Homogenization of the elastic plate equation

We are interested in general homogenization theory for fourth-order elliptic equation describing the Kirchhoﬀ model for pure bending of a thin solid symmetric plate under a transverse load. Such theory is well-developed for second-order elliptic problems, while some results for general elliptic equations were established by Zhikov, Kozlov, Oleinik and Ngoan (1979). We push forward an approach of Antoni´c and Balenovi´c (1999, 2000) by proving a number of properties of H-convergence for stationary plate equation.


Summary
The main goal of this thesis is to study homogenization of the Kirchhoff-Love model for pure bending of a thin symmetric elastic plate, which is described by the fourth order elliptic equation. Homogenization theory is one of the most successful approaches for dealing with optimal design problems (in conductivity or linearized elasticity), which consists of arranging given materials such that obtained body satisfies some optimality criteria, typically expressed mathematically as the minimization of some (integral) functional under some (PDE) constraints. The key role in homogenization theory has H-convergence.
After a brief introduction, in Chapter 1 we prove a number of properties of H-convergence, such as locality, independence of boundary conditions, metrizability of H-topology, convergence of energies and a corrector result. We also discuss smooth dependence of H-limit on a parameter and calculate the H-limit of a periodic sequence of tensors. Moreover, we give special emphasis to calculating the first correction in the small-amplitude homogenization limit of a sequence of periodic tensors.
Using this newly developed theory, in Chapter 2 we put our focus on the composite elastic plate. We show the local character of the set of all possible composites, also called the G-closure, and prove that the set of composites obtained by periodic homogenization is dense in that set. Additionally, we derive explicit expressions for elastic coefficients of composite plate obtained by mixing two materials in thin layers (known as laminated material), and for mixing two materials in the low-contrast regime. Moreover, we derive optimal bounds on the effective energy of a composite material, known as Hashin-Shtrikman bounds. In the case of two-phase isotropic materials, explicit optimal Hashin-Shtrikman bounds are calculated. We show that an analogous results can be derived for the complementary energy of a composite material.

Historical roots and motivation
The theory of homogenization is interesting from both the theoretical and the practical perspective. In order to use its full potential, one first has to develop theoretical results, which might have important applications, for example in optimal design problems. Commonly, optimal design problems do not have solutions (if they exist, such solutions are usually called classical). Therefore, one needs to consider a proper relaxation of the original problem. A relaxation by the homogenization method was introduced in [58], and it consists in introducing generalized composite materials, which are mixtures of original phases on a microscopic scale. Such relaxed problems have solutions, and we call them relaxed or generalized solutions.
The physical idea of homogenization is to average heterogeneous media in order to derive effective properties: we have a fine mixture of some materials and we want to approximate it by a new homogeneous one. Justification for this procedure is that we are not interested in what is happening at every point of the problem domain but rather what is happening on a macroscopic scale. For example, in the model problem of conductivity we are not interested in the pointwise temperature, but in average temperature in some (small) region. The outcomes of this approach are very important, since from a numerical point of view, solving equations will require too much effort if the length scale of heterogeneity is very small. Therefore, rather than considering a simple heterogeneous media with a fixed length scale ε(n), such that ε(n) → 0 as n → +∞, and studying a single problem, we observe a sequence of similar problems: Introduction of composite materials, i.e. mixtures of two or more materials on a microscopic scale. It shows that such mixtures (e.g. steel, carbon fibers) can have much better properties than the components it is made of, so these materials are intensively studied by physicists, engineers and mathematicians [2,22,35,51,54,72,74]. The natural problem is to describe the composite material obtained by the homogenization process. Describing the set of all composite materials obtained by the homogenization process is known as the G-closure problem. Characterization of the G-closure is known for the mixture of two isotropic conductors [48,49], but it is unknown for linearized elasticity system, even for the mixture of two isotropic materials. In the case of an elastic plate, G-closure is known only in some special regimes [50]. It is possible to obtain approximations of the G-closure in the small-amplitude or low contrast regime in the setting of stationary diffusion equation [72], when we mix two materials with similar properties.
By using H-convergence and H-measures as a tool, the small-amplitude homogenization for stationary diffusion equation is developed [68], i.e. the explicit formula for coefficients up to the second order term is derived. In this way, a small-amplitude homogenization result for the periodic case [16] is extended. Using similar techniques, Antonić and Vrdoljak developed the small-amplitude homogenization result for the parabolic equation [12,13]. For the elastic plate equation, the low contrast regime was not studied up to date, and that is one of the goals of this thesis.
In order to derive some effective properties of composite materials, Hashin-Shtrikman bounds are calculated, i.e. bounds on the effective energy of a composite material, which are well known for stationary diffusion equation and elasticity [2]. However, to obtain effective properties and for application in optimal design, it is necessary to calculate them explicitly, as well as the corresponding (sequential) laminates that saturate them [2]. In the case of two-dimensional linearized elasticity this is done in [5], but for the plate equation that is an open problem, which is one of the topics of this thesis.

Introduction
Hashin-Shtrikman bounds on the primal and complementary energy are calculated.
Before reading this thesis, the reader may wish to view the Appendix, since it contains some basic notation and elementary results.

Introduction
We consider a homogeneous Dirichlet boundary value problem for a general fourth-order partial differential equation where Ω ⊆ R d is an open and bounded set, and M is a tensor valued function, which can be understood as a linear operator on the space of all symmetric d × d real matrices, denoted by Sym.
The weak solution u of (1.1) is defined as a function u ∈ H 2 0 (Ω) satisfying The problem is elliptic, if we assume that M is bounded (almost everywhere) and coercive. where β > α > 0 are given, and : stands for the scalar product on the space Sym. The bounds are chosen in this form to ensure their preservation during the homogenization process, as it was shown in the case of stationary diffusion equation [58].
The well-posedness follows by a standard application of the Lax-Milgram lemma. To be precise, the differential operator div div (M∇∇·) : H 2 0 (Ω) −→ H −2 (Ω) is an isomorphism, i.e. a linear and continuous operator with bounded inverse (the bound depending only on Ω and α).
In the two-dimensional case, boundary value problem (1.1) describes the Kirchhoff (also known as Kirchhoff-Love) model for pure bending of a thin, solid symmetric plate clamped at the boundary, under a transverse load f . This model can be derived by taking a limit in 3d elasticity equations with a technique similar to H-convergence [28], or by means of Gamma-convergence [18] (for classical reference see [23]). The plate is assumed to be symmetric with respect to its midplane Ω and a tensor valued function M describes its elastic properties (depending on the material properties and the thickness of the plate). In this model, additional symmetry is present, making the tensor valued function M self-adjoint. This assumption simplifies the theory, since it is equivalent to consider G-convergence [35,51] instead of H-convergence. However, in this chapter we shall present the general theory (in arbitrary space dimension), ignoring this symmetry assumption.
We are interested in the general (non-periodic) homogenization theory for this equation. This theory is well developed for second-order elliptic problems, such as the stationary diffusion equation or the system of linearized elasticity, for which the notion of H-(or G-) convergence has been studied and properties, such as compactness, locality, independence of boundary conditions and convergence of energies, have been established (see [2,72] and references therein). In [77], a homogenization of a general elliptic system of partial differential equations has been considered, and some of the above mentioned properties have been shown in such full generality. However, due to this generality, some of the important properties are missing, while proofs end up being rather complicated.
The results concerning homogenization of the elastic plate equation have already been initiated by Antonić and Balenović [9,10], where, prompted by possible applications in optimal design problems, a more direct approach to the homogenization of the stationary plate equation was considered, and an appropriate variant of H-convergence was defined. Additionally, compactness of H-convergence was established.

Introduction
Definition 1 A sequence of tensor functions (M n ) in M 2 (α, β; Ω) is said to H-converge to M ∈ M 2 (α , β ; Ω) if for any f ∈ H −2 (Ω) the sequence of solutions (u n ) of problems    div div (M n ∇∇u n ) = f u n ∈ H 2 0 (Ω) converges weakly to a limit u in H 2 0 (Ω), while the sequence (M n ∇∇u n ) converges to M∇∇u weakly in the space L 2 (Ω; Sym). If this is the case, then M is called H-limit of the sequence (M n ); note that u solves the boundary value problem .
The sequences (u n ) and (M n ∇∇u n ) in the above definition are bounded in H 2 0 (Ω) and L 2 (Ω; Sym), respectively, and thus converge (on a subsequence). Therefore, H-convergence just makes a connection between their limits. Since the existence of the H-limit M is doubtful, the following compactness theorem justifies the previous definition. Moreover, it shows that the bounds in definition of M 2 (α, β; Ω), which could also be written in many equivalent ways, are chosen in such a way that in the previous definition one actually has α = α and β = β. Theorem 1 (Compactness theorem for H-convergence) Let (M n ) be a sequence in M 2 (α, β; Ω). Then there is a subsequence (M n k ) and a tensor function M ∈ M 2 (α, β; Ω) such that (M n k ) H-converges to M.
In order to proceed with the proof of Theorem 1, we need the following two lemmas [9,10]. The first of them presents the compactness by compensation result and has the key role in proving properties of H-convergence for elastic plate equation. This lemma plays the same role as the div-rot lemma in the theory of homogenization for second-order operators [72].

Lemma 1 (Compactness by compensation result) Let the following convergences be valid:
with an additional assumption that the sequence (div div D n ) is contained in a precompact (for the strong topology) set of the space H −2 loc (Ω). Then we have Proof. Since the sequence (div div D n ) is contained in a precompact (for the strong topology) set of the space H −2 loc (Ω), and div div D n div div D ∞ weakly in H −2 loc (Ω), there is a subsequence (div div D n k ) converging to div div D ∞ in H −2 loc (Ω) strongly. On the other hand, for ϕ ∈ C ∞ c (Ω), the sequence (ϕw n ) converges weakly to ϕw ∞ in H 2 c (Ω), therefore we have (1.2) Integration by parts of the term on the left-hand side of (1.2) yields By using the compactness argument for Sobolev imbeddings, we have ∇w n k −→ ∇w ∞ in L 2 loc (Ω; R d ) and w n k −→ w ∞ in L 2 loc (Ω). Therefore, we can pass to the limit in the first two terms of the above equality. On the other hand, a comparison argument shows that the term Ω D n k : ϕ∇∇w n k dx converges to the limit This gives the statement of the lemma for a subsequence. However, one can easily see that the same holds for any subsequence, with the same limit, and thus for the entire sequence itself.

(1.3)
Proof. Let G = {f 1 , f 2 , . . . } be a countable dense subset of H −2 (Ω). In the sequel, by 1.1. Introduction using a diagonal procedure, we shall construct operators B and R which are well defined on G, and then extend those operators by continuity to linear operators on H −2 (Ω).
More precisely, since A −1 (Ω), and has a weakly convergent subsequence which converges to Bf 1 . We repeat the same procedure with that subsequence for f 2 and denote the cluster point by Bf 2 ; analogously we do for f 3 , etc. Finally, we take a diagonal subsequence (A n k ) so that the following holds: Let us now extend the operator B : G → H 2 0 (Ω) to linear operator in H −2 (Ω). For arbitrary f ∈ H −2 (Ω), we take a sequence (f m ) in G such that f m → f , as m → ∞, in H −2 (Ω), and define Bf := lim m→∞ Bf m . From this construction one can easily conclude that which yields a well defined linear operator B : H −2 (Ω) → H 2 0 (Ω). This operator is bounded by and by taking the limit inferior in k we have: Moreover, it is easy to show that B is coercive with α β 2 : By Lax-Milgram lemma B is invertible, and after denoting A ∞ := B −1 it follows that The sequence (M n k ∇∇u n k ) is bounded in L 2 (Ω; Sym), therefore, by using a diagonal procedure once more, we can construct a subsequence (M n k ) (still denoted by n k ) such that for f ∈ G we have where u n k are solutions to (1.3) for that f . This defines an operator R : G → L 2 (Ω; Sym), which is clearly bounded: since M n k ∇∇u n k = M n k ∇∇(A −1 n k f ), we have Finally, after taking the limit inferior in k, we conclude that R ≤ β α . An analogous construction as in the first part of the proof yields a linear operator R : H −2 (Ω) → L 2 (Ω; Sym), which completes the proof.
Proof of Theorem 1. Let (A n ) and A ∞ as in Lemma 2. First we prove that the operator A ∞ is of the same form as operators A n , in the sense that there is a tensor M ∞ such that A ∞ u = div div (M ∞ ∇∇u). This can be shown by using the method of oscillating test functions [66]. This method consists of constructing a sequence of functions (v n ) in In order to construct the sequence of oscillating test functions, we choose an open set Ω which contains the closure of Ω. For x ∈ Ω \Ω we define the extension of tensor M n (x) := αI 4 , and for a given g ∈ H −2 (Ω ) define (v n ) to be the sequence of solutions to boundary value problems .

Introduction
It remains to show that M ∞ ∈ M 2 (α, β; Ω), i.e. we shall show the equivalent claim Applying the Lemma 1 to the left-hand side of (1.5), gives This implies the coercivity of (M ∞ ) T a. e. in Ω. Since (M n ) T belongs to M 2 (α, β; Ω), it also satisfies Analogously as when showing coercivity of (M n ) T , from (1.6) we obtain: This implies the boundedness of (M ∞ ) T a. e. in Ω, which completes the proof, i.e. (M ∞ ) T ∈ M 2 (α, β; Ω).
As it is already said, in this chapter we are also interested in the small-amplitude homogenization limit of a sequence of periodic tensors. The small-amplitude homogenization procedure of Tartar [72] consists in computing the first correction in the H-limit of a sequence of coefficients, whose difference is proportional to a small parameter. More precisely, after making an asymptotic expansion of the H-limit in terms of the smallamplitude parameter, one wishes to explicitly characterize its first non-vanishing (usually second-order) term. Its physical relevance is in deriving (approximate) effective properties of (conducting or elastic) material that is made by mixing two materials under the so called small-amplitude, small-contrast or small aspect ratio assumption, i.e. that original materials have close coefficients or material properties (for some applications see for example [3,4,39]).
The explicit formula for the correction in the case of second-order elliptic [68] (or parabolic [12]) equation can in general be obtained by using H-measures [69,70] (or their variants [11]). However, in the case of periodic coefficients, the same can be done by using Fourier expansions [13]. In this thesis we use the second approach and explicitly calculate the first correction in the small-amplitude homogenization process for the periodic sequence of tensors. Chapter 1. General homogenization theory for elastic plate equation We are interested in the following expansion of the H-limit: where p is some positive real number, and thus we shall use a variant of Taylor's theorem which is appropriate for Banach spaces. We first recall the notion of Frechet differentiable function [7], which is a natural extension of the usual definition the differential of a map in Euclidean spaces to Banach spaces.
Let X and Y be Banach spaces, U an open subset of X, and we denote Such an operator A is uniquely determined (if it exists) and will be called the (Frechet) differential of F at x 0 , with notation A = F (x 0 ). If F is differentiable at all x 0 ∈ U we say that F is differentiable in U .
To define the n-th differential (n ≥ 2) we can proceed by induction. The n-th differential at a point x 0 ∈ U will be identified with a continuous n-linear map from X × X × · · · × X (n times) to Y , and denoted by F (n) (x 0 ).
In order to state the small-amplitude homogenization results precisely, we need to show that the H-limit of a sequence depending smoothly on a parameter is also smooth. Since continuity is preserved by uniform convergence, we shall use the Arzelà-Ascoli theorem for the purpose of constructing a uniformly converging subsequence.
When dealing with periodic homogenization, we need the notion of a quotient space [44,53].
Let M be a subspace of a vector space X over a field K. We define an equivalence relation on X such that for x, y ∈ X, x ∼ y if and only if x − y ∈ M . For x ∈ X, an equivalence class is defined with The vector space X/M over a field K, with the vector space operations given above, is called the quotient space.
Theorem 5 [53, p. 51-53] Let M be a closed subspace of a normed space X. The quotient norm of X/M is given by the formula and it is a norm on X/M . Additionally, if X is a Banach space, then X/M is also a Banach space.
We are also interested in duals of quotient spaces.

Definition 4
Let X be a normed space and M a subspace of X. We define its annihilator by Obviously, M 0 is a subspace of X .
If p = 2, H k (Ω)/P k−1 is a Hilbert space with the scalar product

Properties of H-convergence
Using Tartar's method of oscillating test functions, we give proofs for the above mentioned properties of H-convergence for the stationary plate equation, and additionally prove a number of results, such as the metrizability and the corrector result. The relationship between H-convergence and some other types of convergence is studied in the following theorem. Theorem 8 Let (M n ) be a sequence of tensors in M 2 (α, β; Ω) that either converges strongly to a limit tensor M in L 1 (Ω; L(Sym, Sym)), or converges to M almost everywhere in Ω. Then, (M n ) also H-converges to M.
Proof. The sequence (M n ) belongs to M 2 (α, β; Ω) and therefore it is bounded in L ∞ (Ω; L(Sym, Sym)). By the Lebesgue dominated convergence theorem (M n ) converges strongly to M in L p (Ω; L(Sym, Sym)), for any 1 ≤ p < ∞. If u n is the solution of , then the sequence (u n ) is bounded in H 2 0 (Ω), and therefore (up to a subsequence) it converges weakly to u ∈ H 2 0 (Ω). Since (M n ) converges strongly to M in L 2 (Ω; L(Sym, Sym)) and (∇∇u n ) converges to ∇∇u weakly in L 2 (Ω; Sym), we conclude that σ n := M n ∇∇u n converges weakly to σ = M∇∇u in L 1 (Ω; Sym), and thus also in L 2 (Ω; Sym), as the sequence (σ n ) is bounded in this space.
The homogenized equation in Definition 1 has a unique solution u ∈ H 2 0 (Ω), so each subsequence of (u n ) converges to the same limit u and this implies that the entire sequence (u n ) converges to u. Since f ∈ H −2 (Ω) is arbitrary, it follows that (M n ) H-converges to

M.
H-convergence is related to the material properties of an elastic plate and it would be desirable that properties of a given material do not depend on boundary conditions, e.g. that it is not important whether the plate is clamped at the boundary or not. The next theorem implies that the notion of H-convergence is not tied to the prescribed boundary conditions: instead of homogeneous Dirichlet boundary conditions in Definition 1 we can take any boundary conditions which ensure well posedness of the boundary value problem.
Theorem 9 (Irrelevance of boundary conditions) Let (M n ) be a sequence of tensors in M 2 (α, β; Ω) that H-converges to M. For any sequence (z n ) such that (Ω), the weak convergence M n ∇∇z n M∇∇z in L 2 loc (Ω; Sym) holds. Proof. Let ω be an open set compactly embedded in Ω. The sequence (z n ) is bounded in H 2 (ω), implying that (M n ∇∇z n ) is bounded in L 2 (ω; Sym). If we denote σ n := M n ∇∇z n , we can extract a weakly convergent subsequence such that σ n σ in L 2 (ω; Sym). Since ω Ω, there exists ϕ ∈ C ∞ c (Ω) such that ϕ| ω = 1. For arbitrary N ∈ Sym, we define Let (w n ) be a sequence of solutions to .
Since (M n ) H-converges to M, the following holds: By coercivity of M n we have which, after passing to the limit and using the compactness by compensation result, becomes (σ − M∇∇w) : (∇∇z − ∇∇w) ≥ 0 a. e. in Ω.
If we consider the previous inequality only in ω, we have: For any joint Lebesgue point x 0 ∈ ω of ∇∇z, σ and M, let N = ∇∇z(x 0 ) + tO, where O ∈ Sym and t ∈ R + are arbitrary. Now (1.7) yields and after dividing this inequality by −t and taking the limit t → 0 + , it follows By arbitrariness of O ∈ Sym, the equality σ(x 0 ) = M(x 0 )∇∇z(x 0 ) easily follows. Due to uniqueness of the limit σ, the entire sequence M n ∇∇z n converges weakly to M∇∇z in L 2 (ω; Sym), which completes the proof.

Remark 1
If we change the assumptions of Theorem 9, such that the weak convergence M n ∇∇z n M∇∇z in L 2 (Ω; Sym) holds.
H-convergence also implies the convergence of energies, as stated in the sequel.
Theorem 10 (Energy convergence) Let (M n ) be a sequence of tensors in M 2 (α, β; Ω) that H-converges to M. For any f ∈ H −2 (Ω), the sequence (u n ) of solutions to where u is the solution of the homogenized equation .
Proof. If we apply Lemma 1, it can easily be seen that From the weak formulation of given homogeneous Dirichlet boundary value problems we get Ω M n ∇∇u n : ∇∇u n dx = H −2 (Ω) f, u n H 2 0 (Ω) , and since (u n ) converges weakly to u in H 2 0 (Ω), we have which concludes the proof. Proof. The proof goes along the same lines as the proof of Theorem 9: since ω is compactly embedded in Ω, there exists ϕ ∈ C ∞ c (Ω) such that ϕ| ω = 1. For arbitrary N ∈ Sym, let us define and let w n be a sequence of solutions to .
Since (M n ) H-converges to M, it follows that For sequence (O n ) we can proceed similarly: for any S ∈ Sym we introduce v(x) : and let (v n ) be a sequence of solutions to Since N is arbitrary, it follows that O ≥ M.
In the following theorem we introduce bounds on homogenized tensor, in the sense of the standard order on symmetric tensors. The bounds are given in terms of weak- * limits, representing the harmonic and arithmetic mean of the corresponding sequence.
Proof. As before, let us take a sequence (w n ) of oscillating test functions satisfying where N ∈ Sym is an arbitrary matrix. Since M n is coercive it follows Similarly, for σ ∈ Sym, the coercivity of (M n ) −1 implies which is equivalent to M n ∇∇w n : ∇∇w n − 2∇∇w n : σ + (M n ) −1 σ : σ ≥ 0.
Passing to the limit as before gives Proof. For f ∈ H −2 (Ω), let (u n ) be the sequence of solutions to .
As sequences (u n ) and ((M n ) T ∇∇u n ) are bounded in H 2 0 (Ω) and L 2 (Ω; Sym), respectively, we can extract a weakly convergent subsequence such that On the other hand, since (M n ) H-converges to M, for g ∈ H −2 (Ω) the sequence (v n ) of Chapter 1. General homogenization theory for elastic plate equation where v is the solution of the homogenized equation For an arbitrary open set ω Ω, there exists ϕ ∈ C ∞ c (Ω) such that ϕ| ω = 1. Choosing g := div div M∇∇ 1 2 Nx · x in ω. Using this, (1.9) becomes MN : ∇∇u = N : σ a. e. in ω , which implies that σ = M T ∇∇u almost everywhere in Ω, by arbitrariness of ω and N. Due to uniqueness of the limit σ, the entire sequence ((M n ) T ∇∇u n ) converges weakly to M T ∇∇u in L 2 (Ω; Sym), which gives the claim of the theorem.
The following result states that H-convergence defines a metrizable topology on the set M 2 (α, β; Ω). is a metric function on M 2 (α, β; Ω) and H-convergence is equivalent to the convergence with respect to d.
Proof. Since M 2 (α, β; Ω) is bounded in L ∞ (Ω; L(Sym, Sym)) and L 2 (Ω; L(Sym, Sym)) is continuously imbedded in H −1 (Ω; L(Sym, Sym)), there exists a constant c > 0 such that Clearly, the same is true if we replace M and (u n ) with tensor O and the corresponding sequence (v n ), which implies that the series in the definition of d converges. In order to verify that d is a metric, we shall only prove that d (Ω) satisfy u = v and M∇∇u = O∇∇v in Ω. Indeed, by definition of d, this immediately follows for f ∈ F , and then for any f ∈ H −2 (Ω) by the density of F in H −2 (Ω) and continuity of the linear mappings f → u and f → v from H −2 (Ω) to H 2 0 (Ω). For a set ω compactly embedded in Ω let us take ϕ ∈ C ∞ c (Ω) such that ϕ| ω = 1. If we take f = div div M∇∇ 1 2 ϕ(x)Sx · x , for arbitrary S ∈ Sym, this yields ∇∇u = ∇∇v = S in ω, implying MS = OS in ω, and finally M = O, by arbitrariness of S and ω.
It remains to prove that H-convergence is equivalent to the convergence in this metric space. Assume that sequence (M m ) in M 2 (α, β; Ω) H-converges to M in M 2 (α, β; Ω), and let (u m n ), (u n ) be the sequences of solutions of  In order to prove the converse statement, let a sequence (M m ) and M belong to M 2 (α, β; Ω) and d(M m , M) → 0. We take an arbitrary f ∈ H −2 (Ω) and a sequence (f n ) ⊆ F strongly converging to f in H −2 (Ω). Let u, u m , u n and u m n be solutions of as m → ∞.
If we subtract the equations for u and u n , we get and similarly for u m and u m n : .
Since (f n ) strongly converges to f , the well-posedness result for these problems ensures that u n → u in H 2 0 (Ω) and thus M∇∇u n → M∇∇u in L 2 (Ω; Sym), as well as u m n → u m in H 2 0 (Ω) and thus M m ∇∇u m n → M m ∇∇u m in L 2 (Ω; Sym), uniformly in m as n → ∞. Here, for the last convergence we have also used boundedness of the sequence (M m ) in L ∞ (Ω; L(Sym, Sym)).

Corrector result
Together with (1.11) this implies (1.12) i.e. (M m ) H-converges to M, by arbitrariness of f . Indeed, for an arbitrary f ∈ H −2 (Ω) and ε > 0, the above (uniform) convergences imply that first and third term on the right-hand side of the inequality can be made ε small for n large enough, i.e.
is valid for every m and n large enough. Taking the limit as m → ∞, from (1.11) and arbitrariness of ε and f we get the first convergence in (1.12), while the second one can be derived similarly.

Corrector result
This section is devoted to the corrector result in dimension d = 2. Its goal is to improve convergence of ∇∇u n by adding correctors, and ending up with strong convergence, instead of the weak one given by the definition of H-convergence.
Definition 5 Let (M n ) be a sequence of tensors in M 2 (α, β; Ω) that H-converges to a limit M. For 1 ≤ i, j ≤ 2 let (w ij n ) n be a sequence of oscillating test functions satisfying where g ij are some elements of H −2 loc (Ω). The tensor W n with components W n ijkm := [∇∇w km n ] ij is called the corrector. It is important to note that functions (w ij n ) 1≤i,j≤2 are not uniquely defined. However, for any other family of such functions, it is easy to see that their difference converges strongly to zero in H 2 (Ω), and similar holds true for the corrector tensors.

Lemma 3
Let (M n ) be a sequence of tensors in M 2 (α, β; Ω) that H-converges to a tensor M. A sequence of correctors (W n ) is unique in the sense that, for any two sequences Chapter 1. General homogenization theory for elastic plate equation of correctors (W n ) and (W n ), their difference (W n −W n ) converges strongly to zero in L 2 loc (Ω; L(Sym, Sym)).
Proof. For 1 ≤ i, j ≤ 2, let (w ij n ) n and ( w ij n ) n be two sequence satisfying (1.13) and let ϕ ∈ C ∞ c (Ω). Using coercivity of M n , and integrating by parts two times we obtain: Each term on the right hand side tends to zero when n → ∞, the first one because of the assumption (1.13), while the second one and the third one converge to zero by the Rellich compactness theorem. Thus, we deduce that ∇∇(w ij n − w ij n ) converges strongly to zero in L 2 loc (Ω; Sym), which proves the statement. Proof. The first convergence is a consequence of the definition of correctors. The second one follows from the definition of H-convergence, and the third one from the compactness by compensation result applied to the components of (W n ) T and M n W n .
In the next theorem we clarify in what sense correctors transform a weak convergence into the strong one. .
Let u be the weak limit of (u n ) in H 2 0 (Ω), i.e., the solution of the homogenized equation

Corrector result
Then, if we denote r n := ∇∇u n − W n ∇∇u, where W n is a corrector, it holds that (r n ) converges strongly to zero in L 1 loc (Ω; Sym).
As n → +∞, the first term on the right hand side converges by Theorem 10, while the second and the third term converge by the compensated compactness result. The last term converges by Lemma 4, leading to If u is smooth (in that case we can choose v m = u), the proof is finished. If u is not smooth, than after taking limit as n → +∞ in the estimate (c is a generic constant below)

H-convergent sequence depending on a parameter
A prerequisite for the small-amplitude homogenization is that H-convergence preserves a smooth (or analytic) dependence with respect to a parameter. More precisely, we shall prove that, if a sequence M n (·, p) depends smoothly on a parameter p, so does the H-limit M(·, p). We shall begin with this simple lemma, whose proof mimics the one in the case of the stationary diffusion equation [12,72] but we present it here for completeness.
Proof. One can easily see that where | · | on the right-hand side of (1.14) denotes the Frobenius norm. This proves the coercivity of M + O.
and the obtained inequality can be rewritten as Finally, Let us now describe a bound for the L ∞ -distance between the H-limits of two sequences M n ∈ M 2 (α, β; Ω) and O n ∈ M 2 (α , β ; Ω), that are nearby in L ∞ (Ω; L(Sym, Sym)).

Lemma 6
Let M n ∈ M 2 (α, β; Ω) and O n ∈ M 2 (α , β ; Ω) be two sequences of tensors that H-converge to the homogenized tensors M and O, respectively. Assume that, Proof. For f, g ∈ H −2 (Ω), let (u n ) and (v n ) be sequences of solutions to By Lemma 1, we have that M n ∇∇u n : ∇∇v n and ∇∇u n : (O n ) T ∇∇v n converge vaguely to M∇∇u : ∇∇v and ∇∇u : O T ∇∇v, respectively. Therefore, for every ϕ ∈ C ∞ c (Ω) one has lim For every ϕ ≥ 0 one can conclude Since for arbitrary a, b ∈ R + such that 4abαα ≥ 1 it holds |∇∇u n ||∇∇v n | ≤ aα|∇∇u n | 2 + bα |∇∇v n | 2 , Chapter 1. General homogenization theory for elastic plate equation Since this inequality is true for every ϕ ∈ C ∞ c (Ω), ϕ ≥ 0, it follows After minimizing the right-hand side of the previous inequality over all a, b satisfying the condition 4abαα ≥ 1 we get An alternative way to obtain the minimum (over a and b) for the right-hand side of (1.16) is to use the arithmetic-geometric mean inequality and ab ≥ 1 4αα . Hence, the desired inequality follows by arbitrariness of u and v, by using an analogous arguments as was done in the proof of Theorem 15.
Let us now prove that when passing to the H-limit in a sequence depending on a parameter, the smoothness is preserved. This result appears to be very important since we want to calculate first correction in the small-amplitude limit.
Theorem 17 Let M n : Ω × P → L(Sym, Sym) be a sequence of tensors, such that M n (·, p) ∈ M 2 (α, β; Ω), for p ∈ P , where P ⊆ R is an open set. Assume that (for some k ∈ N 0 ) a mapping p → M n (·, p) is of class C k from P to L ∞ (Ω; L(Sym, Sym)), with all derivatives up to order k being equicontinuous on every compact set K ⊆ P : Then there is a subsequence (M n k ) such that for every p ∈ P and p → M(·, p) is a C k mapping from P to L ∞ (Ω; L(Sym, Sym)).
Proof. For a countable dense subset Π of P, by the Cantor diagonal method and compactness of M 2 (α, β; Ω) there exists a subsequence (M n k ) such that for every p ∈ Π . For arbitrary compact K ⊆ P , by (1.17), it follows that which implies uniform continuity of p → M(·, p) on K ∩ Π, and thus it can be extended by continuity to the entire set K. In order to prove that (1.18) holds for every p ∈ K, let us suppose the opposite, i.e. that this H-convergence fails for some p ∈ K. Due to the compactness of H-convergence, there exists a subsequence (M n kr ) and N ∈ L ∞ (Ω; L(Sym, Sym)) such that M n kr (·, p) Using the equicontinuity of (M n kr ) and uniform continuity of M over K one can conclude that there exists a δ > 0 such that for every q ∈ K such that |p − q| < δ, it follows From the second inequality, it easily follows while from the first one and Lemma 6 we have for q ∈ Π ∩ K and |p − q| < δ, which is a contradiction. Therefore, (1.18) holds for every p ∈ K and, by arbitrariness of K, the mapping p → M(·, p) is well defined and continuous on P.
Since τ n (p) is an isomorphism (for every p and n), and, by Proposition 1, taking Chapter 1. General homogenization theory for elastic plate equation inverse is a C ∞ mapping, it follows that the mapping p → (τ n (p)) −1 is also a C k mapping. Moreover, one can conclude the following: We shall prove this only for i = 0, since for i ∈ {1, . . . , k} it can be shown analogously. By Theorem 2 and using the notation as in Proposition 1, with X := H 2 0 (Ω), Y := H −2 (Ω), we have: Due to equicontinuity property of the sequence (τ n ), one only has to check that As T ∈ [τ n (p), τ n (q)], τ n (p) = P • M n (·, p), and by using convexity of the set M 2 (α, β; Ω), for some M ∈ M 2 (α, β; Ω) we have T = P • M ∈ L(H 2 0 (Ω), H −2 (Ω)), which is a bounded and coercive operator with constants independent of p, q and n. Thus, it follows that Since the subsequence (M n k (·, p)) H-converges to M(·, p) for every p ∈ P , it follows that where τ (p) = P(M(·, p)). Let us define a family of functions Ψ f,g n : P → R, n ∈ N, and Ψ f,g : P → R as where f, g ∈ H −2 (Ω) are arbitrary nonzero functions. Since p → (τ n (p)) −1 is a C k mapping from P to L(H −2 (Ω), H 2 0 (Ω)), we have that p → Ψ f,g n (p) is of class C k from P to R. Note that due to equicontinuity properties of a sequence ((τ n (p)) −1 ) it follows that Additionally, from (1.20) we have that By the Arzelà-Ascoli theorem it follows that the sequence (Ψ f,g n k ), with all its derivatives, is bounded in C(K), where K ⊆ P is an arbitrary compact set, and pointwise convergence in (1.22) is actually uniform, thus p → Ψ f,g (p) is of class C k from P to R. After passing to the limit in (1.21), one can conclude that Ψ f,g has the same equicontinuity properties as the sequence (Ψ f,g n k ). It follows that p → (τ (p)) −1 is a C k mapping, and using the same reasoning as before, the mapping p → τ (p) is also of class C k .
Let us now consider a sequence Z n : P → L(H 2 0 (Ω); L 2 (Ω; Sym)) defined by Note that, with v n ∈ H 2 0 (Ω) defined as As it is a composition of C k mappings it follows that each Z n is also of class C k . By H-convergence of a subsequence (M n k (·, p)) to M(·, p), one can easily see that Z n k (p)v converges weakly in L 2 (Ω; Sym) to Z(p)v := M(·, p)∇∇v, for arbitrary v ∈ H 2 0 (Ω). Similarly as for (τ (p)) −1 above, one can easily show that p → Z(p) belongs to the class C k (P ; L(H 2 0 (Ω); L 2 (Ω; Sym))).

Chapter 1. General homogenization theory for elastic plate equation
The Lebesgue measure λ is inner regular on R d , hence for every ε > 0 there exists a compact set K ⊆ Ω, such that λ(Ω\K) < ε. For O := K\∂K Ω, and S ∈ Sym, let us take v ∈ H 2 0 (Ω) such that v(x) = 1 2 Sx · x on O. We easily conclude that p → M(·, p)S belongs to the class C k (P ; L ∞ (O; Sym)), and moreover, since λ(Ω\K) < ε, to C k (P ; L ∞ (Ω; Sym)). Due to arbitrariness of S, it follows that p → M(·, p) is of class C k from P to L ∞ (Ω; L(Sym, Sym)), which concludes the proof.

Remark 2
It is easy to see that the above theorem is valid if we take P ⊆ R d an open set. Furthermore, it can be shown that H-convergence also preserves an analytic dependence with respect to a parameter. To be precise, if we assume in the previous theorem that every M n is analytic mapping P −→ L ∞ (Ω; L(Sym, Sym)), then the corresponding H-limit M (after extracting a subsequence) is also analytic. This can be proved using the fact that any weakly converging sequence of analytic functions of operators has a limit which is analytic as well [42], and by following the same technique as in the proof of Theorem 17.

Periodic homogenization
When studying homogenization theory, periodic homogenization [16] appears to be the simplest case. There is a wide range of applications of periodic homogenization, for example in mechanics, physics, chemistry and engineering, in the study of crystalline or polymer structures, nuclear reactor design, etc. If the period of the observed structure is small compared to size of a region in which the system is studied, then asymptotic analysis is used. An example of a periodic domain is given in Figure 1 For the unit cube Y = [0, 1] d in R d and p ∈ [1, ∞], let us consider the following normed spaces of Y -periodic functions [2]: , and the quotient space H 2 # (Y )/R, equipped with the norm ∇∇ · L 2 (Y ) . For simplicity of notation, the class [u] in the quotient space will usually be 1.5. Periodic homogenization identified with a representative of the class u ∈ [u]. If we identify Y with the d-dimensional torus T (by gluing together the opposite faces of Y ), which is a smooth compact manifold without boundary, these spaces are isomorphic to L 2 (T ), H 2 (T ) and H 2 (T )/R, respectively. They can also be defined for vector, matrix or tensor valued functions.
We are interested in what happens in the limit of the periodic case, i.e., we want to derive the explicit formula for the homogenization limit of a periodic sequence of tensors.
has a unique solution in H 2 # (Y )/R. In order to prove that there exists a unique solution of (1.24), we shall check the assumptions of the Lax-Milgram lemma. Obviously, is a bilinear form and it doesn't depend on the choice of representatives of the equivalence classes. Since M ∈ M 2 (α, β; Ω) and ∇∇ · L 2 (Y ) is norm on the space H 2 # (Y )/R, one can easily conclude that bilinear form (1.25) is bounded and coercive: for arbitrary function ϕ ∈ H 2 # (Y )/R. By Theorem 1 there is a subsequence (M n k ) of (M n ) and a tensor valued function M in M 2 (α, β; Ω) such that (M n k ) H-converges to M. Let us define where w ij ∈ H 2 # (Y )/R are unique solutions of (1.27). Since w ij (n·) converges weakly to the average of w ij in H 2 (Ω), we easily conclude the convergence w ij n − However, due to periodicity we have where we used that w ij is a weak solution of (1.27) and took w kl as test function in (1.28).
Since every H-converging subsequence of (M n ) has the same limit, it follows that the entire sequence (M n ) H-converges to M. such that 0 ∈ Cl P , to L ∞ # (Y ; L(Sym, Sym)), for every k ∈ N, thus we have a smooth dependence with respect to a parameter p.

Small
Theorem 17 implies that there is a subsequence (A n k p ) such that for every p ∈ P , A n k p H −− A p in M 2 (α, β; Ω), and p → A p is a C k mapping from P to L ∞ # (Y ; L(Sym, Sym)). By Theorem 18, every H-converging subsequence of (A n p ) has the same limit, thus the entire sequence (A n p ) H-converges to A p . Since p → A p is a C k mapping from P to L ∞ # (Y ; L(Sym, Sym)), by using Taylor's theorem it follows that The goal of small-amplitude homogenization is to obtain the explicit formula for the leading terms B 0 and C 0 in the expansion of the homogenization limit. Theorem 18 implies that By using the integration by parts in (1.30), one easily gets Let us define Note that T (p) may be written as a composition P ). Furthermore, going along the same lines as in the proof of Theorem 17, it follows that the mapping By using this and the definition of w p mn , we conclude that Due to the given boundary value problem (1.31), after comparing expressions corresponding to the same powers of p, it is easy to conclude that w mn 0 = 0: first we insert w p mn in the corresponding boundary value problem By comparing expressions corresponding to the same powers of p, we obtain Uniqueness of the solution of this boundary value problem implies w mn 0 = 0.
If we insert the expression for w p mn in formula (1.32), we have (1.34) In order to fully describe C 0 it remains to calculate w mn 1 , which we shall do in terms of its Fourier coefficients: if we insert pw mn 1 + o(p) instead of w p mn in the corresponding boundary value problem (1.31), by equating powers of p we can easily see that w mn If we choose a representative of this solution satisfying Y w mn 1 (y) dy = 0, its Fourier series expansion takes the form where a mn k , k ∈ J := Z d \{0} are its Fourier coefficients. A simple calculation shows that If B(y) = k∈J B k e 2πik·y is the Fourier series expansion of function B, similarly as above we can calculate the right-hand side of (1.35): and consequently for Fourier coefficients we conclude If we insert the Fourier expansions of B and w mn and by the orthogonality of Fourier basis, we finally conclude The following theorem summarizes the previous results.
Additionally, let p ∈ P , where P ⊆ R is an open set such that 0 ∈ Cl P , and A n p (y) = A 0 + pB n (y).
Then, A n p H-converges to in Ω with coefficient C 0 being (a constant tensor) given by (1.40), for m, n, r, s ∈ {1, 2, · · · , d}, where and B k , k ∈ J, are the Fourier coefficients of B.
On the effective properties of composite elastic plate In this chapter we are interested in the application of the homogenization theory to the modeling of composite elastic plate. After an introductory section, the rest of the chapter is organized as follows: in the second section we show the local character of the set of all possible composites, and prove that the set of composites obtained by periodic homogenization is dense in that set. In the third section we describe the sequential laminates, a particularly interesting class of composite materials. In the fourth section we derive Hashin-Shtrikman bounds on primal energy, which are optimal bounds on the effective energy of a composite material, and, in the next section, we consider an analogous results for the complementary energy of a composite material. After that, we give a characterisation of the G-closure for the Kirchhoff-Love plate in the low contrast or small-amplitude regime. Finally, in the last two sections, we calculate explicit Hashin-Shtrikman bounds on primal and complementary energy for mixtures of two isotropic materials in dimension d = 2.

Introduction
In this chapter we apply previous results to the modeling of a composite elastic plate. Composite materials are heterogeneous materials obtained by mixing several materials on a very fine scale. It is impossible to imagine everyday life without composite materials. They are generally used for buildings, bridges, structures such as boat hulls, storage tanks, swimming pool panels, etc. Some widely used composite materials are wood, which is a natural composite of cellulose fibers in a lignin matrix, and concrete, which is a composite of aggregate (rock, sand or gravel), cement and water.
The main problem with composite materials is to determine their effective (macroscopic) properties. However, homogenization theory allows one to define a composite material as an H-limit. There is an extensive literature on this topic, we refer the interested reader to [2,6,22,46,48,54]. We shall mainly put the focus on composites obtained by mixing only two different materials, i.e. two-phase composite materials. From the physical point of view, these materials are determined with its two phases A and B, their proportions θ and 1 − θ, respectively, and by their microstructure, i.e. by their geometric arrangement in the mixture.
Although it will be defined later for m materials and in a more general setting, let us now define a two-phase composite material [2].

Definition 6
Let χ n ∈ L ∞ (Ω; {0, 1}) be a sequence of characteristic functions and (M n ) be a sequence of tensors defined by where A and B are assumed to be positive definite fourth-order tensors. Assume that there exist θ ∈ L ∞ (Ω; [0, 1]) and M ∈ L ∞ (Ω; L(Sym, Sym)) such that where β > α > 0 are given. The H-limit M is said to be the homogenized tensor of a two-phase composite material obtained by mixing A and B in proportions θ and (1 − θ), respectively, with a microstructure defined by the sequence (χ n ).
Before we proceed further with composite materials, let us introduce some definitions and results which are necessary for better understanding this chapter. Let F : X → R, where X is a Banach space.

Definition 7
Let F be Lipschitz near a given point x ∈ X and let v ∈ X. The upper generalized directional derivative of F at x in the direction v, denoted F 0 (x; v), is defined as follows: where y ∈ X and t ∈ R + .

Definition 8
The generalized gradient of F at x, denoted ∂F (x), is the subset of X given by

Introduction
The following theory can be found in [25]. One may find it helpful in the fourth section, when showing the optimality of Hashin-Shtrikman bounds.
Let f t be a family of functions on a Banach space X, with codomain R, parametrized by t ∈ T , where T is a topological space. Suppose that for some point x ∈ X, each function f t is Lipshitz near x. First, we make the following hypotheses: By M (y) we denote the set {t ∈ T : f t (y) = f (y)}, and for any subset S of T , P [S] signifies the collection of probability Radon measures supported on S.
Theorem 20 [25, p. 86-87] In addition to the hypotheses given above, suppose that X is a vector space of finite dimension. Additionally, assume that each f t is Frechet differentiable on U , and that f t (x) is continuous as a function of (t, x). Then, for each x ∈ U one has Theorem 21 [1,p. 513,Theorem 15.11] If X is a compact metric space, the set P (X) of probability measures on X is compact in the weak- * topology.
The notion of conjugate function appears to be a very useful tool in this chapter.
Remark 5 [17, p. 11-13] Note that ϕ * is convex and lower semicontinuous on E . If we iterate the operation * , we could obtain a function ϕ * * defined on E . Instead of that, we restrict ϕ * * to E, i.e. we define Furthermore, we are going to state some well-known and simple, but also crucial facts which shall be used for calculating explicit Hashin-Shtrikman bounds for mixtures of two isotropic materials in dimension d = 2.

Definition 11
Let K be a closed, convex set. A point e of K is called an extreme point of K if it is not the interior point of a line segment in K. Equivalently, x is not an extreme point of K if there exist y, z ∈ K, y = z, such that The set of extreme points of K is denoted by ExtK.
Theorem 24 [45, p. 195 which are nonempty. Furthermore, ρ − = ν(E − ) > 0, ρ + = ν(E + ) > 0 and ρ − + ρ + = 1. Now, we can decompose the measure ν as a convex combination of probability measures: Using this, the tensor M can also be decomposed as a convex combination of two points in C: problem and it appears to be quite difficult to solve. It was done in the conductivity setting for mixtures of two isotropic conductors [48,72], while in the elasticity setting for two isotropic phases it is still an open problem (for partial results we refer to [55] and references therein), even for the elastic plates [35,51]. One can only obtain bounds that must be satisfied by the effective properties, called Hashin-Shtrikman bounds in their most general form [40]. However, results can be pushed much further under a simplifying assumption of a small-amplitude or low contrast regime [2,3,4,33,38,39,69].
Due to its local character (see Theorem 26 below), the G-closure problem reduces to describing the set G θ of all possible composite materials obtained by mixing M 1 , M 2 , . . . , M m in constant proportion θ ∈ T : An interesting example of composite materials are periodic mixtures, which are obtained in the following way: where the entries of M ∈ G θ are given by (1.26).
For fixed θ ∈ T , by P θ ⊆ G θ we denote the set of all constant homogenized tensors obtained by periodic homogenization, as described above, i.e. by mixing M 1 , M 2 , . . . , M m in the sense of (2.2), for some Y -periodic functions The following theorem asserts the local character of the set of all possible composites, and together with the last theorem in this section, it implies that the set of composites obtained by periodic homogenization is dense in the set of all possible composites. Actually, the statements and the proofs of these two theorems mimic the ones in the case of stationary diffusion equation (see [74] and references therein).

Theorem 26
Let (χ n ), (R n ) be a sequences in L ∞ (Ω; T ) and L ∞ (Ω; SO(R d )), respectively, and (M n ) defined with (2.1) such that We observe (2.7) Our goal is to replace measurable sets E i with open sets U i , such that (2.7) still holds, eventually in some weaker norm. Let C 1 , . . . , C p be a compact sets in R d such that Since C i , i = 1, . . . , p, are pairwise disjoint, there exist pairwise disjoint open sets U i ⊇ C i , i = 1, . . . , p, which cover Ω, up to a set U ε whose measure is less than ε. Next, we define a piecewise constant function and extend it to Ω arbitrarily, such thatθ ε U ε ∈ L ∞ (Ω; R m ) and M ε U ε ∈ L ∞ (Ω; L(Sym, Sym)).
Let us notice that (Ω;L(Sym,Sym)) < 2εµ(Ω), (2.8) which together with (2.7) gives Consequently, on every set U i , i = 1, . . . , p, there exist sequences (χ ε,i n ) and (R ε,i n ) in L ∞ (U i ; T ) and L ∞ (U i ; SO(R d )), respectively, and a sequence (M ε,i n ) defined with (2.1) such that as n → ∞. Let us denote by χ ε n and M ε n functions on Ω whose restrictions on U i are equal to χ ε,i n and M ε,i n , respectively. On the set U ε we can define them arbitrarily (in the permissible set of values). The locality of H-convergence, Banach-Alaoglu theorem [53] and compactness of M 2 (α, β; Ω), imply that on a subsequence one has as n → ∞, where we have predefined functionsθ ε and M ε on the set U ε . Since the measure of U ε is less than ε, (2.8) and (2.9) still hold. By using (2.9), (2.10) and the Cantor diagonal method, we can extract a subsequence (ε(n )) which converges to zero, such that which gives the claim of the theorem.
In the following lemma we use notation of Section 1.5. Proof. For E ∈ Sym and n ∈ N ∪ {∞}, by w n E ∈ H 2 # (Y )/R we denote the solution of boundary value problem The sequence (w n E ) is bounded in H 2 loc (R d ), which implies that there is a subsequence Thus, we deduce that By Theorem 9 it follows M n ∇∇u n − M ∞ ∇∇u ∞ in L 2 loc (Ω; Sym). (2.12) Furthermore, v is a solution of boundary value problem (2.11) for n = ∞ and it coincides with w ∞ E . Therefore, we have (2.13) By (2.12) and (2.13), we can pass to the limit in the following integral: This finishes the proof by arbitrariness of E ∈ Sym.
Proof. In order to show that ClP θ ⊆ G θ , it is enough to prove that G θ is closed (obviously To prove that G θ ⊆ ClP θ , let us take M ∈ G θ . There exist sequences (χ n ) in L ∞ (Y ; T ) and (R n ) in L ∞ (Y ; SO(R d )), such that for M n defined with (2.1), it follows Furthermore, For n ∈ N, we predefine χ n toχ n ∈ L ∞ (Y ; T ) such that Since M n ∈ P θ , the claim of the theorem follows.

Remark 7
In this section, the characterisation of the G-closure, which is the set of all homogenized tensors obtained by mixing m materials in fixed volume fractions such that rotations of materials are allowed, has been given. Since the assumption that rotations of materials are allowed makes further results highly nontrivial to derive (or sometimes even impossible), from now on we shall consider only mixtures with no rotations. Actually, this is a standard setting when dealing with composite materials. It is easy to check that all 2.3. Homogenization of laminated materials results of this section are valid if we replace the sequence (M n ) from (2.1), with a similar one, but with rotations excluded: for χ n = (χ n 1 , . . . , χ n m ) ∈ L ∞ (Ω; T ).

Homogenization of laminated materials
In this section we shall study laminated composite materials which are homogeneous in all directions orthogonal to some fixed unit vector e, which is called the direction of lamination. For simplicity, let us take for the direction of lamination the first canonical basis vector. In this case, the sequence of tensors (M n ) depends only on the first variable x 1 . We are interested in elastic properties of laminated materials, which can be derived by using the following theorem. Its proof goes along the same lines as proof of two-dimensional result [9,10,57] in L ∞ (Ω). For n ∈ N, we denote E n := ∇∇u n , E := ∇∇u and D n := M n E n .
Since div div D n = f , by Lemma 8 D n 11 does not oscillate in x 1 . Furthermore, let us definẽ D n ∈ L ∞ (Ω; Sym) withD Since this argument holds for every accumulation point, it follows that the entire sequence Ψ(M n ) converges to Ψ(M).
In order to prove the reverse implication, by using notation introduced in the first part of the proof, assume that Similarly as before, let (u n ) be the sequence of solutions to for an arbitrary function f ∈ H −2 (Ω). The sequences (u n ) and (M n ∇∇u n ) are bounded in H 2 0 (Ω) and L 2 (Ω; Sym), respectively, and therefore converge weakly on a subsequence: u n − u in H 2 0 (Ω), M n ∇∇u n − σ in L 2 (Ω; Sym).
In order to prove that M n H −− M, we only need to show that σ = M∇∇u.
In the same way as in the first part of the proof, we define matrices G n , O n ∈ L 2 (Ω; Sym), respectively made from good (nonoscillating) and bad (oscillating) components of E n and D n , such that O n = K n G n , (2.15) with K n := Ψ(M n ). By using an analogous arguments, we can pass to a subsequence in (2.15) and obtain where K = Ψ(M). After some calculation, and using also that E n = ∇∇u n − ∇∇u in L 2 (Ω; Sym), from (2.16) it follows that σ = M∇∇u.
The previous theorem shows that H-convergence can be reduced to the weak- * convergence of some combination of entries of the tensors M n , n ∈ N, in the case when they depend only on one variable. Its special case of particular interest is the following lamination formula, which gives the elastic properties of a simple laminate of two materials.

Remark 8
In order to indicate dependence of a function f , defined on Ω, only on x · e, for some e ∈ R d , in the sequel we shall abuse the notation f (x · e) for both, the mapping x → f (x · e), as well as the value of this mapping at the point x ∈ Ω. , (2.17) which also depends only on x 1 .

Corollary 1 Let
Proof. Since χ n (x 1 ) is a sequence of characteristic functions (i.e. it is always equal to 0 or 1), the above formula can be derived by using Theorem 28:

Remark 9
If we take some other unit vector e ∈ R d for the lamination direction, and let θ(x · e) be the weak limit of the sequence χ n (x · e), then the formula (2.17) is still valid, in an analogous form: This approach for deriving the lamination formula was used by Tartar (see e.g. [72]) in the conductivity setting. A different approach was considered in [51] (see also [47]), by using the average values of strain and moment over the given area, which is assumed to be physically small. Then, the lamination formula was derived under the additional assumption that along the boundaries dividing the layers, the conditions of continuity for the normal and tangential component of strain hold.
The process of lamination can be repeated in an iterative way. It is of interest to consider a particular subset of laminated materials, obtained by an iterative process of lamination, where the previous laminate is laminated again with a single pure phase (always the same one). The composite material obtained by this process is called a sequential laminate. Since it is quite difficult to iterate formula (2.22), we shall express it in a different form, by using the following lemma, whose idea is due to Tartar [67].

Lemma 10
If N is an invertible fourth-order tensor, E ∈ Sym and c ∈ R such that 1 + c(NE : E) = 0, then the inverse of N + c(NE) ⊗ (N T E) is Proof. This can be proved by using a variation of the well-known Sherman-Morrison formula [27], but it can also be done straightforwardly: which is the required formula.
Let A * 1 be a simple laminate, obtained from the materials A and B, in proportions θ 1 and (1 − θ 1 ), respectively, in the direction e 1 of lamination: A * 1 can be laminated again with the material B, in proportion θ 2 of material A * 1 and 1 − θ 2 of material B, in the direction of lamination e 2 , to obtain a new laminate A * 2 : i.e. If we continue this iterative process and in p-th step laminate the previously obtained laminate A * p−1 with material B in the lamination direction e p and proportion θ p of A * p−1 , we obtain a composite material called a rank-p sequential laminate with matrix B and core A, which is determined by the following formula: The overall volume fraction of material A in this rank-p sequential laminate is θ = The following result is a simple consequence of (2.24) (for an analogous result in the context of stationary diffusion see [2]). Proof. One should compare formulas (2.24) and (2.25), which gives the following equality: One could interchange the roles of A and B and obtain a symmetric class of sequential laminates, i.e. if we repeat the iterative lamination process p times, in lamination directions (e i ) 1≤i≤p and proportions (θ i ) 1≤i≤p of material A, we obtain a rank-p sequential laminate with matrix A and core B, which is defined by the following formula: (2.28)

Hashin-Shtrikman bounds on the primal energy
In the sequel we want to derive optimal bounds on the effective energy of a twophase composite material, obtained by mixing A, B ∈ Sym 4 in proportions θ and 1 − θ, respectively. These bounds are based on the Hashin-Shtrikman variational principle [40] which applies only when materials are well-ordered, i.e. we assume that Aξ : ξ ≤ Bξ : ξ, ξ ∈ Sym.
Otherwise, when materials are not well-ordered, one could apply the translation method as was done in [5]. The notion of an optimal bound is introduced in the following definition.

Theorem 29
The effective energy of a composite material A * ∈ G θ satisfies the following bounds:

Remark 10
The supremum and infimum in formulas (2.30) and (2.32) can also be taken over R d \{0}. By using the fact that Q · Z d is dense in R d and continuity of functions we obtain An analogous result can be shown for function g.

Remark 11
It follows straightforwardly from the definition of a convex function that the function under the supremum in the definition of g is convex on Sym. This, together with the fact that the maximum of convex functions is also a convex function [8], implies that g is convex, so the lower Hashin-Shtrikman bound is given as the result of a finite-dimensional concave maximization.
On the other hand, it can be shown that h(η) ≤ B −1 η : η: for arbitrary k ∈ R d \{0}, by substituting k⊗k = B − 1 2 X, X ∈ Sym, and by using the Cauchy-Bunyakovsky-Schwarz inequality, we obtain that Proof of Theorem 29. Since ClP θ = G θ , it is enough to prove that these bounds hold for A * obtained by periodic homogenization. We begin with the lower Hashin-Shtrikman bound. By using the weak formulation of the periodic boundary value problem (1.27) introduced in Theorem 18, with M(y) = χ(y)A + (1 − χ(y))B, we have that χ(y))B)(ξ + ∇∇w(y)) : (ξ + ∇∇w(y)) dy, (2.33) where ξ ∈ Sym and χ ∈ L ∞ # (Y ; L(Sym, Sym)) is a characteristic function such that Since B − A ≥ 0, by using the Legendre transform, one can easily conclude that Next, by the Fenchel-Moreau theorem To obtain an explicit formula for g, one can use Fourier analysis since (2.37) represents minimization over periodic functions. Note that the function to be minimized in (2.37) involves only bilinear terms depending on y. Thus, by periodicity, one can write χ and w as These are complex Fourier series [31,37], and since functions χ and w are real valued, their Fourier coefficients must satisfyχ(k) =χ(−k) andŵ(k) =ŵ(−k). The Parseval's relation gives: The minimization over w ∈ H 2 # (Y ) in (2.37) is equivalent to the minimization overŵ(k) ∈ C, for each k ∈ Z d , and the minimization can be performed on each component of the sum (2.38) independently. Note that k = 0 contributes nothing to the sum (2.38). Now, it is easily seen that the above minimum is attained when Therefore where g(η) is given by (2.30). Now, the Parseval's relation implies which yields the lower Hashin-Shtrikman bound. The upper Hashin-Shtrikman bound can be derived analogously.
Let us now prove that the lower bound is optimal in the sense of Definition 13: first, we denote (2.39) As φ is strictly concave in η (see Remark 11) and −φ is coercive, there exists a unique maximum point η * of φ. Since φ is usually not a smooth function, the first-order (both necessary and sufficient) optimality condition for (2.39) reads 0 ∈ ∂φ(η * ), where ∂φ(η * ) is the subdifferential of function φ at η * . To calculate the subdifferential of φ, one should calculate the subdifferential of g.
By simple change k = k e, e ∈ S d−1 , we have Since K is a nonempty, closed, bounded, convex subset of a vector space X of finite dimension n, by Carathéodory's theorem (see Theorem 24) and Lemma 7, every point from K can be represented as a convex combination of at most n + 1 extreme points of for some e i ∈ M (η) and m i ≥ 0, such that n+1 i=1 m i = 1. Now, the first-order optimality condition for maximizing (2.39): η * is optimal if and only if which is equivalent to for some e i ∈ M (η * ) and m i ≥ 0, such that n+1 i=1 m i = 1. Note that in deriving (2.41), the symmetry of A and B is used, otherwise the optimality condition would be given as which would significantly complicate further algebraic calculations.
Taking the inner product of (2.42) with η * gives: Thus, the lower bound (2.29) can be expressed as In order to achieve equality in (2.44), let us consider the sequential laminate given by (2.45) Multiplying (2.45) by η * and using (2.42) gives (2.46) which equals After taking the inner product of (2.47) with ξ, we obtain The previous theorem extends easily to the sum of energies.
Proof. The proof goes along the same lines as the proof for a single effective energy. Let us just highlight the main differences (only for the lower Hashin-Shtrikman bound, as it was done in the proof of Theorem 29). We have: Again, by using Fourier analysis, we obtain where g(η 1 , . . . , η p ) is given with (2.49). This yields the lower Hashin-Shtrikman bound. The upper Hashin-Shtrikman bound can be derived analogously.
Let us now prove the optimality of the lower Hashin-Shtrikman bound on the sum of energies. Similarly as before, we denote: (η 1 , . . . , η p ), which is a strictly concave function in (η 1 , . . . , η p ), such that −φ is coercive, and consequently there exists a unique maximum point (η * 1 , . . . , η * p ) of φ. Furthermore, the same arguments as in the proof of Theorem 29 show that the first-order optimality condition for some e k ∈ M (η * 1 , . . . , η * p ) and m k ≥ 0, such that n+1 k=1 m k = 1. By using this, an analogous reasoning as for the single effective energy implies the claim.
The following corollary is an immediate consequence of the Theorem 29.

Corollary 3 Let
Similarly, an upper Hashin-Shtrikman bound (2.31) is equivalent to Proof. Let us start with the upper bound. First, we denote Obviously, (2.31) is equivalent to ϕ 1 (ξ) ≥ ϕ 2 (ξ), ξ ∈ Sym, and ϕ 1 , ϕ 2 are convex functions. By Fenchel-Moreau theorem, the Legendre transform of ϕ * 2 is again ϕ 2 : where the third equality in (2.51) follows after replacing η by η 2 . Thus, by Remark 6 Similarly, Since ϕ 1 and ϕ 2 are convex and continuous, the inequality In other words, an upper Hashin-Shtrikman bound is equivalent to The assertion for the lower bound can be proved analogously.

Hashin-Shtrikman bounds on the complementary energy and lamination formulas
In this section we want to derive bounds on the complementary energy of a twophase composite material, obtained by mixing A, B ∈ Sym 4 in proportions θ and 1 − θ, respectively. Furthermore, we assume that Aξ : ξ ≤ Bξ : ξ, (2.52) for any ξ ∈ Sym.
Recall that, in the case of the primal energy, when considering bounds on A * ξ : ξ, we concluded that optimality of Hashin-Shtrikman bounds is achieved by a finite-rank sequential laminate. Now, when our focus is on A * −1 σ : σ, first we want to derive lamination formulas in terms of A * Before we proceed further, let us mention that the basic tool for deriving desired lamination formulas will be the following identity:   nonnegative real numbers such that p i=1 m i = 1. Then, there exists a rank-p sequential laminate A * p with core B and matrix A, and with lamination directions e 1 , e 2 , . . . , e p , such that Our next step is to derive bounds for the effective energy written in terms of stress, i.e. bounds on the complementary or dual energy. These bounds, as well as bounds on the primal energy, will pave the way for new results concerning optimal design of a thin, elastic plates.

Remark 12
Analogously as for the Hashin-Shtrikman bounds on the primal energy, one can show that the function is concave, hence the bound (2.60) is given as the result of a finite-dimensional concave maximization, and that the function is convex, which implies that the bound (2.63) is given as the result of a finite-dimensional convex minimization.
Proof of Theorem 31. (2.66) Next, we show that (2.60) is equivalent to Let us first denote and Obviously, (2.60) is equivalent to ϕ 1 (σ) ≤ ϕ 2 (σ), σ ∈ Sym, and ϕ 1 , ϕ 2 : Sym → R are convex functions. By Fenchel-Moreau theorem, since ϕ 1 is continuous, convex and ϕ 1 ≡ +∞, the Legendre transform of ϕ * 1 is again ϕ 1 : By Remark 6, it easily follows that Chapter 2. On the effective properties of composite elastic plate Furthermore, Since ϕ 1 and ϕ 2 are both convex and continuous, which, by symmetry of B −1 , can be written as By using (2.55) and after some calculation, one easily verifies that (2.67) is equivalent to (2.66), which completes this part of the proof. The upper Hashin-Shtrikman bound on the complementary energy can be derived analogously.
Let us now prove that the lower bound (2.60) is optimal in the sense of Definition 13. The proof is analogous as for the optimality of Hashin-Shtrikman bounds on primal energy, hence, we shall just highlight the main parts. We denote It is easy to see that and P [M (η)] is the collection of probability Radon measures on the unit sphere, supported on M (η). By using Carathéodory's theorem, similarly as in the proof of the Theorem 29, the subdifferential of h B (η) is equivalently defined by for some e i ∈ M (η) and m i ≥ 0, such that Furthermore, the optimality condition 0 ∈ ∂φ(η * ) is equivalent to Taking the inner product of (2.69) with η * gives To achieve equality in the lower Hashin-Shtrikman bound on the complementary energy, let us consider the sequential laminate provided by formula (2.58): In order to conclude that optimality is achieved by a finite-rank sequential laminate given with (2.58), it remains to show that φ(η * ) = σ : η * , which easily follows from (2.70). For the upper Hashin-Shtrikman bound on the complementary energy a similar conclusion holds, i.e. the optimality is achieved by a finite-rank sequential laminate (2.59).
Chapter 2. On the effective properties of composite elastic plate

G-closure in the small-amplitude regime
In this section we shall give a characterisation of the G-closure for the Kirchhoff-Love plate in one simplified regime, namely the low-contrast or small-amplitude regime: we mix two materials A, B ∈ Sym 4 , in proportions θ and 1 − θ, respectively, with a microstructure defined by the sequence (χ n ), and additionally we assume that A and B have close properties. More precisely, we assume that there exists a small positive parameter γ and a coercive, symmetric fourth order tensor D such that In order to describe the G-closure in the low-contrast regime, we shall use H-measures. They were introduced independently for weakly convergent sequences in L 2 (R d ) by Tartar [69] and Gérard [34] (under the name microlocal defect measures). An idea is to associate a measure to a weakly converging (sub)sequence, and such measure is called the H-measure associated with that (sub)sequence. The main result which assures existence of such measure is the existence theorem for H-measures [69, Theorem 1.1]. In the case when the sequence is defined with χ n (x) := χ(nx), x ∈ R d , for a Y -periodic function for any f ∈ C(S d−1 ), whereχ(k) are Fourier coefficients of the function χ. Since the sequence (χ n ) is uniquely described with function χ (in this periodic case), we shall say that probability measure ν is the H-measure of χ [2,3].
In the following theorem we give a formula for the composite M * obtained by periodic homogenization, up to a second order in γ, in terms of the H-measure ν of characteristic function χ.

Theorem 32
Let M * be a homogenized tensor in P θ , associated to a characteristic function χ. For any ξ ∈ Sym it follows that Additionally, the remainder is uniform with respect to M * in the sense that there exists a positive constant c, independent of γ and χ, such that Proof. The theorem will be proved in two steps.
I In the periodic case we have an explicit formula for the homogenization limit, i.e. by Theorem 18, for any ξ ∈ Sym and therefore the solution w depends analytically on the small parameter γ, although this fact shall not be used. Let us rather introduce a, b ∈ H 2 # (Y )/R such that After applying the integration by parts in (2.76), one easily gets In order to compute the term of order γ 2 , we use the weak formulation of the boundary value problem (2.74) with a ∈ H 2 # (Y )/R as a test function, which yields: Moreover, using the fact that the solution of (2.74) is also a unique solution of the corresponding minimization problem, we obtain − min The minimum in (2.77) can be computed by using Fourier analysis, in an analogous way as it was done in the proof of Theorem 29, when calculating the function g given by (2.37): − min Introducing the H-measure ν of characteristic function χ and using that D is a symmetric tensor, yields Thus, II To conclude the proof, one only has to show that the term of order γ 3 can be estimated with a positive constant c, independent of γ and χ. Note that the solution a of boundary value problem (2.74) is independent of γ, but that is not the case for the solution b of (2.75). However, since M(y)ξ : ξ ≥ Aξ : ξ ≥ α|ξ| 2 , ξ ∈ Sym, one can show that b satisfies an a priori estimate as will be proved in the sequel.
For simplicity, we identify the unit cube Y with the unit d-dimensional torus T , which can be done by gluing together opposite faces of Y , and in the sequel, due to this identification, a periodic function in Y is actually defined as a function on the unit torus [37]. Moreover, equations (2.74) and (2.75) can be seen as posed in the unit torus, which is a smooth compact manifold without boundary, and thus H 2 0 (T ) = H 2 (T ) [15,41]. From the boundary value problem (2.75), and by using an a priori estimate based on Lax-Milgram lemma, we have: where the last equality in (2.79) follows by Theorem 6. Next, since div div : L 2 (T ; Sym) → H −2 (T ) is linear and continuous, and by definition of the norm on the quotient space H 2 (T )/R, it follows (C is a generic constant below): By using an analogous arguments, from the boundary value problem (2.74), we conclude Therefore, by (2.79), (2.80) and (2.81), Using (2.82), it is easy to conclude that the remainder is uniform with respect to M * , i.e.
Theorem 32 can be extended to any composite tensor M * ∈ G θ .

Theorem 33
Let M * ∈ G θ be the homogenized tensor obtained by mixing two materials A, B ∈ Sym 4 , in proportions θ and 1 − θ, respectively, with a microstructure defined by the sequence χ n , such that there exists a small positive parameter γ and a coercive, symmetric fourth order tensor D so that We take M n :=χ n A + (1 −χ n )B, n ∈ N, to be a sequence in P θ such that M n → M * ((M n ) exists by Theorem 27), and for n ∈ N, let ν n be the H-measure associated to a characteristic functionχ n . Then ν n ν weakly- * in P (S d−1 ) (on a subsequence), and, 2.7. Explicit Hashin-Shtrikman bounds on the primal energy for mixtures of two isotropic materials in dimension d = 2 for any ξ ∈ Sym, A(e ⊗ e) : (e ⊗ e) .
Additionally, the remainder is uniform with respect to M * in the sense that there exists a positive constant c, independent of γ and χ, such that Proof. By Theorem 32, for every n ∈ N, M n is given by for arbitrary ξ ∈ Sym.
From the compactness of the set P (S d−1 ) in the weak- * topology, the claim of the theorem directly follows, i.e. the small amplitude formula of Theorem 32 can be extended to any composite tensor M * ∈ G θ . In the sequel we consider elastic composite materials obtained by mixing two isotropic phases A and B in proportions θ 1 := θ and θ 2 := 1 − θ, respectively. The following results will be stated in dimension d = 2, due to the fact that they all refer to the Kirchhoff model for pure bending of a thin, solid symmetric plate under a transverse load. Isotropic phases A and B are defined by

Explicit
where κ 1 , κ 2 are the bulk moduli, while µ 1 , µ 2 are the shear moduli. Since A and B are assumed to be well-ordered, the following holds for bulk and shear moduli: In order to explicitly calculate the Hashin-Shtrikman bounds, first we have to evaluate functions g and h from Theorem 29.
Since e ∈ S 1 we have and analogously, It is easy to notice that functions under the maximum in (2.85) and minimum in (2.86) are actually the same: The assertion now follows from the fact that η i ≤ η j implies where equalities are achieved by the corresponding eigenvectors of the matrix η.
Proof. Firstly, note that the expression as well as the function g(η) = g(η 1 , η 2 ), depend only on the eigenvalues η 1 and η 2 of the matrix η. Therefore, in order to explicitly compute the lower Hashin-Shtrikman bound, we shall use the classical von Neumann result (see Theorem 23), which implies that the maximum of ξ : η equals 2 i=1 η i ξ i , and it is obtained when η and ξ are simultaneously diagonalizable. This reduces the problem of computing the maximum on the right-hand side of (2.29) to maximization of the concave function over all real numbers η 1 and η 2 . Note that the function F is quadratic by parts. An analogous argument was used in [5], for computing the explicit Hashin-Shtrikman bounds in the context of 2D linearized elasticity. In each of the cases (i) − (iii), and for θ 1 = 0.6, µ 1 = 1, µ 2 = 3, κ 1 = 2, κ 2 = 4, the graph of the function F is given in Figure 2.2.   Due to expression (2.83) for g, we shall consider several cases (note that g is differentiable everywhere except on the lines η 1 = η 2 and η 1 = −η 2 ).
I If |η 1 | > |η 2 |, then the first-order optimality conditions for F are This linear system has a unique solution: If this solution fits the case |η 1 | > |η 2 | then the maximum of F is , and the lower bound is given with The requirement that the solution (2.90) satisfies |η 1 | > |η 2 | is fulfilled if and only if Taking the square and factorizing the inequality (2.92) yields Thus, (2.93) is equivalent to either II The case |η 1 | < |η 2 | is symmetric to the previous one, and one only has to change the roles of ξ 1 and ξ 2 in (2.91), (2.94) and (2.95): if and only if ξ 1 , ξ 2 satisfy Cases I and II, using some standard algebraic calculations, can jointly be written as follows: the bound (2.29) is equivalent to if and only if ξ 1 and ξ 2 satisfy Note that this corresponds to the case (i) in the statement of the theorem.
III If condition (2.99) is not satisfied, then the maximum of F is attained on one of the lines η 1 = η 2 or η 1 = −η 2 .
(ii) If η := η 1 = η 2 , an easy calculation gives us that the maximum of F is reached for If the maximum is attained in this case, the corresponding bound is and the explicit Hashin-Shtrikman bound is given with By the following inequality, which can easily be derived by using elementary algebraic operations, it is easy to check that the maximum is attained on the line η 1 = η 2 if and only if Conversely, the maximum is attained on the line η 1 = −η 2 if and only if It is interesting to see, for some arbitrary parameters θ 1 , µ 1 , µ 2 , κ 1 , κ 2 , how the division of R 2 in conditions (2.87), (2.88) and (2.89) looks like. For µ 1 = 26, κ 1 = 40 (parameters of the first material, i.e. glass), µ 2 = 79, κ 2 = 160 (steel) and θ 1 = 0.4, ξ 1 , ξ 2 ∈ [−1000, 1000], in Figure 2.3 one can see that if (ξ 1 , ξ 2 ) belongs to the blue area then condition (2.87) is satisfied and maximum is attained in one of the cases |η 2 | < |η 1 | or |η 1 | < |η 2 |. If (ξ 1 , ξ 2 ) belongs to the yellow area, then maximum is attained on the line η 1 = −η 2 , and the green area represents (ξ 1 , ξ 2 ) such that maximum is attained on the line η 1 = η 2 .
It remains to explicitly describe the optimal microstructures which saturate the lower Hashin-Shtrikman bound. Theorem 29 assures that the lower Hashin-Shtrikman bound is optimal and that optimality is achieved by a finite-rank sequential laminate. An optimality condition for the maximization problem on the right-hand side of (2.29) is given by (2.42) (2.104) where m i ≥ 0, . (2.105) If g is differentiable at the optimal η, then the optimality condition simplifies to A(e ⊗ e) : (e ⊗ e) η, (2.106) where e is an extremal for (2.105). By using an analogous algebraic calculations as in the proof of Theorem 29, we show that the lower bound (2.29) can be expressed as where η * is the optimal point for the maximization problem on the right-hand side of (2.29). One can conclude that the equality in (2.107) is obtained with laminate A * 1 defined by the formula To be precise, after multiplying (2.108) by η * and using (2.106), we obtain (2.109) which after taking the inner product with ξ gives the claim. Clearly, in this case the optimality is achieved by a rank-one laminate with the lamination direction e, where e is an extremal for g(η * ) given with (2.105). This corresponds to the first case in the Theorem 34, i.e. when the maximum in (2.29) is achieved for |η 1 | = |η 2 |.
It remains to specify the optimal microstructure in the other two cases, i.e. when Additionally, using the fact that ξ and η are simultaneously diagonalizable and symmetric, it follows that there exists orthogonal matrix Q such that Q T ξQ and Q T ηQ are diagonal matrices with eigenvalues of ξ and η as the diagonal entries, respectively. If we multiply (2.104) from the left by Q T and from the right by Q, we conclude that (2.104) is equivalent to This determines m 1 and m 2 : and it is easy to check that m 1 + m 2 = 1 and m 1 , m 2 ≥ 0, as a consequence of the condition which defines this regime. In this case, the bound is obviously achieved by a second rank laminate in the following way: we first layer B with A in volume fractions ρ = 1 − θ 1 m 1 and 1 − ρ respectively, in the direction of lamination v 1 , to get composite C. After that, we layer C with A in volume fractions ρ = θ 2 1 − θ 1 m 1 and 1 − ρ , respectively, in the direction of lamination v 2 , and obtain a composite A * which saturates the lower Hashin-Shtrikman bound.
The following theorem summarizes the previous results.
(i) If then the optimal microstructure for which the bound (2.29) is saturated is a simple laminate with layers orthogonal to the eigenvector associated with an eigenvalue of the largest absolute value, of the extremal η in (2.29).
(ii) If then the optimal microstructure for which the bound (2.29) is saturated is a rank-2 laminate with directions of lamination given with eigenvectors v 1 and v 2 of ξ, and corresponding lamination parameters then the optimal microstructure for which the bound (2.29) is saturated is a rank-2 laminate with directions of lamination given with eigenvectors v 1 and v 2 of the extremal η in (2.29) (which are also eigenvectors of ξ), and corresponding lamination parameters
III Assume η 1 < 0 < η 2 . Similar computation as before gives us which yields when ξ 1 and ξ 2 satisfy In the case η 2 < 0 < η 1 , function F is the same as for η 1 < 0 < η 2 , as well as the obtained bound. An easy computation shows that the bound (2.31) is equivalent to (2.119) if and only if ξ 1 and ξ 2 satisfy This case corresponds to the case (ii) in the statement of the theorem.
IV If ξ 1 and ξ 2 satisfy neither of the conditions (2.118) and (2.121), then the maximum of F is attained on the line η 1 = η 2 . In this case the maximum is reached for and the upper Hashin-Shtrikman bound is given by This case corresponds to the case (iii) in the statement of the theorem.  Figure 2.5 one can see that if (ξ 1 , ξ 2 ) belongs to the blue area then condition (2.110) is satisfied and the maximum is attained in one of the cases 0 ≤ η 1 < η 2 , 0 ≤ η 2 < η 1 , η 1 < η 2 ≤ 0 or η 2 < η 1 ≤ 0 . If (ξ 1 , ξ 2 ) belongs to the yellow area, then the maximum is attained in one of the cases η 2 < 0 < η 1 or η 1 < 0 < η 2 , and the green area represents (ξ 1 , ξ 2 ) such that the maximum is attained on the line η 1 = η 2 .
It remains to explicitly describe the optimal microstructures for which the upper Hashin-Shtrikman bound is saturated. Theorem 29 assures that the upper Hashin-Shtrikman bound is optimal and that it is saturated by a sequentially laminated microstructures. The optimality condition for the maximization problem on the right-hand side of (2.113) can be derived analogously as for (2.29): where e is an extremal for h(η) given with (2.124), for this optimal η. Analogously as for the lower Hashin-Shtrikman bound, one can conclude that in this case the optimality is achieved by a rank-one laminate with the lamination direction e. This is true for every case, except for η 1 = η 2 . It remains to specify the optimal microstructure in that case. Let η := η 1 = η 2 , p = 2 and note that η = ηI 2 , where η is given with (2.122). Obviously, every unit vector is an eigenvector of η and, by Lemma 15, it is also an extremal vector for function (2.124). Thus, for the direction of lamination we can arbitrarily choose unit eigenvectors v 1 and v 2 of ξ, since η and ξ are simultaneously diagonalizable, and we conclude that (2.123) is equivalent to This determines m 1 and m 2 : and it is easy to check that m 1 +m 2 = 1 and m 1 , m 2 ≥ 0, as a consequence of the condition which defines this regime. It follows that the bound is achieved by a second rank laminate in the following way: we first layer A with B in volume fractions ρ = 1 − θ 2 m 1 and 1 − ρ respectively, using layers orthogonal to v 1 , to get composite C. After that, we layer C with B in volume fractions ρ = θ 1 1 − θ 2 m 1 and 1 − ρ , respectively, using layers orthogonal to v 2 , and obtain a composite A * , which saturates the upper Hashin-Shtrikman bound.
The following theorem summarizes the previous results.
(i) If laminate with layers orthogonal to the eigenvector associated with an eigenvalue of the least absolute value, of the extremal η in (2.31).
then the optimal microstructure for which the bound (2.31) is saturated is a simple laminate with layers orthogonal to e, such that e is an extremal for (2.124), where h is function of extremal η in (2.31).
(iii) If then the optimal microstructure for which the bound (2.31) is saturated is a rank-2 laminate with directions of lamination given with eigenvectors v 1 and v 2 of ξ, and corresponding lamination parameters

Explicit Hashin-Shtrikman bounds on the
complementary energy for mixtures of two isotropic materials in dimension d = 2 In the sequel we consider elastic composite materials obtained by mixing two wellordered isotropic phases A and B in proportions θ 1 := θ and θ 2 := 1 − θ, respectively, in an analogous way as it was done when calculating explicit Hashin-Shtrikman bounds on primal energy. The following results will be stated in dimension d = 2. In order to explicitly calculate the Hashin-Shtrikman bounds on the complementary energy, first we have to evaluate functions g A (η) and h B (η). By using that where κ 1 , κ 2 are the bulk moduli, while µ 1 , µ 2 are the shear moduli such that If we label the eigenvalues of η by η 1 and η 2 , it easily follows that the eigenvalues of Aη are equal to (2.125) while eigenvalues of Bη are given in the similar way with By using Lemma 15, for the isotropic phase A, the function g A (η) defined by where ν 1 , ν 2 are given by (2.126).
Proof. Firstly, note that the expression as well as the function g c (η) = g c (η 1 , η 2 ), depend only on the eigenvalues η 1 and η 2 of the matrix η. Accordingly, in order to explicitly compute the lower Hashin-Shtrikman bound on the complementary energy, we shall use the von Neumann result, which implies that the maximum of σ : η is obtained when η and σ are simultaneously diagonalizable and therefore equals This simplifies the problem, which is now equivalent to maximizing the concave function − 2θ 2 µ 2 (η 2 1 + η 2 2 ) − θ 2 (κ 2 − µ 2 )(η 1 + η 2 ) 2 + θ 2 h B (η 1 , η 2 ), in R 2 . Note that the function F is quadratic by parts, analogously as in the case of explicit Hashin-Shtrikman bounds on primal energy.
Due to the expression (2.130) for h B , we shall consider several cases (note that h B is differentiable everywhere except on the line ν 1 = ν 2 ).
Note that cases I and II written jointly, correspond to the case (i) given above.
In the case ν 2 < 0 < ν 1 , the function F is the same as for ν 1 < 0 < ν 2 , as well as the obtained bound. An easy computation shows that the bound (2.60) is equivalent to 2.8. Explicit Hashin-Shtrikman bounds on the complementary energy for mixtures of two isotropic materials in dimension d = 2 (2.141) is equivalent to This determines m 1 and m 2 : and it is easy to check that m 1 +m 2 = 1 and m 1 , m 2 ≥ 0, as a consequence of the condition µ 2 θ 2 (κ 2 − κ 1 )|σ 1 + σ 2 | ≥ (θ 1 µ 2 κ 1 + θ 2 µ 2 κ 2 + κ 1 κ 2 )|σ 1 − σ 2 | which defines this regime. It follows that the bound is achieved by a second rank laminate in the following way: we first layer A with B in volume fractions ρ = 1 − θ 2 m 1 and 1 − ρ respectively, using layers orthogonal to v 1 , to get composite C. After that, we layer C with B in volume fractions ρ = θ 1 1 − θ 2 m 1 and 1 − ρ , respectively, using layers orthogonal to v 2 , to get composite A * which achieves equality in the lower Hashin-Shtrikman bound on the complementary energy.
The previous results are summarized in the following theorem.
(i) If then the optimal microstructure for which the bound (2.60) is saturated is a simple laminate with layers orthogonal to the eigenvector associated with an eigenvalue of the least absolute value, of the extremal η in (2.60).
Cases I and II can jointly be written as: the bound (2.63) is equivalent to if and only if σ 1 and σ 2 satisfy Note that this corresponds to the case (i) in the statement of the theorem.
The following theorem summarizes the previous results.
(i) If then the optimal microstructure for which the bound (2.63) is saturated is a simple laminate with layers orthogonal to the eigenvector associated with an eigenvalue of the largest absolute value, of the extremal η in (2.63).