Junction of elastic plates and beams

Vol. 13, N^o 3, 2007, pp. 419–457 www.edpsciences.org/cocv

DOI: 10.1051/cocv:2007036

JUNCTION OF ELASTIC PLATES AND BEAMS

Antonio Gaudiello

, R´ egis Monneau

, Jacqueline Mossino

, Franc ¸ ois Murat

and Ali Sili

Abstract. We consider the linearized elasticity system in a multidomain ofR³. This multidomain is the union of a horizontal plate with fixed cross section and small thicknessε, and of a vertical beam with fixed height and small cross section of radiusr^ε. The lateral boundary of the plate and the top of the beam are assumed to be clamped. When εand r^ε tend to zero simultaneously, with r^εε², we identify the limit problem. This limit problem involves six junction conditions.

Mathematics Subject Classification. 35B40, 74B05, 74K30.

Received July 14, 2005.

1. Introduction

Letω^a andω^b (afor “above”,b for “below”) be two bounded regular domains in R². In the whole paper, the origin and axes are chosen so that:

ω^ax₁dx₁dx₂=

ω^ax₂dx₁dx₂=

ω^ax₁x₂dx₁dx₂= 0 and 0∈ω^b. (1.1) Letεbe a parameter taking values in a sequence of positive numbers converging to zero, and letr^εbe another positive parameter tending to zero with ε. We introduce the thin multidomain Ω^ε = Ω^aε

J^ε

Ω^bε, where Ω^aε =r^εω^a ×(0,1) represents a vertical beam with fixed height and small cross section, Ω^bε =ω^b×(−ε,0) represents a horizontal plate with small thickness and fixed cross section, andJ^ε=r^εω^a× {0}represents the interface at the junction between the beam and the plate.

Keywords and phrases. Junctions, thin structures, plates, beams, linear elasticity, asymptotic analysis.

1 Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell’Informazione e Matematica Industriale, Universit`a di Cassino, via G. Di Biasio 43, 03043 Cassino (FR), Italia;gaudiell@unina.it

2 CERMICS, ´Ecole Nationale des Ponts et Chauss´ees, 6 et 8 Avenue Blaise Pascal, Cit´e Descartes, 77455 Champs-sur-Marne Cedex 2, France;monneau@cermics.enpc.fr

3 CMLA, ´Ecole Normale Sup´erieure de Cachan, 61 Avenue du Pr´esident Wilson, 94235 Cachan Cedex, France;

mossino@cmla.ens-cachan.fr

4 Laboratoire Jacques-Louis Lions, Universit´e Pierre et Marie Curie, Boˆıte courrier 187, 75252 Paris Cedex 05, France;

murat@ann.jussieu.fr

5D´epartement de Math´ematiques, Universit´e de Toulon et du Var, BP 132, 83957 La Garde Cedex, France;sili@univ-tln.fr c EDP Sciences, SMAI 2007

Article published by EDP Sciences and available at http://www.esaim-cocv.org or http://dx.doi.org/10.1051/cocv:2007036

In this thin multidomain, we consider the displacement U^ε, solution of the three-dimensional linearized elasticity system:

⎧⎪

⎪⎨

⎪⎪

⎩

U^ε∈Y^εand ∀U ∈Y^ε,

Ω^ε

A^εe

U^ε , e(U)

dx=

Ω^εF^ε.Udx+

Ω^ε[G^ε, e(U)] dx+

Σ^aε T^bε

B^bεH^ε.Udσ,

(1.2)

where:

• Y^ε={U ∈(H¹(Ω^ε))³, U = 0 onT^aε=r^εω^a× {1} and on Σ^bε=∂ω^b×(−ε,0)},

• A^ε=A^ε(x) =

⎧⎨

⎩

A^a, ifx∈Ω^aε, k^εA^b, ifx∈Ω^bε,

with k^ε a positive parameter depending on ε and A^a, A^b tensors with constant coefficients A^a_ijkl and A^b_ijkl, i, j, k, l∈ {1,2,3}, satisfying the usual symmetry and coercivity conditions:

A^a_ijkl=A^a_jikl=A^a_ijlk, A^b_ijkl =A^b_jikl=A^b_ijlk,

∃C >0, ∀ξ∈R^3×3_s , [A^aξ, ξ]≥C|ξ|², [A^bξ, ξ]≥C|ξ|², whereR^3×3_s denotes the set of symmetric 3×3-matrices, (A^aξ)_ij=

klA^a_ijklξ_kl, the scalar product [., .] inR^3×3 is defined by [η, ξ] =

ijη_ijξ_ij and|.|is the associated norm; the euclidian scalar product inR³ is denoted by a dot;

• e_ij(U) = 1 2

∂U_i

∂x_j +∂U_j

∂x_i

,

• F^ε∈(L²(Ω^ε))³,

• G^ε∈(L²(Ω^ε))^3×3,

• H^ε ∈(L²(Σ^aε T^bε

B^bε))³, where Σ^aε denotes the lateral boundary of the beam,T^bε and B^bε are respectively the top and the bottom of the plate:

Σ^aε=r^ε∂ω^a×(0,1), T^bε= (ω^b\r^εω^a)× {0}, B^bε=ω^b× {−ε}.

The constraint “U = 0” in the definition ofY^εmeans that the multistructure is clamped on the topT^aεof the beam and on the lateral boundary Σ^bε of the plate. The casek^ε tending to zero or infinity corresponds to very different materials in Ω^aε and Ω^bε (note that breaking the symmetry between Ω^aε and Ω^bε by introducing the coefficient k^ε in front of A^b is not restrictive). In the right-hand side of (1.2), the second term is written in divergence form like in [15, 27, 28]. It is well known that, by means of the Green formula, this second term can contribute to the other ones, giving possibly less regular (not necessarilyL²) volume and surface source terms.

For convenience of the reader, we have chosen to write the three integrals: one recovers the classical formulation by settingG^ε= 0, but the simplest case corresponds toF^ε= 0, H^ε= 0 andG^ε= 0. This case was considered in the short preliminary version [15].

Problem (1.2) admits a unique solution U^ε (see e.g. [29]). The aim of this paper is to describe the limit behaviour of the displacementU^ε, asεtends to zero. We prove that this behaviour depends on the limit of the sequenceq^ε defined by:

q^ε=k^ε ε³ (r^ε)²·

Whenk^εε³ and (r^ε)²have same order (i.e. whenq^ε tends toqwith 0< q <+∞), the limit problem (obtained after suitable rescaling) is a coupled problem between a two-dimensional plate and a one-dimensional beam, with six junction conditions. Ifk^εε³(r^ε)², the multistructure has the limit behaviour of a thin rigid plate and a thin elastic beam which are independent of each other, the beam being clamped at both ends; on the contrary, ifk^εε³ (r^ε)², the structure behaves as a thin rigid beam and a thin elastic plate which are independent of each other, the plate being clamped on its contour and fixed vertically at the junction.

The reader is referred to [1, 3, 4, 6–8, 10–12, 21–23, 25–28, 30, 31] for the derivation of the equations of plates and beams by asymptotic analysis. Junction problems are considered in [5, 9, 13, 14, 16–20]. The present work is a natural follow up of [27, 28], which deal with reduction of dimension for elastic thin cylinders, and of [13, 14], which deal with the diffusion equation in the thin multistructure considered in this paper. Our results were announced in the short note [15].

2. The result 2.1. The rescaled problem

In the sequel, the indexes αandβ take values in the set{1,2}. Moreover,x= (x, x₃) denotes the generic point inR³.

Let Ω^a =ω^a×(0,1), Ω^b =ω^b×(−1,0), T^a =ω^a× {1}, Σ^a =∂ω^a×(0,1) and Σ^b =∂ω^b×(−1,0). The asymptotic behaviour of U^εcan be described by using a convenient rescaling (the reader is referred to Sect. 3.1 for details). This rescaling maps the space Y^ε onto the spaceY^ε defined by:

Y^ε=

u= (u^a, u^b)∈(H¹(Ω^a))³×(H¹(Ω^b))³, u^a= 0 onT^a, u^b= 0 on Σ^b, u^a_α(x,0) =εr^εu^b_α(r^εx,0) and u^a₃(x,0) =u^b₃(r^εx,0), fora.e. x ∈ω^a

.

(2.1)

In particular, we denote byu^ε= (u^aε, u^bε) the rescaling of the solutionU^εof problem (1.2). We set

e^aε(u^a) =

⎛

⎜⎜

⎜⎝ 1

(r^ε)²e_αβ(u^a) 1

r^εe_α3(u^a) 1

r^εe_3α(u^a) e₃₃(u^a)

⎞

⎟⎟

⎟⎠, e^bε(u^b) =

⎛

⎜⎜

⎝

e_αβ(u^b) 1 εe_α3(u^b) 1

εe_3α(u^b) 1

ε²e₃₃(u^b)

⎞

⎟⎟

⎠. (2.2)

Thenu^εis the unique solution of the following problem:

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

u^ε∈ Y^εand ∀u∈ Y^ε,

Ω^a[A^ae^aε(u^aε), e^aε(u^a)] dx+q^ε

Ω^b[A^be^bε(u^bε), e^bε(u^b)] dx

=

Ω^af^aε.u^adx+

Ω^bf^bε.u^bdx+

Ω^a[g^aε, e^aε(u^a)] dx+

Ω^b[g^bε, e^bε(u^b)] dx +

Σ^a

h^aε.u^adσ+

ω^b

h^bε₊.u^b_|x3=0+h^bε₋.u^b_|x3=−1 dx,

(2.3)

whereq^εis defined by:

q^ε=k^ε ε³

(r^ε)², (2.4)

and where the source terms are suitable transforms of (F^ε, G^ε, H^ε) (see Sect. 3.1).

2.2. The setting of the limit problem

For the definition of the limit problem, we introduce the following functional spaces:

U^a =

u^a ∈(H₀²(0,1))²×H¹(Ω^a),∃ζ^a∈H¹(0,1), ζ^a(1) = 0, u^a₃ =ζ^a−x₁du^a₁

dx₃ −x₂du^a₂ dx₃

, V^a=

v^a∈(H¹(Ω^a))²×L²(0,1;H¹(ω^a)), ∃c∈H₀¹(0,1), v₁^a=−c x₂, v^a₂ =c x₁,

ω^a

v₃^a(x, x₃) dx= 0, fora.e. x₃∈(0,1)

, W^a =

w^a∈(L²(0,1;H¹(ω^a)))²× {0},

ω^a

w^a_αdx=

ω^a

(x₁w^a₂−x₂w^a₁) dx= 0, fora.e. x₃∈(0,1)

,

U^b=

u^b∈(H¹(Ω^b))²×H₀²(ω^b), ∃ζ_α^b ∈H₀¹(ω^b), u^b_α=ζ_α^b −x₃∂u^b₃

∂x_α

,

V^b=

v^b∈(L²(ω^b;H¹(−1,0)))²× {0}, ₀

−1v_α^b(x, x₃) dx₃= 0, fora.e. x ∈ω^b

,

W^b =

w^b∈({0})²×L²(ω^b;H¹(−1,0)), ₀

−1

w₃^b(x, x₃) dx₃= 0, fora.e. x∈ω^b

, Z^a=U^a× V^a× W^a, Z^b =U^b× V^b× W^b.

Without loss of generality, we assume thatq^εdefined by (2.4) satisfies:

q^ε→q, with 0≤q≤+∞. (2.5)

According to the value ofq, the functional space for the limit problem is the following one:

Z={z= (z^a, z^b) = ((u^a, v^a, w^a),(u^b, v^b, w^b))∈ Z^a× Z^b, u^a₃(x,0) =u^b₃(0), fora.e. x ∈ω^a},if 0< q <+∞, Z∞={z^a= (u^a, v^a, w^a)∈ Z^a, u^a₃(x,0) = 0, fora.e. x∈ω^a}, ifq= +∞,

Z0={z^b= (u^b, v^b, w^b)∈ Z^b, u^b₃(0) = 0}, ifq= 0.

Let us observe that U^a (resp. U^b) is a Bernoulli-Navier (resp. Kirchhoff-Love) space of displacements. Less classical spaces areV^a,W^a,V^b,W^b, which are introduced in a way similar to [27, 28] (see also App., Sect. 8.1).

As for the boundary conditions, some of them are due to the clamping. These are more or less standard ones:

u^a_α(1) =du^a_α

dx₃(1) =c(1) = 0, u^b₃= 0 and ∂u^b₃

dν = 0 on∂ω^b. In contrast with the other requirements, the six conditions:

u^a_α(0) = du^a_α

dx₃(0) =c(0) = 0, u^a₃(x,0) =u^b₃(0) (respectivelyu^a₃(x,0) = 0 oru^b₃(0) = 0),

which appear in the definition of the above spaces U^a, V^a and Z (respectively Z∞ or Z0), are specific to the junction between the beam and the plate. Note also that, in view of the definition of U^a, the condition u^a₃(x,0) =u^b₃(0) (respectivelyu^a₃(x,0) = 0) reduces toζ^a(0) =u^b₃(0) (respectivelyζ^a(0) = 0).

We finally introduce, for z^a = (u^a, v^a, w^a) in Z^a andz^b= (u^b, v^b, w^b) inZ^b:

e^a(z^a) =

⎛

⎝ e_αβ(w^a) e_α3(v^a) e_3α(v^a) e₃₃(u^a)

⎞

⎠, e^b(z^b) =

⎛

⎝ e_αβ(u^b) e_α3(v^b) e_3α(v^b) e₃₃(w^b)

⎞

⎠. (2.6)

2.3. The main result

We describe the limit behaviour of problem (2.3), asεtends to zero. In the sequel, we assume that

f^aε f^a weakly in (L²(Ω^a))³, (2.7)

f^bε f^b weakly in (L²(Ω^b))³, (2.8)

g^aε g^a weakly in (L²(Ω^a))^3×3, (2.9)

g^bε g^b weakly in (L²(Ω^b))^3×3, (2.10)

h^aε h^a weakly in (L²(Σ^a))³, (2.11)

h^bε₊ h^b₊ and h^bε₋ h^b₋ weakly in (L²(ω^b))³, (2.12) which is not restrictive, as proved in Remark 4 below.

Our main result is the following one:

Theorem 1. Assume that r^ε

ε² tends to +∞ and that (2.5), (2.7) to(2.12) hold true. Then, with the notation e^aε,e^bε defined in (2.2)ande^a,e^b defined in (2.6), one has:

(i) If0< q <+∞, there existsz= (z^a, z^b) = ((u^a, v^a, w^a),(u^b, v^b, w^b))∈ Z such that:

(u^aε, u^bε)(u^a, u^b) weakly in(H¹(Ω^a))³×(H¹(Ω^b))³, (2.13) (e^aε(u^aε), e^bε(u^bε))(e^a(z^a), e^b(z^b))weakly in(L²(Ω^a))^3×3×(L²(Ω^b))^3×3, (2.14) andz is the unique solution of the following problem:

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

z∈ Z and ∀z∈ Z,

Ω^a[A^ae^a(z^a), e^a(z^a)] dx+q

Ω^b[A^be^b(z^b), e^b(z^b)] dx

=

Ω^af^a.u^adx+

Ω^bf^b.u^bdx+

Ω^a[g^a, e^a(z^a)] dx+

Ω^b[g^b, e^b(z^b)] dx +

Σ^ah^a.u^adσ+

ω^b

h^b₊.u^b_|x3=0+h^b₋.u^b_|x3=−1 dx.

(2.15)

Moreover, if the convergences in (2.9), (2.10)are strong, then the convergences in (2.13)and(2.14) are strong.

(ii)If q= +∞, there existsz^a= (u^a, v^a, w^a)∈ Z∞ such that:

u^aε u^a weakly in (H¹(Ω^a))³, u^bε→0 strongly in(H¹(Ω^b))³, (2.16) e^aε(u^aε) e^a(z^a) weakly in(L²(Ω^a))^3×3, e^bε(u^bε)→0strongly in (L²(Ω^b))^3×3, (2.17) andz^a is the unique solution of the following problem:

⎧⎪

⎪⎨

⎪⎪

⎩

z^a∈ Z_∞ and ∀z^a∈ Z_∞,

Ω^a[A^ae^a(z^a), e^a(z^a)] dx=

Ω^af^a.u^adx+

Ω^a[g^a, e^a(z^a)] dx+

Σ^ah^a.u^adσ.

(2.18)

Moreover, if the convergence in (2.9)is strong, then:

u^aε→u^a strongly in(H¹(Ω^a))³, (2.19)

e^aε(u^aε)→e^a(z^a)strongly in(L²(Ω^a))^3×3, √

q^εe^bε(u^bε)→0 strongly in(L²(Ω^b))^3×3. (2.20)

(iii)If q= 0, there existsz^b= (u^b, v^b, w^b)∈ Z₀ such that:

q^εu^aε→0 strongly in(H¹(Ω^a))³, q^εu^bε u^b weakly in(H¹(Ω^b))³, (2.21) q^εe^aε(u^aε)→0 strongly in(L²(Ω^a))^3×3, q^εe^bε(u^bε) e^b(z^b)weakly in(L²(Ω^b))^3×3, (2.22) andz^b is the unique solution of the following problem:

⎧⎪

⎪⎨

⎪⎪

⎩

z^b∈ Z₀ and ∀z^b∈ Z₀,

Ω^b

[A^be^b(z^b), e^b(z^b)] dx=

Ω^b

f^b.u^bdx+

Ω^b

[g^b, e^b(z^b)] dx+

ω^b

h^b₊.u^b_|x3=0+h^b₋.u^b_|x3=−1 dx.

(2.23)

Moreover, if the convergence in (2.10) is strong, then:

q^εu^bε→u^b strongly in (H¹(Ω^b))³, (2.24)

√q^εe^aε(u^aε)→0 strongly in(L²(Ω^b))^3×3, q^εe^bε(u^bε)→e^b(z^b)strongly in(L²(Ω^b))^3×3. (2.25)

Remark 1. The condition that r^ε

ε² tends to +∞ is only used to prove that u^a₃(x,0) = u^b₃(0) andc(0) = 0 (via a convenient Sobolev embedding theorem, as regards the second equality). We do not know if it is just a

technical condition or not.

Remark 2. In the Appendix (Sect. 8.1) we prove that the functionsv^aandw^a(resp. v^b andw^b) which appear in the limit problem are the limits of suitable expressions ofu^aε (resp. u^bε).

2.4. Back to the problem in the thin multidomain

As far as the asymptotic behaviour of the “energy” of the solution of problem (1.2) in the thin multidomain is concerned, we define the following renormalized energy by:

E^ε= λ^ε

r^ε ₂

Ω^ε[A^εe(U^ε), e(U^ε)] dx, (2.26)

whereλ^ε can be made explicit in terms ofε, r^ε, F^ε, G^ε, H^ε(see (3.2) in Sect. 3.1); we also have:

E^ε=

Ω^a[A^ae^aε(u^aε), e^aε(u^aε)] dx+q^ε

Ω^b[A^be^bε(u^bε), e^bε(u^bε)] dx, and from Theorem 1 we deduce the following corollary:

Corollary 1. Assume that r^ε

ε² tends to+∞and that (2.5), (2.7)to(2.12) hold true.

(i) If0< q <+∞ and if the convergences in(2.9), (2.10) are strong, then:

E^ε→ E=

Ω^a

[A^ae^a(z^a), e^a(z^a)] dx+q

Ω^b

[A^be^b(z^b), e^b(z^b)] dx.

(ii)If q= +∞and if the convergence in (2.9)is strong, then:

E^ε→ E_∞=

Ω^a[A^ae^a(z^a), e^a(z^a)] dx.

(iii)If q= 0 and if the convergence in(2.10) is strong, then:

q^εE^ε→ E0=

Ω^b[A^be^b(z^b), e^b(z^b)] dx.

Actually, the proof of Corollary 1 is part of proof of Theorem 1, since the strong convergences ofu^aε tou^a (resp. u^bεtou^b) ande^aε(u^aε) toe^a(z^a) (resp. e^bε(u^bε) toe^b(z^b)) follow from the convergence of the energyE^ε. The following interpretation is a direct consequence of the strong convergences ofe^aε(u^aε) ande^bε(u^bε).

Interpretation. For example, let us consider the particular case of problem (1.2), for which G^ε= 0,H^ε= 0, k^ε= 1 andA^a =A^b=A:

⎧⎪

⎪⎨

⎪⎪

⎩

U^ε∈Y^ε and ∀U ∈Y^ε,

Ω^ε[Ae(U^ε), e(U)] dx=

Ω^εF^ε. Udx, and let us assume thatr^ε=ε^3/2and that:

1 ε⁹

α

F_α^ε2_L2(Ω^aε)+ 1 ε⁶

α

F₃^ε2_L2(Ω^aε)+ 1 ε⁶

α

F_α^ε2_L2(Ω^bε)+ 1 ε⁸

α

F₃^ε2_L2(Ω^bε)= 1. (2.27)

This last condition is not restrictive, since it is just a matter of normalization. One can observe that, in this case, the parameterλ^εintroduced in (3.2), in Section 3.1, has valueε^−3/2. Defining the rescaled force and the rescaled solution by:

f_α^aε(x) = 1

ε³F_α^ε(ε³²x, x₃), f₃^aε(x) = 1

ε³²F₃^ε(ε³²x, x₃), forx∈Ω^a, f_α^bε(x) = 1

ε⁵²F_α^ε(x, εx₃), f₃^bε(x) = 1

ε⁷²F₃^ε(x, εx₃), forx∈Ω^b, u^aε_α(x) =U^ε_α(ε³²x, x₃), u^aε₃ (x) = 1

ε³²U^ε₃(ε³²x, x₃), forx∈Ω^a, u^bε_α(x) = 1

ε⁵²U^ε_α(x, εx₃), u^bε₃(x) = 1

ε³²U^ε₃(x, εx₃), forx∈Ω^b, one can check that u^ε solves the rescaled problem:

⎧⎪

⎪⎨

⎪⎪

⎩

u^ε∈ Y^εand ∀u∈ Y^ε,

Ω^a[Ae^aε(u^aε), e^aε(u^a)] dx+

Ω^b[Ae^bε(u^bε), e^bε(u^b)] dx=

Ω^af^aε.u^adx+

Ω^bf^bε.u^bdx.

Since, thanks to (2.27),

Ω^a|f^aε|²dx+

Ω^b|f^bε|²dx= 1,

it is not restrictive to assume that, for some subsequence of ε, still denoted byε, and for somef^a in L²(Ω^a) andf^b in L²(Ω^b):

f^aε f^a in L²(Ω^a) and f^bε f^b in L²(Ω^b).

Then, Theorem 1 asserts that:

u^aε→u^a strongly in (H¹(Ω^a))³ and u^bε→u^b strongly in (H¹(Ω^b))³, (2.28) e^aε(u^aε)→e^a strongly in (L²(Ω^a))^3×3 and e^bε(u^bε)→e^b strongly in (L²(Ω^b))^3×3, (2.29) wheree^a =e^a(z^a),e^b=e^b(z^b) andz= (z^a, z^b) is the unique solution of the rescaled limit problem:

⎧⎪

⎪⎨

⎪⎪

⎩

z∈ Z and ∀z∈ Z,

Ω^a[Ae^a(z^a), e^a(z^a)] dx+

Ω^b[Ae^b(z^b), e^b(z^b)] dx=

Ω^af^a.u^adx+

Ω^bf^b.u^bdx.

(2.30)

Coming back to the initial domain, we defineE^aε andE^bε by:

E^aε=ε³²e^a x

r^ε, x₃

, forx∈Ω^aε, E^bε=ε⁵²e^b

x,x₃

ε , forx∈Ω^bε,

and we define the relative errors ∆^aε, ∆^bεand ∆^εby:

∆^aε=

Ω^aε|e(U^ε)−E^aε|²dx

Ω^aε|E^aε|²dx

, ∆^bε=

Ω^bε|e(U^ε)−E^bε|²dx

Ω^bε|E^bε|²dx ,

∆^ε=

Ω^aε|e(U^ε)−E^aε|²dx+

Ω^bε|e(U^ε)−E^bε|²dx

Ω^aε|E^aε|²dx+

Ω^bε|E^bε|²dx

·

Assuming thate^a = 0 ande^b= 0, an easy computation gives that:

∆^aε=

Ω^a|e^aε(u^aε)−e^a|²dx

Ω^a|e^a|²dx

, ∆^bε=

Ω^b|e^bε(u^bε)−e^b|²dx

Ω^b|e^b|²dx

·

Hence the strong convergences in (2.29) imply that ∆^aε, ∆^bε, and then ∆^ε, tend to zero withε. These conver- gences of the relative errors mean that the deformation of the original displacement is well described by E^aε andE^bε:

e(U^ε)E^aε in Ω^aε, e(U^ε)E^bε in Ω^bε.

In the same spirit, from the solution z = (z^a, z^b) = ((u^a, v^a, w^a),(u^b, v^b, w^b)) of problem (2.30), we are going to define ˆU^aε and ˆU^bε, which are good approximates of the restrictions of U^ε to Ω^aε and Ω^bε, respectively.

Actually, let us set:

ˆ

u^aε=u^a+r^εv^a+ (r^ε)²w^a=u^a+ε³²v^a+ε³w^a, ˆ

u^bε=u^b+εv^b+ε²w^b, Uˆ_α^aε(x) = ˆu^aε_α

x ε³², x₃

, Uˆ₃^aε(x) =ε³²uˆ^aε₃ x

ε³², x₃

, forx∈Ω^aε,

Uˆ_α^bε(x) =ε⁵²uˆ^bε_α

x,x₃

ε , Uˆ₃^bε(x) =ε³²uˆ^bε₃

x,x₃

ε , forx∈Ω^bε. Assuming that (v^a, w^a), (v^b, w^b) have H¹ regularity, and since:

e^aε(ˆu^aε) =e^a+ε³²

⎛

⎝ 0 e_α3(w^a) e_3β(w^a) e₃₃(v^a)

⎞

⎠, e^bε(ˆu^bε) =e^b+ε

⎛

⎝ e_αβ(v^b) e_α3(w^b) e_3β(w^b) 0

⎞

⎠,

it is clear that, as εtends to zero, e^aε(ˆu^aε) tends to e^a strongly in (L²(Ω^a))^3×3 and that e^bε(ˆu^bε) tends to e^b strongly in (L²(Ω^b))^3×3, and then, from (2.29), that:

Ω^a|e^aε(u^aε−ˆu^aε)|² dx+

Ω^b

e^bε

u^bε−uˆ^bε2dx→0.