η weakly in L2(Ω). Thus

limε→0+

∫

ΩAε

23Fε23 dy =

∫

Ωy1ϑ

′(y3)η(y2, y3) dy = 0.

Let F11 and F22 be, up to subsequences, the weak limits in L2(Ω) of F ε11 and F ε

22,respectively. Summarizing and taking the limit as ε→ 0+ in (41) we obtain (evenfor the whole sequence)

limε→0+

∫

Ω

CEε(aε) · Eε(uε − aε)

ε4dy =

= limε→0+

∫

Ω2µ(Aε

11Fε11 +Aε

22Fε22) + λAε

iiFεαα dy

=

∫

ΩD3v3(F11 + F22)[−2ν(µ+ λ) + λ] dy = 0

because −2ν(µ+ λ) + λ = −λ+ λ = 0, and the proof is concluded. 2

Proof of Theorem 8.1 As already remarked in the proof of Theorem 4.2,

setting vε =(

uε1 − aε

1,uε

2−aε2

ε,uε

3−aε3

ε2

)

we have

‖Evε‖L2(Ω) ≤ ‖Eε(uε − aε)

ε2‖L2(Ω)

and by the application of the standard Korn inequality to vε, and Lemma 8.2, weobtain that

‖vε‖H1(Ω) → 0.

22

From the third equation of (35) applied to the sequence aε follows that(

uε1,uε

2

ε, u

ε

3

ε2

)

→

v strongly in H1(Ω; R3). On the other hand by Theorem 4.2 we have that

limε→0+

∫

Ω|Hε(uε − aε)|2 dy = 0

and since the definition of Hε we deduce from here that

1

ε2D1(u

ε2 − aε

2) → 0,1

εD2(u

ε1 − aε

1) → 0, strongly in L2(Ω).

Thus, from the second equation of (35) applied to the sequence aε follows that

(Wε(uε))12 =1

2εD2(u

ε1 − aε

1) −1

2ε2D1(u

ε2 − aε

2) + (Wε(aε))12 → −ϑ

strongly in L2(Ω). 2

Acknowledgements. The work of L.F. and R.P. has been partially supported bythe INDAM intergroup project GNAMPA-GNFM 2004 “Problemi di Γ-convergenzanella meccanica delle strutture sottili” and by Progetto Cofinanziato 2003 “Model-lazione e tecniche di approssimazione numerica in problemi avanzati nella mecca-nica dei continui e delle strutture” while A.M. has been partially supported byMURST, grant no. 2003082352. Also, R.P. acknowledge the support of ProgettoCofinanziato 2002 “Modelli matematici per la scienza dei materiali” and L.F. ofProgetto Cofinanziato 2002 “Calcolo delle Variazioni: applicazioni all’ottimizzazionedi forma ed a problemi geometrici”.

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