Junction between a plate and a rod of comparable thickness in nonlinear elasticity.

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Junction between a plate and a rod of comparable thickness in nonlinear elasticity.

Dominique Blanchard, Georges Griso

To cite this version:

Dominique Blanchard, Georges Griso. Junction between a plate and a rod of comparable thickness

in nonlinear elasticity.. Journal of Elasticity, Springer Verlag, 2013, 112, pp.79-109. �10.1007/s10659-

012-9401-6�. �hal-00677838v2�

Junction between a plate and a rod of comparable thickness in nonlinear elasticity. Part II

D. Blanchard ¹ , G. Griso ²

1 Universit´e de Rouen, France. DECEASED on February 11th, 2012 ¹ .

2 Laboratoire J.-L. Lions–CNRS, Boˆıte courrier 187, Universit´e Pierre et Marie Curie, 4 place Jussieu, 75005 Paris, France, Email: griso@ann.jussieu.fr

Abstract

We analyze the asymptotic behavior of a junction problem between a plate and a perpendicular rod made of a nonlinear elastic material. The two parts of this multi-structure have small thicknesses of the same order δ. We use the de- composition techniques obtained for the large deformations and the displacements in order to derive the limit energy as δ tends to 0.

KEY WORDS: nonlinear elasticity, junctions, straight rod, plate.

Mathematics Subject Classification (2000): 74B20, 74K10, 74K30.

1 Introduction

In a former paper [12] we derive the limit energy of the junction problem between a plate and a rod under an assumption that couples their respective thicknesses δ and ε to the order of the Lam´e’s coefficients of the materials in the plate and in the rod.

This assumption precludes the case where the thicknesses have the same order and the structure is made of the same material (see equation 1.1 in the introduction of [12]).

The aim of the present paper is to analyze this specific case for a total energy of order δ ⁵ . As in [12], the structure is clamped on a part of the lateral boundary of the plate and it is free on the rest of its boundary.

The main difference here is the behavior in the rod in which, for this level of energy (which is higher than the maximum allowed in [12]), the stretching-compression is of order δ while the bending is of order δ ^1/2 . The most important consequence is that in the limit model for the rod the stretching-compression is actually given by the bending in the rod (through a nonlinear relation) and by the bending in the plate at the junction point

1 We were just finishing this paper when suddenly two days later my friend the Professor Dominique

Blanchard died. We worked seven years together, our collaboration was very successful for both.

(see (6.15) and (6.16)). The bending and torsion models in the rod are the standard linear ones. In the plate the limit model is the Von K´arm´an system in which the action of the rod is modelized by a punctual force at the junction.

Let us emphasize that in order to obtain sharp estimates on the deformations in the junction area, see Lemma 4.2, we use the decomposition techniques in thin domains (see [12],[24], [8], [7]). In order to scale the applied forces which induce a total energy energy of order δ ⁵ , from Lemma 4.2 and [7], we derive a nonlinear Korn’s inequality for the rod (as far as the plate is concerned this type of inequality is already established in [12]). The nonlinear character of these Korn’s inequalities prompt us to adopt smallness assumptions on some components of the forces. Then, we are in a position to study the asymptotic behavior of the Green-St Venant’s strain tensors in the two parts of the structure. At last this allows us to characterize the limit of the rescaled infimum of the 3d energy as the minimum of a functional over a set of limit admissible displacements which includes the nonlinear relation between the stretching-compression and the bending in the rod.

In Section 2 we introduce a few general notations. Section 3 gives a few recalls on the decomposition technique of the deformations in thin structures. In Section 4, we derive first estimates on the terms of the decomposition of a deformation in the rod and sharp estimates in the junction area. In the same section we also obtain Korn’s inequality in the rod. In Section 5 we introduce the elastic energy and the assumptions on the applied forces in order to obtain a total elastic energy of order δ ⁵ . In Section 6 we analyze the asymptotic behavior of the Green-St-Venant’s strain tensors in the plate and in the rod. In Section 7 we prove the main result of the paper namely the characterization of the limit of the rescaled infimum of the 3d energy.

As general references on the theory of elasticity we refer to [2] and [14]. The reader is referred to [1], [32], [22] for an introduction of rods models and to [17], [16], [13], [19], [31] for plate models. As far as junction problems in multi-structures we refer to [15], [16], [28], [29], [30], [3], [26], [27], [23], [20], [21], [4], [5], [6], [25], [10], [11]. For the decomposition method in thin structures we refer to [22], [23], [24], [25], [7], [8], [9], [11].

2 Notations.

Let us introduce a few notations and definitions concerning the geometry of the plate and the rod. Let ω be a bounded domain in R ² with lipschitzian boundary included in the plane (O; e ₁ , e ₂ ) and such that O ∈ ω. The plate is the domain

Ω δ = ω × ] − δ, δ[.

Let γ 0 be an open subset of ∂ω which is made of a finite number of connected components (whose closure are disjoint). The corresponding lateral part of the boundary of Ω δ is

Γ 0,δ = γ 0 × ] − δ, δ[.

The rod is defined by

B δ = D δ × ] − δ, L[, D δ = D(O, δ), D = D(O, 1)

where δ > 0 and where D r = D(O, r) is the disc of radius r and center the origin O.

We assume that D ⊂⊂ ω. The whole structure is denoted S ^δ = Ω δ ∪ B δ

while the junction is

C _δ = Ω _δ ∩ B _δ = D _δ × ] − δ, δ[.

We denote I d the identity map of R ³ . The set of admissible deformations of the structure is

D δ = n

v ∈ H ¹ ( S δ ; R ³ ) | v = I _d on Γ _0,δ o .

The Euclidian norm in R ^k (k ≥ 1) will be denoted | · | and the Frobenius norm of a square matrix will be denoted ||| · ||| .

3 Some recalls.

To any vector F ∈ R ³ we associate the antisymmetric matrix A _F defined by

∀ x ∈ R ³ , A _F x = F ∧ x. (3.1)

From now on, in order to simplify the notations, for any open set O ⊂ R ³ and any field u ∈ H ¹ ( O ; R ³ ), we set

G _s (u, O ) = ||∇ u + ( ∇ u) ^T || L

(O;R

)

and

d(u, O ) = || dist( ∇ u, SO(3)) || ^L

^(O) .

3.1 Recalls on the decompositions of the plate-displacement.

We know (see [23] or [24]) that any displacement u ∈ H ¹ (Ω δ ; R ³ ) of the plate is decom- posed as

u(x) = U (x 1 , x 2 ) + x 3 R (x 1 , x 2 ) ∧ e ₃ + u(x), x ∈ Ω δ (3.2) where U is defined by

U (x 1 , x 2 ) = 1 2δ

Z δ

−δ

u(x 1 , x 2 , x 3 )dx 3 for a.e. x 3 ∈ ω

and where R is also defined via an average involving the displacement u (see [23] or

[24]). The fields U and R belong to H ¹ (ω; R ³ ) and u belongs to H ¹ (Ω δ ; R ³ ). The sum

of the two first terms U e (x) = U (x 1 , x 2 ) + x 3 R (x 1 , x 2 ) ∧ e ₃ is called the elementary displacement associated to u.

The following Theorem is proved in [23] for the displacements in H ¹ (Ω _δ ; R ³ ) and in [24]

for the displacements in W ^1,p (Ω δ ; R ³ ) (1 < p < + ∞ ).

Theorem 3.1. Let u ∈ H ¹ (Ω δ ; R ³ ), there exists an elementary displacement U e (x) = U (x 1 , x 2 ) + x 3 R (x 1 , x 2 ) ∧ e ₃ and a warping u satisfying (3.2) such that

|| u || ^L

^(Ω

^;R

⁾ ≤ CδG s (u, Ω δ ), ||∇ u || ^L

^(Ω

^;R

⁾ ≤ CG s (u, Ω δ ),

∂ R

∂x α

L

(ω;R

) ≤ C

δ ^3/2 G _s (u, Ω δ ),

∂ U

∂x α − R ∧ e _α

L

(ω;R

) ≤ C

δ ^1/2 G _s (u, Ω δ ),

||∇ u − A _R || L

(ω;R

) ≤ CG _s (u, Ω _δ ),

(3.3)

where the constant C does not depend on δ.

The warping u satisfies the following relations Z δ

−δ

u(x 1 , x 2 , x 3 )dx 3 = 0,

Z δ

−δ

x 3 u α (x 1 , x 2 , x 3 )dx 3 = 0 for a.e. (x 1 , x 2 ) ∈ ω.

(3.4) If a deformation v belongs to D δ then the displacement u = v − I _d is equal to 0 on Γ _0,δ . In this case the the fields U , R and the warping u satisfy

U = R = 0 on γ 0 , u = 0 on Γ 0,δ . (3.5)

Then, from (3.3), for any deformation v ∈ D δ the corresponding displacement u = v − I d

verifies the following estimates (see also [23]):

||R|| ^H

^(ω;R

⁾ + ||U ³ || ^H

^(ω) ≤ C

δ ^3/2 G _s (u, Ω δ ),

||R 3 || L

(ω) + ||U α || H

(ω) ≤ C

δ ^1/2 G _s (u, Ω _δ ).

(3.6)

The constants depend only on ω. From the above estimates we deduce the following Korn’s type inequalities for the displacement u

|| u α || L

(Ω

) ≤ C 0 G _s (u, Ω δ ), || u 3 || L

(Ω

) ≤ C 0

δ G _s (u, Ω δ ),

|| u − U|| ^L

^(Ω

^;R

⁾ ≤ C

δ G _s (u, Ω δ ), ||∇ u || ^L

^(Ω

^;R

⁾ ≤ C

δ G _s (u, Ω δ ).

(3.7)

Due to Theorem 3.3 established in [8], the displacement u = v − I d is also decomposed as

u(x) = U (x 1 , x 2 ) + x 3 ( R (x 1 , x 2 ) − I ₃ ) e ₃ + u(x), x ∈ Ω δ (3.8)

where R ∈ H ¹ (ω; R ^3×3 ), u ∈ H ¹ (Ω δ ; R ³ ) and we have the following estimates

|| u || ^L

^(Ω

^;R

⁾ ≤ Cδ d (v, Ω δ ) ||∇ u || ^L

^(Ω

^;R

⁾ ≤ C d (v, Ω δ )

∂ R

∂x α

L

(ω;R

) ≤ C

δ ^3/2 d(v, Ω δ )

∂ U

∂x α − ( R − I ₃ ) e _α

L

(ω;R

) ≤ C

δ ^1/2 d (v, Ω δ ) ∇ v − R

L

(Ω

;R

) ≤ Cd(v, Ω δ )

(3.9)

where the constant C does not depend on δ. The following boundary conditions are satisfied

U = 0, R = I ₃ on γ 0 , u = 0 on Γ 0,δ . (3.10) Due to (3.9) and the above boundary conditions we obtain

|| R − I ₃ || H

(ω;R

) + ||U|| H

(ω;R

) ≤ C

δ ^3/2 d(v, Ω _δ ). (3.11)

3.2 Recall on the decomposition of the rod-deformation.

Now, we consider a deformation v ∈ H ¹ (B δ ; R ³ ) of the rod B δ . This deformation can be decomposed as (see Theorem 2.2.2 of [7])

v(x) = V (x 3 ) + Q(x 3 ) x 1 e ₁ + x 2 e ₂

+ v(x), x ∈ B δ , (3.12) where V (x 3 ) = 1

| D δ | Z

D

v(x)dx 1 dx 2 belongs to H ¹ ( − δ, L; R ³ ), where Q belongs to H ¹ ( − δ, L; SO(3)) and u belongs to H ¹ (B δ ; R ³ ). Let us give a few comments on the above decomposition. The term V gives the deformation of the center line of the rod.

The second term Q(x 3 ) x 1 e ₁ + x 2 e ₂

describes the rotation of the cross section (of the rod) which contains the point (0, 0, x 3 ). The sum of the terms V (x 3 )+Q(x 3 ) x 1 e ₁ +x 2 e ₂ is called an elementary deformation of the rod.

The following theorem (see Theorem 2.2.2 of [7]) gives a decomposition (3.12) of a deformation and estimates on the terms of this decomposition.

Theorem 3.2. Let v ∈ H ¹ (B δ ; R ³ ), there exists an elementary deformation V (x 3 ) + Q(x ₃ ) x ₁ e ₁ + x ₂ e ₂

and a warping v satisfying (3.12) and such that

|| v || ^L

^(B

^;R

⁾ ≤ Cδd(v, B δ ),

||∇ v || L

(B

;R

) ≤ Cd(v, B δ ),

dQ dx ₃

L

(−δ,L;R

) ≤ C

δ ² d(v, B δ ),

d V

dx ₃ − Qe ₃

L

(−δ,L;R

) ≤ C

δ d(v, B δ ), ∇ v − Q

L

(B

;R

) ≤ Cd(v, B δ ),

(3.13)

where the constant C does not depend on δ and L.

4 Preliminaries results

Let v be a deformation in D δ . We set u = v − I _d . We decompose u as (3.2) and (3.8) in the plate and we decompose the deformation v as (3.12) in the rod.

4.1 A complement to the mid-surface bending.

Let us set

H _γ ¹

(ω) = { ϕ ∈ H ¹ (ω) ; ϕ = 0 on γ 0 } .

We define the function U e ³ as the solution of the following variational problem :

 

 



 

 

U e ³ ∈ H _γ ¹

(ω), Z

ω ∇ U e ³ ∇ ϕ = Z

ω

(R − I ₃ )e α · e ₃ ∂ϕ

∂x α

,

∀ ϕ ∈ H _γ ¹

(ω)

(4.1)

where R appears in the decomposition (3.8) of u. Due to (3.9)-(3.11), the function U e ³ belongs to H _γ ¹

(ω) ∩ H ² (D) (remind that D is the disc of radius 1 and center the origin O; and we assumed that D ⊂⊂ ω). The function U e ³ satisfies the estimates:

|| U e ³ || ^H

^(ω) ≤ C

δ ^3/2 d(v, Ω δ ), ||U ³ − U e ³ || ^H

^(ω) ≤ C

δ ^1/2 d(v, Ω δ ),

|| U e ³ || ^H

^(D) ≤ C

δ ^3/2 d(v, Ω δ ), ∂ U e ³

∂x α − (R − I ₃ )e α · e ₃

H

(D) ≤ C

δ ^3/2 d(v, Ω δ ),

| U e ³ (0, 0) | ≤ C

δ ^3/2 d(v, Ω δ ).

(4.2)

The constants do not depend on δ.

4.2 A complement to the rod center-line displacement.

Let V given by (3.12), we consider W (x ₃ ) = V (x ₃ ) − x ₃ e ₃ = 1

| D δ | Z

D

u(x)dx ₁ dx ₂ the rod center-line displacement. From the above Theorem 3.2, the estimate below holds

true d W

dx 3 − Q − I ₃ e ₃

L

(−δ,L;R

) ≤ C

δ d(v, B δ ). (4.3) As in [7] we split the center line displacement W into two parts. The first one W ^(m) stands for the main displacement of the rod which describes the displacement coming from the bending and the second one for the stretching of the rod.

∀ x ₃ ∈ [0, L], W ^(m) (x ₃ ) = W (0) + Z x

0

Q(t) − I ₃ e ₃ dt, W ^(s) (x 3 ) = W (x 3 ) − W ^(m) (x 3 ).

(4.4)

In the lemma below we give estimates on W ^(s) and W ^(m) . Lemma 4.1. We have

||W ^(s) || ^H

^(−δ,L;R

⁾ ≤ C

δ d (v, B δ ), (4.5)

and W α ^(m) − W ^α (0)

H

(−δ,L) ≤ C

δ ² d(v, B δ ) + C ||| Q(0) − I ₃ ||| , (4.6) W 3 ^(m) − W 3 ^(m) (0)

H

(−δ,L) ≤ C δ ⁴

d(v, B δ ) 2

+ C | Q(0) − I ₃

e ₃ · e ₃ | ,

d W 3 ^(m)

dx 3

L

(−δ,δ) ≤ C δ ^5/2

d(v, B δ ) 2

+ Cδ ^1/2 | Q(0) − I ₃

e ₃ · e ₃ | .

(4.7)

The constants do not depend on δ.

Proof. Taking into account the facts that W ^(s) (0) = 0 and d W ^(s) dx 3

= d W

dx 3 − Q − I ₃ e ₃ , the estimate (4.3) leads to (4.5). From the third estimate in (3.13) we obtain

|| Q − Q(0) || L

(−δ,L;R

) ≤ C

δ ² d(v, B δ ), (4.8) Due to the definition (4.4) of W ^(m) and estimate (3.13) ₃ we get

d W ^(m) dx 3

H

(−δ,L) ≤ C

δ ² d(v, B δ ) + C ||| Q(0) − I ₃ |||

and thus (4.6). A straightforward calculation gives d W 3 ^(m)

dx ₃ = Q − I ₃

e ₃ · e ₃ = − 1 2

(Q − I ₃ )e 3

² . (4.9)

Besides we have

d dx 3

(Q − I ₃ )e 3

= dQ dx 3

e ₃ . (4.10)

We recall that for φ ∈ H ¹ (0, L) and η ∈ ]0, L[ we have Z η

0 | φ(t) − φ(0) | ² dt ≤ η ² 2

dφ

dt ²

L

(0,L) ,

Z η

0 | φ(t) − φ(0) | ⁴ dt ≤ η ³ 3

dφ

dt ⁴

L

(0,L) . (4.11) Then, the estimates (3.13) ₃ , (4.11) ₁ and the equality (4.10) give

(Q − I ₃ )e 3 − (Q(0) − I ₃ )e 3

L

(−δ,L;R

) ≤ C

δ ² d(v, B δ ), ( Q − I ₃ ) e ₃ − ( Q (0) − I ₃ ) e ₃

L

(−δ,δ;R

) ≤ C

δ d (v, B δ ).

(4.12)

Now, again (3.13) ₃ and (4.11) ₂ lead to (Q − I ₃ )e 3 − (Q(0) − I ₃ )e 3

L

(−δ,L;R

) ≤ C

δ ² d(v, B δ ), (Q − I ₃ )e ₃ − (Q(0) − I ₃ )e ₃

L

(−δ,δ;R

) ≤ C

δ ^5/4 d(v, B _δ ).

(4.13)

Finally, from (4.9) and the above inequality we obtain (4.7).

4.3 First estimates in the junction area.

Lemma 4.2. We have the following estimate on Q(0) − I ₃ : Q (0) − I ₃

e ₃ · e ₃ ≤ C

δ ^3/2 G _s (u, Ω δ ) + d (v, B δ ) ,

||| Q(0) − I ₃ ||| ≤ C

δ ^7/4 G _s (u, Ω δ ) + C

δ ^3/2 d(v, B δ )

(4.14)

and those about W (0)

|W ^α (0) | ≤ C

δ ^3/4 G _s (u, Ω δ ) + C

δ ^1/2 d(v, B δ ) (4.15) and

|W ³ (0) − U e ³ (0, 0) | ≤ C δ ²

d(v, B δ ) 2

+ C

δ ^1/2 d(v, B δ ) + G _s (u, Ω δ ) + C

δ d(v, Ω δ ),

|W ³ (0) | ≤ C

δ ^3/2 d(v, Ω δ ) + C δ ²

d(v, B δ ) 2

+ C

δ ^1/2 d(v, B δ ) + G _s (u, Ω δ ) .

(4.16)

The constants are independent of δ .

Proof. Step 1. We prove the estimate on Q(0) − I ₃ . We consider the last inequalities in Theorems 3.1 and 3.2. They give

Q − I ₃ − A _R

L

(C

;R

) ≤ C G _s (u, Ω δ ) + d(v, B δ )

. (4.17)

Now, from the third estimate in (3.13), we get Q − Q(0)

L

(−δ,δ;R

) ≤ Cδ dQ dx 3

L

(−δ,δ;R

) ≤ C

δ d(v, B δ ).

Hence

Q(0) − I ₃ − A _R ²

L

(D

;R

) ≤ C δ

G _s (u, Ω δ ) 2

+

d(v, B δ ) 2

. (4.18) We recall that the matrix A _R is antisymmetric, then (4.18) leads to the first estimate in (4.14). Due to (3.6) we have

||R|| ⁴ L

(D

;R

) ≤ Cδ ³ ||R|| ⁴ L

(D

;R

) ≤ Cδ ³ ||R|| ⁴ H

(ω;R

) ≤ C δ ³

G _s (u, Ω δ ) 4

. (4.19)

Then, using the above estimate and (4.18) we deduce the second estimate in (4.14).

Step 2. We prove the estimate (4.15) on W ^α (0).

The two decompositions of u = v − I d ((3.2) and (3.12)) give, for a.e. x ∈ C δ

U (x ₁ , x ₂ ) + x ₃ R (x ₁ , x ₂ ) ∧ e ₃ + u(x)

= W (x 3 ) + (Q(x 3 ) − I ₃ )(x 1 e ₁ + x 2 e ₂ ) + v(x). (4.20) Taking the averages on the cylinder C _δ of the terms in this equality (4.20) give

M ^D

U

= 1

| D δ | Z

D

U (x 1 , x 2 )dx 1 dx 2 = M ^I

W

= 1 2δ

Z δ

−δ W (x 3 )dx 3 . (4.21) Besides, proceeding as for R in (4.19) and from (3.6) we have

||U ^α || ^L

^(D

⁾ ≤ Cδ ^1/4 G _s (u, Ω δ ).

From this estimate we get

|M I

W α

| = |M D

U α

| ≤ C

δ ^3/4 G _s (u, Ω _δ ). (4.22) We set y α (x 3 ) = W ^α (x 3 ) − x 3 (Q(0) − I ₃ )e 3 · e _α . The estimates (4.3) and (4.12) lead to

dy α

dx ₃

L

(−δ,δ) ≤ C

δ d(v, B δ ) which in turn implies y α − y α (0)

L

(−δ,δ) ≤ C d (v, B δ ).

Taking the average, it yields

|M ^I

W ^α

− W ^α (0) | ≤ C

δ ^1/2 d (v, B δ ). (4.23) Finally, from (4.22) and (4.23) we obtain (4.15).

Step 3. We prove the estimate on W ³ (0). Using (4.2) we deduce that

||U ³ − U e ³ || ^L

^(D

⁾ ≤ Cδ ^1/2 ||U ³ − U e ³ || ^L

^(ω)

≤ Cδ ^1/2 ||U ³ − U e ³ || H

(ω) ≤ Cd(v, Ω δ ). (4.24) Then we replace U 3 with U e 3 and W 3 with W 3 ^(m) in (4.21). Taking into account (4.5) we obtain

|M ^D

U e ³

− M ^I

W 3 ^(m)

| ≤ C

δ d(v, Ω δ ) + C

δ ^1/2 d(v, B δ ). (4.25) We carry on by comparing M ^D

U e ³

with U e ³ (0, 0). Let us set r _α = 1

πδ ² Z

D

R(x 1 , x 2 ) − I ₃

e _α · e ₃ dx 1 dx 2

and consider the function Ψ(x 1 , x 2 ) = U e ³ (x 1 , x 2 ) − M ^D

U e ³

− x 1 r ₂ − x 2 r ₁ . Due to (4.2) we first obtain ∂ ² Ψ

∂x α ∂x β

L

(D

,R

) ≤ C

δ ^3/2 d(v, Ω _δ ). (4.26)

Then, applying twice the Poincar´e-Wirtinger inequality in the disc D δ and using (3.3) and the fourth estimate in (4.2) lead to

||∇ Ψ || ² L

(D

,R

) ≤ C δ

d(v, Ω δ ) 2

, || Ψ || ² L

(D

,R

) ≤ Cδ

d(v, Ω δ ) 2

. (4.27)

From the above inequalities (4.26) and (4.27) we deduce that

|| Ψ || ^L

^(D

^,R

⁾ ≤ C

δ ^1/2 d(v, Ω δ ) = ⇒ | Ψ(0, 0) | = | U e ³ (0, 0) − M ^D

U e ³

| ≤ C

δ ^1/2 d(v, Ω δ ).

From this last estimate and (4.25) we obtain

| U e ³ (0, 0) − M ^I

W 3 ^(m)

| ≤ C

δ d(v, Ω δ ) + C

δ ^1/2 d(v, B δ ). (4.28) Then using the second estimate in (4.7) and (4.14) we have

d W 3 ^(m)

dx ₃

L

(−δ,δ) ≤ C δ ^5/2

d(v, B δ ) 2

+ C

δ G _s (u, Ω δ ) + d(v, B δ )

. (4.29) Finally, recalling that W ³ (0) = W 3 ^(m) (0), the above inequality leads to

|M ^I

W 3 ^(m)

−W ³ (0) | ≤ Cδ ^1/2 d W 3 ^(m)

dx 3

L

(−δ,δ) ≤ C δ ²

d(v, B δ ) 2

+ C

δ ^1/2 G _s (u, Ω δ )+d(v, B δ ) which in turn with (4.28) and (4.2) lead to (4.16).

4.4 Global estimates of u : Korn’s type inequality.

Now, we give the last estimates of the displacement u = v − I d in the rod B δ .

Lemma 4.3. For any deformation v in D δ we have the following inequalities for the displacement u = v − I d in the rod B δ :

|| u − W|| L

(B

;R

) ≤ C d(v, B _δ ) + δ ^1/4 G _s (u, Ω _δ ) , W α

L

(−δ,L) + W α ^(m)

L

(−δ,L) ≤ C d(v, B δ )

δ ² + G _s (u, Ω δ ) δ ^7/4

,

W ³

L

(−δ,L) + W 3 ^(m)

L

(−δ,L) ≤ C

d(v, B δ ) 2

δ ⁴ + C δ ^3/2

G _s (u, Ω δ ) + d(v, Ω δ ) + d(v, B δ )

.

(4.30)

The constants do not depend on δ.

Proof. From (4.8) and (4.14) we get Q − I ₃

L

(−δ,L;R

) ≤ C d(v, B _δ )

δ ² + G _s (u, Ω _δ ) δ ^7/4

. (4.31)

Then, from (3.13) and the above inequality we deduce that

|| u − W|| L

(B

;R

) ≤ C d(v, B δ ) + δ ^1/4 G _s (u, Ω δ )

. (4.32)

From (4.6) again (4.14) and (4.15) we obtain W α ^(m)

H

(−δ,L) ≤ C

δ ² d(v, B δ ) + C

δ ^7/4 G _s (u, Ω δ ). (4.33) Then since W = W ^(m) + W ^(s) , (4.5) and (4.33) give the second estimate in (4.30).

From (4.7) and (4.14) we deduce that

d W 3 ^(m)

dx 3

L

(−δ,L) ≤ C δ ⁴

d (v, B δ ) 2

+ C δ ^3/2

G _s (u, Ω δ ) + d (v, B δ ) . which in turn using (4.16) lead to

W 3 ^(m)

L

(−δ,L) ≤ C δ ⁴

d(v, B δ ) 2

+ C

δ ^3/2

G _s (u, Ω δ ) + d(v, Ω δ ) + d(v, B δ ) and then due to (4.5) we get the last estimate in (4.30).

Corollary 4.4. For any deformation v in D δ we have the following Korn’s type inequal- ity for the displacement u = v − I d in the rod B δ :

∇ u

L

(B

;R

) ≤ C

δ d(v, B δ ) + C

δ ^3/4 G _s (u, Ω δ ),

|| u α || ^L

^(B

⁾ ≤ C

δ d (v, B δ ) + C

δ ^3/4 G _s (u, Ω δ ),

|| u 3 || L

(B

) ≤ C

d(v, B δ ) 2

δ ³ + C δ ^1/2

G _s (u, Ω δ ) + d(v, Ω δ ) + d(v, B δ ) .

(4.34)

The constants do not depend on δ.

Proof. From (3.13) and (4.31) we obtain ∇ u

L

(B

;R

) ≤ C

δ d(v, B δ ) + C

δ ^3/4 G _s (u, Ω δ ). (4.35)

The second and third inequalities are immediate consequences of Lemma 4.3.

5 Elastic structure

5.1 Elastic energy.

In this section we assume that the structure S ^δ is made of an elastic material. The associated local energy W c : X ₃ −→ R ⁺ is the following St Venant-Kirchhoff’s law ² (see also [14])

W c (F ) =

( Q(F ^T F − I ₃ ) if det(F ) > 0

+ ∞ if det(F ) ≤ 0. (5.1)

where X ₃ is the space of 3 × 3 symmetric matrices and where the quadratic form Q is given by

Q(E) = λ

8 tr(E) 2

+ µ

4 tr E ²

, (5.2)

and where (λ, µ) are the Lam´e’s coefficients of the material. Let us recall (see e.g. [19]

or [7]) that for any 3 × 3 matrix F such that det(F ) > 0 we have

[tr(F ^T F − I ₃ )] ² = ||| F ^T F − I ₃ ||| ² ≥ dist (F, SO(3)) ² . (5.3)

5.2 Assumptions on the forces and final estimates.

Now we assume that the structure S δ is submitted to applied body forces f _δ ∈ L ² ( S ^δ ; R ³ ) and we define the total energy J δ (v) ³ over D δ by

J _δ (v) = Z

S

W c _δ ( ∇ v )(x)dx − Z

S

f _δ (x) · (v (x) − I _d (x))dx. (5.4) Assumptions on the forces. To introduce the scaling on f _δ , let us consider f _r , g ₁ , g 2 in L ² (0, L; R ³ ) and f p ∈ L ² (ω; R ³ ). We assume that the force f δ is given by

f _δ (x) = δ ^5/2 h

f _r,1 (x ₃ )e ₁ + f _r,2 (x ₃ )e ₂ + 1

δ ^1/2 f _r,3 (x ₃ )e ₃ + x 1

δ ² g ₁ (x ₃ ) + x 2

δ ² g ₂ (x ₃ ) i x ∈ B δ , x 3 > δ,

f δ,α (x) = δ ² f p,α (x 1 , x 2 ), f δ,3 (x) = δ ³ f p,3 (x 1 , x 2 ), x ∈ Ω δ .

(5.5)

We denote

F r,3 (x 3 ) = Z L

x

f r,3 (s)ds, for a. e. x 3 ∈ ]0, L[. (5.6)

2 With a more general assumption on the nonlinear elasticity law (see for example [19] page 1466) we would obtain the same asymptotic behavior as in our case.

3 For later convenience, we have added the term Z

S

f δ (x) · I d (x)dx to the usual standard energy,

indeed this does not affect the minimizing problem for J δ .

Theorem 5.1. There exist two constants C 0 and C 1 , which depend only on ω, L and µ, such that if

|| f p || L

(Ω;R

) ≤ C 0 (5.7) and if either

Case 1: for a. e. x 3 ∈ ]0, L[, F r,3 (x 3 ) ≥ 0, or

Case 2: || f _r,3 || L

(0,L) ≤ C ₁

(5.8) then for δ small enough and for any v ∈ D δ satisfying J δ (v ) ≤ 0 we have

d(v, B δ ) + d(v, Ω δ ) ≤ Cδ ^5/2 (5.9) where the constant does not depend on δ.

Proof. From (3.7) and the assumptions (5.5) on the body forces, we obtain on the one hand for any v ∈ D δ and with u = v − I d

Z

Ω

f δ (x) · u(x)dx ≤ Cδ ^5/2 || f p || ^L

^(ω;R

⁾ G _s (u, Ω δ ). (5.10) As far as the term involving the forces in the rod are concerned we first have

Z

B

f δ (x) · u(x)dx = πδ ^9/2 Z L

δ

f r,α (x 3 ) W ^α (x 3 )dx 3 + πδ ⁴ Z L

δ

f r,3 (x 3 ) W ³ (x 3 )dx 3

+ Z

B

f _δ (x) · (u(x) − W (x ₃ ))dx.

Then, using Lemma 4.3 and (5.5) we first get

Z L

δ

f _r,α (x ₃ ) W α (x ₃ )dx ₃ ≤ C δ ²

X 2 α=1

|| f _r,α || L

(0,L) d(v, B _δ ) + δ ^1/4 G _s (u, Ω _δ ) ,

Z

B

f δ (x) · u(x) − W (x 3 )

dx ≤ Cδ ^5/2 || g 1 || L

(0,L;R

) + || g 2 || L

(0,L;R

)

d(v, B δ ) + δ ^1/4 G _s (u, Ω δ )

(5.11)

Now we estimate Z L

δ

f r,3 W ³ (x 3 )dx 3 . From (4.5) we first obtain

Z L

δ

f r,3 (x 3 ) W 3 ^(s) (x 3 )dx 3

≤ C

δ || f r,3 || ^L

^(0,L) d(v, B δ ). (5.12) Then we have

Z L δ

f _r,3 (x ₃ ) W 3 ^(m) (x ₃ )dx ₃ = F _r,3 (δ) W 3 ^(m) (δ) + Z L

δ

F _r,3 (x ₃ ) d W 3 ^(m)

dx 3

(x ₃ )dx ₃ . (5.13)

Taking to account (4.29) and (4.16) we get

|W 3 ^(m) (δ) | ≤ |W ³ (0) | + δ ^1/2 d W 3 ^(m)

dx 3

L

(−δ,δ)

≤ C

δ ^3/2 d(v, Ω δ ) + C δ ²

d(v, B δ ) 2

+ C

δ ^1/2 G _s (u, Ω δ ) + d(v, B δ ) .

(5.14)

Observe now that due to the expression (4.9) of d W 3 ^(m)

dx 3

, this derivative is nonpositive for a.e. x 3 ∈ ]0, L[ (see (4.9)).

• If we are in Case 1 in (7.34), we have Z L

δ

f r,3 (x 3 ) W ³ (x 3 )dx 3 ≤ C || f r,3 || L

(0,L)

h d(v, Ω δ ) δ ^3/2 +

d(v, B δ ) 2

δ ² + G _s (u, Ω δ ) + d(v, B δ ) δ ^1/2

i .

Hence, we obtain Z

B

f δ (x) · u(x)dx ≤ Cδ ^5/2 X 2 α=1

|| f r,α || L

(0,L) + || g α || L

(0,L;R

)

d(v, B δ ) + δ ^1/4 G _s (u, Ω δ )

+C || f r,3 || L

(0,L)

h δ ^5/2 d(v, Ω δ ) + δ ² [d(v, B δ ) 2

+ δ ^7/2 G _s (u, Ω δ ) + d(v, B δ ) i . We recall that (see [8])

G _s (u, Ω δ ) ≤ Cd(v, Ω δ ) + C δ ^5/2

d(v, Ω δ ) 2

(5.15) where the constant does not depend on δ. Then due to (5.10) and the above inequalities we obtain that

Z

Ω

f δ (x) · u(x)dx ≤ C || f r || ^L

^(ω;R

⁾ δ ^5/2 d (v, Ω δ ) + C ^∗ || f r || ^L

^(ω;R

⁾

d (v, Ω δ ) 2

Z

B

f _δ (x) · u(x)dx ≤ C(f _r , g ₁ , g ₂ )δ ^5/2 d(v, B _δ ) + d(v, Ω _δ ) + C(f r , g 1 , g 2 )δ ^1/4

d(v, Ω δ ) 2

+ C || f r,3 || ^L

^(0,L) δ ²

d(v, B δ ) 2

.

(5.16)

Now, for any v ∈ D δ such that J δ (v) ≤ 0, assumptions (5.1), (5.2),(5.3) and the above estimates lead to

µ 8

d (v, B δ )] ² +

d (v, Ω δ ) 2

≤ Z

S

W c ( ∇ v)(x)dx ≤ Z

S

f δ (x) · u(x)dx

≤ C(f r , g 1 , g 2 )δ ^5/2 d(v, B δ ) + d(v, Ω δ )

+ C(f r , g 1 , g 2 )δ ^1/4

d(v, Ω δ ) 2

+C || f r,3 || ^L

^(0,L) δ ²

d(v, B δ ) 2

+ Cδ ^5/2 || f p || ^ω;R

d(v, Ω δ ) + C ^∗ || f p || ^ω;R

d(v, Ω δ ) 2

.

wich in turn gives µ

8 − C || f r,3 || L

(0,L) δ ²

d(v, B δ )] ² + µ

8 − C ^∗ || f p || ω;R

− C(f r , g 1 , g 2 )δ ^1/4

d(v, Ω δ ) 2

≤ C(f r , g 1 , g 2 )δ ^5/2 d(v, B δ ) + d(v, Ω δ )

+ Cδ ^5/2 || f p || ^ω;R

d(v, Ω δ ).

Indeed the two quantities C || f r,3 || ^L

^(0,L) δ ² and C(f r , g 1 , g 2 )δ ^1/4 tend to 0 as δ tends to 0, then, under the condition C ^∗ || f _p || L

(ω;R

) ≤ µ/32 and for δ small enough we obtain

d(v, B δ ) + d(v, Ω δ ) ≤ Cδ ^5/2 . The constant does not depend on δ.

• If we are in Case 2 in (7.34), from (4.30) we immediately have Z L

δ

f r,3 (x 3 ) W ³ (x 3 )dx 3 ≤ C || f r,3 || ^L

^(0,L) h d(v, B δ ) 2

δ ⁴ + G _s (u, Ω δ ) + d(v, Ω δ ) + d(v, B δ ) δ ^3/2

i .

Then, proceeding as in Case 1 leads to Z

B

f δ (x) · u(x)dx ≤ Cδ ^5/2 X 2 α=1

|| f r,α || ^L

^(0,L) + || g α || ^L

^(0,L;R

⁾

d(v, B δ ) + δ ^1/4 G _s (u, Ω δ ) + C || f r,3 || L

(0,L)

h d(v, B δ ) 2

+ δ ^5/2 G _s (u, Ω δ ) + d(v, Ω δ ) + d(v, B δ ) i . (5.17) Then for δ small enough, we get

µ 8

d(v, B δ )] ² +

d(v, Ω δ ) 2

≤ Z

S

c W ( ∇ v)(x)dx ≤ Z

S

f δ (x) · u(x)dx

≤ Cδ ^5/2 C(f, g) d(v, B δ ) + d(v, Ω δ )

+ C ^∗∗ || f r,3 || L

(0,L)

h d(v, B δ ) 2

+

d(v, Ω δ ) 2 i + Cδ ^5/2 || f p || ^ω;R

d(v, Ω δ ) + C ^∗ || f p || ^ω;R

d(v, Ω δ ) 2

.

Hence, under the conditions C ^∗ || f p || ^L

^(ω;R

⁾ ≤ µ/32 and C ^∗∗ || f r,3 || ^L

^(0,L) ≤ µ/32 we deduce that

d(v, B δ ) + d(v, Ω δ ) ≤ Cδ ^5/2 . In the both cases, we finally obtain (5.9)

As a consequence of Theorem 5.1 and estimates (5.16)-(5.17), we deduce that for δ small enough and for any v ∈ D δ satisfying J δ (v) ≤ 0 we have (u = v − I d )

Z

S

f δ · u ≤ Cδ ⁵ , Z

S

c W δ ( ∇ v)(x)dx ≤ Cδ ⁵ . (5.18) From (5.18) we also obtain for any v ∈ D δ such that J δ (v) ≤ 0

cδ ⁵ ≤ J δ (v) (5.19)

where c is a nonpositive constant which does not depend on δ. We set m δ = inf

v∈D

J δ (v).

As a consequence of (5.19) we have

c ≤ m δ

δ ⁵ ≤ 0. (5.20)

In general, a minimizer of J δ does not exist on D δ .

6 Asymptotic behavior of a sequence of deforma- tions of the whole structure S ^δ .

In this subsection and the following one, we consider a sequence of deformations (v δ ) belonging to D δ and satisfying

d(v δ , B δ ) + d(v δ , Ω δ ) ≤ Cδ ^5/2 (6.1) where the constant does not depend on δ. Setting u δ = v δ − I d , then, due to (6.1) and (5.15) we obtain that

G _s (u δ , Ω δ ) ≤ Cδ ^5/2 . (6.2)

For any open subset O ⊂ R ² and for any field ψ ∈ H ¹ ( O ; R ³ ), we denote γ αβ (ψ) = 1

2 ∂ψ α

∂x β

+ ∂ψ β

∂x α

, (α, β) ∈ { 1, 2 } . (6.3)

6.1 The rescaling operators

Before rescaling the domains, we introduce the reference domain Ω for the plate and the one B for the rod

Ω = ω × ] − 1, 1[, B = D × ]0, L[= D(O, 1) × ]0, L[.

As usual when dealing with thin structures, we rescale Ω δ and B δ using -for the plate- the operator

Π δ (w)(x 1 , x 2 , X 3 ) = w(x 1 , x 2 , δX 3 ) for any (x 1 , x 2 , X 3 ) ∈ Ω

defined for e.g. w ∈ L ² (Ω δ ) for which Π δ (w) ∈ L ² (Ω) and using -for the rod- the operator P δ (w)(X 1 , X 2 , x 3 ) = w(δX 1 , δX 2 , x 3 ) for any (X 1 , X 2 , x 3 ) ∈ B

defined for e.g. w ∈ L ² (B δ ) for which P δ (w) ∈ L ² (B).

6.2 Asymptotic behavior in the plate.

Following Section 2 we decompose the restriction of u δ = v δ − I d to the plate. The Theorem 3.1 gives U ^δ , R ^δ and u δ , then estimates (3.6) lead to the following conver- gences for a subsequence still indexed by δ (see [23] for the detailed proofs of the below convergences and equalities)

1

δ U ^δ,3 −→ U ³ strongly in H ¹ (ω), 1

δ ² U ^δ,α ⇀ U ^α weakly in H ¹ (ω), 1

δ R ^δ ⇀ R weakly in H ¹ (ω; R ³ ), 1

δ ³ Π δ (u δ ) ⇀ u weakly in L ² (ω; H ¹ ( − 1, 1; R ³ ), 1

δ ² ∂ U ^δ

∂x α − R ^δ ∧ e _α

⇀ Z ^α weakly in L ² (ω; R ³ ).

(6.4)

Denoting by A _R the field of antisymmetric matrices associated to R as in Section 2, we also have

1

δ ² Π _δ (u _δ − U δ ) −→ X ₃ R ∧ e ₃ strongly in L ² (Ω; R ³ ), 1

δ Π _δ ( ∇ u _δ ) −→ A _R strongly in L ² (Ω; R ⁹ ).

(6.5) The boundary conditions (3.5) give here

U ³ = 0, U ^α = 0, R = 0 on γ 0 , (6.6)

while (6.4) show that U 3 ∈ H ² (ω) with

∂ U ³

∂x 1

= −R ² , ∂ U ³

∂x 2

= R ¹ . (6.7)

In [8] (see Theorem 7.3) the limit of the Green-St Venant’s strain tensor of the sequence v _δ is also derived. Let us set

u p = u + X 3

2 Z ¹ · e ₃

e ₁ + X 3

2 Z ² · e ₃

e ₂ (6.8)

and

Z ^αβ = γ αβ ( U ) + 1 2

∂ U ³

∂x _α

∂ U ³

∂x _β . (6.9)

Then we have 1

2δ ² Π δ ( ∇ v δ ) ^T ∇ v δ − I ₃

⇀ E _p weakly in L ¹ (Ω; R ⁹ ),

where the symmetric matrix E _p is defined by

E _p =



 

 

 

− X 3

∂ ² U ³

∂x ² ₁ + Z ¹¹ − X 3

∂ ² U ³

∂x 1 ∂x 2

+ Z ¹² 1 2

∂u p,1

∂X 3

∗ − X 3

∂ ² U ³

∂x ² ₂ + Z ²² 1 2

∂u p,2

∂X 3

∗ ∗ ∂u p,3

∂X ₃



 

 

 

. (6.10)

6.3 Asymptotic behavior in the rod.

Now, we decompose the restriction of v δ = u δ + I d to the rod (see Section 2). The Theorem 3.2 gives W ^δ , Q _δ , v δ and thanks to (4.4) we define W δ ^(m) and W δ ^(s) . Then the estimates in Theorem 3.2 and Lemma 4.1 allow to claim that

|| v δ || L

(B

;R

) ≤ Cδ ^7/2 , ||∇ v δ || L

(B

;R

) ≤ Cδ ^5/2 , ||W δ ^(s) || H

(−δ,L;R

) ≤ Cδ ^3/2 ,

||W δ,α ^(m) − W ^δ,α (0) || ^H

^(−δ,L) ≤ Cδ ^1/2 + C ||| Q _δ (0) − I ₃ ||| ,

d W δ,3 ^(m)

dx 3

L

(−δ,L) ≤ Cδ + C | ( Q _δ (0) − I ₃ ) e ₃ · e ₃ | ,

|| Q _δ − Q _δ (0) || H

(−δ,L;R

) ≤ Cδ ^1/2 .

(6.11)

Moreover from Lemma 4.2 we get

||| Q _δ (0) − I ₃ ||| ≤ Cδ ^3/4 , | (Q δ (0) − I ₃ )e 3 · e ₃ | ≤ Cδ,

|W δ,α ^(m) (0) | ≤ Cδ ^7/4 , |W ^δ,3 (0) − U e ^δ,3 (0, 0) | ≤ Cδ ^3/2 ,

|W ^δ,3 (0) | ≤ Cδ.

(6.12)

Finally we obtain the following estimates of the terms W δ,α ^(m) , W δ,3 ^(m) and Q _δ − I ₃ :

|| Q _δ − I ₃ || ^H

^(−δ,L;R

⁾ ≤ Cδ ^1/2 , ||W δ,α ^(m) || ^H

^(−δ,L) ≤ Cδ ^1/2 ,

||W δ,3 ^(m) || H

(−δ,L) ≤ Cδ. (6.13)

Now we are in a position to prove the following lemma:

Lemma 6.1. There exists a subsequence still indexed by δ such that 1

δ ^1/2 W δ,α ^(m) ⇀ W ^α weakly in H ² (0, L), 1

δ ^1/2 W ^δ,α , 1

δ ^1/2 W δ,α ^(m) −→ W ^α strongly in H ¹ (0, L), 1

δ W ^δ,3 , 1

δ W δ,3 ^(m) ⇀ W ³ weakly in H ¹ (0, L), 1

δ ^3/2 W δ ^(s) ⇀ W ^(s) weakly in H ¹ (0, L; R ³ ), 1

δ ^1/2 Q _δ − I ₃ ) ⇀ A _Q weakly in H ¹ (0, L; R ⁹ ), 1

δ ^5/2 P δ (v δ ) ⇀ v weakly in L ² (0, L; H ¹ (D; R ³ )).

(6.14)

We also have W α ∈ H ² (0, L) and for a.e. x ₃ ∈ ]0, L[ we have d W ¹

dx ₃ (x 3 ) = Q ² (x 3 ), d W ²

dx ₃ (x 3 ) = −Q ¹ (x 3 ), d W 3

dx 3

(x ₃ ) + 1 2

h d W 1

dx 3

(x ₃ ) ² + d W 2

dx 3

(x ₃ ) ² i

= 0.

(6.15)

The junction conditions

W ^α (0) = 0, Q (0) = 0, W ^(s) (0) = 0, W ³ (0) = U ³ (0, 0) (6.16) hold true. We have

1

2δ ^3/2 P δ ( ∇ v δ ) ^T ∇ v δ − I ₃

⇀ E _r weakly in L ¹ (B; R ^3×3 ), (6.17) where the symmetric matrix E _r is defined by

E _r =



 

 

 

 

 

 

γ 11 (v r ) γ 12 (v r ) − 1 2 X 2

d Q ³ dx 3

+ 1 2

∂v r,3

∂X 1

+ 1 2

d W 1 ^(s)

dx 3

∗ γ 22 (v r ) 1 2 X 1

d Q ³ dx 3

+ 1 2

∂v r,3

∂X 2

+ 1 2

d W 2 ^(s)

dx 3

∗ ∗ − X 1

d ² W ¹ dx ² ₃ − X 2

d ² W ²

dx ² ₃ + d W 3 ^(s)

dx ₃



 

 

 

 

 

 

. (6.18)

Proof. First, taking into account (6.11), (6.13) and upon extracting a subsequence it follows that the convergences (6.14) hold true. First, due to the definition of W δ ^(m)

and the weak convergence in H ¹ (0, L; R ⁹ ) of the sequence 1

δ ^1/2 Q _δ − I ₃ ) towards the antisymmetric matrix A _Q we deduce that d W

dx 3

= Q∧ e ₃ wich gives the two first equalities

in (6.15). Then, the strong convergence in L ^∞ (0, L; R ³ ) of the sequence 1

δ ^1/2 Q _δ − I ₃ )e 3

towards Q∧ e ₃ , hence 1

δ | (Q δ − I ₃ )e 3 | ² convergences towards |Q∧ e ₃ | ² = d W ¹ dx ₃

² + d W ² dx ₃

² strongly in L ^∞ (0, L). Finally using equality (4.9) we obtain the last equality in (6.15).

The junction conditions on Q and W ^α are immediate consequences of (6.12) and the convergences (6.14).

In order to obtain the junction condition between the bending in the plate and the stretching in the rod note first that the sequence 1

δ U e ^δ,3 converges strongly in H ¹ (ω) to U ³ because of (4.2) and the first convergence in (6.4). Besides this sequence is uniformly bounded in H ² (D), hence it converges strongly to the same limit U ³ in C ⁰ (D). Moreover the weak convergence of the sequence 1

δ W δ,3 ^(m) in H ¹ (0, L), implies the convergence of 1

δ W δ,3 ^(m) (0) = 1

δ W ^δ,3 (0) to W ³ (0). Using the third estimate in (6.12) gives the last condition in (6.16).

Once the convergences (6.14) are established, the limit of the rescaled Green-St Venant strain tensor of the sequence v _δ is analyzed in [7] (see Subsection 3.3) and it gives (6.18).

7 Asymptotic behavior of the sequence m _δ δ ⁵ .

The goal of this section is to establish Theorem 7.2. Let us first introduce a few notations. We set

D 0 = n

( U , W , Q ³ ) ∈ H ¹ (ω; R ³ ) × H ¹ (0, L; R ³ ) × H ¹ (0, L) | U ³ ∈ H ² (ω), W ^α ∈ H ² (0, L), U = 0, ∂ U ³

∂x _α = 0 on γ 0 , d W 3

dx 3

+ 1 2

h d W 1

dx 3

² + d W 2

dx 3

² i

= 0 in ]0, L[, W ³ (0) = U ³ (0, 0), W ^α (0) = d W ^α

dx ₃ (0) = Q ³ (0) = 0 o

(7.1)

Let us notice that D 0 is a closed subset of H ¹ (ω; R ³ ) × H ¹ (0, L; R ³ ) × H ¹ (0, L).

We introduce below the ”limit” elastic energies for the plate and the rod whose expressions are well known for such structures ⁴

4 E, ν are the Young modulus and the Poisson’s ratio of the plate and the rod.

J p ( U ) = E 3(1 − ν ² )

Z

ω

h (1 − ν) X 2 α,β=1

∂ ² U ³

∂x α ∂x β

² + ν ∆ U 3

2 i

+ E

(1 − ν ² ) Z

ω

h

(1 − ν) X 2 α,β=1

Z ^{αβ 2} + ν Z ¹¹ + Z ²² 2 i ,

J ^r ( W ¹ , W ² , Q ³ ) = Eπ 8

Z L 0

h d ² W 1

dx ² ₃

² + d ² W 2

dx ² ₃ ² i

+ µπ 8

Z L 0

d Q 3

dx 3

²

(7.2)

where Z ^αβ is given by

Z αβ = γ _αβ ( U ) + 1 2

∂ U ³

∂x α

∂ U ³

∂x β

.

The total energy of the plate-rod structure is given by the functional J defined over D 0

J ( U , W , Q ³ ) = J ^p ( U ) + J ^r ( W ¹ , W ² , Q ³ ) − L ( U , W , Q ³ ) (7.3) with

L ( U , W , Q ³ ) = 2 Z

ω

f p · U + π Z L

0

f r · W dx 3 + π 2

Z L 0

g α · Q ∧ e _α

dx 3 (7.4) where

Q = − d W ² dx 3

e ₁ + d W ¹ dx 3

e ₂ + Q ³ e ₃ . Below we prove the existence of at least a minimizer of J .

Lemma 7.1. There exist two constants C _p ^∗ , C _r ^∗ such that, if (f p,1 , f p,2 ) satisfies

|| f _p,1 || ² L

(ω) + || f _p,2 || ² L

(ω) < C _p ^∗ (7.5) and if f r,3 satisfies

|| f r,3 || ^L

^(0,L) < C _r ^∗ (7.6)

then the minimization problem

(U,W,Q min

)∈D

J ( U , W , Q ³ ) (7.7)

admits at least a solution.

Proof. Due to the boundary conditions on U 3 in D 0 , we immediately have

||U ³ || ² H

(ω) ≤ C J ^p ( U ). (7.8)

Then we get

X 2 α,β=1

|| γ α,β ( U ) || ² L

(ω) ≤ C J ^p ( U ) + C ∇U ^{3 4}

L

(ω;R

)

≤ C J ^p ( U ) + C[ J ^p ( U )] ² .

(7.9) Thanks to the 2D Korn’s inequality we obtain

||U ¹ || ² H

(ω) + ||U ² || ² H

(ω) ≤ C J ^p ( U ) + C p [ J ^p ( U )] ² . (7.10) Again, due to the boundary conditions on W ^α and Q ³ in D 0 , we immediately have

||W ¹ || ² H

(0,L) + ||W ² || ² H

(0,L) + ||Q ³ || ² H

(0,L) ≤ C J ^r ( W ¹ , W ² , Q ³ ). (7.11) Then, due to the definition of D 0 and (7.11) we get

d W ³

dx 3

²

L

(0,L) ≤ C n d W ¹ dx 3

⁴

L

(0,L) + d W ² dx 3

⁴

L

(0,L)

o ≤ C[ J ^r ( W ¹ , W ² , Q ³ )] ² . (7.12)

From the above inequality and (7.8) we obtain W ^{3 2}

L

(0,L) ≤ C |W ³ (0) | ² + C d W ³ dx 3

²

L

(0,L)

≤ C J ^p ( U ) + C r [ J ^r ( W ¹ , W ² , Q ³ )] ² .

(7.13)

Since J (0, 0, 0) = 0, let us consider a minimizing sequence ( U ^(N) , W ^(N) , Q ^(N 3 ⁾ ) ∈ D 0

satisfying J ( U ^(N) , W ^(N) , Q ^(N 3 ⁾ ) ≤ 0 m = inf

(U ,W,Q

)∈D

J ( U , W , Q ³ ) = lim

N →+∞ J ( U ^{(N )} , W ^{(N )} , Q ^(N) 3 ) where m ∈ [ −∞ , 0].

With the help of (7.8)-(7.13) we get

J ^p ( U ^(N) ) + J ^r ( W 1 ^(N) W 2 ^(N) , Q ^(N) 3 ) ≤ C || f p,3 ||

q

J ^p ( U ^(N) ) + 2 || f p,1 || ² L

(ω) + || f p,2 || ² L

(ω)

1/2

C q

J ^p ( U ^(N) ) + p

C p J ^p ( U ^(N) ) + C

X 2 α=1

|| f r,α || L

(0,L) + || g α || L

(0,L;R

)

q J ^r ( W 1 ^(N) W 2 ^(N) , Q ^(N) 3 )

+ π || f r,3 || ^L

^(0,L) C q

J ^r ( W 1 ^{(N )} W 2 ^{(N )} , Q ^(N) 3 ) + p

C r J ^r ( W 1 ^(N) W 2 ^(N) , Q ^(N 3 ⁾ )

(7.14)

Choosing C _p ^∗ = 1 2C p

and C _r ^∗ = 1 π √

C r

, if the applied forces satisfy (7.5) and (7.6) then the following estimates hold true

||U 3 ^(N) || H

(ω) + ||U 1 ^(N) || H

(ω) + ||U 2 ^(N) || H

(ω) + ||W 1 ^(N) || H

(0,L)

+ ||W 2 ^(N) || ^H

^(0,L) + ||Q ^(N) 3 || ^H

^(0,L) + ||W 3 ^{(N )} || ^H

^(0,L) ≤ C (7.15)

where the constant C does not depend on N .

As a consequence, there exists ( U ^(∗) , W ^(∗) , Q ^(∗) 3 ) ∈ H ¹ (ω; R ³ ) × H ¹ (0, L; R ³ ) × H ¹ (0, L) such that for a subsequence

U 3 ^{(N )} ⇀ U 3 ^(∗) weakly in H ² (ω) and strongly in W ^1,4 (ω), U α ^{(N )} ⇀ U α ^(∗) weakly in H ¹ (ω),

W α ^{(N )} ⇀ W α ^(∗) weakly in H ² (0, L) and strongly in W ^1,4 (0, L), Q ^(N 3 ⁾ ⇀ Q ^(∗) 3 weakly in H ¹ (0, L),

W 3 ^{(N )} ⇀ W 3 ^(∗) weakly in H ¹ (0, L).

Notice that we also get the following convergences:

Z αβ ^(N) ⇀ Z αβ ^(∗) = γ αβ ( U ^(∗) ) + 1 2

∂ U 3 ^(∗)

∂x _α

∂ U 3 ^(∗)

∂x _β weakly in L ² (ω).

The above convergences show that ( U ^(∗) , W ^(∗) , Q ^(∗) 3 ) ∈ D 0 . Finally, since J is weakly sequentially continuous in

H ¹ (ω; R ² ) × H ² (ω) × L ² (ω; R ³ ) × H ² (0, L; R ² ) × H ¹ (0, L; R ² ) with respect to

( U 1 , U 2 , U 3 , Z 11 , Z 12 , Z 22 , W 1 , W 2 , W 3 , Q 3 ) The above weak and strong converges imply that

J ( U ^(∗) , W ^(∗) , Q ^(∗) 3 ) = m = min

(U,W,Q

)∈D

J ( U , W , Q ³ ) which ends the proof of the lemma.

Theorem 7.2. We have lim δ→0

m δ

δ ⁵ = min

(U,W,Q

)∈D

J ( U , W , Q ³ ), (7.16)

where the functional J is defined by (7.3).

Proof. Step 1. In this step we show that

(U,W,Q min

)∈D

J ( U , W , Q ³ ) ≤ lim inf

δ→0

m δ

δ ⁵ . (7.17)

Let (v _δ ) _δ>0 be a sequence of deformations belonging to D δ and such that lim δ→0

J δ (v δ )

δ ⁵ = lim inf

δ→0

m δ

δ ⁵ . (7.18)