mathematical analysis and homogenization of the torsion...

JOURNAL OF c© 2008, Scientific HorizonFUNCTION SPACES AND APPLICATIONS http://www.jfsa.net

Volume 6, Number 2 (2008), 155-176

Mathematical analysis and homogenization

of the torsion problem

Dag Lukkassen, Annette Meidell and Peter Wall

(Communicated by Alois Kufner)

2000 Mathematics Subject Classification. 35B27.

Keywords and phrases. Effective properties, torsion stiffness.

Abstract. In this paper we address the limitations of the classical formulationof the torsion problem and give a self-contained survey of the function spacesand formulations that are more suitable for analysing the torsional behavior ofcomposite materials. We also prove some homogenization results for the torsionproblem in both the periodic and stochastic setting. Our theoretical results areillustrated by numerical examples.

1. Introduction

The first investigation on the torsion problem goes back to the works ofCoulomb and Navier (which included some erroneous conclusions). Themost classical results, including applications to a number of cases oftechnical importance, is mainly due to Saint-Venant. The generalization ofthe Saint-Venant theory to torsion of bars consisting of different materialsjoined along their side surfaces has also been known for a long time (see[13], [14] and [15]). This classical theory, which still is popular especially inthe elasticity community, requires that the surfaces separating the different

156 Mathematical analysis and homogenization of torsion problem

materials are sufficiently smooth. Weak and variational formulations ofthe torsion problem defined on Sobolev spaces enable us to relax thisrequirement, but are also suitable for other reasons, in particular forobtaining bounds on the torsional rigidity and analyzing the limit torsionalbehavior of composite materials when the characteristic length of themicrostructures (e.g. the diameter of the fibres) approaches 0. Thisanalysis is based on the homogenization method, which was initiated byDe Giorgi and Spagnolo in the late 60’s and further developed during thelast 30 years by many well-known mathematicians (for more informationconcerning the homogenization theory see e.g. the books [4], [6] and [17]).Estimating this limit behavior is of great practical importance since anydirect numerical treatment leads to difficulties when the characteristic sizeof the inhomogeneity is small compared with the global geometry (seeFigure 1).

Figure 1. Torsion of composite bar with simply connectedcross section

In this paper we address the limitations of the classical formulation ofthe torsion problem and discuss formulations that are more suitable foranalyzing the mechanical behavior of composites materials (see section 2).We also study some function spaces (see Section 3) which are necessary inconnection with the analysis of the corresponding dual problem. The restof the paper considers homogenization of the torsion problem in both theperiodic and the stochastic setting (see Section 5 and Section 6). In orderto illustrate our theoretical results we even present some concrete numericalcomputations.

For the purpose of making the paper readable to a broader audience,including scientists within the mathematical elasticity community, as well

D. Lukkassen, A. Meidell and P. Wall 157

as pure mathematicians, we have made an effort making the presentationas self-contained as possible starting from the standard equations of elasticequilibrium.

2. Classical and weak formulations

We consider a locally isotropic composite bar with Lamé constantsλ = λ(x, y), μ = (x, y) and cross-section Ω ⊂ R2. Let Ω be an open,bounded and connected set of class C2 with boundary ∂Ω consisting of afinite number of connected components L1, L2 ,...,Lm+1 (disconnected fromeach other) where Lm+1 surrounds the others. The torsion problem consistsof determining the function ϕ(x, y) such that the displacement components(u, v, w) at each point (x, y, z) are given by

(1) u = −τyz, v = τxz, w = τϕ(x, y)

(τ is a constant called the relative twist) under the assumption of no bodyforces present and that the side surface of the bar is free from externalstresses. More precisely, we must satisfy the equations of elastic equilibrium

∂σxx∂x

+∂σxy∂y

+∂σxz∂z

= 0,

∂σyx∂x

+∂σyy∂y

+∂σyz∂z

= 0,

∂σzx∂x

+∂σzy∂y

+∂σzz∂z

= 0,

σxx = λθ + 2μ∂u

∂x, σyy = λθ + 2μ

∂v

∂y, σzz = λθ + 2μ

∂w

∂z,(2)

σyz = μ(∂w

∂y+∂v

∂z

), σzx = μ

(∂w

∂x+∂u

∂z

), σxy = μ

(∂v

∂x+∂u

∂y

),(3)

where

θ =∂u

∂x+∂v

∂y+∂w

∂z,

together with the boundary conditions on the side surface

σxxnx + σxyny = 0

σyxnx + σyyny = 0

σzxnx + σzyny = 0,

where n = (nx, ny) is the outward normal to the boundary ∂Ω. In additionwe have the symmetry property σxy = σyx , σxz = σzx and σyz = σzy .


Inserting (1) into (2) and (3) we obtain the stress components

(4) σxz = μτ(∂ϕ

∂x− y

), σyz = μτ

(∂ϕ

∂y+ x

),

andσxx = σyy = σzz = σxy = 0.

Thus, we observe that the above equations of elastic equilibrium andboundary conditions are satisfied, provided

(5) div (σxz , σyz) = 0,

and

(6) (σxz , σyz) · n = 0 on ∂Ω

(the latter merely tells that the external surface of the bare is free fromexternal forces). Hence, we only have to solve the problem

(7)

⎧⎨⎩div μ(gradϕ+ (−y, x)) = 0 in Ω,μ(∂ϕ∂x − y)nx + μ(∂ϕ∂y + x)ny = 0 on ∂Ω.This problem can only be understood in classical sense in parts of Ω wherethe shear modulus μ = μ(x, y) is sufficiently smooth. However, even pioneerworks in the mathematical theory of elasticity deal with cases where μmay be discontinuous. For example the book Muskhelisvili [15] considersthe case of bars consisting of prismatic (cylindrical) connected componentsmade of different materials and joined along their side surfaces (see also[16]). Assuming that these surfaces are sufficiently smooth, it makes senseto define the normal component (σxz, σyz) · n = σxznx + σyzny of thestress vector (σxz, σyz) on these surfaces. Then, using the condition thatthe forces acting on any surfaces, in particular the surfaces that separatesdifferent materials, are equal in magnitude and opposite in direction, wefind that the normal component (σxz, σyz) · n must be continuous in thedirection n . This implies that (σxz , σyz) · n is equal on opposite sides ofsurfaces separating different materials.

Let Si ⊂ Ω, i = 1, 2, ...,m denote the region of the rod with constantshear modulus μi such that ∪mi=1Si = Ω. By (5)

(8) div (σxz, σyz) = 0 on Si,

in classical sense. On a point of the common boundary between Si and Skwith normal ni (pointing out of Si ) , the normal component (σxz , σyz) · ni


on the set Si, denoted[(σxz, σyz) · ni

]i, must equal on the set Sk, which

is the negative value of the normal component (σxz, σyz) · nk on the set Skwhere nk = −ni (the normal is pointing out of Sk ), i.e.

(9)[(σxz , σyz) · ni

]i= − [(σxz , σyz) · nk]k on ∂Si ∩ ∂Sk.

Multiplying the equation (8) with an arbitrary function v, which is smoothon Ω, we obtain by the Green formula that∫

Si

(σxz, σyz) · gradv dxdy =∫

∂Si

((σxz , σyz) · ni

)v dxdy.

But by (9) and (6) we see that

m∑i=1

∫∂Si

((σxz , σyz) · ni

)v dxdy = 0.

So we conclude that∫Ω

(σxz , σyz) · grad v dxdy =m∑

i=1

∫Si

(σxz, σyz) · grad v dxdy = 0.

This gives us the following weak formulation of the problem: Find ϕ ∈H1(Ω) such that

(10)∫

Ω

μτ

(∂ϕ

∂x− y

))∂v

∂x+ μτ

(∂ϕ

∂y+ x

)∂v

∂ydxdy = 0

for all v ∈ H1(Ω) (the twist-constant may certainly be cancelled) . Thisproblem has a unique solution up to an arbitrary additive constant (whichis easily seen by the Lax-Milgram Lemma), even under much more generalconditions for the shear modulus μ than described above. In fact it isenough to assume that μ : Ω → R is Lebesgue measurable and that thereexist constants α, β such that

(11) 0 < α ≤ μ(x, y) ≤ β


This makes it necessary to understand the concept of divergence andnormal derivative at the boundary in a more abstract way. For any vectorfunction u ∈ L2(Ω) (the set of all u = (u1, u2), ui ∈ L2(Ω)) we definediv u = ∂u1/∂x1 + ∂u2/∂x2 (in sense of distributions). More precisely,div u ∈ H−1(Ω) is defined by

〈div u, v〉 = −∫

Ω

(gradv) · u dxdy, v ∈ H10 (Ω).

In accordance with the terminology of the distribution theory, we say thatdiv u ∈ L2(Ω) if there exists a function f ∈ L2(Ω), such that

〈div u, v〉 =∫

Ω

fv dxdy for all v ∈ H10 (Ω),

in which case we put div u = f (without causing any ambiguity since fnecessarily must be unique).

There exists a linear continuous operator γ0 : H1(Ω) → L2(∂Ω), thetrace operator, such that γ0v is the restriction of v to ∂Ω for all functionsv ∈ H1(Ω) which is twice continuous differentiable in Ω. The space H 12 (∂Ω)denotes the image γ0(H1(Ω)) (which is dense in L2(∂Ω)) equipped withnorm ‖·‖

H12 (∂Ω)

carried from H1(Ω) by γ0, i.e.

‖γ0(u)‖H

12 (∂Ω)

= inf{‖v‖H1(Ω) : γ0(v) = γ0(u)

}.

Moreover, there exists a linear continuous operator lΩ : H12 (∂Ω) → H1(Ω),

the lifting operator, such that γ0lΩ is the identity in H12 (∂Ω). For more

information about traces of functions belonging to Hm(Ω), see Lions andMagenes [8]. Let H−

12 (∂Ω) denote the dual space of H

12 (∂Ω) and let E(Ω)

denote the set of all u ∈ L2(Ω) such that div u ∈ L2(Ω). The following traceresult (see Temam [18, p. 9]) gives a generalized meaning of the normalderivative at the boundary u · n|∂Ω and a generalization of the classicalGreens formula.

Lemma 1. There exists a continuous linear operator γn : E(Ω) →H−

12 (∂Ω) such that γnu is equal to the restriction of u ·n to ∂Ω (in sense

of distributions, i.e. 〈γnu, γ0v〉 =∫

∂Ω (u · n) (γ0v) ds for all v ∈ H1(Ω)) forevery function u which is twice continuous differentiable in Ω . Moreover,the generalized Greens formula

(12)∫

Ω

u · gradw dxdy +∫

Ω

w div u dxdy = 〈γnu, γ0w〉 ,

is valid for all u ∈ E(Ω) and all w ∈ H1(Ω).


In the two dimensional case it is usual to define curlu as follows:

curlu = (−∂u/∂x2, ∂u/∂x1) if u ∈ L2(Ω),curlu = ∂u1/∂x2 − ∂u2/∂x1 if u = (u1, u2) ∈ L2(Ω).

If u is not a differentiable function, these definitions must be understood insense of distributions. More precisely, if u ∈ L2(Ω) (L2(Ω), respectively),then curlu ∈ H−1(Ω) (H−1(Ω), respectively), is defined by

〈curlu, v〉 = −∫

Ω

(curl v) ·u dxdy, for all v ∈ H10(Ω) (H10 (Ω), respectively).

Here, H−1(Ω) denotes the dual of the space H10(Ω) = {v = (v1, v2) : vi ∈H10 (Ω)} . Similarly as for the divergence, we say that curlu ∈ L2(Ω) (L2(Ω),respectively) if there exists a function f ∈ L2(Ω) (L2(Ω), respectively) ,such that

〈curlu, v〉 =∫

Ω

f · v dxdy, for all v ∈ H10(Ω) (H10 (Ω), respectively),

in which case we put curlu = f (without causing any ambiguity since fnecessarily must be unique).

Let V and H be the spaces defined by

V = {u ∈ D(Ω) : div u = 0} ,

(13) H ={u ∈ L2(Ω) : div u = 0, γnu = 0

},

where D(Ω) is the space of smooth functions with compact support in Ω.It turns out that H equals the closure of V in L2(Ω) (see [18]) . Moreover,H⊥, the orthogonal complement of H in L2(Ω), is given by

(14) H⊥ ={u ∈ L2(Ω) : u = grad p, p ∈ H1(Ω)}

(see [18, p. 15]). The space

curl(H1(Ω)

)=

{curl v : v ∈ H1(Ω)}

can be characterized as follows(15)

curl(H1(Ω)

)=

{u ∈ L2(Ω) : div u = 0, 〈γnu, vi〉 = 0, i = 1, 2, ...,m

},

where vi ∈ D(Ω) has the property that vi = 1 on Li and vi = 0 onLj for all j �= i . This result is the same as that found in [18, Appendix


1] with the only difference that 〈γnu, vi〉 = 0 is used instead of∫

Liu · n

ds = 0. In order to see that such a function vi exists, let {Ok}Nk=1 bea finite collections of open disks covering Li but small enough such thatnon of them intersects Lj for all j �= i. Then, due to the theorem ofinfinitely differentiable partitions of unity (see e.g. [1, Theorem 3.14]),we can construct functions ψk ∈ D(R2), with support in Ok, such thatψ(x) :=

∑Nk=1 ψk(x) = 1 for all x ∈ Li. Thus, the function vi = ψ|Ω has

the desired properties.Of independent interest we note that if Ω is simply connected then (15)

reduces tocurl

(H1(Ω)

)=

{u ∈ L2(Ω) : div u = 0} .

This follows directly from (12) by putting w = v1 = 1.

Let us now prove the following characterization of the space H.

Theorem 2. The space H defined in (13) can alternatively be defined asfollows:

H = {curlϕ : ϕ ∈ W} ,where

W ={ϕ ∈ H1(Ω), γ0ϕ = constant on each Li

}.

Proof. By (15) we observe that

V ⊂ H ⊂ curl (H1(Ω)) .Thus for every u ∈ V there exists a function ϕ ∈ H1(Ω) such thatu = curlϕ. By the definition of the space V, u = (−∂ϕ/∂x2, ∂ϕ/∂x1) = 0on a connected open set Oi ⊂ Ω such that Li ⊂ Oi. Thus ϕ =constanton Oi, and therefore γ0ϕ =constant on Li . Since curl (ϕ− 〈ϕ〉) = curlϕ,where 〈ϕ〉 denotes the average value of ϕ (taken over Ω), we may assumethat

∫Ωϕ dx = 0. By the Poincaré inequality we then have that

∫Ω

ϕ2 dxdy ≤ C((∫

Ω

ϕ dxdy

)2+

∫Ω

(gradϕ)2 dxdy

)

≤ C(∫

Ω

(∂ϕ

∂x

)2+

(∂ϕ

∂y

)2dxdy

)

= C(∫

Ω

|curlϕ|2 dxdy)

for some positive constant C, so

(16) ‖ϕ‖H1(Ω) ≤√C + 1 ‖curlϕ‖L2(Ω) .


If u ∈ H then, due to the fact that V is dense in H (with respect to theL2 -norm), we can find a sequence ϕk ∈ H1(Ω),

∫Ωϕk dx = 0, such that

curlϕk → u in L2(Ω). By (16) ϕk is a Cauchy-sequence in H1(Ω), which(by the completeness of H1(Ω)) converges to some function ϕ ∈ H1(Ω).Since

‖curlϕ− curlϕk‖L2(Ω) = ‖curl (ϕ− ϕk)‖L2(Ω) ≤ ‖ϕ− ϕk‖H1(Ω) ,

we obtain that u = curlϕ. The continuity of the trace operator γ0 nowgives that

‖γ0(ϕ) − γ0(ϕk)‖L2(∂Ω) = ‖γ0(ϕ− ϕk)‖L2(∂Ω) ≤ C2 ‖ϕ− ϕk‖H1(Ω) → 0.

Thus, γ0(ϕ) = limk γ0(ϕk), so, since all γ0(ϕk) are constant on each Li,the limit γ0(ϕ) must possess the same property. The proof is complete. �

4. Dual formulation of the torsion problem

When the shear modulus satisfies the general condition (11) we may derivethe weak formulation (10) directly by using the generalized Green formulafrom Lemma 1 by using the assumption that the stress vector

(σxz, σyz) =(μτ

(∂ϕ

∂x− y

), μτ

(∂ϕ

∂y+ x

))∈ H

={u ∈ L2(Ω) : div u = 0, γnu = 0

},

for some ϕ ∈ H1(Ω). The fact that (10) has a solution shows that such astress vector exists. By Theorem 2, (σxz , σyz) = curlψ for some functionψ ∈W . Moreover, we observe that

(17)(

1τμσxz + y,

1τμσyz − x

)= gradϕ ∈ H⊥.

Thus, for all v ∈ V (⊂W ) we have that∫Ω

1τμ

gradψ · gradv dxdy −∫

Ω

(y∂v

∂y+ x

∂v

∂x

)=

∫Ω

(1τμ

(−∂ψ∂y

)+ y,

1τμ

∂ψ

∂x− x

)(− ∂v∂y,∂v

∂x

)dxdy

×∫

Ω

(1τμσxz + y,

1τμσyz − x

)curl v dxdy


=∫

Ω

gradϕ · curl v dxdy.

Since gradϕ ∈ H⊥ by (14) and curl v ∈ H by Theorem 2), this integralvanishes. Hence, putting ψ̃ = ψ/τ (for convenience) we find that thecorresponding weak minimum formulation takes the form: Find ψ ∈ Wsuch that

(18)∫

Ω

1μ

(grad ψ̃ · gradv) dxdy =∫

Ω

(y∂v

∂y+ x

∂v

∂x

)dxdy

for all v ∈ W. Let Ωi denote the region inside the contour Li, and notethat the outward normal to the boundary of Ωi is −n except for i = m+1,where it is n . Then we obtain that∫

Ω

(y∂v

∂y+ x

∂v

∂x

)dxdy

=∫

Ω

(x, y) · gradv dxdy

= −∫

Ω

(div (x, y))v dxdy +∫

∂Ω

((x, y) · n)γ0v ds

= −∫

Ω

2v dxdy +m+1∑i=1

∫Li

((x, y) · n) γ0v|Li ds

= −∫

Ω

2v dxdy +m+1∑i=1

γ0v|Li∫

Li

((x, y) · n) ds

=︸︷︷︸Gauss theorem

−∫

Ω

2v dxdy + γ0v|Lm+1∫

Ωm+1

(div (x, y)) dxdy

−m∑

i=1

γ0v|Li∫

Ωi

(div (x, y)) dxdy

= −∫

Ω

2v dxdy + 2 γ0v|Lm+1 |Ωm+1| − 2m∑

i=1

γ0v|Li |Ωi| .(19)

Assuming that ϕ is twice differentiable in (17), we obtain that

− div 1μ

(grad ψ̃) + 2 =∂

∂y

(1μ

(−∂ψ̃∂y

)+ y

)− ∂∂x

(1μ

∂ψ̃

∂x− x

)

= curl(

1τμσxz + y,

1τμσyz − x

)= curl gradϕ = 0.


Thus, the classical formulation corresponding to (10) takes the form

(20)

⎧⎪⎨⎪⎩div

1μ

(grad ψ̃) = 2 in Ω,

ψ̃ = ki on Li,

where k1 ,...,km+1 are unknown constants (to be determined as part of theproblem). Since the solution of (20) only is determined within an arbitraryadditive constant, we might specify one of the constants k1 ,...,km+1. Forexample, when Ω is simply connected (i.e. m+ 1 = 1, as in Figure 1) theconstant value of each v ∈W along the boundary ∂Ω may be set equal to0. Thus, in this case

(21)∫

Ω

(y∂v

∂y+ x

∂v

∂x

)dxdy = −

∫Ω

2v dxdy,

and (18) takes the form: Find ψ̃ ∈ H10 (Ω) such that∫Ω

1μ

(grad ψ̃ · grad v dxdy = −∫

Ω

2v dxdy,

for all v ∈ H10 (Ω). Hence, (20) reduces to

(22)

⎧⎪⎨⎪⎩div

1μ

(grad ψ̃) = 2 in Ω,

ψ̃ = 0 on ∂Ω.

Note that the resultant torsion moment is given by M = τD, where

(23) D =∫

Ω

(−yμ

(∂ϕ

∂x− y

)+ xμ

(∂ϕ

∂y+ x

))dxdy.

The constant D, which is called the torsional rigidity, is independent of τsince the solution ϕ of (10) is independent of τ . It is possible to show that

D = 2E∗.

where E∗ is the “energy”

(24) E∗ =12

∫Ω

1μ| grad ψ̃|2 dxdy.

In actual calculations of D , using commercially available FE-software, wereplace the above space W by a finite dimensional subspace Wh ⊂W and


compute the value

2E∗h = 2(

12

∫Ω

1μ|gradψh|2 dxdy

)≈ D,

where ψh is the FE-solution , i.e. the solution satisfying the dual problem(18) with W replaced by Wh.

In order to illustrate, let us consider the simple case when Ω = 〈−2, 2〉2and let us compute E∗h by using the software Ansys. We let Wh be afunction space consisting of quadratic polynomials (8 node elements calledplane77 in Ansys with element edge length 0.07). Using this space we obtainthe value E∗h = 17.9939, i.e. D ≈ 35.988. Note that this is close to theexact value (up to three decimals) found by using Fourier-series, which is35.994 (see e.g. [19, p. 313]).

5. Periodic homogenization

Let κ ∈ {1, 2, ...} and consider the case when μ = μκ is of the formμκ(x, y) = G(κx, κy), where G is doubly periodic with respect to a cellY = 〈0, x0〉 × 〈0, y0〉 and Ω = Ωκ ⊃ Ω∞ for some open set Ω∞ such thatthe Lebesgue measure m(Ωκ\Ω∞) → 0 as κ → ∞. Denoting the solutionof (10) by ϕκ, we obtain that

(25)∫

Ωκ

μκ

(∂ϕκ∂x

− y)∂v

∂x+ μκ

(∂ϕκ∂y

+ x)∂v

∂ydxdy = 0

for all v ∈W 1,2(Ωκ). Thus, by (23) the corresponding torsional rigidity Dκis given by

Dκ =∫

Ωκ

μκ (gradϕκ + (−y, x)) · (−y, x) dxdy.

Let b be the constant 2 × 2 matrix, the homogenized matrix, defined by

bξ =∫

Y

G(x, y) (gradu+ ξ) dxdy, ξ ∈ R2,

where u ∈ W 1,2per(Y ) (= members of W 1,2(Y ) with the same trace onopposite faces) is the solution of the so-called cell problem

(26)∫

Y

G(x, y) (gradu+ ξ) · gradv dxdy, for all v ∈ W 1,2per(Y ).


Moreover, we define D∞ by

(27) D∞ =∫

Ω∞b (gradϕ∞ + (−y, x)) · (−y, x) dxdy,

where ϕ∞ ∈ H1(Ω∞) is the solution of the homogenized problem

(28)∫

Ω∞b (gradϕ∞ + (−y, x)) grad v dxdy = 0, for all v ∈ H1(Ω∞).

Theorem 3. It holds that Dκ → D∞ as κ→ ∞ .The proof follows by suitable modifications of well-known homogenization

techniques, which we include below for completeness.

Proof. Using (11) and putting v = ϕκ in (25) it follows that

α

∫Ωκ

(∂ϕκ∂x

− y)2

+(∂ϕκ∂y

+ x)2

dxdy

≤∫

Ωκ

μκ

((∂ϕκ∂x

− y)2

+(∂ϕκ∂y

+ x)2)

dxdy

=∫

Ωκ

μκ

(∂ϕκ∂x

− y)∂ϕκ∂x

+ μκ

(∂ϕκ∂y

+ x)∂ϕκ∂y

dxdy︸︷︷︸=0

+∫

Ωκ

μκ

(∂ϕκ∂x

− y)

(−y) + μκ(∂ϕκ∂y

+ x)x dxdy

≤︸︷︷︸Schwartzinequality

√∫Ωκ

μ2κ (x2 + y2) dxdy

√∫Ωκ

(∂ϕκ∂x

− y)2

+(∂ϕκ∂y

+ x)2

dxdy

≤ β√∫

Ωκ

(x2 + y2) dxdy

√∫Ωκ

(∂ϕκ∂x

− y)2

+(∂ϕκ∂y

+ x)2

dxdy.

Hence, √∫Ωκ

(∂ϕκ∂x

− y)2

+(∂ϕκ∂y

+ x)2

dxdy ≤ Cκ ≤ C,

where

Cκ =β

α

√∫Ωκ

(x2 + y2) dxdy, C =β

α

√∫Q

(x2 + y2) dxdy,


and Q is an open bounded domain such that Q ⊃ Ωκ for all κ . This showsthat {gradϕκ + (−y, x)} and hence {gradϕκ} is bounded in L2(Ω∞). Bysubtracting the average value we can always assume that ϕκ belongs to thespace

V ={v ∈W 1,2(Ω∞) :

∫Ω∞

v dxdy = 0}.

Since ‖D·‖L2(Ω∞) is an equivalent norm on V, by the Poincare inequality,it follows that {ϕκ} is bounded in V. Thus, by the reflexivity of V thereexists a subsequence, still denoted {ϕκ} , such that

ϕκ → ϕ∗ weakly in W 1,2(Ω∞).

Let us now define

ηκ = μκ (gradϕκ + (−y, x)) .

Since ∫Ωκ

|ηκ|2 dxdy =∫

Ωκ

μ2κ |gradϕκ + (−y, x)|2 dxdy

≤ β2∫

Ωκ

|gradϕκ + (−y, x)|2 dxdy

≤ β2C,

(29)

it follows that {ηκ} is bounded in L2(Ω∞). Thus, there exists asubsequence, still denoted by {ηκ} , and some function η∗ ∈ L2(Ω∞) suchthat

(30) ηκ → η∗ weakly in L2(Ω∞).

If g ∈ L2(Q) then ∫Ω∞

ηκg dxdy →∫

Ω∞η∗g dxdy.

Moreover, Schwartz inequality and (29) gives that∣∣∣∣∣∫

Ωκ\Ω∞ηκg dxdy

∣∣∣∣∣ ≤√∫

Ωκ\Ω∞|ηκ|2 dxdy

√∫Ωκ\Ω∞

|g|2 dxdy

≤ β√C

√∫Q

χΩκ\Ω∞ |g|2 dxdy,


which vanishes as κ → ∞, since ∫QχΩκ\Ω∞ |g|2 dxdy → 0 by Lebesgue

dominated convergence theorem. This shows that

(31)∫

Ωκ

ηκg dxdy =∫

Ωκ\Ω∞ηκg dxdy +

∫Ω∞

ηκg dxdy →∫

Ω∞η∗g dxdy.

Thus, since ∫Ωκ

ηκ gradv dxdy = 0

for any v ∈W 1,2(Q) by (25), we obtain from (31) that

(32)∫

Ω∞η∗ gradv dxdy = lim

κ→∞

∫Ω∞

ηκ gradv dxdy = 0.

If we could prove that

(33) η∗ = b (gradϕ∗ + (−y, x)) ,

then (32) would show that ϕ∗ satisfies (32). Moreover, it is possible to showthat b satisfies the same ellipticity conditions (11) as μκ (see e.g. [4] or[17]), which implies that the solution of (32) is unique within an arbitraryconstant K , and it follows that ϕ∗ = ϕ∞ + K. Putting g = (−y, x) into(31) we therefore obtain that

Dκ =∫

Ωκ

μκ (gradϕκ + (−y, x)) · (−y, x) dxdy

=∫

Ωκ

ηκg dxdy →∫

Ω∞η∗g dxdy

=∫

Ω∞b (gradϕ∞ + (−y, x)) · (−y, x) dxdy

= D∞,

and we are done. Therefore we only have to prove (33). Let w = gradu+ξ,where u is the solution of the cell problem (26). For the Y -periodicextension of w we have that divGw = 0 in sense of distributions (for theproof, see e.g. [6, p. 6]). Hence, div (μκwκ) = 0 in Q , where wκ = w(κ (·)),i.e.

(34)∫

Q

μκwκ · gradv dxdy, for all v ∈W 1,20 (Q).

Recall that for any Y -periodic function f ∈ Lploc(Rn), p ≥ 1, it holds that

(35) f(κ (·)) → 〈f〉


weakly in Lp(Q) as κ→ ∞, where 〈f〉 denotes the mean value of f. Hence,wκ → ξ and μκwκ → bξ weakly in L2(Q). If pκ, p ∈ L1(Ω∞), we writepκ

∗⇀ p, if pκ is bounded in L1(Ω∞) and∫

Ω∞pκϕdxdy →

∫Ω∞

pϕ dxdy

for all ϕ ∈ C∞0 (Ω∞). This limit is unique. According to the compensatedcompactness lemma of Murat and Tartar (see e.g. [4, p. 245], it holdsthat if rκ → r , sκ → s weakly in L2(Ω∞), where div rκ → f0 strongly inH−1(Ω∞) (i.e. ∫

Ω∞rκ gradv dxdy → f0(v)

for all v ∈ W 1,20 (Q)) and curl sκ is bounded in L2(Ω∞), then rκsκ ∗⇀ rs.We have that div ηκ → 0 strongly in H−1(Ω∞) by (32). Moreover,curlwκ = 0, div (μκwκ) = 0 and curl(gradϕκ + (−y, x)) = 2. Thus, sinceηκ → η∗, wκ → ξ, μκwκ → bξ and gradϕκ + (−y, x) → gradϕ∗ + (−y, x)weakly in L2(Q), each pair ηκ,wκ and μκwκ, gradϕκ + (−y, x) satisfythe conditions of the compensated compactness lemma. Hence, using thislemma on each side of the identity,

wκ · ηκ = μκwκ · (gradϕκ + (−y, x)) ,

separately, and using the uniqueness of the ∗⇀ limit, we obtain the identity

ξ · η∗ = bξ · (gradϕ∗ + (−y, x)) ,

i.e. η∗ = b (gradϕ∗ + (−y, x)) , and we are done. �

6. Stochastic homogenization

Let (M,F , λ) be a probability space and Tz (z = (x, y) ∈ R2 ) anergodic dynamic system on M . A vector field f ∈ [L2(M)]2 is said tobe potential (respectively solenoidal) if its generic realization f(Tzω) is apotential (respectively solenoidal) vector field defined on R2 . We denoteby L2pot(M) (respectively L2sol(M)) the subspace of

[L2(M)]2 formed by

potential (respectively solenoidal) vector fields. We can now define thefollowing space of vector fields with vanishing mean value

V 2pot(M) ={f ∈ L2pot(M) :

∫Mf dλ = 0

}.


Let a = a(ω) be a measurable function such that

0 < α ≤ a(ω) ≤ β


Roughly speaking the proof of this theorem relies on mimicing theproof above in the periodic setting for fixed realizations (with suitableadjustments) and replacing convergences of the type (35) with thoseobtained from the use of Birkhoff ergodicity theorem (in the form whichis usual in connection with stochastic homogenization, see e.g. [6]).

The homogenized matrix b given by (37) is usually impossible to calculatedirectly. Fortunately, using results given by Bourgeat and Piatnitski [2] weare able to find b approximately by using so-called periodic approximation.Indeed, let the function μω given by

μω(x, y) = a(T(x,y)ω)

be restricted to the cube Yδ = [0, δ]2 and then extended by periodicityto R2 . We denote this extension μδω . The homogenized matrix b

δω

corresponding to this periodic structure is then given by

bδωξ =1

|Yδ|∫

Yδ

μδω(graduδω + ξ

)dxdy, ξ ∈ R2,

where u ∈W 1,2per(Yδ) is the solution of the problem

(40)∫

Y

μδω(graduδω + ξ

) · grad v dxdy, for all v ∈ W 1,2per(Yδ),Note also that since bδω is symmetric it is also possible to find this matrixby using that

(41) ξ · bδωξ =1

|Yδ| minv∈W 1,2per(Yδ)

∫Yδ

μδω |gradv + ξ|2 dxdy.

Concerning this fact, see e.g. [6]. According to [2]

(42) bδω → b as δ → ∞,

for almost every ω ∈ M . This remarkable fact makes it possible to relatestochastic homogenization to periodic homogenization. For each fixed δ > 0let bδ denote the expectation of bδω, i.e.

(43) ξ · bδξ =∫Mξ · bδωξ dλ.

By using that α ≤ μδω ≤ β we obtain from (41) that

α |ξ|2 ≤ ξ · bδωξ ≤ β |ξ|2 .


Hence, by Lebesque dominated convergence theorem, (42) and (43) we findthat

(44) bδ → b as δ → ∞.

Figure 2. A typical realization for the case when Ωκ =Ω∞ = [0, 1]2 for κ = 17

As an example, let 0 < ρ < 1/2 and consider the probability space(M,F , λ) defined as follows. Each ω ∈ M is identified as a union of discsin R2 with radius ρ distributed in such a way that there exists a 1-periodiclattice (with period equal to 1 in both directions) where each square containsexactly one disc. It is easy to see that M can be identified with the spaceΓ = [0, 1]2 × ∏∞n=1 [0, 1]2 and λ with the product measure on Γ, i.e. themeasure λ = m × ∏∞n=1m, where m denotes the Lebesgue measure on[0, 1]2 . Moreover, we define the dynamic system Tz : M → M by lettingTzω be the translated set Tzω = ω + z . By results from ergodic theory itis possible to show that this dynamic system is ergodic (see e.g. the book[5], page 180). Now, let a = a(ω) be defined by

a(ω) =

{α if 0 ∈ ω,β if 0 /∈ ω.

The realization

μκ,ω(x, y) = a(T(κx,κy)ω),


can then be written as

(45) μκ,ω = αχω + βχR2\ω,

where χA denotes the characteristic function (indicator function) of theset A . It is clear from (45) that each ω ∈ M corresponds to a fixed two-component composite material with shear modulus equal to α on the discs(fibers) and β on the remaining part (the matrix). The matrix bδ can befound approximately by first computing bδω (numerically using the finiteelement method) for a large number N of different realizations and thentaking the average of these values. By letting ρ = 0.3, α = 1, β = 1000and using this way of calculating bδ for N = 400 realizations, it appearsthat b2 ≈ b4 ≈ b6 ≈ b8 ≈ (1.8) I , where I is the identity matrix. Thevariations between these four matrixes are found to be in third decimal.Therefore it is reasonable to use this matrix as an approximation also for thehomogenized matrix, i.e. putting b ≈ (1.8) I . This matrix can now be usedto find an approximate value D∞ for torsional rigidity Dκ,ω by solving theglobal problem (39) and inserting this solution into (38). This is certainlya much simpler problem to solve than calculating Dκ,ω directly, especiallywhen the number of fibres is large. As an example we may consider thecase when Ωκ = Ω∞ = [0, 1]

2. Using that the torsional rigidity for a unit

square with unit shear modulus is 2.2496 (see [19, p. 313]), we obtain thatD∞ = 2.2496 · (1.8) = 4.05. Thus, by the above homogenization resultwe may assume that Dκ,ω ≈ 4.05 when κ (or equivalently the numberof fibers κ2 ) is sufficiently large. In Figure 2, we have illustrated onesuch structure for κ = 17 which is randomly generated by using a simplecomputer program according to the above description of M . For similarexamples of numerical computations of random structures we refer to [3].

References

[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1978.

[2] A. Bourgeat and A. Piatnitski, Approximation of effective coefficientsin stochastic homogenization, Ann. Inst. H. Poincare, Prob. Statistics,40(2)(2004), 153-165.

[3] J. Byström, J. Dasht and P. Wall, A numerical study of the convergencein stochastic homogenization, J. Anal. Appl., 2(3)(2004), 159–171.

[4] D. Cioranescu and P. Donato, An Introduction to Homogenization,Oxford Lecture Series in Mathematics and its Applications, 17, OxfordUniversity Press, New York, 1999.


[5] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, SpringerVerlag, Berlin, 1982.

[6] V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization ofDifferential Operators and Integral Functionals, Springer-Verlag,Berlin, 1994.

[7] J.-L. Lions, D. Lukkassen, L.-E. Persson and P. Wall, Reiteratedhomogenization of nonlinear monotone operators, Chin. Ann. Math.,Ser. B, 22(1)(2001), 1–12.

[8] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary ValueProblems and Applications I, Springer-Verlag Berlin Heidelberg NewYork, 1972.

[9] D. Lukkassen, Some sharp estimates connected to the homogenized, p-Laplacian equation, Z. Angew. Math. Mech, 76(S2)(1996), 603–604.

[10] D. Lukkassen, Formulae and bounds connected to optimal design andhomogenization of partial differential operators and integral functionals,Ph.D. thesis, Dept. of Math., Tromsö University, Norway, 1996.

[11] D. Lukkassen, A new reiterated structure with optimal macroscopicbehaviour, SIAM J. Appl. Math., 59(5)(1999), 1825–1842.

[12] D. Lukkassen, P. Wall, On weak convergence of locally periodicfunctions, J. Nonl. Math. Phys. 9(1)(2002), 42–57.

[13] N. I. Muskhelishvili, Sur le problème de torsion des poutres élastiquescomposées, Comptes Rendus, Paris, 194(p)(1932), 1435.

[14] N. I. Muskhelishvili, On the problem of torsion and bending of elasticand compound bars, Izv. A. N. S.S.S.R., (1932), 907–945.

[15] N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theoryof Elasticity. Fundamental Equations, Plane Theory of Elasticity,Torsion and Bending, P. Noordhoff, Ltd., Groningen, 1963.

[16] J. Nečas, I. Hlavácěk, Mathematical Theory of Elastic and Elastico-Plastic Bodies, An Introduction. Elsevier, Amsterdam, 1981.

[17] L.-E. Persson, L. Persson, N. Svanstedt and J. Wyller, TheHomogenization Method: An Introduction, Studentlitteratur, Lund,1993.

[18] R. Temam, Navier-Stokes equations. Theory and Numerical AnalysisNorth-Holland, 3rd Edition, Amsterdam, 1984.

[19] S.P. Timoshenko and J.N. Goodier, Theory of Elasticity, Third edition,McGraw-Hill, New York, 1987.


Narvik University CollegeN-8505 NarvikNorwayandNorut Narvik, P.O. Box 250N-8504 NarvikNorway(E-mail : [email protected])

Narvik University CollegeN-8505 NarvikNorway

Department of MathematicsLule̊aUniversity of TechnologySE-971 87 Lule̊aSweden(E-mail : [email protected])

(Received : June 2007 )

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