- Email: [email protected]
Homogenization of an elasticity problem in domains with a net of slender bars near surface
C. R. Mecanique 331 (2003) 829–834
Homogenization of an elasticity problem in domains with a net of slender bars near surface Mariya Goncharenko a,b , Leonid Pankratov a,b a Laboratoire Jacques-Louis Lions de l’Université Pierre et Marie Curie, 4, place Jussieu, 75252 Paris cedex 05, France b Département de mathématiques, B.Verkin institut des basses températures (FTINT), 47, av. Lénine, 61103 Kharkov, Ukraine
Received 11 June 2003; accepted 23 September 2003 Presented by Évariste Sanchez-Palencia
Abstract We consider an elasticity problem in a domain Ω (ε) = Ω \ F (ε) , where Ω is an open bounded domain in R3 , F (ε) is a connected nonperiodic set in Ω like a net of slender bars, and ε is a parameter characterizing the microstructure of the domain. We consider the case of a surface distribution of the set F (ε) , i.e., for sufficiently small ε, the set F (ε) is concentrated in arbitrary small neighbourhood of a surface Γ . Under a hypothesis on the asymptotic behaviour of the energy functional, we obtain the macroscopic (homogenized) model. To cite this article: M. Goncharenko, L. Pankratov, C. R. Mecanique 331 (2003). 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved. Résumé Homogénéisation d’un problème d’élasticité dans des domaines avec un réseau de barres fines proche de la surface. On étudie un problème d’élasticité dans un domaine Ω (ε) = Ω \ F (ε) , où Ω est un ouvert borné dans R3 , F (ε) est un ensemble non-périodique connexe dans Ω de type un réseau de barres fines et ε est un paramétre qui caractérise la microstructure du domaine. On considère le cas d’une distribution surfacique de l’ensemble F (ε) , c’est-á-dire, lorsque ε → 0, cet ensemble se concentre dans un voisinage d’une surface Γ aussi petit que l’on veut. Sous une hypothèse sur le comportement asymptotique de la fonctionelle d’énergie on obtient le modèle macroscopique (homogénéisé). Pour citer cet article : M. Goncharenko, L. Pankratov, C. R. Mecanique 331 (2003). 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved. Keywords: Computational solid mechanics; Elasticity problem Mots-clés : Mécanique de solides numérique ; Problème d’élasticité
E-mail addresses: [email protected], [email protected] (M. Goncharenko), [email protected], [email protected] (L. Pankratov). 1631-0721/$ – see front matter 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved. doi:10.1016/j.crme.2003.09.005
830
M. Goncharenko, L. Pankratov / C. R. Mecanique 331 (2003) 829–834
1. Introduction We study a homogenization problem for the elasticity equation which describes the equilibrium of an elastic medium with rigid inclusions, such as a nonperiodic net of slender bars concentrated in an arbitrary small neigbourhood of a smooth surface. We assume that the inclusions are strongly fastened on the board (see the boundary condition (2) below). Most of the studies on the subject are using periodic homogenization (see, e.g., [1,2,4], and the bibliography herein), except [3], where a nonperiodic case of a bulk distribution of inclusions was considered. In this Note, instead of the classical periodicity assumption on the set of inclusions, we impose a certain condition (see (H1) in Theorem 2.1) on the local energy functional (4). We obtain a global homogenized model giving a general setting for various types of surface distribution of nets formed by slender bars. In Section 3 we show that (H1) is satisfied, in particular, in the case of a periodic net of bars having the diameter exponentially small with respect to the periodicity of the net. In this case the coefficients of the homogenized model are calculated in the explicit form. Notation. We denote by C a generic constant independent of ε and we adopte the Einstein convention of summation over repeated indices.
2. Problem statement and formulation of the main result Let Ω be a fixed domain in R3 with a smooth boundary ∂Ω and F (ε) be closed connected set in Ω, such as a net of slender bars. Suppose that F (ε) depends on a parameter ε such that the distance between the bars and their diameters tends to zero as ε → 0. We consider the case of a surface distribution of the set F (ε) , i.e., for sufficiently small ε, the inclusions F (ε) are concentrated in an arbitrary small neighbourhood of a fixed smooth surface Γ ⊂ Ω. In the domain Ω (ε) , Ω (ε) = Ω \ F (ε) , we consider the boundary value problem: −µ u ε − (λ + µ)grad div u ε = K u = 0 ε
on ∂Ω
in Ω (ε)
(ε)
(1) (2)
where K ∈ [L2 (Ω)]3 is a given vector-function, and λ, µ are Lame’s constants. We study the asymptotic behaviour of u ε (x) solution of problem (1), (2) as ε → 0. Let us introduce quantitative characteristics of set F (ε) . Let S be an arbitrary part of surface Γ and T (S, δ) = {y + αny , y ∈ S, |α| δ}, where ny is the normal vector to S at the point y. Thus T (S, δ) is a layer of thikness 2δ with the central surface S. Denote by Sδ+ and Sδ− the bases of the layer and by T (ε)(S, δ) = T (S, δ) \ F (ε) . Consider the functional Qa (ε, S, δ) = inf W ( v ε ) dx (3) ε v
T (ε) (S,δ)
Here the infimum is taken in the class of vector functions v ε ∈ [H 1 (T (ε) (S, δ))]3 equal zero on F (ε) and a on Sδ± , where a is an arbitrary vector in R3 ; W ( u) = W ( u, u ) is given by 1 ∂ui ∂uk W ( u, v) = Ciklm Eik ( u)Elm ( v ), Eik ( u) = + 2 ∂xk ∂xi where Ciklm are the constants appearing in Hook’s law. The functional Qa (ε, S, δ) can be represented in the form Qa (ε, S, δ) = q ik (ε, S, δ)ai ak
M. Goncharenko, L. Pankratov / C. R. Mecanique 331 (2003) 829–834
with
831
q ik (ε, S, δ) =
W ( v k,ε , v i,ε ) dx
(4)
T (ε) (S,δ)
where v k,ε is the vector-function which minimizes (3) when a = e k (e k is the unit vector of the axis xk ). The main result of the Note is the following: Theorem 2.1. Assume that for any S ⊂ Γ the following condition is fulfilled: lim lim q ik (ε, S, δ) = lim lim q ik (ε, S, δ) = q ik (x) dS δ→0 ε→0
δ→0 ε→0
(H1)
S
where q ik (x) is a continuous function on S. Then u ε solution of problem (1), (2), extended by zero on F (ε) , converges in [L2 (Ω)]3 to u solution of − u − (λ + µ) grad div u = K +
in Ω \ Γ
(5)
−
u) on Γ ( u) = ( − + t ( u u) = Q u) − t (
(6) on Γ
where Q = Q(x) is a matrix with the elements
(7) q ik
= q ik (x)
and
t( u) = Ciklm Elm ( u) cos( n, e i )e k is the stress vector. Here the signs ± indicate the values of the functions u and t ( u) on the different sides of the surface Γ .
3. Sketch of the proof of Theorem 2.1 The solution u ε of problem (1), (2) minimizes the functional u ε ) dx J ε ( u ε) = W ( u ε ) − 2(K,
(8)
Ω (ε)
in the class of vector-functions u ε ∈ [H01 (Ω (ε))]3 . We extend u ε by zero on F (ε) keeping for these functions the same notation. Then Korn’s and Friedrichs inequalities imply [L2 (Ω)]3 u ε [H 1 (Ω)]3 CK
(9)
Therefore the sequence of functions { u ε } is a weakly compact set in [H 1 (Ω)]3 and one can extract a ε subsequence { u , ε = εk → 0} weakly convergent to a function u ∈ [H 1(Ω)]3 . Let us show that u is the solution of problem (5)–(7). Let Γ = i Si with Si ∩ Sj = ∅ (the boundary li of Si is assumed to be piecewise smooth). We assume that the diameters of Si are of order h, where ε h 1. Let δ be a small parameter such that δ < 2δ . Let T (Si , δ ) be a layer of thickness 2δ with center surface Si . Consider the function ϕδ ∈ W21 ( i T (Si , δ )) such that ϕδ ≡ 0 in the 2δ -neighbourhood of curve l = i li and ϕδ ≡ 1 outside of δ/2-neighbourhood of curve l, 0 ϕδ 1 everywhere and |∇ϕδ |2 dx → 0 lim δ →0
Ω
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Let w be a vector-function from the space [C02 (Ω)]3 . Consider a function in Ω \ i T (Si , δ) w(x) w ε (x) = w(x)ϕ δ (x) in i (T (Si , δ)) \ i (T (Si , δ )) wk v k,i,ε ϕδ in i T (ε) (Si , δ )
(10)
where v k,i,ε is a vector-function minimizing (3) when a = e k in T (Si , δ ). Since u ε is a function minimazing (8), we have J ε ( u ε ) J ε (w ε)
(11)
Let us estimate the right-hand side of (11). It follows from the conditions of Theorem 2.1 and the definitions of w ε and ϕδ that def w) W (w) − 2(K, dx + q lk wl wk dΓ lim J ε ( u ε ) Jhom (w) = ε→0
Ω
Let us show now that if
u ε
converges weakly in
(12)
Γ
[H 1(Ω)]3
to u ∈ [H 1 (Ω)]3 then
Jhom ( u) lim J ε ( u ε)
(13)
ε=εk →0
Let u σ ∈ C02 be a vector-function such that u − u σ [H 1 (Ω)]3 < σ σ = u − u σ instead of w. It is easy to see that, for sufficiently small We construct the function w σ,ε by (10) with w ε, σ [H 1 (Ω)]3 w σ,ε [H 1 (Ω)]3 Cw and w σ,ε converges weakly in [H 1 (Ω)]3 to w σ . σ,ε ε σ,ε . Then it is evident that Let u = u − w u − u σ [H 1 (Ω)]3 u σ,ε − u ε [H 1 (Ω)]3 Let Γ = i Si with Si ∩ Sj = ∅ (i = j ) and let T (Si , δ) be the layer of thikness 2δ with the center surface Si . Define Ωσ,A = {x: x ∈ Ω, |uσi | > A}, where A > 0. In the set Ωσ,A we consider the vector-function V i with the components ε uσj (x i ) uσ,ε (x) σ i u , j = 1, 2, 3 (14) (x ) + χ (x) u (x) − u (x) δ j j uσj (x) j uσj (x) where xi ∈ Si and χδ ∈ C0∞ (Ω) such that χδ (x) = 0 for x ∈ i T (Si , δ ) and χδ (x) = 1 for x ∈ Ω \ i T (Si , δ). The function V i equals u σ (x i ) for x ∈ Ω \ i T (Si , δ) and zero for x ∈ F (ε) . Therefore, W (V i ) dx Quσ (x i ) (s, Si , δ) (15) Vji (x) =
T (Si ,δ)
It follows now from (15) that σ,ε W ( u ) dx T (Si ,δ)
W ( u σ ) dx + Quσ (x i ) (s, Si , δ ) + Ei (s, Si , δ, uσ )
(16)
T (Si ,δ)
where lim Ei (s, Si , δ, uσ ) = O(hδ 1/2 σ )
ε=εk →0
(17)
M. Goncharenko, L. Pankratov / C. R. Mecanique 331 (2003) 829–834
833
Therefore, (15)–(17) imply that
ε σ,ε u ) W ( u σ ) dx + q j k (s, Si , δ )uσj (x i )uσk (x i ) J ( Ωσ,A
−2
i
u σ,ε ) dx + O(δ ) + O(δ 1/2 h−1 ε1/2 ) (K, 2
Ω
Passing to the limit in this inequality as ε → 0 and δ → 0 and then as A → 0 and σ → 0 we get (13). Finally, it follows from (11), (13) that u) Jhom (w) Jhom ( is a vector-function from the space where u is the weak limit of the solutions u ε of problem (1), (2) and w ∈ [H01 (Ω)]3 . Since the solution of problem (5)–(7) is unique, all the [C02 (Ω)]3 . Then it remains true for any w sequence of solutions of problem (1), (2) extended by zero on F (ε) converges to u solution of (5)–(7). Theorem 2.1 is proved. ✷ 4. Periodic example In a general case, the elements of the matrix Q in (7) are given by the complicated characteristic (3) of the set F (ε) . However, in some special cases they can be calculated explicitely. Suppose that the set F (ε) consists of a large number of thin cylinders whose axes belong to the surface Γ = {x = (x1 , x2 , x3 ), x3 = 0}. We denote by d ε = e−a/ε the diameters of the cylinders. The axes of the cylinders are parallel to the axes x1 and x2 . The distance between the cylinder axes we denote by l ε = lε. We represent Γ as a union of squares Ki , Ki = K(x i , h), centered at x i and the side length h > 0. We suppose that the sides of the squares Ki are parallel to the axes of the cylinders. We denote by T (Ki , δ) the layer with the center surface Ki . Let us calculate q 11 . To this end we introduce the functions 1 e , r < dε r d ε −1 ε r (18) V 1 = ln ε ln ε 1 , d ε < r < r1ε ψ ε e r r r 1 1 1 (0, 0, 0), r > r1ε e 1 , r < dε r V 2 = (ξ(r, ϕ), 0, η(r, ϕ))ψ ε ε , d ε < r < r1ε , 0 ϕ 2π (19) r1 (0, 0, 0), r > r1ε Here ψ ε ∈ C02 (Ω) such that ψ ε (t) = 1 for 0 t 1/2 and ψ ε (t) = 0 for t 3/4; r1ε = εβ (β > 3/2); the functions ξ(r, ϕ) and η(r, ϕ) are the components of the vector-function g = (ξ, η) which is the solution of the following auxiliary problem: g − (λ + µ) grad div g = 0, A0 g = −µ g = e , 1
r =d ; ε
Consider the functions
uε = e 1 − (T (ε) (K,δ))1
g = 0, V 1 ,
r
d ε < r < r1ε ,
0 ϕ 2π
= r1ε
uˆ ε = e 1 −
(T (ε) (K,δ))2
V 2
(20)
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M. Goncharenko, L. Pankratov / C. R. Mecanique 331 (2003) 829–834
Here the indices 1 and 2 mean that the summation is taken over the cylinders that parallel to the axes x1 and x2 , respectively. We introduce the function w ε by setting (21) w ε (x) = uˆ ε1 uε1 , 0, uε1 uˆ ε3 It is clear that w ε belongs to the same class that the function minimizing (4), therefore, for sufficiently small ε, h2 λ + 2µ q 11 ε, T (ε) (K, δ), δ W (w ε ) dx = 2πµ ε 1 + 2 + o(δ) (22) l | ln d ε | λ + 3µ T (ε) (K,δ)
as δ → 0. Using now the explicit form of the function w ε one can show that q 11 (ε, T (ε) (K, δ), δ) can be also estimated as follows h2 λ + 2µ (23) 2πµ ε 1 + 2 + o(δ) q 11 ε, T (ε) (K, δ), δ ε l | ln d | λ + 3µ as δ → 0. Then from (22) and (23) we finally get 1 λ + 2µ 11 q = 2πµ 1+2 la λ + 3µ In a similar way we obtain that 22 = q 11 = 2πµ 1 1 + 2 λ + 2µ q la λ + 3µ 1 λ + 2µ q 33 = 4πµ la λ + 3µ ik q = 0 (i = k) Acknowledgements This paper was completed while M. Goncharenko was visiting the University Paris-6. Her research was supported by a NATO grant. M. Goncharenko is grateful to Professor F. Murat for his attention and hospitality. The authors are grateful to Professor E. Khruslov for fruitful discussions.
References [1] N.S. Bakhvalov, G.P. Panasenko, Homogenization: Averaging Processes in Periodic Media, Kluwer, Dordrecht–Boston–London, 1989. [2] D. Cioranescu, J. Saint Jean Paulin, Homogenization of Reticulated Structures, in: Applied Mathematical Sciences, Vol. 136, SpringerVerlag, New York–Berlin–Heidelberg, 1999. [3] E. Khruslov, V. Marchenko, Boundary Value Problems in Domains with Fine-Grained Boundaries, Naukova Dumka, 1974 (in Russian). [4] É. Sanchez–Palencia, Nonhomogeneous Media and Vibration Theory, in: Lecture Notes in Phys., Vol. 127, Springer-Verlag, New York, 1980.
Homogenization of an elasticity problem in domains with a net of slender bars near surface Mariya Goncharenko a,b , Leonid Pankratov a,b a Laboratoire Jacques-Louis Lions de l’Université Pierre et Marie Curie, 4, place Jussieu, 75252 Paris cedex 05, France b Département de mathématiques, B.Verkin institut des basses températures (FTINT), 47, av. Lénine, 61103 Kharkov, Ukraine
Received 11 June 2003; accepted 23 September 2003 Presented by Évariste Sanchez-Palencia
Abstract We consider an elasticity problem in a domain Ω (ε) = Ω \ F (ε) , where Ω is an open bounded domain in R3 , F (ε) is a connected nonperiodic set in Ω like a net of slender bars, and ε is a parameter characterizing the microstructure of the domain. We consider the case of a surface distribution of the set F (ε) , i.e., for sufficiently small ε, the set F (ε) is concentrated in arbitrary small neighbourhood of a surface Γ . Under a hypothesis on the asymptotic behaviour of the energy functional, we obtain the macroscopic (homogenized) model. To cite this article: M. Goncharenko, L. Pankratov, C. R. Mecanique 331 (2003). 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved. Résumé Homogénéisation d’un problème d’élasticité dans des domaines avec un réseau de barres fines proche de la surface. On étudie un problème d’élasticité dans un domaine Ω (ε) = Ω \ F (ε) , où Ω est un ouvert borné dans R3 , F (ε) est un ensemble non-périodique connexe dans Ω de type un réseau de barres fines et ε est un paramétre qui caractérise la microstructure du domaine. On considère le cas d’une distribution surfacique de l’ensemble F (ε) , c’est-á-dire, lorsque ε → 0, cet ensemble se concentre dans un voisinage d’une surface Γ aussi petit que l’on veut. Sous une hypothèse sur le comportement asymptotique de la fonctionelle d’énergie on obtient le modèle macroscopique (homogénéisé). Pour citer cet article : M. Goncharenko, L. Pankratov, C. R. Mecanique 331 (2003). 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved. Keywords: Computational solid mechanics; Elasticity problem Mots-clés : Mécanique de solides numérique ; Problème d’élasticité
E-mail addresses: [email protected], [email protected] (M. Goncharenko), [email protected], [email protected] (L. Pankratov). 1631-0721/$ – see front matter 2003 Académie des sciences. Published by Elsevier SAS. All rights reserved. doi:10.1016/j.crme.2003.09.005
830
M. Goncharenko, L. Pankratov / C. R. Mecanique 331 (2003) 829–834
1. Introduction We study a homogenization problem for the elasticity equation which describes the equilibrium of an elastic medium with rigid inclusions, such as a nonperiodic net of slender bars concentrated in an arbitrary small neigbourhood of a smooth surface. We assume that the inclusions are strongly fastened on the board (see the boundary condition (2) below). Most of the studies on the subject are using periodic homogenization (see, e.g., [1,2,4], and the bibliography herein), except [3], where a nonperiodic case of a bulk distribution of inclusions was considered. In this Note, instead of the classical periodicity assumption on the set of inclusions, we impose a certain condition (see (H1) in Theorem 2.1) on the local energy functional (4). We obtain a global homogenized model giving a general setting for various types of surface distribution of nets formed by slender bars. In Section 3 we show that (H1) is satisfied, in particular, in the case of a periodic net of bars having the diameter exponentially small with respect to the periodicity of the net. In this case the coefficients of the homogenized model are calculated in the explicit form. Notation. We denote by C a generic constant independent of ε and we adopte the Einstein convention of summation over repeated indices.
2. Problem statement and formulation of the main result Let Ω be a fixed domain in R3 with a smooth boundary ∂Ω and F (ε) be closed connected set in Ω, such as a net of slender bars. Suppose that F (ε) depends on a parameter ε such that the distance between the bars and their diameters tends to zero as ε → 0. We consider the case of a surface distribution of the set F (ε) , i.e., for sufficiently small ε, the inclusions F (ε) are concentrated in an arbitrary small neighbourhood of a fixed smooth surface Γ ⊂ Ω. In the domain Ω (ε) , Ω (ε) = Ω \ F (ε) , we consider the boundary value problem: −µ u ε − (λ + µ)grad div u ε = K u = 0 ε
on ∂Ω
in Ω (ε)
(ε)
(1) (2)
where K ∈ [L2 (Ω)]3 is a given vector-function, and λ, µ are Lame’s constants. We study the asymptotic behaviour of u ε (x) solution of problem (1), (2) as ε → 0. Let us introduce quantitative characteristics of set F (ε) . Let S be an arbitrary part of surface Γ and T (S, δ) = {y + αny , y ∈ S, |α| δ}, where ny is the normal vector to S at the point y. Thus T (S, δ) is a layer of thikness 2δ with the central surface S. Denote by Sδ+ and Sδ− the bases of the layer and by T (ε)(S, δ) = T (S, δ) \ F (ε) . Consider the functional Qa (ε, S, δ) = inf W ( v ε ) dx (3) ε v
T (ε) (S,δ)
Here the infimum is taken in the class of vector functions v ε ∈ [H 1 (T (ε) (S, δ))]3 equal zero on F (ε) and a on Sδ± , where a is an arbitrary vector in R3 ; W ( u) = W ( u, u ) is given by 1 ∂ui ∂uk W ( u, v) = Ciklm Eik ( u)Elm ( v ), Eik ( u) = + 2 ∂xk ∂xi where Ciklm are the constants appearing in Hook’s law. The functional Qa (ε, S, δ) can be represented in the form Qa (ε, S, δ) = q ik (ε, S, δ)ai ak
M. Goncharenko, L. Pankratov / C. R. Mecanique 331 (2003) 829–834
with
831
q ik (ε, S, δ) =
W ( v k,ε , v i,ε ) dx
(4)
T (ε) (S,δ)
where v k,ε is the vector-function which minimizes (3) when a = e k (e k is the unit vector of the axis xk ). The main result of the Note is the following: Theorem 2.1. Assume that for any S ⊂ Γ the following condition is fulfilled: lim lim q ik (ε, S, δ) = lim lim q ik (ε, S, δ) = q ik (x) dS δ→0 ε→0
δ→0 ε→0
(H1)
S
where q ik (x) is a continuous function on S. Then u ε solution of problem (1), (2), extended by zero on F (ε) , converges in [L2 (Ω)]3 to u solution of − u − (λ + µ) grad div u = K +
in Ω \ Γ
(5)
−
u) on Γ ( u) = ( − + t ( u u) = Q u) − t (
(6) on Γ
where Q = Q(x) is a matrix with the elements
(7) q ik
= q ik (x)
and
t( u) = Ciklm Elm ( u) cos( n, e i )e k is the stress vector. Here the signs ± indicate the values of the functions u and t ( u) on the different sides of the surface Γ .
3. Sketch of the proof of Theorem 2.1 The solution u ε of problem (1), (2) minimizes the functional u ε ) dx J ε ( u ε) = W ( u ε ) − 2(K,
(8)
Ω (ε)
in the class of vector-functions u ε ∈ [H01 (Ω (ε))]3 . We extend u ε by zero on F (ε) keeping for these functions the same notation. Then Korn’s and Friedrichs inequalities imply [L2 (Ω)]3 u ε [H 1 (Ω)]3 CK
(9)
Therefore the sequence of functions { u ε } is a weakly compact set in [H 1 (Ω)]3 and one can extract a ε subsequence { u , ε = εk → 0} weakly convergent to a function u ∈ [H 1(Ω)]3 . Let us show that u is the solution of problem (5)–(7). Let Γ = i Si with Si ∩ Sj = ∅ (the boundary li of Si is assumed to be piecewise smooth). We assume that the diameters of Si are of order h, where ε h 1. Let δ be a small parameter such that δ < 2δ . Let T (Si , δ ) be a layer of thickness 2δ with center surface Si . Consider the function ϕδ ∈ W21 ( i T (Si , δ )) such that ϕδ ≡ 0 in the 2δ -neighbourhood of curve l = i li and ϕδ ≡ 1 outside of δ/2-neighbourhood of curve l, 0 ϕδ 1 everywhere and |∇ϕδ |2 dx → 0 lim δ →0
Ω
832
M. Goncharenko, L. Pankratov / C. R. Mecanique 331 (2003) 829–834
Let w be a vector-function from the space [C02 (Ω)]3 . Consider a function in Ω \ i T (Si , δ) w(x) w ε (x) = w(x)ϕ δ (x) in i (T (Si , δ)) \ i (T (Si , δ )) wk v k,i,ε ϕδ in i T (ε) (Si , δ )
(10)
where v k,i,ε is a vector-function minimizing (3) when a = e k in T (Si , δ ). Since u ε is a function minimazing (8), we have J ε ( u ε ) J ε (w ε)
(11)
Let us estimate the right-hand side of (11). It follows from the conditions of Theorem 2.1 and the definitions of w ε and ϕδ that def w) W (w) − 2(K, dx + q lk wl wk dΓ lim J ε ( u ε ) Jhom (w) = ε→0
Ω
Let us show now that if
u ε
converges weakly in
(12)
Γ
[H 1(Ω)]3
to u ∈ [H 1 (Ω)]3 then
Jhom ( u) lim J ε ( u ε)
(13)
ε=εk →0
Let u σ ∈ C02 be a vector-function such that u − u σ [H 1 (Ω)]3 < σ σ = u − u σ instead of w. It is easy to see that, for sufficiently small We construct the function w σ,ε by (10) with w ε, σ [H 1 (Ω)]3 w σ,ε [H 1 (Ω)]3 Cw and w σ,ε converges weakly in [H 1 (Ω)]3 to w σ . σ,ε ε σ,ε . Then it is evident that Let u = u − w u − u σ [H 1 (Ω)]3 u σ,ε − u ε [H 1 (Ω)]3 Let Γ = i Si with Si ∩ Sj = ∅ (i = j ) and let T (Si , δ) be the layer of thikness 2δ with the center surface Si . Define Ωσ,A = {x: x ∈ Ω, |uσi | > A}, where A > 0. In the set Ωσ,A we consider the vector-function V i with the components ε uσj (x i ) uσ,ε (x) σ i u , j = 1, 2, 3 (14) (x ) + χ (x) u (x) − u (x) δ j j uσj (x) j uσj (x) where xi ∈ Si and χδ ∈ C0∞ (Ω) such that χδ (x) = 0 for x ∈ i T (Si , δ ) and χδ (x) = 1 for x ∈ Ω \ i T (Si , δ). The function V i equals u σ (x i ) for x ∈ Ω \ i T (Si , δ) and zero for x ∈ F (ε) . Therefore, W (V i ) dx Quσ (x i ) (s, Si , δ) (15) Vji (x) =
T (Si ,δ)
It follows now from (15) that σ,ε W ( u ) dx T (Si ,δ)
W ( u σ ) dx + Quσ (x i ) (s, Si , δ ) + Ei (s, Si , δ, uσ )
(16)
T (Si ,δ)
where lim Ei (s, Si , δ, uσ ) = O(hδ 1/2 σ )
ε=εk →0
(17)
M. Goncharenko, L. Pankratov / C. R. Mecanique 331 (2003) 829–834
833
Therefore, (15)–(17) imply that
ε σ,ε u ) W ( u σ ) dx + q j k (s, Si , δ )uσj (x i )uσk (x i ) J ( Ωσ,A
−2
i
u σ,ε ) dx + O(δ ) + O(δ 1/2 h−1 ε1/2 ) (K, 2
Ω
Passing to the limit in this inequality as ε → 0 and δ → 0 and then as A → 0 and σ → 0 we get (13). Finally, it follows from (11), (13) that u) Jhom (w) Jhom ( is a vector-function from the space where u is the weak limit of the solutions u ε of problem (1), (2) and w ∈ [H01 (Ω)]3 . Since the solution of problem (5)–(7) is unique, all the [C02 (Ω)]3 . Then it remains true for any w sequence of solutions of problem (1), (2) extended by zero on F (ε) converges to u solution of (5)–(7). Theorem 2.1 is proved. ✷ 4. Periodic example In a general case, the elements of the matrix Q in (7) are given by the complicated characteristic (3) of the set F (ε) . However, in some special cases they can be calculated explicitely. Suppose that the set F (ε) consists of a large number of thin cylinders whose axes belong to the surface Γ = {x = (x1 , x2 , x3 ), x3 = 0}. We denote by d ε = e−a/ε the diameters of the cylinders. The axes of the cylinders are parallel to the axes x1 and x2 . The distance between the cylinder axes we denote by l ε = lε. We represent Γ as a union of squares Ki , Ki = K(x i , h), centered at x i and the side length h > 0. We suppose that the sides of the squares Ki are parallel to the axes of the cylinders. We denote by T (Ki , δ) the layer with the center surface Ki . Let us calculate q 11 . To this end we introduce the functions 1 e , r < dε r d ε −1 ε r (18) V 1 = ln ε ln ε 1 , d ε < r < r1ε ψ ε e r r r 1 1 1 (0, 0, 0), r > r1ε e 1 , r < dε r V 2 = (ξ(r, ϕ), 0, η(r, ϕ))ψ ε ε , d ε < r < r1ε , 0 ϕ 2π (19) r1 (0, 0, 0), r > r1ε Here ψ ε ∈ C02 (Ω) such that ψ ε (t) = 1 for 0 t 1/2 and ψ ε (t) = 0 for t 3/4; r1ε = εβ (β > 3/2); the functions ξ(r, ϕ) and η(r, ϕ) are the components of the vector-function g = (ξ, η) which is the solution of the following auxiliary problem: g − (λ + µ) grad div g = 0, A0 g = −µ g = e , 1
r =d ; ε
Consider the functions
uε = e 1 − (T (ε) (K,δ))1
g = 0, V 1 ,
r
d ε < r < r1ε ,
0 ϕ 2π
= r1ε
uˆ ε = e 1 −
(T (ε) (K,δ))2
V 2
(20)
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Here the indices 1 and 2 mean that the summation is taken over the cylinders that parallel to the axes x1 and x2 , respectively. We introduce the function w ε by setting (21) w ε (x) = uˆ ε1 uε1 , 0, uε1 uˆ ε3 It is clear that w ε belongs to the same class that the function minimizing (4), therefore, for sufficiently small ε, h2 λ + 2µ q 11 ε, T (ε) (K, δ), δ W (w ε ) dx = 2πµ ε 1 + 2 + o(δ) (22) l | ln d ε | λ + 3µ T (ε) (K,δ)
as δ → 0. Using now the explicit form of the function w ε one can show that q 11 (ε, T (ε) (K, δ), δ) can be also estimated as follows h2 λ + 2µ (23) 2πµ ε 1 + 2 + o(δ) q 11 ε, T (ε) (K, δ), δ ε l | ln d | λ + 3µ as δ → 0. Then from (22) and (23) we finally get 1 λ + 2µ 11 q = 2πµ 1+2 la λ + 3µ In a similar way we obtain that 22 = q 11 = 2πµ 1 1 + 2 λ + 2µ q la λ + 3µ 1 λ + 2µ q 33 = 4πµ la λ + 3µ ik q = 0 (i = k) Acknowledgements This paper was completed while M. Goncharenko was visiting the University Paris-6. Her research was supported by a NATO grant. M. Goncharenko is grateful to Professor F. Murat for his attention and hospitality. The authors are grateful to Professor E. Khruslov for fruitful discussions.
References [1] N.S. Bakhvalov, G.P. Panasenko, Homogenization: Averaging Processes in Periodic Media, Kluwer, Dordrecht–Boston–London, 1989. [2] D. Cioranescu, J. Saint Jean Paulin, Homogenization of Reticulated Structures, in: Applied Mathematical Sciences, Vol. 136, SpringerVerlag, New York–Berlin–Heidelberg, 1999. [3] E. Khruslov, V. Marchenko, Boundary Value Problems in Domains with Fine-Grained Boundaries, Naukova Dumka, 1974 (in Russian). [4] É. Sanchez–Palencia, Nonhomogeneous Media and Vibration Theory, in: Lecture Notes in Phys., Vol. 127, Springer-Verlag, New York, 1980.