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Asymptotic Analysis 47 (2006) 139–169 IOS Press
139
Compensated compactness for homogenization and reduction of dimension: The case of elastic laminates Björn Gustafsson a and Jacqueline Mossino b a
Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden C.M.L.A., Ecole Normale Supérieure de Cachan, 61, Avenue du Président Wilson, 94235 Cachan cedex, France
b
Abstract. The aim of this paper is to extend, to the linear elasticity system, the asymptotic analysis by compensated compactness previously developed by the authors for the linear diffusion equation. For simplicity, we restrict ourselves to stratified media. In the case of sole homogenization we recover the classical result of W.H. Mc Connel, deriving explicitly the effective elasticity tensor for stratified media. Here we give a new proof of his result, based on compensated compactness and on a technique of decomposing matrices. As for the case of simultaneous homogenization and reduction of dimension, we perform the asymptotic analysis, as the thickness tends to zero, of a three-dimensional laminated thin plate having an anisotropic, rapidly oscillating elasticity tensor. The limit problem is presented in three different ways, the final formulation being a fourth-order problem on the two-dimensional plate, with explicitly given elasticity tensors and effective source terms. Keywords: nonperiodic homogenization, reduction of dimension, theory of plates, compensated compactness, laminated or stratified materials
1. Introduction The aim of this paper is to extend, to the linear elasticity system, the compensated compactness method developed by the authors in [9,10] for the asymptotic analysis by homogenization and reduction of dimension of the linear diffusion equation. For simplicity, we restrict ourselves to the three-dimensional Euclidean space and to stratified (or laminated) materials. The paper is divided into three parts: • In the first part (Section 2), we give some preliminaries of linear algebra, showing the analogy between the linear diffusion equation and the linear elasticity system. In particular, we write a 3×3×3×3 tensor as a 3 × 3 matrix, the elements of which are themselves 3 × 3 matrices. (This gives a planar representation of the elasticity tensor.) In the same spirit, we write a 3 × 3 matrix as a three-component vector. (This gives a linear representation of the matrix of symmetrized gradients.) In some sense, this allows us to consider the symmetrized gradient as an ordinary gradient and the elasticity tensor as a conductivity matrix. In the case of stratified or laminated media, one of the space variables, say x3 , plays a particular role. In this case, we give a well suited planar representation of a general elasticity tensor A, writing A=
A
L
C , A33
0921-7134/06/$17.00 2006 – IOS Press and the authors. All rights reserved
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and we show that A admits the same decomposition as the one that we have previously introduced in the diffusion case, writing A = (M )−1 P , with M=
I
−C(A33 )−1 (A33 )−1
0
P =
and
A − C(A33 )−1 L
(A33 )−1 L
0 . I
• In the second part (Section 3), we use the above decomposition and the compensated compactness method introduced by Murat and Tartar (see [14] and [15]) to pass to the homogenized elasticity system in the case of a stratified material occupying a fixed domain. In doing this, we give a new proof of an old result by Mc Connel [13], by extending to elasticity the proof of [7] (see also [3–5]). Briefly speaking, we prove that, in the framework of elasticity, the H-limit of Aε = (M ε )−1 P ε , with M ε and P ε defined from Aε as above, is A = (M )−1 P , with M and P being the weak -L∞ limits of M ε and P ε . In this part, it is crucial that div P ε and curl M ε are relatively compact in a suitable sense, with div P ε and curl M ε defined by
ε = div P ε = div Pijk
∂pεijkl
∂xl
l
ε
curl M =
ε curl Mijk
=
,
ε where Pijk = pεijkl ,
∂mεijks ∂mεijkt − , ∂xt ∂xs
(1.1)
ε where Mijk = mεijkl .
(1.2)
In other words, div and curl are taken with respect to the last index. Our methods in principle allow for extensions far beyond the “stratified case”, since, in our particular case, div P ε and curl M ε actually vanish. However, we leave this as a programme for the future. • In the third part (Section 4), we consider the case of laminated plates, for which the two phenomena of homogenization and reduction of dimension occur simultaneously. In this case, we use the analogy that has been established in Section 2 between the linear diffusion equation and the linear elasticity system, in order to extend to elasticity the results of [9,10]. First, we derive, in the “big cylinder” obtained by rescaling the thin plate, a second-order limit problem for two unknown functions, the limit u of the rescaled displacement and an additional function y. The limit problem involves the H-limit A of the rescaled elasticity tensor, which is explicitly given by our study in Section 3. The source terms of this limit problem, say (f , g, h), are natural limits of the rescaled source terms of the original problem. Then, by eliminating y in favor of u, we prove that u itself solves a second-order limit problem in the “big cylinder”, involving a new tensor derived from the previous H-limit A, and involving new source terms derived from (f , g, h) and A. Finally, from the previous second-order limit problem in u, we derive the fourth-order limit problem for the Kirchhoff–Love displacement field ζ, defined over the limit two-dimensional plate. In this part, it is crucial not only that div P ε and curl M ε are relatively compact in a suitable sense, but also that M ε and P ε have some constant column vectors (compare [6] and [10]). Again, this should allow for many extensions beyond the “stratified case”. We emphasize that, in the diffusion case, nonlinear extensions of this method of compensated compactness are considered in [4,5] for the homogenization theory, and in [6] for simultaneous homogenization and reduction of dimension. But we are very far from extending to nonlinear elasticity, the compensated compactness method developed in this paper.
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141
2. Preliminary: from diffusion to elasticity 2.1. Planar representation of 3 × 3 × 3 × 3 tensors Consider a tensor A = (aijkl )i,j ,k,l=1,2,3 . We shall introduce a way of giving a planar representation of A, namely by considering it as a matrix of matrices. On the major level A will be a matrix in the second and fourth indices (j and l above). The entries (coefficients) of this major matrix are themselves matrices, namely in the first and third indices (i and k). We may think of j and l as exterior indices and i and k as interior ones. Thus we write
A11 A = A21 A31
A12 A22 A32
A13 A23 = Ajl j ,l=1,2,3 , A33
where, for fixed j and l, Ajl = (aijkl )i,k=1,2,3 . Since the x3 -coordinate will play a special role in this paper we shall further arrange our matrices (at least on the major level) on block form by grouping the index values as {1, 2} plus {3}. This means that we write A on the form A=
A
L
C a = iβkδ ai3kδ A33
aiβk3 , ai3k3
(2.1)
where the blocks are given by A =
A11 A21
A12 = (aiβkδ ), A22
C=
A13 = (aiβk3 ), A23
L = (A31
A32 ) = (ai3kδ ).
Here we have used the following convention, valid throughout the paper. Notational convention. Latin indices (i, j, k, l, . . . ) always run over {1, 2, 3} while Greek indices (α, β, γ, δ, . . . ) only run over {1, 2}. In addition, we shall constantly use the Einstein summation convention: whenever an index occurs twice in a term, it is summed over (from 1 to 3 if it is a Latin index, from 1 to 2 if it is a Greek index). 2.2. Multiplication of 3 × 3 × 3 × 3 tensors, by multiplication of their planar representation Multiplication of 3 × 3 × 3 × 3 tensors is defined so that it becomes ordinary matrix multiplication, on each level, in the above matrix of matrices representation. On the component level, this means that if aijkl ), then the components of the product A = AA are given by a ˜ijkl = aijst a ¯stkl . A = (aijkl ), A = (¯
On the block representation level, this means that Ajl = Ajt Atl . The two levels correspond to first summing over the interior index s and then over the exterior one, t.
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Writing the matrices on block form, A=
A
L
C , A33
A A= L
C , A33
it is easy to check that the multiplication scheme applies, giving
A A + CL A C + CA33 . AA = LA + A33 L LC + A33 A33 We note that the identity tensor I with respect to the above multiplication has entries (I)ijkl = δik δjl = 1 if i = k and j = l, 0 otherwise. 2.3. Linear representation of 3 × 3 matrices Let ξ = (ξij ) be a 3×3 matrix. Just as above we consider j as an exterior index and i as interior, and we get a linear (or vectorial) representation of ξ by considering it as a column vector in j, the components of which are themselves column vectors, in i. Writing it on block form it becomes
ξ , ξ3
ξ=
with ξ = (ξiβ ), ξ3 = (ξi3 ). If A = (aijkl ) is a 3 × 3 × 3 × 3 tensor, as in Subsection 2.1, the product Aξ is defined by (Aξ)ij = aijkl ξkl . As in the previous subsection, it is easy to check that this product can be done in the block representation, giving Aξ =
A
L
C A33
ξ
ξ3
=
A ξ + Cξ3 . Lξ + A33 ξ3
Scalar products between matrices considered as vectors is denoted by [·, ·]: [ξ, η] := ξij ηij = ξiβ ηiβ + ξi3 ηi3 = [ξ , η ] + ξ3 · η3 . However, the ordinary scalar product in R3 is denoted by a “dot”. In both cases the corresponding norms are denoted | · |. 2.4. Decomposition of 3 × 3 × 3 × 3 tensors If A=
A
L
C , A33
with A33 invertible (which always holds true in this paper), then we consider (A33 )−1 = R = (rik )
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143
as part of a 3 × 3 × 3 × 3 tensor by identifying it with R = (ri3k3 ), where ri3k3 = rik . Similarly, it is natural to define CR, RL and CRL as the following tensors: (CR)iβk3 = aiβs3 rs3k3 , (RL)i3kδ = ri3s3 as3kδ , (CRL)iβkδ = aiβs3 rs3t3 at3kδ . Thus we can define 3 × 3 × 3 × 3 tensors M and P by: M=
I
−CR , R
0
P =
A − CRL
RL
0 , I
(2.2)
with (I )iβkδ = δik δβδ , I = I33 = (δik δ33 ) = (δik ) = identity matrix in R3 . It is easy to check that M is invertible with respect to the multiplication introduced in Subsection 2.2, with inverse M
−1
=
I
0
C , A33
(2.3)
and that A = M −1 P.
(2.4)
3. Homogenization of laminates Let Ω be a fixed bounded domain in R3 . We consider the displacement field uε , solution of the threedimensional linear elasticity system: 1 3 ε 1 3 ε u ∈ H (Ω) , u = 0 on Γ0 , and ∀v ∈ H (Ω) , v = 0 on Γ0 , ε ε ε ε A (x )e u , e(v) dx = f · v dx + g , e(v) dx + 3 Ω
Ω
Here • Γ0 ⊂ ∂Ω; • γ denotes surface measure on ∂Ω;
Ω
∂Ω\Γ0
hε · v dγ.
(3.1)
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• Aε = Aε (x3 ) is a tensor with coefficients aεijkl (x3 ), i, j, k, l ∈ {1, 2, 3}, satisfying the following symmetry, boundedness and coercivity conditions: aεijkl = aεjikl = aεijlk ,
(3.2)
∃α, β > 0, for a.e. x ∈ Ω, ∀ξ ∈ R3s×3 , ε A (x3 )ξ, ξ α|ξ|2
• • • •
and
ε A (x3 )ξ β|ξ|,
(3.3)
where R3s×3 denotes the set of symmetric 3 × 3 matrices. ∂v eij (v) = 12 ( ∂∂xvji + ∂xji ) is the symmetrized gradient of v; f ε ∈ (L2 (Ω))3 ; g ε ∈ (L2 (Ω))3s×3 , the set of symmetric 3 × 3 matrices with L2 -coefficients; hε ∈ (L2 (∂Ω \ Γ0 ))3 .
We remark that the upper bound in (3.3) is equivalent to [Aε ξ, η] β|ξ||η| (∀ξ, η ∈ R3s×3 ). This implies [Aε ξ, ξ] β|ξ|2 , but the converse implication only holds under additional symmetry assumptions on Aε , namely that aεijkl = aεklij . The physical interpretation of (3.1) and the quantities involved is as follows: Ω is a region in space filled with a (linear) elastic material subject to some small deformations and subsequent stresses. The vector field uε describes the (infinitesimal) displacement of the body and its symmetrized gradient e(uε ) = (eij (uε )) is the strain tensor, which measures the rate of deformation. There is a corresponding ε ) which is related to e(uε ) by Hooke’s law (in general form) stress tensor σ ε = (σij
σ ε = Aε e u ε . With this relation between σ ε and e(uε ), (3.1) is the variational formulation of Newton’s law of balance of forces on an infinitesimal level, namely −div σ ε = f ε − div g ε
in Ω
or, in component form, −
ε ε ∂σij ∂gij = fiε − ∂xj ∂xj
in Ω,
together with the boundary conditions uε = 0 on Γ0 , σ ε n = g ε n + hε
on ∂Ω \ Γ0 .
Above the divergence is taken with respect to the last index and n denotes the outward normal unit vector on ∂Ω. The fact that the elasticity tensor Aε = Aε (x3 ) depends only on x3 means that the body Ω is composed of materials which are stratified (or laminated) along horizontal planes. We allow that Aε is a “full
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
145
tensor”, which means that the material may be anisotropic (its behavior is not governed only by the two Lamé coefficients). The constraint uε = 0 on Γ0 means that the body is clamped on that part, Γ0 , of its boundary. On the remaining parts of ∂Ω there act surface forces represented by hε , while the fields f ε and g ε represent body forces. The presence of g ε allows for taking into account more singular body forces than with just f ε . It is well known that problem (3.1) admits a unique solution uε (see, e.g., [1]). We are interested in the limit behavior of uε as ε tends to zero. This question has been solved by Mc Connel in [13]. Here we give a different proof, by using compensated compactness and arguments similar to those introduced by Courilleau and ourselves in [3,9,10,6], see also [7,4,5], namely by using a decomposition of Aε such that Aε = (M ε )−1 P ε , with curl M ε and div P ε vanishing (see (1.1) and (1.2)). 3.1. A priori estimates We have the following a priori estimates. Lemma 1. Assume (3.3) and assume that {f ε }ε is bounded in (L2 (Ω))3 , {g ε }ε is bounded in (L2 (Ω))3×3 and {hε }ε is bounded in (L2 (∂Ω \ Γ0 ))3 . Then {e(uε )}ε is bounded in (L2 (Ω))3×3 , which implies that {uε }ε is bounded in (H 1 (Ω))3 and {σ ε = Aε e(uε )}ε is bounded in (L2 (Ω))3×3 . Proof. Taking v = uε as test function in (3.1), we get
ε ε ε A e u , e u dx = Ω
Ω
f ε · uε dx +
ε ε g , e u dx +
Ω
∂Ω\Γ0
hε · uε dγ
f ε (L2 (Ω ))3 uε (L2 (Ω ))3 g ε (L2 (Ω ))3×3 e uε (L2 (Ω ))3×3
+ hε (L2 (∂Ω\Γ0 ))3 uε (L2 (∂Ω\Γ0 ))3 . Now, by the Korn inequality, ε u
(L2 (Ω ))3
uε (H 1 (Ω ))3 C e uε (L2 (Ω ))3×3 ,
and, by continuity of the trace mapping, ε u
(L2 (∂Ω\Γ0 ))3
C uε (H 1 (Ω ))3 C e uε (L2 (Ω ))3×3 .
Moreover, by coercivity (see (3.3)) Ω
ε ε ε A e u , e u dx α
ε 2 e u dx,
Ω
so that
αe uε (L2 (Ω ))3×3 C f ε (L2 (Ω ))3 + g ε (L2 (Ω ))3×3 + hε (L2 (∂Ω\Γ0 ))3 . By boundedness of {Aε }ε (see (3.3)), it follows that ε σ
(L2 (Ω ))3×3
β e uε (L2 (Ω ))3×3
is bounded, when ε tends to zero.
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3.2. Compactness From Lemma 1, classical compactness arguments immediately give: Lemma 2. Assume (3.3) and that, for some f in (L2 (Ω))3 , g in (L2 (Ω))3s×3 and h in (L2 (∂Ω \ Γ0 ))3 ,
3
3×3
fε f
weakly in L2 (Ω) ,
gε g
weakly in L2 (Ω)
hε h
(3.4) ,
(3.5)
3
weakly in L2 (∂Ω \ Γ0 ) .
(3.6)
Then, for some subsequence of ε, still denoted by ε, and for some u in (H 1 (Ω))3 such that u = 0 on Γ0 , uε u
3
weakly in H 1 (Ω) ,
(3.7)
implying that the following three convergences hold true:
3
uε → u strongly in L2 (Ω) , uε|∂Ω\Γ0 → u|∂Ω\Γ0
(3.8)
3
strongly in L2 (∂Ω \ Γ0 ) ,
e uε e(u) weakly in L2 (Ω)
3×3
,
(3.9) (3.10)
and for some σ in (L2 (Ω))3s×3 ,
σ ε = Aε e u ε σ
weakly in L2 (Ω)
3×3
.
(3.11)
Moreover, σ satisfies 1 3 ∀v ∈ H (Ω) , v = 0 on Γ0 , σ − g, e(v) dx = f · v dx + Ω
Ω
∂Ω\Γ0
h · v dγ.
(3.12)
It remains to relate σ to u. In the next two subsections we shall show that σ = Ae(u), for the matrix A which was obtained by Mc Connel in [13]. 3.3. The H-limit matrix Here we introduce the matrix A which will turn out be the H-limit of {Aε (x3 )}ε .
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147
Lemma 3. Assume that {Aε = Aε (x)}ε satisfies aεijkl = aεjikl = aεijlk ,
(3.13)
∃α, β > 0, for a.e. x ∈ Ω, ∀ξ ∈ R3s×3 , ε A (x)ξ, ξ α|ξ|2 and Aε (x)ξ β|ξ|,
(3.14)
and write, as in (2.1),
(Aε ) Aε = Lε
aεiβkδ Cε = ε A33 aεi3kδ
aεiβk3 . aεi3k3
Then Aε33 is invertible, with inverse Rε having L∞ -coefficients. Moreover, there exists a subsequence of ε (which we still denote by ε) and a 3 × 3 × 3 × 3 tensor A with bounded coefficients, in block form A=
A
L
C a = iβkδ A33 ai3kδ
aiβk3 , ai3k3
with A33 invertible, such that
Rε = Aε33
−1
R = (A33 )−1
weakly in L∞ (Ω)
3×2×3
3×3×2
weakly in L∞ (Ω)
C ε Rε CR
Rε Lε RL weakly in L∞ (Ω) ε A − C ε Rε Lε A − CRL
3×3
,
(3.15)
,
(3.16)
,
(3.17)
weakly in L∞ (Ω)
3×2×3×2
.
(3.18)
Proof. • Let us first prove that Aε33 is invertible. For this, it is enough to prove that ε α A33 η · η |η|2 ,
2
∀η ∈ R3 ,
(3.19)
that is, aεi3k3 ηi ηk
α ηi ηi . 2
We know that, for every ξ = (ξij ) ∈ R3s×3 , aεijkl ξij ξkl αξij ξij . Choosing here ξαβ = 0, ξα3 = ξ3α = 12 ηα , ξ33 = η3 , i.e., 1 ξij = (ηi δj 3 + ηj δi3 ) 2
(3.20)
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(which is in R3s×3 ) and using the symmetries of aεijkl we get 1 aεi3k3 ηi ηk = aεijkl ηi δj 3 ηk δl3 = aεijkl (ηi δj 3 + ηj δi3 )(ηk δl3 + ηl δk3 ) = aεijkl ξij ξkl 4 α α α αξij ξij = (ηi δj 3 + ηj δi3 )(ηi δj 3 + ηj δi3 ) = (2η1 η1 + 2η2 η2 + 4η3 η3 ) ηi ηi , 4 4 2 proving (3.19). Actually, the coercivity constant α2 is optimal, as is seen by taking aεijkl = symmetrized identity matrix. • Let us denote by Rε the inverse of Aε33 and let us next prove that ∀η ∈ R3 ,
ε α R η ·η |η|2 2
and
2β
ε R η 2 |η|.
1 2 (δik δjl
+ δil δjk ), the
(3.21)
α
In general, if a matrix B in R3×3 satisfies ∀η ∈ R3 ,
(Bη) · η α|η|2
and
|Bη| β|η|,
then it is invertible and its inverse B −1 satisfies ∀η ∈ R3 ,
−1 α B η · η 2 |η|2
and
β
|B −1 η|
1 |η| α
(3.22)
(see, e.g., [10, p. 289]). In our case we have the lower estimate (3.19), and from the upper bound in (3.14) we easily get, using the recipe (3.20) to pass from R3 to R3s×3 , |Aε33 η| β|η|. Now (3.21) follows. • As {Rε }ε is bounded in (L∞ (Ω))3×3 , we have, up to a subsequence, Rε R
weakly in L∞ (Ω)
3×3
for some R satisfying ∀η ∈ R3 ,
(Rη) · η
α |η|2 2β 2
and
|Rη|
2 |η|. α
Applying again the general result (3.22) mentioned above, R is invertible, with inverse A33 satisfying ∀η ∈ R3 ,
(A33 η) · η
α3 2 |η| 8β 2
and
|A33 η|
2β 2 |η|. α
(Here the coercivity constant is not optimal. It will follow from Theorem 1 and (3.19) that the best coercivity constant is α/2.) • By now we have proved that (up to a subsequence)
Rε = Aε33
−1
R = (A33 )−1
weakly in L∞ (Ω)
3×3
,
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149
with A33 having L∞ -coefficients. From the boundedness of {Aε }ε , in the sense of (3.3), we also have boundedness in the usual sense, i.e., when acting on not necessarily symmetric vectors: ε A ζ β|ζ|.
∀ζ ∈ R3×3 ,
To see this, just apply |Aε ξ| β|ξ| with ξ ∈ R3s×3 given by ξij = 12 (ζij + ζji ). Then, by (3.13), |Aε ζ| = |Aε ξ| β|ξ| β|ζ|. (On the other hand, Aε is not invertible, and hence not coercive, on R3×3 , because it annihilates all antisymmetric tensors.) From the above it is clear that C ε Rε , Rε Lε and (Aε ) − C ε Rε Lε are uniformly bounded with bounds only depending on α and β, so that, up to a subsequence,
weakly in L∞ (Ω)
C ε Rε X
3×2×3
,
3×3×2 Rε Lε Y weakly in L∞ (Ω) , ε 3×2×3×2 A − C ε Rε Lε Z weakly in L∞ (Ω) ,
for some X, Y , Z. From these one obtains C, L and A by setting X = CR
C = XA33 ,
or
Y = RL or
L = A33 Y ,
A = Z + CRL = Z + XA33 Y.
Z = A − CRL or
Clearly, the resulting tensor A is bounded from above with bounds only depending on α and β. This finishes the proof of the lemma. 3.4. The limit problem We are now ready to formulate our version (Theorem 1 below) of Mc Connel’s theorem [13]. Defining the matrices M ε , P ε , M and P as in Section 2.4, by ε
M =
M=
I
0
I
0
(Aε ) − C ε Rε Lε P = Rε Lε
−C ε Rε , Rε
ε
−CR , R
P =
A − CRL
RL
0 , I
0 , I
the weak -L∞ convergences (3.15) to (3.18) in Lemma 3 can be summarized in matrix form as Mε M
and
Pε P
weakly in L∞ (Ω)
3×3×3×3
.
(3.23)
The main point in Theorem 1 is that when Aε depends only on x3 , then the convergences (3.23) are exactly what is needed to ensure that, for arbitrary boundary and source term data (f , g, h, etc.), the weak limits σ = lim σ ε and e(u) = lim e(uε ) obtained in Lemma 2 are related by σ = Ae(u).
(3.24)
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Then the fact that σ satisfies (3.12) means that u solves the appropriate limit version of (3.1), namely problem (3.29) below. Thus (3.23) (or (3.15) to (3.18)) characterizes H-convergence Aε → A in the stratified case Aε = Aε (x3 ). Theorem 1. Assume (3.2), (3.3), (3.4), (3.6) and moreover
gε → g
strongly in L2 (Ω)
3×3
,
(3.25)
and finally that {Aε }ε H-converges to a matrix A in the sense that (3.23) holds. Then, first of all, A depends only on x3 , it has the symmetries aijkl = ajikl = aijlk ,
(3.26)
and it satisfies the uniform conditions for a.e. x ∈ Ω, ∀ξ ∈ R3s×3 ,
[Aξ, ξ] α|ξ|2
and
|Aξ|
β2 |ξ|, α
(3.27)
where α and β are the coercivity constants in (3.3). Next, for the whole sequence ε, one has convergences (3.7) to (3.10), as well as
Aε e uε Ae(u) weakly in L2 (Ω)
3×3
,
(3.28)
with u being the unique solution of 1 3 1 3 u ∈ H (Ω) , u = 0 on Γ0 , and ∀v ∈ H (Ω) , v = 0 on Γ0 , A(x3 )e(u), e(v) dx = f · v dx + g, e(v) dx + h · v dγ. Ω
Ω
Ω
(3.29)
∂Ω\Γ0
Finally, the energies converge in the sense that Ω
ε A (x3 )e uε , e uε dx →
Ω
A(x3 )e(u), e(u) dx.
Proof. In order to make the proof more transparent we divide it into several steps. • Step 0: Generalities. The proof relies on the “div–curl lemma” of compensated compactness. We recall that this lemma in general says that if f ε , g ε , f , g ∈ (L2 (Ω))3 are vector fields such that f ε f , g ε g weakly in (L2 (Ω))3 and such that div f ε and the components of curl g ε are all contained in a compact subset of H −1 (Ω), then f ε · g ε f · g weakly as distributions. Let us spell out the matrices M ε = (mεijkl ) and P ε = (pεijkl ) in components. We have ε
M =
I
0
δ δ −C ε Rε = ik βδ Rε 0
−aεiβs3 rsε3k3 , riε3k3
(3.30)
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
(Aε ) − C ε Rε Lε Pε = Rε Lε
aεiβkδ − aεiβs3 rsε3t3 aεt3kδ 0 = I riε3s3 aεs3kδ
151
0 . δik
(3.31)
Similarly for M = (mijkl ) and P = (pijkl ), without the superscript ε. One circumstance that will complicate the proof of the theorem is that the matrices M ε and P ε do not enjoy the same symmetries (3.2) of Aε . We only have that P ε is symmetric in the last two indices, as a direct consequence of P ε = M ε Aε . From this we can however conclude that P = lim P ε , and hence A = M −1 P , are symmetric in the last two indices. The fact that the limit matrix A is symmetric in the first two indices will come out later on (Step 3). • Step 1: Proof that a subsequence of uε converges to a solution of (3.29). This is the most essential part of the proof, and it can be conceived as a continuation of Lemma 2 and Lemma 3, with now the information that Aε depends only on x3 being used. The issue is to show that when (3.23) holds and Aε = Aε (x3 ), then the weak limits σ = lim σ ε = lim Aε e(uε ) and e(u) = lim e(uε ) obtained in Lemma 2 are related by (3.24). So select a subsequence ε and u, σ as in Lemma 2. Due to the symmetry of A with respect to the last two indices, Eq. (3.24) (to be proved) is equivalent to σ = A∇u,
(3.32)
where ∇u is the nonsymmetrized gradient of u: (∇u)ij = the limit in
∂ui ∂x j .
Eq. (3.24) will be proved by passing to
σ ε = Aε ∇uε .
(3.33)
Using the decomposition Aε = (M ε )−1 P ε , the latter relation becomes M ε σ ε = P ε ∇uε .
(3.34)
This way of writing (3.33) is the key to everything, and the main conclusions are obtained from it by applying the “div–curl lemma” to (3.34). ◦ Passing to the limit in P ε ∇uε . According to (1.1), we notice that div P ε = 0 in the sense that, for every i, j, k ∈ {1, 2, 3}, ∂pεijkl = 0. ∂xl Indeed, by (3.31),
pεiβkδ (x3 ) 0 . P = pεi3kδ (x3 ) δik ε
Hence ∂pεijkδ ∂pεijkl = (x3 ) = 0. ∂xl ∂xδ
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Now ε ∂uε ε P ∇uε ij = pεijkl ∇uε kl = pεijkl kε = Pijk · ∇uεk , ∂u l
ε = (pε ) with Pijk ijkl l=1,2,3 . We can apply classical compensated compactness, since, for every i, j, k,
ε Pijk Pijk
3
weakly in L2 (Ω) ,
ε = 0, div Pijk
uεk uk
weakly in H 1 Ω .
ε ∇uε P ∇u in the sense of distributions and then, by boundedness, weakly in It follows that Pijk ijk k k 2 L (Ω). This means that
P ε ∇uε P ∇u weakly in L2 (Ω)
3×3
.
◦ Passing to the limit in M ε σ ε . ε = (mε ) According to (1.2), we notice that curl M ε = 0 in the sense that, if we set Mijk ijkl l=1,2,3 , then, ε for every i, j, k, curl Mijk = 0. Indeed, by (3.30),
δ δ M = ik βδ 0 ε
mεiβk3 (x3 ) , mεi3k3 (x3 )
so that
ε Mijk = (mijkδ )δ=1,2 , mεijk3 , ε has the form V ε = (V , V ε (x )) and with mijkδ independent of ε and x. In other words, Mijk 3 δ 3
curl V ε
δα
=
∂Vδ ∂Vα − = 0, ∂xα ∂xδ
curl V ε
δ3
=
∂Vδ ∂V ε − 3 = 0. ∂x3 ∂xδ
ε = M ε · σ ε , with σ ε = (σ ε ) Now (M ε σ ε )ij = mεijkl σkl ijk k k kl l=1,2,3 . We can apply compensated compactness, as soon as we have proved that div σkε is relatively compact in H −1 (Ω), and we get that M ε σ ε M σ in (D (Ω))3×3 and then, by boundedness, in (L2 (Ω))3×3 . ◦ Now we prove that div σkε is relatively compact in H −1 (Ω). Clearly, from (3.1), as σ ε − g ε ∈ (L2 (Ω))3s×3 ,
Ω
ε σ − g ε , ∇v dx =
so that −
∂ ε ε σkl − gkl = fkε ∂xl
Ω
f ε · v dx,
3
∀v ∈ D(Ω) ,
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
153
∂g ε
and div σkε is relatively compact in H −1 (Ω) as soon as the same holds true for div gkε = ∂xkll (recall that, by compactness, fkε → fk , strongly in H −1 (Ω)). But this holds true, since, by assumption, g ε → g strongly in (L2 (Ω))3×3 . This finishes the proof of (3.32) and, hence, of (3.24). • Step 2: Proof of the coercivity bound in (3.27). Choose ξ ∈ R3s×3 , and consider it as a constant vector in (L2 (Ω))3s×3 . Since M is invertible, we can define ξ , ξ ε ∈ (L2 (Ω))3×3 and ξ¯ε ∈ (L2 (Ω))3s×3 by ξ = tM ξ ,
1 ξ¯ε = ξ ε +t ξ ε . 2
ξ ε = tM ε ξ ,
Here t M (etc.) denotes the transpose in the sense of interchanging the first pair of indices with the second pair: if M = (mijkl ) then t M = (mklij ). (For ξ = (ξij ) the transpose is defined as usual: tξ = (ξji ).) These definitions give immediately that t
ε ε ε t ε t ε A ξ ,ξ = M ξ , P ξ .
[Aξ, ξ] = M ξ ,t P ξ ,
Due to the symmetries (3.2) of Aε we also have ε ε ε ε ε ε A ξ¯ , ξ¯ = A ξ , ξ . , and since M , M ε , ξ only depend on x and since the mε Now (t M ε ξ )ij = mεklij ξkl 3 klij are constants t ε when j = 3, it follows that the only nonzero contributions to curl( M ξ ) are terms of the form
∂ ε ∂ξ mkliβ ξkl = mkliβ kl , ∂x3 ∂x3 which are fixed (i.e., independent of ε) elements of H −1 (Ω). Thus the components of curl(t M ε ξ ) stay in a compact subset of H −1 (Ω) as ε → 0. For similar reasons, the components of div(t P ε ξ ) stay in a compact subset of H −1 (Ω). Therefore, the above identities combined with compensated compactness show that ε ε ε A ξ¯ , ξ¯ [Aξ, ξ]
weakly as distributions. We also have that ξ¯ε ξ weakly in (L2 (Ω))3s×3 , which by a lower semicontinuity argument and (3.3) gives
|ξ| ϕ dx α lim inf
2
α Ω
Ω
ε 2 ξ¯ ϕ dx lim inf
ε ε ε A ξ¯ , ξ¯ ϕ dx = Ω
[Aξ, ξ]ϕ dx, Ω
for smooth test functions ϕ 0. From this the coercivity estimate in (3.27) follows. • Step 3: Proof of symmetry of A with respect to the first two indices. We are going to show that aijkl = ajikl , i.e., that the first equality in (3.26) holds. The idea is that the limit equation σ = Ae(u) holds for very many fields u (by varying the data f , g, h) and that therefore the symmetry of σ, which is a consequence of σ = lim σ ε and the symmetry of σ ε , forces A to be symmetric
154
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
in the first two indices. To achieve this we shall simply prescribe the limit field u = uλ (λ a parameter) on an arbitrary relatively compact subdomain of Ω. So let ω Ω be a relatively compact subdomain of Ω and let φ ∈ D(Ω) be a smooth test function with compact support in Ω such that φ = 1 in ω. Given λ ∈ R3×3 define uλ ∈ (D(Ω))3 and g λ ∈ (L2 (Ω))3s×3 by uλi (x) = λij xj φ(x) and g λ = 12 (Ae(uλ ) +t (Ae(uλ )). Using that e(v) is symmetric it is clear that uλ solves uλ ∈ H01 (Ω) 3 and ∀v ∈ H01 (Ω) 3 , λ A(x3 )e uλ , e(v) dx = g , e(v) dx. Ω
Ω
Notice that this is the limit problem (3.29) for the data f = 0, g = g λ , Γ0 = ∂Ω (hence there is no h). Moreover the above problem has only one solution, by virtue of the coercivity of A proved in Step 2. Now we consider the solution uελ of uελ ∈ H01 (Ω) 3 and ∀v ∈ H01 (Ω) 3 , ε λ A (x3 )e uελ , e(v) dx = g , e(v) dx. Ω
Ω
By Step 1 of the proof we have, at least for a subsequence, σ ελ = Aε ∇uελ σ λ = A∇uλ weakly in (L2 (Ω))3×3 . Hence the symmetry of σ ελ implies the symmetry of σ λ and we have, in ω,
aijkl λkl = aijkl ∇uλ
kl
λ λ = σij = σji = ajikl ∇uλ
kl
= ajikl λkl .
Choosing λkl = δkr δls , we get aijrs = ajirs in ω. As this holds true for any ω, the desired symmetry of A follows. • Step 4: Proof of the upper bound in (3.27). Since we now know that the tensor (or matrix) A = A(x3 ) has the symmetries (3.26) we can consider it, for each x3 , as a linear map A : R3s×3 → R3s×3 . The coercivity proved in Step 2 shows that as such a map it is invertible (whereas it is not invertible as a map R3×3 → R3×3 ). The same statements hold for tA. We also know (see Subsection 2.4) that M , and hence tM , is invertible as a map M : R3×3 → R3×3 . It follows that tP =t AtM maps R3×3 onto R3s×3 . Therefore, given ξ ∈ R3s×3 we can define ζ ∈ (L2 (Ω))3×3 and ξ ε in (L2 (Ω))3s×3 by ξ =t P ζ,
ξ ε =t P ε ζ.
(The symmetry of ξ ε is due to the symmetry of P ε in the last two indices.) It is easy to check that −1 t A ξ, ξ = P ζ,t M ζ , ε −1 ε ε t ε t ε A ξ , ξ = P ζ, M ζ .
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
155
Again compensated compactness gives that ε −1 ε ε A ξ , ξ A−1 ξ, ξ
weakly as distributions. Now, the statement (3.22) about the bounds for the inverse of a linear operator applies also to the vector space R3s×3 if B : R3s×3 → R3s×3 satisfies ∀ξ ∈ R3s×3 ,
[Bξ, ξ] α|ξ|2
and
|Bξ| β|ξ|,
then it is invertible on R3s×3 and the inverse B −1 satisfies ∀ξ ∈ R3s×3 ,
−1 α B ξ, ξ 2 |ξ|2 .
β
Consequently it follows from (3.3) that ε −1 ε ε α 2 A ξ , ξ 2 ξ ε .
β
Hence passing to the limit ε → 0 as before gives −1 α A ξ, ξ 2 |ξ|2 .
β
2
Combining with the Cauchy–Schwarz inequality, we get |ξ| βα |A−1 ξ|, which is equivalent to the upper bound in (3.27). • Step 5: Uniqueness of solution for the limit problem and convergence for the entire sequence. In Step 1 we obtained one solution u of the limit problem (3.29) as the limit of a subsequence uε of solutions of (3.1), with also e(uε ) and σ ε = Aε e(uε ) converging to e(u) and σ = Ae(u), respectively. From the coercivity and symmetry properties of A follows on the other hand uniqueness of solutions of the limit problem. Thus u above is this unique solution of (3.29), and standard arguments then show that we actually have the above convergences for the entire sequence ε. • Step 6: Convergence of energies. Finally we prove convergence of energies. We have
ε ε ε A e u , e u dx = Ω
Ω
f ε · uε dx +
ε ε g , e u dx + Ω
∂Ω\Γ0
hε · uε dγ.
It follows from (3.4), (3.6), (3.8), (3.9), (3.10) and (3.25), that Ω
ε ε ε A e u , e u dx →
= Ω
Ae(u), e(u) dx.
Ω
f · u dx +
Ω
g, e(u) dx + ∂Ω\Γ0
h · u dγ
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B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
4. Homogenization and reduction of dimension for laminated plates Let ω be a fixed domain in R2 and let ε > 0 be a small parameter. The domain Ω ε = ω × (− ε2 , ε2 ) represents a thin horizontal plate. We are given F ε ∈ (L2 (Ω ε ))3 , Gε ∈ (L2 (Ω ε ))3s×3 , the set of symmetric ε , H ε ∈ (L2 (ω))3 and a 3 × 3 × 3 × 3 tensor Aε = Aε (x ) 3 × 3 matrices with L2 -coefficients, H+ 3 − ε ε defined in (− 2 , 2 ), satisfying the same symmetry, boundedness and coercivity conditions (with α and β independent of ε) as in Section 3. Here again we allow that Aε is a “full tensor”, which means that the material may be anisotropic (its behavior is not governed only by the two Lamé coefficients). The fact that Aε = Aε (x3 ) means that the plate is laminated, along horizontal planes. We consider the three-dimensional linear elasticity system in Ω ε : 1 ε 3 ε 3 ε ε ε ε U ∈ H (Ω ) , U = 0 on Σ = ∂ω × − , , and ∀U ∈ H 1 Ω ε , U = 0 on Σ ε , 2 2 ε ε ε ε (4.1) A (x )e U , e(U ) dx = F · U dx + G , e(U ) dx 3 Ωε Ωε Ωε ε ε + H ε · U x , + H ε · U x , − dx . ∂ω
+
2
−
2
The physical interpretation of (4.1) and the quantities involved is the same as for (3.1). We are interested in the limit behavior of U ε , when ε runs through a sequence of values converging to zero. This limit behavior is established in the sequel, by using again compensated compactness arguments. The scalar case for a linear equation was solved in such a way in [9,10]; for a more general nonlinear equation of the form −div((M ε (x))−1 ϕ(P ε (x)∇uε )) = f ε , it was solved in [6], under the assumption (which is satisfied for lamination) that curl M ε and div P ε are relatively compact in a convenient meaning and provided that M ε and P ε have special forms, in the sense that some of their components do not depend on ε. For the asymptotic behavior of plates made of a homogeneous isotropic material, the reader is referred to [2] for linear elasticity, and to [8,11] (or [12] for the shell model) for nonlinear elasticity. 4.1. Rescaling problem (4.1) We adopt the standard technique (see, e.g., [2]) of rescaling the problem so that everything takes place in a fixed domain. This we take to be Ω = ω × (− 12 , 12 ) and, from U : Ω ε → R3 we get u : Ω → R3 defined by: uα (x) = uα (x , x3 ) = Uα (x , εx3 ),
u3 (x) = u3 (x , x3 ) = εU3 (x , εx3 ).
Then ∂Uα ∂uα (x) = (x , εx3 ), ∂xβ ∂xβ ∂u3 ∂U3 (x) = ε (x , εx3 ), ∂xα ∂xα so that
ε
e(U ) = e (u) :=
eαβ (u) 1 ε e3β (u)
∂uα ∂Uα (x) = ε (x , εx3 ), ∂x3 ∂x3 ∂U3 ∂u3 (x) = ε2 (x , εx3 ), ∂x3 ∂x3
1 ε eα3 (u) 1 e (u) ε2 33
.
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
157
We also set Aε (x3 ) = Aε (εx3 ), which defines a 3 × 3 × 3 × 3 tensor in Ω having the same properties as in (3.2), (3.3), namely aεijkl = aεjikl = aεijlk ,
(4.2)
3×3 ∃α, x ∈ Ω, ε β > 0, ∀ξ ∈ Rs 2 , for a.e. A (x3 )ξ, ξ α|ξ| and Aε (x3 )ξ β|ξ|.
(4.3)
Then the variational equation in (4.1) is transformed into Ω
ε A (x3 )eε uε , eε (u) dx =
Ω
Fαε (x , εx3 )uα (x) dx +
+ Ω
Ω
1 ε F (x , εx3 )u3 (x) dx ε 3
ε G (x , εx3 ), eε (u) dx
+ ∂ω
1 ε 1 1 ε 1 H uα x , + 2 H+ 3 u3 x , ε +α 2 ε 2
1 ε 1 + H−α uα x , − ε 2
1 ε 1 + 2 H− 3 u3 x , − ε 2
dx .
Hence the natural scaling in U induces corresponding scalings of the source terms: 1 f3ε (x) = F3ε (x , εx3 ), ε
fαε (x) = Fαε (x , εx3 ), g ε (x) = Gε (x , εx3 ), 1 ε hε±α (x ) = H±α (x ), ε
hε±3 (x ) =
1 ε H (x ), ε2 ±3
and problem (4.1) becomes, in terms of the new variables: 1 3 ε 3 1 1 ε u ∈ H (Ω) , u = 0 on Σ = ∂ω × − , and ∀w ∈ H 1 (Ω) , w = 0 on Σ, , 2 2 ε ε ε ε ε ε ε A (x3 )e u , e (w) dx = f · w dx + g , e (w) dx Ω Ω Ω 1 ε 1 ε + h (x ) · w x , + h (x ) · w x , − dx . ω
+
−
2
(4.4)
2
We are going to pass to the limit in problem (4.4), when ε tends to zero. 4.2. A priori estimates Lemma 4. Assume (4.3) and assume that {f ε }ε is bounded in (L2 (Ω))3 , {g ε }ε is bounded in (L2 (Ω))3×3 and {hε± }ε is bounded in (L2 (ω))3 . Then {eε (uε )}ε is bounded in (L2 (Ω))3×3 , which implies that {e(uε )}ε is bounded in (L2 (Ω))3×3 , {uε }ε is bounded in (H 1 (Ω))3 and {σ ε = Aε eε (uε )}ε is bounded in (L2 (Ω))3×3 . Moreover, defining y ε by: yαε =
uεα −u ˜εα − ε
1 2
− 12
ε u
α
ε
−u ˜εα dx3 ,
with u ˜εα (x) = −
x3 1 ∂uε3 − 12
ε ∂xα
(x , t) dt,
(4.5)
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B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
1 = 2 uε3 − ε
y3ε
1 2
− 12
1 ε u dx3 , ε2 3
(4.6)
1 1 then {y ε }ε is bounded in (L2 (ω; Hm (− 12 , 12 )))3 , where Hm is the subset of H 1 of functions having mean value zero.
Proof. Taking uε as test function in (4.4), we get Ω
ε A (x3 )eε uε , eε uε dx
f · u dx + ε
= Ω
+ ω
ε ε ε g , e u dx
ε
Ω
hε+ (x ) · uε x ,
1 2
+ hε− (x ) · uε x , −
1 2
dx
f ε (L2 (Ω ))3 uε (L2 (Ω ))3 + g ε (L2 (Ω ))3×3 eε uε (L2 (Ω ))3×3 ε ε ε ε 1 1 + h+ (L2 (ω))3 u ·, + h− (L2 (ω))3 u ·, − 2 2 3 2
.
(L2 (ω ))3
(L (ω ))
By the Korn inequality, for ε < 1, ε u
(L2 (Ω ))3
uε (H 1 (Ω ))3 C e uε (L2 (Ω ))3×3 C eε uε (L2 (Ω ))3×3
and, by continuity of the trace mapping, ε u ·, ± 1 2
(L2 (ω ))3
C e uε (L2 (Ω ))3×3 C eε uε (L2 (Ω ))3×3 .
Thus, by using the coercivity of {Aε }ε (see (4.3)), we get
αeε uε (L2 (Ω ))3×3 C f ε (L2 (Ω ))3 + g ε (L2 (Ω ))3×3 + hε+ (L2 (ω))3 + hε− (L2 (ω))3 . Hence {eε (uε )}ε is bounded in (L2 (Ω))3×3 and, by boundedness of {Aε }ε (see (4.3)), it follows that {σ ε }ε is bounded in (L2 (Ω))3×3 . Moreover, by definition of y ε (see (4.5), (4.6)), ∂yαε 1 ∂uεα 1 ∂uε3 2 = + = eα3 uε = 2eεα3 uε ∂x3 ε ∂x3 ε ∂xα ε
is bounded in L2 (Ω) and ∂y3ε 1 ∂uε 1 = 2 3 = 2 e33 uε = eε33 uε ∂x3 ε ∂x3 ε
is also bounded in L2 (Ω). As y ε has mean value zero with respect to x3 , it follows from the Poincaré– 1 Wirtinger inequality that {y ε }ε is bounded in (L2 (ω; Hm (− 12 , 12 )))3 .
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
159
4.3. Compactness Lemma 5. Assume (4.3) and assume that, for some f in (L2 (Ω))3 , g in (L2 (Ω))3s×3 and h± in (L2 (ω))3 , fε f
3
3×3
weakly in L2 (Ω) ,
gε g
weakly in L2 (Ω)
,
(4.8)
weakly in L2 (ω) .
(4.9)
hε± h±
(4.7)
3
Then, for some subsequence of ε, still denoted by ε, for some u in the Kirchhoff–Love space
U = v ∈ H (Ω) 1
2
H02 (ω), ∃ζ
×
∈
2 H01 (ω) , vα
∂v3 = ζα − x3 , α = 1, 2 , ∂xα
1 (− 12 , 12 )))3 and for some σ in (L2 (Ω))3s×3 , for some y in (L2 (ω; Hm
3
uε ·, ±
1 2
→ u ·, ±
1 2
3
y (defined in Lemma 4) y
ε ε eα3 (u )
1
1
ε e3β
ε2
e33
ω; Hm
= e(uε , y ε ) :=
(uε )
1 ∂y α 2 ∂x 3
1 ∂y β 2 ∂x 3
∂y 3 ∂x 3
σ ε = Aε eε uε σ
1
eαβ (u)
e(u, y) :=
1
(4.11)
2
weakly in L
eαβ (uε ) (uε )
(4.10)
strongly in L2 (ω) ,
ε
eε uε =
3
uε → u strongly in L2 (Ω) ,
uε u weakly in H 1 (Ω) ,
3×3
3
,
eαβ (uε )
ε 1 ∂y α 2 ∂x 3
ε 1 ∂y β 2 ∂x 3
∂y3ε ∂x 3
weakly in L2 (Ω)
weakly in L2 (Ω)
1 1 − , 2 2
3×3
(4.12)
,
(4.13)
.
(4.14)
Moreover, σ satisfies 3 1 1 1 ∀(v, z) ∈ U × L2 ω; Hm − , , 2 2 1 1 σ − g, e(v, z) dx = f · v dx + h+ · v x , + h− · v x , − dx . Ω
Ω
ω
2
(4.15)
2
Proof. The convergences of uε , y ε and σ ε follow from Lemma 4 and from classical compactness arguments. But the weak convergence of uε in (H 1 (Ω))3 implies that, for every α, β ∈ {1, 2},
eαβ uε eαβ (u) weakly in L2 (Ω),
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B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
1 while the weak convergence of y ε in (L2 (ω; Hm (− 12 , 12 )))3 implies that
2 ε ∂y ε ∂yα eα3 u = 2eεα3 uε = α weakly in L2 (Ω), ε ∂x3 ∂x3 ∂y3ε ∂y3 1 ε ε ε e u = e u = weakly in L2 (Ω), 33 33 2 ε ∂x3 ∂x3
so that (4.13) holds true. Finally, for (v, z) ∈ U × (C 1 (Ω))3 , z = 0 on Σ, we define v ε = (vαε , v3ε ) by vαε = vα + εzα , ε v3 = v3 + ε2 z3 . It is clear that v ε ∈ (H 1 (Ω))3 , v ε = 0 on Σ, and, since eα3 (v) = 0 and e33 (v) = 0, ε ε
e v
=
1 ∂z α 2 ∂x 3
eαβ (v + εz) ∂z 3 1 ∂z β 2 ∂x 3 + ε ∂x β
→ e(v, z)
strongly in L2 (Ω)
+ ε ∂∂xzα3
= e(v, z) + ε
∂z 3 ∂x 3
3×3
eαβ (z) 1 ∂z 3 2 ∂x β
1 ∂z 3 2 ∂x α
0
.
Moreover, vε → v
3
strongly in H 1 (Ω) .
Taking v ε as test function in (4.4), we have that Ω
ε σ − g ε , eε v ε dx =
f · v dx + ε
Ω
ε
ω
hε+ (x )
·v
ε
1 1 x, + hε− (x ) · v ε x , − 2 2
dx
and, passing to the limit, we get that
Ω
σ − g, e(v, z) dx =
Ω
f · v dx +
ω
1 1 h+ (x ) · v x , + h− (x ) · v x , − 2 2
dx ,
for any v ∈ U and z ∈ (C 1 (Ω))3 , z = 0 on Σ. By density of {z ∈ C 1 (Ω), z = 0 on Σ} in L2 (ω; H 1 (− 12 , 12 )) and by continuity, this is true also for any v ∈ U and z ∈ (L2 (ω; H 1 (− 12 , 12 )))3 (a fortiori for z having mean value zero with respect to x3 ). 4.4. Second-order limit problem whose solution is a pair The following theorem is the most central result of the paper. It gives the convergence properties under simultaneous homogenization and reduction of dimension to a limit problem stated for a pair of functions, u and y, which together represent the displacement field in the limit. The function u belongs to the Kirchhoff–Love space U, which means that it has quite simple behavior in the scaled direction x3 (the first two components are linear in x3 and the third component does not depend on x3 at all), while y can be thought of as representing the fine behavior in that direction.
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
161
Theorem 2. Assume symmetry, uniform boundedness and coercivity of Aε as in (4.2) and (4.3), assume (4.7), (4.9) of Lemma 5 and, moreover,
gε → g
strongly in L2 (Ω)
3×3
,
(4.16)
and assume finally that the Aε H-converge to a matrix A in the sense that (3.23) holds. Then, for the whole sequence ε, the convergences (4.10) to (4.13), as well as
Aε eε uε Ae(u, y)
weakly in L2 (Ω)
3×3
,
(4.17)
hold true, where (u, y) is the unique solution of 3 3 1 1 1 1 2 1 2 1 , , u ∈ U, y ∈ L ω; H − , and ∀v ∈ U, z ∈ L ω; H − , m m 2 2 2 2 A(x )e(u, y), e(v, z) dx = f · v dx + g, e(v, z) dx 3 Ω Ω Ω 1 1 + h+ (x ) · v x , + h− (x ) · v x , − dx .
2
ω
(4.18)
2
Moreover the energies converge in the sense that Ω
ε A (x3 )eε uε , eε uε dx →
Ω
A(x3 )e(u, y), e(u, y) dx.
Proof. In view of Lemma 5 and the uniqueness of the solution of (4.18), the proof of Theorem 2 will be complete (apart from convergence of energies) whenever we have proved that σ = Ae(u, y) for the limit functions u, y, σ obtained in Lemma 5. By symmetry of A, this is equivalent to proving that σ = A∇(u, y). We start from eαβ (uε ) ε ε ε ε ε σ =A e u =A ∂y ε
ε 1 ∂y α 2 ∂x 3
∂y3ε ∂x 3
β 1 2 ∂x 3
= Aε e uε , y ε = Aε ∇(uε , y ε ).
We are going to pass to the limit in this equation, written as in Section 2.4
M ε σ ε = P ε ∇ uε , y ε , with ε
M =
I
0
(Aε ) − C ε Rε Lε P = Rε Lε ε
δik δβδ −C ε Rε = ε R 0 0 I
−aεiβs3 rsε3k3 , riε3k3
aεiβkδ − aεiβs3 rsε3t3 aεt3kδ = riε3s3 aεs3kδ
0 . δik
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B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
• Passing to the limit in P ε ∇(uε , y ε ). We have P ε ∇ uε , y ε =
pεiβkδ pεi3kδ
∂uεα
0 ∂x β δik 0
ε ∂y α ∂x 3 , ∂y3ε ∂x 3
ε ε ε ∂uε ∂y ε P ∇ u , y ij = pεijkl ∇kl uε , y ε = pεijαβ α + pεijk3 k . ∂xβ ∂x3
We are going to pass to the limit successively in each term of the right member above. ◦ For the first term, pεijαβ
∂uεα ∂uα ∂ ε ∂ = pijαβ uεα (pijαβ uα ) = pijαβ , ∂xβ ∂xβ ∂xβ ∂xβ
in the sense of distributions, since the convergences uεα → uα
strongly in L2 (Ω) and
pεijαβ → pijαβ
weakly in L∞ (Ω)
imply that pεijαβ uεα pijαβ uα , in the sense of distributions. Then, by boundedness, pεijαβ
∂uεα ∂uα pijαβ ∂xβ ∂xβ
weakly in L2 (Ω).
◦ For the second term, pεijk3
∂ykε ∂y ε = pijk3 k , ∂x3 ∂x3
with pijk3 = 0 or δik , independent of ε. As pεijk3
∂ykε ∂yk pijk3 ∂x3 ∂x3
∂ykε ∂x 3
∂y k ∂x 3
weakly in L2 (Ω), we have
weakly in L2 (Ω).
◦ Hence
P ε ∇ uε , y ε P ∇(u, y),
weakly in L2 (Ω)
3×3
.
• Passing to the limit in M ε σ ε . Here we have
M εσε
ij
ε ε ε = mεijkl σkl = mεijkα σkα + mεijk3 σkε3 = mijkα σkα + mεijk3 σkε3 ,
ε mijkα σkα mijkα σkα
weakly in L2 (Ω).
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163
Moreover, ε mεijk3 σkε3 = Mijk ·σ ˜kε ,
ε with Mijk = mεijkl
l=1,2,3
and with σ ˜kε = 0, 0, σkε3 .
ε = 0 (see Section 3.4), so we can apply compensated compactWe have already checked that curl Mijk ness, and get that
M εσε M σ
weakly in L2 (Ω)
3×3
,
as soon as we have checked that div σ ˜kε is relatively compact in H −1 (Ω). • Now we prove that div σ ˜kε is relatively compact in H −1 (Ω). Clearly, from (4.4), Ω
ε σ − g ε , eε (w) dx =
Ω
f ε · w dx,
3
∀w ∈ D(Ω) ,
that is, as σ ε − g ε ∈ (L2 (Ω))3s×3 , ε ∂wα ∂wα ∂w3 ∂w3 1 1 1 σ − g ε αβ + σ ε − g ε α3 + σ ε − g ε 3β + 2 σ ε − g ε 33 dx Ω
= Ω
ε
∂xβ
∂x3
ε
∂xβ
ε
ε fα wα + f3ε w3 dx,
i.e., in the sense of distributions, ∂ ε 1 ∂ ε σ − g ε αβ − σ − g ε α3 = fαε , − ∂xβ ε ∂x3 ε 1 1 ∂ ε ∂ ε − σ − g 3β − 2 σ − g ε 33 = f3ε .
ε ∂xβ
We have to prove that
ε ∂x3
ε ∂σk3 ∂x 3
is relatively compact in H −1 (Ω). But
∂σαε 3 ∂g ε ∂ ε ∂gα3 ε = α3 − ε fαε + σαβ − gαβ → ∂x3 ∂x3 ∂xβ ∂x3 since, by virtue of (4.16), ∂gαε 3 ∂gα3 → ∂x3 ∂x3
strongly in H −1 (Ω),
and since fαε +
∂ ε ε σαβ − gαβ ∂xβ
is bounded in H −1 (Ω).
strongly in H −1 (Ω),
∂x3
164
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
In the same way,
ε ∂σ33 ∂g ε 1 ∂ ε ∂g33 = 33 − ε2 f3ε + σ3β − g3εβ → ∂x3 ∂x3 ε ∂xβ ∂x3
strongly in H −1 (Ω).
• Finally, let us prove convergence of energies. We have, due to (4.7), (4.9), (4.10), (4.11), (4.13) and (4.16),
ε ε ε ε ε A e u , e u dx Ω
= →
Ω
Ω
=
ε ε ε g , e u dx +
ε
f · u dx +
f · u dx + ε
Ω
ω
ω
hε+
·u
ε
1 x, 2
+
hε−
·u
ε
1 x ,− 2
1 1 h+ · u x , + h− · uε x , − 2 2
g, e(u, y) dx +
Ω
dx
dx
Ae(u, y), e(u, y) dx.
Ω
4.5. Reduced second-order limit problem Here we decouple the limit problem of Theorem 2, so that we get one problem for u alone (the reduced second-order problem), plus one equation which recovers y from u. Theorem 3. Let (u, y) be the pair defined in Theorem 2. Setting g = (g , g3 ) with g = (gkδ ) and g3 = (gk3 ), B = A − CRL = (biαkβ ) ) l = g − CRg3 = (lkδ
e (v) = ekδ (v) ,
and
and
biαkβ = aiαkβ − aiαs3 rs3t3 at3kβ ,
lkδ = gkδ − akδs3 rs3t3 gt3 ,
ekδ (v) = 1 − δk3
∂vk
∂xδ
(4.19) (4.20)
,
the function u is the unique solution of the following reduced second-order limit problem u ∈ U, and ∀v ∈ U, B e (u), e (v) dx = f · v dx + l , e (v) dx Ω Ω Ω 1 1 + h+ (x ) · v x , + h− (x ) · v x , − dx ω
2
(4.21)
2
1 (− 12 , 12 )))3 of and the function y is the unique solution in (L2 (ω; Hm
∂y = (A33 )−1 g3 − Le (u) = R g3 − Le (u) . ∂x3
(4.22)
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
165
Setting
= (b B αβγδ ),
˜l = (l ), αβ
∇ u =
∂uα , ∂xβ
(4.23)
another formulation of (4.21) is u ∈ U, and ∀v ∈ U, B∇ 1
u, ∇ v dx = ˜ l, ∇ v dx + f · v dx + h+ (x ) · v x , 2 Ω Ω Ω ω 1 dx . + h− (x ) · v x , −
(4.24)
2
Proof. By virtue of the symmetry property (3.26) of A,
Ω
A(x3 )e(u, y), e(v, z) dx =
Ω
A(x3 )∇(u, y), ∇(v, z) dx,
with
∇(v, z) =
∂vα ∂x β
0
∂z α ∂x 3 . ∂z 3 ∂x 3
Hence, as also g is symmetric, the limit variational equation can be written
Ω
A(x3 )∇(u, y), ∇(v, z) dx =
Ω
f · v dx +
+ ω
Ω
g, ∇(v, z) dx
h+ (x ) · v x ,
1 2
+ h− (x ) · v x , −
1 2
dx .
By using the symmetry properties of A and g, the limit variational equation obtained in Theorem 2 can be written as
Ω
A∇(u, y), ∇(v, z) dx =
Ω
f · v dx +
+ ω
∇(v, z) =
α
∂x β
0
∂z α ∂x 3 ∂z 3 ∂x 3
,
or, in vectorial notation (see Section 2.3), ∇(v, z) =
e (v)
e3 (z)
,
Ω
g, ∇(v, z) dx
h+ (x ) · v x ,
with ∂v
1 1 + h− (x ) · v x , − 2 2
dx ,
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B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
with
e (v) = ekδ (v) ,
e3 (z) = ek3 (z) ,
ekδ (v) = (1 − δk3 )
∂vk , ∂xδ
ek3 (z) =
∂zk . ∂x3
Then, writing also g as a vector g = (g , g3 ), we get (see Section 2.3)
g, ∇(v, z) = g , e (v) + g3 · e3 (z)
and A∇(u, y) =
A
C A33
L
e (u)
e3 (y)
A e (u) + Ce3 (y) , Le (u) + A33 e3 (y)
=
A∇(u, y), ∇(v, z) = A e (u) + Ce3 (y), e (v) + Le (u) + A33 e3 (y) e3 (z).
It follows that the limit variational equation obtained in Theorem 2 has the formulation: Ω
A e (u) + Ce3 (y) − g , e (v) dx +
= Ω
f · v dx +
ω
h+ (x ) · v x ,
Ω
Le (u) + A33 e3 (y) − g3 · e3 (z) dx
1 1 + h− (x ) · v x , − 2 2
dx ,
which decouples into Ω
A e (u) + Ce3 (y) − g , e (v) dx
= Ω
f · v dx +
ω
h+ (x ) · v x ,
1 1 + h− (x ) · v x , − 2 2
dx
(4.25)
and
Ω
Le (u) + A33 e3 (y) − g3 · e3 (z) dx = 0,
with
e3 (z) =
∂z1 ∂z2 ∂z3 , , . ∂x3 ∂x3 ∂x3
As the last integral equation holds true for any z in (L2 (ω; H 1 (− 12 , 12 )))3 , it is equivalent to
Ω
Le (u) + A33 e3 (y) − g3 · ψ dx = 0,
for any ψ in L2 (Ω)3 , which allows to eliminate y in favor of u, resulting in (4.22). Then an easy computation shows that (4.25) reduces to (4.21) or to (4.24).
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
167
4.6. Reduced fourth-order limit problem Recalling the definition of the Kirchhoff–Love space,
2
3
U = v ∈ H 1 (Ω)
2
× H02 (ω), ∃η ∈ H01 (ω) , vα = ηα − x3
= v ∈ H 1 (Ω) , ∃η ∈ H01 (ω)
2
× H02 (ω), vα = ηα − x3
∂v3 , α = 1, 2 , ∂xα
∂η3 , α = 1, 2, v3 = η3 , ∂xα
we see that each function v in U is parametrized by a function η defined on ω, called its Kirchhoff–Love displacement field. In this subsection we translate problem (4.24) for u ∈ U into a fourth-order problem for its Kirchoff–Love displacement field ζ = ζ(x ).
= B(x
3 ) be defined in (4.19), (4.23), that is, Theorem 4. Let B
= (b B αβγδ )
with bαβγδ = aαβγδ − aαβs3 rs3t3 at3γδ
and let K1 , K2 , K3 and K be defined by
K1 =
1 2
− 12
2 dx3 , Bx 3
K1 K= −K2
K2 =
1 2
− 12
3 dx3 , Bx
K3 =
1 2
− 12
dx3 , B
−K2 . K3
Let ˜l be defined in (4.20), (4.23), that is, ˜l = (l ) with l = gαβ − aαβs3 rs3t3 gt3 αβ αβ and let L1 = L1 (x ), L2 = L2 (x ), L = L(x ), f 0 = f 0 (x ) and l0 = l0 (x ) be defined by L1 = −
f0 =
1 2
− 12 1 2
− 12
l0 =
1 2
− 12
˜lx3 dx3 ,
L2 =
1 2
− 12
˜l dx3 ,
L = (L1 , L2 ),
f dx3 + h+ + h− , 1 f x3 dx3 + (h+ − h− ), 2
with f = (fα ), h± = (h±α ).
Let u be the limit function given in Theorem 2 and Theorem 3. Then u = (ζα − x3 ∂∂xζα3 , ζ3 ) where ζ = ζ(x ) is the unique solution of the fourth-order variational problem: 2 2 ζ ∈ H01 (ω) × H02 (ω), and ∀η ∈ H01 (ω) × H02 (ω), f 0 · η dx + [L, η] dx − l0 · ∇η3 dx , [Kζ, η] dx = ω
ω
ω
ω
(4.26)
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B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
with the second-order linear operator defined by
η = D2 η3 , ∇η ,
D2 η3
αβ
=
∂2 η3 ∂ηα , (∇η )αβ = . ∂xα ∂xβ ∂ηβ
Remark 1. Problem (4.26) reads: 1 2 1 2 2 ζ ∈ H (ω) × H (ω), and ∀η ∈ H (ω) × H02 (ω), 0 0 0 0 2 2 2
f3 η3 + L1 , D η3 − l0 · ∇η3 dx ,
K1 D ζ3 − K2 ∇ζ , D η3 dx =
ω ω 0 2 −K2 D ζ3 + K3 ∇ζ , ∇η dx = f · η + [L2 , ∇η ] dx , ω
(4.27)
ω
that is, ζ ∈ (H01 (ω))2 × H02 (ω) is the unique solution, in the sense of distributions, of the following equations, in the two-dimensional plate ω: (K1 )αβγδ
∂4 ζ3 ∂3 ζγ − (K2 )αβγδ = ϕ3 , ∂xα ∂xβ ∂xγ ∂xδ ∂xα ∂xβ ∂xδ ∂3 ζ3 ∂2 ζγ (K2 )αβγδ − (K3 )αβγδ =ϕα , ∂xβ ∂xγ ∂xδ ∂xβ ∂xδ
(4.28)
with 1 ∂ (h+α − h−α ) + h+3 + h−3 + ϕ3 = 2 ∂xα
ϕα = h+α + h−α +
1 2
− 12
1 2
− 12
∂2 ˜lαβ ∂fα − x3 + f3 dx3 , ∂xα ∂xα ∂xβ
∂˜lαβ fα − dx3 . ∂xβ
Proof of Theorem 4. For any v in U, we define η = η(v) by ηα = vα + x3
∂v3 , ∂xα
η3 = v3 ,
so that (∇ v)αβ =
∂2 η3 ∂vα ∂ηα = − x3 = (∇η )αβ − x3 D2 η3 αβ ∂xβ ∂xβ ∂xα ∂xβ
and, writing similarly ζ = ζ(u), problem (4.24) reads as the following fourth-order variational problem: 2 2 ζ = ζ(u) ∈ H01 (ω) × H02 (ω), and ∀η ∈ H01 (ω) × H02 (ω), ∂η3 2 2 B ∇ζ − x3 D ζ3 , ∇η − x3 D η3 dx = fα ηα − x3 dx f3 η3 dx ∂xα Ω Ω Ω 1 ∂η3 2 ˜ l, ∇η − x3 D η3 dx + (h+ h− ) · η + (h−α − h+α ) dx , + Ω
ω
2
∂xα
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
169
which, as an easy computation shows, is equivalent to (4.26), that is (4.27), or (4.28), in the sense of distributions. Acknowledgements The first author is grateful to the Swedish Research Council for basic support and to Ecole Normale Supérieure de Cachan for visiting support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
P.G. Ciarlet, Mathematical Elasticity, Vol. I: Three Dimensional Elasticity, North-Holland, 1988. P.G. Ciarlet, Mathematical Elasticity, Vol. II: Theory of Plates, North-Holland, Amsterdam, 1997. P. Courilleau, Homogénéisation et compacité par compensation, C. R. Acad. Sci. Paris Ser. I 332 (2001), 991–994. P. Courilleau, S. Fabre and J. Mossino, Homogénéisation de problèmes non linéaires dans des milieux quasi-stratifiés, C. R. Acad. Sci. Paris Ser. I 328 (1999), 401–406. P. Courilleau, S. Fabre and J. Mossino, Homogenization of some nonlinear problems with specific dependence upon coordinates, Boll. Un. Math. Ital. B (8) 4 (2001), 711–729. P. Courilleau and J. Mossino, Compensated compactness for nonlinear homogenization and reduction of dimension, Calc. Var. 20 (2004), 65–91. S. Fabre and J. Mossino, H-convergence of multiplicable matrices, Calc. Var. 7 (1998), 125–139. G. Friesecke, R. James and S. Mueller, Rigorous derivation of nonlinear plate theory and geometric rigidity, C. R. Acad. Sci. Paris, Ser. I 334 (2002), 173–178. B. Gustafsson and J. Mossino, Explicit limits for nonperiodic homogenization and reduction of dimension, C. R. Acad. Sci. Paris, Ser. I 334 (2002), 977–982. B. Gustafsson and J. Mossino, Nonperiodic explicit homogenization and reduction of dimension: the linear case, IMA J. Appl. Math. 68 (2003), 269–298. H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. 74 (1995), 549–578. H. Le Dret and A. Raoult, The membrane shell model in nonlinear elasticity, J. Nonlinear Sci. 6 (1996), 58–84. W.H. Mc Connel, On the approximation of elliptic operators with discontinuous coefficients, Ann. Sc. Norm. Sup. Pisa (4) 3(1) (1976), 123–137. F. Murat, H-convergence, in: Séminaire d’Analyse Fonctionnelle et Numérique, University of Alger, 1977–1978. L. Tartar, Homogénéisation et compacité par compensation, Cours Peccot, Collège de France, partially written in [14].
139
Compensated compactness for homogenization and reduction of dimension: The case of elastic laminates Björn Gustafsson a and Jacqueline Mossino b a
Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden C.M.L.A., Ecole Normale Supérieure de Cachan, 61, Avenue du Président Wilson, 94235 Cachan cedex, France
b
Abstract. The aim of this paper is to extend, to the linear elasticity system, the asymptotic analysis by compensated compactness previously developed by the authors for the linear diffusion equation. For simplicity, we restrict ourselves to stratified media. In the case of sole homogenization we recover the classical result of W.H. Mc Connel, deriving explicitly the effective elasticity tensor for stratified media. Here we give a new proof of his result, based on compensated compactness and on a technique of decomposing matrices. As for the case of simultaneous homogenization and reduction of dimension, we perform the asymptotic analysis, as the thickness tends to zero, of a three-dimensional laminated thin plate having an anisotropic, rapidly oscillating elasticity tensor. The limit problem is presented in three different ways, the final formulation being a fourth-order problem on the two-dimensional plate, with explicitly given elasticity tensors and effective source terms. Keywords: nonperiodic homogenization, reduction of dimension, theory of plates, compensated compactness, laminated or stratified materials
1. Introduction The aim of this paper is to extend, to the linear elasticity system, the compensated compactness method developed by the authors in [9,10] for the asymptotic analysis by homogenization and reduction of dimension of the linear diffusion equation. For simplicity, we restrict ourselves to the three-dimensional Euclidean space and to stratified (or laminated) materials. The paper is divided into three parts: • In the first part (Section 2), we give some preliminaries of linear algebra, showing the analogy between the linear diffusion equation and the linear elasticity system. In particular, we write a 3×3×3×3 tensor as a 3 × 3 matrix, the elements of which are themselves 3 × 3 matrices. (This gives a planar representation of the elasticity tensor.) In the same spirit, we write a 3 × 3 matrix as a three-component vector. (This gives a linear representation of the matrix of symmetrized gradients.) In some sense, this allows us to consider the symmetrized gradient as an ordinary gradient and the elasticity tensor as a conductivity matrix. In the case of stratified or laminated media, one of the space variables, say x3 , plays a particular role. In this case, we give a well suited planar representation of a general elasticity tensor A, writing A=
A
L
C , A33
0921-7134/06/$17.00 2006 – IOS Press and the authors. All rights reserved
140
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
and we show that A admits the same decomposition as the one that we have previously introduced in the diffusion case, writing A = (M )−1 P , with M=
I
−C(A33 )−1 (A33 )−1
0
P =
and
A − C(A33 )−1 L
(A33 )−1 L
0 . I
• In the second part (Section 3), we use the above decomposition and the compensated compactness method introduced by Murat and Tartar (see [14] and [15]) to pass to the homogenized elasticity system in the case of a stratified material occupying a fixed domain. In doing this, we give a new proof of an old result by Mc Connel [13], by extending to elasticity the proof of [7] (see also [3–5]). Briefly speaking, we prove that, in the framework of elasticity, the H-limit of Aε = (M ε )−1 P ε , with M ε and P ε defined from Aε as above, is A = (M )−1 P , with M and P being the weak -L∞ limits of M ε and P ε . In this part, it is crucial that div P ε and curl M ε are relatively compact in a suitable sense, with div P ε and curl M ε defined by
ε = div P ε = div Pijk
∂pεijkl
∂xl
l
ε
curl M =
ε curl Mijk
=
,
ε where Pijk = pεijkl ,
∂mεijks ∂mεijkt − , ∂xt ∂xs
(1.1)
ε where Mijk = mεijkl .
(1.2)
In other words, div and curl are taken with respect to the last index. Our methods in principle allow for extensions far beyond the “stratified case”, since, in our particular case, div P ε and curl M ε actually vanish. However, we leave this as a programme for the future. • In the third part (Section 4), we consider the case of laminated plates, for which the two phenomena of homogenization and reduction of dimension occur simultaneously. In this case, we use the analogy that has been established in Section 2 between the linear diffusion equation and the linear elasticity system, in order to extend to elasticity the results of [9,10]. First, we derive, in the “big cylinder” obtained by rescaling the thin plate, a second-order limit problem for two unknown functions, the limit u of the rescaled displacement and an additional function y. The limit problem involves the H-limit A of the rescaled elasticity tensor, which is explicitly given by our study in Section 3. The source terms of this limit problem, say (f , g, h), are natural limits of the rescaled source terms of the original problem. Then, by eliminating y in favor of u, we prove that u itself solves a second-order limit problem in the “big cylinder”, involving a new tensor derived from the previous H-limit A, and involving new source terms derived from (f , g, h) and A. Finally, from the previous second-order limit problem in u, we derive the fourth-order limit problem for the Kirchhoff–Love displacement field ζ, defined over the limit two-dimensional plate. In this part, it is crucial not only that div P ε and curl M ε are relatively compact in a suitable sense, but also that M ε and P ε have some constant column vectors (compare [6] and [10]). Again, this should allow for many extensions beyond the “stratified case”. We emphasize that, in the diffusion case, nonlinear extensions of this method of compensated compactness are considered in [4,5] for the homogenization theory, and in [6] for simultaneous homogenization and reduction of dimension. But we are very far from extending to nonlinear elasticity, the compensated compactness method developed in this paper.
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141
2. Preliminary: from diffusion to elasticity 2.1. Planar representation of 3 × 3 × 3 × 3 tensors Consider a tensor A = (aijkl )i,j ,k,l=1,2,3 . We shall introduce a way of giving a planar representation of A, namely by considering it as a matrix of matrices. On the major level A will be a matrix in the second and fourth indices (j and l above). The entries (coefficients) of this major matrix are themselves matrices, namely in the first and third indices (i and k). We may think of j and l as exterior indices and i and k as interior ones. Thus we write
A11 A = A21 A31
A12 A22 A32
A13 A23 = Ajl j ,l=1,2,3 , A33
where, for fixed j and l, Ajl = (aijkl )i,k=1,2,3 . Since the x3 -coordinate will play a special role in this paper we shall further arrange our matrices (at least on the major level) on block form by grouping the index values as {1, 2} plus {3}. This means that we write A on the form A=
A
L
C a = iβkδ ai3kδ A33
aiβk3 , ai3k3
(2.1)
where the blocks are given by A =
A11 A21
A12 = (aiβkδ ), A22
C=
A13 = (aiβk3 ), A23
L = (A31
A32 ) = (ai3kδ ).
Here we have used the following convention, valid throughout the paper. Notational convention. Latin indices (i, j, k, l, . . . ) always run over {1, 2, 3} while Greek indices (α, β, γ, δ, . . . ) only run over {1, 2}. In addition, we shall constantly use the Einstein summation convention: whenever an index occurs twice in a term, it is summed over (from 1 to 3 if it is a Latin index, from 1 to 2 if it is a Greek index). 2.2. Multiplication of 3 × 3 × 3 × 3 tensors, by multiplication of their planar representation Multiplication of 3 × 3 × 3 × 3 tensors is defined so that it becomes ordinary matrix multiplication, on each level, in the above matrix of matrices representation. On the component level, this means that if aijkl ), then the components of the product A = AA are given by a ˜ijkl = aijst a ¯stkl . A = (aijkl ), A = (¯
On the block representation level, this means that Ajl = Ajt Atl . The two levels correspond to first summing over the interior index s and then over the exterior one, t.
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Writing the matrices on block form, A=
A
L
C , A33
A A= L
C , A33
it is easy to check that the multiplication scheme applies, giving
A A + CL A C + CA33 . AA = LA + A33 L LC + A33 A33 We note that the identity tensor I with respect to the above multiplication has entries (I)ijkl = δik δjl = 1 if i = k and j = l, 0 otherwise. 2.3. Linear representation of 3 × 3 matrices Let ξ = (ξij ) be a 3×3 matrix. Just as above we consider j as an exterior index and i as interior, and we get a linear (or vectorial) representation of ξ by considering it as a column vector in j, the components of which are themselves column vectors, in i. Writing it on block form it becomes
ξ , ξ3
ξ=
with ξ = (ξiβ ), ξ3 = (ξi3 ). If A = (aijkl ) is a 3 × 3 × 3 × 3 tensor, as in Subsection 2.1, the product Aξ is defined by (Aξ)ij = aijkl ξkl . As in the previous subsection, it is easy to check that this product can be done in the block representation, giving Aξ =
A
L
C A33
ξ
ξ3
=
A ξ + Cξ3 . Lξ + A33 ξ3
Scalar products between matrices considered as vectors is denoted by [·, ·]: [ξ, η] := ξij ηij = ξiβ ηiβ + ξi3 ηi3 = [ξ , η ] + ξ3 · η3 . However, the ordinary scalar product in R3 is denoted by a “dot”. In both cases the corresponding norms are denoted | · |. 2.4. Decomposition of 3 × 3 × 3 × 3 tensors If A=
A
L
C , A33
with A33 invertible (which always holds true in this paper), then we consider (A33 )−1 = R = (rik )
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as part of a 3 × 3 × 3 × 3 tensor by identifying it with R = (ri3k3 ), where ri3k3 = rik . Similarly, it is natural to define CR, RL and CRL as the following tensors: (CR)iβk3 = aiβs3 rs3k3 , (RL)i3kδ = ri3s3 as3kδ , (CRL)iβkδ = aiβs3 rs3t3 at3kδ . Thus we can define 3 × 3 × 3 × 3 tensors M and P by: M=
I
−CR , R
0
P =
A − CRL
RL
0 , I
(2.2)
with (I )iβkδ = δik δβδ , I = I33 = (δik δ33 ) = (δik ) = identity matrix in R3 . It is easy to check that M is invertible with respect to the multiplication introduced in Subsection 2.2, with inverse M
−1
=
I
0
C , A33
(2.3)
and that A = M −1 P.
(2.4)
3. Homogenization of laminates Let Ω be a fixed bounded domain in R3 . We consider the displacement field uε , solution of the threedimensional linear elasticity system: 1 3 ε 1 3 ε u ∈ H (Ω) , u = 0 on Γ0 , and ∀v ∈ H (Ω) , v = 0 on Γ0 , ε ε ε ε A (x )e u , e(v) dx = f · v dx + g , e(v) dx + 3 Ω
Ω
Here • Γ0 ⊂ ∂Ω; • γ denotes surface measure on ∂Ω;
Ω
∂Ω\Γ0
hε · v dγ.
(3.1)
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• Aε = Aε (x3 ) is a tensor with coefficients aεijkl (x3 ), i, j, k, l ∈ {1, 2, 3}, satisfying the following symmetry, boundedness and coercivity conditions: aεijkl = aεjikl = aεijlk ,
(3.2)
∃α, β > 0, for a.e. x ∈ Ω, ∀ξ ∈ R3s×3 , ε A (x3 )ξ, ξ α|ξ|2
• • • •
and
ε A (x3 )ξ β|ξ|,
(3.3)
where R3s×3 denotes the set of symmetric 3 × 3 matrices. ∂v eij (v) = 12 ( ∂∂xvji + ∂xji ) is the symmetrized gradient of v; f ε ∈ (L2 (Ω))3 ; g ε ∈ (L2 (Ω))3s×3 , the set of symmetric 3 × 3 matrices with L2 -coefficients; hε ∈ (L2 (∂Ω \ Γ0 ))3 .
We remark that the upper bound in (3.3) is equivalent to [Aε ξ, η] β|ξ||η| (∀ξ, η ∈ R3s×3 ). This implies [Aε ξ, ξ] β|ξ|2 , but the converse implication only holds under additional symmetry assumptions on Aε , namely that aεijkl = aεklij . The physical interpretation of (3.1) and the quantities involved is as follows: Ω is a region in space filled with a (linear) elastic material subject to some small deformations and subsequent stresses. The vector field uε describes the (infinitesimal) displacement of the body and its symmetrized gradient e(uε ) = (eij (uε )) is the strain tensor, which measures the rate of deformation. There is a corresponding ε ) which is related to e(uε ) by Hooke’s law (in general form) stress tensor σ ε = (σij
σ ε = Aε e u ε . With this relation between σ ε and e(uε ), (3.1) is the variational formulation of Newton’s law of balance of forces on an infinitesimal level, namely −div σ ε = f ε − div g ε
in Ω
or, in component form, −
ε ε ∂σij ∂gij = fiε − ∂xj ∂xj
in Ω,
together with the boundary conditions uε = 0 on Γ0 , σ ε n = g ε n + hε
on ∂Ω \ Γ0 .
Above the divergence is taken with respect to the last index and n denotes the outward normal unit vector on ∂Ω. The fact that the elasticity tensor Aε = Aε (x3 ) depends only on x3 means that the body Ω is composed of materials which are stratified (or laminated) along horizontal planes. We allow that Aε is a “full
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tensor”, which means that the material may be anisotropic (its behavior is not governed only by the two Lamé coefficients). The constraint uε = 0 on Γ0 means that the body is clamped on that part, Γ0 , of its boundary. On the remaining parts of ∂Ω there act surface forces represented by hε , while the fields f ε and g ε represent body forces. The presence of g ε allows for taking into account more singular body forces than with just f ε . It is well known that problem (3.1) admits a unique solution uε (see, e.g., [1]). We are interested in the limit behavior of uε as ε tends to zero. This question has been solved by Mc Connel in [13]. Here we give a different proof, by using compensated compactness and arguments similar to those introduced by Courilleau and ourselves in [3,9,10,6], see also [7,4,5], namely by using a decomposition of Aε such that Aε = (M ε )−1 P ε , with curl M ε and div P ε vanishing (see (1.1) and (1.2)). 3.1. A priori estimates We have the following a priori estimates. Lemma 1. Assume (3.3) and assume that {f ε }ε is bounded in (L2 (Ω))3 , {g ε }ε is bounded in (L2 (Ω))3×3 and {hε }ε is bounded in (L2 (∂Ω \ Γ0 ))3 . Then {e(uε )}ε is bounded in (L2 (Ω))3×3 , which implies that {uε }ε is bounded in (H 1 (Ω))3 and {σ ε = Aε e(uε )}ε is bounded in (L2 (Ω))3×3 . Proof. Taking v = uε as test function in (3.1), we get
ε ε ε A e u , e u dx = Ω
Ω
f ε · uε dx +
ε ε g , e u dx +
Ω
∂Ω\Γ0
hε · uε dγ
f ε (L2 (Ω ))3 uε (L2 (Ω ))3 g ε (L2 (Ω ))3×3 e uε (L2 (Ω ))3×3
+ hε (L2 (∂Ω\Γ0 ))3 uε (L2 (∂Ω\Γ0 ))3 . Now, by the Korn inequality, ε u
(L2 (Ω ))3
uε (H 1 (Ω ))3 C e uε (L2 (Ω ))3×3 ,
and, by continuity of the trace mapping, ε u
(L2 (∂Ω\Γ0 ))3
C uε (H 1 (Ω ))3 C e uε (L2 (Ω ))3×3 .
Moreover, by coercivity (see (3.3)) Ω
ε ε ε A e u , e u dx α
ε 2 e u dx,
Ω
so that
αe uε (L2 (Ω ))3×3 C f ε (L2 (Ω ))3 + g ε (L2 (Ω ))3×3 + hε (L2 (∂Ω\Γ0 ))3 . By boundedness of {Aε }ε (see (3.3)), it follows that ε σ
(L2 (Ω ))3×3
β e uε (L2 (Ω ))3×3
is bounded, when ε tends to zero.
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3.2. Compactness From Lemma 1, classical compactness arguments immediately give: Lemma 2. Assume (3.3) and that, for some f in (L2 (Ω))3 , g in (L2 (Ω))3s×3 and h in (L2 (∂Ω \ Γ0 ))3 ,
3
3×3
fε f
weakly in L2 (Ω) ,
gε g
weakly in L2 (Ω)
hε h
(3.4) ,
(3.5)
3
weakly in L2 (∂Ω \ Γ0 ) .
(3.6)
Then, for some subsequence of ε, still denoted by ε, and for some u in (H 1 (Ω))3 such that u = 0 on Γ0 , uε u
3
weakly in H 1 (Ω) ,
(3.7)
implying that the following three convergences hold true:
3
uε → u strongly in L2 (Ω) , uε|∂Ω\Γ0 → u|∂Ω\Γ0
(3.8)
3
strongly in L2 (∂Ω \ Γ0 ) ,
e uε e(u) weakly in L2 (Ω)
3×3
,
(3.9) (3.10)
and for some σ in (L2 (Ω))3s×3 ,
σ ε = Aε e u ε σ
weakly in L2 (Ω)
3×3
.
(3.11)
Moreover, σ satisfies 1 3 ∀v ∈ H (Ω) , v = 0 on Γ0 , σ − g, e(v) dx = f · v dx + Ω
Ω
∂Ω\Γ0
h · v dγ.
(3.12)
It remains to relate σ to u. In the next two subsections we shall show that σ = Ae(u), for the matrix A which was obtained by Mc Connel in [13]. 3.3. The H-limit matrix Here we introduce the matrix A which will turn out be the H-limit of {Aε (x3 )}ε .
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Lemma 3. Assume that {Aε = Aε (x)}ε satisfies aεijkl = aεjikl = aεijlk ,
(3.13)
∃α, β > 0, for a.e. x ∈ Ω, ∀ξ ∈ R3s×3 , ε A (x)ξ, ξ α|ξ|2 and Aε (x)ξ β|ξ|,
(3.14)
and write, as in (2.1),
(Aε ) Aε = Lε
aεiβkδ Cε = ε A33 aεi3kδ
aεiβk3 . aεi3k3
Then Aε33 is invertible, with inverse Rε having L∞ -coefficients. Moreover, there exists a subsequence of ε (which we still denote by ε) and a 3 × 3 × 3 × 3 tensor A with bounded coefficients, in block form A=
A
L
C a = iβkδ A33 ai3kδ
aiβk3 , ai3k3
with A33 invertible, such that
Rε = Aε33
−1
R = (A33 )−1
weakly in L∞ (Ω)
3×2×3
3×3×2
weakly in L∞ (Ω)
C ε Rε CR
Rε Lε RL weakly in L∞ (Ω) ε A − C ε Rε Lε A − CRL
3×3
,
(3.15)
,
(3.16)
,
(3.17)
weakly in L∞ (Ω)
3×2×3×2
.
(3.18)
Proof. • Let us first prove that Aε33 is invertible. For this, it is enough to prove that ε α A33 η · η |η|2 ,
2
∀η ∈ R3 ,
(3.19)
that is, aεi3k3 ηi ηk
α ηi ηi . 2
We know that, for every ξ = (ξij ) ∈ R3s×3 , aεijkl ξij ξkl αξij ξij . Choosing here ξαβ = 0, ξα3 = ξ3α = 12 ηα , ξ33 = η3 , i.e., 1 ξij = (ηi δj 3 + ηj δi3 ) 2
(3.20)
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(which is in R3s×3 ) and using the symmetries of aεijkl we get 1 aεi3k3 ηi ηk = aεijkl ηi δj 3 ηk δl3 = aεijkl (ηi δj 3 + ηj δi3 )(ηk δl3 + ηl δk3 ) = aεijkl ξij ξkl 4 α α α αξij ξij = (ηi δj 3 + ηj δi3 )(ηi δj 3 + ηj δi3 ) = (2η1 η1 + 2η2 η2 + 4η3 η3 ) ηi ηi , 4 4 2 proving (3.19). Actually, the coercivity constant α2 is optimal, as is seen by taking aεijkl = symmetrized identity matrix. • Let us denote by Rε the inverse of Aε33 and let us next prove that ∀η ∈ R3 ,
ε α R η ·η |η|2 2
and
2β
ε R η 2 |η|.
1 2 (δik δjl
+ δil δjk ), the
(3.21)
α
In general, if a matrix B in R3×3 satisfies ∀η ∈ R3 ,
(Bη) · η α|η|2
and
|Bη| β|η|,
then it is invertible and its inverse B −1 satisfies ∀η ∈ R3 ,
−1 α B η · η 2 |η|2
and
β
|B −1 η|
1 |η| α
(3.22)
(see, e.g., [10, p. 289]). In our case we have the lower estimate (3.19), and from the upper bound in (3.14) we easily get, using the recipe (3.20) to pass from R3 to R3s×3 , |Aε33 η| β|η|. Now (3.21) follows. • As {Rε }ε is bounded in (L∞ (Ω))3×3 , we have, up to a subsequence, Rε R
weakly in L∞ (Ω)
3×3
for some R satisfying ∀η ∈ R3 ,
(Rη) · η
α |η|2 2β 2
and
|Rη|
2 |η|. α
Applying again the general result (3.22) mentioned above, R is invertible, with inverse A33 satisfying ∀η ∈ R3 ,
(A33 η) · η
α3 2 |η| 8β 2
and
|A33 η|
2β 2 |η|. α
(Here the coercivity constant is not optimal. It will follow from Theorem 1 and (3.19) that the best coercivity constant is α/2.) • By now we have proved that (up to a subsequence)
Rε = Aε33
−1
R = (A33 )−1
weakly in L∞ (Ω)
3×3
,
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with A33 having L∞ -coefficients. From the boundedness of {Aε }ε , in the sense of (3.3), we also have boundedness in the usual sense, i.e., when acting on not necessarily symmetric vectors: ε A ζ β|ζ|.
∀ζ ∈ R3×3 ,
To see this, just apply |Aε ξ| β|ξ| with ξ ∈ R3s×3 given by ξij = 12 (ζij + ζji ). Then, by (3.13), |Aε ζ| = |Aε ξ| β|ξ| β|ζ|. (On the other hand, Aε is not invertible, and hence not coercive, on R3×3 , because it annihilates all antisymmetric tensors.) From the above it is clear that C ε Rε , Rε Lε and (Aε ) − C ε Rε Lε are uniformly bounded with bounds only depending on α and β, so that, up to a subsequence,
weakly in L∞ (Ω)
C ε Rε X
3×2×3
,
3×3×2 Rε Lε Y weakly in L∞ (Ω) , ε 3×2×3×2 A − C ε Rε Lε Z weakly in L∞ (Ω) ,
for some X, Y , Z. From these one obtains C, L and A by setting X = CR
C = XA33 ,
or
Y = RL or
L = A33 Y ,
A = Z + CRL = Z + XA33 Y.
Z = A − CRL or
Clearly, the resulting tensor A is bounded from above with bounds only depending on α and β. This finishes the proof of the lemma. 3.4. The limit problem We are now ready to formulate our version (Theorem 1 below) of Mc Connel’s theorem [13]. Defining the matrices M ε , P ε , M and P as in Section 2.4, by ε
M =
M=
I
0
I
0
(Aε ) − C ε Rε Lε P = Rε Lε
−C ε Rε , Rε
ε
−CR , R
P =
A − CRL
RL
0 , I
0 , I
the weak -L∞ convergences (3.15) to (3.18) in Lemma 3 can be summarized in matrix form as Mε M
and
Pε P
weakly in L∞ (Ω)
3×3×3×3
.
(3.23)
The main point in Theorem 1 is that when Aε depends only on x3 , then the convergences (3.23) are exactly what is needed to ensure that, for arbitrary boundary and source term data (f , g, h, etc.), the weak limits σ = lim σ ε and e(u) = lim e(uε ) obtained in Lemma 2 are related by σ = Ae(u).
(3.24)
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Then the fact that σ satisfies (3.12) means that u solves the appropriate limit version of (3.1), namely problem (3.29) below. Thus (3.23) (or (3.15) to (3.18)) characterizes H-convergence Aε → A in the stratified case Aε = Aε (x3 ). Theorem 1. Assume (3.2), (3.3), (3.4), (3.6) and moreover
gε → g
strongly in L2 (Ω)
3×3
,
(3.25)
and finally that {Aε }ε H-converges to a matrix A in the sense that (3.23) holds. Then, first of all, A depends only on x3 , it has the symmetries aijkl = ajikl = aijlk ,
(3.26)
and it satisfies the uniform conditions for a.e. x ∈ Ω, ∀ξ ∈ R3s×3 ,
[Aξ, ξ] α|ξ|2
and
|Aξ|
β2 |ξ|, α
(3.27)
where α and β are the coercivity constants in (3.3). Next, for the whole sequence ε, one has convergences (3.7) to (3.10), as well as
Aε e uε Ae(u) weakly in L2 (Ω)
3×3
,
(3.28)
with u being the unique solution of 1 3 1 3 u ∈ H (Ω) , u = 0 on Γ0 , and ∀v ∈ H (Ω) , v = 0 on Γ0 , A(x3 )e(u), e(v) dx = f · v dx + g, e(v) dx + h · v dγ. Ω
Ω
Ω
(3.29)
∂Ω\Γ0
Finally, the energies converge in the sense that Ω
ε A (x3 )e uε , e uε dx →
Ω
A(x3 )e(u), e(u) dx.
Proof. In order to make the proof more transparent we divide it into several steps. • Step 0: Generalities. The proof relies on the “div–curl lemma” of compensated compactness. We recall that this lemma in general says that if f ε , g ε , f , g ∈ (L2 (Ω))3 are vector fields such that f ε f , g ε g weakly in (L2 (Ω))3 and such that div f ε and the components of curl g ε are all contained in a compact subset of H −1 (Ω), then f ε · g ε f · g weakly as distributions. Let us spell out the matrices M ε = (mεijkl ) and P ε = (pεijkl ) in components. We have ε
M =
I
0
δ δ −C ε Rε = ik βδ Rε 0
−aεiβs3 rsε3k3 , riε3k3
(3.30)
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
(Aε ) − C ε Rε Lε Pε = Rε Lε
aεiβkδ − aεiβs3 rsε3t3 aεt3kδ 0 = I riε3s3 aεs3kδ
151
0 . δik
(3.31)
Similarly for M = (mijkl ) and P = (pijkl ), without the superscript ε. One circumstance that will complicate the proof of the theorem is that the matrices M ε and P ε do not enjoy the same symmetries (3.2) of Aε . We only have that P ε is symmetric in the last two indices, as a direct consequence of P ε = M ε Aε . From this we can however conclude that P = lim P ε , and hence A = M −1 P , are symmetric in the last two indices. The fact that the limit matrix A is symmetric in the first two indices will come out later on (Step 3). • Step 1: Proof that a subsequence of uε converges to a solution of (3.29). This is the most essential part of the proof, and it can be conceived as a continuation of Lemma 2 and Lemma 3, with now the information that Aε depends only on x3 being used. The issue is to show that when (3.23) holds and Aε = Aε (x3 ), then the weak limits σ = lim σ ε = lim Aε e(uε ) and e(u) = lim e(uε ) obtained in Lemma 2 are related by (3.24). So select a subsequence ε and u, σ as in Lemma 2. Due to the symmetry of A with respect to the last two indices, Eq. (3.24) (to be proved) is equivalent to σ = A∇u,
(3.32)
where ∇u is the nonsymmetrized gradient of u: (∇u)ij = the limit in
∂ui ∂x j .
Eq. (3.24) will be proved by passing to
σ ε = Aε ∇uε .
(3.33)
Using the decomposition Aε = (M ε )−1 P ε , the latter relation becomes M ε σ ε = P ε ∇uε .
(3.34)
This way of writing (3.33) is the key to everything, and the main conclusions are obtained from it by applying the “div–curl lemma” to (3.34). ◦ Passing to the limit in P ε ∇uε . According to (1.1), we notice that div P ε = 0 in the sense that, for every i, j, k ∈ {1, 2, 3}, ∂pεijkl = 0. ∂xl Indeed, by (3.31),
pεiβkδ (x3 ) 0 . P = pεi3kδ (x3 ) δik ε
Hence ∂pεijkδ ∂pεijkl = (x3 ) = 0. ∂xl ∂xδ
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Now ε ∂uε ε P ∇uε ij = pεijkl ∇uε kl = pεijkl kε = Pijk · ∇uεk , ∂u l
ε = (pε ) with Pijk ijkl l=1,2,3 . We can apply classical compensated compactness, since, for every i, j, k,
ε Pijk Pijk
3
weakly in L2 (Ω) ,
ε = 0, div Pijk
uεk uk
weakly in H 1 Ω .
ε ∇uε P ∇u in the sense of distributions and then, by boundedness, weakly in It follows that Pijk ijk k k 2 L (Ω). This means that
P ε ∇uε P ∇u weakly in L2 (Ω)
3×3
.
◦ Passing to the limit in M ε σ ε . ε = (mε ) According to (1.2), we notice that curl M ε = 0 in the sense that, if we set Mijk ijkl l=1,2,3 , then, ε for every i, j, k, curl Mijk = 0. Indeed, by (3.30),
δ δ M = ik βδ 0 ε
mεiβk3 (x3 ) , mεi3k3 (x3 )
so that
ε Mijk = (mijkδ )δ=1,2 , mεijk3 , ε has the form V ε = (V , V ε (x )) and with mijkδ independent of ε and x. In other words, Mijk 3 δ 3
curl V ε
δα
=
∂Vδ ∂Vα − = 0, ∂xα ∂xδ
curl V ε
δ3
=
∂Vδ ∂V ε − 3 = 0. ∂x3 ∂xδ
ε = M ε · σ ε , with σ ε = (σ ε ) Now (M ε σ ε )ij = mεijkl σkl ijk k k kl l=1,2,3 . We can apply compensated compactness, as soon as we have proved that div σkε is relatively compact in H −1 (Ω), and we get that M ε σ ε M σ in (D (Ω))3×3 and then, by boundedness, in (L2 (Ω))3×3 . ◦ Now we prove that div σkε is relatively compact in H −1 (Ω). Clearly, from (3.1), as σ ε − g ε ∈ (L2 (Ω))3s×3 ,
Ω
ε σ − g ε , ∇v dx =
so that −
∂ ε ε σkl − gkl = fkε ∂xl
Ω
f ε · v dx,
3
∀v ∈ D(Ω) ,
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153
∂g ε
and div σkε is relatively compact in H −1 (Ω) as soon as the same holds true for div gkε = ∂xkll (recall that, by compactness, fkε → fk , strongly in H −1 (Ω)). But this holds true, since, by assumption, g ε → g strongly in (L2 (Ω))3×3 . This finishes the proof of (3.32) and, hence, of (3.24). • Step 2: Proof of the coercivity bound in (3.27). Choose ξ ∈ R3s×3 , and consider it as a constant vector in (L2 (Ω))3s×3 . Since M is invertible, we can define ξ , ξ ε ∈ (L2 (Ω))3×3 and ξ¯ε ∈ (L2 (Ω))3s×3 by ξ = tM ξ ,
1 ξ¯ε = ξ ε +t ξ ε . 2
ξ ε = tM ε ξ ,
Here t M (etc.) denotes the transpose in the sense of interchanging the first pair of indices with the second pair: if M = (mijkl ) then t M = (mklij ). (For ξ = (ξij ) the transpose is defined as usual: tξ = (ξji ).) These definitions give immediately that t
ε ε ε t ε t ε A ξ ,ξ = M ξ , P ξ .
[Aξ, ξ] = M ξ ,t P ξ ,
Due to the symmetries (3.2) of Aε we also have ε ε ε ε ε ε A ξ¯ , ξ¯ = A ξ , ξ . , and since M , M ε , ξ only depend on x and since the mε Now (t M ε ξ )ij = mεklij ξkl 3 klij are constants t ε when j = 3, it follows that the only nonzero contributions to curl( M ξ ) are terms of the form
∂ ε ∂ξ mkliβ ξkl = mkliβ kl , ∂x3 ∂x3 which are fixed (i.e., independent of ε) elements of H −1 (Ω). Thus the components of curl(t M ε ξ ) stay in a compact subset of H −1 (Ω) as ε → 0. For similar reasons, the components of div(t P ε ξ ) stay in a compact subset of H −1 (Ω). Therefore, the above identities combined with compensated compactness show that ε ε ε A ξ¯ , ξ¯ [Aξ, ξ]
weakly as distributions. We also have that ξ¯ε ξ weakly in (L2 (Ω))3s×3 , which by a lower semicontinuity argument and (3.3) gives
|ξ| ϕ dx α lim inf
2
α Ω
Ω
ε 2 ξ¯ ϕ dx lim inf
ε ε ε A ξ¯ , ξ¯ ϕ dx = Ω
[Aξ, ξ]ϕ dx, Ω
for smooth test functions ϕ 0. From this the coercivity estimate in (3.27) follows. • Step 3: Proof of symmetry of A with respect to the first two indices. We are going to show that aijkl = ajikl , i.e., that the first equality in (3.26) holds. The idea is that the limit equation σ = Ae(u) holds for very many fields u (by varying the data f , g, h) and that therefore the symmetry of σ, which is a consequence of σ = lim σ ε and the symmetry of σ ε , forces A to be symmetric
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B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
in the first two indices. To achieve this we shall simply prescribe the limit field u = uλ (λ a parameter) on an arbitrary relatively compact subdomain of Ω. So let ω Ω be a relatively compact subdomain of Ω and let φ ∈ D(Ω) be a smooth test function with compact support in Ω such that φ = 1 in ω. Given λ ∈ R3×3 define uλ ∈ (D(Ω))3 and g λ ∈ (L2 (Ω))3s×3 by uλi (x) = λij xj φ(x) and g λ = 12 (Ae(uλ ) +t (Ae(uλ )). Using that e(v) is symmetric it is clear that uλ solves uλ ∈ H01 (Ω) 3 and ∀v ∈ H01 (Ω) 3 , λ A(x3 )e uλ , e(v) dx = g , e(v) dx. Ω
Ω
Notice that this is the limit problem (3.29) for the data f = 0, g = g λ , Γ0 = ∂Ω (hence there is no h). Moreover the above problem has only one solution, by virtue of the coercivity of A proved in Step 2. Now we consider the solution uελ of uελ ∈ H01 (Ω) 3 and ∀v ∈ H01 (Ω) 3 , ε λ A (x3 )e uελ , e(v) dx = g , e(v) dx. Ω
Ω
By Step 1 of the proof we have, at least for a subsequence, σ ελ = Aε ∇uελ σ λ = A∇uλ weakly in (L2 (Ω))3×3 . Hence the symmetry of σ ελ implies the symmetry of σ λ and we have, in ω,
aijkl λkl = aijkl ∇uλ
kl
λ λ = σij = σji = ajikl ∇uλ
kl
= ajikl λkl .
Choosing λkl = δkr δls , we get aijrs = ajirs in ω. As this holds true for any ω, the desired symmetry of A follows. • Step 4: Proof of the upper bound in (3.27). Since we now know that the tensor (or matrix) A = A(x3 ) has the symmetries (3.26) we can consider it, for each x3 , as a linear map A : R3s×3 → R3s×3 . The coercivity proved in Step 2 shows that as such a map it is invertible (whereas it is not invertible as a map R3×3 → R3×3 ). The same statements hold for tA. We also know (see Subsection 2.4) that M , and hence tM , is invertible as a map M : R3×3 → R3×3 . It follows that tP =t AtM maps R3×3 onto R3s×3 . Therefore, given ξ ∈ R3s×3 we can define ζ ∈ (L2 (Ω))3×3 and ξ ε in (L2 (Ω))3s×3 by ξ =t P ζ,
ξ ε =t P ε ζ.
(The symmetry of ξ ε is due to the symmetry of P ε in the last two indices.) It is easy to check that −1 t A ξ, ξ = P ζ,t M ζ , ε −1 ε ε t ε t ε A ξ , ξ = P ζ, M ζ .
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
155
Again compensated compactness gives that ε −1 ε ε A ξ , ξ A−1 ξ, ξ
weakly as distributions. Now, the statement (3.22) about the bounds for the inverse of a linear operator applies also to the vector space R3s×3 if B : R3s×3 → R3s×3 satisfies ∀ξ ∈ R3s×3 ,
[Bξ, ξ] α|ξ|2
and
|Bξ| β|ξ|,
then it is invertible on R3s×3 and the inverse B −1 satisfies ∀ξ ∈ R3s×3 ,
−1 α B ξ, ξ 2 |ξ|2 .
β
Consequently it follows from (3.3) that ε −1 ε ε α 2 A ξ , ξ 2 ξ ε .
β
Hence passing to the limit ε → 0 as before gives −1 α A ξ, ξ 2 |ξ|2 .
β
2
Combining with the Cauchy–Schwarz inequality, we get |ξ| βα |A−1 ξ|, which is equivalent to the upper bound in (3.27). • Step 5: Uniqueness of solution for the limit problem and convergence for the entire sequence. In Step 1 we obtained one solution u of the limit problem (3.29) as the limit of a subsequence uε of solutions of (3.1), with also e(uε ) and σ ε = Aε e(uε ) converging to e(u) and σ = Ae(u), respectively. From the coercivity and symmetry properties of A follows on the other hand uniqueness of solutions of the limit problem. Thus u above is this unique solution of (3.29), and standard arguments then show that we actually have the above convergences for the entire sequence ε. • Step 6: Convergence of energies. Finally we prove convergence of energies. We have
ε ε ε A e u , e u dx = Ω
Ω
f ε · uε dx +
ε ε g , e u dx + Ω
∂Ω\Γ0
hε · uε dγ.
It follows from (3.4), (3.6), (3.8), (3.9), (3.10) and (3.25), that Ω
ε ε ε A e u , e u dx →
= Ω
Ae(u), e(u) dx.
Ω
f · u dx +
Ω
g, e(u) dx + ∂Ω\Γ0
h · u dγ
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4. Homogenization and reduction of dimension for laminated plates Let ω be a fixed domain in R2 and let ε > 0 be a small parameter. The domain Ω ε = ω × (− ε2 , ε2 ) represents a thin horizontal plate. We are given F ε ∈ (L2 (Ω ε ))3 , Gε ∈ (L2 (Ω ε ))3s×3 , the set of symmetric ε , H ε ∈ (L2 (ω))3 and a 3 × 3 × 3 × 3 tensor Aε = Aε (x ) 3 × 3 matrices with L2 -coefficients, H+ 3 − ε ε defined in (− 2 , 2 ), satisfying the same symmetry, boundedness and coercivity conditions (with α and β independent of ε) as in Section 3. Here again we allow that Aε is a “full tensor”, which means that the material may be anisotropic (its behavior is not governed only by the two Lamé coefficients). The fact that Aε = Aε (x3 ) means that the plate is laminated, along horizontal planes. We consider the three-dimensional linear elasticity system in Ω ε : 1 ε 3 ε 3 ε ε ε ε U ∈ H (Ω ) , U = 0 on Σ = ∂ω × − , , and ∀U ∈ H 1 Ω ε , U = 0 on Σ ε , 2 2 ε ε ε ε (4.1) A (x )e U , e(U ) dx = F · U dx + G , e(U ) dx 3 Ωε Ωε Ωε ε ε + H ε · U x , + H ε · U x , − dx . ∂ω
+
2
−
2
The physical interpretation of (4.1) and the quantities involved is the same as for (3.1). We are interested in the limit behavior of U ε , when ε runs through a sequence of values converging to zero. This limit behavior is established in the sequel, by using again compensated compactness arguments. The scalar case for a linear equation was solved in such a way in [9,10]; for a more general nonlinear equation of the form −div((M ε (x))−1 ϕ(P ε (x)∇uε )) = f ε , it was solved in [6], under the assumption (which is satisfied for lamination) that curl M ε and div P ε are relatively compact in a convenient meaning and provided that M ε and P ε have special forms, in the sense that some of their components do not depend on ε. For the asymptotic behavior of plates made of a homogeneous isotropic material, the reader is referred to [2] for linear elasticity, and to [8,11] (or [12] for the shell model) for nonlinear elasticity. 4.1. Rescaling problem (4.1) We adopt the standard technique (see, e.g., [2]) of rescaling the problem so that everything takes place in a fixed domain. This we take to be Ω = ω × (− 12 , 12 ) and, from U : Ω ε → R3 we get u : Ω → R3 defined by: uα (x) = uα (x , x3 ) = Uα (x , εx3 ),
u3 (x) = u3 (x , x3 ) = εU3 (x , εx3 ).
Then ∂Uα ∂uα (x) = (x , εx3 ), ∂xβ ∂xβ ∂u3 ∂U3 (x) = ε (x , εx3 ), ∂xα ∂xα so that
ε
e(U ) = e (u) :=
eαβ (u) 1 ε e3β (u)
∂uα ∂Uα (x) = ε (x , εx3 ), ∂x3 ∂x3 ∂U3 ∂u3 (x) = ε2 (x , εx3 ), ∂x3 ∂x3
1 ε eα3 (u) 1 e (u) ε2 33
.
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
157
We also set Aε (x3 ) = Aε (εx3 ), which defines a 3 × 3 × 3 × 3 tensor in Ω having the same properties as in (3.2), (3.3), namely aεijkl = aεjikl = aεijlk ,
(4.2)
3×3 ∃α, x ∈ Ω, ε β > 0, ∀ξ ∈ Rs 2 , for a.e. A (x3 )ξ, ξ α|ξ| and Aε (x3 )ξ β|ξ|.
(4.3)
Then the variational equation in (4.1) is transformed into Ω
ε A (x3 )eε uε , eε (u) dx =
Ω
Fαε (x , εx3 )uα (x) dx +
+ Ω
Ω
1 ε F (x , εx3 )u3 (x) dx ε 3
ε G (x , εx3 ), eε (u) dx
+ ∂ω
1 ε 1 1 ε 1 H uα x , + 2 H+ 3 u3 x , ε +α 2 ε 2
1 ε 1 + H−α uα x , − ε 2
1 ε 1 + 2 H− 3 u3 x , − ε 2
dx .
Hence the natural scaling in U induces corresponding scalings of the source terms: 1 f3ε (x) = F3ε (x , εx3 ), ε
fαε (x) = Fαε (x , εx3 ), g ε (x) = Gε (x , εx3 ), 1 ε hε±α (x ) = H±α (x ), ε
hε±3 (x ) =
1 ε H (x ), ε2 ±3
and problem (4.1) becomes, in terms of the new variables: 1 3 ε 3 1 1 ε u ∈ H (Ω) , u = 0 on Σ = ∂ω × − , and ∀w ∈ H 1 (Ω) , w = 0 on Σ, , 2 2 ε ε ε ε ε ε ε A (x3 )e u , e (w) dx = f · w dx + g , e (w) dx Ω Ω Ω 1 ε 1 ε + h (x ) · w x , + h (x ) · w x , − dx . ω
+
−
2
(4.4)
2
We are going to pass to the limit in problem (4.4), when ε tends to zero. 4.2. A priori estimates Lemma 4. Assume (4.3) and assume that {f ε }ε is bounded in (L2 (Ω))3 , {g ε }ε is bounded in (L2 (Ω))3×3 and {hε± }ε is bounded in (L2 (ω))3 . Then {eε (uε )}ε is bounded in (L2 (Ω))3×3 , which implies that {e(uε )}ε is bounded in (L2 (Ω))3×3 , {uε }ε is bounded in (H 1 (Ω))3 and {σ ε = Aε eε (uε )}ε is bounded in (L2 (Ω))3×3 . Moreover, defining y ε by: yαε =
uεα −u ˜εα − ε
1 2
− 12
ε u
α
ε
−u ˜εα dx3 ,
with u ˜εα (x) = −
x3 1 ∂uε3 − 12
ε ∂xα
(x , t) dt,
(4.5)
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B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
1 = 2 uε3 − ε
y3ε
1 2
− 12
1 ε u dx3 , ε2 3
(4.6)
1 1 then {y ε }ε is bounded in (L2 (ω; Hm (− 12 , 12 )))3 , where Hm is the subset of H 1 of functions having mean value zero.
Proof. Taking uε as test function in (4.4), we get Ω
ε A (x3 )eε uε , eε uε dx
f · u dx + ε
= Ω
+ ω
ε ε ε g , e u dx
ε
Ω
hε+ (x ) · uε x ,
1 2
+ hε− (x ) · uε x , −
1 2
dx
f ε (L2 (Ω ))3 uε (L2 (Ω ))3 + g ε (L2 (Ω ))3×3 eε uε (L2 (Ω ))3×3 ε ε ε ε 1 1 + h+ (L2 (ω))3 u ·, + h− (L2 (ω))3 u ·, − 2 2 3 2
.
(L2 (ω ))3
(L (ω ))
By the Korn inequality, for ε < 1, ε u
(L2 (Ω ))3
uε (H 1 (Ω ))3 C e uε (L2 (Ω ))3×3 C eε uε (L2 (Ω ))3×3
and, by continuity of the trace mapping, ε u ·, ± 1 2
(L2 (ω ))3
C e uε (L2 (Ω ))3×3 C eε uε (L2 (Ω ))3×3 .
Thus, by using the coercivity of {Aε }ε (see (4.3)), we get
αeε uε (L2 (Ω ))3×3 C f ε (L2 (Ω ))3 + g ε (L2 (Ω ))3×3 + hε+ (L2 (ω))3 + hε− (L2 (ω))3 . Hence {eε (uε )}ε is bounded in (L2 (Ω))3×3 and, by boundedness of {Aε }ε (see (4.3)), it follows that {σ ε }ε is bounded in (L2 (Ω))3×3 . Moreover, by definition of y ε (see (4.5), (4.6)), ∂yαε 1 ∂uεα 1 ∂uε3 2 = + = eα3 uε = 2eεα3 uε ∂x3 ε ∂x3 ε ∂xα ε
is bounded in L2 (Ω) and ∂y3ε 1 ∂uε 1 = 2 3 = 2 e33 uε = eε33 uε ∂x3 ε ∂x3 ε
is also bounded in L2 (Ω). As y ε has mean value zero with respect to x3 , it follows from the Poincaré– 1 Wirtinger inequality that {y ε }ε is bounded in (L2 (ω; Hm (− 12 , 12 )))3 .
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
159
4.3. Compactness Lemma 5. Assume (4.3) and assume that, for some f in (L2 (Ω))3 , g in (L2 (Ω))3s×3 and h± in (L2 (ω))3 , fε f
3
3×3
weakly in L2 (Ω) ,
gε g
weakly in L2 (Ω)
,
(4.8)
weakly in L2 (ω) .
(4.9)
hε± h±
(4.7)
3
Then, for some subsequence of ε, still denoted by ε, for some u in the Kirchhoff–Love space
U = v ∈ H (Ω) 1
2
H02 (ω), ∃ζ
×
∈
2 H01 (ω) , vα
∂v3 = ζα − x3 , α = 1, 2 , ∂xα
1 (− 12 , 12 )))3 and for some σ in (L2 (Ω))3s×3 , for some y in (L2 (ω; Hm
3
uε ·, ±
1 2
→ u ·, ±
1 2
3
y (defined in Lemma 4) y
ε ε eα3 (u )
1
1
ε e3β
ε2
e33
ω; Hm
= e(uε , y ε ) :=
(uε )
1 ∂y α 2 ∂x 3
1 ∂y β 2 ∂x 3
∂y 3 ∂x 3
σ ε = Aε eε uε σ
1
eαβ (u)
e(u, y) :=
1
(4.11)
2
weakly in L
eαβ (uε ) (uε )
(4.10)
strongly in L2 (ω) ,
ε
eε uε =
3
uε → u strongly in L2 (Ω) ,
uε u weakly in H 1 (Ω) ,
3×3
3
,
eαβ (uε )
ε 1 ∂y α 2 ∂x 3
ε 1 ∂y β 2 ∂x 3
∂y3ε ∂x 3
weakly in L2 (Ω)
weakly in L2 (Ω)
1 1 − , 2 2
3×3
(4.12)
,
(4.13)
.
(4.14)
Moreover, σ satisfies 3 1 1 1 ∀(v, z) ∈ U × L2 ω; Hm − , , 2 2 1 1 σ − g, e(v, z) dx = f · v dx + h+ · v x , + h− · v x , − dx . Ω
Ω
ω
2
(4.15)
2
Proof. The convergences of uε , y ε and σ ε follow from Lemma 4 and from classical compactness arguments. But the weak convergence of uε in (H 1 (Ω))3 implies that, for every α, β ∈ {1, 2},
eαβ uε eαβ (u) weakly in L2 (Ω),
160
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
1 while the weak convergence of y ε in (L2 (ω; Hm (− 12 , 12 )))3 implies that
2 ε ∂y ε ∂yα eα3 u = 2eεα3 uε = α weakly in L2 (Ω), ε ∂x3 ∂x3 ∂y3ε ∂y3 1 ε ε ε e u = e u = weakly in L2 (Ω), 33 33 2 ε ∂x3 ∂x3
so that (4.13) holds true. Finally, for (v, z) ∈ U × (C 1 (Ω))3 , z = 0 on Σ, we define v ε = (vαε , v3ε ) by vαε = vα + εzα , ε v3 = v3 + ε2 z3 . It is clear that v ε ∈ (H 1 (Ω))3 , v ε = 0 on Σ, and, since eα3 (v) = 0 and e33 (v) = 0, ε ε
e v
=
1 ∂z α 2 ∂x 3
eαβ (v + εz) ∂z 3 1 ∂z β 2 ∂x 3 + ε ∂x β
→ e(v, z)
strongly in L2 (Ω)
+ ε ∂∂xzα3
= e(v, z) + ε
∂z 3 ∂x 3
3×3
eαβ (z) 1 ∂z 3 2 ∂x β
1 ∂z 3 2 ∂x α
0
.
Moreover, vε → v
3
strongly in H 1 (Ω) .
Taking v ε as test function in (4.4), we have that Ω
ε σ − g ε , eε v ε dx =
f · v dx + ε
Ω
ε
ω
hε+ (x )
·v
ε
1 1 x, + hε− (x ) · v ε x , − 2 2
dx
and, passing to the limit, we get that
Ω
σ − g, e(v, z) dx =
Ω
f · v dx +
ω
1 1 h+ (x ) · v x , + h− (x ) · v x , − 2 2
dx ,
for any v ∈ U and z ∈ (C 1 (Ω))3 , z = 0 on Σ. By density of {z ∈ C 1 (Ω), z = 0 on Σ} in L2 (ω; H 1 (− 12 , 12 )) and by continuity, this is true also for any v ∈ U and z ∈ (L2 (ω; H 1 (− 12 , 12 )))3 (a fortiori for z having mean value zero with respect to x3 ). 4.4. Second-order limit problem whose solution is a pair The following theorem is the most central result of the paper. It gives the convergence properties under simultaneous homogenization and reduction of dimension to a limit problem stated for a pair of functions, u and y, which together represent the displacement field in the limit. The function u belongs to the Kirchhoff–Love space U, which means that it has quite simple behavior in the scaled direction x3 (the first two components are linear in x3 and the third component does not depend on x3 at all), while y can be thought of as representing the fine behavior in that direction.
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
161
Theorem 2. Assume symmetry, uniform boundedness and coercivity of Aε as in (4.2) and (4.3), assume (4.7), (4.9) of Lemma 5 and, moreover,
gε → g
strongly in L2 (Ω)
3×3
,
(4.16)
and assume finally that the Aε H-converge to a matrix A in the sense that (3.23) holds. Then, for the whole sequence ε, the convergences (4.10) to (4.13), as well as
Aε eε uε Ae(u, y)
weakly in L2 (Ω)
3×3
,
(4.17)
hold true, where (u, y) is the unique solution of 3 3 1 1 1 1 2 1 2 1 , , u ∈ U, y ∈ L ω; H − , and ∀v ∈ U, z ∈ L ω; H − , m m 2 2 2 2 A(x )e(u, y), e(v, z) dx = f · v dx + g, e(v, z) dx 3 Ω Ω Ω 1 1 + h+ (x ) · v x , + h− (x ) · v x , − dx .
2
ω
(4.18)
2
Moreover the energies converge in the sense that Ω
ε A (x3 )eε uε , eε uε dx →
Ω
A(x3 )e(u, y), e(u, y) dx.
Proof. In view of Lemma 5 and the uniqueness of the solution of (4.18), the proof of Theorem 2 will be complete (apart from convergence of energies) whenever we have proved that σ = Ae(u, y) for the limit functions u, y, σ obtained in Lemma 5. By symmetry of A, this is equivalent to proving that σ = A∇(u, y). We start from eαβ (uε ) ε ε ε ε ε σ =A e u =A ∂y ε
ε 1 ∂y α 2 ∂x 3
∂y3ε ∂x 3
β 1 2 ∂x 3
= Aε e uε , y ε = Aε ∇(uε , y ε ).
We are going to pass to the limit in this equation, written as in Section 2.4
M ε σ ε = P ε ∇ uε , y ε , with ε
M =
I
0
(Aε ) − C ε Rε Lε P = Rε Lε ε
δik δβδ −C ε Rε = ε R 0 0 I
−aεiβs3 rsε3k3 , riε3k3
aεiβkδ − aεiβs3 rsε3t3 aεt3kδ = riε3s3 aεs3kδ
0 . δik
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B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
• Passing to the limit in P ε ∇(uε , y ε ). We have P ε ∇ uε , y ε =
pεiβkδ pεi3kδ
∂uεα
0 ∂x β δik 0
ε ∂y α ∂x 3 , ∂y3ε ∂x 3
ε ε ε ∂uε ∂y ε P ∇ u , y ij = pεijkl ∇kl uε , y ε = pεijαβ α + pεijk3 k . ∂xβ ∂x3
We are going to pass to the limit successively in each term of the right member above. ◦ For the first term, pεijαβ
∂uεα ∂uα ∂ ε ∂ = pijαβ uεα (pijαβ uα ) = pijαβ , ∂xβ ∂xβ ∂xβ ∂xβ
in the sense of distributions, since the convergences uεα → uα
strongly in L2 (Ω) and
pεijαβ → pijαβ
weakly in L∞ (Ω)
imply that pεijαβ uεα pijαβ uα , in the sense of distributions. Then, by boundedness, pεijαβ
∂uεα ∂uα pijαβ ∂xβ ∂xβ
weakly in L2 (Ω).
◦ For the second term, pεijk3
∂ykε ∂y ε = pijk3 k , ∂x3 ∂x3
with pijk3 = 0 or δik , independent of ε. As pεijk3
∂ykε ∂yk pijk3 ∂x3 ∂x3
∂ykε ∂x 3
∂y k ∂x 3
weakly in L2 (Ω), we have
weakly in L2 (Ω).
◦ Hence
P ε ∇ uε , y ε P ∇(u, y),
weakly in L2 (Ω)
3×3
.
• Passing to the limit in M ε σ ε . Here we have
M εσε
ij
ε ε ε = mεijkl σkl = mεijkα σkα + mεijk3 σkε3 = mijkα σkα + mεijk3 σkε3 ,
ε mijkα σkα mijkα σkα
weakly in L2 (Ω).
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
163
Moreover, ε mεijk3 σkε3 = Mijk ·σ ˜kε ,
ε with Mijk = mεijkl
l=1,2,3
and with σ ˜kε = 0, 0, σkε3 .
ε = 0 (see Section 3.4), so we can apply compensated compactWe have already checked that curl Mijk ness, and get that
M εσε M σ
weakly in L2 (Ω)
3×3
,
as soon as we have checked that div σ ˜kε is relatively compact in H −1 (Ω). • Now we prove that div σ ˜kε is relatively compact in H −1 (Ω). Clearly, from (4.4), Ω
ε σ − g ε , eε (w) dx =
Ω
f ε · w dx,
3
∀w ∈ D(Ω) ,
that is, as σ ε − g ε ∈ (L2 (Ω))3s×3 , ε ∂wα ∂wα ∂w3 ∂w3 1 1 1 σ − g ε αβ + σ ε − g ε α3 + σ ε − g ε 3β + 2 σ ε − g ε 33 dx Ω
= Ω
ε
∂xβ
∂x3
ε
∂xβ
ε
ε fα wα + f3ε w3 dx,
i.e., in the sense of distributions, ∂ ε 1 ∂ ε σ − g ε αβ − σ − g ε α3 = fαε , − ∂xβ ε ∂x3 ε 1 1 ∂ ε ∂ ε − σ − g 3β − 2 σ − g ε 33 = f3ε .
ε ∂xβ
We have to prove that
ε ∂x3
ε ∂σk3 ∂x 3
is relatively compact in H −1 (Ω). But
∂σαε 3 ∂g ε ∂ ε ∂gα3 ε = α3 − ε fαε + σαβ − gαβ → ∂x3 ∂x3 ∂xβ ∂x3 since, by virtue of (4.16), ∂gαε 3 ∂gα3 → ∂x3 ∂x3
strongly in H −1 (Ω),
and since fαε +
∂ ε ε σαβ − gαβ ∂xβ
is bounded in H −1 (Ω).
strongly in H −1 (Ω),
∂x3
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In the same way,
ε ∂σ33 ∂g ε 1 ∂ ε ∂g33 = 33 − ε2 f3ε + σ3β − g3εβ → ∂x3 ∂x3 ε ∂xβ ∂x3
strongly in H −1 (Ω).
• Finally, let us prove convergence of energies. We have, due to (4.7), (4.9), (4.10), (4.11), (4.13) and (4.16),
ε ε ε ε ε A e u , e u dx Ω
= →
Ω
Ω
=
ε ε ε g , e u dx +
ε
f · u dx +
f · u dx + ε
Ω
ω
ω
hε+
·u
ε
1 x, 2
+
hε−
·u
ε
1 x ,− 2
1 1 h+ · u x , + h− · uε x , − 2 2
g, e(u, y) dx +
Ω
dx
dx
Ae(u, y), e(u, y) dx.
Ω
4.5. Reduced second-order limit problem Here we decouple the limit problem of Theorem 2, so that we get one problem for u alone (the reduced second-order problem), plus one equation which recovers y from u. Theorem 3. Let (u, y) be the pair defined in Theorem 2. Setting g = (g , g3 ) with g = (gkδ ) and g3 = (gk3 ), B = A − CRL = (biαkβ ) ) l = g − CRg3 = (lkδ
e (v) = ekδ (v) ,
and
and
biαkβ = aiαkβ − aiαs3 rs3t3 at3kβ ,
lkδ = gkδ − akδs3 rs3t3 gt3 ,
ekδ (v) = 1 − δk3
∂vk
∂xδ
(4.19) (4.20)
,
the function u is the unique solution of the following reduced second-order limit problem u ∈ U, and ∀v ∈ U, B e (u), e (v) dx = f · v dx + l , e (v) dx Ω Ω Ω 1 1 + h+ (x ) · v x , + h− (x ) · v x , − dx ω
2
(4.21)
2
1 (− 12 , 12 )))3 of and the function y is the unique solution in (L2 (ω; Hm
∂y = (A33 )−1 g3 − Le (u) = R g3 − Le (u) . ∂x3
(4.22)
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
165
Setting
= (b B αβγδ ),
˜l = (l ), αβ
∇ u =
∂uα , ∂xβ
(4.23)
another formulation of (4.21) is u ∈ U, and ∀v ∈ U, B∇ 1
u, ∇ v dx = ˜ l, ∇ v dx + f · v dx + h+ (x ) · v x , 2 Ω Ω Ω ω 1 dx . + h− (x ) · v x , −
(4.24)
2
Proof. By virtue of the symmetry property (3.26) of A,
Ω
A(x3 )e(u, y), e(v, z) dx =
Ω
A(x3 )∇(u, y), ∇(v, z) dx,
with
∇(v, z) =
∂vα ∂x β
0
∂z α ∂x 3 . ∂z 3 ∂x 3
Hence, as also g is symmetric, the limit variational equation can be written
Ω
A(x3 )∇(u, y), ∇(v, z) dx =
Ω
f · v dx +
+ ω
Ω
g, ∇(v, z) dx
h+ (x ) · v x ,
1 2
+ h− (x ) · v x , −
1 2
dx .
By using the symmetry properties of A and g, the limit variational equation obtained in Theorem 2 can be written as
Ω
A∇(u, y), ∇(v, z) dx =
Ω
f · v dx +
+ ω
∇(v, z) =
α
∂x β
0
∂z α ∂x 3 ∂z 3 ∂x 3
,
or, in vectorial notation (see Section 2.3), ∇(v, z) =
e (v)
e3 (z)
,
Ω
g, ∇(v, z) dx
h+ (x ) · v x ,
with ∂v
1 1 + h− (x ) · v x , − 2 2
dx ,
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B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
with
e (v) = ekδ (v) ,
e3 (z) = ek3 (z) ,
ekδ (v) = (1 − δk3 )
∂vk , ∂xδ
ek3 (z) =
∂zk . ∂x3
Then, writing also g as a vector g = (g , g3 ), we get (see Section 2.3)
g, ∇(v, z) = g , e (v) + g3 · e3 (z)
and A∇(u, y) =
A
C A33
L
e (u)
e3 (y)
A e (u) + Ce3 (y) , Le (u) + A33 e3 (y)
=
A∇(u, y), ∇(v, z) = A e (u) + Ce3 (y), e (v) + Le (u) + A33 e3 (y) e3 (z).
It follows that the limit variational equation obtained in Theorem 2 has the formulation: Ω
A e (u) + Ce3 (y) − g , e (v) dx +
= Ω
f · v dx +
ω
h+ (x ) · v x ,
Ω
Le (u) + A33 e3 (y) − g3 · e3 (z) dx
1 1 + h− (x ) · v x , − 2 2
dx ,
which decouples into Ω
A e (u) + Ce3 (y) − g , e (v) dx
= Ω
f · v dx +
ω
h+ (x ) · v x ,
1 1 + h− (x ) · v x , − 2 2
dx
(4.25)
and
Ω
Le (u) + A33 e3 (y) − g3 · e3 (z) dx = 0,
with
e3 (z) =
∂z1 ∂z2 ∂z3 , , . ∂x3 ∂x3 ∂x3
As the last integral equation holds true for any z in (L2 (ω; H 1 (− 12 , 12 )))3 , it is equivalent to
Ω
Le (u) + A33 e3 (y) − g3 · ψ dx = 0,
for any ψ in L2 (Ω)3 , which allows to eliminate y in favor of u, resulting in (4.22). Then an easy computation shows that (4.25) reduces to (4.21) or to (4.24).
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
167
4.6. Reduced fourth-order limit problem Recalling the definition of the Kirchhoff–Love space,
2
3
U = v ∈ H 1 (Ω)
2
× H02 (ω), ∃η ∈ H01 (ω) , vα = ηα − x3
= v ∈ H 1 (Ω) , ∃η ∈ H01 (ω)
2
× H02 (ω), vα = ηα − x3
∂v3 , α = 1, 2 , ∂xα
∂η3 , α = 1, 2, v3 = η3 , ∂xα
we see that each function v in U is parametrized by a function η defined on ω, called its Kirchhoff–Love displacement field. In this subsection we translate problem (4.24) for u ∈ U into a fourth-order problem for its Kirchoff–Love displacement field ζ = ζ(x ).
= B(x
3 ) be defined in (4.19), (4.23), that is, Theorem 4. Let B
= (b B αβγδ )
with bαβγδ = aαβγδ − aαβs3 rs3t3 at3γδ
and let K1 , K2 , K3 and K be defined by
K1 =
1 2
− 12
2 dx3 , Bx 3
K1 K= −K2
K2 =
1 2
− 12
3 dx3 , Bx
K3 =
1 2
− 12
dx3 , B
−K2 . K3
Let ˜l be defined in (4.20), (4.23), that is, ˜l = (l ) with l = gαβ − aαβs3 rs3t3 gt3 αβ αβ and let L1 = L1 (x ), L2 = L2 (x ), L = L(x ), f 0 = f 0 (x ) and l0 = l0 (x ) be defined by L1 = −
f0 =
1 2
− 12 1 2
− 12
l0 =
1 2
− 12
˜lx3 dx3 ,
L2 =
1 2
− 12
˜l dx3 ,
L = (L1 , L2 ),
f dx3 + h+ + h− , 1 f x3 dx3 + (h+ − h− ), 2
with f = (fα ), h± = (h±α ).
Let u be the limit function given in Theorem 2 and Theorem 3. Then u = (ζα − x3 ∂∂xζα3 , ζ3 ) where ζ = ζ(x ) is the unique solution of the fourth-order variational problem: 2 2 ζ ∈ H01 (ω) × H02 (ω), and ∀η ∈ H01 (ω) × H02 (ω), f 0 · η dx + [L, η] dx − l0 · ∇η3 dx , [Kζ, η] dx = ω
ω
ω
ω
(4.26)
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B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
with the second-order linear operator defined by
η = D2 η3 , ∇η ,
D2 η3
αβ
=
∂2 η3 ∂ηα , (∇η )αβ = . ∂xα ∂xβ ∂ηβ
Remark 1. Problem (4.26) reads: 1 2 1 2 2 ζ ∈ H (ω) × H (ω), and ∀η ∈ H (ω) × H02 (ω), 0 0 0 0 2 2 2
f3 η3 + L1 , D η3 − l0 · ∇η3 dx ,
K1 D ζ3 − K2 ∇ζ , D η3 dx =
ω ω 0 2 −K2 D ζ3 + K3 ∇ζ , ∇η dx = f · η + [L2 , ∇η ] dx , ω
(4.27)
ω
that is, ζ ∈ (H01 (ω))2 × H02 (ω) is the unique solution, in the sense of distributions, of the following equations, in the two-dimensional plate ω: (K1 )αβγδ
∂4 ζ3 ∂3 ζγ − (K2 )αβγδ = ϕ3 , ∂xα ∂xβ ∂xγ ∂xδ ∂xα ∂xβ ∂xδ ∂3 ζ3 ∂2 ζγ (K2 )αβγδ − (K3 )αβγδ =ϕα , ∂xβ ∂xγ ∂xδ ∂xβ ∂xδ
(4.28)
with 1 ∂ (h+α − h−α ) + h+3 + h−3 + ϕ3 = 2 ∂xα
ϕα = h+α + h−α +
1 2
− 12
1 2
− 12
∂2 ˜lαβ ∂fα − x3 + f3 dx3 , ∂xα ∂xα ∂xβ
∂˜lαβ fα − dx3 . ∂xβ
Proof of Theorem 4. For any v in U, we define η = η(v) by ηα = vα + x3
∂v3 , ∂xα
η3 = v3 ,
so that (∇ v)αβ =
∂2 η3 ∂vα ∂ηα = − x3 = (∇η )αβ − x3 D2 η3 αβ ∂xβ ∂xβ ∂xα ∂xβ
and, writing similarly ζ = ζ(u), problem (4.24) reads as the following fourth-order variational problem: 2 2 ζ = ζ(u) ∈ H01 (ω) × H02 (ω), and ∀η ∈ H01 (ω) × H02 (ω), ∂η3 2 2 B ∇ζ − x3 D ζ3 , ∇η − x3 D η3 dx = fα ηα − x3 dx f3 η3 dx ∂xα Ω Ω Ω 1 ∂η3 2 ˜ l, ∇η − x3 D η3 dx + (h+ h− ) · η + (h−α − h+α ) dx , + Ω
ω
2
∂xα
B. Gustafsson and J. Mossino / Compensated compactness for homogenization and reduction of dimension
169
which, as an easy computation shows, is equivalent to (4.26), that is (4.27), or (4.28), in the sense of distributions. Acknowledgements The first author is grateful to the Swedish Research Council for basic support and to Ecole Normale Supérieure de Cachan for visiting support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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