In this thin multidomain we introduce a bulk energy density of the kind W(D2U), whereW is a convex function with growthp ∈ ]1

c World Scientific Publishing Company

JUNCTION IN A THIN MULTIDOMAIN FOR A FOURTH ORDER PROBLEM

ANTONIO GAUDIELLO

DAEIMI, Universit`a degli Studi di Cassino, via G. Di Biasio 43, 03043 Cassino (FR), Italia

gaudiell@unina.it

ELVIRA ZAPPALE

DIIMA, Universit`a degli Studi di Salerno, via Ponte Don Melillo, 84084 Fisciano (SA), Italia

zappale@diima.unisa.it

Received 3 October 2005 Revised 30 November 2005 Communicated by F. Brezzi

We consider a thin multidomain of R^N, N ≥ 2, consisting (e.g. in a 3D setting) of a vertical rod upon a horizontal disk. In this thin multidomain we introduce a bulk energy density of the kind W(D²U), whereW is a convex function with growthp ∈ ]1,+∞[, andD²Udenotes the Hessian tensor of a scalar (or vector-valued) functionU. By assuming that the two volumes tend to zero with the same rate, under suitable boundary conditions, we prove that the limit model is well-posed in the union of the limit domains, with dimensions, respectively, 1 andN−1. Moreover, we show that the limit problem is uncoupled if 1< p≤ ^N−1₂ , “partially” coupled if ^N−1₂ < p≤N−1, and coupled ifN−1< p. The main result is applied in order to derive the equilibrium configuration of two joint beams, T-shaped, clamped at the three endpoints and subject to transverse loads. The main result is also applied in order to describe the equilibrium configuration of a wire upon a thin film with contact at the origin, when the thin structure is filled with a martensitic material.

Keywords: Fourth order operator; junction in thin multidomains; beam; wire, thin film;

martensitic material; dimension reduction; Γ-convergence.

AMS Subject Classification: 74B20, 74K10, 74K20, 74K30, 74K35, 78M30, 78M35

1. Introduction

Recently, there has been a growing interest in the thin structures theory, since these structures arise in many applications (see, for instance, Ref. 6 and references therein). The usual approach consists of dimensional reduction, through asymptotic analysis, from high dimensions to lower ones, and in finding

1887 Math. Models Methods Appl. Sci. 2006.16:1887-1918. Downloaded from www.worldscientific.com by PIERRE & MARIE CURIE UNIV on 05/15/17. For personal use only.

Ω Ω

ω ω

1

-h r

a

b n n

n n

Fig. 1. The thin multidomain.

precise relations between the initial models and the limit ones (see for example Refs. 1, 9, 10, 12, 16, 18, 19, 31–35).

In the past few years, a new insight in dimensional reduction problems has been given in Refs. 22–25, where the authors analyze the junction of two thin cylinders (for a survey on junction problems see also Refs. 7,8,11,15,27,29 and 30). The model, described in Ref. 22 through its integral energy and in Ref. 23 through the related constitutive equations, is a quasilinear Neumann second-order scalar prob- lem in a thin multidomain ofR^N,N ≥2. Precisely, the thin multidomain consists of two vertical cylinders, one placed upon the other: the first one with constant height and small cross-section, the second one with small thickness and constant cross-section (see Fig. 1). The authors derive the limit problem, by assuming that the volumes of the two cylinders tend to zero with the same rate. An analogous analysis is performed in Refs. 24 and 25, in order to derive the limit problem in the case of the linearized elasticity system inR³.

In the same spirit, we consider an analogous thin multidomain of R^N with a bulk energy density of the kind W(D²U), whereW is a convex function with growthp ∈]1,+∞[, and D²U denotes the Hessian tensor of a scalar function U. By assuming that the volumes of the two cylinders tend to zero with the same rate, under suitable boundary conditions on the top of the vertical cylinder and on the lateral surface of the horizontal cylinder, we derive the limit energy, and the limit junction conditions in the origin. More precisely, we prove that the limit problem is well-posed in the union of the limit domains, with dimensions 1 and N−1, respectively, and involves 6 limit functions. Moreover, we show that the limit problem is uncoupled if 1< p≤ ^N−1₂ , “partially” coupled if ^N−1₂ < p≤N−1, and coupled if N−1< p (see the limit space (3.8), Theorem 3.1 and Corollary 3.1).

Furthermore, if ^N₂⁻¹ < p, the minimizers of the limit problem depend also on the Math. Models Methods Appl. Sci. 2006.16:1887-1918. Downloaded from www.worldscientific.com by PIERRE & MARIE CURIE UNIV on 05/15/17. For personal use only.

limit of the ratio between the volumes of the two cylinders. They do not depend on it, if 1< p≤ ^N−1₂ .

We do not take explicitly into account lower order terms, volume forces and surface forces which can be easily treated (see Remarks 3.3 and 3.4).

Since the considered energies are convex, we just study the problem in the scalar case. The main results can be extended, without effort, to the vectorial case with R^M-valued functions U (M > 1), always for convex energies W(D²U) (see Remark 3.4).

By starting from the two-dimensional plate energy of Kirchhoff–Love (see Ref. 12), the main result is applied in order to obtain a rigorous derivation of the equilibrium configuration of two joint elastic beams, T-shaped, clamped at the three endpoints, and subject to transverse loads (see Sec. 4.1). In this case the limit problem is coupled. The flexion and the twist of the T-structure are explicitly calculated and discussed in the case of some particular loads.

Our result is also applied in order to describe the equilibrium configuration of a martensitic multistructure consisting of a wire upon a thin film with contact at the origin (see Sec. 4.2). The fact that the limit problem is “partially” coupled allows us to build up a model where there is a surprising sharp phase transition, without transition layers, from an austenite phase to a martensitic phase, with bulk interfacial energy not identically null. For a survey on martensitic materials see, for instance, Refs. 3–6,28,31,32,36 and 37 and the large bibliography quoted therein.

Our results could also be considered in order to describe “non-simple materials of grade 2” (see Refs. 14,26,38,39 and references therein).

In the following section, after having introduced the problem in a thin mul- tidomain, we reformulate it on a fixed domain through appropriate rescalings of the kind proposed by Ciarlet and Destuynder in Ref. 13. Section 3 is devoted to describe the main result. Some mechanical applications of this result are described in Sec. 4. The proof of the main result is developed in several steps and it is based on the Γ-convergence method introduced by E. De Giorgi (see Ref. 17). A density result for the limit space is given in Sec. 5, a recovery sequence for the Γ-limit is built in Sec. 6 and a compactness argument is presented in Sec. 7. Finally, the proof is completed in Sec. 8.

2. The Original Problem and the Rescalings

LetN ≥2 be an integer number. In the sequel, x= (x₁, . . . , x_N−1, x_N) = (x, x_N) denotes the generic point of R^N, R^k×k_s (for k = N, N −1) the set of symmetric k×k-matrices. Moreover,D_x and D_x²,D_xN and D²_x

N stand for the gradient and the Hessian tensor with respect to the first N −1 variables, for the first and the second derivative with respect to the last variable, respectively. Then, according to these notations,D²_x,xN stands for (D_xN)_x.

Letω⊂R^N−1 be a bounded open connected set such that the origin inR^N−1, denoted by 0, belongs to ω, and let {r_n}n∈N, {h_n}n∈N⊂]0,1[ be two sequences Math. Models Methods Appl. Sci. 2006.16:1887-1918. Downloaded from www.worldscientific.com by PIERRE & MARIE CURIE UNIV on 05/15/17. For personal use only.

such that

limn h_n = 0 = lim

n r_n. (2.1)

For everyn ∈N, consider the thin multidomain Ω_n = Ω^a_n∪Ω^b_n (a for “above”,b for “below”) union of two vertical cylinders, one placed upon the other: Ω^a_n = r_nω×[0,1[ with small cross-sectionr_nω and constant height, Ω^b_n = ω×]−h_n,0[

with small thickness h_n and constant cross-section (see Fig. 1). Moreover, set Ω =ω×]−1,1[.

In the thin multidomain introduce a convex bulk energy density of the kind W(D²U). Precisely, let

W :M ∈R^N×N_s →W(M)∈R (2.2)

be a function satisfying the following assumptions:

W is convex; (2.3)

a+α|M|^p≤W(M)≤b+β|M|^p, ∀ M ∈R^N_s^×N; for some a, b∈R, α, β∈]0,+∞[ and 1< p <+∞,

(2.4)

where, ifM= (m_i,j)i,j=1,...,N,|M|= _i,j=1,N, m²_i,j¹₂

. Moreover, in the sequel, for a givenA∈R^(Ns ^{−1)×(N−1)},B∈R^N−1andC∈R,W

A B^T

B C

meansW(M), where M = (m_i,j)i,j=1,...,N and (m_i,j)i,j=1,...,N−1 = A, (m_N,j)_{j=1,...,N−1}=B, (m_i,N)i=1,...,N−1=B^T andm_N,N =C.

For everyn∈N, consider a function U_n which minimizes the energy

U_n→

Ωn

W(D²U_n)dx=

Ωn

W

D²_xU_n (D²_x,x

NU_n)^T D²_x,x

NU_n D_x²

NU_n

dx, (2.5)

among all the functionsU_n∈W^2,p(Ω_n) realizing the Dirichlet boundary condition c^a +d^a ·x on top of Ω^a_n, and f^b+g^bx_N on the lateral surface of Ω^b_n, for some c^a ∈R,d^a ∈R^N−1andf^b, g^b∈W^2,p(ω).

As usual, one tries to reformulate the problem on a fixed domain through appro- priate rescalings which map Ω_n into Ω (see Fig. 2). Namely, by setting

u_n(x) =

u^a_n(x, x_N) =U_n(r_nx, x_N), (x, x_N) a.e. in Ω^a=ω×]0,1[;

u^b_n(x, x_N) =U_n(x, h_nx_N), (x, x_N) a.e. in Ω^b=ω×]−1,0[;

(2.6)

it is easily seen that u^a_n ∈ W^2,p(Ω^a) assumes the rescaled Dirichlet boundary condition c^a + r_nd^a · x on top of Ω^a, u^b_n ∈ W^2,p(Ω^b) assumes the rescaled Math. Models Methods Appl. Sci. 2006.16:1887-1918. Downloaded from www.worldscientific.com by PIERRE & MARIE CURIE UNIV on 05/15/17. For personal use only.

1

-1 ω

Ω

Ω

a

b

Ω

Fig. 2. The fixed domain.

Dirichlet boundary conditionf^b+h_ng^bx_N on the lateral boundary of Ω^b. Moreover, u_n= (u^a_n, u^b_n) satisfies the following junction conditions:

















u^a_n(x,0) =u^b_n(r_nx,0), x a.e. inω;

1

r_nD_xu^a_n(x,0) = (D_xu^b_n)(r_nx,0), x a.e. inω;

D_xNu^a_n(x,0) = 1

h_nD_xNu^b_n(r_nx,0), x a.e. inω;

and minimizes the rescaled energy (divided throughr^N_n⁻¹):

u_n= (u^a_n, u^b_n)→

Ω^a

W











 1 r²_nD²_xu^a_n

1

r_nD²_x,xNu^a_{n T}

1

r_nD²_x,xNu^a_n D_x²

Nu^a_n











dx

+ h_n r^N_n⁻¹

Ω^b

W













D²_xu^b_n

1

h_nD²_x,xNu^b_{n T}

1

h_nD²_x,xNu^b_n 1 h²_nD_x²

Nu^b_n











dx,

among all the functionsu_n subject to the same conditions ofu_n.

The aim of this paper consists of describing the limit energy, asn→+∞, when the volumes of Ω^a_n and Ω^b_n tend to zero with same rate, that is

limn

h_n

r^N_n⁻¹ =q∈]0,+∞[. (2.7)

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3. Main Results

For everyn∈N, let (u^a_n, u^b_n)∈ U_n be a solution of the following problem:

K_n^a(u^a_n) + h_n

r_n^N−1K_n^b(u^b_n) = min

(u^a_n,u^b_n)∈Un

K_n^a(u^a_n) + h_n

r^N_n⁻¹K_n^b(u^b_n)

, (3.1)

where

K_n^a:u^a∈W^2,p(Ω^a)→

Ω^a

W











 1 r_n²D²_xu^a

1

r_nD²_x,xNu^a _T 1

r_nD²_x,xNu^a D_x²

Nu^a











dx, (3.2)

K_n^b :u^b∈W^2,p(Ω^b)→

Ω^b

W













D²_xu^b

1

h_nD²_x,xNu^b _T 1

h_nD²_x,xNu^b 1 h²_nD²_xNu^b











dx, (3.3)

and

Un ={(u^a, u^b)∈(c^a+r_nd^a·x+W_a^2,p(Ω^a))×(f^b+h_ng^bx_N +W_b^2,p(Ω^b)) u^a(x,0) =u^b(r_nx,0), x a.e. inω;

1

r_nD_xu^a(x,0) = (D_xu^b)(r_nx,0), x a.e. inω;

D_xNu^a(x,0) = 1

h_nD_xNu^b(r_nx,0), x a.e. inω}, (3.4) with r_n, h_n, c^a, d^a, f^b and g^b as defined in Sec. 2, W_a^2,p(Ω^a) the closure, with respect to W^2,p-norm, of {u^a ∈ C^∞(Ω^a) : u^a = 0 in a neighborhood of ω× {1}}andW_b^2,p(Ω^b) the closure, with respect toW^2,p-norm, of{u^b∈C^∞(Ω^b) : u^b = 0 in a neighborhood of ∂ω×]−1,0[}. Without loss of generality, one can assume that

f^b= 0 =g^b a.e. inB, (3.5)

for some (N−1)-dimensional ballB such that 0∈B ⊂⊂ω .

To describe the limit energy of the sequence in (3.1), as n → +∞, when the volumes of Ω^a_n and Ω^b_n tend to zero with the same rate, introduce the limit func- tionalsK^a,K^b, and the limit spacesV^p (we point out the strong dependence onp for the limit junction conditions):

K^a: (u^a, ξ^a, z^a)∈W^2,p(Ω^a)×(W^1,p(Ω^a))^N−1×L^p(]0,1[;W^2,p(ω))

→

Ω^a

W

D²_xz^a (D_xNξ^a)^T D_xNξ^a D²_xNu^a

dx, (3.6)

Math. Models Methods Appl. Sci. 2006.16:1887-1918. Downloaded from www.worldscientific.com by PIERRE & MARIE CURIE UNIV on 05/15/17. For personal use only.

K^b: (u^b, ξ^b, z^b)∈W^2,p(Ω^b)×W^1,p(Ω^b)×L^p(ω;W^2,p(]−1,0[))

→

Ω^b

W

D²_xu^b

D_xξ^b_T D_xξ^b D²_x

Nz^b

dx, (3.7)

and

V^p=































U×Ξ×Z, ifp≤ N−1

2 ; {((u^a, u^b),(ξ^a, ξ^b),(z^a, z^b))∈ U×Ξ×Z:

u^a(0) =u^b(0)}, if N−1

2 < p≤N−1;

{((u^a, u^b),(ξ^a, ξ^b),(z^a, z^b))∈ U×Ξ×Z: u^a(0) =u^b(0), ξ^a(0) =D_xu^b(0),

D_xNu^a(0) =ξ^b(0)} ifN−1< p;

(3.8)

where

U = (c^a+W_a^2,p(]0,1[))×(f^b+W₀^2,p(ω)), Ξ = (d^a+ (W_a^1,p]0,1[)^N−1)×(g^b+W₀^1,p(ω)),

Z =L^p(]0,1[;W_m^2,p(ω))×L^p(ω;W_m^2,p(]−1,0[)), (3.9) (W_a^1,p(]0,1[) = {u ∈ W^1,p(]0,1[) : u(1) = 0}, W_a^2,p(]0,1[) = {u ∈ W^2,p(]0,1[) : u(1) = 0 = Du(1)}, and for any subset A ⊂ R^k, W_m^2,p(A) = {v ∈ W^2,p(A) :

Avd(x₁, . . . , x_k) = 0,

ADvd(x₁, . . . , x_k) = 0}).

This paper is devoted to prove the following result

Theorem 3.1. LetW be a function satisfying(2.2)–(2.4),and let,for everyn∈N, (u^a_n, u^b_n) ∈ Un be a solution of Problem (3.1). Let K^a, K^b andV^p be as in (3.6)–

(3.8),respectively. Assume that (2.1)and(2.7)hold.

Then, there exist an increasing sequence of positive integers {n_i}_i∈N and ((u^a, u^b),(ξ^a, ξ^b),(z^a, z^b)) ∈ V^p, depending possibly on the selected subsequence {n_i}i∈N, such that

u^a_ni u^a weakly in W^2,p(Ω^a),

u^b_ni u^b weakly inW^2,p(Ω^b), (3.10)







 1

r_niD_xu^a_ni ξ^a weakly in(W^1,p(Ω^a))^N−1, 1

h_niD_xNu^b_ni ξ^b weakly inW^1,p(Ω^b),

(3.11)









 1

r_n²_iD_x²u^a_ni D²_xz^a weakly in(L^p(Ω^a))^{(N−1)×(N−1)}, 1

h²_n

i

D_x²_Nu^b_ni D_x²_Nz^b weakly in L^p(Ω^b),

(3.12) Math. Models Methods Appl. Sci. 2006.16:1887-1918. Downloaded from www.worldscientific.com by PIERRE & MARIE CURIE UNIV on 05/15/17. For personal use only.

asi→+∞,and((u^a, u^b),(ξ^a, ξ^b),(z^a, z^b))is a solution of the following problem:

K^a(u^a, ξ^a, z^a) +qK^b(u^b, ξ^b, z^b)

= min

((u^a,u^b),(ξ^a,ξ^b),(z^a,z^b))∈V^p{K^a(u^a, ξ^a, z^a) +qK^b(u^b, ξ^b, z^b)}. (3.13) Moreover, the energies converge in the sense that

limn

K_n^a(u^a_n) + h_n

r^N_n⁻¹K_n^b(u^b_n)

=K^a(u^a, ξ^a, z^a) +qK^b(u^b, ξ^b, z^b). (3.14) Furthermore,ifW is strictly convex,Problem(3.13)admits a unique solution. Con- sequently convergences (3.10)–(3.12),hold true for the whole sequence.

Remark 3.1. Let us point out that the limit problem is uncoupled if 1< p≤ ^N−1₂ , partially coupled by the junction condition:u^a(0) =u^b(0) if ^N−1₂ < p≤N −1, and coupled by the previous junction condition for uand by the junction conditions:

ξ^a(0) = D_xu^b(0), D_xNu^a(0) = ξ^b(0) if N−1< p. Moveover, if ^N−1₂ < p, the minimizers of the limit problem depend also on the limit of the ratio between the volumes of the beam and the plate. They do not depend on it, if 1< p≤^N₂⁻¹. Remark 3.2. IfW(D²U) =|D²U|^p, then it is evident thatz^a = 0 =z^b. Remark 3.3. If one considers volume forces of the kind

Ωn

J_nU_ndx,

withJ_n∈L^p−1p (Ω_n), then in the rescaled problem (divided throughr_n^N−1) one has terms of the kind:

Ω^a

j_n^au^a_ndx+ h_n r^N_n⁻¹

Ω^b

j_n^bu^b_ndx, where

j_n(x) =

j_n^a(x, x_N) =J_n(r_nx, x_N), (x, x_N) a.e. in Ω^a; j_n^b(x, x_N) =J_n(x, h_nx_N), (x, x_N) a.e. in Ω^b. Then, by assuming that

j_n^a j^a weakly inL^p−1p (Ω^a) andj_n^b j^b weakly inL^p−1p (Ω^b), the additional term

Ω^a

j^au^adx+q

Ω^b

j^bu^bdx Math. Models Methods Appl. Sci. 2006.16:1887-1918. Downloaded from www.worldscientific.com by PIERRE & MARIE CURIE UNIV on 05/15/17. For personal use only.

will appear in the limit problem. Similarly, it is possible to add surface forces on the lateral surface of Ω^a_n and on the basis of Ω^b_n.

As regards the original problem proposed in the previous section, from the rescaling (2.6) and Theorem 3.1, the result below follows:

Corollary 3.1. LetW be a function satisfying(2.2)–(2.4),and let,for everyn∈N, U_n be a minimizer of the original problem introduced in Sec.2. Let K^a, K^b and V^p be as in(3.6)–(3.8),respectively. Assume that(2.1)and(2.7)hold.

Then, there exist an increasing sequence of positive integer numbers {n_i}_i∈N and((u^a, u^b),(ξ^a, ξ^b),(z^a, z^b))∈V^p,depending possibly on the selected subsequence {n_i}i∈N, such that











−−

r_niω

U_ni(x, x_N)dx u^a weakly inW^2,p(]0,1[),

−−

₀

−h_ni

U_ni(x, x_N)dx_N u^b weakly inW^2,p(ω),











−−

r_niω

D_xU_ni(x, x_N)dx ξ^a weakly in (W^1,p(]0,1[))^N−1,

−−

₀

−h_ni

D_xNU_ni(x, x_N)dx_N ξ^b weakly inW^1,p(ω),











−−

r_niω

D²_xU_ni(x, x_N)dx−−

ω

D²_xz^a(x, x_N)dx weakly in(L^p(]0,1[))^(N−1)2,

−−

₀

−h_ni

D_x²_NU_ni(x, x_N)dx_{N 0}

−1

D²_xNz^b(x, x_N)dx_N weakly inL^p(ω),

as i → +∞, and ((u^a, u^b),(ξ^a, ξ^b),(z^a, z^b)) is a solution of Problem (3.13).

Moreover, the energies converge in the sense that

limn

1 r^N_n⁻¹

Ωn

W

D²_xU_n D²_x,x

NU_nT D²_x,x

NU_n D²_x

NU_n

dx

=K^a(u^a, ξ^a, z^a) +qK^b(u^b, ξ^b, z^b).

Furthermore,ifW is strictly convex,Problem(3.13)admits a unique solution. Con- sequently the previous convergences hold true for the whole sequence.

Remark 3.4. LetN≥2 andM ≥1 be two integers and W :M ∈(R^N×N_s )^M →W(M)∈R,

be a convex function satisfying (2.4) with M ∈(R^N×N_s )^M. For every n ∈ N, let U_n: Ω_n →R^M be a minimizer of the energy (2.5) among all the admissible vectors Math. Models Methods Appl. Sci. 2006.16:1887-1918. Downloaded from www.worldscientific.com by PIERRE & MARIE CURIE UNIV on 05/15/17. For personal use only.

U_n∈

W^2,p(Ω_n)_M

realizing the Dirichlet boundary conditionc^a+d^a·xon top of Ω^a_n, andf^b+g^bx₃ on the lateral surface of Ω^b_n, for somec^a∈(W^2,p(]0,1[))^M, some (M ×(N−1))-matrixd^a andf^b, g^b∈

W^2,p(ω)_M

, where·reads as a row column product.

Then, it is easily seen that the statements of Theorem 3.1 and Corollary 3.1 still hold true, with the limit functions in (3.10)–(3.12) belonging to the following spaces:

u^a ∈c^a+ (W_a^2,p(]0,1[))^M, u^b∈f^b+

W₀^2,p(ω)_M , ξ^a ∈d^a+ (W_a^1,p(]0,1[;R^N−1))^M, ξ^b∈g^b+

W₀^1,p(ω)_M , z^a ∈(L^p(]0,1[;W_m^2,p(ω)))^M, z^b ∈(L^p(ω;W_m^2,p(]−1,0[)))^M. Moreover, if we add to (2.5) a term of the type

ΩnΦ(DU_n)dx, where Φ :M ∈(R^N)^M →Φ(M)∈R

is a continuous function satisfyingp- growth and coercivity assumptions, then it is evident that the additional term:

|ω| ₁

0

Φ (ξ^a, D_x3u^a)dx₃+q

ω

Φ

D_xu^b, ξ^b dx₁dx₂

will appear in the limit problem. Let us point out that the presence of the leading energy depending on the Hessian tensor allows us to avoid of assuming Φ convex.

4. Applications

4.1. Junction of two beams (coupled limit problem)

In this section we derive rigorously the equilibrium configuration of two joint elastic beams, T-shaped and clamped at the three endpoints (see Fig. 3). We assume that the material of the beams is isotropic and homogeneous, and that the forces applied to the beams are transverse to the T-structure.

-1 1

1

0 Fig. 3.

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rn

rn Fig. 4.

We derive our model via an asymptotic analysis based on a dimensional reduc- tion of a Kirchhoff–Love^aplate (see Ref. 12). Namely, for everyn∈N, we introduce an approximating T-shaped plate Ω_n = Ω^a_n

Ω^b_n = (]−r_n, r_n[×[0,1[)

(]−1,1[× ]−r_n,0[) (see Fig. 4) with the following flexural energy:

c

Ωn

(∆U_n)²+ 2(1−ν)((D_x²_1x2U_n)²−D²_x1U_nD²_x2U_n)dx₁dx₂

−

Ωn

J_nU_ndx₁dx₂, (4.1)

whereν ∈]0,¹₂[ is the Poisson ratio, 2c >0 represents the flexural rigidity modulus of the plate (precisely, c = _12(1−ν^Eh3₂₎, with E >0 Young modulus and h denoting the small thickness of the plate), and J_n represent the transverse forces to the plate.

For everyn∈N, letU_n : Ω_n →Rbe the transverse displacement of the plate, which minimizes the energy (4.1) among all the kinematically admissible fields U_n∈H²(Ω_n) such thatU_n and its normal derivative vanish on ]−r_n, r_n[× {1}and on{−1,1} ×]−r_n,0[. Letj^aandj^bbe the weak-L²limits of the rescaled forces (see Remark 3.3).

In order to derive our model, we pass to the limit, asn→+∞, in (4.1). Then, by applying Corollary 3.1, it is easily seen that









 1 2r_n

_rn

−rn

U_n(x₁, x₂)dx₁ u^a weakly inH²(]0,1[), 1

h_{n 0}

−hn

U_n(x₁, x₂,)dx₂ u^bweakly inH²(]–1,1[),









 1 2r_n

_rn

−rn

D_x1U_n(x₁, x₂)dx₁ ξ^a weakly inH¹(]0,1[)), 1

h_{n 0}

−hn

D_x2U_n(x₁, x₂,)dx₂ ξ^b weakly inH¹(]–1,1[),

aIn the Kirchhoff–Love theory, a plate, being a three-dimensional solid with a dimension (the thickness) very small with respect to the others, is described by a two-dimensional object.

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