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https://www.eduzhai.net International Journal of Composite Materials 2014, 4(2): 45-51 DOI: 10.5923/j.cmaterials.20140402.01 The Effective Behavior of Laminated Composite Materials in the Case of Debonded Folds Yahya Berrehili Equipe de Modélisation et Simulation Numérique, Université Mohamed 1er, Ecole Nationale des Sciences Appliquées, Oujda, 60000, Maroc Abstract This paper is devoted to the study the effective behavior of laminated composites whose folds are debonded (but still in contact inter them). The aim is to show that the macroscopic behavior of such structures is a generalized behavior. By using the homogenization theory of periodic media, we show that the macroscopic kinematic is described not only by the usual macroscopic displacement field but also another field describing the sliding of the stiff layers with respect to the soft ones. Accordingly, new homogenized tensors and new coupled equilibrium equations appear. Keywords Homogenization, Laminated composite, Debonded folds, Modeling, Behavior 1. Introduction If the results on modeling of the behavior of composite materials are well established using the homogenization theory of periodic media [1], certain situations, as this paper shows, deserve to rethink completely the modeling approach [2] [3]. Indeed, by considering a laminated composite structure in which was developed a damage by debonding, we show that the behavior of such structure is not that of a simple material, where the results are classical and well known [4] [5] [6], but that of a micro-structured media: one must add to the classical macroscopic displacement field, a field of planar vectors, representing the relative sliding of stiff layers with respect to the soft ones. And consequently, new homogenized tensors and new coupled equilibrium equations appear [2] [3]. It is the purpose of this paper. Specifically, the paper is organized as follows. The next section is devoted to the setting of the problem: one consider a laminated composite structure, occupying the open bounded domain Ω of IR3 constituted by a periodic distribution of stiff elastic layers embedded and stacked in the direction e3(see Figure 1) in an elastic matrix(soft layers). In a part noted Ωc, the stiff layers are assumed perfectly bonded to the matrix while in the complementary part noted Ωd, they are assumed to be debonded but still in contact without friction with the matrix. The number of folds n is assumed large enough (so that the microstructure parameter 1/n is small enough [1] [7] [8] [9] [10] [11]). The problem consist in finding the displacement field u0 and the * Corresponding author: yberrehili@ensa.ump.ma (Yahya Berrehili) Published online at https://www.eduzhai.net Copyright © 2014 Scientific & Academic Publishing. All Rights Reserved associated stress field σ0 (limits respectively of un and σn, when n goes to infinity) solutions of the real problem (1)-(4) which can be written in variational form (5)-(7). The third section is devoted to a brief review of the homogenization theory of periodic media and the writing of equations governing the fields u0, u1 and u2 (three first terms of asymptotic expansion of un [3]) whose goal to determine the displacement field limit u0. The fourth section is divided into three sub-sections. The sub-section 4.1 is devoted to the determination of the form of the displacement field u0. This displacement form is given by the expression (15) which valid in the two part Ωc(χd(y3)=0) and Ωc(χd(y3)=1) [2] [3]. In this last part we remark the appearance of a new macroscopic field (noted δ) forgotten in the existent literature (see [4] [12] [13] [14] [15] and [16] [17] [18] [19] [20]), interpreted as the relative sliding between the stiff and soft layers [3]. This is due to the fact that the layers considered, in the part Ωd, are completely debonded. In sub-section 4.2, we simply explicit the variational equations, capable to express u1 in terms of u0. We obtain the classical ones in Ωc, given by (20), and other ones in Ωd, given by (27). The equations obtained in Ωd, contain additional terms that dependent of the derivatives of δ with respect to the macroscopical coordinates x1, x2 and x3. The sub-section 4.3 is devoted to the writing of the homogenized problem. Exploiting the form of displacement field u0 given by (15) and the integral equations, linking the fields u0 and u1, given by (21)-(23) and (28)-(30), we determine the macroscopic problem governing the displacement field u0. This problem is given in variational form by (47) or distribution form by (48). We will remark the appearance of new homogenized tensors K and Σ which are interpreted respectively as the stiffness tensor to the relative deformation of debonded folds(stiff and soft) and the tensor of internal stresses generated in the cell by an extension of 46 Yahya Berrehili et al.: The Effective Behavior of Laminated Composite Materials in the Case of Debonded Folds these folds. And we finally conclude in the last section. Divxσn=0,σn=Aεx(un), εx(un)=1/2(▼un+▼Tun) in Ω\Γdn,(1) 2. Position of the Problem and Notations un=0 sur Γc, σnn=F on Γs (2) [un].n=0, [σn]n=0, on Γcn (3) We work in the framework of linear elasticity and we [un].n=0, [σn]n=0, σnn^n=0 on Γdn (4) consider a laminated composite structure whose natural reference configuration is the open bounded domain Ω of IR3 with a smooth boundary ∂Ω. We denote by (e1, e2, e3) the canonical basis of IR3 and (x1, x2, x3) the coordinates of a point x of Ω. The structure is assumed constituted by two materials: stiff layers qualified of reinforcing and soft layers(matrix) playing the role of binder. The two constituents of the composite structure are assumed elastics, homogeneous and isotropic whose Lamé coefficients are where the last line translates the continuity of normal displacement field un3e3, the continuity of the stress vector σne3= σn13+ σn23+ σn33 and the nullity of the shear (σn13=σn23=0) on the debonded interfaces Γdn. This problem is written in distribution form. We can write it in variational form: it consists in finding un in Cn such that ∫ . Aεx (un ) εx (v)(= x)dx Fn (v), ∀v ∈ Cn (5) Ω / Γdn λ(x) and µ(x). The number of folds n is assumed large enough for that the period of the microstructure 1/n is small enough. In a part Ωc of Ω the stiff layers are assumed perfectly bonded to the matrix while in the complementary part Ωd they are assumed debonded but still in contact with with . = Fn (v) ∫ F (x) v(x)dΓ(x) Γs (6) the matrix. The structure is assumed submitted to an ex= ternal v loading applied on the boundary ∂Ω. Specifically, we fix a part Γc and we apply a surface force density F on the Cn= v3 (v1,v2 ,v3) / ∈ H1(Ω), =v (v1, v2 ) ∈ 0 on Γc H 1(Ω / Γ dn ), complementary part Γs of boundary ∂Ω (see Figure 1). The . (7) body force density is assumed negligible. 3. Application of Homogenization Theory Following the classical two-scale procedure in homogenization theory of periodic media [3] [4], we assume that un can be expanded as follows: un(x)=u0(x,y3)+ 1 n u1(x,y3)+ 1 n2 u2(x,y3)+… (8) Figure 1. The laminated composite and the two unite cells Y and Y\Γ We assume that in the part Ωd, and during the deformation, the stiff layers remain in contact with the matrix and can slip without friction. This expresses then that the normal displacement field is continuous and the shear vanish on the debonded interfaces Γdn of the part Ωd. In Ωc by cons, the displacement and the stress fields are continuous on the bonded interfaces Γcn. Denoting by A(x) the linearized elasticity tensor into a point x, Divx and εx the divergence and the symmetrized gradient with respect to x, the real elastic problem consists in seeking for the couple (un, σn) checking the following static equilibrium equations: where y3=nx3 is the microscopic variable, describing the cell Y or Y\Γ according to x is in Ωc or in Ωd, with Y=[1/2,+1/2] and Γ={-α/2, +α/2} (α denotes the volume fraction of the stiff folds into the matrix). And the ui=(ui1, ui2, ui3), i ≥ 0, are the Y-periodic fields with respect to the variable microscopic y3. By substituting the development postulated (8) of un into the variational problem (5) and by identifying formally the terms of same power of n, we obtain a sequence of interrelated problems whose the unknowns are the fields ui(x,y3), i ≥ 0. The determination of the first term u0(x,y3) in the asymptotic expansion of un(x) provides the effective behavior sought of the microstructure Ω. We write thus only the three first problems of order, n2, n1 and n0. And this sufficient for the determination of u0. (i) At order n2: ∫ . A(y3)εy (u0 ) εy (v)(x,y3)dxdy3 = 0 (9) Ω×Y (ii) At order n1: ∫ . A(y3)(εy (u1) + εx (u0 )) εy (v)(x,y3)dxdy3 = 0 Ω×Y International Journal of Composite Materials 2014, 4(2): 45-51 47 (10) respectively to Ωd and Yr defined by: (iii) At order n0: ∫ . A(y3)(εy (u2 ) + εx (u1)) εy (v)(x,y3)dxdy3 + 1 χd(x) = 0 si x ∈Ωd , si x ∈Ωc 1 χr(y3) = 0 si y3 ∈Yr si y3 ∈ym (16) ∫ . Ω×Y A(y3)(εy (u1) + εx (u0 )) εx (v)(x,y3 )dxdy3 = F0 (v) (11) disTphlaecemmeanctruo0scisopthicen strain field given by: associated to the Ω×Y εx(u0)(x)= ε(u)(x)+χd(x)χr(y3)ε(δ)(x) ∀ x∈Ω (17) where . ∫ F0(v) = F (x) v(x,y3)(x)d Γ (x) (12) Γs ×Y Remark: The integrals on ΩxY must be understood as the sum of integrals on ΩcxY and on Ωdx(Y\Γ). where ε(δ) denotes the strain tensor of the sliding field δ given by: ε(v) = 1 2 2δ 1,1 − sym (δ1,2 2δ − + δ2,1) 2, 2 0 δ δ1, 3 2, 3 . (18) 4. Resolution and Asymptotic Results We agreed here the simplistic notation φ,α for α=1,2,3, as derivative of the scalar field φ(x) with respect to xα. 4.1. Form of the Displacement Field u0 The equation (9) allows having the form of the unknown displacement field u0. Indeed, by choosing as function test v(x,y3)=u0(x,y3) in the equation (9) and owing the positivity of the elasticity tensor A, we deduce that the microscopic strain tensor of u0 vanish, i.e. εy(u0)(x,y3)=0 in Y or in Y\Γ according to x is in Ωc or in Ωd. Therefore u0, considered as function of y3, is a rigid displacement. Therefore ● The rigid displacements of the cell Y, associated at the bonded part Ωc, are translations because Y is a connected part of IR. We find the classical and known result: u0(x,y3)=u(x) if x∈Ωc. (13) 4.2. Expression of u1 in Term of u0 Assuming for the moment that the fields u and δ are known, the equation (10), connecting u1 to u0, will allow us to determine u1 in terms of the gradient of u and δ. Indeed, taking into account (15) and coefficients of the elasticity tensor A, Aijkl, 0 ≤ i, j, k, l ≤ 3, given by Aijkl(y2)=λ(y2)δijδkl+µ(y2)(δikδjl+δilδjk), (19) (where δij denote the Kronecker symbol equal to 1 if i=j and 0 otherwise) and by choosing functions tests well defined, of the form v(x,y3)=v(x)φ(y3), we can express u1(x,y3), in terms of u and δ. In always distinguishing Ωc and Ωd, we obtain: u being the classical displacement field in the (i) In Ωc, the equation (10) becomes: ∫ homogenization theory of periodic media. ● The rigid displacements of the cell Y\Γ, associate= d at 0 the debonded part Ωd, are also translations, but since Y\Γ is 1 2Y μ( ∂u11 ∂y3 + 2ε13 (u )) ∂φ1 ∂y3 dy3 the union of two connected parts Ym and Yr, each part has its own translation. So, we get a relative translation of Yr compared to Ym. And the displacement field u0 can be written then, for x∈Ωd, as follow: u0 ( x, y3 ) = u( u( x) x) + if δ( y3 x) ∈Ym if y3 ∈Yr (14) with δ(x)= δ1(x)e1+ δ2(x)e2. It should be noted that we find once again the classical displacement field u which interpreted as displacement field of the matrix. By cons there is birth of a new field δ forgotten in the existent literature [12] [13] [14] [15]. It modeling the relative sliding of stiff layers with respect to the soft ones. ∫ + 1 2Y μ( ∂u12 ∂y3 + 2ε 23 (u )) ∂φ2 ∂y3 dy3 ∫ + Y λ(ε11(u) + ε22 (u)) ∂φ3 ∂y3 dy3 ∫+ ((λ + 2μ) ∂u13 Y ∂y3 + (λ + 2µ )ε33 (u)) ∂φ3 ∂y3 dy3 (20) This is obtained in choosing v(x)∈D(Ωc) and φ∈H1(Y) with φ Y-periodic. D(Ωc) is the space of infinitely differentiable functions with compact support in Ωc. We then deduce from (20) that Remark: We can have a single expression (instead of two (13) and (14)) of the displacement field u0 valid in Ω= Ωc∪Ωd, defined as follow: μ( ∂u11 ∂y3 + 2ε13 (u )) = 2C1 (21) u0(x,y3)= u(x)+χd(x)χr(y3)δ(x) (15) where χd and χr are the characteristic functions, associated μ( ∂u12 ∂y3 + 2ε 23 (u )) = 2C2 (22) 48 Yahya Berrehili et al.: The Effective Behavior of Laminated Composite Materials in the Case of Debonded Folds (λ + 2µ )( ∂u13 ∂y3 + ε33 (u )) + λ(ε11(u) + ε22 (u)) = C3 (23) ∫ where C'3 is given by ( since Y ∂u13 ∂y3 dy3 =0) ∫ where C1, defined. C2 and Indeed, C3 are constants denoting by withre=specf= t(tyo3y)3dayn3d wtehlel C3' < < λ 1 / / (λ (λ + + 2μ) 2μ) > > (ε11 (u ) + ε 22 (u)) Y ∫ mean value of f over the cell Y we get, since Y ∂u1i ∂y3 dy3 = 0 , 1 ≤ i ≤ 3 (because of the Y-periodicity): C1 = < 1 1/μ > ε13 (u ) (24) C2 = 1 < 1/μ > ε 23 (u ) (25) = C3 < λ/(λ 1/(μ + 2μ) + 2μ > (ε11(u) + ε22 (u)) + < 1 1/(μ + 2μ > ε33 (u) (26) + < 1 1/(μ + 2μ > ε33 (u) + < 1/(λ 1 + 2μ) > ε33 (u ) + < λχr /(λ + 2μ) > < 1/(λ + 2μ) > (ε11 (δ ) + ε 22 (δ)) (31) 5. Macroscopic Homogenized Problem From the equations (9) and (10) we found the form of displacement field u0 in terms of u and δ (given by (15)) and we obtained equations connecting this fields to u1 (given by (21)-(23) and (28)-(30)). Now, using equation (11), we should obtain a variational equation governing only the fields u and δ. Let consider for that, a particular test field v(x, y3) verifying εy(v) = 0, i.e. (ii) In Ωd we obtain, in the same manner, a variational equation which is valid for any Y-periodic field φ∈H1 (Y\Γ) verifying [φ(±α/2)]e3 ≡ [ φ3(±α/2)]=0, i.e. ∫ =0 1 2Y μ( ∂u11 ∂y3 + 2ε13 (u ) + 2χ r (y3 )ε13 (δ )) ∂φ1 ∂y3 dy3 ∫ + 1 2Y μ( ∂u12 ∂y3 + 2ε 23 (u ) + 2χ r (y3 )ε 23 (δ )) ∂φ2 ∂y3 dy3 ∫+ (λ Y + 2μ)( ∂u13 ∂y3 + ε33 (u )) ∂φ3 ∂y3 dy3 ∫ + Y λ(ε11(u) + ε22 (u)) ∂φ3 ∂y3 dy3 ∫ + Y λχ r (y3 )(ε11(δ ) + ε22 (δ )) ∂φ3 ∂y3 dy3 (27) v(x,y3)=u*(x)+χd(x)χr(y3)δ*(x), (32) with δ*(x)=δ*1(x)e1+δ*2(x)e2. The and the displacement of order 2, equation u2(x,y3), (11) is simplified disappear in this equation and we can write it: ∫ σ •e(x,y= 3)dxdy3 f (u*, δ*) ≡ f 0 (v), (33) Ω×Y with = f (u*,δ *) ∫ F (x)u*(x)dΓ(x) Γ s ∫ +α (F1(x)δ1*(x) + F2 (x)δ*2 (x)) dΓ (x) (34) Γs where we have posed, σ ≡ Aεx(v)(x,y3)=Aε(u*)+χd(x)χr(y3)Αε(δ*)(x) (35) and from which we deduce that(because of the possibility of a tangential discontinuity of the displacement and the nullity e ≡ εy(u1)+εx(u0)= εy(u1)+ ε(u)+χd(x)χr(y3)ε(δ) (36) of the shear): Multiply these two last expressions of tensors σ and e and 0 =∂u11 ∂y3 + 2ε13 (u ) + 2χ r (y3)ε13(δ) (28) 0 =∂u12 ∂y3 + 2ε23(u) + 2χ r (y3)ε23(δ) (29) C3' = (λ + 2μ)( ∂u13 ∂y3 + ε33 (u )) + λ(ε11(u) + ε22 (u)) +λχ r (y3)(ε11(δ) + ε22 (δ)) (30) integrating the result obtained over the cell Y, taking into account the relations connecting u1 to u and δ(given by (21)-(23) in Ωc and (28)-(30) in Ωd) we obtain, in distinguishing always Ωc and Ωd, the following results: (i) In Ωc, ∫ < σ.e >≡ σ.e(x,y3)dy3 = Acε(u)ε(u*)(x) (37) Y Ac denotes the homogenized stiffness tensor of the bonded laminated composite part. The macroscopic relation stress-strain is given by the following matrix representation: International Journal of Composite Materials 2014, 4(2): 45-51 49 σ11 σ 22 σ 33 σ 23 σ13 σ12 = AcT λcT λ3c AcT λ3c A3c − Sym − 0 0 0 2μ c 3 0 0 0 0 2μ c 3 0 0 0 0 0 2μTc ε11 ε 22 ε 33 ε 23 ε13 ε12 (38) with, ATc =< 4μ(λ+ μ) / (λ+ 2μ) > + < λ/ (λ+ 2μ) >2 < 1/ (λ+ 2μ) > A3c = < 1 / 1 (λ+ 2 μ) > λTc =< 2 λμ/ (λ+ 2 μ) > + < λ/ (λ+ 2μ) >2 < 1/ (λ+ 2μ) > (39) λ3c = < < λ/ 1/ (λ+ (λ+ 2 2 μ) μ) > > μcT =< μ >, μ3c = < 1 1/ μ > , where 2μ=cT AcT − λcT . Five coefficients are independent in the expression of tensor Ac, the homogenized structure is thus transversely isotropic. (ii) In Ωd, After some simplifications which we give not the details here (you can see the details in [3]), we get: ∫ < σ.e >≡ σ • e(x,y3)dy3 (40) Y/Γ =Adε(u)ε(u*)+Kε(δ).ε (δ*)+Σε(u).ε(δ*)+Σε(δ*)ε(u) where, ● Ad denotes the homogenized stiffness tensor of the debonded laminated part, but without deformation of the stiff layers, given by AdT λdT λ3d 000 AdT λ3d 0 0 0 Ad = A3d − Sym − 000 00 0 (41) 0 0 2μ d T with, AdT = ATc , A3d = A3c , λTd = λTc , λ3d = λ3c , µTd = µTc (42) with, 2μ=dT AdT − λdT . Four independent coefficients only, the terms Ad1313 and Ad2323 are null because of the nullity of shear on the debonded interfaces. The homogenized structure obtained is also transversely isotropic. ● K is interpreted as the rigidity tensor to the relative plane deformation of the debonded stiff layers (but in contact with the matrix). It is given by K T K1 KT 0 0 0 0 0 0 0 0 K = 00 0 0 (43) 000 − Sym − 0 0 4K 2 with, KT = < 4μ(λ+ μ) χr / (λ+ 2μ) > + < λχ r / (λ+ 2 μ) > < < λ/ 1/ (λ+ (λ+ 2 μ) 2 μ) > > A3c = < 1 / 1 (λ+ 2 μ) > K1 = < 2 λμχr / (λ+ 2μ) > + < λχ r / (λ+ 2 μ) > < λ/ <1/ (λ+ (λ+ 2 μ) 2 μ) > > (44) + < λχ r / (λ+ 2 μ) > < < λ/ 1/ (λ+ (λ+ 2 μ) 2 μ) > > K2 = < μχ r > with KT = K1 + 2K2 ● Σ is interpreted as the stress tensor resulting of internal stresses generated in the cell by an internal extension of the debonded stiff layers (but in contact with the matrix). It is given by ΣT Σ1 Σ3 0 0 0 ΣT Σ3 0 0 0 Σ = 0000 (45) 0 0 0 − Sym − 0 0 2Σ2 50 Yahya Berrehili et al.: The Effective Behavior of Laminated Composite Materials in the Case of Debonded Folds ΣT =< 4 μ(λ+ μ) χr / (λ+ 2 μ) > + < λχ r / (λ+ 2 μ) > < < λ/ 1/ (λ+ (λ+ 2 μ) 2 μ) > > Σ1 =< 2 λμχ r / (λ+ 2 μ) > + < λχ r / (λ+ 2 μ) > < < λ/ 1/ (λ+ (λ+ 2 μ) 2 μ) > > Σ2 = < 2μχr > Σ3 = < λ < χf 1/ / (λ+ 2μ) (λ+ 2μ) > > with ΣT = Σ1 + Σ2 (46) Remark: One can show that the tensors, Ac and Ad, are symmetric and definite positive. And Ac is greater than Ad in sense of quadratic forms [3] [21]. K is also symmetric. Σ by cons is not symmetrical: as operator, it acts on two types of spaces of test functions, associated at fields u and δ [3]. From (33), (37) and (40), we obtain finally that the macroscopic displacement fields u and δ are solutions of the following variational effective problem: ∫ ∫ Adε(u)ε(u*) + Kε(δ).ε(δ*) Ω d Ω d ∫ ∫ + Σε(u).ε(δ*)+ Σε(δ*)ε (u) = F(u*) + G(δ*) (47) Ω d Ω d for all displacement fields, u* and δ*, kinematically admissible. Remark: The expressions of K and Σ show that only the derivatives, with respect to x1 and x2, of sliding δ=δ1e1+δ2e2 appear in the variational formulation (47). δ is thus seeking for in a space of functions f=(f1,f2) square integrable over Ωd, and whose only the derivatives fα,β, 1≤α,β≤2, square integrable on Ωd. But this pose problem of verification of the boundary conditions, since such fields does not admit necessarily a trace on the boundary. In fact, we can define f at a point x of a surface Γ, provided that the components n1 or n2 of the normal n of Γ at this point is nonzero. Thus we will write (n1+n2)δ = 0 on Γ and on ∂Ωc∩∂Ωd. One must note also that there is a coupling, via the stress tensor Σ, between the displacement field of stiff layers δ and the displacement field of the matrix u. Let us write now the homogenized problem which deduced from the variational problem (47) above. It consists to finding a displacement field u and a stress field σ, such that DivxAcε(u) = 0 in Ωc, Divx [Adε(u)+Σε(δ)] = 0 in Ωd, Divx [Kε(δ)+Σε(u)] = 0 in Ωd, [Kε(δ)+Σε(u)]n = α(F1e1+F2e2) on Γs∩∂Ωd, [Adε(u)+Σε(δ)]n = F on Γs∩∂Ωd, (48) ((Ac-Ad) ε(u) +Σε(δ))n=0 on ∂Ωc∩∂Ωd, Acε(u)n = F on Γs∩∂Ωd, u=0 on Γc, (n1+n2)δ=0 on Γc∪(∂Ωc∩∂Ωd). The three first equilibrium equations must be understood in the sense of distributions when the loading is not sufficiently smooth. The two first ones are three-dimensional equations valid respectively in Ωc and Ωd while the third one is a bidimensional equations family of plane type, indexed by x3( Kε(u) representing the normal force). We see that in the second equation the term Σε(δ) play the role of a pre-stressed field of the medium, while in the third equation, the term Σε(u) play the role of a pre-stressed field of a planar medium. This system of equations is completed by boundary conditions that we deduce also from (47). 6. Conclusions The three-dimensional study made on the effective behavior of a laminated composite material, whose stiff folds are perfectly bonded to the matrix in a part of this structure and debonded in the complementary part but still in contact with the matrix, shows that: ● In the perfectly bonded part, the results obtained are the classical properties of homogenization theory which stands that the leading term of the asymptotic displacement field expansion given by (13) does not depend on the microscopic coordinates, and in the homogenized problem appear the classical homogenized stiffness tensor given by (38)(39). However, these properties hold true only when the folds are perfectly bonded to the matrix. ● By cons in the debonded complementary part, the result obtained differs from the usual property of the homogenization theory. Indeed, because of debonding of the stiff folds from the matrix, the leading term of the asymptotic displacement field expansion depends here on the microscopic coordinates. Moreover a new macroscopic vector field enters in the effective kinematic of the laminated composite. Specifically, we obtain a classical vector field representing the macroscopic displacement of the matrix and an additional vector field representing the relative slip of the stiff layers with respect to the matrix given by (14). And the homogenized problem (48) obtained is not classic. It contains new homogenized tensors coupling those two vector fields, given by (43)(44) and (45)(46), ignored in the existent literature. 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