Limit analysis of an elastic thin oscillating layer

The problem of inclusion of an elastic thin layer between two elastic bodies, where the thin layer obeys to a nonlinear elastic law of power type, is widely studied in many works. Here we can mention some authors who treated the elasticity and thermal problems on the structure containing a thin plate layer, in one hand the thermal case was treated by Brillard and Sanchez-Palencia et al. in [7,12]. In the other hand the elasticity case was studied by Ait Moussa et al., Brillard et al., Geymonat et al. and Lenci et al. in [2,9,10,11]. The plastic plate case was treated by Messaho et al. in [4]. From the mechanical point of view for the thin layer and the paper of Acerbi et al. [1], we are motivated to studying the elasticity problems on a structure containing a thin oscillating layer. It is, therefore of interest to study the limit behavior of thin layer with an oscillating boundary, with a small enough periodicity parameter, between the two adherents when the thickness, rigidity and


Introduction
The problem of inclusion of an elastic thin layer between two elastic bodies, where the thin layer obeys to a nonlinear elastic law of power type, is widely studied in many works.Here we can mention some authors who treated the elasticity and thermal problems on the structure containing a thin plate layer, in one hand the thermal case was treated by Brillard and Sanchez-Palencia et al. in [7,12].In the other hand the elasticity case was studied by Ait Moussa et al., Brillard et al., Geymonat et al. and Lenci et al. in [2,9,10,11].The plastic plate case was treated by Messaho et al. in [4].From the mechanical point of view for the thin layer and the paper of Acerbi et al. [1], we are motivated to studying the elasticity problems on a structure containing a thin oscillating layer.It is, therefore of interest to study the limit behavior of thin layer with an oscillating boundary, with a small enough periodicity parameter, between the two adherents when the thickness, rigidity and periodicity parameters depending on a small enough parameter intended to tend towards 0 where the rigidity parameter is great enough.
In this present work, we consider a structure containing a thin oscillating layer of thickness, rigidity and periodicity parameter depending on ε being a parameter intended to tend towards 0. In a such structure we have treated the scalar case for a thermal conductivity problem in Messaho et al in [3].The aim of this work is to study the limit behavior of an elasticity problem where a convex energy functional defined 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 notations 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 devoted to the determination of the limits problems and our main result.

Statement of the problem
We consider a structure, occupying a bonded domain Ω ⊂ R 3 with lipschitzian boundary ∂Ω.It is constituted of two elastics bodies joined together by a thin layer with oscillating boundary (see figure 1), the latter obeys to a nonlinear elastic law of power type.More precisely the stress field is related to the field of displacement by The structure occupying the domain Ω is subjected to a density of forces of volume f , 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 ε : Ω → R 3 are Where ε being a positive parameter intended to tend towards zero, p > 1, α ≥ 0, a ijkh being the elasticity coefficients and R 9 S is the vector space of the square symmetrical matrices of three order .e ij (u) components of the linearized tensor of deformation e(u).
ϕ ε being 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 (2.4) Figure 1: The domain Ω.

Notations
Here is the notation that will be used in the sequel: ) and r ′ is the conjugate of r (satisfies to 1 r + 1 r ′ = 1).In the following C will denote any constant with respect to ε.Also we use the convention 0. + ∞ = 0.

Functional setting
First, we introduce the following space : we show easily that V ε is a Banach space with the following norm V p 0 (Σ) and V C (Σ) are two Banach spaces, provided with the norm of H 1 0 (Ω, R 3 ).V p (Σ) and V 1,p (Σ) are two Banach spaces, provided respectively with the following norms

It was known that
D α,p = G α,p .
Our goal in this work is to study the problem (2.1), and its limit behavior when ε tends to zero.

Study of the problem (2.1)
We remark that, the problem (2.1) is equivalent to the minimization problem In order to study the problem (2.1), we will interest to the study of the minimization problem (4.1).The existence of the solution of (4.1) is given in the following proposition.
The proof of this proposition is based on classical convexity arguments see for example [6].Lemma 4.2.Under the hypothesis (2.4) and for f ∈ L r ′ (Ω, R 3 ), the solution of (4.1) u ε satisfies e(u ε ) ) Proof: Since u ε is the solution of the problem (4.1) , we have In particular for v = u ε , we obtain According to the inequalities of Hölder and Young, we have ) So that e(u ε ) Therefore, we will have (4.2) and (4.3).Since r ≤ min(2, p) and according to (4.2) and (4.3), so for a small enough ε the solution ( We give some lemmas that will be used in the sequel.
From the Green formula we have the following lemme ) and u ∈ D(Σ, R 3 ), so we have ) Lemma 4.5.Let u be a regular function defined in a neighborhood of Σ, then This lemma is a consequence of [1, proposition 2].Let us Lemma 4.6.The solution u ε of the problem (4.1) possess a cluster point u * in W 1,r 0 (Ω, R 3 ) with respect to the weak topology, such that u * |Σ is a weak cluster point of w ε in L r (Σ, R 3 ).
Proof: According to the lemma 4.2, (4.2) and (4.3), for a small enough ε, u ε is bounded in W 1,r 0 (Ω, R 3 ), so for a subsequences of u ε , still denoted by u ε , there We have Thanks to the lemma 4.2 and the Korn's inequality, so we have Lemma 4.7.The sequence v ε possess a weak cluster point The proof of this lemma is based on the same technic used in the proof of the lemma 4.6.
Note by e T (u) = (I − e 3 ⊗ e 3 )δu the tangential part of the tensor e(u).In order to apply the epiconvergence 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 Proposition 4.8.The solution u ε of the problem (4.1), possess a weak cluster point Proof: 1. Thanks to the lemma 4.6, for a subsequence of u ε , still denoted by u ε , there exists u * ∈ W 1,r 0 (Ω, R 3 ) such that S ), so for a subsequence of χ Ωε e(u ε ), still denoted by χ Ωε e(u ε ), there exists S ), such that Passing to the limit, we have S ), and according to the classical result (see, [13, proposition 1.2, p. 16]), we have u * ∈ H 1 0 (Ω, R 3 ).

If α = 1, let us pose
we have thanks to the inequality of Hölder and a 1 ≤ ϕ ≤ a 2 , (4.9) becomes 3), we have ), according to the lemma 4.5 we obtain to simplify the writing, note Thanks to the lemma 4.4, we have thanks to the lemma 4.6 and ϕ ε → m(ϕ) in L r ′ (Σ), so passing to limit, we obtain Now, to continue our proof, we need to establish the following lemma Lemma 4.9.for all g ∈ D(Σ, R 9 S ), we have

The proof of this lemma
Proof: Indeed, we have First, show that β ε = Σ g δϕ ε ⊗ U ε → 0 and the second term will be shown with the same way.We have , ).We have , so there exists h ε,n → 0 when n → +∞, such that , since ∂ϕ ∂xi is bonded on Σ, so for a greater enough n, for each x ′ ∈ Σ we have according to the Hölder inequality, we have So for each γ > 0, there exists ε 0 such that for ε < ε 0 we have .
We show easily that lim ε→0 a ε,l = 0, so passing to the limit in γ to 0, n to +∞, then in ε to 0, we obtain β ε → 0.
Hence the proof of the lemma 4.9 is complete.✷ So thanks to the lemma 4.9, we obtain S ), and according to S ), so it follows that e T u * |Σ ∈ L p (Σ, R 9 S ).
According to the lemma 4.5, we find that we show easily, like in the proof of the lemma 4.9 and thanks to the lemma 4.6, that and Let us show that lim ε→0 I 2 = 0, indeed According to the lemma 4.5, and the fact that 1 ε εϕ ε −εϕ ε (e(u ε )e 3 − e 33 (u ε )e 3 ) converges to 0 in L p (Σ, R 3 ), and while redoing the same way like in the proof of the lemma 4.9, we obtain
To study the limit behavior of the problem (4.1), we'll use the epi-convergence method , (see Annex, definition 6.1).

Limit behavior
We design by τ f the weak topology on the space W 1,r 0 (Ω, R 3 ).In sequel, we shall characterize, according to the values of α, the epi-limit of the functional energy given by (5.1) in the following theorem.
Theorem 5.1.According to the values of α, there exists a functional where F α is given by 1.If α = 1 and α = p + 1 : 2. If α = 1 : Proof: a) We are now in position to determine the upper epi-limit Let u ∈ G α,p ⊂ W 1,r 0 (Ω, R 3 ), so there exists a sequence (u n ) in D α,p such that u n → u in G α,p when n → +∞, so u n → u in W 1,r 0 (Ω, R 3 ).Let θ be a regular function satisfies we define we have u ε,n ∈ V ε and we prove easily that u ε,n ⇀ u n in G α,p when ε −→ 0. As It implies that Since ϕ ε is bounded, so it follows that we have with ω ε → 0 when ε → 0. So passing to the upper limit, we obtain , so passing to the upper limit we have 3. If α > p + 1 (5.4) becomes so passing to the upper limit we obtain Since u n → u in G α,p when n −→ +∞, therefore according to a classic result, diagonalization's lemma , (see, [5, Lemma 1.15 p. 32]), there exists a function n(ε) : R + → N increasing to +∞ when ε → 0 such that u ε,n(ε) ⇀ u in G α,p when ε → 0. and while n → +∞, consequently we have 3. If α = 1 and α = p + 1, we have a ijkh e kh (u)e ij (u).
For u ∈ W 1,r 0 (Ω, R 3 ) \ G α,p , so for any sequence b) We are now in position to determine the lower epi-limit Let u ∈ G α,p and (u ε ) ⊂ V ε such that u ε ⇀u in W 1,r 0 (Ω, R 3 ), then χ Ωε e(u ε ) ⇀ e(u) in L r (Ω, R 9 S ) (5.5) From the subdifferentiability's inequality of u → 1 2 Ωε a ijkh e kh (u)e ij (u), and passing to the lower limit, we obtain a ijkh e kh (u)e ij (u).

If α = 1
If lim inf ε→0 F ε (u ε ) = +∞, there is nothing to prove, because otherwise, 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 F ε (u ε ) ≤ C, which implies that Let v ε be the sequence defined in the proof of the proposition 4.8, according to the last and from (5.6), we obtain We have we have thanks to the lemma 6.4, we have 1 (Σ) and the fact that m( 1 ϕ p−1 ) m(ϕ) p−1 ≥ 1, so from (5.5) and passing to the limit, we deduce that otherwise, 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 F ε (u ε ) ≤ C, which implies that Let ξ ε be the sequence defined in the proof of the proposition 4.8, according to this last and from (5.7), we have To simplify the writing let us pose ξ = 2 p+1 m(ϕ p+1 )δδu 3| Σ , which implies that According to the subdifferentiability's inequality of We have from the lemma 6.4, we have 1 (Σ) and the fact that m( 1 ϕ p 2 −1 ) m(ϕ p+1 ) p−1 ≥ 1, so from (5.5) and passing to the limit, consequently we have So there exists a constant C > 0 and a subsequence of F ε (u ε ), still denoted by F ε (u ε ), such that So u ε verifies the following evaluations (4.2) and (4.3), as u ε ⇀ u in W 1,r 0 (Ω, R 3 ), thanks to the remark 4.10, we have u ∈ G α,p , what contradicts the fact that u ∈ W 1,r 0 (Ω, R 3 ) \ G α,p , consequently we have lim inf Hence the proof of the theorem 5.1 is complete.✷ In the sequel, we are interested to the limit problem determination linked to the problem (4.1), when ε approaches to zero.Thanks to the epi-convergence results, (see Annex, theorem 6.3, proposition 6.2) and the theorem 5.1, according to the τ f -continuity of the functional G in W 1,r 0 (Ω, R 3 ), we have ). Proposition 5.2.For any f ∈ L r ′ (Ω, R 3 ) and according to the values of the parameter α, there exists u Proof: Thanks to the lemma 4.2, the family u ε is bounded in W 1,r 0 (Ω, R 3 ), therefore it possess a τ f −cluster point u * in W 1,r 0 (Ω, R 3 ).And thanks to a classical epi-convergence result, (see Annex, theorem 6.3), it follows that u * is a solution of the limit problem (5.9) As F α equals +∞ on W 1,r 0 (Ω, R 3 ) \ G α,p , so (5.9) becomes According to the uniqueness of solutions of the problem (5.9), so u ε admits an unique τ f -cluster point u * , and therefore u ε ⇀ u * in W 1,r 0 (Ω, R 3 ).✷

Conclusion.
We showed that the structure, constituted of two elastic bodies joined together by an elastic thin oscillating layer of thickness and rigidity and periodicity parameter depending on a small enough parameter ε, obeying to a nonlinear elastic law whose parameters depend on the negative powers of ε, behaves at the limit like an elastic body embedded on the boundary and subjected to a density of forces of volume f, according to the powers of ε, the layer behaves like a rather rigid nonlinear elastic material surface with membrane effect, too rigid inextensible material surface, a material surface with effect of inflection or the structure is embedded on the interface Σ.
Note the following stability result of the epi-convergence.