Asymptotic modeling of thin plastic oscillating layer

The inclusion of a very thin layer of very rigid material into a given elastic body has been widely considered, and in the classic literature. For more details, we can refer to [6], [7], [10] and [11]. In general, the computation of solution using numerical methods is very difficult. In one hand, this is because the thickness of the adhesive requires a fine mesh, which in turn implies an increase of the degrees of flexible than the adherents, and this produces numerical instabilities in the stiffness matrix. To overcame this difficulties, thanks to Goland and Reissner [12] find a limit problem in which the adhesive is treated on this theoretical approach, see for example A. Ait Moussa and J. Messaho [1], Acerbi, Buttazzo and Perceivable [2], Licht and Michail [4] and A. Ait Moussa and L. Zlaïji [8]. In this present work, we consider a structure containing a plastic thin oscillating layer of thickness, rigidity, and periodicity parameter depending on ε, where ε is


Introduction
The inclusion of a very thin layer of very rigid material into a given elastic body has been widely considered, and in the classic literature.For more details, we can refer to [6], [7], [10] and [11].In general, the computation of solution using numerical methods is very difficult.In one hand, this is because the thickness of the adhesive requires a fine mesh, which in turn implies an increase of the degrees of flexible than the adherents, and this produces numerical instabilities in the stiffness matrix.To overcame this difficulties, thanks to Goland and Reissner [12] find a limit problem in which the adhesive is treated on this theoretical approach, see for example A. Ait Moussa and J. Messaho [1], Acerbi, Buttazzo and Perceivable [2], Licht and Michail [4] and A. Ait Moussa and L. Zlaïji [8].
In this present work, we consider a structure containing a plastic thin oscillating layer of thickness, rigidity, and periodicity parameter depending on ε, where ε is a parameter intended to tend towards 0. In a such structure, we have treated the scalar case for a thermal conductivity problem in [3].The aim of this work is to study the limit behavior of an elasticity problem with a convex energy functional posed in a such structure.
This paper is organized in the following way.In section 2, we express the problem to study, and we give some notation and we define functional spaces for this study in the section 3.In the section 4, we study the problem (4.1).The section 5 is reserved to the determination of the limits problems and our main result.

Statement of the problem
We consider a structure, occupying a bonded domain Ω in R 3 with Lipschitzian boundary ∂Ω.It is constituted of two elastic bodies joined together by a rigid thin layer with oscillating boundary, the latter obeys to nonlinear elastic law of power type.More precisely, the stress field is related to the displacement's field by The structure occupies the regular domain and Ω ε = Ω \ B ε represent the regions occupied by the thin plate and the two elastic bodies, see Figure 1, ε being a positive parameter intended to approach 0, and Σ = {x = (x ′ , x 3 ) / |x 3 | = 0}.The structure is subjected to a density of forces of volume f : Ω → R 3 , and it is fixed on the boundary ∂Ω.Equations which relate the stress field σ ε , σ ε : Ω → R 9 S , and the field of displacement u ε , u where a ijkh are the elasticity coefficients, and R 9 S the vector space of the square symmetrical matrices of order three, e ij (u) are the components of the linearized tensor of deformation e(u).ϕ ε is a bounded real function and ]0, ε[ 2 -periodic.In the sequel, we assume that the elasticity coefficients a ijkh satisfy to the following hypotheses : We begin by introducing some notation which is used throughout the paper x = (x ′ , x 3 ), where ).
We set In the following C will denote any constant with respect to ε and [v] is the jump of displacement field v through Σ.

Functions
First, we introduce the following space : ε is the jump of u on Σ ± ε defined by s ), and u = 0 on ∂Ω} we easily show that V ε is a Banach space with respect to the norm Our goal in this work is to study the problem (P ε ), and its limit behavior when ε tends to zero.

Study of Problem
The problem P ε is equivalent of the minimization problem To study problem P ε , we will study the minimization problem (4.1).The existence and uniqueness of solutions to (4.1) is given in the following proposition.Proof: From (2.1) and (2.3), we show easily that the energy functional in (4.1) is weakly lower semicontinuous, strictly convex and coercive over V ε , Since V ε is not reflexive, so we may not apply directly result given in Dacorogna [13], but we can follow our proof by using the compact imbedding of Sobolev for the LD 0 (Ω) space in the reflexivity space L q (Ω), or q ∈]1, 3  2 ] for more information see Temam ( [5] p.117).On the other hand, let u n be a minimizing sequence for (4.1), to simplify the writing let Using the coercivity of F ε , we may then deduce that there exists a constant C > 0, independent of n, such that then u n bounded in L q , therefore a subsequence of u n , still denoted by u n , there exists u 0 ∈ V ε such that u n ⇀ u 0 in V ε .The weak lower semi-continuity and the strict convexity of F ε imply then the result.✷ Lemma 4.2.Assuming that for any sequence Then, taking advantage of the fact that u ε vanishes on ∂Ω : otherwise since LD 0 ֒→ L q (Ω, R 3 ) for all q ∈ [1, 3  2 ], in particular for q 0 = 3 2 , we denote by q ′ 0 the conjugate of q 0 , by Hölder inequality, we obtain since u ε = 0 on ∂B ε , one has, according to Pioncaré's type inequality see [5], such as ϕ ε is Y -periodic and for a small enough ε, than we have : According to (4.4), and using (4.5), (4.6), then we obtain Therefore, we will have (4.2) and (4.3).According to (4.2), (4.3) and for a small enough ε the sequence u ε is bounded in LD 0 (Ω).✷ We give some lemmas that will be used in the sequel.
). Lemma 4.4.Let u be a regular function defined in a neighborhood of Σ, then This lemme is a consequence of ( [2] Proposition 2).
To apply the epi-convergence method, we need to characterize the topological spaces containing any cluster point of the solution of the problem (4.1) with respect to the used topology, therefore the weak topology to use is insured by the Lemma 4.2.So the topological spaces characterization is given in the following proposition.
Let us The solution u ε of the problem (4.1) possess a cluster point u * in LD 0 (Ω), with respect to the weak topology and u * |Σ is a weak cluster point of w ε in LD 0 (Σ, R 3 ).
Proof: According to a (4.2), (4.3) and for a small ε, the solution u ε is bounded in LD 0 (Ω), then It's relatively compact in L 1 (Ω), this is consequence of ( [5], Theorem 1.4 p.117), and e(u ε ) so for a subsequences of u ε , still denoted by u ε , there exists there exists u * ∈ L 1 (Ω), such that Thanks to Lemma 4.2 and the Young's inequality, so we have Then lim Remark 4.6.The Proposition 4.5 remains true for any weak cluster point u of a sequence u ε in LD 0 (Ω, R 3 ) satisfies (4.2) and ( 4.3).
To study the limit behavior of the solution of the problem (4.1), we will use the epi-convergence method, (see Annex, definition ).

Limit Behavior
In this section, we are interested to the asymptotic behavior of the solution of the problem (4.1) when ε close to zero.In the sequel, we consider the following functionals We design by τ f the weak topology on the space.In the sequel, we shall characterize, the epi-limit of the energy functional given by (5.1) in the following theorem : Theorem 5.1.Under (2.1), (2.2), (2.3) and for f ∈ L ∞ (Ω, R 3 ), there exists a functional F : where F is given by .
We are now in position to determine the upper epi-limit.Let u ∈ LD 0 (Ω), as C ∞ (Ω) is dense in LD 0 (Ω) see ( [5], p.116), so there exists a sequence Let us consider the sequence where θ is a regular function satisfies : we have As ϕ ε is bounded, therefore Otherwise, ϕ ε → m(ϕ) in L 1 (Ω) see Annex, so by passing to the upper limit, we obtain : Since u n → u in C ∞ (Ω), there fore according to the classic result diagonalization's Lemma see [9], there exists a real function n(ε) : We are now in position to determine the lower epi-limit.Let as 2 Ω a ijhk e ij (u)e hk (u)dx + m(ϕ) Otherwise, we suppose lim inf ε→0 F ε (u ε ) < +∞, there exists a subsequence of F ε (u ε ), still denoted by F ε (u ε ) and a constant C > 0, such that Then χ Ωε e(u ε ) is bounded in L 2 (Ω), so for a subsequence of χ Ωε e(u ε ), still denoted by χ Ωε e(u ε ) we then show easily, like in the proof of the above proposition, that From the subdifferentiability's inequality of u → 1 2 Ωε a ijhk e ij (u)e hk (u)dx and passing to the lower limit, we obtain According to the diagonalization's Lemma ( [9], Lemma 1.15 p.32), there exists a function η(ε) : R + → R + decreasing to 0 when ε → 0 such that According to a Lemma 4.4, and let w ε be the sequence define before the Proposition 4.5, we have where according to the Lemma 4.3, let g ∈ D(Σ, R 9 ) we have thanks to a Proposition 4.5 and ϕ ε → m(ϕ) in L 1 (Σ)(see Lemma 7.1 Annex), so passing to limit, we obtain ). (5.5) By passing to the limit (η → 0) in (5.4) we have From the definition of B η with (5.3), we deduce that Asymptotic modeling of thin plastic oscillating layer 105 after (5.2) and (5.6), So there exists a constant C > 0 and a subsequence of F ε (u ε ), still de noted by Hence the proof of the Theorem 5.1 is complete.✷ In the sequel, we determine the limit problem linked to (4.1), when ε approaches to zero.Thanks to the epi-convergence results, see Annex Theorem According to the uniqueness of solutions of problem (5.6), so u ε admits an unique τ f -cluster point u * , and therefore u ε ⇀ u * in LD 0 (Ω) ✷ Since ϕ is bounded in Σ, so for evry s ≥ 1, there existes a constant C > 0, such that