thin notched beams

Journal of Elasticity 64: 157–178, 2001.© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

157

Thin Notched Beams

ELIO CABIB1, LORENZO FREDDI2, ANTONINO MORASSI1

and DANILO PERCIVALE3

1Dipartimento di Ingegneria Civile, via delle Scienze 208, 33100 Udine, Italy.E-mail: [email protected]; [email protected] di Matematica e Informatica, via delle Scienze 206, 33100 Udine, Italy.E-mail: [email protected] di Metodi e Modelli Matematici, piazzale J.F. Kennedy 1, Fiera del Mare Pad. D,16129 Genova, Italy. E-mail: [email protected]

Received 19 February 2001; in revised form 11 October 2001

Abstract. The three-dimensional elasticity energy for a notched rod under axial loads is shown toconverge in a variational sense to a class of one-dimensional models with a localized compliancewhen the beam becomes very slender and when the ratio between the depth and the width of eachnotch vanishes in a suitable way. Also the case of notched thermal conductors is considered, in aparallel treatment, because it falls in a similar but simpler framework.

Mathematics Subject Classifications (2000): 49J45, 73C, 73K05, 73V25.

Key words: elasticity, variational convergence, dimension reduction.

1. Introduction

Structural behavior of slender elastic beams is altered by geometrical discontinu-ities resulting from notches. The main hindrance in describing the effect of suchdiscontinuities in one-dimensional models is due to the fact that classical Saint-Venant’s principle, starting-point for most of beam theories, cannot possibly betrue in these cases, see [15]. So, several mechanical models have been proposed inthe technical literature to include the effect of a notch in one-dimensional systems,and an account of them can be found in [13]. A recurring assumption in modelling anotch is that its effects essentially have a local character, inducing a beam stiffnessreduction in a region close to the discontinuity, and rapidly decaying far from thenotch [7]. Usually, physical experiments are carried out in order to characterizequantitatively the stiffness perturbation produced by real notches of various widthsand depths [14].

A different approach to notch modelization in linear elastic straight beams sub-ject to longitudinal loads is presented here. The basic idea comes from the papersby Acerbi et al. [1], Anzellotti et al. [3] and Percivale [12], where it is shownthat classical one-dimensional models of slender beams with smooth longitudinal

158 E. CABIB ET AL.

profile can be obtained as variational limit of the corresponding three-dimensionalelasticity problems when the radius of the cross-section goes to zero. By adopting aflow of ideas which is typical of the �-convergence of functionals defined on mea-sures, in this paper we extend some of the above results to include the occurrenceof abrupt geometrical changes on the longitudinal profile of the beam such as thoseinduced by a notch.

It turns out that the three-dimensional elasticity problem for a notched rod un-der axial loads converges to a class of one-dimensional models with a localizedcompliance when the beam becomes very slender and when the ratio between thedepth and the width of each notch vanishes in a suitable way. Roughly speaking, inthe limit family the classical one-dimensional model for axial deformations holdsfor the two segments of beam adjacent to the notch, and the notch is representedby means of a translational elastic spring located at the damaged cross-section.This macroscopic description of localized discontinuities in beams is well knownin the technical field and it is usually assumed by engineers for studying damagedetection problems via dynamic methods [2, 11]. As a matter of fact, experimentsshow that the analytical model with a localized compliance describes accuratelythe behavior of a notched beam under axial loads as well as the classical modeldoes for a homogeneous beam [10].

A similar dimension reduction problem can be considered also for thin thermalconductors with abrupt changes in the cross section. In fact, the scalar character ofthe thermal energy functional allows to simplify proofs and considerably reducestechnicalities while, on the other hand, the same basic ideas can be used as well asin the elastic case. This is the main motivation for our parallel presentation of thetwo cases.

From the mathematical point of view, Continuum Mechanics is one of the mostfruitful field of application of �-convergence in Calculus of Variations. Since thiskind of convergence of functionals preserves in general minima and minimizers, itcan be viewed as a reasonable setting to study the asymptotic behaviour of struc-tures with spatial rapidly oscillating material properties, such as microstructures,and many other situations, like optimal design or optimal shape design for conduct-ing or elastic media, when the material properties of the structure are unknown.

This paper is devoted to some progress of recent ideas in �-convergence withvarying domains for dimensional reduction problems. More precisely, we consider,as a sequence of functionals, the stored energy of a cylindrical isotropic linearelastic rod, whose cross section has a small radius ε. Making this parameter goto zero, we need to recall and to use a suitable notion of �-convergence whenthe functionals are defined on a corresponding sequence of vector spaces, eachrelated to a different material domain for the admissible fields of displacement.Once the �-limit has been identified, this functional can be viewed as the elasticenergy of the bar after being reduced to a one-dimensional structure. Its domainis a “limit space” of possible generalized functions which are defined on the lineinterval representing the axis itself.

THIN NOTCHED BEAMS 159

Asymptotic behaviour of energy functionals in dimension reduction problemshas been studied within this setting by Anzellotti et al. in [3] for beams withcircular cross section having constant radius, and by Percivale in [12] for beamswith arbitrary but constant cross section. However, we extend here the analysisto the more general situation when the cross section varies along the axis, bothfor the scalar case, in conductivity, and for the vector one of elasticity. Referringto the latter, in this brief presentation, for its more relevant interest, we let somecylindrical portion of the beam to shrink, either radially or axially, in accordancewith powers of ε of higher order with respect to the complement. We then expectthat the limit model should provide discontinuous displacements, whose jumps aredue to the arising of pointwise concentrations of elastic rigidities, such as springslocated in the thinner regions.

2. Notation

Let us denote by C0(a, b) the space of continuous functions which vanish at aand b while the elements of C0[a, b) are the continuous functions which vanishat b. Their dual M(a, b) = C0(a, b)

∗ and M[a, b) = C0[a, b)∗ are the spaces ofbounded Borel measures on the interval (a, b) and [a, b), respectively. They can beendowed by the weak∗ topology induced by the duality; let’s recall that a sequenceµn weakly∗ converges to µ in M(I ) (where I denotes the interval (a, b) or [a, b))if and only if∫

I

ϕ dµn →∫I

ϕ dµ

for every ϕ ∈ C0(I ). If f ∈ L1(a, b) then it can be identified with a measure inM[a, b) (hence in M(a, b)) which will be denoted by f (x1) dx1; in particular dx1

is the Lebesgue measure on [a, b). As usual, given two measures λ and m, dλ/dmwill denote the Radon–Nikodym derivative, that is the density, of λ with respectto m. Finally, BV (a, b) will denote the space of functions with bounded variationon the open interval (a, b), that is all L1(a, b) functions with first distributionalderivative belonging to M(a, b); a sequence vn is said to weakly∗ converge to afunction v in BV (a, b) if and only if it converges in the norm of L1(a, b) while thederivatives v′

n weakly∗ converge to v′ in M(a, b).

3. Main Theorems

3.1. THIN THERMAL CONDUCTORS

In this section we consider the problem of reducing the thermal energy functionalfrom three dimensions to one. Let a, b ∈ R, a < b and � = [a, b); we shall takethe subdomains of R

3

�ε = {x = (x1, x2, x3) ∈ R

3: x1 ∈ �, (x2, x3) ∈ Sε(x1)}

160 E. CABIB ET AL.

where the cross-sections Sε(x1) are two-dimensional open balls with Lipschitzcontinuous radii satisfying Cε � rε(x1) � cε > 0 for suitable contants C andcε. Our aim is to prove that, if rε converges to 0 when ε → 0+, then the minimizersof the energies of �ε converge to the minimizer of the energy of a possibly in-homogeneous one-dimensional limit medium, whose conductivity depends on therate of convergence of the radii. But, of course, the minimum value of the energiestends to zero as ε → 0+. Anyway, the desired convergence of such minima can berecovered by considering for every ε > 0 the functional Fε − Lε describing the“scaled” total energy of a thermal homogeneous conductive medium occupying aregion of space �ε under given distributed heat sources and boundary conditions.The functionals Fε are defined on W 1,2(�ε) by

Fε(u) = 1

2ε2

∫�ε

|∇u(x)|2dx + χa(u)

where the function χa(u) takes the value 0 if u = 0 on Sε(a) and +∞ otherwise.On the other hand, the functionals Lε are the scaled potential energies of the heatsources and are given by

Lε(u) = 1

ε2

∫�ε

g(x)u(x) dx

where, for the sake of simplicity, the given bounded function g is taken to dependon x1 only, although more general cases can be considered as well. The storedenergy of the one-dimensional limit structure will be found to be of the followingform

F (v) =

π

2

(∫ b

a

∣∣∣∣dv′

dm

∣∣∣∣2

dm+ |v(a+)|2m({a})

)if v ∈ BV (a, b), v′ � m,

+∞ elsewhere in L1(a, b)

(3.1)

where m denotes the measure on [a, b) defined by (3.5) and where we have set byconvention that |v(a+)|2/m({a}) = 0 if m({a}) = 0. Finally, the limit potentialenergy is given by

L(v) =∫ b

a

g(x1)v(x1) dx1. (3.2)

Our result can be more easily stated by firstly performing a suitable change ofvariables which trasforms the ε-depending domain �ε into the fixed domain " =[a, b)× S1. Afterwards, setting

v(x1, x2, x3) = u(x1, rε(x1)x2, rε(x1)x3),

then v is defined on " and

Fε(u) = F "ε (v), Lε(u) = L"

ε (v),


where F "ε ,L

"ε are the functionals defined on W 1,2(") by

F "ε (v) = 1

2ε2

∫"

[(rεvx1 − rεx2vx2 − rεx3vx3)

2 + v2x2

+ v2x3

]dx + χa(v),

L"ε (v) = 1

ε2

∫"

r2ε gv dx.

(3.3)

Our main goal, in the thermal case, consists in proving the following theoremwhere the convergence of minimizers will be expressed in terms of convergence inL1(a, b) of their averages on the cross section, q: W 1,2(") → W 1,2(a, b), definedby

q(v)(x1) = 1

π

∫S1

v(x1, x2, x3) dx2 dx3. (3.4)

THEOREM 3.1. Let us assume that

ε2

r2ε (x1)

dx1 → m weakly∗ in M[a, b), (3.5)

ε2

r2ε (x1)

r2ε (x1) dx1 → 0 weakly∗ in M[a, b). (3.6)

If vε minimizes F "ε − L"

ε then

(i) if (εn) is a sequence such that limn→∞ εn = 0 and if q(vεn) converges to somev in L1(a, b), then v is a minimizer of F − L and

limn→∞

[F "εn(vεn)− L"

εn(vεn)

] = F (v)− L(v);(ii) there is a sequence εn such that limn→∞ εn = 0 and a minimizer v of F − L

such that q(vεn) converges to v weakly∗ in BV (a, b), hence strongly inL1(a, b).

REMARK 3.1. By the uniform coercivity with respect to the transverse deriva-tives, the limit v of minimizing sequences vε cannot depend on x2 and x3. This factmakes the use of the averaging operator q uninfluential but, on the other hand, itimproves readability particularly in view of its application to the elastic case where,on the contrary, it plays an important role.

EXAMPLE 3.1. Let us consider now the case of a conductor shaped as in Fig-ure 1, the main peculiarity being that the radius of the cross section vanishes,within a small interval, with a higher order than outside it. In this case we take� = [−1, 1) and we assume that the radius rε be given by

rε(x1) =

ε if |x1| > εα−2 + εβ

2k,

ε − εα/2

εβ

(2k|x1| − εα−2

) + εα/2 ifεα−2

2k< |x1| � εα−2 + εβ

2k,

εα/2 if |x1| � εα−2

2k

162 E. CABIB ET AL.

Figure 1.

with α > 2 and k > 0. In order that (3.5) and (3.6) hold we have to assumerespectively that α and β satisfy the following inequalities

α

2− 1 � β < 3 − α

2. (3.7)

Then it is easy to see that the limit measure is m = dx1 + (1/k)δ0, hence the limitenergy turns out to be the functional

F (v) =π

2

∫ 1

−1|v′(x1)|2 dx1 + kπ

2[v]2

0 if v ∈ W 1,2(�\{0}), v(−1) = 0,

+∞ elsewhere in L1(�)

where [v]0 = |v(0+)−v(0−)| is the jump of v at 0. Let us remark that if the secondinequality in (3.7) is not satisfied (that is for instance if the slope of the linear jointexceeds a certain critical value) then (3.6) may be not satisfied letting the integrals∫ 1−1(ε

2/r2ε (x1))r

2ε (x1) dx1 (and therefore also

∫ 1−1(1/aε(x1)) dx1) to be unbounded,

and the family of minimizers may lose compactness (see Section 6 below).Concerning the physical meaning of the limit functional, we can notice that

the first term is just the classical stored thermal energy distributed on the one-dimensional limit structure. The remaining term expresses, actually, a concentra-tion of conductivity k at the point 0 which allows the minimizer to jump.

3.2. THIN ELASTIC BEAMS

Let, in this section, Fε be the functionals defined on W 1,2(�ε; R3) by

Fε(u) =∫�ε

f (e(u)) dx + χa(u) (3.8)


where

e(u) = (eij (u))i,j=1,2,3, eij = ui,j + uj,i

2

is the linearized strain tensor associated to the displacement vector u and f (e) isthe deformation energy density. Moreover we shall make the assumption that thebody is made of linear elastic, isotropic and homogeneous material, so that

f (e) = µ|e|2 + λ

2|tr(e)|2

where λ � 0, µ > 0 are the Lamé moduli.We assume that the structure be loaded only by axial forces which are expressed

by a vector field g of the form (g(x1), 0, 0), so that their scaled potential energiesare given by

Lε(u) = 1

ε2

∫�ε

g(x1)u1(x) dx. (3.9)

The stored energy of the one-dimensional limit structure will be the functional

F (v) =πE

2

(∫ b

a

∣∣∣∣dv′

dm

∣∣∣∣2

dm+ |v(a+)|2m({a})

)if v ∈ BV (a, b), v′ � m,


(3.10)

where E = µ(3λ+ 2µ)/(λ+ µ) is the Young’s modulus of the material. Finally,the limit potential energy is given by

L(v) =∫ b

a

g(x1)v1(x1) dx1. (3.11)

By performing the same change of variables as in the thermal case and denoting by

vi(x1, x2, x3) = ui(x1, rε(x1)x2, rε(x1)x3), i = 1, 2, 3

the transformed functional F "ε : W 1,2("; R

3) → R is given by

F "ε (v) = µ

ε2

∫"

[|rεv1,1 − rεx2v1,2 − rεx3v1,3|2 + |v2,2|2 + |v3,3|2

+ 1

2|v1,2 + rεv2,1 − rεx2v2,2 − rεx3v2,3|2

+ 1

2|v1,3 + rεv3,1 − rεx2v3,2 − rεx3v3,3|2 + 1

2|v2,3 + v3,2|2

]dx

+ λ

2ε2

∫"

[|rεv1,1 − rεx2v1,2 − rεx3v1,3|2 + |v2,2|2 + |v3,3|2

+ (rεv1,1 − rεx2v1,2 − rεx3v1,3)(v2,2 + v3,3)+ 2v2,2v3,3]

dx

+χa(u) (3.12)

164 E. CABIB ET AL.

while

L"ε (v) = 1

ε2

∫"

r2ε (x1)g(x1)v1(x) dx. (3.13)

Let q denote the same averaging map introduced in the thermal case.

THEOREM 3.2. Let us assume that

ε2

r2ε (x1)

dx1 → m weakly∗ in M[a, b), (3.14)

ε2

r2ε (x1)

r2ε (x1) dx1 → 0 weakly∗ in M[a, b). (3.15)

If vε minimizes F "ε − L"

ε then

(i) if (εn) is a sequence such that limn→∞ εn = 0 and if q(vεn1 ) converges to some vin L1(a, b) then v is a minimizer of F − L and

limn→∞

[F "εn(vεn)− L"

εn(vεn)

] = F (v)− L(v);

(ii) there is a sequence εn such that limn→∞ εn = 0 and a minimizer v of F − Lsuch that q(vεn1 ) converges to v weakly∗ inBV (a, b), hence strongly inL1(a, b).

4. Sequential �-Convergence

For the purposes of the paper we need of a kind of variational convergence whichallows to treat families of functionals Fε: X → R defined on a space which may bedifferent from the domain of the limiting functional. It is a variant of De Giorgi’s�-convergence, and has been introduced by Anzellotti et al. in [3] in order to studydimension reduction problems in mechanics.

Let X be a set, let (Y, τ) be a topological space and let q: X → Y . Given asequence of functionals Fn: X → R and a point v ∈ Y , let us denote by

�(q, τY−) lim infn→∞ Fn(v) := inf

{lim infn→∞ Fn(xn): q(xn)

τ→ v},

�(q, τY−) lim supn→∞

Fn(v) := inf{lim supn→∞

Fn(xn): q(xn)τ→ v

},

which are called, respectively, the �−-lower and the �−-upper limit at the point v.

DEFINITION 4.1. Given a sequence (εn) such that limn→∞ εn = 0, we say that asequence Fεn : X → R (sequentially) �(q, τY−)-converges to a functional F : Y→ R at a point v ∈ Y , and we write

�(q, τY−) limn→∞Fεn(v) = F(v)


if

�(q, τY−) lim infn→∞ Fεn(v) = �(q, τY−) lim sup

n→∞Fεn(v) = F(v). (4.1)

We say that the family of functionals Fε: X → R (sequentially) �(q, τY−)-converges to a functional F : Y → R at a point v ∈ Y , and we write

�(q, τY−) limε→0+ Fε(v) = F(v) (4.2)

if for any sequence (εn) such that limn→∞ εn = 0 we have that

�(q, τY−) limn→∞Fεn(v) = F(v).

We say that a family of functionals �(q, τY−)-converges on a set if it �(q, τY−)-converges at every point of the set.

REMARK 4.1. Let us remark that (4.1) holds if and only if the following twoconditions are satisfied

(1) for every sequence xn ∈ X such that q(xn)τ→ v one has

lim infn→∞ Fεn(xn) � F(v);

(2) there exists a sequence xn ∈ X such that q(xn)τ→ v and

limn→∞Fεn(xn) = F(v).

If X = Y and q is the identity map, then the �-limits defined above are theclassical De Giorgi’s sequential �-limits which will be denoted by the classicalnotation

�(τX−) limε→0+ Fε = F.

Moreover, it is easy to see that, setting for every n ∈ N

Iε(v) = inf{Fε(x): q(x) = v}then (4.2) holds true if and only if

�(τY−) limε→0+ Iε(v) = F(v).

Besides the notion of sequential �-convergence, also a more general concept of�-convergence, associated with a topology on X, can be introduced (see, for in-stance, [8]) and the two kinds of �-convergence coincide on first countable topo-logical spaces. Hence when the topological space (Y, τ) is first countable, andthe family Fε is equi-coercive then the �(q, τY−)-convergence has the variationalcharacter which is typical of �-convergence, that is, it ensures lower semicontinuityto the limit which, moreover, turns out to be coercive when uniform coercivityassumptions are made on the family (Fε). Moreover, roughly speaking, it preservesconvergence of minima and of minimizers. This is summarized by the followingproposition.

166 E. CABIB ET AL.

PROPOSITION 4.1. Let us assume that �(q, τY−) limε→0+ Fε = F on Y and thatthe family Fε be equi-coercive in the sense that for every real numberM there existsa τ -compact and τ -closed subset KM of Y such that {q(x): Fε(x) � M} ⊆ KM

for any ε > 0. Then we have:

(i) F is τ -lower semicontinuous;(ii) F is τ -coercive;

(iii) if xε ∈ X satisfy lim infε→0+ Fε(xε) = lim infε→0+ infFε (e.g., if xε mini-mizes Fε) then

(a) if (εn) is a sequence such that limn→∞ εn = 0 and if q(xεn)τ→ v then v

is a minimizer of F on Y and limn→∞ Fεn(xεn) = F(v);(b) there is a sequence εn such that limn→∞ εn = 0 and a minimizer v of F

on Y such that q(xεn)τ→ v.

REMARK 4.2. We have to observe, because it will be subsequently used, thatthe semicontinuity property does hold not only for the �-lower but also for the�-upper limit of a sequence or a family of functionals (see, for instance, [8, Propo-sition 6.8]).

�-limits of sums behave differently from usual limits because, generally, theyare not additive, except in particular cases like that considered in the followingproposition (see [8, Proposition 6.20]).

PROPOSITION 4.2. Let Fε: X → R and Gε: X → R be two families offunctionals. Let us assume that

1. there exists the �-limit �(q, τY−) limε→0+ Fε = F on Y ;2. Gε converges continuously to a real valued limit functional G defined on Y ,

that is

q(xε)τ→ v �⇒ Gε(xε) → G(v).

Then �(q, τY−) limε→0+(Fε +Gε) = F +G.

5. �-limits of Functionals Defined on Measures

Let us consider the sequence of functionals defined on L1(a, b) by

Fn(u) =∫ b

a

an(x)|u(x)|2 dx

where an: (a, b) → [0,+∞) are given functions. It is well known, and a simplefact, that if the coefficients an satisfy the equi-uniform ellipticity condition

an(x) � α > 0 for every n ∈ N and every x ∈ (a, b) (5.1)


and 1/an → 1/a in the weak∗ topology of L∞(a, b), then the sequence (Fn) �-converges in the weak topology of L1(a, b) to the limit functional

F(u) =∫ b

a

a(x)|u(x)|2 dx.

On the contrary, if condition (5.1) is not satisfied, then the existence of �-converg-ing subsequences of (Fn) in the weak topology of L1 may fail. Nevertheless, underthe weaker condition∫ b

a

1

an(x)dx � C for every n ∈ N (5.2)

compactness can be recovered, for instance, in the larger space M[a, b) for thesequence of extended functionals

Fn(u) =

∫ b

a

an(x)|u(x)|2 dx if u ∈ L1(a, b),

+∞ elsewhere in M[a, b)and there exists a subsequence which �-converges in the weak∗ sense of M[a, b).The choice of including the left extreme into the interval [a, b) allows us to takeinto account a boundary datum which will be eventually prescribed on a; also theextreme b must be included if one wants to prescribe a datum also at this point.

The �-limit has been completely identified by Bouchitté [4] (see also [6]). Thisresult becomes more expressive by writing Fn in the following equivalent form

Fn(λ) =

∫ b

a

an(x)

∣∣∣∣dλ

dx(x)

∣∣∣∣2

dx if λ � dx,

+∞ elsewhere in M[a, b)where dx is the Lebesgue measure on [a, b) and d/dx denotes the Radon–Nikodymderivative with respect to dx (that is the absolutely continuous part of λwith respectto the Lebesgue measure).

THEOREM 5.1 (Buttazzo and Freddi [6, Theorem 2.2]). Let us assume that themeasures (an(x))−1dx weakly∗ converge to a measure m in the space M[a, b).Then, for every λ ∈ M[a, b) we have

�(w∗M[a, b)−)

limn→∞Fn(λ) = F(λ)

where

F(λ) =

∫[a,b)

∣∣∣∣ dλ

dm

∣∣∣∣2

dm if λ � m,

+∞ elsewhere in M[a, b).

168 E. CABIB ET AL.

5.1. APPLICATION

Theorem 5.1 can be used to characterize the asymptotic behaviour of a sequenceof functionals defined on W 1,2(a, b) by

Fn(u) =∫ b

a

an(x)|u′(x)|2 dx,

where an: (a, b) → [0,+∞) are given functions, under the weak ellipticity con-dition (5.2). The sequence of extended functionals

Fn(u) =

∫ b

a

an(x)|u′(x)|2 dx if u ∈ W 1,1(a, b),


turns out to be indeed equi-coercive in the norm topology of L1(a, b). The follow-ing theorem provides a complete description of the �-limit.

THEOREM 5.2 (Buttazzo and Freddi [6, Theorem 3.3]). Let us assume that themeasures (an(x))−1dx weakly∗ converge to a measure m in the space M[a, b).Then, for every u ∈ L1(a, b) we have

�(L1(a, b)−

)limn→∞[Fn + χa](u) = F(u)

where

F(u) =

∫(a,b)

∣∣∣∣du′

dm

∣∣∣∣2

dm+ |u(a+)|2m({a}) if u ∈ BV (a, b), u′ � m,

+∞ elsewhere in L1(a, b).

6. Reducing the Thermal Energy

In this section we consider the problem, already stated in Section 3.1 of reducingthe thermal energy functional from three dimensions to one, and we give the proofof Theorem 3.1. We shall prove, in fact, the �-convergence of the family of theenergies of the loaded conductors in the strongest topology under which the familyof minimizers turns out to be compact. Theorem 3.1 follows then by the variationalproperty of �-convergence stated in Proposition 4.1.

6.1. COERCIVITY OF ENERGIES AND COMPACTNESS OF MINIMIZERS

Let us write the functional F "ε defined in 3.3 in the form

F "ε (v) = 1

2ε2

∫"

⟨Aε(x1, x2, x3)∇v,∇v

⟩dx + χa(v)


where the symmetric matrix Aε(x1, x2, x3) is given by

r2ε (x1) −rε(x1)rε(x1)x2 −rε(x1)rε(x1)x3

−rε(x1)rε(x1)x2 1 + r2ε (x1)x

22 r2

ε (x1)x2x3

−rε(x1)rε(x1)x3 r2ε (x1)x2x3 1 + r2

ε (x1)x23

.

The eigenvalues of Aε turn out easily to be

λ1 = 1 + r2ε (x1)+ r2

ε (x1)(x22 + x2

3)

2

−√

[1 + r2ε (x1)+ r2

ε (x1)(x22 + x2

3 )]2 − 4r2ε (x1)

2,

λ2 = 1 + r2ε (x1)+ r2

ε (x1)(x22 + x2

3)

2

+√

[1 + r2ε (x1)+ r2

ε (x1)(x22 + x2

3 )]2 − 4r2ε (x1)

2,

λ3 = 1,

and therefore, recalling also that x22 + x2

3 � 1, we have

λi � r2ε (x1)

1 + r2ε (x1)+ r2

ε (x1), i = 1, 2, 3. (6.1)

Then, setting

aε(x1) = r2ε (x1)

ε2[1 + r2ε (x1)+ r2

ε (x1)] (6.2)

we can prove the following theorem.

THEOREM 6.1. If there exists a constant K > 0 such that∫ b

a

1

aε(x1)dx1 � K (6.3)

then

(i) the family of the energy functionals {F "ε − L"

ε }ε is equi-coercive in L1(a, b),in the sense that, for every M ∈ R there is a compact subset KM of L1(a, b)

such that {q(v): F "ε (v)− L"

ε (v) � M} ⊆ KM for every ε > 0;(ii) the family {q(vε)}ε, where vε are minimizers of F "

ε −L"ε , is weakly∗ relatively

compact in BV (a, b).

REMARK 6.1. Condition (6.3) is satisfied under assumptions (3.5) and (3.6).

170 E. CABIB ET AL.

Proof. Let us prove (i). Let v be a function in W 1,2(") such that F "ε (v) −

L"ε (v) � M for every ε > 0; then

F "ε (v) � M + ∣∣L"

ε (v)∣∣ � M + π‖g‖∞‖q(v)‖1 ∀ ε > 0. (6.4)

By definition of F "ε , (6.4) implies that q(v)(a) = 0, and the Poincaré inequality

gives∫ b

a

|q(v)| dx1 � Cp

∫ b

a

|q(v)′| dx1 = Cp

π

∫ b

a

∣∣∣∣∫S1

vεx1dx2 dx3

∣∣∣∣ dx1

� Cp

π

∫"

|∇v| dx. (6.5)

On the other hand, by Hölder’s inequality∫"

|∇v| dx =∫"

|∇v|√aε1√aε

dx

�(∫

"

aε(x1)|∇v|2 dx

)1/2(∫"

1

aε(x1)dx

)1/2

(6.6)

for every ε > 0, while, from (6.1) and (6.2) we get∫"

aε(x1)|∇v|2 dx � 2F "ε (v). (6.7)

By (6.4) and using the fact that rε(x1) � Cε, together with (6.6), (6.7) and assump-tion (6.3), we have

‖∇v‖21," � Kπ

(M + π‖g‖∞‖q(v)‖1

)and, by (6.5), there are two positive constants C and D such that ‖∇v‖2

1," � C +D‖∇v‖1,". Hence ‖∇v‖1," � C which can be used in (6.5) to conclude that thereexists a constant R > 0, depending only on M, such that

F "ε (v)− L"

ε (v) � M ∀ ε > 0 �⇒ |q(v)‖1 � Cp‖q(v)′‖1 � R.

Then, for every ε > 0 the sets {q(v): F "ε (v)−L"

ε (v) � M} are all contained intoa same closed ball of W 1,1(a, b) which is a weakly∗ compact subset of BV (a, b),hence strongly compact in L1(a, b), and the family of the total energy functionalsturns out to be equi-coercive.

(ii) follows by (i) and by the fact that, if vε minimizes F "ε −L"

ε , then F "ε (v

ε)−L"ε (v

ε) � 0. Let us remark that the weak∗ convergence in BV (a, b) stated in (ii) isonly apparently stronger then the norm convergence inL1(a, b) because, as pointedout at the end of the proof of point (i), any family q(vε) such that the correspondingtotal energies are uniformly bounded from above is weakly∗ relatively compact inBV (a, b). ✷


In the sequel, it will be convenient to deal with the extension of the functionalF "ε to L1(") by putting F "

ε (v) = +∞ if v ∈ L1(") \ W 1,2("). It is clear thatTheorem 6.1 still holds for the extended functionals.

6.2. �-CONVERGENCE OF THE THERMAL ENERGY

THEOREM 6.2. Let the convergence assumptions (3.5) and (3.6) of Theorem 3.1hold true. Then, for every v ∈ L1(a, b)

�(q,L1(a, b)−

)limε→0+ F "

ε (v) = F (v)

where F is the functional defined in (3.1).Proof. Let us observe, first of all, that assumptions (3.5) and (3.6) imply that

(see (6.2))

1

aε(x1)dx1 → m weakly∗ in M[a, b). (6.8)

Let v ∈ L1(a, b) and let (εn) be a sequence such that limn→∞ εn = 0; according toRemark 4.1 (1) and (2), we have to prove the following two statements:

(1) for any sequence (vn) of functions inL1(") such that q(vn) → v inL1(a, b)

we have

lim infn→∞ F "

εn(vn) � F (v), (6.9)

(2) there exists a sequence of functions vn ∈ L1(") such that q(vn) → v inL1(a, b) and

limn→∞ F "

εn(vn) = F (v).

Let us prove (1). To avoid trivialities we may assume that the left-hand side of (6.9)be finite and that the lim inf be a limit, which implies that vn ∈ W 1,2(") for everyn large enough and there exists a positive constant C such that

supn∈N

F "εn(vn) � C. (6.10)

From inequality (6.7) we have

F "εn(vn) � 1

2

∫"

aεn(x1)∣∣vnx1

(x1, x2, x3)∣∣2

dx (6.11)

and therefore, by Jensen’s inequality, we get

lim infn→∞ F "

εn(vn) � 1

2lim infn→∞

∫"

aεn(x1)|vnx1(x1, x2, x3)|2 dx

= π

2lim infn→∞

∫ b

a

aεn(x1)

π

(∫S1

|vnx1(x1, x2, x3)|2 dx2 dx3

)dx1

� π

2lim infn→∞

∫ b

a

aεn(x1)|q(vn)′(x1)|2 dx1.

172 E. CABIB ET AL.

By metrizability of the weak∗ topology on bounded sets, we have that q(vn)′weakly∗ converges in M[a, b) to v′ so that, by Theorem 5.1, we get

lim infn→∞

∫ b

a

aεn(x1)|q(vn)′(x1)|2 dx1 �

∫[a,b)

∣∣∣∣dv′

dm

∣∣∣∣2

dm if v′ � m,


and, finally,

lim infn→∞ F "

εn(vn) � F (v)

which proves (1).It remains now to prove (2). By Theorem 5.2, there exists a sequence wn ∈

W 1,2(a, b), with wn(a) = 0, such that

wn → v in L1(a, b)

and ∫ b

a

r2εn(x1)

ε2n

∣∣wn′∣∣2dx1 → 2

F (v)

π.

Let us define

vn(x1, x2, x3) = wn(x1)

for every n ∈ N. Then vn satisfy the boundary condition, converge to v in L1("),and

F "εn(vn) = π

2ε2n

∫ b

a

r2εn(x1)

∣∣wn′∣∣2dx1

which tends, of course, to F (v). ✷COROLLARY 6.1. Let the convergence assumptions (3.5) and (3.6) of Theo-rem 3.1 hold true. Then, for every v ∈ L1(a, b)

�(q,L1(a, b)−

)limε→0+

[F "ε − L"

ε

](v) = F (v)− L(v)

where F and L are the functionals defined in (3.1) and (3.2).Proof. It is enough to observe that the real valued functionals L"

ε continuouslyconverge to L and to apply Proposition 4.2. ✷

Proof of Theorem 3.1. By the �-convergence Theorem 6.2 and the compact-ness Theorem 6.1, Proposition 4.1 applies and (i) and (ii) of Theorem 3.1 followsimmediately by (a) and (b) in Proposition 4.1. ✷


7. Reducing the Elastic Energy

In this section we consider the problem, already stated in Section 3.2, of dimen-sional reduction of the functionals Fε − Lε defined in (3.8) and (3.9) which, afterhaving changed variables take the forms F "

ε − L"ε displayed in (3.12) and (3.13).

7.1. COERCIVITY OF ENERGIES AND COMPACTNESS OF MINIMIZERS

The averaging map q, introduced in (3.4), is now considered as defined on thespace W 1,2("; R

3) instead than W 1,2("), but acting only on the first component v1

of v ∈ W 1,2("; R3); for this reason we will not distinguish, in the sequel, between

q(v) and q(v1).Let aε be the function of x1 defined in (6.2). We can prove the following theo-

rem.

THEOREM 7.1. If there exists a constant K > 0 such that∫ b

a

1

aε(x1)dx1 � K (7.1)

then

(i) the family of the energy functionals {F "ε − L"

ε }ε is equi-coercive in L1(a, b),in the sense that, for every M ∈ R there is a compact subset KM of L1(a, b)

such that {q(v1): F "ε (v)− L"

ε (v) � M} ⊆ KM for every ε > 0;(ii) the family {q(vε1)}ε, where vε are minimizers of F "

ε −L"ε , is weakly∗ relatively

compact in BV (a, b).

REMARK 7.1. Condition (7.1) is satisfied under assumptions (3.14) and (3.15).Proof. Let us prove (i). Let v ∈ W 1,2("; R

3) be such that F "ε (v)−L"

ε (v) � M

for every ε > 0; then

F "ε (v) � M + ∣∣L"

ε (v)∣∣ � M + π‖g‖∞‖q(v)‖1 ∀ ε > 0. (7.2)

Let us write the functional F "ε into the form

F "ε (v) = 1

ε2

∫"

〈Aε(x)∇v,∇v〉 dx + χa(v)

with ∇v = (v1,1, v1,2, v1,3, v2,1, v2,2, v2,3, v3,1, v3,2, v3,3)T, Aε = µSε + (λ/2)Tε

and Sε and Tε are the 9 × 9 positive semi-definite symmetric matrices associated tothe corresponding quadratic forms. We remark that, since f (e(u)) = 0 if and onlyif u is a rigid displacement of the body �ε, that is if and only if u = a + b ∧ x

where a and b are suitable constants, then

〈Aε(x)∇v,∇v〉 = 0

174 E. CABIB ET AL.

if and only if

v = a + (b2rεx3 − b3rεx2, b3x1 − b1rεx3, b1rεx2 − b2x1)

that is if and only if ∇v(x) belongs to the space Eε spanned by

nε1 = (0, 0, 0, −rεx3, 0, −rε, rεx2, rε, 0),

nε2 = (−rεx3, 0, −rε, 0, 0, 0, 1, 0, 0),

nε3 = (−rεx2, −rε, 0, 1, 0, 0, 0, 0, 0).

Now, the gradient ∇v can be written as the sum of a term, ∇Eεv, belonging to thenull eigenspace Eε and another term, ∇E⊥

εv, belonging to the orthogonal space E⊥

ε

∇v = ∇Eεv + ∇E⊥εv.

Since Aε∇Eεv = 0, we have∫"

〈Aε∇v,∇v〉 dx =∫"

〈Aε∇E⊥εv,∇E⊥

εv〉 dx.

The eigenvalues of the matrix Sε can be explicitly computed and the least positiveone among them is

sε(x1, x2, x3) = 1 + r2ε (x1)+ r2

ε (x1)(x22 + x2

3)

2

−√

[1 + r2ε (x1)+ r2

ε (x1)(x22 + x2

3)]2 − 4r2ε (x1)

2

and, as in the thermal case, it can be estimated from below by ε2aε (see (6.2)),hence∫

"

aε(x1)|∇E⊥εv|2 dx � F "

ε (v). (7.3)

Concerning the component of the gradient along Eε, since we have v1,1|Eε =bε2 rεx3 − bε3 rεx2 for suitable constants bε2 and bε3, then

q(v1,1|Eε ) = 1

π

∫S1

v1,1|Eε dx2 dx3 = 0 (7.4)

so that

q(v1)′ = q(v1,1) = q(v1,1|Eε )+ q(v1,1|E⊥

ε) = q(v1,1|E⊥

ε) (7.5)

and therefore, by the Poincaré inequality,∫ b

a

|q(v1)| dx1 � Cp

∫ b

a

|q(v1)′| dx1

= Cp

∫ b

a

|q(v1,1|E⊥ε)| dx1 � Cp

π

∫"

|∇E⊥εv| dx. (7.6)


On the other hand, by the Holder’s inequality,∫"

|∇E⊥εv| dx �

(∫"

aε(x1)|∇vE⊥ε|2 dx

)1/2(∫"

1

aε(x1)dx

)1/2

. (7.7)

By (7.2) and using the fact that rε(x1) � Cε, together with (7.3), (7.7) and assump-tion (7.1), we have

‖∇E⊥εv‖2

1," � Kπ(M + π‖g‖∞‖q(v1)‖1

)and, by (7.6), there are two positive constants C and D such that ‖∇E⊥

εv‖2

1," �C + D‖∇E⊥

εv‖1,". Hence ‖∇E⊥

εv‖1," � C which can be used in (7.6), as in the

thermal case, to conclude the proof.(ii) follows by the fact that, if vε is a minimizer of F "

ε − L"ε , then F "

ε (vε) −

L"ε (v

ε) � 0. ✷

7.2. �-CONVERGENCE OF THE ELASTIC ENERGY

THEOREM 7.2. Let the convergence assumptions (3.14) and (3.15) of Theo-rem 3.2 hold true. Then, for every v ∈ L1(a, b)

�(q,L1(a, b)−

)limε→0+ F "

ε (v) = F (v)

where F (v) is the functional defined in (3.10).Proof. Let (εn) be a sequence such that limn→∞ εn = 0. The claim will be

attained by proving the following two statements:

(1) for any sequence (vn) of functions in L1("; R3) such that q(vn1 ) → v in

L1(a, b) we have

lim infn→∞ F "

εn(vn) � F (v); (7.8)

(2) for every v ∈ L1(a, b)

F (v) � �(L1(a, b)−

)lim supn→∞

F "εn(v). (7.9)

Let us prove (1). Avoiding trivialities we can assume that the left-hand sideof (7.8) be finite and that the lower limit be a limit, which implies that vn ∈W 1,2("; R

3) for every n large enough and that there exists a positive constant Csuch that

supn∈N

F "εn(vn) � C. (7.10)

We have

F "εn(vn) = 1

ε2n

∫"

⟨Aεn∇E⊥

εnvn,∇E⊥

εnvn

⟩dx

176 E. CABIB ET AL.

whereAε = µSε+(λ/2)Tε. The only non zero eigenvalue of the rank one matrix Tεis

tε(x1, x2, x3) = 2 + r2ε (x1)+ r2

ε (x1)(x2

2 + x23

)and it is easy to see that

µsε + λ

2tε � E

2ε2aε

for any ε > 0 small enough, uniformly with respect to x1, x2, x3. Then, by Jensen’sinequality, we have

F "εn(vn) � E

2

∫"

aεn(x1)|∇E⊥εnvn|2 dx � E

2

∫"

aεn(x1)|vn1,1|E⊥εn

|2 dx

� πE

2

∫ b

a

aεn(x1)

(1

π

∫S1

vn1,1|E⊥εn

dx2 dx3

)2

dx1

= πE

2

∫ b

a

aεn(x1)|q(vn1,1|Eε⊥n)|2 dx1

= πE

2

∫ b

a

aεn(x1)|q(vn1 )′|2 dx1 (7.11)

where in the last equality we used the fact that q(vn1,1|Eε⊥n) = q(vn1 )

′ (see (7.5)).

Since

q(vn1 )′ → v′ weakly∗ in M[a, b),

by Theorem 5.1 we have

lim infn→∞

∫ b

a

aεn(x1)∣∣q(vn1 )′(x1)

∣∣2dx1 �

∫[a,b)

∣∣∣∣dv′

dm

∣∣∣∣2

dm if v′ � m,


and therefore we get

lim infn→∞ F "

εn(vn) � F (v)

which proves (1).Let us prove (2). Let v ∈ L1(a, b) be such that v′ � m, and moreover assume

that the function θ(x1) = (dv′/dm)(x1) belong to C10 [a, b). Let

wε(x1, x2, x3) = ε2∫ x1

a

θ(t)

r2ε (t)

dt.

Then wε ∈ W 1,2(") and, by assumptions (3.14) and (3.15),

q(wε) → v in L1(a, b)

q(wε)′ → v′ weakly∗ in M[a, b)


and moreover

limε→0+

∫ b

a

r2ε

ε2|q(wε)

′|2 dx1 =∫

[a,b)

∣∣∣∣dv′

dm

∣∣∣∣2

dm.

Suppose

vε =(wε,− λ

2(λ+ µ)x2rεq(wε)

′,− λ

2(λ+ µ)x3rεq(wε)

′).

Then

F "ε (v

ε) = πE

2

∫ b

a

r2ε

ε2|q(wε)

′|2 dx1 + µλ2

8(λ+ µ)2

∫ b

a

r4ε

ε2|q(wε)

′′|2 dx1.

Since

q(wε)′′ = θ ′ ε

2

r2ε

− 2θε2

r2ε

r2ε

we have

limε→0+ F "

ε (vε) = π

E

2

∫[a,b)

∣∣∣∣dv′

dm

∣∣∣∣2

dm = F (v).

Then, letting

F + = �(L1(a, b)−

)lim supε→0+

F "ε

we can say that

F +(v) � F (v) ∀v ∈ L1(a, b):dv′

dm∈ C1

0 [a, b).In order to prove (7.9) we can assume F (v) < +∞ which implies v ∈ L1(a, b)

and the function θ(x1) = (dv′/dm)(x1) belongs to L2([a, b),m). Then there existsa sequence θh ∈ C1

0 [a, b) such that θh → θ in the space L2([a, b),m) and setting

vh(x1) =∫

[a,x1)

θh(t) dm(t)

we have vh → v in L1(a, b) and, by the lower semicontinuity of F + we get

F +(v) � lim infh→∞ F +(vh) � lim inf

h→∞ F (vh)

= lim infh→∞

πE

2

∫[a,b)

|θh(t)|2 dm(t) = πE

2

∫[a,b)

|θ(t)|2 dm(t)

= πE

2

∫[a,b)

∣∣∣∣dv′

dm

∣∣∣∣2

dm = F (v)

which concludes the proof. ✷

178 E. CABIB ET AL.

COROLLARY 7.1. Let the convergence assumptions (3.14) and (3.15) of Theo-rem 3.2 hold true. Then, for every v ∈ L1(a, b)

�(q,L1(a, b)−

)limε→0+

[F "ε − L"

ε

](v) = F (v)− L(v)

where F and L are the functionals defined in (3.10) and (3.11).Proof. It is enough to observe that the real valued functionals L"

ε continuouslyconverge to L and to apply Proposition 4.2. ✷REMARK 7.2. The proof of Theorem 3.2 is analogous to that of Theorem 3.1,already given at the end of Section 6.

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thin notched beams

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