
The graph
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
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
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
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
Vh+1=Vh∪(2−hA2+Vh)∪(2−hA3+Vh). | (1) |
Let us set
V∞=∪h∈NVh. | (2) |
The Sierpinski gasket, which is denoted here by
Σ=¯V∞. | (3) |
We define the graph
The edges which belong to
Let
Σ∩∂ω=V0. | (4) |
Let
limh→∞εh2h=0. | (5) |
We define
Tkh=(ω∩Skh)×(−εh,εh) | (6) |
and set
Th=∪k∈IhT,kh, | (7) |
where
|Th|=εh3h+12h. | (8) |
Let
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
σhij(u)=λhemm(u)δij+2μheij(u) ; i,j=1,2,3, |
with
λh=ηhλ0 and μh=ηhμ0, | (10) |
where
ηh=1εh(56)h. | (11) |
The special scaling (10) and (11) of the Lamé -coefficients depend on the structural constants of
We suppose that a perfect adhesion occurs between
Fh(u)={∫Ω∖Thσij(u)eij(u)dx+∫Thσhij(u)eij(u)dsdx3 if u∈H10(Ω,R3)∩H1(Th,R3),+∞ otherwise, | (12) |
where
minu∈L2(Ω,R3)∩L2(Th,R3){Fh(u)−2∫Ωf.udx}. | (13) |
We use
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
F∞(u,v)={∫Ωσij(u)eij(u)dx+μ0∫ΣdLΣ(¯v)+πμγHd(Σ)(ln2)2∫ΣA(s)(u−v).(u−v)dHd(s) if (u,v)∈H10(Ω,R3)×DΣ,E×L2Hd(Σ),+∞ otherwise, | (15) |
where
d=ln3/ln2, | (16) |
A(s)={Diag(1,2(1+κ),2(1+κ)) if n(s)=±(0,1),(7+κ4(1+κ)√3(κ−1)4(1+κ)0√3(κ−1)4(1+κ)3κ+54(1+κ)0002(1+κ))if n(s)=±(−√32,12),(7+κ4(1+κ)√3(1−κ)4(1+κ)0√3(1−κ)4(1+κ)3κ+54(1+κ)0002(1+κ))if n(s)=±(√32,12), | (17) |
where
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
{ [σα3|x3=0]Σ=πμγHd(Σ)(ln2)2Aαβ(s)(Uβ−Vβ)Hd on Σ,πμγ(ln2)2Aαβ(s)(Uβ−Vβ)=−μ0ΔΣVα in Σ; α,β=1,2, | (18) |
where
[σα3|x3=0]Σ=σα3|Σ×{0+}−σα3|Σ×{0−}; α=1,2, | (19) |
is the jump of
If
F∞(u)={∫Ωσij(u)eij(u)dx+μ0∫ΣdLΣ(¯u) if u∈H10(Ω,R3)∩(DΣ,E×L2Hd(Σ)), +∞ otherwise. | (20) |
If
F0(u)={∫Ωσij(u)eij(u)dx if u∈H10(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
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
EhΣ(w)=(53)h∑p,q∈Vh|p−q|=2−h|w(p)−w(q)|2. | (22) |
Let us define the energy
EΣ(z)=limh→∞EhΣ(z), | (23) |
with domain
D={z∈C(Σ,R2):EΣ(z)<∞}, | (24) |
where
DE=¯D‖.‖DE, | (25) |
where
‖z‖DE={EΣ(z)+‖z‖2L2Hd(Σ,R2)}1/2. | (26) |
The space
DΣ,E={z∈DE:z(A1)=z(A2)=z(A3)=0}. | (27) |
We denote
EΣ(w,z)=12(EΣ(w+z)−EΣ(w)−EΣ(z)), ∀w,z∈DΣ,E. | (28) |
One can see that
EΣ(w,z)=limh→∞EhΣ(w,z), | (29) |
where
EhΣ(w,z)=(53)h∑p,q∈Vh|p−q|=2−h(w(p)−w(q)).(z(p)−z(q)). | (30) |
The form
1. (local property)
2. (regularity)
The second property implies that
Now, applying [29,Chap. 6], we have the following result:
Lemma 2.1. There exists a unique self-adjoint operator
DΔΣ={w=(w1w2)∈L2Hd(Σ,R2):ΔΣw=(ΔΣw1ΔΣw2)∈L2Hd(Σ,R2)}⊂DΣ,E |
dense in
EΣ(w,z)=−∫Σ(ΔΣw).zdHdHd(Σ). |
Let us consider the sequence
νh=1Card(Vh)∑p∈Vhδp, | (31) |
where
Lemma 2.2. The sequence
ν=1Σ(s)dHd(s)Hd(Σ), |
where
Proof. Let
limh→∞∫Σφ(x)dνh=limh→∞∑p∈Vhφ(p)Card(Vh)=1Hd(Σ)∫Σφ(s,0)dHd(s). |
We note that the approximating form
EhΣ(w,z)=∫Σ∇hw.∇hz dνh, | (32) |
where
∇hw.∇hz(p)=∑q: |p−q|=2−h(w(p)−w(q))|p−q|ϰ/2.(z(p)−z(q))|p−q|ϰ/2, |
where
Proposition 1. For every
LhΣ(w,z)(A)=∫A∩Σ∇hw.∇hzdνh,∀A⊂Σ, |
weakly converges in the topological dual
EΣ(w,z)=∫ΣdLΣ(w,z),∀w,z∈DΣ,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
∫ΣφdLhΣ(w)=EhΣ(φw,w)−12EhΣ(φe1,|w|2e1), | (33) |
we deduce, taking into account the regularity of the form
limh→∞∫ΣφdLhΣ(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)=∫ΣdLΣ(w), | (35) |
where
∫ΣφdLΣ(w)=EΣ(φw,w)−12EΣ(φe1,|w|2e1), ∀φ∈DΣ,E∩C(Σ). |
Thus, combining with (34), the sequence
LhΣ(w,z)=12(LhΣ(w+z)−LhΣ(w)−LhΣ(z)), |
we deduce that the sequence
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
1.
2.
Proof. 1. Observing that
Fh(uh)≥∫Ωσij(uh)eij(uh)dx, |
we have, using Korn's inequality (see for instance [43]), that
suph∫Ω|∇uh|2dx<+∞. | (36) |
2. Let
(ss⊥)=(1/2√3/2−√3/21/2)(x1x2) if Skh⊥(−√3/2,1/2), | (37) |
by
(ss⊥)=(−1/2√3/2√3/21/2)(x1x2) if Skh⊥(√3/2,1/2), | (38) |
and by
(ss⊥)=(1001)(x1x2) if Skh⊥(0,1), | (39) |
where the symbol
ukh(x1(s),x2(s),r,θ)=uh(s,rsinθ+sk2h,rcosθ). | (40) |
Then, according to (36), we have, for
∑k∈Ih∫Skh∫r2r1|∂ukh(x1(s),x2(s),r,θ)∂r|2rdrds≤C. | (41) |
Solving the Euler equation of the following one dimensional minimization problem:
min{∫r2r1(ψ′)2rdr: ψ(r1)=0, ψ(r2)=1}, |
we deduce that, for every
lnr2r1∫r2r1|∂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
∑k∈Ih∫Skh∫2π0|ukh(x1(s),x2(s),r2,θ)−ukh(x1(s),x2(s),r1,θ)|2dθds≤Clnr2r1. | (43) |
Let us define
ϝ(r,θ)=∑k∈Ih∫Skh|ukh(x1(s),x2(s),r,θ)|2ds. | (44) |
We deduce from the inequality (43) that, for
∫2π0ϝ(r1,θ)dθ≤C(∫2π0ϝ(r2,θ)dθ+lnr2r1). | (45) |
Observing that, for
|ukh(x1(s),x2(s),r,θ)−ukh(x1(s),x2(s),r,θ0)|2=|∫θθ0∂ukh∂θ(x1(s),x2(s),r,ϕ)dϕ|2≤Cr∫2π0|1r∂ukh∂θ(x1(s),x2(s),r,ϕ)|2rdϕ, |
we deduce that
∑k∈Ih∫Skh∫2π0∫εh0|ukh(x′(s),r,θ)−ukh(x′(s),r,θ0)|2drdθds ≤ Cεh∑k∈Ih∫Ckh|∇uh|2dx1dx2dx3≤Cεh∫Ω|∇uh|2dx, | (46) |
where
∫εh0ϝ(ρ,θ0)drdθ≤C{∫2π0∫εh0ϝ(r,θ)drdθ+εh}. | (47) |
Now, using (45) and (47), we deduce, by setting
mh∫Th|uh|2dsdx3=mh∫εh0ϝ(r,0)dr+mh∫εh0ϝ(r,π)dr≤Cmh(∫2π0∫εh0ϝ(r,θ)drdθ+εh)≤C(mh∫εh0(∫2π0ϝ(r2,θ)dθ+lnr2r)dr+2h3h+1)≤C2h3h+1(∫2π0ϝ(r2,θ)dθ+lnr2εh+1)≤C((23)h/2r2∫2π0ϝ(r2,θ)dθ+2h3h+1(−lnεh+1)). |
Integrating with respect to
1|Th|∫Th|uh|2dsdx3≤C(∫bhah∫2π0ϝ(r,θ)rdrdθ−2h3h+1lnεh)≤C{‖uh‖2L2(Ω,R3)−2h3h+1lnεh}. |
Let
Lemma 3.2. Let
suph1|Th|∫Th|uh|2dsdx3<+∞. |
Then, there exists a subsequence of
uh1Th(x)|Th|dsdx3∗⇀h→∞v1Σ(s)dHd(s)⊗δ0(x3)Hd(Σ)inM(R3), |
with
Proof. Let us consider the sequence of Radon measures
ϑh=1Th(x)|Th|dsdx3. |
Let
limh→∞∫R3φ(x)dϑh=limh→∞∑k∈Ih23h+1φ(xkh,0)=limh→∞∑k∈Ih1Nhφ(xkh,0)=1Hd(Σ)∫Σφ(s,0)dHd(s), |
from which we deduce that
ϑ=1Σ(s)dHd(s)⊗δ0(x3)Hd(Σ). |
Let
suph1|Th|∫Th|uh|2dsdx3<+∞. |
As
|∫R3uhdϑh|2≤∫R3|uh|2dϑh=1|Th|∫Th|uh|2dsdx3, |
from which we deduce that the sequence
liminfh→∞12∫R3|uh|2dϑh≥liminfh→∞(∫R3uh.φdϑh−12∫R3|φ|2dϑh)≥⟨χ,φ⟩−12∫R3|φ|2dϑ. |
As the left hand side of this inequality is bounded, we deduce that
sup{⟨χ,φ⟩; φ∈C0(R3,R3), ∫Σ|φ|2(s,0)dHd(s)≤1}<+∞, |
from which we deduce, according to Riesz' representation theorem, that there exists
Proposition 2. Let
1.
2. If
uh1Th(x)|Th|dsdx3∗⇀h→∞v(s,0)1Σ(s)dHd(s)Hd(Σ), |
with
3. If
liminfh→∞∫Thσhij(uh)eij(uh)dsdx3≥μ0EΣ(¯v). |
Proof. 1. Thanks to Lemma 3.1
2. If
uh1Th(x)|Th|dsdx3∗⇀h→∞v(s,0)1Σ(s)dHd(s)Hd(Σ), |
with
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((e11(uh))2+2(e12(uh))2+(e22(uh))2)ds=∫Skh(14(∂uk1,h∂s)2+38(∂uk2,h∂s)2)ds≥14∫Skh((∂uk1,h∂s)2+(∂uk2,h∂s)2)ds. | (49) |
For
∫Skh((e11(uh))2+2(e12(uh))2+(e22(uh))2)ds=∫Skh(∂uk1,h∂x1)2+12(∂uk2,h∂x1)2ds≥14∫Skh((∂uk1,h∂x1)2+(∂uk2,h∂x1)2)ds. | (50) |
According to (48) and (10), we deduce from (49) and (50) that
∫Thσhij(uh)eij(uh)dsdx3≥μh2∫Th(∂uk1,h∂s)2+(∂uk2,h∂s)2dsdx3 ≥2hεhμh∑k∈Ih12εh∫εh−εh(uα,h(pk,x3)−uα,h(qk,x3))2dx3 =2hεhηhμ0∑k∈Ih12εh∫εh−εh(uα,h(pk,x3)−uα,h(qk,x3))2dx3 ≥μ0(53)h∑p,q∈Vh|p−q|=2−h(12εh∫εh−εh(uα,h(p,x3)−uα,h(q,x3))dx3)2. | (51) |
Let us set
We define the function
min{Eh+1Σ(w); w:Vh+1⟶R2, w=˜¯uh on Vh}. | (52) |
Then
Hm˜¯uh=Hm(Hm−1(...(Hh+1˜¯uh))). |
For every
EmΣ(Hm˜¯uh)=EhΣ(˜¯uh). | (53) |
Now we define, for a fixed
H˜¯uh(p)=Hm˜¯uh(p). | (54) |
As
suphEΣ(H˜¯uh)=suphEhΣ(˜¯uh)<+∞, | (55) |
from which we deduce, using Section
EΣ(¯u∗)≤ liminfh→∞EΣ(H˜¯uh)≤ liminfh→∞EhΣ(˜¯uh). | (56) |
On the other hand, using Lemma 3.2, we have, for every
limh→∞1Hd(Σ)∫ΣH˜¯uh.φdHd(s)=limh→∞∫R3¯uh.φdυh=1Hd(Σ)∫Σ¯v(s,0).φdHd(s) , |
which implies that
liminfh→∞ ∫Thσhij(uh)eij(uh)dsdx3≥μ0EΣ(¯v). |
In this Section we state the main result of this work. According to Proposition 2 we introduce the following topology
Definition 4.1. We say that a sequence
{uh⇀h→∞u in H1(Ω,R3)-weak,uh1Th(x)|Th|dsdx3∗⇀h→∞v1Σ(s)dHd(s)⊗δ0(x3)Hd(Σ) in M(R3), |
with
Our main result in this work reads as follows:
Theorem 4.2. If
1. (
limsuph→∞Fh(uh)≤F∞(u,v), |
where
2. (
liminfh→∞Fh(uh)≥F∞(u,v). |
Before proving this Theorem, let us write the homogenized problem obtained at the limit as
Corollary 1. Problem (13) admits a unique solution
{−σij,j(U)=fiinΩ,−μ0Δα,Σ(Vα)=πμγ(ln2)2Aαβ(s)(Uβ−Vβ);α,β=1,2,inΣ,[σα3|x3=0]Σ=πμHd(Σ)(ln2)2Aαβ(s)(Uβ−Vβ)HdonΣ,U3=V3onΣ,U=0on∂Ω,Vα=0;α=1,2,onV0. | (57) |
Proof. One can easily check that problem (13) has a unique solution
Fh(Uh)−2∫Ωf.Uhdx≤Fh(0)=0, |
we deduce, using the fact that
∫Ω|∇Uh|2dx≤∫Ωσij(Uh)eij(Uh)dx+∫Thσhij(Uh)eij(Uh)dsdx3≤2‖f‖L2(Ω,R3)‖Uh‖L2(Ω,R3)≤C‖∇Uh‖L2(Ω,R9), |
from which we deduce that
min(ξ,ζ)∈V{∫Ωσij(ξ)eij(ξ)dx+μ0∫ΣdLΣ(ζ)+πμγHd(Σ)(ln2)2∫ΣA(s)(ξ−ζ).(ξ−ζ)dHd(s)−2∫Ωf.ξdx }, | (58) |
where
∫Σ|ψ(x)|2dHd(x)+∫Σ∫Σ|x−y|<1|ψ(x)−ψ(y)|2|x−y|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
∫Ω(−σij,j(U)−fi)ξidx−μ0Hd(Σ)∫Σ(Δα,Σ¯V)ζαdHd(s)+πμγHd(Σ)(ln2)2∫ΣA(s)(U−V).(ξ−ζ)dHd(s)−⟨[σi3|x3=0]Σ,ξi⟩B2−d(Σ,R3),B2d(Σ,R3)=0, | (60) |
−μ0Δα,Σ(Vα)=πμγ(ln2)2Aαβ(s)(Uβ−Vβ); 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
{σij,j(wm)(y)=0∀y∈R2+; i=1,2,wm(y1,0)=em∀y1∈]−1,1[,σi2(wm)(y1,0)=0∀y1∈R∖]−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
w11(y)=14πμ∫1−1ξ(t)(−(1+κ)ln(√(y1−t)2+(y2)2)+2(y2)2(y1−t)2+(y2)2)dt,w12(y)=14πμ∫1−1ξ(t)(−(1−κ)arctan(y2y1−t)+2y2(y1−t)(y1−t)2+(y2)2)dt | (62) |
and
w21(y)=14πμ∫1−1ξ(t)((1−κ)arctan(y2y1−t)+2y2(y1−t)(y1−t)2+(y2)2)dt,w22(y)=14πμ∫1−1ξ(t)(−(1+κ)ln(√(y1−t)2+(y2)2)−2(y2)2(y1−t)2+(y2)2)dt, | (63) |
where
ξ(t)={4μ(1+κ)ln21√1−t2if t∈]−1,1[,0otherwise. | (64) |
One can check that
R2−={y=(y1,y2)∈R2 ; y2<0}. |
We introduce the scalar problem
{Δw(y)=0∀y∈R2+; i=1,2,w(y1,0)=1∀y1∈]−1,1[,∂w∂y2(y1,0)=0∀y1∈R∖]−1,1[,w(y)=−ln|y|ln2 as |y|→∞, y2>0. | (65) |
The solution of (65) is given by
w(y)=−1πln2∫1−1ln(√(y1−t)2+(y2)2)√1−t2dt. | (66) |
Observe that
Proposition 3. ([18,Proposition 7]). One has
1.
2.
Let
limh→∞2hrh=limh→∞εhrh=0. | (67) |
We define the rotation
R(xkh)={IdR3if nk=±(0,1),(1/2√3/20−√3/21/20001)if nk=±(−√3/2,1/2),(−1/2√3/20√3/21/20001)if nk=±(√3/2,1/2), | (68) |
where
φkh(x)={4(r2h−R2k,h(x))3r2hif rh/2≤Rkh(x)≤rh,1if Rkh(x)≤rh/2,0if Rkh(x)≥rh, | (69) |
where
Dkh(rh)={((x−xkh).nk,x3)∈R2; Rkh(x)<rh, ∀x∈R3} | (70) |
and the cylinder
Zkh=R(xkh)Skh×Dkh(rh). | (71) |
We then set
Zh=⋃k∈IhZkh. | (72) |
We define, the function
w1kh(x)=φkh(x)R(xkh)(e1−1lnεh(1−w(x3εh,(x′−xkh).nkεh)00)), | (73) |
w2kh(x)=φkh(x)R(xkh)(e2−1lnεh(01−w11(x3εh,(x′−xkh).nkεh)w12(x3εh,(x′−xkh).nkεh))) | (74) |
and
w3kh(x)=φkh(x)R(xkh)(e3−1lnεh(0w21(x3εh,(x′−xkh).nkεh)1−w22(x3εh,(x′−xkh).nkεh))), | (75) |
where
wmh(x)=wmkh(x), ∀k∈Ih, ∀x∈Ω. | (76) |
We have the following result:
Lemma 5.1. If
limh→∞∫Zhσij(wmhΦm)eij(wlhΦl)dx=πμγHd(Σ)(ln2)2∫ΣA(s)Φ(s).Φ(s)dHd(s), |
where
Proof. Let us introduce the change of variables
{y1=x3εh,y2=(x′−xkh).nkεh, |
on
limh→∞∫Zhσij(wmhΦm)eij(wlhΦl)dx=limh→∞∑k∈Ih∫Zkhσij(wmkh)eij(wlkh)ΦmΦldx=limh→∞3h+12hln2εh∫D(0,rhεh)∖D(0,1)σij(wm)eij(wl)dy1dy2×(∑k∈Ih1Nh(R(xkh)Φ)m(R(xkh)Φ)l(xk1h,xk2h,0))=πμγHd(Σ)ln2∫Σ(BR(s)Φ(s))m(R(s)Φ(s))ldHd(s)=πμγHd(Σ)(ln2)2∫ΣRt(s)BR(s)Φ(s).Φ(s)dHd(s), |
where
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
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
{ζ1,kh(x′)=2√μhs+pkh,1/2−pkh,2√3/2√λh+2μh,ζ2,kh(x′)=2(s+pkh,1/2−pkh,2√3/2)√3, | (78) |
using the local coordinates (38) for
{ζ1,kh(x′)=√2√μhs−pkh,1/2−pkh,2√3/2√λh+2μh,ζ2,kh(x′)=√2(s−pkh,1/2−pkh,2√3/2)√3, |
and, using the local coordinates (39) for
{ζ1,kh(x′)=√μh2(x1−pkh,1)√λh+2μh ,ζ2,kh(x′)=(x1−pkh,1)√2. |
Let us now introduce the intervals
Skh∩Jpkhh=[pkh,pkh+sh), Skh∩Jqkhh=(qkh−sh,qkh], | (79) |
where
ψkh={1on Skh∖Jpkhh∪Jqkhh,0on Jpkhh∪Jqkhh∖((pkh,pkh+sh)∪(qkh−sh,qkh)). | (80) |
We define the test-function
vh=ψkhvkh, ∀k∈Ih. | (81) |
We have the following convergences:
Lemma 5.2. We have
1.
2.
Proof. 1. Let
limh→∞∫R3φ(x).vh(x′)1Th(x)|Th|dsdx3=limh→∞∑k∈Ih2v(xk1h,xk2h)3h+1.φ(xk1h,xk2h,0)+Climh→∞∑α=1,2i=1,2,3k∈Ih23h+1|vα(pkh)−vα(qkh)|φi(xk1h,xk2h,0), |
where
limh→∞∑k∈Ih23h+1v(xk1h,xk2h).φ(xk1h,xk2h,0)=limh→∞∑k∈Ih1Nhv(xkh).φ(xkh,0)=1Hd(Σ)∫Σv(s).φ(s,0)dHd(s). |
On the other hand, since
|vα(pkh)−vα(qkh)|≤C|pkh−qkh| |
and
limh→∞∑α=1,2i=1,2,3k∈Ih23h+1|vα(pkh)−vα(qkh)|φi(xk1h,xk2h,0)=0. |
2. Computing tensors in local coordinates (37) and (38), we obtain, for
σhij(vkh)eij(vkh)=(λh+2μh)4(∂vk1,h∂s)2+3μh4(∂vk2,h∂s)2, |
and if
σhij(vkh)eij(vkh)=(λh+2μh)(∂vk1,h∂x1)2+μh(∂vk2,h∂x1)2. |
Thus, according to (77)-(78), we obtain on each
σhij(vkh)eij(vkh)=μh22h{|v1(pkh)−v1(qkh)|2+|v2(pkh)−v2(qkh)|2}, |
which implies that
limh→∞∫Thσhij(vh)eij(vh)dsdx3=μ0limh→∞ηh∑k∈Ih,α=1,2εh2h|vα(pkh)−vα(qkh)|2=μ0limh→∞(53)h∑k∈Ih,α=1,2|vα(pkh)−vα(qkh)|2=μ0limh→∞(53)h∑α=1,2p,q∈Vh|p−q|=2−h|vα(p)−vα(q)|2. |
We prove here the lim-sup condition of the
Proposition 4. If
limsuph→∞Fh(uh)≤F∞(u,v). |
Proof. Let
u0n,h=un−wmh((un)m−(vn,h)m), | (82) |
where
We have
Fh(u0n,h)=∫Ω∖Zhσij(u0n,h)eij(u0n,h)dx +∫Zhσij(u0n,h)eij(u0n,h)dx+∫Thσhij(vn,h)eij(vn,h)dsdx3. | (83) |
We immediately obtain
limh→∞∫Ω∖Zhσij(u0n,h)eij(u0n,h)dx=∫Ωσij(un)eij(un)dx. |
Using Lemma 5.1, it follows that
limh→∞∫Zhσij(u0n,h)eij(u0n,h)dx=limh→∞∫Zhσij(wmh((un)m−(vn,h)m))eij(wmh((un)m−(vn,h)m))=πμγHd(Σ)ln2∫ΣA(s)(un−vn).(un−vn)dHd(s) | (84) |
and, using Lemma 5.2 and Proposition 1, we obtain
limh→∞∫Thσhij(vn,h)eij(vn,h)dsdx3=μ0limh→∞(53)h∑α=1,2p,q∈Vh|p−q|=2−h|vα,n(p,0)−vα,n(q,0)|2=μ0EΣ(¯vn)=μ0∫ΣdLΣ(¯vn). |
This yields
limh→∞Fh(u0n,h)=∫Ωσij(un)eij(un)dx+μ0∫ΣdLΣ(¯vn) +πμγHd(Σ)ln2∫ΣA(s)(un−vn).(un−vn)dHd(s) =F∞(un,vn). | (85) |
The continuity of
limsuph→∞Fh(uh)≤F∞(u,v). |
In this Subsection we prove the second assertion of Theorem 4.2.
Proposition 5. If
liminfh→∞Fh(uh)≥F∞(u,v). |
Proof. Let
liminfh→∞ ∫Thσhij(uh)eij(uh)dsdx3≥μ0EΣ(¯v)=μ0∫ΣdLΣ(¯v). | (86) |
Let
un⟶n→∞u H1(Ω,R3)−strong, |
∫Zhσij(uh)eij(uh)dx≥∫Zhσij(u0n,h)eij(u0n,h)dx +2∫Zhσij(u0n,h)eij(uh−u0n,h)dx. | (87) |
Due to the structure of the sequence
(88) |
Since
(89) |
Using the definition of the perturbation
(90) |
where
(91) |
We deduce from (84) that
(92) |
On the other hand, as
(93) |
We deduce from (86)-(93) that
Letting
which is equivalent to
This ends the proof of Theorem 4.2.
1. | Mustapha El Jarroudi, Mhamed El Merzguioui, Mustapha Er-Riani, Aadil Lahrouz, Jamal El Amrani, Dimension reduction analysis of a three-dimensional thin elastic plate reinforced with fractal ribbons, 2023, 0956-7925, 1, 10.1017/S0956792523000025 | |
2. | Haifa El Jarroudi, Mustapha El Jarroudi, Asymptotic behavior of a viscous incompressible fluid flow in a fractal network of branching tubes, 2024, 16, 2836-3310, 655, 10.3934/cam.2024030 | |
3. | Mustapha El Jarroudi, A second and third gradient material with torsion resulting from the homogenization of a highly contrasted rigid fibre-reinforced composite, 2024, 36, 0935-1175, 471, 10.1007/s00161-024-01278-4 |
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