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1 This is an Open Access document downloaded from ORCA, Cardiff University's institutional repository: This is the author s version of a work that was submitted to / accepted for publication. Citation for final published version: Cherdantsev, Mikhail, Cherednichenko, Kirill and Neukamm, Stefan 7. High contrast homogenisation in nonlinear elasticity under small loads. Asymptotic Analysis 4 -, pp /ASY-743 file Publishers page: < Please note: Changes made as a result of publishing processes such as copy-editing, formatting and page numbers may not be reflected in this version. For the definitive version of this publication, please refer to the published source. You are advised to consult the publisher s version if you wish to cite this paper. This version is being made available in accordance with publisher policies. See for usage policies. Copyright and moral rights for publications made available in ORCA are retained by the copyright holders.

2 High contrast homogenisation in nonlinear elasticity under small loads Mikhail Cherdantsev, Kirill Cherednichenko, and Stefan Neukamm 3 School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF4 4AG, UK Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA 7AY, UK 3 TU Dresden, Fachrichtung Mathematik, Institut für Wissenschaftliches, Rechnen, D-6 Dresden, Germany June 9, 7 Abstract We study the homogenisation of geometrically nonlinear elastic composites with high contrast. The composites we analyse consist of a perforated matrix material, which we call the stiff material, and a soft material that fills the remaining pores. We assume that the pores are of size < ε and are periodically distributed with period ε. We also assume that the stiffness of the soft material degenerates with rate ε γ, γ >, so that the contrast between the two materials becomes infinite as ε. We study the homogenisation limit ε in a low energy regime, where the displacement of the stiff component is infinitesimally small. We derive an effective two-scale model, which, depending on the scaling of the energy, is either a quadratic functional or a partially quadratic functional that still allows for large strains in the soft inclusions. In the latter case, averaging out the small scale-term justifies a single-scale model for high-contrast materials, which features a non-linear and non-monotone effect describing a coupling between microscopic and the effective macroscopic displacements. Keywords: High-contrast homogenisation; Nonlinear elasticity; Two-scale Γ-convergence. Contents Introduction. Notation Geometric and constitutive setup 8 3 Main results 9 3. Compactness and two-scale convergence Convergence in the small strain regime m ε = oε γ Convergence in the finite strain regime m ε = ε γ

3 4 Proofs 5 4. Proofs of Lemma, Lemma, and Lemma 3: a priori estimate, compactness and approximation Proof of Theorems, and Propositions, : small strain regime Proof of Theorem 3: finite strain regime Introduction We consider a geometrically nonlinear elastic composite material that consists of a stiff matrix material and periodically distributed pores filled by a soft material: for ε > and a fixed scaling parameter γ > we consider the energy functional of non-linear elasticity I ε u := ε γ W uχ ε +W u χ ε dx f ε udx, u H. Here denotes a Lipschitz domain in R d the reference domain of the elastic body and u : R d is a deformation satisfying clamped boundary conditions: ux = x on. We denote by f ε : R d the density of the applied body forces, W and W are frameindifferent, non-degenerate energy densities see Section below for the precise assumptions, and χ ε denotes the indicator function of the pores, i.e. of the domain occupied by the soft material component. As will be made precise in Section, we assume that the pores are of size ε and are periodically distributed in the interior of with period ε. As can be seen from, in the homogenisation limit ε the stiffness of the soft material degenerates with rate ε γ γ >, while the stiffness of the stiff material remains unchanged. Hence, the contrast between the soft material occupying the pores and the stiff material occupying the perforated matrix becomes infinite in the limit ε. We therefore refer to the corresponding limit procedure as high-contrast homogenisation. Our goal is to identify the effective behaviour of the minimisation problem associated with I ε by studying its limit under a proper rescaling. Summary and discussion of our result. To illustrate our result, here in the introduction we restrict ourselves to the special case γ =. If we assume that the density of the body forces is small in magnitude, in the sense that f ε = ε α f for some α, and has vanishing first moment, i.e. fx xdx =, then can be expressed as Iε α ϕ := ε αi εu = ε α W I +ε α ϕχ ε + ε αw I +ε α ϕ χ ε dx f ϕdx, where ϕx = ux x ε α, x, ϕ H ;R d, 3 denotes the scaled displacement, and I stands for the identity matrix in R d d. In this paper we analyse the asymptotics of the minimisation problem associated with I α ε in the limit ε by appealing to the concept of Γ-convergence. The latter goes back to De Giorgi e.g. see [9] for a standard reference. In a metric setting it is defined as follows: Definition Γ-convergence. Let X, d denote a metric space. A sequence of functionals I ε : X [, ] Γ-converges to a functional I : X [, ], if

4 a lower bound. For every x X and every x ε x in X we have liminfi ε x ε I x. ε b recovery sequence. For every x X there exists a sequence x ε x in X such that lim ε I εx ε = I x. In that case we call I the Γ-limit of the sequence I ε. A fundamental property of Γ-convergence is the following fact: If a sequence of functionals Γ-converges and the functionals are equicoercive, then the associated sequence of minima resp. minimisers converge up to a subsequence to the minimum resp. a minimiser of the Γ-limit, and any minimiser of the Γ-limit can be obtained as a limit of a minimizing sequence of the original functionals. Thanks to this property, Γ-convergence is especially useful for the study of the asymptotics of parametrised minimisation problems. The Γ-limit, if it exists, is unique; yet, the question weather a sequence of functionals Γ-converges or not, and the form of the Γ-limit depend on the topology of X. In particular, a sequence of functionals is more likely to Γ-converge in a stronger topology, while it is more likely to be equicoercive in a weaker topology. Therefore, it is natural to consider the strongest notion of convergence on X for which the functionals remain equicoercive. In the situation we are interested in, namely the asymptotics of the functionals Iε α defined on X = H, it turns out that due to the presence of high-contrast, a natural and appropriate notion of convergence on H is a variant of two-scale convergence that we discuss next. Let us first recall the standard notion of two-scale convergence from [3] and []: Definition. We say that a sequence f ε L weakly two-scale converges to f L Y if the sequence f ε is bounded and f ε xϕx,x/εdx = fx,yϕx,ydxdy lim for all ϕ Cc,C # Y, where C# Y denotes the space of Y :=,d -periodic functions in C R d. We say that a sequence f ε L, strongly two-scale converges to f L Y, if the sequence f ε weakly two-scale converges to f and one has f ε L f L Y as ε. For vector-valued functions two-scale convergence is defined component-wise. In the study of the asumptotics of Iε α we work with a variant of this definition that is tailor made to capture the effects of high-contrast. For γ = it can be summarised as follows for details and the general case see Section 3.: Given a sequence of displacements ϕ ε in H we consider the unique decomposition ϕ ε = gε + gε into a harmonic function gε H ε and a remainder gε H. We then write ϕ ε g,g and say that ϕ ε converges to a pair g,g with g L ;H Y and g H, if g ε g weakly in H, and y g weakly two-scale. The component g of a limit pair g,g describes gε g, ε gε the scaled macroscopic displacement of the body, and g is a two-scale function describing the scaled microscopic displacement on the pores relative to the deformed matrix material. As a main result see Theorem and Theorem 3 we prove in fact in a slightly more general situation that Iε α Γ-converges w.r.t. the type of convergence introduced above. It turns out that two different regimes emerge for α > and α =. In the small strain regime α >, the strain ε α ϕ becomes infinitesimally small in the entire domain, and the limit behaviour is expressed by a linearised, two-scale energy Y 3

5 I small : g L ;H Y H ;Rd R, I small g,g := Q y g x,y dy +Q hom g x dx 4 Y g x,ydy +g x fxdx. Y 5 Here Q and Q are the quadratic forms of the quadratic expansions of W and W at the identity, and Q hom denotes the homogenised energy density obtained from Q, see and 3 for details. The functional I small is the two-scale Γ-limit of the sequence I α ε in the sense that cf. Theorem : a lower bound. For every g,g and every ϕ ε g,g we have liminf I small g,g. I α ε ϕ ε b recovery sequence. For every g,g there exists a sequence ϕ ε g,g such that lim I α ε ϕ ε I small g,g. In addition, we prove that the functional I α ε ε> are equicoercive and deduce convergence of the associated minimisation problems see Theorem and Proposition. In Theorem we establish a two-scale expansion showing that if ϕ ε is an almost minimiser of I α ε, then ϕ ε x g x+εψx,x/ε+g x,x/ε, 6 where g,g is a minimiser of I small, and ψ denotes a corrector function that only depends on g. Finally, we illustrate that by averaging out the fast variable y, the limit I small can be further simplified. In fact, Proposition shows that Iε α Γ-converges w.r.t. weak convergence in L to the functional Īsmall : L R {+ } given by { } Ī small ϕ := min Q hom g +QG : g H, G L with g +G = ϕ f ϕdx, 7 with a positive definite quadratic form Q : R d [, defined by { QG := min Q g y } dy : g HY, g dy = G. Y Y Q captures the influence of the pores and their geometry on the effective behavior. The minimiser ϕ L to Īsmall takes the form ϕ = g +G with G := Y g,ydy. In view of 6 the field G can be interpreted as the gap between the macroscopic displacement and the microscopic displacements in the pores. In the finite strain regime, which corresponds to α =, the displacement gradient ε ϕ ε becomes infinitesimally small only in the stiff component, while large strains still may occur in the soft pores. Therefore, the Γ-limit is a non-convex partially linearised functional of the form I finite g,g := QW I + y g x,y dy +Q hom g x dx Y g x,ydy +g x fxdx, Y 4

6 where QW denotes the quasiconvex envelope of W. Similarly to the small strain regime, one can average out the fast scale y and obtain Γ-convergence of Iε to the functional Īfinite : L R {+ } given by { Ī finite ϕ := min f ϕdx, Q hom g +VG : g H, G L with g +G = ϕ with non-convex potential V : R d [, defined by { VG := min QW I + g y } dy : g HY, g dy = G, Y Y } 8 see Remark. In contrast to the small strain regime, where Q is quadratic, the potential V is non-convex and expresses a nonlinear and non-monotone coupling between the macroscopic and microscopic displacement. Connection to acoustic wave propagation in high-contrast materials. The sequence of functionals ε α I ε, in either of the two regimes described above, occupies an intermediate position between a fully nonlinearly elastic composite and fully linearised models, as ε. Notably, linear models with high contrast, which are suitable for the description of small displacement fields that often occur, say, in acoustic wave propagation already exhibit a coupling between the macroscopic part g and microscopic part g of the minimiser of I small, which in our case is obtained as a limit in the small-strain regime α >. This can be seen by considering the time-harmonic solitons to the equations of elastodynamics with the elastic part of the energy given by 4, away from the sources of the elastic motion. In this case the function f in 5 which in our analysis we assume to be independent of the fast variable x/ε for simplicity, an assumption that can be relaxed with no changes in the proofs needed has to be replaced by the sum g + g, with the integration in 5 carried over Y and Q at the same time, i.e. the work of the external forces 5 is replaced by the expression for the work of self-forces ω g x,y+g x g x,y+g x dydx, Y where ω is the frequency. The solution to the Euler-Lagrange equation for the resulting functional is a coupled system of equations for g, g, so that when the equation for g is solved in terms of g and substituted into the second equation, it takes the form away from the sources: A hom g = ω βω g, 9 for some non-negative self-adjoint differential operator A hom and a special nonlinear function β, which takes positive and negative values on alternating intervals of the real axis leading to lacunae, or band gaps in the spectrum of the corresponding operator and is obtained from the spectral decomposition of g and the subsequent averaging over Y, see [38]. From this point of view, the non-quadratic finite-strain functional I finite is a matching, partially quadratic, homogenised model corresponding, e.g., to finite-amplitude, rather than smallamplitude, wave motions that can no longer be treated using a quadratic model such as I small but can still be used in place of models of nonlinear elasticity where the elastic energy terms on both components of the composite stiff and soft are non-quadratic. 5

7 Methods and previous results. In this paper we appeal to analytic methods that have been developed in the last two decades in the areas of nonlinear elasticity and homogenisation. Among these are the notion of two-scale convergence introduced in [3], [] and periodic unfolding see [8] and references therein. The convergence statements of our main results are expressed in the language of Γ-convergence see [9] and references therein. In order to treat the geometric nonlinearity of the considered functional, we make use of the geometric rigidity estimate see [3]. Since we consider a low energy regime, linearisation and homogenisation take place at the same time. The simultaneous treatment of both effects is inspired by recent works [8], [9], [3], [3], [5] of the third author, where various problems involving simultaneous homogenisation, linearisation and dimension reduction are studied. The homogenisation of the kind of high-contrast composites that we study is related to the homogenisation for periodically perforated domains e.g. see [33], []. For instance, we make use of extensions across the pores. As a side result we prove a version of the geometric rigidity estimate for perforated domains see Lemma 4 below. We would like to remark that while the present work is one of the few papers, along with [7], [8], that treat the fully nonlinear high-contrast case, during the last decade there has been a significant amount of literature devoted to the mathematical analysis of phenomena associated with, or modelled by, a high degree of contrast between the properties of the materials constituting a composite, in the linearised setting. The first contributions in this direction are due to Zhikov [38], and Bouchitté and Felbacq [6], following an earlier paper by Allaire [] and the collection of papers by Hornung et al. [4] see also the references therein, where the special role of high-contrast elliptic PDE was pointed out albeit not studied in detail. These works demonstrated that the behaviour of the field variable in such models is of a two-scale type in the homogenisation limit, i.e. the limit model cannot be reduced to a one-scale formulation and fields that depend on the fast variable remain in the effective model. They also noticed that the spectrum of such materials has a band-gap structure, as in 9, and indicated how this fact could be exploited for high-resolution imaging and cloaking. It has since been an adopted approach to the theoretical construction of negative refraction media, or more generally metamaterials, which is now a hugely popular area of research in physics see e.g. [34] and references therein. On the analytical side, a number of further works followed, in particular [3], [4], [], [3], [4], [6], [6], [5], [35], [7], [9], where various consequences of high contrast or, mathematically speaking, the property of non-uniform ellipticity in the underlying equations have been explored. Among these are the non-locality and micro-torsion effects in materials with high-contrast inclusions in the shape of fibres extending in one or more directions, the partial band-gap wave propagation due to the high degree of anisotropy of one of the constituent media, and the localisation of energy in high-contrast media with a defect photonic crystal fibres, all of which can be thought of as examples of non-standard, or non-classical, behaviour in composites, which is not available in the usual moderate-contrast materials. In the present paper we aim to develop further a rigorous high-contrast theory in the context of finite elasticity, where the underlying model is nonlinear. With this paper we continue the multiscale theme initiated in[7], where the regime of large deformation gradients in the soft component of the composite was considered. Let us emphasise two points that contrast our contribution to some earlier work within the related field. First, we note that, apart from [7], [8], a number of other articles e.g. [5], [], [7] have treated high-contrast periodic composites in the nonlinear context. However, the related results are of limited relevance to nonlinear elasticity, due to the convexity or monotonicity assumptions made in these works. In the present paper we study a class of functionals subject to the requirement of material fame indifference see assumption W in Section, which makes 6

8 our analysis fit the fully nonlinear elasticity framework, as opposed to the works mentioned. Second, as was discussed above, the analysis of composites with soft inclusions within a stiff matrix cannot be reduced to a decoupled model where the perforated medium obtained by removing the inclusions is considered first and the displacement within the inclusions is found independently, which from the physics perspective can be viewed as a kind of resonance phenomenon; cf. 9 in the linearisation regime, for which an inherent energy coupling, in the limit as ε, between the soft and stiff components of the composite is essential. On a related note, the proof of the key compactness statement Lemma involves the simultaneous analysis of the displacements on the two components. We would also like to highlight the fact that in [7] the order of the relative scaling of the displacements on the soft and stiff components of the composite are assumed from the outset, while in the present work it is the result of the above compactness argument itself. Organisation of the paper. In Section we state the assumptions on the geometry of the composite and the material law. In Section 3 we present the main results, starting with results regarding two-scale compactness, convergence results in the small strain regime and finally the convergence result in the finite strain regime. All proofs are presented in Section 4.. Notation Here we list some notation that we use throughout the text. Additional items will be introduced whenever they are first used in the text. d is the integer dimension of the space occupied by the material. p is the exponent in the notation L p for a Lebesgue space. Y :=, d the reference period cell; Y is an open Lipschitz set whose closure is contained in Y, and Y := Y \Y., ε and ε denote the reference domains of the composite, the set occupied by the pore material, and the domain occupied by the matrix material, respectively, see Section for precise definition. Unless stated otherwise, all function spaces L, H, H, etc. consist of functions taking values in R d. Function spaces whose notation contains subscript c consist of functions that vanish outside a compact set. The function spaces H #, H Y, and AY are introduced in Section 3.. We write and : for the canonical inner products in R d and R d d, respectively. SOd denotes the set of rotations in R d d. stands for up to a multiplicative constant that only depends on d, Y,, and on p if applicable. 7

9 Geometric and constitutive setup The pore geometry. The set Y defined above describes the pores contained within the cell Y. Note that Y is an open, bounded, connected set with Lipschitz boundary. Therefore, to each ϕ H Y we can associate see e.g. [33] a unique harmonic extension g H Y characterised by g = ϕ in Y, g : ζdy = ζ HY. Y For this extension the inequality g L Y C ϕ L Y holds with a constant C that only depends on Y. For a given domain R d and ε >, we define the sets ε and ε as follows: ε := } {εξ +Y ξ Z d, εξ +Y, ε := \ ε. Note that by construction ε is a Lipschitz domain. In particular, it is connected and ε. We denote by χ ε the indicator function of the set of pores: {, x ε, χ ε x :=, x R d \ ε. The composite. The two materials are described by energy densities W i : R d d [,+ ], i =,. Unless stated otherwise, we assume that for i =, : W W i is frame-indifferent, i.e. W i RF = W i F for all R SOd and all F R d d ; W The identity matrix I R d d is a natural state, i.e. W i I =, and W i is nondegenerate, i.e. for all W i F c dist F,SOd, F R d d, c >. W3 W i has a quadratic expansion at I, i.e. there exists a non-negative quadratic form Q i on R d d and an increasing function r i : [, [, ] with lim s r i s =, such that W i I +G Q i G G r i G G R d d. As shown in [3, Lemma.7] the quadratic form Q i associated with W i via W3 satisfies c symg Q i G = Q i symg c symg G R d d, c >. 3 In the finite strain regime we consider a different set of assumptions for W, which are listed in Section 3.3. The scaling parameter γ. Throughout the paper γ > denotes a fixed scaling parameter. It is a quantitative measure of the relative contrast between the two components of the composite. Energy functional. We define the elastic energy as a functional of the displacement, as follows: E ε u = ε γ W I + uχ ε +W I + u χ ε dx, u H,R d. 4 8

10 3 Main results 3. Compactness and two-scale convergence We first present an a priori estimate and a two-scale compactness statement for sequences ϕ ε H whose energy is equi-bounded in the sense that where Φ γ εv := lim supφ γ εv ε <, 5 dist I + vx,sod ε γ χ ε + χ ε dx. Notethat, byvirtueofthenon-degeneracyassumptionwthefunctionalφ γ ε boundsbelow E ε id+, where idx = x, x. As we shall see in the upcoming Lemma, the inequality 5 implies that the sequence ϕ ε is bounded in H, and thus weakly converges up to extracting a subsequence to a limit displacement ϕ H. For our purpose we require a precise understanding of the oscillations that emerge along that limit. We achieve this by combining two concepts: We write a representation for ϕ ε in the spirit of an asymptotic decomposition as ε. We study the convergence properties of the terms in this decomposition by appealing to two-scale convergence. In the following lemma we address the first item above. Lemma. Let ϕ ε H and < ε. a There exists a unique pair of functions gε H ε and gε H such that i ϕ ε = gε +ε γ gε, ii gε : ζ = ζ H ε. ε 6 b There exists a positive constant C that only depends on,y such that g ε L + ε g ε L + g ε H CΦγ εϕ ε, 7 where g ε, g ε and ϕ ε are related to each other as in a. As already explained in the introduction, for our purpose it is convenient to appeal to two-scale convergence, see Definition. We use the following shorthand notation: f ε f : f ε strongly converges to f in L, f ε f : f ε weakly converges to f in L, f ε f : fε weakly two-scale converges to f in L Y, f ε f : fε strongly two-scale converges to f in L Y. The upcoming lemma states a two-scale compactness result for the displacements g ε and g ε that appear in the representation 6. Due to the differential constraint satisfied by g ε, the corresponding two-scale limits automatically satisfy certain structural properties, which can be captured with the help of the following function spaces: 9

11 H # is the space of [,d -periodic functions in H loc Rd. H Y is the closed subspace of H # consisting of functions ψ H # with ψ = on Y. AY is the closed subspace of H# consisting of functions ψ H # identity y ψ : y ζdy = ζ HY. Y that satisfy the Lemma. Consider a sequence ϕ ε H and let g ε,g ε be associated with ϕ ε via 6. Suppose that there exists a sequence of positive numbers m ε such that Then there exist lim supm ε Φ γ εm ε ϕ ε < and m ε = Oε γ. g L,H Y, g H, ψ L,AY 8 such that, up to selecting a subsequence, one has g ε g, ε g ε y g, g ε g weakly in H and g ε g + y ψ. 9 The identification obtained in the previous lemma is sharp, in the sense of the following statement. Lemma 3. Let g L,H Y, g H and ψ L,AY. Let c ε be an arbitrary sequence of positive numbers converging to zero. Then there exist function sequences g ε H ε, g ε H such that g ε,g ε is related to ϕ ε := g ε +ε γ g ε as in 6, and g ε g, ε g ε y g, gε g weakly in H and gε g + y ψ, lim supc ϕε L ε =. Our main result is formulated in terms of the notion of convergence described in the above lemmas. For convenience we use the following notation: Given ϕ ε H we write gε +ε γ gε functions are related to ϕ ε as in 6. 6 := ϕ ε, if g ε H, g ε H ε, and both We write ϕ ε g,g, if g ε +ε γ g ε 6 := ϕ ε and g ε g, ε g ε y g, g ε g weakly in H. We write ϕ ε g,g, if g ε +ε γ g ε 6 := ϕ ε and g ε g, ε g ε y g, g ε g weakly in H.

12 3. Convergence in the small strain regime m ε = oε γ Throughout this section we assume that the densities W and W satisfy the conditions W W3. We show that in the small strain regime the limit functional is given by where E small : L,H Y H [,, E small g,g := Y More precisely, the following theorem holds. Q y g x,y +Q hom g x dx Q hom F := min Q F + y ψy dy. 3 ψ AY Y Theorem. Let m ε be a sequence of positive numbers and assume that m ε = oε γ as ε. a Compactness. Suppose that ϕ ε H satisfy lim supm ε E ε m ε ϕ ε <. Then, up to a subsequence, one has ϕ ε g,g for some g L,H Y and g H. b Lower bound. Consider ϕ ε H and suppose that ϕ ε L,H Y and g H. Then the estimate holds. lim inf m ε E ε m ε ϕ ε E small g,g g,g for some g c Recovery sequence. For all g L,H Y and g H there exists a sequence ϕ ε H such that ϕ ε g,g and limm ε E ε m ε ϕ ε = E small g,g. In the next result we consider a minimisation problem that involves the density of the body forces l ε L. We study the variational limit of the scaled total energy I ε ϕ := m E ε m ε ϕ l ε m ε ϕdx 4 ε where the scaling factor m ε is determined by the body forces via m ε := ε γ l ε L ε + l ε L. 5a In the small strain regime we assume that the body forces are small in the sense that m ε = oε γ, ε. 5b

13 Moreover, we assume that the scaled body-force densities converge, as ε, in the following way: m ε ε γ χ ε l ε l, m ε l ε l. 5c It follows from Theorem that the variational limit of the total energy 4 is given by the functional I small g,g := E small g,g l g dy +l g dx. 6 Y Proposition. Assume that 5a 5c hold. b Convergence of infima. One has lim inf I ε ϕ = mini small g,g, ϕ H where the minimum on the right-hand side is taken over all g L,H Y and g H. Moreover, the minimum is attained for a unique pair g,g. b Convergence of minimisers. Let ϕ ε H be a sequence of almost minimisers, i.e. Then I ε ϕ ε inf ϕ H I ε ϕ+o, ε. 7 ϕ ε g,g and g ε g + y ψ where ψ L,AY denotes the unique corrector characterised by Q hom g x = Q g x+ y ψ x,y dy, Y for almost every x. Next, we prove that almost minimisers ϕ ε satisfy the asymptotic relation Y ψ x,ydy =, 8 ϕ ε = g,εx+ε γ g,εx+o, ε, in W,p p <, 9 where g,ε and g,ε formally obey the ansatz g,εx formally = g x,x/ε, g,εx formally = g x+εψ x,x/ε. 3 Here g,g and ψ denote the minimising pair and corrector from Proposition. Since the functionsontheright-handsidesin3areingeneralnotsmoothenoughtodefineg,ε andg,ε by 3 directly, we use instead the approximation associated with g,g,ψ via Lemma 3. In addition to the properties of Y assumed in Section, we require the following assumption on the regularity of Y : Assumption. There exist an exponent p < and a constant C such that for all ϕ H Y and g H Y related via we have g L p Y C ϕ L p Y. Note that Assumption is satisfied if Y can be written as the disjoint union of a finite number of Lipschitz domains Y,...Y N with Y i Yj = Ø for i j.

14 Theorem. Assume that 5a 5c hold, and let Assumption be satisfied. Let ϕ ε H be a sequence of almost minimisers, i.e. I ε ϕ ε inf ϕ H I ε ϕ+o, ε. Let g,g be the minimiser of I small and let ψ be defined through 8. Let g,ε,g,ε and ϕ,ε = g,ε + ε γ g,ε be associated with g,g,ψ as in Lemma 3, i.e. ϕ,ε g,g and g,ε g + y ψ. Then for gε +ε γ gε gε g,ε L p ε + ε gε ε g,ε 6 := ϕ ε one has L p ε + gε g,ε W,p as ε. 3 Remark. To illustrate the result of Theorem, consider the case γ = with l ε := m ε lx, x ε, where lx,y is smooth both in x and y and is periodic in y. If the domain and the pore set Y are sufficiently regular, the minimisers g,g and ψ are smooth, by the classical elliptic regularity theory, see e.g. []. In that case we may set g,εx := g x,x/ε and g,ε := g x+εψ x,x/ε, and the asymptotic formula for ϕ ε reads where R ε W,p as ε. ϕ ε = g x,x/ε+g x+εψ x,x/ε+r ε x, In the remainder of this section, we restrict to the special case of the introduction in the small strain regime, i.e. we assume that α >, γ =, m ε = ε α and l ε = ε α f for some f L, so that the functionals I α ε and I ε defined in and 4 are identical. Hence, Theorem and Proposition prove two-scale Γ-convergence of I α ε and the convergence of the associated minimisation problems as claimed in the introduction, and Theorem yields the two-scale expansion 6. We argue that the functionals I α ε Γ-converge to the single scale limit Ī small : Proposition. For α > and f L consider Iε α and Īsmall defined in and 7. We extend Iε α to a functional on L by setting Iε α := + on L \H. Then: a Compactness. Suppose that ϕ ε L satisfy lim supiε α ϕ ε <. Then, up to a subsequence, we have ϕ ε ϕ weakly in L. b Lower bound. For every ϕ ε L with ϕ ε ϕ weakly in L we have lim inf I α ε ϕ ε Īsmallϕ c Upper bound. For every ϕ L we can find ϕ ε ϕ weakly in L such that lim I α ε ϕ ε = Īsmallϕ 3

15 d Convergence of the minimisation problem. Let ϕ ε H denote an infimizing sequence of Iε α, i.e. Iε α ϕ ε inf ϕ L Iα ε ϕ+o, ε. Let g,g denote the unique minimiser of I small and ϕ denote the unique minimiser of Ī small. Then inf L Iε α min L Ī, ϕ ε ϕ weakly in L, and ϕ = g + g dy, Ī small ϕ = I small g,g. Y 3.3 Convergence in the finite strain regime m ε = ε γ Throughout this section we assume that W satisfies the conditions W W3. W : R d d [, is continuous and satisfies the growth condition c dist F,SOd W F c + F F R d d, 3a and the local Lipschitz condition W F +G W F c + F + G G F,G R d d. 3b We prove that in the finite strain regime the limit functional E finite : L,H Y H [, is given by E finite g,g := QW I + y g x,y dydx+ Y Q hom g x dx, where QW denotes the quasiconvex envelope of W see e.g. []. The associated limit of the total energy I ε, see 4, is given by cf. 6 I finite g,g := E finite g,g l g dy +l g dx, Y where l, l are defined in the same way as in 5c. Theorem 3. a Lower bound. Consider a sequence ϕ ε H and the associated decomposition ε γ gε +gε 6 := ϕ ε. If ϕ ε g,g and gε g, then lim inf ε γ E ε ε γ ϕ ε E finite g,g. b Recovery sequence. For any g L,H Y and g H there exists a sequence ϕ ε H such that ϕ ε g,g and limε γ E ε ε γ ϕ ε = E finite g,g. 4

16 c Suppose that the force densities l ε L satisfy 5a and 5c with m ε = ε γ. Then the infima converge, i.e. lim inf I ε ϕ = infi finite g,g, 33 ϕ H where the infimum on the right-hand side is taken over all functions g L,H Y and g H. Moreover, there exist a minimising pair g,g and a recovery sequence ϕ ε H with ϕ ε g,g such that I ε ϕ ε I finite g,g = mini finite g,g as ε. 34 Remark Example in Section. If we consider Theorem 3 and Proposition in the case γ =, m ε = ε and l ε = εf for some f L, then we recover the special case in the finite strain regime presented in the introduction. In particular, we deduce that the functionals I α ε two-scale Γ-converge in the sense of Theorem 3 to I finite. Arguing as in the small-strain regime, cf. Proposition, we deduce that I α ε Γ-converges with respect to the weak topology in L to Īfinite, cf. 8. We leave the details to the readers. 4 Proofs We start by proving the auxiliary results discussed in Section 3.. Sections 4. and 4.3 contain the proofs of the main statements in the small strain and finite strain cases, respectively. 4. Proofs of Lemma, Lemma, and Lemma 3: a priori estimate, compactness and approximation A key ingredient in the proof of Lemma is the geometric rigidity estimate by Friesecke et al. [3]: Theorem 4 Geometric rigidity estimate, see [3]. Let U be an open, bounded Lipschitz domain in R d, d. There exists a constant CU with the following property: for each v H U there is a rotation R SOd such that vx R dx CU dist vx,sod dx. U Moreover, the constant CU is invariant under uniform scaling of U. In fact, we need the following modified version, which is adapted to perforated domains. Lemma 4. There exists a constant C > that only depends on and Y such that for all ε > and v H satisfying v : ζdx = ζ H ε, 35 the estimates ε U dist v,sod L C dist v,sod L ε, 36 v R L ε C dist v,sod L ε, 37 hold for some R SOd, which may depend on v. In addition, if vx = x+c on for some constant c, then we may set R = I. 5

17 Proof of Lemma 4. Step. The proof of the inequality 36. Let ε := { εξ +Y ξ Z d, εξ +Y } denote the union of ε-cells that are completely contained in. Since \ ε ε, it suffices to prove 36 for replaced by ε, respectively. In fact we shall prove the following stronger estimate: for all ξ Z d with εξ +Y we have dist v,sod dx dist v,sod dx. 38 εξ+y εξ+y For the argument fix an admissible ξ Z d. Application of Theorem 4 with U = εξ + Y yields a rotation R SOd such that v R dx dist v,sod dx. 39 εξ+y εξ+y Note that the multiplicative constant in the estimate above only depends on Y, since εξ+y is a dilation and translation of Y. On the other hand, since εξ +Y x vx Rx is harmonic, we have cf. : dist v,sod dx v R v R. εξ+y Combined with 39, inequality 38 follows. εξ+y εξ+y Step. The proof of the rigidity estimate 37. From 36 and Theorem 4 applied with U = we deduce that for some R SOd: v R L dist v,sod L ε, 4 which in particular implies 37. Finally we argue that one can set R = I, if v = x+c on. In view of 4, it suffices to show that v I dx v R dx for all R SOd. This inequality can be seen as follows: Consider ϕx := vx x c and note that ϕ vanishes on, so that v I dx = ϕ dx ϕ + I R dx = ϕ+i R dx = v R dx, which in fact holds for an arbitrary matrix R. We are now in position to present the proofs of Lemmas and. Proof of Lemma. In the following, the symbol stands for up to a multiplicative constant that only depends on Y and. Step. Existence of the decomposition 6 and derivation of the estimate for gε. Let gε denote the unique function in H characterised by gε = ϕ ε in ε and 6 ii. Since ε is Lipschitz, we deduce that gε := ε γ ϕ ε gε H ε. This proves the existence of the decomposition. We claim that gε dx ϕ ε dx = gε dx. 4 ε ε Since ε is defined as the union of the sets εξ+y with ξ Z ε := {ξ Z d : εξ+y }, it suffices to prove εξ+y g ε dx εξ+y ϕ ε dx. The latter follows from by a 6 ε

18 scaling argument, since the rescaled functions y ϕ ε εξ +y and y g ε εξ +y satisfy. Next, we prove 7. Consider v ε x := x+gεx and note that v ε x satisfies 35. Hence, 4 and Lemma 4 yield gε dx g ε dx v ε I 37 C dist I + ϕ ε x,sod dx Φ γ εϕ ε. ε ε 4 Since gε vanishes on the boundary of, the estimate upgrades by Poincaré s inequality to H CΦγ εϕ ε. g ε Step. Derivation of the estimate for g ε. Since we have an improved Poincaré inequality see e.g. [4, Lemma.6]: g H ε : g L ε ε g L ε, 43 it suffices to prove ε g L ε Φγ εϕ ε. ε To this end, notice that since ϕ ε vanishes on the boundary of, we have ϕ ε L = min I + ϕ ε R Theorem 4 L R SOd dist I + ϕ ε x,sod dx ε γ Φ γ εϕ ε. 44 Thanks to the first identity in 6, we get by triangle inequality: ε γ gε L ε ϕ ε L + gε L. Combined with 4 and 44 we finally get ε gε L ε = εγ ε γ gε L ε Φγ εϕ ε. Proof of Lemma. Step. A priori estimate and basic compactness. From Lemma we deduce that lim sup gε L + ε gε L + gε C limsupm H ε Φ γ εm ε ϕ ε <. 45 Hence, by standard results concerning two-scale convergence cf. [, Proposition.4] and [36, Proposition 4.], there exist g H, ψ L,H # and g L,H # such that, up to a subsequence, one has g ε g weakly in H, g ε g + y ψ, g ε g, ε g ε y g. Step. The proof of the inclusion ψ L,AY. By a density argument, it suffices to show that Y y ψx,y : y ζ xζ y dxdy = 46 7

19 for all scalar functions ζ C c, and all ζ C c Y. To this end, we identify ζ with its unique Y-periodic extension to R d that vanishes on Y, and set Thanks to 6 we have ζ ε x := εζ xζ x/ε, x. g ε : ζ ε dx =. As can be easily checked, we have ζ ε ζ x y ζ y, so that = lim gε : ζ ε dx = g x+ y ψx,y : ζ x y ζ y dxdy Y = y ψx,y : ζ x y ζ y dxdy, Y where the last identity holds thanks to the periodicity of ζ. This proves 46. Step 3. The proof of the inclusion g L,H Y. By a density argument, it suffices to show that g x,y ζ xζ y dxdy = 47 Y forallscalarfunctionsζ C c andallζ H # withζ = ony. Wearguebyconsidering the function ζ ε x := ζ xζ x ε, x, the support of which is contained in ε for ε. Since ζ ε ζ xζy, and since gε is supported in ε, we deduce that = lim g εx ζ ε xdx = g x,y ζ xζ y dxdy. This completes the argument. In the proof of Lemma 3 we appeal to the construction of a diagonal sequence that is due to Attouch, see []: Lemma 5. For any h : [, [,+ ], there exists a mapping, ε δε, such that limδε = and limsup Y h ε,δε limsup δ lim suphε,δ. Proof of Lemma 3. Step. Characterisation of strong two-scale convergence via unfolding. For f ε : R and f : Y R define d ε f ε,f := fε ε x/ε +εy fx,y dydx R d Y where f ε denotes the extension by zero of f ε to R d, f denotes the extension by zero of f to R d Y, and z denotes the unique element in Z d with z z [, d. We recall from [36] that f ε f dε f ε,f. 48 8

20 The characterisation extends in the obvious way to vector-valued functions. Step. Construction of gε. We claim that there exists a sequence gε in H whose elements satisfy 6ii and gε g weakly in H, gε g + y ψ, limsupc ε g ε L =. 49 Indeed, by a density argument there exist g,δ Cc, δ,, and ψ Cc,C # Y such that g,δ g H + y ψ δ y ψ L Y δ δ. For ε >, δ,, define g,δ ε x := g,δ x+εψ δ x,x/ε, and set d δ ε := d ε g,δ ε, g + y ψ + g,δ ε g L +c ε g,δ ε L. By construction, we have lim δ limsup d δ ε =, and Lemma 5 yields a function ε δε with lim d δε ε =. In view of Step, this implies that the diagonal sequence g ε := gε,δε satisfies 49. Now, for each ε >, let gε denote the function satisfying 6ii and such that gε = g ε on ε. To conclude the argument, we only need to show that gε satisfies 49. Consider the difference η ε := g ε gε. Since η ε is bounded in H and η ε = in ε, we have η ε in H, and, up to a subsequence, η ε y ϕ for some ϕ L,H Y. On the other hand, since gε satisfies 6 ii and g ε satisfies 49, we deduce that η ε L = η ε : η ε dx = g ε : η ε dx g + y ψ : y ϕdxdy. Y ε ε Since g is independent of y, and because ψ L,AY, the integral on the right-hand side vanishes. Hence, η ε L, and thus g ε satisfies 49. Step 3. Conclusion. As can be shown by appealing to a combination of a density argument and a diagonalsequence argument, similar to Step, there exists a sequence g ε H ε such that g ε g, ε g ε y g, limsup c ε ε γ gε L =. Now define ϕ ε x := ε γ g ε+g ε, and note that g ε,g ε satisfy 6. In view of the convergence of g ε and g ε, the sequence ϕ ε has the required properties. 4. Proof of Theorems, and Propositions, : small strain regime As a preliminary remark, we note that two different effects play a role when passing to the limit ε in the small strain regime: The non-convex energy functional is linearised at identity map which is a stress-free state for E ε this corresponds to the passage from nonlinear to linearised elasticity. The obtained linearised, still oscillating, convex-quadratic energy is homogenised. 9

21 The following lemma is used to treat both effects simultaneously. Its proof combines convex homogenisation methods e.g. [37, Proposition.3] with a careful Taylor expansion in the spirit of [3, Proof of Theorem 6.]. For notational convenience, we introduce two linearised functionals: For G ε = G ε,g ε L,R d d L,R d d set Q ε G ε = Q ε G ε,g ε := Q G εx dx+ ε ε Q G εx dx. For G = G,G L Y,R d d L Y,R d d set QG = QG,G := Q G x,y dxdy + Q G x,y dxdy. Y Y Lemma 6. Consider sequences gε,g ε H that satisfy lim sup ε gε L + g L ε <. Set ϕ ε := ε γ g ε +g ε and G ε = G ε,g ε := ε g ε, g ε. a If G ε G and mε = oε γ as ε, then lim inf m ε E ε m ε ϕ ε liminfq ε θ ε G ε QG, 5 where θ ε : {,} is defined by {, if ϕ ε m ε ε γ /, θ ε x :=, otherwise. b If G ε G, mε = oε γ as ε, and lim sup m ε ϕ ε L =, 5 then limm ε E ε m ε ϕ ε = QG. Remark 3. In Lemma 6, following [3], the function θ ε is introduced, in order to truncate the peaks of G ε. This is needed for exploiting the quadratic expansion W3. Since both ε g ε and g ε are assumed to be bounded sequences in L, we deduce, from the definition of θ ε, the fact that m ε ε γ = o and the Chebyshev inequality, that r < θ ε L r, and θ ε L. 5 In the proof of Lemma 6 we need to pass to the limit in products of the form f ε θ ε, where θ ε satisfies 5, or f ε χ ε, where χ ε denotes the indicator of ε. This is done by appealing to the next two lemmas, the proofs of which are elementary and left to the reader.

22 Lemma 7. Let f ε,θ ε be sequences in L and assume that θ ε satisfies 5, then the following implications are valid: lim sup f ε L < θ ε f ε f ε L p p <, f ε f θ ε f ε f, f ε f θε f ε f, { } fε f fε θ ε f ε f ε L equi-integrable. Lemma 8. Suppose that f ε be a sequence in L and, as above, let χ ε denote the set indicator function of ε. Then the following implications hold: f ε f χε f ε χyfx,y, f ε f χε f ε χyfx,y, where χ denotes the indicator function of Y. Proof of Lemma 6. Step. Linearisation. We claim that the following statement holds for i =,: Let F ε denote a sequence in L,R d d, and let c ε be a sequence of positive numbers converging to zero, such that Then the convergence lim c ε lim supc ε F ε L =. 53 ε W i I +c ε F ε dx Q i F ε x dx = 54 holds. Indeed, thanks to W3 we have c ε W i I +c ε F ε Q i F ε F ε r i c ε F ε F ε r i c ε F ε L Thanks to 53, and since F ε is bounded in L, the right-hand side converges to zero in L, and 54 follows. Step. Proof of part a. Since the energy densities W, W are minimised at the identity, cf. W, we have m ε E ε ϕ ε m ε ε γ W I+m ε ε γ θ ε Fεx dx+m ε W I+m ε θ ε Fεx dx, 55 where Fε := χ ε G ε +ε γ G ε, F ε x := χ ε G ε. 56 Thanks to the definition of θ ε we have m ε ε γ θ ε Fε L + m ε θ ε Fε L, so that we may apply 54 to the right-hand side in 55. We get lim inf m ε E ε ϕ ε liminfq ε θε Fε,θ ε F ε = liminf Q ε θε G ε,θ ε G ε, a.e.

23 where for the last identity we used the facts that Fε = G ε on ε and Fε G ε L ε = ε γ gε L ε. It remains to argue that lim inf Q ε θ ε G ε QG. In order to show this, notice that Q ε θ ε G ε = Q θ ε χ ε G ε dx+ Q θ ε χ ε G ε dx. 57 From G ε G we deduce, using Lemma 7, Remark 3 and Lemma 8, that θ ε χ ε G ε χyg x,y, θ ε χ ε G ε χy G x,y. By appealing to the lower semicontinuity of convex integral functionals with respect to weak two-scale convergence cf. [37, Proposition.3], we deduce that the lim inf of the right-hand side in 57 is bounded below by QG. This completes the argument. Step 3. Proof of part b. We claim that Note that lim m ε m ε E ε ϕ ε = m ε ε γ ε WE ε m ε ϕ ε Q ε ε gε, g ε =. 58 W I +m ε ε γ Fεx dx+m ε W I +m ε Fεx dx L + m ε Fε L where Fε and Fε are defined in 56. By 5 we have m ε ε γ Fε and 54 yields m ε E ε m ε ϕ ε Q ε F ε,fε as ε. Since G ε G we have, thanks to Lemma 8: χ ε F ε χyg x,y, χ ε F ε χy G x,y. Hence, the continuity of convex integral functionals with respect to strong two-scale convergence cf. [36] yields Q ε F ε,f ε QG, which completes the argument. We are now in a position to prove the Γ-convergence statement for the energies E ε. Proof of Theorem. Step. Part a Compactness. Thanks to W we have m ε E ε m ε ϕ ε c m ε Φ γ εm ε ϕ ε. Hence, the claim of Theorem a directly follows from Lemma. Step. Part b Lower bound.

24 Without loss of generality we assume that lim inf m ε E ε m ε ϕ ε = limsup m ε E ε m ε ϕ ε <. Furthermore, thanks to Lemma, we can assume in addition that gε ψ L,AY, so that G ε := ε g ε, g ε y g, g + y ψ =: G. Applying Lemma 6a yields lim infm ε E ε m ε ϕ ε Q y g x,y dxdy + Y g + y ψ for some Y Q g x+ y ψx,y dxdy. This completes the argument, since the right-hand side is bounded from below by E small g,g. Step 3. Part c Upper bound. Choose ψ L,AY such that Q g x+ y ψx,y dxdy = Y Q hom g x dx. 59 Let ϕ ε denote the sequence associated with g,g and ψ via Lemma 3 with c ε := m ε. In view of, applying Lemma 6 b yields limm ε E ε m ε ϕ ε = Q y g x,y dxdy + Q g x+ y ψx,y dxdy. Y Y It follows from 59 that the right-hand side equals E small g,g. Proof of Proposition. Step. A priori estimate. We claim that for every sequence ϕ ε in H the following implication holds: lim sup Indeed, we have l ε m ε ϕ ε dx m ε l ε L I ε ϕ ε < limsupm ε E ε m ε ϕ ε <. 6 g ε 5a m ε m ε gε L + m ε gε W m ε m ε E ε m ε ϕ ε. Combining this with the definition of I ε we get which implies 6. Step. The proof of parts a and b. m ε E ε m ε ϕ ε I ε ϕ ε + L +ε γ l ε L ε gε 3 L m ε E ε ϕ ε, L ε 7 m ε m ε φ γ εm ε ϕ ε

25 The existence of a minimiser to I small follows by the direct method. The minimiser g,g is unique, since the implication Q y g x,y dxdy + Q hom g x dx = g = g = Y holds for all g L,H Y and g H. The remaining claims of Proposition follow from the standard Γ-convergence arguments cf. [9, Corollary 7.], provided the functionals I ε, ε >, are equi-coercive and Γ-converge to I small. Indeed, thanks to 5c, it is easy to check that ϕ ε g,g implies lim m l ε m ε ϕ ε dx = l g dy +l g dx, 6 ε Y since the integral on the left-hand side only involves products of weakly and strongly two-scale convergent factors, cf. [36, Proposition.8]. In combination with Theorem, this implies that I ε Γ-converges to I small. In addition, the trivial inequality inf I ε ϕ I ε =, ϕ H combined with 6 and Lemma, proves that the functionals I ε are equi-coercive. For the proof of Theorem we make use of the following lemma: Lemma 9 Decomposition Lemma, see [], [7]. Let v H and let v ε H be a sequence with v ε v in H. Then there exists a sequence V ε H such that the following properties hold for a subsequence of v ε not relabelled: a V ε v in H ; b V ε = v ε in a neighbourhood of ; c V ε is equi-integrable; d {x : v ε x V ε x} as ε. Proof of Theorem. It suffices to prove the theorem for a subsequence. Throughout the proof we write G ε := ε g ε, g ε, G := y g, g + y ψ. Furthermore, we make use of the functionals Q ε and Q introduced at the beginning of Section 4.. Recall that E small g,g = QG. Step. Convergence of ϕ ε and of the corresponding energy values. We claim that, as ε, one has G ε G, 6 I ε ϕ ε I small g,g, 63 m ε E ε m ε ϕ ε E small g,g. 64 4

26 Indeed, from Proposition we immediately deduce that ϕ ε g,g and 63. Furthermore, in view of the continuity of the loading term, cf. 6, this implies 64. For 6, it remains to argue that gε g + y ψ. Thanks to ϕ ε g,g and Lemma we have, up to a subsequence, gε g + y ψ for some ψ L,AY. Furthermore, from 64 and Lemma 6 a we infer that E small g,g = liminfm ε E ε m ε ϕ ε Q y g dxdy + Q g + y ψ dxdy. Y Y This, in particular, implies Q g + y ψ dxdy = Y In view of 8 we conclude that ψ = ψ and 6 follows. Q hom g dx. Step. Equi-integrable decomposition. We claim that for a subsequence not relabelled there exist sequences ḡε,g ε H such that G ε := ε G ε, G ε satisfies G ε G ε, Gε G ε L p, 65 and Q ε G ε E small g,g. 66 To show the above, notice that thanks to Lemma 9 there exist sequences G ε,g ε H such that ε G ε and ḡ ε are equi-integrable, the indicator function θ ε defined by {, if G θ ε x := εx = gεx and G εx = gεx,, otherwise, 67 satisfies 5. Since G ε G ε = θ ε G ε G ε and p <, the convergence 65 follows from the boundedness of the sequence G ε G ε in L, Lemma 7, and Hölder s inequality. We prove 66. Thanks to 6 we have G ε G, so that due to the lower semicontinuity of convex integral functionals with respect to weak two-scale convergence, cf. [37, Proposition.3]: lim infq ε G ε QG = E small g,g. Hence, for 66 it suffices to prove the opposite estimate, i.e. limsup Q ε G ε E small G, which, thanks to 5 and 64, follows from lim sup Qε G ε Q ε θ ε G ε. 68 5

27 In order to show 68 notice that since the supports of θ ε and θ ε are disjoint, and because θ ε θ ε G ε = θ ε θ ε G ε cf. 67, an expansion of the squares yields Q ε G ε Q ε θ ε G ε = Q ε θ ε G ε +Q ε θε G ε Qε θ ε θ ε G ε Q ε θε θ ε G ε = Q ε θε θ ε G ε +Qε θε G ε Qε θε θ ε G ε Q ε θε θ ε G ε +Qε θε G ε. It is easy to check that θ ε θ ε and θ ε converge to zero in L r for all r <. Hence, since G ε isequi-integrable, Lemma7impliesthattheright-handsideofthepreviousestimate converges to zero, and 68 follows. Step 3. Error estimate. We claim that εsym ḡε g,ε dx+ ε ε sym ḡ ε g,ε dx as ε. 69 For the argument set G,ε := ε g,ε, g,ε. In view of 3 it suffices to argue that The latter can be seen as follows: We have Q ε G ε G,ε as ε. Q ε G ε G,ε = Q ε G ε Q ε G,ε +B ε G,ε ;G,ε G ε, where B ε denotes the bilinear form associated with Q ε. The difference of the two quadratic terms on the right-hand side converges to zero, since G,ε is associated with a recovery sequence, and thanks to 66. On the other hand, since G,ε strongly two-scale converges, and G,ε Ḡε by 65, we deduce that limb ε G,ε ;G,ε G ε =, as B ε G,ε ;G,ε G ε only involves products between a weakly and a strongly two-scale convergent factor cf. [36, Proposition.8]. Step 4. Conclusion Proof of 3. We split the estimate into gε g,ε p + ε gε ε g p,ε dx as ε, 7 gε g,ε p + gε g,ε p dx as ε. 7 Thanks to 65 and Step 3 we have εsym gε g,ε p dx+ ε ε sym g ε g,ε p dx as ε. 7 Argument for 7: Set ηε := gε g,ε. Since ηε H ε H, Korn s inequality yields ηε p dx C sym ηε p dx = C sym ηε p dx, 6 ε