Homogenisation of thin periodic frameworks with high-contrast inclusions

We analyse a problem of two-dimensional linearised elasticity for a two-component periodic composite, where one of the components consists of disjoint soft inclusions embedded in a rigid framework. We consider the case when the contrast between the elastic properties of the framework and the inclusions, as well as the ratio between the period of the composite and the framework thickness increase as the period of the composite becomes smaller. We show that in this regime the elastic displacement converges to the solution of a special two-scale homogenised problem, where the microscopic displacement of the framework is coupled both to the slowly-varying"macroscopic"part of the solution and to the displacement of the inclusions. We prove the convergence of the spectra of the corresponding elasticity operators to the spectrum of the homogenised operator with a band-gap structure.


Introduction
The multi-scale extension of the notion of the weak L 2 -limit was proposed in [11], [2], where a general theorem about two-scale compactness of L 2 -bounded sequences was proved and a corrector-type result for the uniformly elliptic periodic homogenisation problem was established. Multi-scale convergence has proved to be an effective tool in the analysis of the behaviour of periodic composite media under minimal spatial regularity assumptions on the material properties of the composite, e.g. measurability and boundedness. Further, in problems where solutions do not converge in the strong L 2 -sense, for example in the presence of degeneracies, see e.g. [14], the related techniques have the additional benefit of capturing the multi-scale structure of the limit, by providing a suitable generalised notion of strong convergence. As opposed to the uniformly elliptic case, where the limit function only depends on the macroscopic variable and is a solution to a single boundary-value problem, the multi-scale limit for degenerate homogenisation problems satisfies a coupled system of equations for the macroscopic and microscopic parts of the limit solution. This happens to be the case for periodic "thin structures", which are the subject of the present work.
We define a thin structure as an arrangement of rods of thickness a > 0 joined together at a number of junction points ("nodes"). Fig. 1 shows an example of a thin structure, where the two panels show rods and the "singular" structure obtained by taking the mid-lines of the rods (right). In the literature, equations of elasticity on thin structures are studied by treating it as a parameter a(ε) linked to the typical rod length ε. In the context of homogenisation, the rods are often assumed to be arranged periodically with period ε, and the asymptotic behaviour of the structure is studied as ε → 0. The use of two-scale convergence for the study of periodic singular structures has been proposed in [16,4], where the two-scale approach of [11,2] was extended to the setting of general Borel measures, and conditions on the measure sufficient for passing to the two-scale limit were determined.
The use of multi-scale convergence techniques for the analysis of periodic thin structures was initiated in the work [16], which showed that if the thickness of the rods a = a(ε) is a function of the period ε of the network, such that lim ε→0 a(ε) = 0, then the overall limit behaviour of the framework depends on the asymptotics of the ratio a/ε 2 as ε → 0. In particular, in the case when lim ε→0 a/ε 2 = θ > 0, sequences of symmetric gradients of the solutions are, in general, not compact with respect to strong two-scale convergence. As a consequence, the equation describing the limit energy balance is no longer obtained by setting the test function to be the solution of the homogenised equation for the corresponding "singular structure", obtained by considering the mid-lines of the rods with the measure induced by the thin structure (cf. Fig. 1). This problem was addressed by [20], where the correct form of the energy equality was determined and the limit system of equations was derived. This study was followed by the analysis of Sobolev spaces for a variable measure [18,13], Korn inequalities for periodic frames [19], and gaps in the spectrum of the elasticity operator on a high-contrast periodic structure [21] with non-vanishing volume fraction of the components as ε → 0. In the paper [21], which can be viewed as the development of the results of [15] to the high-contrast elasticity context, the band-gap nature of the spectrum of the limit operator is analysed and the convergence of the spectra of the heterogeneous problems to the limit spectrum is proved. Notably, as was first observed by [22], the spectrum of the limit problem for a thin structure in the case of the above "critical" scaling lim ε→0 a/ε 2 = θ > 0 shows a remarkable similarity to the limit spectrum for the high-contrast, fixed-volume-fraction case of [21]. Some reasons for this similarity have been found in a recent work [6], which uses operator-theoretic tools to show that the resolvents of both models are operator-norm close to a limit Kronig-Penney model of the so-called "δ -type".
In the present work we consider a two-component periodic composite where the region occupied by the main material ("matrix") is a framework with a/ε 2 → θ > 0, and the complementary part of the space consisting of disjoint "inclusions" is filled by a less rigid material, so that the ratio between the stiffness of inclusions and the matrix is of the order O(ε 2 ). In other words, in addition to the assumption of high contrast, cf. [15], we assume that the stiff component is a thin structure so that its volume fraction is of the order O(ε). While our analysis uses some elements of both multi-scale approaches to thin structures of [16,20] and high-contrast structures of [15,21], the proofs of our results, namely homogenisation (Theorem 3.1) and spectral convergence (Theorem 4.1), require new tools that link the behaviour of solutions to the original sequence of problems with rapidly oscillating coefficients on the matrix and on the inclusions. The limit functions for the restrictions of the solutions to each of the two components are coupled together as described in Section 3.1, and lead to a homogenised system of equations of a new kind.

Problem formulation and main result
We consider a periodic rod framework ("stiff" component of the composite) filled by a different material ("soft" component). We assume that the rod thickness a > 0 is a function of the period ε > 0, and consider the regime when lim ε→0 a/ε 2 = θ > 0. The ratio of the elastic moduli of the soft and stiff component is assumed to be of the order O(ε 2 ). Denote by F h 1 the domain occupied by the scaled rods of thickness h := a/ε in the scaled structure of period Q := [0, 1] 2 and by F 1 the corresponding singular structure, also of period Q, obtained in the limit h → 0. The original rod framework is then the h d=2: The ''matrix'' and its ''skeleton''   In what follows, we consider equations of two-dimensional elasticity in R 2 . These are obtained from the full system of linearised elasticity in three dimensions when there is a direction, say x 3 , along which material properties are constant, assuming that the displacement does not depend on x 3 . At each point x ∈ R 2 , the fourth-order tensor of the elastic moduli of the medium is set to be given by where A 0 and A 1 are constant positive definite fourth-order tensors: 1 A j ξ · ξ ≥ c j ξ 2 , c j > 0, j = 0, 1, for all symmetric matrices ξ.
where H d , d = 1, 2, is the d-dimensional Hausdorff measure (see e.g. [9]), and extended to R 2 by Q-periodicity. Clearly, the weak convergence λ h λ holds as h → 0, i.e. one has 2 Similarly, for the "composite" measures µ := (1/2)dx + (1/2)λ and µ h := (1/2)dx + (1/2)λ h , where dx is the plane Lebesgue measure, one has µ h µ as h → 0. Further, we consider the "scaled" measure Define a bilinear form B h ε (·, ·) and a linear form L h ε (·) by (1.2) Notice that B h ε is coercive and continuous, and L h ε is continuous on It is a consequence of the Lax-Milgram lemma (see e.g. [8,Chapter 6]) that (1.1) has a unique solution u h ε . In what follows we aim to describe the structure of the limit problem for the weak two-scale limit of the function u h ε as ε → 0.
Our main result, Theorem 3.1, states that in the case when a/ε 2 → θ > 0 as ε → 0, the solutions u h ε to the problems (1.1), where h = a/ε, converge is an appropriate two-scale sense (see Section 2) to a function u(x, y), x ∈ Ω, y ∈ Q, whose trace on F 1 ∩ Q has the form (1.5) and satisfies an equation involving F 1 -transversal components of the y-gradient of the function u = u(x, y). In addition, the function u(x, ·) − u 0 (x) =: U(x, ·), x ∈ Ω, belongs to the space H 1 per (Q) 2 a.e. x ∈ Ω and satisfies an elliptic equation that couples its values to the solution u 0 of (1.6), where the average Q u(·, y)dλ(y) is replaced by Q u(·, y)dµ(y). More precisely, for each link I of the network F 1 , let τ and ν be unit tangent and normal vectors that form a positively orientated system. Then all vectors v ∈ R 2 are written as In Section 3 the vectors U and χ are shown to satisfy a system of equations of the form where A 0 is a second-order differential operator in Q expressed in terms of the tensor A 0 only, L τ is a fourth-order differential operator in the "longitudinal" direction τ , and T ν is a first-order differential operator in the "transverse" direction ν corresponding to each link I.

Two-scale structure of solution sequences
In this section we establish the structure of various two-scale limits on the soft and stiff components. This is achieved by taking the limits, as ε → 0, of the integrals entering the identity (1.1), with suitably chosen test functions ϕ.

Two-scale convergence: definition and properties
We first recall the notion of weak and strong two-scale convergence and their basic properties, see [16]. Within this section d = 2 or d = 3, and the measure sequence µ h ε ("limit" measure µ) can be replaced by the sequence λ h ε ("limit" measure λ).
compact with respect to weak two-scale convergence.
We say that a function u = u(x, y) ∈ The next proposition shows that the converse also holds. Proposition 2.4. For any arbitrary a ∈ L ∞ (Q), the weak (resp. strong) two-scale convergence of u h ε to u(x, y) implies the weak (resp. strong) two-scale convergence of a(·/ε)u h ε to a(y)u(x, y).

Two-scale compactness of solutions to (1.1)
Consider the equation (1.1) with ϕ = u h ε : Using ellipticity estimates on the left-hand side and the inequality 2ab ≤ a 2 + b 2 , a, b ∈ R, on the right-hand side yields where c 0 , c 1 are the ellipticity constants of A 0 , A 1 . Hence, the following a priori bounds hold.

Rigid displacements, potential and solenoidal matrices
We denote the set of periodic rigid displacements by R, omitting the reference to the measure λ.
We assume (see e.g. [16] for relevant examples of periodic frameworks) that any u ∈ R has a unique representation u(y) = c + χ(y), y ∈ Q, where c ∈ R 2 and χ is a periodic transverse displacement, i.e. on each link of the singular network F 1 it is orthogonal to the link. Thus R is the direct sum of R 2 and the set of transverse displacements, which we denote by R. The next definition characterises transverse displacements that occur in the study of rod networks with a/ε 2 → θ > 0 as ε → 0.
(C2) The first derivative along the link is continuous across each node: The norm in R 0 is defined to be the sum of the H 2 -norms of χ · ν over all the links.
Definition 2.5. For a given Borel measure κ on Q, we define the space V κ pot of κ-potential matrices as the closure of the set e(u) Denoting by V κ sol the set of κ-solenoidal matrices, we can write (see e.g. [16]) L 2 per (Q, dκ) 3 = V κ pot ⊕V κ sol . It follows that the orthogonal decomposition L 2 (Ω×Q, dx×dκ) 3 = L 2 (Ω, V κ pot )⊕L 2 (Ω, V κ sol ) holds, where the two-scale L 2 -spaces of κ-potential and κ-solenoidal vector fields are the closures of the linear spans of matrices we(u), w ∈ C ∞ 0 (Ω), u ∈ C ∞ per (Q) 2 and wv, w ∈ C ∞ 0 (Ω), v ∈ V κ sol , with respect to the norm of L 2 (Ω × Q, dx × dκ) 3 . When κ is the Lebesgue measure on Q, we simply write

Convergence on the stiff component
We first study the relationship between the limit functions u(x, y) and u(x, y), see Section 2.2.
1. We say that the sequence ψ h ε weakly converges to ψ ∈ L 2 per (Q, dµ) 2 , and write ψ h where the test function ξ is extended to R 2 by Q-periodicity.
2. We say that ψ h ε strongly converge to a function ψ ∈ L 2 per (Q, dµ) holds. Notice further that, by the Hölder inequality, one has The weak two-scale convergence of u h ε and the relation (2.7) imply that The claim now follows by choosing an approximation sequence ζ = ζ k such that ζ k → ψ in L 2 per (Q, dµ) 2 .
Theorem 2.1. The function u is the trace of u on F 1 , in the sense that u(x, y) = u(x, y) a.e. x ∈ Ω, λ-a.e. y ∈ F 1 .
Proof. For all functions ψ ∈ L 2 per (Q, dλ) 2 and h > 0, we define where the summation is carried out over all links I j of F 1 ∩Q, and for each link I j we set ψ h j (y) = ψ(y * ) whenever y is in the h-neighboughood of I j and |y−y * | = dist(y, I j ), y * ∈ I j , and ψ h j (y) = 0 otherwise. Notice that for all ϕ ∈ C ∞ 0 (Ω) one has Due to the fact that [ψ] h ε → µ h ε ψ, the following convergence holds: Similarly, for the first integral on the right-hand side of (2.9), we obtain It follows that the limits in (2.10) and (2.11) coincide, as required.
The description of the structure of the two-scale limit of χ ε,h 1 u h ε is a consequence of several statements proved in [20]. Combining this with Theorem 2.1, we obtain the following result (cf. [20, Theorem 3.1]). Theorem 2.3. In the formula u(x, y) = u 0 (x) + χ(x, y), the transverse displacement χ is an element of the space L 2 (Ω, R 0 ).
Proof. For each δ > 0, consider a C ∞ -domain Q δ such that F 2δ 0 ∩ Q ⊂ Q δ ⊂ F δ 0 ∩ Q and the set where n is the unit normal to ∂Q δ . For all b ∈ X δ , a = div b in Q δ , consider the functions extended to R 2 by Q-periodicity. Then for sufficiently small ε > 0 (recall that h → 0 as ε → 0) the following identity holds: (2.14) Passing to the limit in the last identity as ε → 0 and using the fact that a, b vanish in Q \ Q δ , we obtain As ϕ ∈ C ∞ 0 (Ω) is arbitrary, it follows that u(x, y) · a(y) dy a.e. x ∈ Ω.
(2.15) Taking divergence-free fields b ∈ X δ in (2.15) we infer (see e.g. [7]) the existence of v ∈ L 2 (Ω, H 1 (Q δ )) 2 such that p(x, y) = e y v(x, y) , y ∈ Q δ , which implies Using the density in L 2 (Q δ ) 2 of vector functions a having the above representation implies that v(x, y) and u(x, y) differ by a constant for y ∈ Q δ , hence p = e y (v) = e y (u), a.e. y ∈ Q δ . By virtue of the arbitrary choice of the parameter δ, we conclude that p = e y (u) for a.e. y ∈ Q.

Homogenisation theorem
In what follows, we consider the case of the framework F 1 shown in Fig. 2 ("model framework"). However, the analysis presented is readily extended to any framework such that the representation (2.6) for periodic rigid displacements, with obvious modifications in the statements.

Homogenised system of equations
Definition 3.1. We denote by V the energy space consisting of vectors For a given f ∈ L 2 (Ω × Q, dx × dµ) 2 , we refer to u ∈ V as the solution of the homogenised problem where A hom λ is given by (2.12), and K 1 is a function on the network F 1 ∩ Q, defined on each link of the network by where τ is the tangent to the current link, and the prime denotes the tangential derivative, as in Definition 2.4, e.g. χ := (τ · ∇χ)ν. The identity (3.1) is equivalent to a system of partial differential equations, which is obtained by considering various classes of test functions in (3.1). First, taking functions of the form ϕ(x, y) where the angle brackets denote microscopic averaging, e.g. U := Q U(·, y)dµ(y).
Further, we set ϕ(x, y) = ϕ(x)Ψ(y), with ϕ ∈ C ∞ 0 (Ω), Ψ ∈ V , where the space V consists of functions in H 1 per (Q) 2 whose trace on F 1 ∩ Q coincides with a rigid-body motion λ-a.e. Assume for simplicity that the tensor A 0 is isotropic, i.e. for all ξ ∈ Sym 2 one has A 0 ξ = 2M 0 ξ + L 0 (tr ξ)I, with M 0 , L 0 > 0, Taking first functions Ψ ∈ C ∞ 0 (F 0 ∩ Q) 2 we obtain where P H is the orthogonal projection operator from L 2 (Ω × Q, dx × dµ) 2 onto V. Finally, taking arbitrary Ψ ∈ V yields additional equations coupling the framework F 1 ∩Q and the inclusion component F 0 ∩ Q. For example, on those links that are parallel to the y 2 -axis, we obtain For a general periodic framework F 1 , on each link there is a positively orientated pair of vectors τ , ν with τ pointing along the link and ν orthogonal to the link. The corresponding version of the equation (3.5) on each link of F 1 is as follows: where ∂ τ , ∂ ν denote differentiation along the link and in the direction normal to the link, and the superscript (ν) is understood in the sense of the notation introduced at the end of Section 1.

Extension theorem
Before proving the main result, we recall the description of a class of functions that extend periodic rigid displacements in R 0 on the framework F 1 to the rod network F h 1 , introduced in [20]. The following two statements are proved in [20].
Proposition 3.1. The set D is dense in the space R 0 with the respect to the norm of L 2 per (Q, dλ) 2 .
Proposition 3.2. For each g ∈ D, there exists a smooth extension g h = g h (y) to the network F h 1 with the following properties: 1. For each node O k of F 1 ∩ Q, the symmetric gradient e y (g h ) is zero in B δ k (O k ) for some δ k > 0.

For each h > 0 and for each link
where τ , ν are the unit tangent and normal to the link I, I h is the h-neighbourhood of I, O is either of the two end-points of I, and the union is taken over all nodes of F 1 ∩ Q. Consider also the "network-to-rod extension" (g · ν) K 1 h of the function (g · ν) K 1 , as in the second formula in (2.8).

Convergence of solutions
Suppose that G ∈ C ∞ per (Q) 2 , g ∈ D are such that G(y) = g(y) for all y ∈ F 1 ∩ Q. We approximate where g h is the extension described in Proposition 3.2. This is achieved, e.g., by setting and χ h is the convolution of the characteristic function of the set F Proof. Note first that since G, G h , g h are smooth and therefore their L 2 -norms on F h 1 and F 2h Further, since G h − G = (ḡ h − G)(1 − χ h ) and by the same argument as above one has To this end, notice that ∇χ h = O(h −1 ), and since G = g h on F 1 ∩ Q one has uniformly in y. It follows that from which the claim follows.
We denote the four terms on the left-hand side of (3.10) by I j (ε), j = 1, 2, 3, 4. It follows from the L 2boundedness of the sequence εe(u h ε ) and the fact that A 1 e(u 0 (x)) + v(x, y) is pointwise orthogonal to the matrix g(y) ⊗ ∇w(x), for y ∈ F 1 ∩ Q, x ∈ Ω, (see [15,Lemma 5.3]) that the terms I 4 (ε) and I 2 (ε) converge to zero as ε → 0. The convergence results on the soft component discussed in Section 2.5 imply that The following statement is a consequence of [20,Lemma 3.5].
holds, where σ h is the function defined by (3.7).
Taking into account the asymptotics (3.8), it follows from the above proposition that the convergence holds. Finally, passing to the limit in (3.10) as ε → 0 we obtain Adding together the identities (3.9) and (3.13) and denoting ϕ(x, y) = ϕ 0 (x)+Φ(x, y), the homogenised formulation (3.1) follows. In order to prove the strong convergence of solutions when f h ε 2 → f , consider another version of problem (1.1) with right-hand sides g h (3.14) Setting ϕ = u h ε in the above, ϕ = v h ε in the original problem (1.1) with f = f h ε , and then subtracting one from the other yields where where v solves the homogenised equation with the right-hand side g.
Finally, in order to show the convergence of energies, we set ϕ = u h ε in (1.1) with f = f h ε and use the definition of strong two-scale convergence as well as the identity (3.1), as follows:

Convergence of spectra
Here we establish the convergence of the spectra of the operators associated with (1.1) to the spectrum given by the limit problem (3.1).

Spectrum of the limit operator
Consider the bilinear forms (cf. where the space V is defined in Section 3.1, see the paragraph preceding (3.3). The spectral problem associated with (3.1) can be written in the form Let {φ n } n∈N ⊂ V be an orthonormal set of eigenvectors with non-zero average for the bilinear form b micro with corresponding set of eigenvalues {ω n } n∈N : Assuming that the value s is outside the spectrum Sp(b micro ) of the form b micro , the function U(x, y) is written as a series in terms of eigenfunctions {φ n } n∈N : Substituting this expansion for U into (4.3), we obtain (4.6) Versions of the function β appear in the study of scalar [15] and vector ( [14], [21], [22]) homogenisation problems. The following statement is a straightforward modification of a result in [22].  Proof. Suppose that s belongs to the right-hand side of (4.8). We argue that the problem has a solution for every f ∈ H, given that s satisfies the required assumptions of the lemma. Since s / ∈ Sp(b micro ), it follows that U can be written in the form (4.5) with u 0 replaced by su 0 +f . Substituting this into the first equation of (4.9) yields (4.10) Since all eigenvalues of β(s) are in ρ(b macro ), the operator induced by the bilinear form on the left-hand side of (4.10) is invertible and thus the identity (4.10) has a unique solution.
Conversely, one has ρ(A) ⊂ ρ(b micro ) and if s ∈ ρ(A) then β(s) has no eigenvalues in Sp(b macro ), for otherwise the problem (4.9) would not be uniquely solvable for any f ∈ H.
In the case of the model framework, the matrix β is proportional to the identity matrix I. Indeed, if the set F 1 ∩ Q is invariant with respect to a rotation R, i.e. one has Ry : y ∈ F 1 ∩ Q = F 1 ∩ Q, then for an eigenfunction φ of the bilinear form b micro , the vector Rφ is an eigenvector with the same eigenvalue, hence one has Rβ(s)R −1 = β(s), in view of the definition of β, see (4.6). Taking R to be the rotation through π/2 yields the required claim, namely β(s) = b(s)I for a scalar function b. Let {γ n } n∈N denote the increasing sequence of zeros of the function b and let {δ n } n∈N be the increasing sequence of all eigenvalues in the set {ω n } n∈N , counting multiple eigenvalues only once. The spectrum of the limit operator A has the "band" form:

Proof of spectral convergence
Here we show that the spectra of the original problems converge to the spectrum of the limit problem (3.1).
Definition 4.1. We say that a sequence of sets X ε ⊂ R, ε > 0, converges in the sense of Hausdorff to X ⊂ R if the following two statements hold: (H1) For each ω ∈ X , there exists a sequence ω ε ∈ X ε such that ω ε → ω; (H2) For all sequences ω ε ∈ X ε such that ω ε → ω ∈ R, it follows that ω ∈ X .