Two-scale homogenization of piezoelectric perforated structures
aa r X i v : . [ m a t h . NA ] S e p Two-scale homogenization of piezoelectricperforated structures
Houari Mechkour ∗ Abstract
We are interested in the homogenization of elastic-electric couplingequation, with rapidly oscillating coefficients, in periodically perforatedpiezoelectric body. We justify the two first terms in the usual asymptoticdevelopment of the problem solution. For the main convergence resultsof this paper, we use the notion of two-scale convergence . A two-scalehomogenized system is obtained as the limit of the periodic problem.While in the static limit the method provides homogenized electroelasticcoefficients whicht coincide with those deduced from other homogenizationtechniques (asymptotic homogenization [5], Γ-convergence [16]).
Key words.
Homogenization; Piezoelectricity; Perforations
Composites and perforated (lattice) materials are widely used in many practi-cal applications, such as aircraft, civil engineering, electrotechnics, and manyothers. These materials are with a large number of heterogeneities (inclusionsor holes), and in strong contrast to continum materials, their behavior is defini-tively influenced by micromechanical events.The first goal of this work we study the homogenization of the equation ofthe elastic-electric coupling with rapidly oscillating coefficients in a periodicallyperforated domain. The homogenized of this problem for a fixed domain has al-ready 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 thesame model by using homogenization technique of “ two-scale convergence ” ,which permits us to conclude the limit problem, the approximation of final stateis altrered by a constant named as the volum fraction which depends on the pro-portion of material in the perforated domain and is equal to 1 when there areno holes. ∗ Centre de Mathmatiques Appliques (UMR 7641) ´Ecole Polytechnique, 91128 Palaiseau,France. ([email protected]).
Throughout this paper L (Ω) in the Sobolev space of real-valued functions thatare 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 thatare periodic of Y . Then, L ♯ ( Y ) (respectively, H ♯ ( Y )) is the completion for thenorm of L ( Y ) (respectively, H ( Y )) of C ∞ ♯ ( Y ). Let Ω ⊂ R be a bounded three dimensional domain with the boundary Γ = ∂ Ω.We denote x the macroscopic variable and by y = xε the microscopic variable.Let us define Ω ε of periodically perforated subdomains of a bounded open setΩ. The period of Ω ε is εY ∗ , where Y ∗ is a subset of the unit cube Y = (0 , ,which represents the solid or material domain, S ∗ obtainded by Y -periodicityfrom Y ∗ , is a smooth connected (the material is in one piece) open set in R .Denoting by χ ( y ) the caracteristic function of S ∗ (Y-periodic), in Ω ε well bedefined analytically by Ω ε = (cid:8) x ∈ Ω , χ ( xε ) = 1 (cid:9) We adopt the convention of Einstein for the summation of repeated indices, weuse Latin indices, understood from 1 to 3, we note by u ε the fields of displace-ment in elastic, and by ϕ ε of electric potentiel. The equations of equilibriumand Gauss’s law of electrostatics in the absence of free charges, written as (cid:26) − div σ ε ( u ε , ϕ ε ) = f in Ω ε , − div D ε ( u ε , ϕ ε ) = 0 in Ω ε , (1)we complete the boundary conditions, ( u ε , ϕ ε ) = ( ,
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 , (2)2here f ∈ L (Ω ε ). The second-order stress tensor σ ε = ( σ εij ), and the elec-tric displacement vector D ε = ( D εi ), are linearly related to the second-orderstrain tensor s kl ( u ) = ( ∂ k u l + ∂ l u k ) and the electric field vector ∂ k ϕ ε by theconstitutive law (cid:26) σ εij ( u ε , ϕ ε ) = c εijkl s kl ( u ε ) + e εkij ∂ k ϕ ε in Ω ε ,D εi ( u ε , ϕ ε ) = − e εikl s kl ( u ε ) + d εij ∂ j ϕ ε in Ω ε . (3)1 ≤ i, j, k, l ≤ , where ( div σ ε ) j = ∂ i σ εij , div D ε = ∂ i D εi , ∂ i = ∂∂x i , x = ( x i ) ∈ Ω. And thematerial proprieties are given by the fourth-order stiffness tensor c εijkl measuredat constant electric field, the elastic coefficients satisfy the following symmetriesand ellipticity uniformily in ε , c εijkl ( x ) = c ijkl ( x, xε ) ,c εijkl = c εjikl = c εklij = c εijlk ,c ijkl ( x, y ) ∈ L ∞ (Ω; C ♯ ( Y )) , ∃ α c = α c ( ε ) > c εijkl X ij X kl ≥ α c X ij X ij , ∀ X ij = X ji ∈ R . (4)The third-order piezoelectric tensor e εijk (the coupled tensor), verify the follow-ing symmetry, e εijk = e ijk ( x, xε ) ,e εijk = e εikj ,e ijk ( x, y ) ∈ L ∞ (Ω; C ♯ ( Y )) . (5)The second-order electric tensor d εij (dielectric permittivity), measured at con-stant strain, verify the conditions of symmetric and ellipticity uniformily by ε , d εij = d ij ( x, xε ) ,d εij = d εji ,d ij ( x, y ) ∈ L ∞ (Ω; C ♯ ( Y )) , ∃ α d = α d ( ε ) > d εij X i X j ≥ α d X i X i , ∀ X i ∈ R . (6) Introducing the two Hilbert spaces V ε (Ω ε ) = { v ∈ H (Ω ε ) , v = on ∂ Ω } W ε (Ω ε ) = { ψ ∈ H (Ω ε ) , ψ = 0 on ∂ Ω } With two norms : k . k V ε (Ω ε ) = k . k H (Ω) , k . k W ε (Ω ε ) = k . k H (Ω) . Thevariational problem is defined by : Find ( u ε , ϕ ε ) ∈ V ε (Ω ε ) × W ε (Ω ε ) , such that a ε (( u ε , ϕ ε ) , ( v , ψ )) = L ε ( v , ψ ) ∀ ( v , ψ ) ∈ V ε (Ω ε ) × W ε (Ω ε ) , (7)3here a ε (( u ε , ϕ ε ) , ( v , ψ )) = Z Ω ε { [ c εijkl s kl ( u ε ) + e εkij ∂ k ϕ ε ] s ij ( v )+ [ − e εikl s kl ( u ε ) + d εij ∂ j ϕ ε ] ∂ i ψ } dxL ε ( v , ψ ) = Z Ω ε f i v i dx (8)It is pointed out that under assumptions (4)-(5)-(6), the variational problem(7)-(8) have a unique solution ( u ε , ϕ ε ) ∈ V ε (Ω ε ) × W ε (Ω ε ), corresponding thesaddle point of this functional (see [12]) :( v , ψ ) → Z Ω ε ( c ε ( v , v ) + 2 e ε ( u , ψ ) − d ε ( ψ, ψ )) dx − Z Ω ε f v dx, where c ε ( u , v ) = c εijkl s ij ( u ) s kl ( v ) e ε ( u , ψ ) = e εikl s kl ( u ) ∂ i ψd ε ( ψ, ψ ) = d εij ∂ i ψ ∂ j ψ 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 estimatesfor displacement field and the electric potentiel. The second step we use therelatively compact property with the classical procedure of prolongation (wichis the extension by 0 from Ω ε to Ω). Finally we pass the limit ε →
0, in orderto obtain the homogenized and the local problems in same time.
Proposition 1
By using the two equivalent norms of V ε (Ω ε ) , W ε (Ω ε ) , for anysequence of solution ( u ε , ϕ ε ) ε ⊂ V ε (Ω ε ) × W ε (Ω ε ) of variational problem (7)-(8). Then, this solution is bounded, and we have this a priori estimate uniformlyby ε k u ε k H (Ω ε ) + k ϕ ε k H (Ω ε ) ≤ C, (9) where C is constant strictement positive and independent by ε . Proof.
By choosing v = u ε and ψ = ϕ ε in variational formulae (7)-(8), and by usingthe Korn’s and Poincar´e’s inequalities in perforated domains ( see Oleinik et al. [13] for the Korn’s inequality and Allaire-Murat [2] for Poincar´e’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]. 4 Two-scale convergence
We denote by ∼ . the extension by zero in the holes Ω − Ω ε . The sequence ofsolution ( u ε , ϕ ε ) ε ⊂ V ε (Ω ε ) × W ε (Ω ε ) of variational problem (7)-(8) verify (9)and, in this case, by adding the relatively compact property and elementaryproperties of two-scale convergence, imply Lemma 1
1. There exists u ( x ) ∈ H (Ω) and ϕ ( x ) ∈ H (Ω) such that, thetwo sequences ( ∼ u ε ) ε , ( ∼ ϕ ε ) ε two-scale converge to χ ( y ) u ( x ) , χ ( y ) ϕ ( x ) , re-spectively.2. There exists u ( x, y ) ∈ L [Ω; H ♯ ( Y ∗ ) / R ] , ϕ ( x, y ) ∈ L [Ω; H ♯ ( Y ∗ ) / R ] such that, ∼ ∇ u ε → χ ( y )[ ∇ x u ( x ) + ∇ y u ( x, y )] in two-scale sense ∼ ∇ ϕ ε → χ ( y )[ ∇ x ϕ ( x ) + ∇ y ϕ ( x, y )] in two-scale sense3. We have ∼ s ( u ε ) → χ ( y )[ s x ( u ( x )) + s y ( u ( x, y ))] in two-scale senseindex x or y means that the derivatives are with respect to the variable. Proof.
For details see [1], [10], [11] and [12].
Corollary 1
The sequence ( ∼ u ε ) ε> ( resp . ( ∼ ϕ ε ) ε> ) , converge weakly to a limit θ u ( resp . θϕ ) in L (Ω) ( resp . L (Ω)) . Remark 1
Let ρ ∈ L ♯ ( Y ) , define ρ ε ( x ) = ρ ( xε ) , and ( v ε ) ε ⊂ L (Ω) two-scaleconverge to a limit v ∈ L (Ω × Y ) . Then ( ρ ε v ε ) ε two-scale converges to a limit ρv (see [12]). From last results, we can state next theorem
Theorem 1
The sequences ( ∼ u ε ) ε , ( ∼ s ( u ε )) ε , ( ∼ ϕ ε ) ε and ( ∼ ∇ ϕ ε ) ε two-scale con-verge to χ ( y ) u ( x ) , χ ( y )[ s x ( u ) + s y ( u )] , χ ( y ) ϕ ( x ) and χ ( y )[ ∇ x ϕ + ∇ y ϕ ] re-spectively, where ( u ( x ) , u ( x, y ) , ϕ ( x ) , ϕ ( x, y )) are the unique solutions in H (Ω) × L [Ω; H ♯ ( Y ∗ ) / R ] × H (Ω) × L [Ω; H ♯ ( Y ∗ ) / R ] of the following two- cale homogenized system; − ∂∂x j [ Z Y ∗ { c ijkl ( x, y )[ s kl,x ( u ) + s kl,y ( u )] + e kij ( x, y )[ ∂ k,x ϕ + ∂ k,y ϕ ] } dy ] = θf i ( x ) in Ω , − ∂∂x i [ Z Y ∗ {− e ikl ( x, y )[ s kl,x ( u ) + s kl,y ( u )] + d ij ( x, y )[ ∂ j,x ϕ + ∂ j,y ϕ ] } dy ] = 0 in Ω , − ∂∂y j { c ijkl ( x, y )[ s kl,x ( u ) + s kl,y ( u )] + e kij ( x, y )[ ∂ k,x ϕ + ∂ k,y ϕ ] } = 0 in Ω × Y ∗ , − ∂∂y i {− e ikl ( x, y )[ s kl,x ( u ) + s kl,y ( u )] + d ij ( x, y )[ ∂ j,x ϕ + ∂ j,y ϕ ] } = 0 in Ω × Y ∗ , (10) and we have this boundary conditions u ( x ) = on ∂ Ω ,ϕ ( x ) = 0 on ∂ Ω , { c ijkl ( x, y )[ s kl,x ( u ) + s kl,y ( u )] + e kij ( x, y )[ ∂ k,x ϕ + ∂ k,y ϕ ] } .n j = 0 on ∂Y ∗ − ∂Y, {− e ikl ( x, y )[ s kl,x ( u ) + s kl,y ( u )] + d ij ( x, y )[ ∂ j,x ϕ + ∂ j,y ϕ ] } .n i = 0 on ∂Y ∗ − ∂Y. (11) y → u ( x, y ) is Y − periodic ,y → ϕ ( x, y ) is Y − periodic , (12) where θ is the volum fraction of material ( i.e. θ = < χ > = R Y χ ( y ) dy = | Y ∗ | ) , | Y | denote mesure of Y . The equations (10)-(11)-(12) are referred to as the two-scale homogenizedsystem . Proof.
From the idea of G.Nguetseng [10], the test functions in (7)-(8) arechossed on the form v ε ( x ) = v ( x, xε ) = v ( x ) + ε v ( x, xε ) ,ψ ε ( x ) = ψ ( x, xε ) = ψ ( x ) + εψ ( x, xε ) , v ∈ C ∞ (Ω) , ψ ∈ C ∞ (Ω) , v ∈ C ∞ (Ω; C ∞ ♯ ( Y )) and ψ ∈ C ∞ (Ω; C ∞ ♯ ( Y )),we obtain Z Ω ε { [ c εijkl s kl ( u ε ) + e εkij ∂ k ϕ ε ][ s ij,x ( v )( x ) + { s ij,y ( v ) + εs ij,x ( v ) } ( x, xε )] − [ − e εkij s kl ( u ε ) + d εij ∂ j ϕ ε ][ ∂ i,x ψ ( x ) + { ∂ i,y ψ + ε∂ i,x ψ } ( x, xε )] } dx = Z Ω ε f i ( x ) [ v i ( x ) + εv i ( x, xε )] dx Under the precedent hypotheses, and passing to the two-scale limit, yields Z Ω Z Y [ c ijkl ( x, y ) χ ( y )( s kl,x ( u ) + s kl,y ( u )) + e kij ( x, y ) χ ( y )( ∂ k,x ϕ + ∂ k,y ϕ )] χ ( y )[ s kl,x ( v ) + s kl,y ( v )] dx dy − Z Ω Z Y [ − e ikl ( x, y ) χ ( y )( s kl,x ( u ) + s kl,y ( u )) + d ij ( x, y ) χ ( y )( ∂ j,x ϕ + ∂ j,y ϕ )] χ ( y )[ ∂ i,x ψ + ∂ i,y ψ ] dx dy = Z Ω Z Y f i ( x ) χ ( y ) v i ( x ) dx dy. (13)By definition of χ , we have Z Ω Z Y ∗ [ c ijkl ( x, y )( s kl,x ( u ) + s kl,y ( u )) + e kij ( x, y )( ∂ k,x ϕ + ∂ k,y ϕ )][ s kl,x ( v ) + s kl,y ( v )] dx dy + Z Ω Z Y ∗ [ − e ikl ( x, y )( s kl,x ( u ) + s kl,y ( u )) + d ij ( x, y )( ∂ j,x ϕ + ∂ j,y ϕ )][ ∂ i,x ψ + ∂ i,y ψ ] dx dy = θ Z Ω f i ( x ) v i ( x ) dx, (14)where θ = R Y χ ( y ) dy , by density of spaces from which we chose the test func-tions, the equation (14) holds true for any v ∈ H (Ω) , ψ ∈ H (Ω), and forany v ∈ L [Ω; H ♯ ( Y ∗ ) / R ], ψ ∈ L [Ω; H ♯ ( Y ∗ ) / R ],Integrating by parts, shows that (14) is variational formulation associated tothe two-scale homogenized system − ∂ j (cid:2) Z Y ⋆ { c ijkl ( x, y )[ s kl,x ( u ) + s kl,y ( u )] + e kij ( x, y )[ ∂ k,x ϕ + ∂ k,y ϕ } dy (cid:3) = θf i ( x ) − ∂ i (cid:2) Z Y ⋆ {− e ikl ( x, y )[ s kl,x ( u ) + s kl,y ( u )] + d ij ( x, y )[ ∂ j,x ϕ + ∂ j,y ϕ ] } dy (cid:3) = 0(15)We complete (15) by the boundary conditions (11)-(12). To prove existenceand uniqueness in (14), by application of the Lax-Milgram lemma, let focus on7he coercivity in H (Ω) × L [Ω; H ♯ ( Y ∗ ) / R ] × H (Ω) × L [Ω; H ♯ ( Y ∗ ) / R ] of thebilinear form defined by the left-hand side of (14) (For a complete demonstrationsee [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 , ϕ, ϕ ) , each dependenton 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 ofequations. Also it is clear that the two-scale homogenized problem has the sameform as the original equation. The object of new paragraph is to give another form of theorem which is moresuitable for further physical interpretations. Indeed, we shall eliminate themicroscopic variable y (one doesn’t want to solve the small scale structure), anddecouple the two-scale homogenized problem (10)-(11)-(12) in homogenized andcell equations. However, it is preferable, from a physical or numerical point ofview ( see [12]). Due to the linearity of the original problem, and assuming the regularity invariation of the coefficients, we take u ( x, y ) = s mh,x ( u ( x )) w mh ( y ) + ∂ϕ ( x ) ∂x n q n ( y ) , (16) ϕ ( x, y ) = s mh,x ( u ( x )) ϕ mh ( y ) + ∂ϕ ( x ) ∂x n ψ n ( y ) , (17)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 ∗ − ∂∂y j n c ijkl ( x, y ) h τ klmh + s kl,y ( w mh ) i + e kij ( x, y ) ∂ϕ mh ∂y k o = 0 in Y ∗ , − ∂∂y i n − e ikl ( x, y ) h τ klmh + s kl,y ( w mh ) i + d ij ( x, y ) ∂ϕ mh ∂y j o = 0 in Y ∗ , w mh , ϕ mh Y ∗ − periodics , (18)where τ klmh = 12 [ δ km δ lh + δ kh δ lm ] 1 ≤ k, m, l, h ≤ . − ∂∂y j n c ijkl ( x, y ) s kl,y ( q n ) + e kij ( x, y ) h δ kn + ∂ψ n ∂y k io = 0 in Y ∗ , − ∂∂y i n − e ikl ( x, y ) s kl,y ( q n ) + d ij ( x, y ) h δ jn + ∂ψ n ∂y j io = 0 in Y ∗ ,ϕ n , ψ n Y ∗ − periodics . (19)However, in general a relation (16)-(17) like this does not exist, if we have’tlinearity of problem.Now substitue the expansions (16) and (17) in this equation − ∂∂y j { c ijkl ( x, y )[ s kl,x ( u )+ s kl,y ( u )]+ e kij ( x, y )[ ∂ k,x ϕ + ∂ k,y ϕ ] } = 0 in Ω × Y ∗ , we obtain − ∂s mh,x ( u ) ∂y j n c ijkl ( x, y ) h τ klmh + s kl,y ( w mh ) i + e kij ( x, y ) ∂ϕ mh ∂y k o + ∂ϕ∂x n n c ijkl ( x, y ) s kl,y ( q n ) + e kij ( x, y ) h δ kn + ∂ψ n ∂y k io = 0 . (20)Calling τ mh the basic of symmetric second order tensors τ klmh = [ δ km δ lh + δ kh δ lm ], where δ ij is the Kronecker symbol.Analogoulosy, we substitue the expansions (16) and (17) in this equation − ∂∂y i {− e ikl ( x, y )[ s kl,x ( u ) + s kl,y ( u )] + d ij ( x, y )[ ∂ j,x ϕ + ∂ j,y ϕ ] } = 0 in Ω × Y ∗ , we obtain − ∂s mh,x ( u ) ∂y i n − e ikl ( x, y ) h τ klmh + s kl,y ( w mh ) i + d ij ( x, y ) ∂ϕ mh ∂y j o + ∂ϕ∂x n n − e ikl ( x, y ) s kl,y ( q n ) + d ij ( x, y ) h δ jn + ∂ψ n ∂y j io = 0 . (21)From the relation (20)-(21), after lenghty calculations we arrive at the homog-enized (effective) coefficients : c Hijmh = D c ijkl ( x, y ) h τ klmh + s kl,y ( w mh ) i + e kij ( x, y ) ∂ϕ mh ∂y k E , (22) e Hnij = D c ijkl ( x, y ) s kl,y ( q n ) + e kij ( x, y ) h δ kn + ∂ψ n ∂y k iE , (23) f Himh = D e ikl ( x, y ) h τ klmh + s kl,y ( w mh ) i − d ij ( x, y ) ∂ϕ mh ∂y j E , (24) d Hin = D − e ikl ( x, y ) s kl,y ( q n ) + d ij ( x, y ) h δ jn + ∂ψ n ∂y j iE , (25)9here h h i = R Y ∗ h ( y ) dy, the measurements on Y ∗ of function h .Now we give the results concerning some properties of elasticity homogenizedtensor. Proposition 2
The coefficients of elasticity homogenized tensor C H = ( c Hijkl ) defined by (22) satisfy:a) c Hijkl = c Hklij = c Hijlk = c Hjilk , ∀ ≤ i, j, k, l ≤ , b) There exists α Hc > , such that for all ξ , symmetric tensor ( ξ ij = ξ ji ), c Hijkl ξ ij ξ kl ≥ α Hc ξ ij ξ ij Proof.
The part of the symmetry of these coefficients is evident c Hijmh = c Hjimh = c Hjihm . We are interesing in the proof of c Hijmh = c Hmhij
Following the ideas, we transform the above expression to obtain a symmetricform.We defined the tensor of second order Σ, by Σ kl = ( y k ~e l + y l ~e k ), we define3 × H kl matrix by H kl = s y (Σ kl ), it is evident the coefficients of this matrixits defined by[ H kl ] mh = τ klmh = 12 [ δ km δ lh + δ kh δ lm ] 1 ≤ k, l, m, h ≤ . If we use this new notation, we can rewrite the problem (18), as the form givenas under − ∂∂y j n c ijkl ( x, y ) s kl,y (cid:16) Σ mh + w mh (cid:17) + e kij ( x, y ) ∂ϕ mh ∂y k o = 0 in Y ∗ , − ∂∂y i n − e ikl ( x, y ) s kl,y (cid:16) Σ mh + w mh (cid:17) + d ij ( x, y ) ∂ϕ mh ∂y j o = 0 in Y ∗ , w mh , ϕ mh Y ∗ − periodics . (26)We introduce the problem functions ( w ij , q ij ), solutions of the problem − ∂∂y j n c kjαβ ( x, y ) s αβ,y (cid:16) Σ ij + w ij (cid:17) + e αkj ( x, y ) ∂ϕ ij ∂y α o = 0 in Y ∗ , − ∂∂y k n − e kαβ ( x, y ) s αβ,y (cid:16) Σ ij + w ij (cid:17) + d kj ( x, y ) ∂ϕ ij ∂y j o = 0 in Y ∗ , w ij , ϕ ij Y ∗ − periodics . (27)10he coefficient of elasticity tensor can be rewritten as c Hijmh = Z Y ∗ c ijkl ( x, y ) s kl,y (cid:16) Σ mh + w mh (cid:17) dy + Z Y ∗ e kij ( x, y ) ∂ϕ mh ∂y k dy. (28)The second integral of the right-hand side of precedent expression, is evaluatedas follows Z Y ∗ e kij ( x, y ) ∂ϕ mh ∂y k dy = Z Y ∗ e kαβ ( x, y ) ∂ϕ mh ∂y k δ αi δ βj dy, = 12 Z Y ∗ e kαβ ( x, y ) ∂ϕ mh ∂y k (cid:16) δ αi δ βj + δ αj δ βi (cid:17) dy, = − Z Y ∗ e kαβ ( x, y ) ∂ϕ mh ∂y k s αβ,y ( w ij ) dy + Z Y ∗ e kαβ ( x, y ) ∂ϕ mh ∂y k s αβ,y (Σ ij + w ij ) dy. (29)We use the variationnal formulation of the first equation of problem (26), andtaking the test function v = w ij , we obtain Z Y ∗ e kαβ ( x, y ) ∂ϕ mh ∂y k s αβ,y ( w ij ) dy = Z Y ∗ c αβkl ( x, y ) s kl,y (Σ mh + w mh ) s αβ,y ( w ij ) dy. (30)Multiplying the second equation by ϕ mh , and integrating by parts, we have Z Y ∗ e kαβ ( x, y ) ∂ϕ mh ∂y k s αβ,y (Σ ij + w ij ) dy = Z Y ∗ d kα ( x, y ) ∂ϕ mh ∂y k ∂ϕ ij ∂y α dy. (31)Regrouping these results, and using the definition (28), we derive c Hijmh = Z Y ∗ c ijkl ( x, y ) s kl,y (Σ mh + w mh ) dy + Z Y ∗ e kij ( x, y ) ∂ϕ mh ∂y k dy, = Z Y ∗ c ijkl ( x, y ) s kl,y (Σ mh + w mh ) dy + Z Y ∗ c αβkl ( x, y ) s kl,y (Σ mh + w mh ) s αβ,y ( w ij ) dy + Z Y ∗ d kα ( x, y ) ∂ϕ mh ∂y k ∂ϕ ij ∂y α dy, = Z Y ∗ c αβkl ( x, y ) s kl,y (Σ mh + w mh ) s αβ,y (Σ ij ) dy + Z Y ∗ d kα ( x, y ) ∂ϕ mh ∂y k ∂ϕ ij ∂y α dy + Z Y ∗ c αβkl ( x, y ) s kl,y (Σ mh + w mh ) s αβ,y ( w ij ) dy, = Z Y ∗ c αβkl ( x, y ) s kl,y (Σ mh + w mh ) s αβ,y (Σ ij + w ij ) dy + Z Y ∗ d kα ( x, y ) ∂ϕ mh ∂y k ∂ϕ ij ∂y α dy. (32)11t is immediate from above, that the coefficients of elasticity tensor satisfies c Hijmh = c Hmhij
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 Hijkl is elliptic, if for all the second order tensor X ij symetric ( X ij = X ji ), wehave ∃ α Hc > , c Hijkl X ij X kl ≥ α Hc X ij X ij Consider expression (22) of the tensor c Hijmh , we have c Hijmh X ij X mh = Z Y ∗ c ijmh ( x, y ) X ij X mh dy + Z Y ∗ c ijkl ( x, y ) s kl,y ( w mh ) X ij X mh dy + Z Y ∗ e kij ( x, y ) ∂ϕ mh ∂y k X ij X mh dy, = Z Y ∗ c ijmh ( x, y ) X ij X mh dy + Z Y ∗ c ijkl ( x, y ) s kl,y ( w mh X mh ) X ij dy + Z Y ∗ e kij ( x, y ) ∂ ( ϕ mh X mh ) ∂y k X ij dy, = Z Y ∗ c ijkl ( x, y ) (cid:16) s kl,y ( w ) + P kl (cid:17) P ij dy + Z Y ∗ e kij ( x, y ) ∂ζ∂y k P ij dy, where w = w mh X mh , ζ = ϕ mh X mh and P ij = τ ijmh X mh = X ij . Therfore thecouple ( w , ζ ) is a saddle point of the functional J defined by( v , ψ ) → J ( v , ψ ) J ( v , ψ ) = 12 Z Y ∗ n c ijkl (cid:16) s ij,y ( v )+ P ij (cid:17)(cid:16) s kl,y ( v )+ P kl (cid:17) +2 e kij ∂ψ∂y k (cid:16) s ij,y ( v )+ P ij (cid:17) − d ij ∂ψ∂y i ∂ψ∂y j o dy, By definition of the saddle point, we have J ( w , ψ ) ≤ J ( w , ζ ) ≤ J ( v , ζ ) ∀ ( v , ψ ) periodic functions.Or J ( w ,
0) = 12 Z Y ∗ c ijkl ( x, y ) (cid:16) s ij,y ( w ) + P ij (cid:17)(cid:16) s kl,y ( w ) + P kl (cid:17) dy. But, if we use the first equation of system (18), we obtain c Hijmh X ij X mh = 2 J ( w , ζ ) , ≥ J ( w , , = 12 Z Y ∗ c ijkl ( x, y ) (cid:16) s ij,y ( w ) + P ij (cid:17)(cid:16) s kl,y ( w ) + P kl (cid:17) dy,> , We have the homogenized elastcity tensor C H = ( c Hijkl ), which is elliptic.Now we give a results concerning some properties of dielectric homogenizedtensor. 12 roposition 3
The coefficients of dielectric homogenized tensor D H = ( d Hin ) defined by (25) satisfy:a) d Hin = d Hni , ∀ ≤ i, n ≤ , b) There exists α Hd > , such that for all vector ξ , d Hin ξ i ξ n ≥ α Hd ξ i ξ i Proof.
By analgy, we transform these coefficients to obtain a symmetric form d Hin = d Hni
The problem (19), can be rewritten as − ∂∂y j n c ijkl ( x, y ) s kl,y ( q n ) + e kij ( x, y ) ∂ ( y n + ψ n ) ∂y k o = 0 in Y ∗ , − ∂∂y i n − e ikl ( x, y ) s kl,y ( q n ) + d ij ( x, y ) ∂ ( y n + ψ n ) ∂y j o = 0 in Y ∗ , q n , ψ n Y ∗ − periodics . (33)Introducing ( q i , ψ i ), in the solution of this local problem − ∂∂y j n c ijkl ( x, y ) s kl,y ( q i ) + e kij ( x, y ) ∂ ( y i + ψ i ) ∂y k o = 0 in Y ∗ , − ∂∂y i n − e ikl ( x, y ) s kl,y ( q i ) + d ij ( x, y ) ∂ ( y i + ψ i ) ∂y j o = 0 in Y ∗ , q i , ψ i Y ∗ − periodics . (34)We can rewrite these coefficients of electric tensor in form given as d Hin = Z Y ∗ − e ikl ( x, y ) s kl,y ( q n ) dy + Z Y ∗ d ij ( x, y ) ∂ ( y n + ψ n ) ∂y j dy. (35)The first term of the second integral of precedent expression, is evaluated asfollows Z Y ∗ − e ikl ( x, y ) s kl,y ( q n ) dy = Z Y ∗ − e αkl ( x, y ) s kl,y ( q n ) δ αi dy, = Z Y ∗ − e αkl ( x, y ) s kl,y ( q n ) ∂y i ∂y α dy, = Z Y ∗ − e αkl ( x, y ) s kl,y ( q n ) ∂ψ i ∂y α dy − Z Y ∗ − e αkl ( x, y ) s kl,y ( q n ) ∂ ( y i + ψ i ) ∂y α dy. (36)13sing the variationnal formulation of the second equation of system (33), andchossing a test function ϕ = ψ i , we obtain Z Y ∗ − e αkl ( x, y ) s kl,y ( q n ) ∂ψ i ∂y α dy = Z Y ∗ d αj ( x, y ) ∂ ( y n + ψ n ) ∂y j ∂ψ i ∂y α dy. Let us now consider the second integral in (36). Multiplying the first equationof system (34) by φ n , and integrating by parts, we have Z Y ∗ − e αkl ( x, y ) s kl,y ( q n ) ∂ ( y i + ψ i ) ∂y α dy = − Z Y ∗ c klαβ ( x, y ) s αβ,y ( q i ) s kl,y ( ϕ n ) dy. Finality, we regroup these lasts results, and using the definition (35), we obtain d Hin = Z Y ∗ − e ikl ( x, y ) s kl,y ( q n ) dy + Z Y ∗ d ij ( x, y ) ∂ ( y n + ψ n ) ∂y j dy, = Z Y ∗ d αj ( x, y ) ∂ ( y n + ψ n ) ∂y j ∂ψ i ∂y α dy − Z Y ∗ c klαβ ( x, y ) s αβ,y ( q i ) s kl,y ( q n ) dy + Z Y ∗ d ij ( x, y ) ∂ ( y n + ψ n ) ∂y j dy, = Z Y ∗ d αj ( x, y ) ∂ ( y n + ψ n ) ∂y j ∂ψ i ∂y α dy − Z Y ∗ c klαβ ( x, y ) s αβ,y ( q i ) s kl,y ( q n ) dy + Z Y ∗ d αj ( x, y ) ∂ ( y n + ψ n ) ∂y j ∂y i ∂y α dy, = Z Y ∗ d αj ( x, y ) ∂ ( y n + ψ n ) ∂y j ∂ ( y i + ψ i ) ∂y α dy − Z Y ∗ c klαβ ( x, y ) s αβ,y ( q i ) s kl,y ( q n ) dy. (37)It is clear from that the coefficients of electric tensor is symmetric.Now we are interested in the ellipticity of this tensor, recall d Hin is elliptic, if forall vector X i , we have ∃ α Hd > , d Hin X i X n ≥ α Hd X i X i . We consider the expression (25) of tensor d Hin , we derive d Hin X i X n = Z Y ∗ d in ( x, y ) X i X n dy − Z Y ∗ e ikl ( x, y ) s kl,y ( q n ) X i X n dy + Z Y ∗ d ij ( x, y ) ∂ψ n ∂y j X i X n dy, = Z Y ∗ d in ( x, y ) X i X n dy − Z Y ∗ e ikl ( x, y ) s kl,y ( q n X n ) X i dy + Z Y ∗ d ij ( x, y ) ∂ ( ψ n X n ) ∂y j X i dy, = Z Y ∗ d ij ( x, y ) (cid:16) Q j + ∂ξ∂y j (cid:17) Q i dy − Z Y ∗ e ikl ( x, y ) s kl,y ( ς ) Q i dy, 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 asaddle point of the functionnal G defined by :( v , ψ ) → G ( v , ψ ) G ( v , ψ ) = 12 Z Y ∗ n c ijkl s ij,y ( v ) s kl,y ( v )+2 e kij s ij,y ( v ) (cid:16) Q k + ∂ψ∂y k (cid:17) − d ij (cid:16) Q i + ∂ψ∂y i (cid:17)(cid:16) Q j + ∂ψ∂y j (cid:17)o dy. By definition of the saddle point, we have G ( ξ, ψ ) ≤ G ( ξ, ς ) ≤ G ( v , ς ) ∀ ( v , ψ ) periodic functions.Or G (0 , ς ) = − Z Y ∗ d ij ( x, y )( Q i + ∂ς∂y i )( Q j + ∂ς∂y j ) dy But, if we use the second equation of system (19), we have G (0 , ς ) = − Z Y ∗ d ij ( x, y ) (cid:16) Q j + ∂ς∂y j (cid:17) Q i dy − Z Y ∗ d ij ( x, y ) (cid:16) Q j + ∂ς∂y j (cid:17) ∂ς∂y i dy, = − Z Y ∗ d ij ( x, y ) (cid:16) Q j + ∂ς∂y j (cid:17) Q i dy + 12 Z Y ∗ e ikl ( x, y ) s kl,y ( ς ) X i dy, = − d Hin X i X n ,< . We have the dielectric homoginized tensor which is elliptic.Now we give the results concerning some properties of piezoelectric homogenizedtensor.
Proposition 4
The coefficients of piezoelectric homogenized tensor E H = ( e Hnij ) defined by (23) satisfy: e Hnij = e Hnji
Moreover, we have the identity e Hnij = f Hnij
Proof.
By definition of coefficients e Hnij (using the fact that c ijkl ( x, y ) = c jikl ( x, y ), e kji ( x, y ) = e kij ( x, y )), we have e Hnij = Z Y ∗ n c ijkl ( x, y ) s kl,y ( q n ) + e kij ( x, y ) (cid:16) δ kn + ∂ψ n ∂y k (cid:17)o dy, = Z Y ∗ n c jikl ( x, y ) s kl,y ( q n ) + e kji ( x, y ) (cid:16) δ kn + ∂ψ n ∂y k (cid:17)o dy, = e Hnji . (38)15e can rewrite the coefficients e Hnij , as given as under e Hnij = Z Y ∗ n c ijkl ( x, y ) s kl,y ( q n ) + e kij ( x, y ) (cid:16) δ kn + ∂ψ n ∂y k (cid:17)o dy, = Z Y ∗ n e nij ( x, y ) + e kij ( x, y ) ∂ψ n ∂y k + c ijkl ( x, y ) s kl,y ( q n ) o dy. (39)Same, we can rewrite the coefficients f Hnij , as form givens under f Hnij = Z Y ∗ n e nkl ( x, y ) (cid:16) τ klij + s kl,y ( w ij ) (cid:17) − d nt ( x, y ) ∂ϕ ij ∂y t o dy, = Z Y ∗ n e nij ( x, y ) + e nkl ( x, y ) s kl,y ( w ij ) − d nt ( x, y ) ∂ϕ ij ∂y t o dy. (40)Using the two variationnals formulations corresponding of problems (18) and(19), and chossing the appropriete test functions, we can directly prove as e Hnij = f Hnij .Finallity, using the three last propositions, we can purpose the altarnative formof 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 − div σ H ( u , ϕ ) = θ f in Ω , − div D H ( u , ϕ ) = 0 in Ω , (41) where the boundary conditions u ( x ) = on ∂ Ω ,ϕ ( x ) = 0 on ∂ Ω , (42) σ Hij and D Hi are defined by the homogenized constitutive law σ Hij ( u , ϕ ) = c Hijmh s mh,x ( u ) + e Hnij ∂ϕ∂x n ,D Hi ( u , ϕ ) = − e Himh s mh,x ( u ) + d Hin ∂ϕ∂x n , (43) the homogenized coefficients c Hijkl , e Hnij and d Hij are defined respectively by (22),(23) and (25).
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 the16sual 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 lim ε → Z Ω ψ ( x, xε ) dx = Z Ω Z Y ψ ( x, y ) dx dy (44)Here we recall the Allaire’s lemma Lemma 2 (Allaire [1]) :Let the function ψ ( x, y ) ∈ L (Ω; C ♯ ( Y )) , then ψ ( x, y ) is an admissible test func-tion in the sense of Definition 1. Using this lemma, we obtain the following proposition.
Proposition 5
The two functions s ij,y ( u ( x, y )) and ∂ i,y ϕ ( x, y ) are admissi-ble test functions in the sense of Definition 1. Proof.
By definition we have u ( x, y ) = s mh,x ( u ( x )) w mh ( y ) + ∂ϕ ( x ) ∂x n q n ( y ) ,ϕ ( x, y ) = s mh,x ( u ( x )) ϕ mh ( y ) + ∂ϕ ( x ) ∂x n ψ n ( y ) , we obtain s ij,y ( u ( x, y )) = s mh,x ( u ( x )) s ij,y ( w mh ( y )) + ∂ϕ ( x ) ∂x n s ij,y ( q n ( y )) ,∂ i,y ϕ ( x, y ) = s mh,x ( u ( x )) ∂ i,y ϕ mh ( y ) + ∂ϕ ( x ) ∂x n ∂ i,y ψ n ( y ) . Using Lemma 2, s ij,y ( u ( x, y )) and ∂ i,y ϕ ( x, y ) are the admissible test functionsin the sense of Definition 1. Theorem 2
We have these two strong convergence results ∼ u ε ( x ) − χ ( xε )[ u ( x ) + u ( x, xε )] → strongly in H (Ω) ∼ ϕ ε ( x ) − χ ( xε )[ ϕ ( x ) + ϕ ( x, xε )] → strongly in H (Ω)17 roof. We consider the variational formulation under the following form Z Ω ε { [ c εijkl s kl ( u ε )+ e εkij ∂ k ϕ ε ] s ij ( v )+[ − e εikl s kl ( u ε )+ d εij ∂ j ϕ ε ] ∂ i ψ } dx = Z Ω ε f i ( x ) v i ( x ) dx (45)Chosing v = u ε and ψ = ϕ ε in (45), we obtain Z Ω ε { [ c εijkl s kl ( u ε )+ e εkij ∂ k ϕ ε ] s ij ( u ε )+[ − e εikl s kl ( u ε )+ d εij ∂ j ϕ ε ] ∂ i ϕ ε } dx = Z Ω ε f i ( x ) u εi ( x ) dx By simplification, we get Z Ω ε { c εijkl ( x ) s ij ( u ε )( x ) s kl ( u ε )( x )+ d εij ( x ) ∂ i ϕ ε ( x ) ∂ j ϕ ε ( x ) } dx = Z Ω ε f i ( x ) u εi ( x ) dx (46)By applying (46), we can write Z Ω c εijkl n ∼ s ij ( u ε ) − χ ( xε ) h s ij,x ( u ) + s ij,y ( u ) ion ∼ s kl ( u ε ) − χ ( xε ) h s kl,x ( u ) + s kl,y ( u ) io dx + Z Ω d εij ( x ) n ∼ ∂ i ϕ ε ( x ) − χ ( xε ) h ∂ i,x ϕ + ∂ i,y ϕ ion ∼ ∂ j ϕ ε ( x ) − χ ( xε ) h ∂ j,x ϕ + ∂ j,y ϕ io dx = Z Ω f i ( x ) ∼ u εi ( x ) dx + Z Ω c εijkl ( x ) χ ( xε ) h s ij,x ( u ( x )) + s ij,y ( u ( x, xε )) ih s kl,x ( u ( x )) + s kl,y ( u ( x, xε )) i dx + Z Ω d εij ( x ) χ ( xε ) h ∂ i,x ϕ ( x ) + ∂ i,y ϕ ( x, xε ) ih ∂ j,x ϕ ( x ) + ∂ j,y ϕ ( x, xε ) i dx − Z Ω c εijkl ( x ) χ ( xε ) ∼ s ij ( u ε ) h s kl,x ( u ( x )) + s kl,y ( u ( x, xε )) i dx − Z Ω d εij ( x ) χ ( xε ) ∼ ∂ i ϕ ε ( x ) h ∂ j,x ϕ ( x ) + ∂ j,y ϕ ( x, xε ) i dx. c εijkl ) and dielectric ( d εij ) tensors,we get α c k ∼ s ij ( u ε ) − χ ( xε ) s ij,x ( u ( x )) − χ ( xε ) s ij,y ( u ( x, xε )) k L (Ω) + α d k ∼ ∂ i ϕ ε ( x ) − χ ( xε ) ∂ i,x ϕ ( x ) − χ ( xε ) ∂ i,y ϕ ( x, xε ) k L (Ω) ≤ Z Ω f i ( x ) ∼ u εi ( x ) dx + Z Ω c εijkl ( x ) χ ( xε ) h s ij,x ( u ( x )) + s ij,y ( u ( x, xε )) ih s kl,x ( u ( x )) + s kl,y ( u ( x, xε )) i dx + Z Ω d εij ( x ) χ ( xε ) h ∂ i,x ϕ ( x ) + ∂ i,y ϕ ( x, xε ) ih ∂ j,x ϕ ( x ) + ∂ j,y ϕ ( x, xε ) i dx − Z Ω c εijkl ( x ) χ ( xε ) ∼ s ij ( u ε ) h s kl,x ( u ( x )) + s kl,y ( u ( x, xε )) i dx − Z Ω d εij ( x ) χ ( xε ) ∼ ∂ i ϕ ε ( x ) h ∂ j,x ϕ ( x ) + ∂ j,y ϕ ( x, xε ) i dx. Using the fact that s ij,y ( u ( x, y )) and ∂ i,y ϕ ( x, y ) are the admissible test func-tions and taking the limit in the sense of the two-scale convergence in the secondright-hand side of the inequality, we obtain α c lim ε → k ∼ s ij ( u ε ) − χ ( xε ) { s ij,x ( u ( x )) − s ij,y ( u ( x, xε )) } k L (Ω) + α d lim ε → k ∼ ∂ i ϕ ε ( x ) − χ ( xε ) { ∂ i,x ϕ ( x ) − ∂ i,y ϕ ( x, xε ) } k L (Ω) ≤ Z Ω Z Y ∗ f i ( x ) u i ( x ) dx dy − Z Ω Z Y ∗ c ijkl ( x, y )[ s ij,x ( u ( x )) + s ij,y ( u ( x, y ))][ s kl,x ( u ( x )) + s kl,y ( u ( x, y ))] dx dy − Z Ω Z Y ∗ d ij ( x, y )[ ∂ i,x ϕ ( x ) + ∂ i,y ϕ ( x, y )][ ∂ j,x ϕ ( x ) + ∂ j,y ϕ ( x, y )] dx dy (47)Recalling the form of the two-scale homogenized problem (10)-(11)-(12), weobserve that the right-hand side of the inequality (47) vanishes, so tha, weobtain lim ε → k ∼ s ij ( u ε ) − χ ( xε ) { s ij,x ( u ( x )) − s ij,y ( u ( x, xε )) } k L (Ω) = 0and lim ε → k ∼ ∂ i ϕ ε ( x ) − χ ( xε ) { ∂ i,x ϕ ( x ) − ∂ i,y ϕ ( x, xε ) } k L (Ω) = 0 . Conclusion
In this work, we have given the new convergence results, and the explicite formsof the elastic, piezoelectric and dielectric homogenized coefficients. The two-scale convergence is applied to our problem yields the strong convergence resulton the correctors. This technique of two-scale convergence can handle also otherhomogenization problems, in medium which has periodic structure for examplethe 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]).
Acknowledgment.
This work has been supported in part by the Ministryfor higher education and scientific research of Algeria (University of Oran, De-partement of Mathematics). The author is grateful to Professor BernadetteMiara for helpful discussions.
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