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High contrast homogenisation in nonlinear elasticity under small loads

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 0<ε1 and are periodically distributed with period ε. We also assume that the stiffness of the soft material degenerates with rate ε2γ, γ>0, so that the contrast between the two materials becomes infinite as ε0. We study the homogenisation limit ε0 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.

1.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>0 and a fixed scaling parameter γ>0 we consider the energy functional of non-linear elasticity

(1)Iε(u):=Ω(ε2γW0(u)χε+W1(u)(1χε))dxΩfε·udx,uH1(Ω).
Here Ω denotes a Lipschitz domain in Rd (the reference domain of the elastic body) and u:ΩRd is a deformation satisfying clamped boundary conditions: u(x)=x on Ω. We denote by fε:ΩRd the density of the applied body forces, W0 and W1 are frame-indifferent, non-degenerate energy densities (see Section 2 below for the precise assumptions), and χε 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 2, we assume that the pores are of size ε and are periodically distributed in the interior of Ω with period ε. As can be seen from (1), in the homogenisation limit ε0 the stiffness of the “soft” material degenerates with rate ε2γ (γ>0), 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 ε0. 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 Iε 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=1. If we assume that the density of the body forces is small in magnitude, in the sense that fε=εαf for some α1, and has vanishing first moment, i.e. Ωf(x)·xdx=0, then (1) can be expressed as

Iεα(φ):=1ε2αIε(u)(2)=Ω(1ε2(α1)W0(I+εαφ)χε+1ε2αW1(I+εαφ)(1χε))dxΩf·φdx,
where
(3)φ(x)=u(x)xεα,xΩ,φH01(Ω;Rd),
denotes the scaled displacement, and I stands for the identity matrix in Rd×d. In this paper we analyse the asymptotics of the minimisation problem associated with Iεα in the limit ε0 by appealing to the concept of Γ-convergence. The latter goes back to De Giorgi (e.g. see [20] for a standard reference). In a metric setting it is defined as follows:

Definition 1

Definition 1(Γ-convergence).

Let (X,d) denote a metric space. A sequence of functionals Iε:X[,] Γ-converges to a functional I:X[,], if

  • (a) (lower bound). For every x0X and every xεx0 in X we have lim infε0Iε(xε)I0(x0).

  • (b) (recovery sequence). For every x0X there exists a sequence xεx0 in X such that limε0Iε(xε)=I0(x0).

In that case we call I0 the Γ-limit of the sequence Iε.

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 whether 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 Iεα defined on X=H01(Ω), it turns out that due to the presence of high-contrast, a natural and appropriate notion of convergence on H01(Ω) is a variant of two-scale convergence that we discuss next. Let us first recall the standard notion of two-scale convergence from [32] and [1]:

Definition 2.

We say that a sequence fεL2(Ω) weakly two-scale converges to fL2(Ω×Y) if the sequence fε is bounded and

limε0Ωfε(x)φ(x,x/ε)dx=Ω×Yf(x,y)φ(x,y)dxdy
for all φL2(Ω,C#(Y)), where C#(Y) denotes the Banach space of continuous, Y:=(0,1)d-periodic functions on Rd. We say that a sequence fεL2(Ω) strongly two-scale converges to fL2(Ω×Y) if the sequence fε weakly two-scale converges to f and one has fεL2(Ω)fL2(Ω×Y) as ε0. For vector-valued functions two-scale convergence is defined component-wise.

In the study of the asymptotics of Iεα we work with a variant of this definition that is tailor-made to capture the effects of high-contrast. For γ=1 it can be summarised as follows (for details and the general case see Section 3.1): Given a sequence of displacements φε in H01(Ω) we consider the unique decomposition φε=gε0+gε1 into a function gε0H01(Ωε0) and a contribution gε1H01(Ω) that is harmonic in Ωε0. We then write φε2(g0,g1) and say that φε converges to a pair (g0,g1) with g0L2(Ω;H01(Y0)) and g1H01(Ω), if gε1g1 weakly in H1(Ω), and gε02g0, εgε02yg0 weakly two-scale. The component g1 of a limit pair (g0,g1) describes the (scaled) macroscopic displacement of the body, and g0 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 1 and Theorem 3) we prove (in fact in a slightly more general situation) that Iεα Γ-converges with respect to the type of convergence introduced above. It turns out that two different regimes emerge for α>1 and α=1.

In the small strain regime (α>1), the strain εαφ becomes infinitesimally small in the entire domain Ω, and the limit behaviour is expressed by a linearised, two-scale energy Ismall:L2(Ω;H01(Y0))×H01(Ω;Rd)R,

(4)Ismall(g0,g1):=Ω×YQ0(yg0(x,y))dy+Qhom1(g1(x))dx(5)Ω(Y0g0(x,y)dy+g1(x))·f(x)dx.
Here Q0 and Q1 are the quadratic forms of the quadratic expansions of W0 and W1 at the identity, and Qhom1 denotes the homogenised energy density obtained from Q1, see (12) and (23) for details. The functional Ismall is the two-scale Γ-limit of the sequence Iεα in the sense that (cf. Theorem 1):
  • (a) (lower bound). For every (g0,g1) and every φε2(g0,g1) we have lim infε0Iεα(φε)Ismall(g0,g1).

  • (b) (recovery sequence). For every (g0,g1) there exists a sequence φε2(g0,g1) such that limε0Iεα(φε)Ismall(g0,g1).

In addition, we prove that the functionals (Iεα)ε>0 are equicoercive and deduce convergence of the associated minimisation problems (see Theorem 1 and Proposition 1). In Theorem 2 we establish a two-scale expansion showing that if φε is an (almost) minimiser of Iεα, then
(6)φε(x)g1(x)+εψ(x,x/ε)+g0(x,x/ε),
where (g0,g1) is a minimiser of Ismall, and ψ denotes a corrector function that only depends on g1. Finally, we illustrate that by averaging out the fast variable y the limit Ismall can be further simplified. In fact, Proposition 2 shows that Iεα Γ-converges (with respect to weak convergence in L2(Ω)) to the functional I¯small:L2(Ω)R{+} given by
I¯small(φ):=min{ΩQhom1(g1)+Q(G):g1H01(Ω),GL2(Ω) with g1+G=φ}(7)Ωf·φdx,
with a (positive definite) quadratic form Q:Rd[0,) defined by
Q(G):=min{Y0Q0(g0(y))dy:g0H01(Y0),Y0g0dy=G}.
Q captures the influence of the pores (and their geometry) on the effective behavior. The minimiser φL2(Ω) to I¯small takes the form φ=g1+G with G:=Y0g0(·,y)dy. 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=1, the displacement gradient εφε 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

Ifinite(g0,g1):=Ω×YQW0(I+yg0(x,y))dy+Qhom1(g1(x))dxΩ(Y0g0(x,y)dy+g1(x))·f(x)dx,
where QW0 denotes the quasiconvex envelope of W0. Similarly to the small strain regime, one can average out the fast scale y and obtain Γ-convergence of Iε0 to the functional I¯finite:L2(Ω)R{+} given by
(8)I¯finite(φ):=min{ΩQhom1(g1)+V(G):g1H01(Ω),GL2(Ω) with g1+G=φ}Ωf·φdx,
with non-convex potential V:Rd[0,) defined by
V(G):=min{Y0QW0(I+g0(y))dy:g0H01(Y0),Y0g0dy=G},
see Remark 2. 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 ε2αIε, in either of the two regimes described above, occupies an intermediate position between a fully nonlinearly elastic composite and fully linearised models, as ε0. 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 g0 and microscopic part g1 of the minimiser of Ismall, which in our case is obtained as a limit in the small-strain regime α>1. 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 g0+g1, with the integration in (5) carried over Y0 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”

ω2ΩY0(g0(x,y)+g1(x))·(g0(x,y)+g1(x))dydx,
where ω is the frequency. The solution to the Euler–Lagrange equation for the resulting functional is a coupled system of equations for g0, g1, so that when the equation for g0 is solved in terms of g1 and substituted into the second equation, it takes the form (away from the sources):
(9)Ahomg1=ω2β(ω2)g1,
for some non-negative self-adjoint differential operator Ahom 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 g0 and the subsequent averaging over Y0, see [38]. From this point of view, the non-quadratic finite-strain functional Ifinite is a “matching”, “partially quadratic”, homogenised model corresponding, e.g., to finite-amplitude, rather than small-amplitude, wave motions that can no longer be treated using a quadratic model such as Ismall 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.

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 [1,32] and periodic unfolding (see [18] and references therein). The convergence statements of our main results are expressed in the language of Γ-convergence (see [20] and references therein). In order to treat the geometric nonlinearity of the considered functional, we make use of the geometric rigidity estimate (see [23]). 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 [24,2831] 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 [8,33]). 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 [9,16], 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 [1] and the collection of papers by Hornung [25] (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 [3,4,7,10,11,1315,17,26,35], 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 [16], 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 [9,16], a number of other articles (e.g. [5,7,12]) 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 (W1) in Section 2), which makes 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 ε0, between the soft and stiff components of the composite is essential. On a related note, the proof of the key compactness statement (Lemma 1) involves the simultaneous analysis of the displacements on the two components. We would also like to highlight the fact that in [16] 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 2 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.

1.1.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.

  • d2 is the (integer) dimension of the space occupied by the material.

  • p1 is the exponent in the notation Lp for a Lebesgue space.

  • Y:=(0,1)d the reference period cell; Y0 is an open Lipschitz set whose closure is contained in Y, and Y1:=YY0.

  • Ω, Ωε0 and Ωε1 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 2 for precise definition.

  • Unless stated otherwise, all function spaces L2(Ω), H1(Ω), H01(Ω), etc. consist of functions taking values in Rd.

  • Function spaces whose notation contains subscript “c” consist of functions that vanish outside a compact set.

  • The function spaces H#1, H01(Y0), and A(Y0) are introduced in Section 3.1.

  • We write · and : for the canonical inner products in Rd and Rd×d, respectively.

  • SO(d) denotes the set of rotations in Rd×d.

  • ≲ stands for ⩽ up to a multiplicative constant that only depends on d, Y1, Ω, and on p if applicable.

2.Geometric and constitutive setup

The pore geometry. The set Y0 defined above describes the “pores” contained within the cell Y. Note that Y1 is an open, bounded, connected set with Lipschitz boundary. Therefore, to each φH1(Y1) we can associate (see e.g. [33]) a unique harmonic extension g1H1(Y) characterised by

(10)g1=φin Y1,Y0g1:ζdy=0ζH01(Y0).
For this extension the inequality
(11)g1L2(Y0)CφL2(Y1)
holds with a constant C that only depends on Y1.

For a given domain ΩRd and ε>0, we define the sets Ωε0 and Ωε1 as follows:

Ωε0:={ε(ξ+Y0)|ξZd,ε(ξ+Y)Ω},Ωε1:=ΩΩε0.
Note that by construction Ωε1 is a Lipschitz domain. In particular, it is connected and ΩΩε1. We denote by χε the indicator function of the set of pores:
χε(x):=1,xΩε0,0,xRdΩε0.

The composite. The two materials are described by energy densities Wi:Rd×d[0,+], i=0,1. Unless stated otherwise, we assume that for i=0,1:

  • (W1) Wi is frame-indifferent, i.e. Wi(RF)=Wi(F) for all RSO(d) and all FRd×d;

  • (W2) The identity matrix IRd×d is a “natural state”, i.e. Wi(I)=0, and Wi is non-degenerate:

    Wi(F)c0dist2(F,SO(d)),FRd×d,c0>0.

  • (W3) Wi has a quadratic expansion at I, i.e. there exists a non-negative quadratic form Qi on Rd×d and an increasing function ri:[0,)[0,] with lims0ri(s)=0, such that

    (12)|Wi(I+G)Qi(G)||G|2ri(|G|)GRd×d.

As shown in [30, Lemma 2.7] the quadratic form Qi associated with Wi via (W3) satisfies
(13)c1|symG|2Qi(G)=Qi(symG)c11|symG|2GRd×d,c1>0.
In the finite strain regime we consider a different set of assumptions for W0, which are listed in Section 3.3.

The scaling parameter γ. Throughout the paper γ>0 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:

(14)Eε(u)=Ω(ε2γW0(I+u)χε+W1(I+u)(1χε))dx,uH01(Ω,Rd).

3.Main results

3.1.Compactness and two-scale convergence

We first present an a priori estimate and a two-scale compactness statement for sequences φεH01(Ω) whose energy is equi-bounded in the sense that

(15)lim supε0Φεγ(vε)<,
where
Φεγ(v):=Ωdist2(I+v(x),SO(d))(ε2γχε+(1χε))dx.
Note that, by virtue of the non-degeneracy assumption (W2) the functional Φεγ(·) bounds below Eε(id+·), where id(x)=x, xΩ. As we shall see in the upcoming Lemma 1, the inequality (15) implies that the sequence φε is bounded in H1(Ω), and thus weakly converges (up to extracting a subsequence) to a limit displacement φH01(Ω). 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 φε in the spirit of an asymptotic decomposition as ε0.

  • 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 1.

Let φεH01(Ω) and 0<ε1.

  • (a) There exists a unique pair of functions gε0H01(Ωε0) and gε1H01(Ω) such that

    (16)(i)φε=gε1+ε1γgε0,(ii)Ωε0gε1:ζ=0ζH01(Ωε0).

  • (b) There exists a positive constant C that only depends on Ω,Y0 such that

    (17)gε0L2(Ω)2+εgε0L2(Ω)2+gε1H1(Ω)2CΦεγ(φε),
    where gε0, gε1 and φε 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 2. We use the following shorthand notation:

fεf0:fε strongly converges to f0 in L2(Ω),fεf0:fε weakly converges to f0 in L2(Ω),fε2f:fε weakly two-scale converges to f in L2(Ω×Y),fε2f:fε strongly two-scale converges to f in L2(Ω×Y).
The upcoming lemma states a two-scale compactness result for the displacements gε0 and gε1 that appear in the representation (16). Due to the differential constraint satisfied by gε0, the corresponding two-scale limits automatically satisfy certain structural properties, which can be captured with the help of the following function spaces:
  • H#1 is the space of [0,1)d-periodic functions in Hloc1(Rd).

  • H01(Y0) is the closed subspace of H#1 consisting of functions ψH#1 with ψ=0 on Y1.

  • A(Y0) is the closed subspace of H#1 consisting of functions ψH#1 that satisfy the identity

    Y0yψ:yζdy=0ζH01(Y0).

Lemma 2.

Consider a sequence φεH01(Ω) and let (gε0,gε1) be associated with φε via (16). Suppose that there exists a sequence of positive numbers mε such that

lim supε0mε2Φεγ(mεφε)<andmε=O(εγ).
Then there exist
(18)g0L2(Ω,H01(Y0)),g1H01(Ω),ψL2(Ω,A(Y0))
such that, up to selecting a subsequence, one has
(19)gε02g0,εgε02yg0,gε1g1weakly in H1(Ω)andgε12g1+yψ.

The identification obtained in the previous lemma is sharp, in the sense of the following statement.

Lemma 3.

Let g0L2(Ω,H01(Y0)), g1H01(Ω) and ψL2(Ω,A(Y0)). Let cε be an arbitrary sequence of positive numbers converging to zero. Then there exist function sequences gε0H01(Ωε0), gε1H01(Ω) such that (gε0,gε1) is related to φε:=gε1+ε1γgε0 as in (16), and

(20)gε02g0,εgε02yg0,gε1g1weakly in H1(Ω)andgε12g1+yψ,lim supε0cεφεL(Ω)=0.

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 φεH1(Ω) we write gε1+ε1γgε0:=(16)φε, if gε1H1(Ω), gε0H01(Ωε0), and both functions are related to φε as in (16).

  • We write φε2(g0,g1), if gε1+ε1γgε0:=(16)φε and

    (21)gε02g0,εgε02yg0,gε1g1weakly in H1(Ω).

  • We write φε2(g0,g1), if gε1+ε1γgε0:=(16)φε and

    (22)gε02g0,εgε02yg0,gε1g1weakly in H1(Ω).

3.2.Convergence in the small strain regime mε=o(εγ)

Throughout this section we assume that the densities W0 and W1 satisfy the conditions (W1)–(W3). We show that in the small strain regime the limit functional

Esmall:L2(Ω,H01(Y0))×H01(Ω)[0,),
is given by
Esmall(g0,g1):=Ω×Y(Q0(yg0(x,y))+Qhom1(g1(x)))dx,
where
(23)Qhom1(F):=minψA(Y0)Y1Q1(F+yψ(y))dy.
More precisely, the following theorem holds.

Theorem 1.

Let mε be a sequence of positive numbers and assume that mε=o(εγ) as ε0.

  • (a) (Compactness). Suppose that φεH01(Ω) satisfy

    lim supε0mε2Eε(mεφε)<.
    Then, up to a subsequence, one has φε2(g0,g1) for some g0L2(Ω,H01(Y0)) and g1H01(Ω).

  • (b) (Lower bound). Consider φεH01(Ω) and suppose that φε2(g0,g1) for some g0L2(Ω,H01(Y0)) and g1H01(Ω). Then the estimate

    lim infε0mε2Eε(mεφε)Esmall(g0,g1)
    holds.

  • (c) (Recovery sequence). For all g0L2(Ω,H01(Y0)) and g1H01(Ω) there exists a sequence φεH01(Ω) such that φε2(g0,g1) and

    limε0mε2Eε(mεφε)=Esmall(g0,g1).

In the next result we consider a minimisation problem that involves the density of the “body forces” εL2(Ω). We study the variational limit of the (scaled) total energy

(24)Iε(φ):=1mε2(Eε(mεφ)Ωε·(mεφ)dx),
where the scaling factor mε is determined by the body forces via
(25a)mε:=ε1γεL2(Ωε0)+εL2(Ω).
In the small strain regime we assume that the body forces are small in the sense that
(25b)mε=o(εγ),ε0.
Moreover, we assume that the (scaled) body-force densities converge, as ε0, in the following way:
(25c)mε1ε1γχεε20,mε1ε1.
It follows from Theorem 1 that the variational limit of the total energy (24) is given by the functional
(26)Ismall(g0,g1):=Esmall(g0,g1)Ω(Y00·g0dy+1·g1)dx.

Proposition 1.

Assume that (25a)–(25c) hold.

  • (a) (Convergence of infima). One has

    limε0infφH01(Ω)Iε(φ)=minIsmall(g0,g1),
    where the minimum on the right-hand side is taken over all g0L2(Ω,H01(Y0)) and g1H01(Ω). Moreover, the minimum is attained for a unique pair (g0,g1).

  • (b) (Convergence of minimisers). Let φεH01(Ω) be a sequence of almost minimisers, i.e.

    (27)Iε(φε)infφH01(Ω)Iε(φ)+o(1),ε0.
    Then
    φε2(g0,g1)andgε12g1+yψ,
    where ψL2(Ω,A(Y0)) denotes the unique “corrector” characterised by
    (28)Qhom1(g1(x))=Y1Q1(g1(x)+yψ(x,y))dy,Yψ(x,y)dy=0,
    for almost every xΩ.

Next, we prove that almost minimisers φε satisfy the asymptotic relation

(29)φε=g,ε1(x)+ε1γg,ε0(x)+o(1),ε0,in W1,p(Ω)(p<2),
where g,ε1 and g,ε0 formally obey the “ansatz”
(30)g,ε0(x)=formallyg0(x,x/ε),g,ε1(x)=formallyg1(x)+εψ(x,x/ε).
Here (g0,g1) and ψ denote the minimising pair and corrector from Proposition 1. Since the functions on the right-hand sides in (30) are in general not smooth enough to define g,ε0 and g,ε1 by (30) directly, we use instead the approximation associated with (g0,g1,ψ) via Lemma 3.

In addition to the properties of Y0 assumed in Section 2, we require the following assumption on the regularity of Y0:

Assumption 1.

There exist an exponent p<2 and a constant C such that for all φH1(Y1) and g1H1(Y) related via (10) we have g1Lp(Y0)CφLp(Y1).

Note that Assumption 1 is satisfied if Y0 can be written as the disjoint union of a finite number of Lipschitz domains Y10,,YN0 with Yi0Yj0= for ij.

Theorem 2.

Assume that (25a)–(25c) hold, and let Assumption 1 be satisfied. Let φεH01(Ω) be a sequence of almost minimisers, i.e.

Iε(φε)infφH01(Ω)Iε(φ)+o(1),ε0.
Let (g0,g1) be the minimiser of Ismall and let ψ be defined through (28). Let (g,ε0,g,ε1) and φ,ε=g,ε1+ε1γg,ε0 be associated with (g0,g1,ψ) as in Lemma 3, i.e. φ,ε2(g0,g1) and g,ε12g1+yψ. Then for gε1+ε1γgε0:=(16)φε one has
(31)gε0g,ε0Lp(Ωε0)+εgε0εg,ε0Lp(Ωε0)+gε1g,ε1W1,p(Ω)0as ε0.

Remark 1.

To illustrate the result of Theorem 2, consider the case γ=1 with ε:=mε(x,xε), where (x,y) is smooth both in x and y and is periodic in y. If the domain Ω and the pore set Y0 are sufficiently regular, the minimisers (g0,g1) and ψ are smooth, by the classical elliptic regularity theory, see e.g. [21]. In that case we may set

g,ε0(x):=g0(x,x/ε)andg,ε1:=g1(x)+εψ(x,x/ε),
and the asymptotic formula for φε reads
φε=g0(x,x/ε)+g1(x)+εψ(x,x/ε)+Rε(x),
where RεW1,p(Ω)0 as ε0.

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>1, γ=1, mε=εα and ε=εαf for some fL2(Ω), so that the functionals Iεα and Iε defined in (2) and (24) are identical. Hence, Theorem 1 and Proposition 1 prove two-scale Γ-convergence of Iεα and the convergence of the associated minimisation problems (as claimed in the Introduction), and Theorem 2 yields the two-scale expansion (6). We argue that the functionals Iεα Γ-converge to the (single scale) limit I¯small:

Proposition 2.

For α>1 and fL2(Ω) consider Iεα and I¯small defined in (2) and (7). We extend Iεα to a functional on L2(Ω) by setting Iεα:=+ on L2(Ω)H01(Ω). Then:

  • (a) (Compactness). Suppose that φεL2(Ω) satisfy

    lim supε0Iεα(φε)<.
    Then, up to a subsequence, we have φεφ weakly in L2(Ω).

  • (b) (Lower bound). For every φεL2(Ω) with φεφ weakly in L2(Ω) we have

    lim infε0Iεα(φε)I¯small(φ)

  • (c) (Upper bound). For every φL2(Ω) we can find φεφ weakly in L2(Ω) such that

    limε0Iεα(φε)=I¯small(φ)

  • (d) (Convergence of the minimisation problem). Let φεH01(Ω) denote an infimizing sequence of Iεα, i.e.

    Iεα(φε)infφL2(Ω)Iεα(φ)+o(1),ε0.
    Let (g0,g1) denote the unique minimiser of Ismall and φ denote the unique minimiser of I¯small. Then infL2(Ω)IεαminL2(Ω)I¯small, φεφ weakly in L2(Ω), and
    φ=g1+Y0g0dy,I¯small(φ)=Ismall(g0,g1).

3.3.Convergence in the finite strain regime mε=εγ

Throughout this section we assume that

  • W1 satisfies the conditions (W1)–(W3).

  • W0:Rd×d[0,) is continuous and satisfies the growth condition

    (32a)c0dist2(F,SO(d))W0(F)c01(1+|F|2)FRd×d,
    and the local Lipschitz condition
    (32b)|W0(F+G)W0(F)|c01(1+|F|+|G|)|G|F,GRd×d.

We prove that in the finite strain regime the limit functional
Efinite:L2(Ω,H01(Y0))×H01(Ω)[0,)
is given by
Efinite(g0,g1):=Ω×Y0QW0(I+yg0(x,y))dydx+ΩQhom1(g1(x))dx,
where QW0 denotes the quasiconvex envelope of W0 (see e.g. [19]). The associated limit of the total energy Iε, see (24), is given by (cf. (26))
Ifinite(g0,g1):=Efinite(g0,g1)Ω(Y00·g0dy+1·g1)dx,
where 0, 1 are defined in the same way as in (25c).
Theorem 3.

  • (a) (Lower bound). Consider a sequence φεH01(Ω) and the associated decomposition ε1γgε0+gε1:=(16)φε. If φε2(g0,g1) and gε02g0, then

    lim infε0ε2γEε(εγφε)Efinite(g0,g1).

  • (b) (Recovery sequence). For any g0L2(Ω,H01(Y0)) and g1H01(Ω) there exists a sequence φεH01(Ω) such that φε2(g0,g1) and

    limε0ε2γEε(εγφε)=Efinite(g0,g1).

  • (c) Suppose that the force densities εL2(Ω) satisfy (25a) and (25c) with mε=εγ. Then the infima converge, i.e.

    (33)limε0infφH01(Ω)Iε(φ)=infIfinite(g0,g1),
    where the infimum on the right-hand side is taken over all functions g0L2(Ω,H01(Y0)) and g1H01(Ω). Moreover, there exist a minimising pair (g0,g1) and a recovery sequence φεH01(Ω) with φε2(g0,g1) such that
    (34)Iε(φε)Ifinite(g0,g1)=minIfinite(g0,g1)as ε0.

Remark 2

Remark 2(Example in Section 1).

If we consider Theorem 3 and Proposition 1 in the case γ=1, mε=ε and ε=εf for some fL2(Ω), then we recover the special case (in the finite strain regime) presented in the introduction. In particular, we deduce that the functionals Iεα two-scale Γ-converge (in the sense of Theorem 3) to Ifinite. Arguing as in the small-strain regime, cf. Proposition 2, we deduce that Iεα Γ-converges (with respect to the weak topology in L2(Ω)) to I¯finite, cf. (8). We leave the details to the readers.

4.Proofs

We start by proving the auxiliary results discussed in Section 3.1. Sections 4.2 and 4.3 contain the proofs of the main statements in the small strain and finite strain cases, respectively.

4.1.Proofs of Lemma 1, Lemma 2, and Lemma 3: A priori estimate, compactness and approximation

A key ingredient in the proof of Lemma 1 is the geometric rigidity estimate by Friesecke et al. [23]:

Theorem 4

Theorem 4(Geometric rigidity estimate, see [23]).

Let U be an open, bounded Lipschitz domain in Rd, d2. There exists a constant C(U) with the following property: for each vH1(U) there is a rotation RSO(d) such that

U|v(x)R|2dxC(U)Udist2(v(x),SO(d))dx.
Moreover, the constant C(U) 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>0 that only depends on Ω and Y1 such that for all ε>0 and vH1(Ω) satisfying

(35)Ωε0v:ζdx=0ζH01(Ωε0),
the estimates
(36)dist(v,SO(d))L2(Ω)Cdist(v,SO(d))L2(Ωε1),(37)vRL2(Ωε1)Cdist(v,SO(d))L2(Ωε1),
hold for some RSO(d), which may depend on v. In addition, if v(x)=x+c on Ω for some constant c, then we may set R=I.

Proof of Lemma 4.

Step 1. The proof of the inequality (36).

Let Ωˆε:={ε(ξ+Y)|ξZd,ε(ξ+Y)Ω} denote the union of ε-cells that are completely contained in Ω. Since ΩΩˆεΩε1, it suffices to prove (36) for Ω replaced by Ωˆε, respectively. In fact we shall prove the following stronger estimate: for all ξZd with ε(ξ+Y)Ω we have

(38)ε(ξ+Y)dist2(v,SO(d))dxε(ξ+Y1)dist2(v,SO(d))dx.
For the argument fix an admissible ξZd. Application of Theorem 4 with U=ε(ξ+Y1) yields a rotation RSO(d) such that
(39)ε(ξ+Y1)|vR|2dxε(ξ+Y1)dist2(v,SO(d))dx.
Note that the multiplicative constant in the estimate above only depends on Y1, since ε(ξ+Y1) is a dilation and translation of Y1. On the other hand, since ε(ξ+Y0)x(v(x)Rx) is harmonic, we have (cf. (11)):
ε(ξ+Y)dist2(v,SO(d))dxε(ξ+Y)|vR|2ε(ξ+Y1)|vR|2.
Combined with (39), inequality (38) follows.

Step 2. The proof of the rigidity estimate (37).

From (36) and Theorem 4 (applied with U=Ω) we deduce that for some RSO(d):

(40)vRL2(Ω)dist(v,SO(d))L2(Ωε1),
which in particular implies (37). Finally we argue that one can set R=I, if v=x+c on Ω. In view of (40), it suffices to show that Ω|vI|2dxΩ|vR|2dx for all RSO(d). This inequality can be seen as follows: Consider φ(x):=v(x)xc and note that φ vanishes on Ω, so that
Ω|vI|2dx=Ω|φ|2dxΩ|φ|2+|IR|2dx=Ω|φ+(IR)|2dx=Ω|vR|2dx,
which in fact holds for an arbitrary matrix R. □

We are now in position to present the proofs of Lemmas 1 and 2.

Proof of Lemma 1.

In the following, the symbol ≲ stands for ⩽ up to a multiplicative constant that only depends on Y1 and Ω.

Step 1. Existence of the decomposition (16) and derivation of the estimate for gε1.

Let gε1 denote the unique function in H1(Ω) characterised by gε1=φε in Ωε1 and (16)(ii). Since Ωε0 is Lipschitz, we deduce that gε0:=εγ1(φεgε1)H01(Ωε0). This proves the existence of the decomposition. We claim that

(41)Ωε0|gε1|2dxΩε1|φε|2dx=Ωε1|gε1|2dx.
Since Ωε0 is defined as the union of the sets ε(ξ+Y0) with ξZε:={ξZd:ε(ξ+Y)Ω}, it suffices to prove ε(ξ+Y0)|gε1|2dxε(ξ+Y1)|φε|2dx. The latter follows from (11) by a scaling argument, since the rescaled functions yφε(ε(ξ+y)) and ygε1(ε(ξ+y)) satisfy (10).

Next, we prove (17). Consider vε(x):=x+gε1(x) and note that vε(x) satisfies (35). Hence, (41) and Lemma 4 yield

Ω|gε1|2dxΩε1|gε1|2dxΩ|vεI|2(37)CΩε1dist2(I+φε(x),SO(d))dx(42)Φεγ(φε).
Since gε1 vanishes on the boundary of Ω, the estimate upgrades (by Poincaré’s inequality) to gε1H1(Ω)CΦεγ(φε).

Step 2. Derivation of the estimate for gε0.

Since we have an improved Poincaré inequality (see e.g. [25, Lemma 1.6]):

(43)gH01(Ωε0):gL2(Ωε0)εgL2(Ωε0),
it suffices to prove
εgε0L2(Ωε0)2Φεγ(φε).
To this end, notice that since φε vanishes on the boundary of Ω, we have
φεL2(Ω)2=minRSO(d)I+φεRL2(Ω)2Theorem 4Ωdist2(I+φε(x),SO(d))dx(44)ε2γΦεγ(φε).
Thanks to the first identity in (16), we get by triangle inequality:
ε1γgε0L2(Ωε0)φεL2(Ω)+gε1L2(Ω).
Combined with (42) and (44) we finally get
εgε0L2(Ωε0)2=ε2γε1γgε0L2(Ωε0)2Φεγ(φε).

Proof of Lemma 2.

Step 1. A priori estimate and basic compactness.

From Lemma 1 we deduce that

(45)lim supε0(gε0L2(Ω)2+εgε0L2(Ω)2+gε1H1(Ω)2)Clim supε0mε2Φεγ(mεφε)<.
Hence, by standard results concerning two-scale convergence (cf. [1, Proposition 1.14] and [36, Proposition 4.2]), there exist g1H01(Ω), ψL2(Ω,H#1) and g0L2(Ω,H#1) such that, up to a subsequence, one has
gε1g1weakly in H1(Ω),gε12g1+yψ,gε02g0,εgε02yg0.

Step 2. The proof of the inclusion ψL2(Ω,A(Y0)).

By a density argument, it suffices to show that

(46)Ω×Y0yψ(x,y):y(ζ1(x)ζ2(y))dxdy=0
for all scalar functions ζ1Cc(Ω), and all ζ2Cc(Y0). To this end, we identify ζ2 with its unique Y-periodic extension to Rd that vanishes on Y1, and set
ζε(x):=εζ1(x)ζ2(x/ε),xΩ.
Thanks to (16) we have
Ωgε1:ζεdx=0.
As can be easily checked, we have ζε2ζ1(x)yζ2(y), so that
0=limε0Ωgε1:ζεdx=Ω×Y0(g1(x)+yψ(x,y)):(ζ1(x)yζ2(y))dxdy=Ω×Y0yψ(x,y):(ζ1(x)yζ2(y))dxdy,
where the last identity holds thanks to the periodicity of ζ2. This proves (46).

Step 3. The proof of the inclusion g0L2(Ω,H01(Y0)).

By a density argument, it suffices to show that

(47)Ω×Yg0(x,y)·(ζ1(x)ζ2(y))dxdy=0
for all scalar functions ζ1Cc(Ω) and all ζ2H#1 with ζ2=0 on Y0. We argue by considering the function ζε(x):=ζ1(x)ζ2(xε), xΩ, the support of which is contained in Ωε1 for ε1. Since ζε2ζ1(x)ζ(y), and since gε0 is supported in Ωε0, we deduce that
0=limε0Ωgε0(x)·ζε(x)dx=Ω×Yg0(x,y)·(ζ1(x)ζ2(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 [2]:

Lemma 5.

For any h:[0,)2[0,+], there exists a mapping (0,1)εδ(ε)(0,1) such that

limε0δ(ε)=0andlim supε0h(ε,δ(ε))lim supδ0lim supε0h(ε,δ).

Proof of Lemma 3.

Step 1. Characterisation of strong two-scale convergence via unfolding.

For fε:ΩR and f:Ω×YR define

dε(fε,f):=RdY|f˜ε(εx/ε+εy)f˜(x,y)|2dydx,
where f˜ε denotes the extension by zero of fε to Rd, f˜ denotes the extension by zero of f to Rd×Y, and z denotes the unique element in Zd with zz[0,1)d. We recall from [36] that
(48)fε2fdε(fε,f)0.
The characterisation extends in the obvious way to vector-valued functions.

Step 2. Construction of gε1.

We claim that there exists a sequence gε1 in H01(Ω) whose elements satisfy (16)(ii) and

(49)gε1g1weakly in H1(Ω),gε12g1+yψ,lim supε0cεgε1L(Ω)=0.
Indeed, by a density argument there exist g1,δCc(Ω), δ(0,1), and ψCc(Ω,C#(Y)) such that
g1,δg1H1(Ω)+yψδyψL2(Ω×Y)δδ.
For ε>0, δ(0,1), define
gε1,δ(x):=g1,δ(x)+εψδ(x,x/ε),
and set
dεδ:=dε(gε1,δ,g1+yψ)+gε1,δg1L2(Ω)+cεgε1,δL(Ω).
By construction, we have limδ0lim supε0dεδ=0, and Lemma 5 yields a function εδ(ε) with limε0dεδ(ε)=0. In view of Step 1, this implies that the diagonal sequence g˜ε1:=gε1,δ(ε) satisfies (49). Now, for each ε>0, let gε1 denote the function satisfying (16)(ii) and such that gε1=g˜ε1 on Ωε1. To conclude the argument, we only need to show that gε1 satisfies (49). Consider the difference ηε:=g˜ε1gε1. Since ηε is bounded in H1(Ω) and ηε=0 in Ωε1, we have ηε0 in H1(Ω), and, up to a subsequence, ηε2yφ for some φL2(Ω,H01(Y0)). On the other hand, since gε1 satisfies (16)(ii) and g˜ε1 satisfies (49), we deduce that
ηεL2(Ω)2=Ωε0ηε:ηεdx=Ωε0g˜ε1:ηεdxΩ×Y0(g1+yψ):yφdxdy.
Since g1 is independent of y, and because ψL2(Ω,A(Y0)), the integral on the right-hand side vanishes. Hence, ηεL2(Ω)20, and thus gε1 satisfies (49).

Step 3. Conclusion.

As can be shown by appealing to a combination of a density argument and a diagonal-sequence argument, similar to Step 1, there exists a sequence gε0H01(Ωε0) such that

gε02g0,εgε02yg0,lim supε0cεε1γgε0L(Ω)=0.
Now define φε(x):=ε1γgε0+gε1, and note that (gε0,gε1) satisfy (16). In view of the convergence of gε0 and gε1, the sequence φε has the required properties. □

4.2.Proof of Theorems 1, 2 and Propositions 1, 2: Small strain regime

As a preliminary remark, we note that two different effects play a role when passing to the limit ε0 in the small strain regime:

  • The non-convex energy functional is linearised at identity map (which is a stress-free state for Eε) – this corresponds to the passage from nonlinear to linearised elasticity.

  • The obtained linearised, still oscillating, convex-quadratic energy is homogenised.

The following lemma is used to treat both effects simultaneously. Its proof combines convex homogenisation methods (e.g. [37, Proposition 1.3]) with a “careful Taylor expansion” in the spirit of [23, Proof of Theorem 6.2]. For notational convenience, we introduce two “linearised” functionals:
  • For Gε=(Gε0,Gε1)L2(Ω,Rd×d)×L2(Ω,Rd×d) set

    Qε(Gε)=Qε(Gε0,Gε1):=Ωε0Q0(Gε0(x))dx+Ωε1Q1(Gε1(x))dx.

  • For G=(G0,G1)L2(Ω×Y,Rd×d)×L2(Ω×Y,Rd×d) set

    Q(G)=Q(G0,G1):=Ω×Y0Q0(G0(x,y))dxdy+Ω×Y1Q1(G1(x,y))dxdy.

Lemma 6.

Consider sequences gε0,gε1H1(Ω) that satisfy

lim supε0(εgε0L2(Ω)+gε1L2(Ω))<.
Set φε:=ε1γgε0+gε1 and Gε=(Gε0,Gε1):=(εgε0,gε1).
  • (a) If Gε2G and mε=o(εγ) as ε0, then

    (50)lim infε0mε2Eε(mεφε)lim infε0Qε(θεGε)Q(G),
    where θε:Ω{0,1} is defined by
    θε(x):=1,if |φε|(mεεγ)1/2,0,otherwise.

  • (b) If Gε2G, mε=o(εγ) as ε0, and

    (51)lim supε0mεφεL(Ω)=0,
    then
    limε0mε2Eε(mεφε)=Q(G).

Remark 3.

In Lemma 6, following [23], the function θε is introduced, in order to truncate the peaks of Gε. This is needed for exploiting the quadratic expansion (W3). Since both εgε0 and gε1 are assumed to be bounded sequences in L2(Ω), we deduce, from the definition of θε, the fact that mεεγ=o(1) and the Chebyshev inequality, that

(52)r<θε1Lr(Ω)0,andθεL(Ω)1.

In the proof of Lemma 6 we need to pass to the limit in products of the form fεθε, where θε satisfies (52), or fεχε, where χε denotes the indicator of Ωε0. This is done by appealing to the next two lemmas, the proofs of which are elementary and left to the reader.

Lemma 7.

Let fε,θε be sequences in L2(Ω) and assume that θε satisfies (52), then the following implications are valid:

lim supε0fεL2(Ω)<θεfεfεLp(Ω)0p<2,fεfθεfεf,fε2fθεfε2f,{fεf(|fε|2) equi-integrable}θεfεfεL2(Ω)0.

Lemma 8.

Suppose that fε be a sequence in L2(Ω) and, as above, let χε denote the set indicator function of Ωε0. Then the following implications hold:

fε2fχεfε2χ(y)f(x,y),fε2fχεfε2χ(y)f(x,y),
where χ denotes the indicator function of Y0.

Proof of Lemma 6.

Step 1. Linearisation.

We claim that the following statement holds for i=1,2: Let Fε denote a sequence in L2(Ω,Rd×d), and let cε be a sequence of positive numbers converging to zero, such that

(53)lim supε0cεFεL(Ω)=0.
Then the convergence
(54)limε0|cε2ΩWi(I+cεFε)dxΩQi(Fε(x))dx|=0
holds. Indeed, thanks to (W3) we have
|cε2Wi(I+cεFε)Qi(Fε)||Fε|2ri(cε|Fε|)|Fε|2ri(cεFεL(Ω))a.e.
Thanks to (53), and since Fε is bounded in L2(Ω), the right-hand side converges to zero in L1(Ω), and (54) follows.

Step 2. Proof of part (a).

Since the energy densities W0, W1 are minimised at the identity, cf. (W2), we have

(55)mε2Eε(φε)(mεεγ)2ΩW0(I+mεεγθεFε0(x))dx+mε2ΩW1(I+mεθεFε1(x))dx,
where
(56)Fε0:=χε(Gε0+εγGε1),Fε1(x):=(1χε)Gε1.
Thanks to the definition of θε we have mεεγθεFε0L(Ω)+mεθεFε1L(Ω)0, so that we may apply (54) to the right-hand side in (55). We get
lim infε0mε2Eε(φε)lim infε0Qε(θεFε0,θεFε1)=lim infε0Qε(θεGε1,θεGε0),
where for the last identity we used the facts that Fε1=Gε1 on Ωε1 and
Fε0Gε0L2(Ωε0)=εγgε1L2(Ωε0)0.
It remains to argue that
lim infε0Qε(θεGε)Q(G).
In order to show this, notice that
(57)Qε(θεGε)=ΩQ0(θεχεGε0)dx+ΩQ1(θε(1χε)Gε1)dx.
From Gε2G we deduce, using Lemma 7, Remark 3 and Lemma 8, that
θεχεGε02χ(y)G0(x,y),θε(1χε)Gε12(1χ(y))G1(x,y).
By appealing to the lower semicontinuity of convex integral functionals with respect to weak two-scale convergence (cf. [37, Proposition 1.3]), we deduce that the lim inf of the right-hand side in (57) is bounded below by Q(G). This completes the argument.

Step 3. Proof of part (b).

We claim that

(58)limε0|1mε2Eε(mεφε)Qε(εgε0,gε1)|=0.
Note that
mε2Eε(φε)=(mεεγ)2ΩW0(I+mεεγFε0(x))dx+mε2ΩW1(I+mεFε1(x))dx,
where Fε0 and Fε1 are defined in (56). By (51) we have mεεγFε0L(Ω)+mεFε1L(Ω)0 and (54) yields
|mε2Eε(mεφε)Qε(Fε0,Fε1)|0as ε0.
Since Gε2G we have, thanks to Lemma 8:
χεFε02χ(y)G0(x,y),(1χε)Fε12(1χ(y))G1(x,y).
Hence, the continuity of convex integral functionals with respect to strong two-scale convergence (cf. [36]) yields
Qε(Fε0,Fε1)Q(G),
which completes the argument. □

We are now in a position to prove the Γ-convergence statement for the energies Eε.

Proof of Theorem 1.

Step 1. Part (a) (Compactness).

Thanks to (W2) we have

mε2Eε(mεφε)c0mε2Φεγ(mεφε).
Hence, the claim of Theorem 1(a) directly follows from Lemma 2.

Step 2. Part (b) (Lower bound).

Without loss of generality we assume that

lim infε0mε2Eε(mεφε)=lim supε0mε2Eε(mεφε)<.
Furthermore, thanks to Lemma 2, we can assume in addition that gε12g1+yψ for some ψL2(Ω,A(Y0)), so that
Gε:=(εgε0,gε1)2(yg0,g1+yψ)=:G.
Applying Lemma 6(a) yields
lim infε0mε2Eε(mεφε)Ω×Y0Q0(yg0(x,y))dxdy+Ω×Y1Q1(g1(x)+yψ(x,y))dxdy.
This completes the argument, since the right-hand side is bounded from below by Esmall(g0,g1).

Step 3. Part (c) (Upper bound).

Choose ψL2(Ω,A(Y0)) such that

(59)Ω×Y1Q(g1(x)+yψ(x,y))dxdy=ΩQhom1(g1(x))dx.
Let φε denote the sequence associated with g0,g1 and ψ via Lemma 3 with cε:=mε. In view of (20), applying Lemma 6(b) yields
limε0mε2Eε(mεφε)=Ω×YQ0(yg0(x,y))dxdy+Ω×Y1Q1(g1(x)+yψ(x,y))dxdy.
It follows from (59) that the right-hand side equals Esmall(g0,g1). □

Proof of Proposition 1.

Step 1. A priori estimate.

We claim that for every sequence φε in H01(Ω) the following implication holds:

(60)lim supε0Iε(φε)<lim supε0mε2Eε(mεφε)<.
Indeed, we have
|Ωε·mεφεdx|mε(εL2(Ω)gε1L2(Ω)+ε1γεL2(Ωε0)gε0L2(Ωε0))(25a)mε(mεgε1L2(Ω)+mεgε0L2(Ω))(17)mε2mε2ϕεγ(mεφε)(W2)mε2mε2Eε(mεφε).
Combining this with the definition of Iε we get
mε2Eε(mεφε)Iε(φε)+mε2Eε(φε),
which implies (60).

Step 2. The proof of parts (a) and (b).

The existence of a minimiser to Ismall follows by the direct method. The minimiser (g0,g1) is unique, since the implication

Ω×YQ0(yg˜0(x,y))dxdy+ΩQhom1(g˜1(x))dx=0g˜0=g˜1=0
holds for all g˜0L2(Ω,H01(Y0)) and g˜1H01(Ω).

The remaining claims of Proposition 1 follow from the standard Γ-convergence arguments (cf. [20, Corollary 7.20]), provided the functionals Iε, ε>0, are equi-coercive and Γ-converge to Ismall. Indeed, thanks to (25c), it is easy to check that φε2(g0,g1) implies

(61)limε01mε2Ωε·mεφεdx=Ω(Y00·g0dy+1·g1)dx,
since the integral on the left-hand side only involves products of weakly and strongly two-scale convergent factors, cf. [36, Proposition 2.8]). In combination with Theorem 1, this implies that Iε Γ-converges to Ismall. In addition, the trivial inequality
infφH01(Ω)Iε(φ)Iε(0)=0,
combined with (60) and Lemma 2, proves that the functionals Iε are equi-coercive. □

For the proof of Theorem 2 we make use of the following lemma:

Lemma 9

Lemma 9(Decomposition Lemma, see [22,27]).

Let vH01(Ω) and let vεH1(Ω) be a sequence with vεv in H1(Ω). Then there exists a sequence VεH1(Ω) such that the following properties hold for a subsequence of vε (not relabelled):

  • (a) Vεv in H1(Ω);

  • (b) Vε=vε in a neighbourhood of Ω;

  • (c) (|Vε|2) is equi-integrable;

  • (d) |{xΩ:vε(x)Vε(x)}|0 as ε0.

Proof of Theorem 2.

It suffices to prove the theorem for a subsequence. Throughout the proof we write

Gε:=(εgε0,gε1),G:=(yg0,g1+yψ).
Furthermore, we make use of the functionals Qε and Q introduced at the beginning of Section 4.2. Recall that
Esmall(g0,g1)=Q(G).

Step 1. Convergence of φε and of the corresponding energy values.

We claim that, as ε0, one has

(62)Gε2G,(63)Iε(φε)Ismall(g0,g1),(64)mε2Eε(mεφε)Esmall(g0,g1).
Indeed, from Proposition 1 we immediately deduce that φε2(g0,g1) and (63). Furthermore, in view of the continuity of the loading term, cf. (61), this implies (64). For (62), it remains to argue that gε12g1+yψ. Thanks to φε2(g0,g1) and Lemma 2 we have, up to a subsequence, gε12g1+yψ for some ψL2(Ω,A(Y0)). Furthermore, from (64) and Lemma 6(a) we infer that
Esmall(g0,g1)=lim infε0mε2Eε(mεφε)Ω×Y0Q0(yg0)dxdy+Ω×Y1Q1(g1+yψ)dxdy.
This, in particular, implies
Ω×Y1Q1(g1+yψ)dxdy=ΩQhom1(g1)dx.
In view of (28) we conclude that ψ=ψ and (62) follows.

Step 2. Equi-integrable decomposition.

We claim that for a subsequence (not relabelled) there exist sequences g¯ε0,Gε1H01(Ω) such that Gε:=(εGε0,Gε1) satisfies

(65)GεGε20,GεGεLp(Ω)0,
and
(66)Qε(Gε)Esmall(g0,g1).
To show the above, notice that thanks to Lemma 9 there exist sequences Gε0,Gε1H01(Ω) such that
  • (ε2|Gε0|2) and (|g¯ε1|2) are equi-integrable,

  • the indicator function θ¯ε defined by

    (67)θ¯ε(x):=1,if Gε0(x)=gε0(x) and Gε1(x)=gε1(x),0,otherwise,
    satisfies (52).

Since GεGε=(1θ¯ε)(GεGε) (and p<2), the convergence (65) follows from the boundedness of the sequence (GεGε) in L2(Ω), Lemma 7, and Hölder’s inequality.

We prove (66). Thanks to (62) we have Gε2G, so that (due to the lower semicontinuity of convex integral functionals with respect to weak two-scale convergence, cf. [37, Proposition 1.3]):

lim infε0Qε(Gε)Q(G)=Esmall(g0,g1).
Hence, for (66) it suffices to prove the opposite estimate, i.e. lim supε0Qε(Gε)Esmall(G), which, thanks to (50) and (64), follows from
(68)lim supε0(Qε(Gε)Qε(θεGε))0.
In order to show (68) notice that since the supports of θ¯ε and (1θ¯ε) are disjoint, and because θ¯εθεGε=θ¯εθεGε (cf. (67)), an expansion of the squares yields
Qε(Gε)Qε(θεGε)=Qε(θ¯εGε)+Qε((1θ¯ε)Gε)Qε(θ¯εθεGε)Qε((1θ¯ε)θεGε)=Qε(θ¯ε(1θε)Gε)+Qε((1θ¯ε)Gε)Qε((1θ¯ε)θεGε)Qε(θ¯ε(1θε)Gε)+Qε((1θ¯ε)Gε).
It is easy to check that θ¯ε(1θε) and 1θ¯ε converge to zero in Lr(Ω) for all r<. Hence, since |Gε|2 is equi-integrable, Lemma 7 implies that the right-hand side of the previous estimate converges to zero, and (68) follows.

Step 3. Error estimate.

We claim that

(69)Ωε0ε|sym(g¯ε0g,ε0)|2dx+Ωε1|sym(g¯ε1g,ε1)|2dx0as ε0.
For the argument set G,ε:=(εg,ε0,g,ε1). In view of (13) it suffices to argue that
Qε(GεG,ε)0as ε0.
The latter can be seen as follows: We have
Qε(GεG,ε)=Qε(Gε)Qε(G,ε)+2Bε(G,ε;G,εGε),
where Bε denotes the bilinear form associated with Qε.

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,εG¯ε20 by (65), we deduce that

limε0Bε(G,ε;G,εGε)=0,
as Bε(G,ε;G,εGε) only involves products between a weakly and a strongly two-scale convergent factor (cf. [36, Proposition 2.8]).

Step 4. Conclusion (Proof of (31)).

We split the estimate into

(70)Ω(|gε0g,ε0|p+|εgε0εg,ε0|p)dx0as ε0,(71)Ω(|gε1g,ε1|p+|gε1g,ε1|p)dx0as ε0.
Thanks to (65) and Step 3 we have
(72)Ωε0ε|sym(gε0g,ε0)|pdx+Ωε1|sym(gε1g,ε1)|pdx0as ε0.
Argument for (70): Set ηε0:=gε0g,ε0. Since ηε0H01(Ωε0)H01(Ω), Korn’s inequality yields
Ω|ηε0|pdxCΩ|symηε0|pdx=CΩε0|symηε0|pdx,
where C>0 only depends on Ω, p, and d. Combined with the improved Poincaré inequality (43) and (72), (70) follows.

Argument for (71): We claim that (71) follows from

(73)symηε1Lp(Ωε0)0,
where ηε1:=gε1g,ε1. Indeed, since ηε1 vanishes on Ω, (73), (72) and Korn’s first inequality yield (71).

Thanks to the definition of Ωε0, the argument for (73) can be reduced to the following statement: For all ξZε:={ξZd:ε(ξ+Y)Ω} we have

(74)ε(ξ+Y0)|symηε|pdxε(ξ+Y1)|symηε|pdx.
For the argument consider the rescaled function
ηˆε:YRd,ηˆε(y):=ηε(ε(ξ+y))Sy+c,
where SRskewd×d and cRd are chosen such that the Poincaré and Korn inequalities yield
(75)Y1(|ηˆε|p+|ηˆε|p)dyY1|symηˆε|pdy.
Since both gε1 and g,ε1 satisfy (16)(ii), we have ηˆε=0 in Y0 in the distributional sense. Hence, thanks to Assumption 1 and (75), we have
Y0|ηˆε|pdyY1|symηˆε|pdy,
and thus
ε(ξ+Y0)|symηε|pdx=εdpY0|symηˆε|pdyεdpY1|symηˆε|pdy=ε(ξ+Y1)|symηε|pdx.

Proof of Proposition 2.

Step 1. Proof of (a).

Arguing as in Step 1 in the proof of Proposition 1 we find that lim supε0ε2αEε(εαφε)<, and thus the compactness part of Theorem 1 (and the fact that two-scale convergence implies weak convergence) yields (for a subsequence, which we  do not relabel):

(76)φε2(g0,g1)andφεφ:=g1+Y0g0dy.

Step 2. Proof of (b).

We may restrict to the case

lim infε0Iεα(φε)=lim supε0Iεα(φε)<.
Thanks to Step 1 we may assume without loss of generality that (76) holds. From the lower bound part of Theorem 1 (and the fact that Ωf·φεdxΩf·φdx) we thus deduce that
lim infε0Iεα(φε)Ismall(g0,g1).
With the definition of Q from the introduction and G:=Y0g0dy we get the inequality:
Ismall(g0,g1)ΩQhom1(g1(x))(g1(x)+G(x))·f(x)dx+Ωmin{Y0Q0(yg˜0(y)):g˜0H01(Y0),Y0g˜0(y)=G(x)}=ΩQhom1(g1(x))+Q(G(x))(g1(x)+G(x))·f(x)dx.
Since φ=g1+G¯ by (76), the right-hand side is bounded from below by I¯small(φ), which completes the argument for (b).

Step 3. Proof of (c).

Let φL2(Ω). It suffices to argue that there exist g0L2(Ω,H01(Y0)) and g1H01(Ω) with φ=g1+Y0g0dy and

(77)I¯small(φ)=Ismall(g0,g1).
Indeed, in that case, we can find by part (c) of Theorem 1 a sequence φεH01(Ω) such that φε2(g0,g1) and limε0Iεα(φε)=Ismall(g0,g1). Since φε2(g0,g1) implies φεφ=g1+Y0g0dy weakly in L2(Ω), we deduce from (77) that φε is the sought for recovery sequence. In order to prove (77), let ρi (i=1,2,3) denote the unique minimiser in H01(Y0) to the functional
H01(Y0)ρY0Q0(ρ(y))dysubject to Y0ρdy=ei,
and set
q¯i:=Y0Q0(ρi(y))dy.
Then it is easy to check that
Q(G)=i=13Gi2q¯ifor all GR3,
and since q¯1,,q¯3>0, we deduce that Q is a positive definite quadratic form. Hence, since Qhom1(F)c|symF|2 for some c>0, we deduce that we can find a unique function g1H01(Ω) that minimizes the functional
(78)H01(Ω)gΩQhom1(g(x))+Q(φ(x)g(x))dx.
Setting g0(x,y):=i=13((φ(x)g(x))·ei)ρi(y) we deduce that ΩQ(φ(x)g(x))dx=Ω×Y0Q0(yg0(x,y))dy, and thus (thanks to the definition of I¯small) (77) follows.

Step 4. Proof of (d).

By Proposition 1 we have φε2(g0,g1) and thus φεφ˜:=g1+Y0g0dy. By definition of I¯small we have

(79)Ismall(g0,g1)I¯small(φ˜)infL2(Ω)I¯small.
On the other hand, the map L2(Ω)φgH01(Ω) with g minimizing the functional in (78) is linear and bounded; hence, we deduce that I¯small is quadratic and strictly convex. It thus admits a unique minimiser φL2(Ω). By Step 3 (cf. (77)) we associated with φ a pair (g0,g1) such that I¯small(φ)=Ismall(g0,g1). Combined with (79) we get
minL2(Ω)I¯small=I¯small(φ)=Ismall(g0,g1)Ismall(g0,g1)I¯small(φ˜)minL2(Ω)I¯small.
Hence, equality holds everywhere and the claimed identities follow from the strict convexity of I¯small. The convergence of infL2(Ω)IεαminL2(Ω)I¯small follows from part (b) and (c) by standard arguments from theory of Γ-convergence. □

4.3.Proof of Theorem 3: Finite strain regime

We define, for gε0H01(Ωε0), g0L2(Ω,H01(Y0)), and gε1,g1H01(Ω), the following functionals:

Iε0(gε0):=Ωε0(W0(I+εgε0(x))ε12γε·gε0)dx,I00(g0):=Ω×Y0(QW0(I+yg0(x,y))0·g0)dxdy,Iε1(gε1):=ε2γΩε1(W1(I+εγgε1(x))Ωεγε·gε1)dx,I01(g1):=Ω(Qhom1(g1(x))1·g1)dx.
Thanks to the Lipschitz condition (32b), we can decompose Iε into the sum Iε0+Iε1 at the expense of a small error. More precisely, the following lemma holds.

Lemma 10.

Suppose that mε=εγ. There exists a constant C>0 such that for all ε>0 and φεH01(Ω), ε1γgε0+gε1:=(16)φε we have

|Iε(φε)(Iε0(gε0)+Iε1(gε1))|Cεγ(1+Φεγ(φε)).

Proof.

Note that

|Iε(φε)(Iε0(gε0)+Iε1(gε1))|=|Ωε0W0(I+εγ(ε1γgε0+gε1))W0(I+εgε0)dx|.
In view of (32b) and (17), the statement follows. □

The following lemma is a simple consequence of [16, Lemma 21, Lemma 22] and (25c):

Lemma 11.

Assume (25a), (25c) and mε=eγ.

  • (a) Consider a sequence gε0H01(Ωε0). If gε02g0 and εg012yg0, then

    lim infε0Iε0(gε0)I00(g0).

  • (b) For all g0L2(Ω,H01(Y0)) there exists a sequence gε0H01(Ωε0) such that gε02g0, εgε02yg0, and

    limε0Iε0(gε0)=I00(g0).

For the stiff part one can prove (similar to Lemma 6) the following lemma:

Lemma 12.

Assume that (25a)–(25c) hold.

  • (a) Consider gε1H01(Ω). If gε1g1 weakly in H1(Ω), then

    lim infε0Iε1(gε1)I01(g1).

  • (b) For all g1H01(Ω) there exists a sequence gε1H01(Ω) such that

    gε1g1weakly in H1(Ω),andlimε0Iε1(gε1)=I01(g1).

We proceed to the proof of Theorem 3.

Proof of Theorem 3.

Step 1. The proof of parts (a) and (b).

Statement (a) and (b) directly follow from Lemma 10, Lemma 11 and Lemma 12.

Step 2. Proof of (33): convergence of the minima.

For brevity set

eε:=infφH01(Ω)ε2γIε(φε),e0:=infg0L2(Ω,H01(Y0))g1H01(Ω)Ifinite(g0,g1).
We prove (33) in the form of the two inequalities
(80)lim supε0eεe0,lim infε0eεe0.
The argument for the first inequality in (80) is standard: for δ>0 choose (g0,g1) with Ifinite(g0,g1