Asymptotic analysis of an elastic material reinforced with thin fractal strips

  • Received: 01 February 2021 Revised: 01 August 2021
  • 35B40, 28A80, 35J20

  • We study the asymptotic behavior of a three-dimensional elastic material reinforced with highly contrasted thin vertical strips constructed on horizontal iterated Sierpinski gasket curves. We use Γ-convergence methods in order to study the asymptotic behavior of the composite as the thickness of the strips vanishes, their Lamé constants tend to infinity, and the sequence of the iterated curves converges to the Sierpinski gasket in the Hausdorff metric. We derive the effective energy of the composite. This energy contains new degrees of freedom implying a nonlocal effect associated with thin boundary layer phenomena taking place near the fractal strips and a singular energy term supported on the Sierpinski gasket.

    Citation: Mustapha El Jarroudi, Youness Filali, Aadil Lahrouz, Mustapha Er-Riani, Adel Settati. Asymptotic analysis of an elastic material reinforced with thin fractal strips[J]. Networks and Heterogeneous Media, 2022, 17(1): 47-72. doi: 10.3934/nhm.2021023

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  • We study the asymptotic behavior of a three-dimensional elastic material reinforced with highly contrasted thin vertical strips constructed on horizontal iterated Sierpinski gasket curves. We use Γ-convergence methods in order to study the asymptotic behavior of the composite as the thickness of the strips vanishes, their Lamé constants tend to infinity, and the sequence of the iterated curves converges to the Sierpinski gasket in the Hausdorff metric. We derive the effective energy of the composite. This energy contains new degrees of freedom implying a nonlocal effect associated with thin boundary layer phenomena taking place near the fractal strips and a singular energy term supported on the Sierpinski gasket.



    A Reinforced material is a composite building material consisting of two or more materials with different properties. The main objective of studies of reinforced materials is the prediction of their macroscopic behavior from the properties of their individual components as well as from their microstructural characteristics.

    The theory of ideal fiber-reinforced composites was initiated by Adkins and Rivlin [4] who studied the deformation of a structure reinforced with thin, flexible and inextensible cords, which lie parallel and close together in smooth surfaces. This theory was further developed by the authors in [44], [1], [2], [3], [45].

    The homogenization of elastic materials reinforced with highly contrasted inclusions has been considered by several authors in the two last decades (see for instance [6], [10], [17], [21], [18], and the references therein). The main result is that the materials obtained by the homogenization procedure have new elastic properties.

    The homogenization of structures reinforced with fractal inclusions has been considered by various authors, among which [39], [31], [40], [41], [12], [42], [13], and [14]. Lancia, Mosco and Vivaldi studied in [31] the homogenization of transmission problems across highly conductive layers of iterated fractal curves. In [40], Mosco and Vivaldi dealt with the asymptotic behavior of a two-dimensional membrane reinforced with thin polygonal strips of large conductivity surrounding a pre-fractal curve obtained after n-iterations of the contractive similarities of the Sierpinski gasket. In [39], they considered an analogous problem with the Koch curve. The same authors considered in [41] a two-dimensional domain reinforced by an increasing number of thin conductive fibers developing a fractal geometry and studied the spectral asymptotic properties of conductive layered-thin-fibers of fractal nature in [42].

    The homogenization of insulating fractal surfaces of Koch type approximated by three-dimensional insulating layers has been performed by Capitanelli et al. in [12], [13], and [14]. Due to the physical characteristics of the inclusions, singular energy forms containing fractal energies are obtained in these articles as the limit of non-singular full-dimensional energies. On the other hand, the effective properties of elastic materials fixed on rigid thin self-similar micro-inclusions disposed along two and three dimensional Sierpinski carpet fractals have been recently obtained in [20].

    In the present work, we consider the deformation of a three-dimensional elastic material reinforced with highly contrasted thin vertical strips constructed on horizontal iterated Sierpinski gasket curves. Our main purpose is to describe the macroscopic behavior of the composite as the width of the strips tends to zero, their material coefficients tend to infinity, and the sequence of the iterated Sierpinski gasket curves converges to the Sierpinski gasket in the Hausdorff metric.

    The asymptotic analysis of problems of this kind was previousely studied in [11], [26], [9], and [5], where the authors considered media comprising low dimensional thin inclusions or thin layers of higher conductivity or higher rigidity. The limit problems consist in second order transmission problems. Problems involving thin highly conductive fractal inclusions have been addressed in a series of papers (see for instance [39], [31], [12], [14], and [19]). The obtained mathematical models are elliptic or parabolic boundary value problems involving transmission conditions of order two on the interfaces. The homogenization of three-dimensional elastic materials reinforced by highly rigid fibers with variable cross-section, which may have fractal geometry, has been carried out in [21]. The authors showed that the geometrical changes induced by the oscillations along the fiber-cross-sections can provide jumps of displacement fields or stress fields on interfaces, including fractal ones, due to local concentrations of elastic rigidities. Note that the numerical approximation of second order transmission problems across iterated fractal interfaces has been considered in some few papers among which [32] and [15].

    Let us first consider the points A1=(0,0), A2=(1,0) and A3=(1/2,3/2) of the xy-plane. Let V0={A1A2A3} be the set of vertices of the equilateral triangle A1A2A3 of side one. We define inductively

    Vh+1=Vh(2hA2+Vh)(2hA3+Vh). (1)

    Let us set

    V=hNVh. (2)

    The Sierpinski gasket, which is denoted here by Σ, is then defined (see for instance [30]) as the closure of the set V, that is,

    Σ=¯V. (3)

    We define the graph Σh=(Vh,Sh), where Sh is the set of edges [p,q]; p,qVh, such that |pq|=2h, where |pq| is the Euclidian distance between p and q. The graph Σh is then the standard aproximation of the Sierpinski gasket, which means that the sequence (Σh)h converges, as h tends to , to the Sierpinski gasket Σ in the Hausdorff metric.

    The edges which belong to Sh can be rearranged as Skh; k=1,2,...,Nh, where Nh=3h+1.

    Let ω be a bounded domain in R2 with Lipschitz continuous boundary ω such that Σ¯ω and

    Σω=V0. (4)

    Let (εh)hN be a sequence of positive numbers, such that

    limhεh2h=0. (5)

    We define

    Tkh=(ωSkh)×(εh,εh) (6)

    and set

    Th=kIhT,kh, (7)

    where Ih={1,2,...,Nh}. Denoting |Th| the 2-dimensional measure of Th, one can see that

    |Th|=εh3h+12h. (8)

    Let Ω=ω×(1,1). We suppose that ΩTh is the reference configuration of a linear, homogeneous and isotropic elastic material with Lamé coefficients μ>0 and λ0. This means that the deformation tensor e(u)=(eij(u))i,j=1,2,3, with eij(u)=12(uixj+ujxi) for some displacement u, is linked to the stress tensor σ(u)=(σij(u))i,j=1,2,3 through Hooke's law

    σij(u)=λemm(u)δij+2μeij(u) ; i,j=1,2,3, (9)

    where the summation convention with respect to repeated indices has been used and will be used in the sequel, and δij denotes Kronecker's symbol. We suppose that Th is the reference configuration of a linear, homogeneous and isotropic elastic material with σh(u)=(σhij(u))i,j=1,2,3:

    σhij(u)=λhemm(u)δij+2μheij(u) ; i,j=1,2,3,

    with

    λh=ηhλ0 and μh=ηhμ0, (10)

    where λ0 and μ0 are positive constants and

    ηh=1εh(56)h. (11)

    The special scaling (10) and (11) of the Lamé -coefficients depend on the structural constants of Th. The choice of ηh is dictated by the lower bound inequality of assertion 3 of Proposition 6, which will play a crucial role in the asymptotic behavior of the energy Fh.

    We suppose that a perfect adhesion occurs between ΩTh and Th along their common interfaces. We suppose that the material in Ω is submitted to volumic forces with density fL2 (Ω,R3) and is held fixed on Ω. We define the energy functional Fh on L2(Ω,R3) through

    Fh(u)={ΩThσij(u)eij(u)dx+Thσhij(u)eij(u)dsdx3                  if  uH10(Ω,R3)H1(Th,R3),+             otherwise, (12)

    where ds is the one-dimensional Lebesgue measure on the line segments Skh; k=1,2,...,Nh. The equilibrium state in Ω is described by the minimization problem

    minuL2(Ω,R3)L2(Th,R3){Fh(u)2Ωf.udx}. (13)

    We use Γ-convergence methods (see for instance [5] and [16]) in order to describe the asymptotic behavior of problem (13) as h goes to . According to the critical term

    γ=limh(3h+12hlnεh), (14)

    which is associated with the size of the boundary layers taking place in the neighbourhoods of the fractal strips, we prove that if γ(0,+) then the effective energy of the composite is given by

    F(u,v)={Ωσij(u)eij(u)dx+μ0ΣdLΣ(¯v)+πμγHd(Σ)(ln2)2ΣA(s)(uv).(uv)dHd(s)               if  (u,v)H10(Ω,R3)×DΣ,E×L2Hd(Σ),+         otherwise, (15)

    where ¯v=(v1,v2), LΣ(¯v) is a quadratic measure-valued gradient form supported on Σ (see Proposition 1 in the next Section), Hd is the d-dimensional Hausdorff measure; d being the fractal dimension of Σ with

    d=ln3/ln2, (16)

    DΣ,E is the domain of the energy supported on the fractal Σ (see (27) in the next Section), and

    A(s)={Diag(1,2(1+κ),2(1+κ)) if n(s)=±(0,1),(7+κ4(1+κ)3(κ1)4(1+κ)03(κ1)4(1+κ)3κ+54(1+κ)0002(1+κ))if n(s)=±(32,12),(7+κ4(1+κ)3(1κ)4(1+κ)03(1κ)4(1+κ)3κ+54(1+κ)0002(1+κ))if n(s)=±(32,12), (17)

    where κ=3μ+λμ+λ and n(s) is the unit normal on sΣ.

    The effective energy (15) contains new degrees of freedom implying nonlocal effects associated with thin boundary layer phenomena taking place near the fractal strips and a singular energy term supported on the Sierpinski gasket Σ. The equilibrium of the fractal Σ is asymptotically described by a generalized Laplace equation which is related to the discontinuity of the effective stresses through the following relations (see Corollary 1):

    { [σα3|x3=0]Σ=πμγHd(Σ)(ln2)2Aαβ(s)(UβVβ)Hd on Σ,πμγ(ln2)2Aαβ(s)(UβVβ)=μ0ΔΣVα in Σα,β=1,2 (18)

    where ΔΣ is the Laplace operator on the Sierpinski gasket, that is, the second order operator in L2Hd(Σ,R2) defined by the form EΣ in Lemma 2.1 in the next Section under the Dirichlet condition Vα=0 on V0=Σω; α=1,2, μ0 is the effective shear modulus of the material occupying the fractal Σ,

    [σα3|x3=0]Σ=σα3|Σ×{0+}σα3|Σ×{0}α=1,2, (19)

    is the jump of σα3|x3=0 on ΣTm; {Tm}mN being the network of the interiors of the triangles which are contained in the Sierpinski gasket Σ (see Figure 2).

    Figure 1. 

    The graph Σh for h=0,1,2,3

    .
    Figure 2. 

    The network {Tm}mN where σα3|Σ×{0+} is the outward normal stress on ΣTm and σα3|Σ×{0} is the inward normal stress

    .

    If γ=+ then, for every (u,v)H10(Ω,R3)×DΣ,E×L2Hd(Σ), F(u,v)<u=v on Σ. In this case the energy supported on the structure is given by

    F(u)={Ωσij(u)eij(u)dx+μ0ΣdLΣ(¯u)              if  uH10(Ω,R3)(DΣ,E×L2Hd(Σ))+         otherwise. (20)

    If γ=0 the displacements u and v are independent. In this case the effective energy of the structure turns out to be

    F0(u)={Ωσij(u)eij(u)dx if  uH10(Ω,R3)+                    otherwise. (21)

    The paper is organized as follows: in Section 2 we introduce the energy form and the notion of a measure-valued local energy on the Sierpinski gasket Σ. Section 3 is devoted to compactness results which is useful for the proof of the main result. In Section 4 we formulate the main result of this work. Section 5 is consacred to the proof of the main result. This proof is developed in 3 Subsections: in the first Subsection we study the boundary layers at the interface matrix/strips, in the second Subsection we establish the first condition of the Γ-convergence property, and in the third Subsection we prove the second condition of the Γ -convergence property.

    In this Section we introduce the energy form and the notion of a measure-valued local energy (or Lagrangian) on the Sierpinski gasket. For the definition and properties of Dirichlet forms and measure energies we refer to [24], [35], and [37].

    For any function w:VR2 we define

    EhΣ(w)=(53)hp,qVh|pq|=2h|w(p)w(q)|2. (22)

    Let us define the energy

    EΣ(z)=limhEhΣ(z), (23)

    with domain D={z:VR2:EΣ(z)<}. Every function zD can be uniquely extended to be an element of C(Σ,R2), still denoted z. Let us set

    D={zC(Σ,R2):EΣ(z)<}, (24)

    where EΣ(z)=EΣ(zV). Then DC(Σ,R2)L2Hd(Σ,R2). We define the space DE as

    DE=¯D.DE, (25)

    where .DE is the intrinsic norm

    zDE={EΣ(z)+z2L2Hd(Σ,R2)}1/2. (26)

    The space DE is injected in L2Hd(Σ,R2) and is an Hilbert space with the scalar product associated to the norm (26).Let us now define the space

    DΣ,E={zDE:z(A1)=z(A2)=z(A3)=0}. (27)

    We denote EΣ(.,.) the bilinear form defined on DΣ,E×DΣ,E by

    EΣ(w,z)=12(EΣ(w+z)EΣ(w)EΣ(z))w,zDΣ,E. (28)

    One can see that

    EΣ(w,z)=limhEhΣ(w,z), (29)

    where

    EhΣ(w,z)=(53)hp,qVh|pq|=2h(w(p)w(q)).(z(p)z(q)). (30)

    The form EΣ(.,.) is a closed Dirichlet form in the Hilbert space L2Hd(Σ,R2) and, according to [25,Theorem 4.1], EΣ(.,.) is a local regular Dirichlet form in L2Hd(Σ,R2), which means that

    1. (local property) w,zDΣ,E with supp[w] and supp[z] are disjoint compact sets EΣ(w,z)=0,

    2. (regularity) DΣ,EC0(Σ,R2) is dense both in C0(Σ,R2) (the space of functions of C(Σ,R2) with compact support) with respect to the uniform norm and in DΣ,E with respect to the intrinsic norm (26).

    The second property implies that DΣ,E is not trivial (that is DΣ,E is not made by only the constant functions). Moreover, every function of DΣ,E possesses a continuous representative. Indeed, according to [36,Theorem 6.3. and example 71], the space DΣ,E is continuously embedded in the space Cβ(Σ,R2) of Hölder continuous functions with β=ln53/ln4.

    Now, applying [29,Chap. 6], we have the following result:

    Lemma 2.1. There exists a unique self-adjoint operator ΔΣ on L2Hd(Σ,R2) with domain

    DΔΣ={w=(w1w2)L2Hd(Σ,R2):ΔΣw=(ΔΣw1ΔΣw2)L2Hd(Σ,R2)}DΣ,E

    dense in L2Hd(Σ,R2), such that, for every wDΔΣ and zDΣ,E,

    EΣ(w,z)=Σ(ΔΣw).zdHdHd(Σ).

    Let us consider the sequence (νh)h of measures defined by

    νh=1Card(Vh)pVhδp, (31)

    where Card(Vh))=3h+1+32 is the number of verticies of Vh and δp is the Dirac measure at the point p. We have the following result:

    Lemma 2.2. The sequence (νh)h weakly converges in C(Σ) to the measure

    ν=1Σ(s)dHd(s)Hd(Σ),

    where C(Σ) is the topological dual of the space C(Σ).

    Proof. Let φC(Σ). Then, according to the ergodicity result of [22,Theorem 6.1],

    limhΣφ(x)dνh=limhpVhφ(p)Card(Vh)=1Hd(Σ)Σφ(s,0)dHd(s).

    We note that the approximating form EhΣ(.,.) can be written as

    EhΣ(w,z)=Σhw.hz dνh, (32)

    where νh is the measure defined in (31) and

    hw.hz(p)=q: |pq|=2h(w(p)w(q))|pq|ϰ/2.(z(p)z(q))|pq|ϰ/2,

    where ϰ is the unique positive number for which the sequence (EhΣ(.,.))h has a non trivial limit (see [38] for more details). We note that, according to equality (22), ϰ=ln53/ln2. We have the following result:

    Proposition 1. For every w,zDΣ,E, the sequence of measures (LhΣ(w,z))h defined by

    LhΣ(w,z)(A)=AΣhw.hzdνh,AΣ,

    weakly converges in the topological dual C(Σ,R2) of the space C(Σ,R2) to a signed finite Radon measure LΣ(w,z) on Σ, called Lagrangian measure on Σ. Moreover,

    EΣ(w,z)=ΣdLΣ(w,z),w,zDΣ,E.

    Proof. The proof follows the lines of the proof of [23,Proposition 2.3] for the von Koch snowflake. Let us set, for every wDΣ,E, LhΣ(w)=LhΣ(w,w). We deduce from (23), (29), and (32) that (LhΣ(w)(Σ))h is a uniformly bounded sequence. Then, observing that, for every wDΣ,E and every φe1DΣ,EC(Σ,R2), with e1=(1,0),

    ΣφdLhΣ(w)=EhΣ(φw,w)12EhΣ(φe1,|w|2e1), (33)

    we deduce, taking into account the regularity of the form EΣ(.,.), that

    limhΣφdLhΣ(w)=EΣ(φw,w)12EΣ(φe1,|w|2e1). (34)

    On the other hand, according to [33,Proposition 1.4.1], the energy form EΣ(w), which is a Dirichlet form of diffusion type, admits the following integral representation:

    EΣ(w)=ΣdLΣ(w), (35)

    where LΣ(w) is a positive Radon measure which is uniquely determined by the relation

    ΣφdLΣ(w)=EΣ(φw,w)12EΣ(φe1,|w|2e1)φDΣ,EC(Σ).

    Thus, combining with (34), the sequence (LhΣ(w))h converges in the sense of measures to the measure LΣ(w). Now, observing that

    LhΣ(w,z)=12(LhΣ(w+z)LhΣ(w)LhΣ(z)),

    we deduce that the sequence (LhΣ(w,z))h weakly converges in C(Σ,R2) to the measure LΣ(w,z).

    In this Section we establish the compactness results which is very useful for the proof of the main homogenization result.

    Lemma 3.1. For every sequence (uh)h; uhH10(Ω,R3)H1(Th,R3), such that suphFh(uh)<+, we have

    1. suphuhH10(Ω,R3)<+,

    2. 1|Th|Th|uh|2dsdx3C{uh2L2(Ω,R3)2h3h+1lnεh}, where C is a positive constant independent of h.

    Proof. 1. Observing that

    Fh(uh)Ωσij(uh)eij(uh)dx,

    we have, using Korn's inequality (see for instance [43]), that

    suphΩ|uh|2dx<+. (36)

    2. Let nk be the unit normal to Skh; kIh. Then nk=±(3/2,1/2), nk=±(3/2,1/2) or nk=±(0,1). Let us denote s,s the local coordinates defined by

    (ss)=(1/23/23/21/2)(x1x2) if Skh(3/2,1/2), (37)

    by

    (ss)=(1/23/23/21/2)(x1x2) if Skh(3/2,1/2), (38)

    and by

    (ss)=(1001)(x1x2) if Skh(0,1), (39)

    where the symbol represents the direction normal to the edge Skh. Let skh=(sk1h,sk2h) denotes the center of Skh; kIh, in the new coordinates. We define, for θ[0,2π) and r(0,εh/2),

    ukh(x1(s),x2(s),r,θ)=uh(s,rsinθ+sk2h,rcosθ). (40)

    Then, according to (36), we have, for r1r2<εh/2 and θ[0,2π),

    kIhSkhr2r1|ukh(x1(s),x2(s),r,θ)r|2rdrdsC. (41)

    Solving the Euler equation of the following one dimensional minimization problem:

    min{r2r1(ψ)2rdrψ(r1)=0ψ(r2)=1},

    we deduce that, for every θ[0,2π),

    lnr2r1r2r1|ukh(x1(s),x2(s),r,θ)r|2rdr      |ukh(x1(s),x2(s),r2,θ)ukh(x1(s),x2(s),r1,θ)|2. (42)

    Then, using (41) and (42), we obtain that

    kIhSkh2π0|ukh(x1(s),x2(s),r2,θ)ukh(x1(s),x2(s),r1,θ)|2dθdsClnr2r1. (43)

    Let us define

    ϝ(r,θ)=kIhSkh|ukh(x1(s),x2(s),r,θ)|2ds. (44)

    We deduce from the inequality (43) that, for r1r2<εh/2,

    2π0ϝ(r1,θ)dθC(2π0ϝ(r2,θ)dθ+lnr2r1). (45)

    Observing that, for kIh and θ0 fixed in [0,2π),

    |ukh(x1(s),x2(s),r,θ)ukh(x1(s),x2(s),r,θ0)|2=|θθ0ukhθ(x1(s),x2(s),r,ϕ)dϕ|2Cr2π0|1rukhθ(x1(s),x2(s),r,ϕ)|2rdϕ,

    we deduce that

    kIhSkh2π0εh0|ukh(x(s),r,θ)ukh(x(s),r,θ0)|2drdθds     CεhkIhCkh|uh|2dx1dx2dx3CεhΩ|uh|2dx, (46)

    where x(s)=(x1(s),x2(s)), Ckh is the cylinder of radius εh around the edge Skh. This estimate implies that

    εh0ϝ(ρ,θ0)drdθC{2π0εh0ϝ(r,θ)drdθ+εh}. (47)

    Now, using (45) and (47), we deduce, by setting mh=1εh2h3h+1, that, for θ0=0 and π, and for every r2[ah,bh]; ah=143(23)h/2 and bh=2ah,

    mhTh|uh|2dsdx3=mhεh0ϝ(r,0)dr+mhεh0ϝ(r,π)drCmh(2π0εh0ϝ(r,θ)drdθ+εh)C(mhεh0(2π0ϝ(r2,θ)dθ+lnr2r)dr+2h3h+1)C2h3h+1(2π0ϝ(r2,θ)dθ+lnr2εh+1)C((23)h/2r22π0ϝ(r2,θ)dθ+2h3h+1(lnεh+1)).

    Integrating with respect to r2 over the interval [ah,bh], we obtain that

    1|Th|Th|uh|2dsdx3C(bhah2π0ϝ(r,θ)rdrdθ2h3h+1lnεh)C{uh2L2(Ω,R3)2h3h+1lnεh}

    Let M(R3) be the space of Radon measures on R3. We have the following result:

    Lemma 3.2. Let uhL2(Ω,R3)L2(Th,R3), such that

    suph1|Th|Th|uh|2dsdx3<+.

    Then, there exists a subsequence of (uh)h, still denoted (uh)h, such that

    uh1Th(x)|Th|dsdx3hv1Σ(s)dHd(s)δ0(x3)Hd(Σ)inM(R3),

    with v(s,0)L2Hd(Σ,R3).

    Proof. Let us consider the sequence of Radon measures (ϑh)h on R3 defined by

    ϑh=1Th(x)|Th|dsdx3.

    Let xkh=(xk1h,xk2h) denotes the center of Skh; kIh, in Cartesian coordinates. Then, using the ergodicity result of [22,Theorem 6.1], we have, for every φC0(R3),

    limhR3φ(x)dϑh=limhkIh23h+1φ(xkh,0)=limhkIh1Nhφ(xkh,0)=1Hd(Σ)Σφ(s,0)dHd(s),

    from which we deduce that ϑhhϑ, with

    ϑ=1Σ(s)dHd(s)δ0(x3)Hd(Σ).

    Let uhL2(Ω,R3)L2(Th,R3), such that

    suph1|Th|Th|uh|2dsdx3<+.

    As ϑh (R3)=1|Th|Thdsdx3=1, we have

    |R3uhdϑh|2R3|uh|2dϑh=1|Th|Th|uh|2dsdx3,

    from which we deduce that the sequence (uhϑh)h is uniformly bounded in variation, hence -weakly relatively compact. Possibly passing to a subsequence, we can suppose that the sequence (uhϑh)h converges to some χ. Let φC0(R3,R3). Then, using Fenchel's inequality (also known as the Fenchel-Young inequality, see for instance [7]), we have

    liminfh12R3|uh|2dϑhliminfh(R3uh.φdϑh12R3|φ|2dϑh)χ,φ12R3|φ|2dϑ.

    As the left hand side of this inequality is bounded, we deduce that

    sup{χ,φφC0(R3,R3)Σ|φ|2(s,0)dHd(s)1}<+,

    from which we deduce, according to Riesz' representation theorem, that there exists v such that χ=v(s,x3)ϑ and v(s,0)L2Hd(Σ,R3).

    Proposition 2. Let (uh)h; uhH10(Ω,R3)H1(Th,R3), be a sequence, such that suphFh(uh)<+. There exists a subsequence, still denoted (uh)h, such that

    1. uhhu H10(Ω,R3)-weak,

    2. If γ(0,+) then

    uh1Th(x)|Th|dsdx3hv(s,0)1Σ(s)dHd(s)Hd(Σ),

    with v(s,0)L2Hd(Σ,R3).

    3. If γ(0,+) then, with ηh given in (11), we have ¯v(s,0)DΣ,E and

    liminfhThσhij(uh)eij(uh)dsdx3μ0EΣ(¯v).

    Proof. 1. Thanks to Lemma 3.11, one immediately obtains that, up to some subsequence, uhhu  H10(Ω,R3)-weak.

    2. If γ(0,+) then, according to Lemma 3.12 and Lemma 3.2, one has, up to some subsequence,

    uh1Th(x)|Th|dsdx3hv(s,0)1Σ(s)dHd(s)Hd(Σ),

    with v(s,0)L2Hd(Σ,R3).

    3. One can easily check that

    Thσhij(uh)eij(uh)dsdx3    2μh(Th((e11(uh))2+2(e12(uh))2+(e22(uh))2)dsdx3). (48)

    Computing the strain tensor in the local coordinates (37) and (38), observing that for Skh(3/2,1/2) or Skh(3/2,1/2) the covariant derivative ukα,hs=0 on Skh; α=1,2, we obtain

    Skh((e11(uh))2+2(e12(uh))2+(e22(uh))2)ds=Skh(14(uk1,hs)2+38(uk2,hs)2)ds14Skh((uk1,hs)2+(uk2,hs)2)ds. (49)

    For Skh(0,1), since ukα,hx2=0 on Skh; α=1,2, we have

    Skh((e11(uh))2+2(e12(uh))2+(e22(uh))2)ds=Skh(uk1,hx1)2+12(uk2,hx1)2ds14Skh((uk1,hx1)2+(uk2,hx1)2)ds. (50)

    According to (48) and (10), we deduce from (49) and (50) that

    Thσhij(uh)eij(uh)dsdx3μh2Th(uk1,hs)2+(uk2,hs)2dsdx3 2hεhμhkIh12εhεhεh(uα,h(pk,x3)uα,h(qk,x3))2dx3 =2hεhηhμ0kIh12εhεhεh(uα,h(pk,x3)uα,h(qk,x3))2dx3 μ0(53)hp,qVh|pq|=2h(12εhεhεh(uα,h(p,x3)uα,h(q,x3))dx3)2. (51)

    Let us set ¯uh=(u1,h,u2,h) and ˜¯uh=12εhεhεh¯uh(.,x3)dx3. We introduce the harmonic extension of ˜¯uhVh obtained by the decimation procedure (see for instance [30,Proposition 1] and [8,Corollary1]):

    We define the function Hh+1˜¯uh:Vh+1R2 as the unique minimizer of the problem

    min{Eh+1Σ(w)w:Vh+1R2w=˜¯uh on Vh}. (52)

    Then Eh+1Σ(Hh+1˜¯uh)=EhΣ(˜¯uh). For m>h, we define the function Hm˜¯uh from Vm into R2 by

    Hm˜¯uh=Hm(Hm1(...(Hh+1˜¯uh))).

    For every m>h we have Hm˜¯uhVh=˜¯uhVh and

    EmΣ(Hm˜¯uh)=EhΣ(˜¯uh). (53)

    Now we define, for a fixed hN, the function H˜¯uh on V as follows. For pV, we choose mh such that pVm and set

    H˜¯uh(p)=Hm˜¯uh(p). (54)

    As suphThσhij(uh)eij(uh)dsdx3<, we have, according to (51) and (53),

    suphEΣ(H˜¯uh)=suphEhΣ(˜¯uh)<+, (55)

    from which we deduce, using Section 2, that H˜¯uh has a unique continuous extension on Σ, still denoted H˜¯uh, and that the sequence (H˜¯uh)h is bounded in DΣ,E. Therefore, there exists a subsequence, still denoted (H˜¯uh)h, weakly converging to some ¯uDΣ,E, with

    EΣ(¯u) liminfhEΣ(H˜¯uh) liminfhEhΣ(˜¯uh). (56)

    On the other hand, using Lemma 3.2, we have, for every φC0(Σ,R2),

    limh1Hd(Σ)ΣH˜¯uh.φdHd(s)=limhR3¯uh.φdυh=1Hd(Σ)Σ¯v(s,0).φdHd(s) ,

    which implies that ¯u(s)=¯v(s,0). Therefore ¯v(s,0)DΣ,E and, according to (51) and (56),

    liminfh Thσhij(uh)eij(uh)dsdx3μ0EΣ(¯v).

    In this Section we state the main result of this work. According to Proposition 2 we introduce the following topology τ:

    Definition 4.1. We say that a sequence (uh)h; uhH10(Ω,R3)H1(Th,R3), τ-converges to (u,v) if

    {uhhu in  H1(Ω,R3)-weak,uh1Th(x)|Th|dsdx3hv1Σ(s)dHd(s)δ0(x3)Hd(Σ) in M(R3),

    with vv(s,0)L2Hd(Σ,R3).

    Our main result in this work reads as follows:

    Theorem 4.2. If γ(0,+) then

    1. (limsup inequality) for every (u,v) H10(Ω,R3)×DΣ,E×L2Hd(Σ) there exists a sequence (uh)h; uhH10(Ω,R3)H1(Th,R3) , such that (uh)h τ-converges to (u,v) and

    limsuphFh(uh)F(u,v),

    where F is the functional defined in (15),

    2. (liminf inequality) for every sequence (uh)h ; uhH10(Ω,R3)H1(Th,R3), such that (uh)h τ -converges to (u,v), we have ¯vDΣ,E and

    liminfhFh(uh)F(u,v).

    Before proving this Theorem, let us write the homogenized problem obtained at the limit as h.

    Corollary 1. Problem (13) admits a unique solution Uh which, under the hypothesis of Theorem 4.2, τ-converges to (U,V)H10(Ω,R3)×DΔΣ×L2Hd(Σ) solution of the problem

    {σij,j(U)=fiinΩ,μ0Δα,Σ(Vα)=πμγ(ln2)2Aαβ(s)(UβVβ);α,β=1,2,inΣ,[σα3|x3=0]Σ=πμHd(Σ)(ln2)2Aαβ(s)(UβVβ)HdonΣ,U3=V3onΣ,U=0onΩ,Vα=0;α=1,2,onV0. (57)

    Proof. One can easily check that problem (13) has a unique solution UhH10(Ω,R3)H1(Th,R3). Now, observing that

    Fh(Uh)2Ωf.UhdxFh(0)=0,

    we deduce, using the fact that limhηh=+, the Korn inequality, and the Poincaré inequality, that

    Ω|Uh|2dxΩσij(Uh)eij(Uh)dx+Thσhij(Uh)eij(Uh)dsdx32fL2(Ω,R3)UhL2(Ω,R3)CUhL2(Ω,R9),

    from which we deduce that suphFh(Uh)<+. Then, using Proposition 2 and Theorem 4.2, we deduce, according to [16,Theorem 7.8]), that the sequence (Uh)h τ-converges to the solution (U,V) of the problem

    min(ξ,ζ)V{Ωσij(ξ)eij(ξ)dx+μ0ΣdLΣ(ζ)+πμγHd(Σ)(ln2)2ΣA(s)(ξζ).(ξζ)dHd(s)2Ωf.ξdx }, (58)

    where V=H10(Ω,R3)×DΔΣ×L2Hd(Σ). On the other hand, according to [27,Theorem 6], the trace of ξH1(Ω,R3) on ωΣ exists for Hd-almost-every xωΣ and belongs to the Besov space B2d(Σ,R3) of functions ψ:ΣR3 such that

     Σ|ψ(x)|2dHd(x)+ΣΣ|xy|<1|ψ(x)ψ(y)|2|xy|2ddHd(x)dHd(y)<+. (59)

    Then, according to Lemma 2.1, we obtain from (58), using for example [46,Theorems 3.1 and 3.3], that ¯vDΔΣ and for every (ξ,ζ)H10(Ω,R3)×DΣ,E×L2Hd(Σ),

    Ω(σij,j(U)fi)ξidxμ0Hd(Σ)Σ(Δα,Σ¯V)ζαdHd(s)+πμγHd(Σ)(ln2)2ΣA(s)(UV).(ξζ)dHd(s)[σi3|x3=0]Σ,ξiB2d(Σ,R3),B2d(Σ,R3)=0, (60)

    B2d(Σ,R3) being the dual space of B2d(Σ,R3) (see [28,p. 291]). Since (V1,V2)DΣ,EL2Hd(Σ,R2), the trace of U on Σ belongs to B2d(Σ,R3)L2Hd(Σ,R3), and, according to Lemma 2.1, Δα,Σ; α=1,2, is a second order operator in L2Hd(Σ) defined by the form EΣ under the Dirichlet condition Vα=0 on V0; α=1,2, the transmission condition

    μ0Δα,Σ(Vα)=πμγ(ln2)2Aαβ(s)(UβVβ)α,β=1,2, in Σ,

    in problem (57) is well posed.

    The proof of Theorem 4.2 is given in three steps.

    We consider here a local problem associated with boundary layers in the vicinity of the strips. We denote wm; m=1,2, the solution of the following boundary value problem:

    {σij,j(wm)(y)=0yR2+i=1,2,wm(y1,0)=emy1]1,1[,σi2(wm)(y1,0)=0y1R]1,1[,wmm(y)=ln|y|ln2as |y|y2>0 ,|wmp|(y)Cfor {p=2 if m=1,p=1 if m=2, (61)

    where R2+={y=(y1,y2)R2y2>0} and em=(δ1m,δ2m); m=1,2. The displacement wm; m=1,2, which belongs to the space H1loc(R2+,R2), is given (see for instance [34] and [18]) by

    w11(y)=14πμ11ξ(t)((1+κ)ln((y1t)2+(y2)2)+2(y2)2(y1t)2+(y2)2)dt,w12(y)=14πμ11ξ(t)((1κ)arctan(y2y1t)+2y2(y1t)(y1t)2+(y2)2)dt (62)

    and

    w21(y)=14πμ11ξ(t)((1κ)arctan(y2y1t)+2y2(y1t)(y1t)2+(y2)2)dt,w22(y)=14πμ11ξ(t)((1+κ)ln((y1t)2+(y2)2)2(y2)2(y1t)2+(y2)2)dt, (63)

    where

    ξ(t)={4μ(1+κ)ln211t2if t]1,1[,0otherwise. (64)

    One can check that wm(y); m=1,2, is also the solution of problem (61) posed in the half-plane R2:

    R2={y=(y1,y2)R2 ; y2<0}.

    We introduce the scalar problem

    {Δw(y)=0yR2+i=1,2,w(y1,0)=1y1]1,1[,wy2(y1,0)=0y1R]1,1[,w(y)=ln|y|ln2 as |y|y2>0. (65)

    The solution of (65) is given by

    w(y)=1πln211ln((y1t)2+(y2)2)1t2dt. (66)

    Observe that w(y) is also the solution of problem (65) posed in the half-plane R2. We now state the following preliminary result in this section:

    Proposition 3. ([18,Proposition 7]). One has

    1. limR1lnRB(0,R)R2±σij(wm)eij(wl)dy=δml2μπ(1+κ)(ln2)2; m,l=1,2,

    2. limR1lnRB(0,R)R2±|w|2dy=π(ln2)2, where B(0,R) is a disc of radius R centred at the origin.

    Let rh be a positive parameter, such that

    limh2hrh=limhεhrh=0. (67)

    We define the rotation R(xkh); xkh=(xk1h,xk2h) being the center of Skh in Cartesian coordinates, by

    R(xkh)={IdR3if nk=±(0,1),(1/23/203/21/20001)if nk=±(3/2,1/2),(1/23/203/21/20001)if nk=±(3/2,1/2), (68)

    where nk is the unit normal on Skh and IdR3 is the 3×3 identity marix. Let φkh; kIh, be the truncation function defined on R2 by

    φkh(x)={4(r2hR2k,h(x))3r2hif rh/2Rkh(x)rh,1if Rkh(x)rh/2,0if Rkh(x)rh, (69)

    where Rkh(x)=((xxkh).nk)2+x23 with x=(x1,x2). We define, for kIh,

    Dkh(rh)={((xxkh).nk,x3)R2Rkh(x)<rhxR3} (70)

    and the cylinder

    Zkh=R(xkh)Skh×Dkh(rh). (71)

    We then set

    Zh=kIhZkh. (72)

    We define, the function wmkh(x); kIh and m=1,2,3, by

    w1kh(x)=φkh(x)R(xkh)(e11lnεh(1w(x3εh,(xxkh).nkεh)00)), (73)
    w2kh(x)=φkh(x)R(xkh)(e21lnεh(01w11(x3εh,(xxkh).nkεh)w12(x3εh,(xxkh).nkεh))) (74)

    and

    w3kh(x)=φkh(x)R(xkh)(e31lnεh(0w21(x3εh,(xxkh).nkεh)1w22(x3εh,(xxkh).nkεh))), (75)

    where em=(δ1m,δ2m,δ3m); m=1,2,3. We define the local perturbation wmε; m=1,2,3, on Ω by

    wmh(x)=wmkh(x)kIhxΩ. (76)

    We have the following result:

    Lemma 5.1. If γ(0,+) then, for every ΦC1(¯Ω,R3), we have

    limhZhσij(wmhΦm)eij(wlhΦl)dx=πμγHd(Σ)(ln2)2ΣA(s)Φ(s).Φ(s)dHd(s),

    where A(s) is the material matrix defined in (17).

    Proof. Let us introduce the change of variables

    {y1=x3εh,y2=(xxkh).nkεh,

    on Zkh; kIh. Then, using the smoothness of Φ and Proposition 3, we have

    limhZhσij(wmhΦm)eij(wlhΦl)dx=limhkIhZkhσij(wmkh)eij(wlkh)ΦmΦldx=limh3h+12hln2εhD(0,rhεh)D(0,1)σij(wm)eij(wl)dy1dy2×(kIh1Nh(R(xkh)Φ)m(R(xkh)Φ)l(xk1h,xk2h,0))=πμγHd(Σ)ln2Σ(BR(s)Φ(s))m(R(s)Φ(s))ldHd(s)=πμγHd(Σ)(ln2)2ΣRt(s)BR(s)Φ(s).Φ(s)dHd(s),

    where B=Diag(1,2(1+κ),2(1+κ)) and R(s) is the rotation matrix defined by R(s) =IdR3 on the faces of Σ which are perpendicular to the vectors ±(0,1), by R(s)=(1/23/203/21/20001) on the faces of Σ which are perpendicular to the vectors ±(3/2,1/2), and (1/23/203/21/20001) on the faces of Σ which are perpendicular to the vectors ±(3/2,1/2). Then observing that

    Rt(s)BR(s)=R(s)BR(s)=A(s),

    we have the result.

    In this Subsection we prove the lim-sup condition of the Γ -convergence property stated in Theorem 4.2. Let pkh=(pkh,1,pkh,2), qkh=(qkh,1,qkh,2) be the extremities of the line segment Skh. Let vC1c(ω,R3). Then, we build the following sequence:

    vk1,h(x)=v1(xk1h,xk2h)+2hζ1,kh(x)|v1(pkh)v1(qkh)|,vk2,h(x)=v2(xk1h,xk2h)+2hζ2,kh(x)|v2(pkh)v2(qkh)|,vk3,h(x)=v3(xk1h,xk2h), (77)

    for every xω, where, using the local coordinates (37) for Skh(3/2,1/2),

    {ζ1,kh(x)=2μhs+pkh,1/2pkh,23/2λh+2μh,ζ2,kh(x)=2(s+pkh,1/2pkh,23/2)3, (78)

    using the local coordinates (38) for Skh(3/2,1/2),

    {ζ1,kh(x)=2μhspkh,1/2pkh,23/2λh+2μh,ζ2,kh(x)=2(spkh,1/2pkh,23/2)3,

    and, using the local coordinates (39) for Skh(0,1),

    {ζ1,kh(x)=μh2(x1pkh,1)λh+2μh ,ζ2,kh(x)=(x1pkh,1)2.

    Let us now introduce the intervals Jpkhh and Jqkhh centred at the points pkh and qkh respectively, such that

    SkhJpkhh=[pkh,pkh+sh)SkhJqkhh=(qkhsh,qkh], (79)

    where sh=(shsh), such that limh2hsh=0. Let ψkh be a Cc(SkhJpkhhJqkhh) test-function, such that

    ψkh={1on  SkhJpkhhJqkhh,0on JpkhhJqkhh((pkh,pkh+sh)(qkhsh,qkh)). (80)

    We define the test-function vh by

    vh=ψkhvkhkIh. (81)

    We have the following convergences:

    Lemma 5.2. We have

    1. vh1Th(x)|Th|dsdx3hv1Σ(s)dHd(s)Hd(Σ),

    2. limhThσhij(vh)eij(vh)dsdx3=μ0limh(53)hα=1,2p,qVh|pq|=2h|vα(p)vα(q)|2.

    Proof. 1. Let φC0(R3,R3). We have

    limhR3φ(x).vh(x)1Th(x)|Th|dsdx3=limhkIh2v(xk1h,xk2h)3h+1.φ(xk1h,xk2h,0)+Climhα=1,2i=1,2,3kIh23h+1|vα(pkh)vα(qkh)|φi(xk1h,xk2h,0),

    where C is a positive constant independent of h. On the one hand we have

    limhkIh23h+1v(xk1h,xk2h).φ(xk1h,xk2h,0)=limhkIh1Nhv(xkh).φ(xkh,0)=1Hd(Σ)Σv(s).φ(s,0)dHd(s).

    On the other hand, since

    |vα(pkh)vα(qkh)|C|pkhqkh|

    and |pkhqkh|=2h, we have

    limhα=1,2i=1,2,3kIh23h+1|vα(pkh)vα(qkh)|φi(xk1h,xk2h,0)=0.

    2. Computing tensors in local coordinates (37) and (38), we obtain, for Skh(3/2,1/2) or Skh(3/2,1/2),

    σhij(vkh)eij(vkh)=(λh+2μh)4(vk1,hs)2+3μh4(vk2,hs)2,

    and if Skh(0,1),

    σhij(vkh)eij(vkh)=(λh+2μh)(vk1,hx1)2+μh(vk2,hx1)2.

    Thus, according to (77)-(78), we obtain on each Skh; kIh,

    σhij(vkh)eij(vkh)=μh22h{|v1(pkh)v1(qkh)|2+|v2(pkh)v2(qkh)|2},

    which implies that

    limhThσhij(vh)eij(vh)dsdx3=μ0limhηhkIh,α=1,2εh2h|vα(pkh)vα(qkh)|2=μ0limh(53)hkIh,α=1,2|vα(pkh)vα(qkh)|2=μ0limh(53)hα=1,2p,qVh|pq|=2h|vα(p)vα(q)|2.

    We prove here the lim-sup condition of the Γ-convergence property stated in Theorem 4.21.

    Proposition 4. If γ(0,+) then, for every (u,v) H10(Ω,R3)×DΣ,E×L2Hd(Σ), there exists a sequence (uh)h, such that uhH10(Ω,R3)H1(Th,R3), (uh)h τ -converges to (u,v), and

    limsuphFh(uh)F(u,v).

    Proof. Let (u,v) H10(Ω,R3)×DΣ,E×L2Hd(Σ). Let (un,vn)n be a sequence in the space C1c(Ω,R3)×(C1c(Ω,R3)DΣ,E×L2Hd(Σ)) such that unnu  H1(Ω,R3)-strong, ¯vnn¯v strongly with respect to the norm (26), and v3,nnv3 strongly with respect to the norm of L2Hd(Σ). We define the sequence (u0n,h)h,n by

    u0n,h=unwmh((un)m(vn,h)m), (82)

    where vn,h is the test-function (81) associated with vn, and wmh is the perturbation defined in (76). Then u0n,hH10(Ω,R3)H1(Th,R3) and, using Lemma 5.1, Lemma 5.2, and the fact that the measure |Zh| of the set Zh tends to zero as h tends to , that (u0n,h)h τ-converges to (un,vn) as h tends to .

    We have

    Fh(u0n,h)=ΩZhσij(u0n,h)eij(u0n,h)dx    +Zhσij(u0n,h)eij(u0n,h)dx+Thσhij(vn,h)eij(vn,h)dsdx3. (83)

    We immediately obtain

    limhΩZhσij(u0n,h)eij(u0n,h)dx=Ωσij(un)eij(un)dx.

    Using Lemma 5.1, it follows that

    limhZhσij(u0n,h)eij(u0n,h)dx=limhZhσij(wmh((un)m(vn,h)m))eij(wmh((un)m(vn,h)m))=πμγHd(Σ)ln2ΣA(s)(unvn).(unvn)dHd(s) (84)

    and, using Lemma 5.2 and Proposition 1, we obtain

    limhThσhij(vn,h)eij(vn,h)dsdx3=μ0limh(53)hα=1,2p,qVh|pq|=2h|vα,n(p,0)vα,n(q,0)|2=μ0EΣ(¯vn)=μ0ΣdLΣ(¯vn).

    This yields

    limhFh(u0n,h)=Ωσij(un)eij(un)dx+μ0ΣdLΣ(¯vn)           +πμγHd(Σ)ln2ΣA(s)(unvn).(unvn)dHd(s)           =F(un,vn). (85)

    The continuity of F implies that limnlimhFh(u0n,h)=F(u,v). Then, using the diagonalization argument of [5,Corollary 1.18], we prove the existence of a sequence (uh)h=(u0n(h),h)h: limhn(h)=+, such that

    limsuphFh(uh)F(u,v).

    In this Subsection we prove the second assertion of Theorem 4.2.

    Proposition 5. If γ(0,+), then, for every sequence (uh)h, such that uhH10(Ω,R3)H1(Th,R3) and (uh)h τ-converges to (u,v), we have ¯vDΣ,E and

    liminfhFh(uh)F(u,v).

    Proof. Let (uh)h; uhH10(Ω,R3)H1(Th,R3), such that (uh)h τ-converges to (u,v). We suppose that suphFh(uh)<+, otherwise the liminf inequality is trivial. Then, owing to Proposition 2 and Proposition 1, we have that ¯vDΣ,E and

    liminfh Thσhij(uh)eij(uh)dsdx3μ0EΣ(¯v)=μ0ΣdLΣ(¯v). (86)

    Let (un,vn)n C1c(Ω,R3)×(C1c(Ω,R3)DΣ,E×L2Hd(Σ)), such that

    unnu H1(Ω,R3)strong,

    ¯vnn¯v strongly with respect to the norm (26), and v3,nnv3 strongly with respect to the norm of L2Hd(Σ). Let (u0n,h)h,n be the corresponding sequence defined in (82). We have from the definition of the subdifferentiability of convex functionals

    Zhσij(uh)eij(uh)dxZhσij(u0n,h)eij(u0n,h)dx                      +2Zhσij(u0n,h)eij(uhu0n,h)dx. (87)

    Due to the structure of the sequence , we have

    (88)

    Since tends to zero as tends to , it follows that

    (89)

    Using the definition of the perturbation and the expressions (62), (63) and (66), we get

    (90)

    where is a positive constant which may depend of . Then, using Lemma 5.1, we obtain that

    (91)

    We deduce from (84) that

    (92)

    On the other hand, as tends to zero as tends to , we have

    (93)

    We deduce from (86)-(93) that

    Letting tend to in the right hand side of the above inequality, we deduce that

    which is equivalent to

    This ends the proof of Theorem 4.2.



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