Two-scale homogenization of piezoelectric perforated structures

We are interested in the homogenization of elastic-electric coupling equation, with rapidly oscillating coefficients, in periodically perforated piezoelectric body. We justify the two first terms in the usual asymptotic development of the problem solution. For the main convergence results of this paper, we use the notion of {\it two-scale convergence}. A two-scale homogenized system is obtained as the limit of the periodic problem. While in the static limit the method provides homogenized electroelastic coefficients whicht coincide with those deduced from other homogenization techniques (asymptotic homogenization, $\Gamma$-convergence).


Introduction
Composites and perforated (lattice) materials are widely used in many practical applications, such as aircraft, civil engineering, electrotechnics, and many others. These materials are with a large number of heterogeneities (inclusions or holes), and in strong contrast to continum materials, their behavior is definitively influenced by micromechanical events.
The first goal of this work we study the homogenization of the equation of the elastic-electric coupling with rapidly oscillating coefficients in a periodically perforated domain. The homogenized of this problem for a fixed domain has already been studied, by the author (Feng and Wu. [9], Castillero and all. [5], Ruan and all. [14]). But in this work we give new convergence results concerning the same model by using homogenization technique of " two-scale convergence ", which permits us to conclude the limit problem, the approximation of final state is altrered by a constant named as the volum fraction which depends on the proportion of material in the perforated domain and is equal to 1 when there are no holes.
The second goal of this paper is to apply the technique of a formal asymptotic homogenization, to determine the effective elastic, piezoelectric and dielectric modulous of periodic medium. The final formulae for the effective parameters are given in a relatively simpler closed form.
The third goal of this paper is to also establish a corrector-type theorem, which permits to replace the sequence by its "two-scale" limit using the result on the strong convergence, and permits to justify the two first terms in the usual asymptotic expansion of the solution. In the last section we treat of the enregy aspect of our problem.

Homogenization problem
Throughout this paper L 2 (Ω) in the Sobolev space of real-valued functions that are measurable and square summable in Ω with respect of the Lebesgue measure. We denote by C ∞ ♯ (Y ) the space of infinitely differentiable functions in R 3 that are periodic of Y . Then, L 2 ♯ (Y ) (respectively, H 1 ♯ (Y )) is the completion for the norm of L 2 (Y ) (respectively, H 1 (Y )) of C ∞ ♯ (Y ).

Geometric of the medium
Let Ω ⊂ R 3 be a bounded three dimensional domain with the boundary Γ = ∂Ω.
We denote x the macroscopic variable and by y = x ε the microscopic variable. Let us define Ω ε of periodically perforated subdomains of a bounded open set Ω. The period of Ω ε is εY * , where Y * is a subset of the unit cube Y = (0, 1) 3 , which represents the solid or material domain, S * obtainded by Y -periodicity from Y * , is a smooth connected (the material is in one piece) open set in R 3 . Denoting by χ(y) the caracteristic function of S * (Y-periodic), in Ω ε well be defined analytically by

Model problem
We adopt the convention of Einstein for the summation of repeated indices, we use Latin indices, understood from 1 to 3, we note by u ε the fields of displacement in elastic, and by ϕ ε of electric potentiel. The equations of equilibrium and Gauss's law of electrostatics in the absence of free charges, written as we complete the boundary conditions,    (u ε , ϕ ε ) = (0, 0) on ∂Ω, σ ε (u ε , ϕ ε ).n ε = 0 on the boundary of holes ∂Ω ε − ∂Ω, D ε (u ε , ϕ ε ).n ε = 0 on the boundary of holes ∂Ω ε − ∂Ω, where f ∈ L 2 (Ω ε ). The second-order stress tensor σ ε = (σ ε ij ), and the electric displacement vector D ε = (D ε i ), are linearly related to the second-order strain tensor s kl (u) = 1 2 (∂ k u l + ∂ l u k ) and the electric field vector ∂ k ϕ ε by the constitutive law And the material proprieties are given by the fourth-order stiffness tensor c ε ijkl measured at constant electric field, the elastic coefficients satisfy the following symmetries and ellipticity uniformily in ε, The third-order piezoelectric tensor e ε ijk (the coupled tensor), verify the following symmetry, The second-order electric tensor d ε ij (dielectric permittivity), measured at constant strain, verify the conditions of symmetric and ellipticity uniformily by ε,

A priori estimates
In order to prove the main convergence results of this paper we use the notion of two-scale convergence which was introduced in [10] and developed further in [1]. The idea of this convergence is based in first step by taking a priori estimates for displacement field and the electric potentiel. The second step we use the relatively compact property with the classical procedure of prolongation (wich is the extension by 0 from Ω ε to Ω). Finally we pass the limit ε → 0, in order to obtain the homogenized and the local problems in same time.

Proof.
By choosing v = u ε and ψ = ϕ ε in variational formulae (7)- (8), and by using the Korn's and Poincaré's inequalities in perforated domains (see Oleinik et al. [13] for the Korn's inequality and Allaire-Murat [2] for Poincaré's inequality), we see that u ε and ϕ ε are bounded, by a constant which does not depend on ε.
For other details see [12].

Two-scale convergence
We denote by ∼ . the extension by zero in the holes Ω − Ω ε . The sequence of (7)-(8) verify (9) and, in this case, by adding the relatively compact property and elementary properties of two-scale convergence, imply x or y means that the derivatives are with respect to the variable.
From last results, we can state next theorem and we have this boundary conditions where θ is the volum fraction of material The equations (10)-(11)-(12) are referred to as the two-scale homogenized system.
Proof. From the idea of G.Nguetseng [10], the test functions in (7) Under the precedent hypotheses, and passing to the two-scale limit, yields By definition of χ, we have where θ = Y χ(y) dy, by density of spaces from which we chose the test functions, the equation (14) holds true for any v 0 ∈ H 1 0 (Ω), ψ 0 ∈ H 1 0 (Ω), and for , Integrating by parts, shows that (14) is variational formulation associated to the two-scale homogenized system We complete (15) by the boundary conditions (11)- (12). To prove existence and uniqueness in (14), by application of the Lax-Milgram lemma, let focus on /R] of the bilinear form defined by the left-hand side of (14) (For a complete demonstration see [12]).
Remark 2 It is evident that the two-scale homogenized problem (10)-(11)- (12) is a system of four equation, four unknown (u, u 1 , ϕ, ϕ 1 ), each dependent on both space variables x and y (i.e. the macroscopic and microscopic scales) which are mixed. Although seems to be complicated, it is well-posed system of equations. Also it is clear that the two-scale homogenized problem has the same form as the original equation.
The object of new paragraph is to give another form of theorem which is more suitable for further physical interpretations. Indeed, we shall eliminate the microscopic variable y (one doesn't want to solve the small scale structure), and decouple the two-scale homogenized problem (10)-(11)-(12) in homogenized and cell equations. However, it is preferable, from a physical or numerical point of view (see [12]).

Derivation of the homogenized coefficients
Due to the linearity of the original problem, and assuming the regularity in variation of the coefficients, we take u 1 (x, y) = s mh,x (u(x))w mh (y) + ∂ϕ(x) ∂x n q n (y), where w mh , ϕ n , q mh and ψ n are Y * -periodic functions in y, independent of x, solutions of these two locals problems in Y * where However, in general a relation (16)-(17) like this does not exist, if we have't linearity of problem. Now substitue the expansions (16) and (17) in this equation Calling τ mh the basic of symmetric second order tensors τ kl mh = 1 2 [δ km δ lh + δ kh δ lm ], where δ ij is the Kronecker symbol. Analogoulosy, we substitue the expansions (16) and (17)  we obtain − ∂s mh,x (u) ∂y i − e ikl (x, y) τ kl mh + s kl,y (w mh ) + d ij (x, y) ∂ϕ mh ∂y j + ∂ϕ ∂x n − e ikl (x, y)s kl,y (q n ) + d ij (x, y) δ jn + ∂ψ n ∂y j = 0.
From the relation (20)-(21), after lenghty calculations we arrive at the homogenized (effective) coefficients : e H nij = c ijkl (x, y)s kl,y (q n ) + e kij (x, y) δ kn + ∂ψ n ∂y k , d H in = − e ikl (x, y)s kl,y (q n ) + d ij (x, y) δ jn + ∂ψ n ∂y j , where h = Y * h(y) dy, the measurements on Y * of function h. Now we give the results concerning some properties of elasticity homogenized tensor.

Proposition 2
The coefficients of elasticity homogenized tensor C H = (c H ijkl ) defined by (22) satisfy: Proof.
The part of the symmetry of these coefficients is evident We are interesing in the proof of Following the ideas, we transform the above expression to obtain a symmetric form.
We defined the tensor of second order Σ, by Σ kl = 1 2 (y k e l + y l e k ), we define 3 × 3 H kl matrix by H kl = s y (Σ kl ), it is evident the coefficients of this matrix its defined by If we use this new notation, we can rewrite the problem (18), as the form given as under We introduce the problem functions (w ij , q ij ), solutions of the problem The coefficient of elasticity tensor can be rewritten as The second integral of the right-hand side of precedent expression, is evaluated as follows We use the variationnal formulation of the first equation of problem (26), and taking the test function v = w ij , we obtain (30) Multiplying the second equation by ϕ mh , and integrating by parts, we have Regrouping these results, and using the definition (28), we derive It is immediate from above, that the coefficients of elasticity tensor satisfies This is the end of the proof of the first section of proposition i.e of symmetry. We now study the ellipticity of the coefficients of the elasticity tensor, we recall c H ijkl is elliptic, if for all the second order tensor X ij symetric ( where w = w mh X mh , ζ = ϕ mh X mh and P ij = τ ij mh X mh = X ij . Therfore the couple (w, ζ) is a saddle point of the functional J defined by By definition of the saddle point, we have Or J(w, 0) = 1 2 Y * c ijkl (x, y) s ij,y (w) + P ij s kl,y (w) + P kl dy.
But, if we use the first equation of system (18), we obtain We have the homogenized elastcity tensor C H = (c H ijkl ), which is elliptic. Now we give a results concerning some properties of dielectric homogenized tensor.

Proposition 3 The coefficients of dielectric homogenized tensor
By analgy, we transform these coefficients to obtain a symmetric form The problem (19), can be rewritten as Introducing (q i , ψ i ), in the solution of this local problem We can rewrite these coefficients of electric tensor in form given as The first term of the second integral of precedent expression, is evaluated as follows Using the variationnal formulation of the second equation of system (33), and chossing a test function ϕ = ψ i , we obtain Let us now consider the second integral in (36). Multiplying the first equation of system (34) by φ n , and integrating by parts, we have Finality, we regroup these lasts results, and using the definition (35), we obtain It is clear from that the coefficients of electric tensor is symmetric. Now we are interested in the ellipticity of this tensor, recall d H in is elliptic, if for all vector X i , we have We consider the expression (25) of tensor d H in , we derive where ξ = ψ n X n , ς = ϕ n X n and Q i = δ in X n = X i . Or the couple (ξ, ς) is a saddle point of the functionnal G defined by : By definition of the saddle point, we have Or But, if we use the second equation of system (19), we have We have the dielectric homoginized tensor which is elliptic. Now we give the results concerning some properties of piezoelectric homogenized tensor.
Using the two variationnals formulations corresponding of problems (18) and (19), and chossing the appropriete test functions, we can directly prove as e H nij = f H nij . Finallity, using the three last propositions, we can purpose the altarnative form of the principal convergence theorem Theorem-Bis (the altarnative form) Set (u, ϕ) solution of the two-scale homogenized problem (10)-(11)- (12), then (u, ϕ) is defined by that the solution of this homogenized problem where the boundary conditions

Correctors result
The corrector results are obtained easily by the two-scale convergence method. The objective of the next theorem justify rigorously the two first terms in the usual asymptotic expansion of the solution. Following the idea of Allaire [1], we introduce the following definition Definition 1 We call ψ(x, y) an admissible test function, if it is Y -periodic, and satisfies the following relation Here we recall the Allaire's lemma Lemma 2 (Allaire [1]) : Let the function ψ(x, y) ∈ L 2 (Ω; C ♯ (Y )), then ψ(x, y) is an admissible test function in the sense of Definition 1.
Using this lemma, we obtain the following proposition.

Proposition 5
The two functions s ij,y (u 1 (x, y)) and ∂ i,y ϕ 1 (x, y) are admissible test functions in the sense of Definition 1.
Using Lemma 2, s ij,y (u 1 (x, y)) and ∂ i,y ϕ 1 (x, y) are the admissible test functions in the sense of Definition 1.

Conclusion
In this work, we have given the new convergence results, and the explicite forms of the elastic, piezoelectric and dielectric homogenized coefficients. The twoscale convergence is applied to our problem yields the strong convergence result on the correctors. This technique of two-scale convergence can handle also other homogenization problems, in medium which has periodic structure for example the laminated piezocomposite materials or fiber materials (see [5] [9] [11] [14] [12]). Numerical implementation for perforated, laminated and fiber structures, will be presented in forthcoming publications (see [12]).