Homogenization of layered materials with rigid components in single-slip finite crystal plasticity

We determine the effective behavior of a class of composites in finite-strain crystal plasticity, based on a variational model for materials made of fine parallel layers of two types. While one component is completely rigid in the sense that it admits only local rotations, the other one is softer featuring a single active slip system with linear self-hardening. As a main result, we obtain explicit homogenization formulas by means of Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-convergence. Due to the anisotropic nature of the problem, the findings depend critically on the orientation of the slip direction relative to the layers, leading to three qualitatively different regimes that involve macroscopic shearing and blocking effects. The technical difficulties in the proofs are rooted in the intrinsic rigidity of the model, which translates into a non-standard variational problem constraint by non-convex partial differential inclusions. The proof of the lower bound requires a careful analysis of the admissible microstructures and a new asymptotic rigidity result, whereas the construction of recovery sequences relies on nested laminates.


Introduction
The search for new materials with desirable mechanical properties is one of the key tasks in materials science.As suitable combinations of different materials may exceed their individual constituents with regard to important characteristics, like strength, stiffness or ductility, composites play an important role in material design, e.g.[33,25,42].In this pursuit, the following question is of fundamental interest: Given the arrangement and geometry of the building blocks on a mesoscopic level, as well as the deformation mechanisms inside the homogeneous components, can we predict the macroscopic material response of a sample under some applied external load?
By now there are various homogenization methods available that help to give answers.A substantial body of literature has emerged in materials science, engineering, and mathematics, see for instance [24,33] and the references therein, or more specifically, [37,41,20,26] for heterogeneous plastic materials, and [30,2,32] for fiber-reinforced materials, and [9,5] for highcontrast composites, to mention just a few references.A rigorous analytical approach that has proven successful for variational models based on energy minimization principles rests on the concept of Γ-convergence introduced by de Giorgi and Franzoni [17,18].By letting the length scale of the heterogeneities tend towards zero, one passes to a limit energy, which gives rise to the effective material model.In this paper, we follow along these lines and study a variational model for reinforced bilayered materials in the context of geometrically nonlinear plasticity.The model is set in the plane and we assume that the material consists of periodically alternating strips of rigid components and softer ones that can be deformed plastically by single-slip.As this problem is highly anisotropic, considering the layered structure and the distinguished orientation of the slip system, there are interesting interactions to be observed.
Let Ω ⊂ R 2 be a bounded Lipschitz domain, modeling the reference configuration of an elastoplastic body in two space dimensions, and let u : Ω → R 2 be a deformation field.For describing the periodic material heterogeneities, we take the unit cell Y = [0, 1) 2 , and define for λ ∈ (0, 1) the subsets which correspond to the softer and rigid component, respectively.Throughout this paper, we identify the sets Y rig and Y soft with their Y -periodic extensions to R 2 .To provide a measure for the length scale of the oscillations between the material components, we introduce the parameter ε > 0, which describes the thickness of two neighboring layers.With these notations, the sets εY rig ∩ Ω and εY soft ∩ Ω refer to the stiff and softer layers.For an illustration of the geometric set-up see Figure 1.
Following the classic work by Kröner and Lee [28,27] on finite-strain crystal plasticity, we use the multiplicative decomposition of the deformation gradient ∇u = F e F p as a fundamental assumption.Here, the elastic part F e describes local rotation and stretching of the crystal lattice, and the inelastic part F p captures local plastic deformations resulting from the movement of dislocations.Recent progress on a rigorous derivation of the above splitting as the continuum limit of micromechanically defined elastic and plastic components has been made in [38,39].
In this model, proper elastic deformations are excluded by requiring F e to be (locally) a rotation, i.e.F e ∈ SO(2) pointwise.This lack of elasticity makes the overall material fairly rigid.For the plastic part, we impose F p = I on εY rig ∩ Ω, reflecting that there is no plastic deformation in the stiff layers.In the softer layers εY soft ∩ Ω, plastic glide can occur along one active slip system (s, m) with slip direction s ∈ R 2 with |s| = 1 and slip plane normal m = s ⊥ , so that integration of the plastic flow rule yields F p = I + γs ⊗ m, where γ ∈ R corresponds to the amount of slip, for more details see [14, Section 2].Altogether, we observe that the deformation gradient ∇u is restricted pointwise to the set and in the stiff components even to SO (2).
As regards relevant energy expressions, the latter entails that the energy density in the rigid layers is given by W rig (F ) = 0 if F ∈ SO(2) and W rig (F ) = ∞ otherwise in R 2×2 .Moreover, adopting the homogeneous single-slip model with linear self-harding introduced in [12] (cf.also [13]) gives rise to the condensed energy density in the softer layers We combine the energy contributions in the two components to obtain the heterogeneous density which is periodic with respect to the unit cell Y and reflects the bilayered structure of the material.Here, ½ U is the symbol for the characteristic function of a set U ⊂ R 2 .
According to [36,8], the dynamical behavior of plastic materials under deformation can be well approximated by incremental minimization, that is by a time-discrete variational approach (for earlier work in the context of fracture and damage see [21,22]).Note that in this paper, we discuss only the first time step.This simplification suppresses delicate issues of microstructure evolution.As system energy of the first incremental problem we consider the energy functional Bilayered elastoplastic material with periodic structure; rigid components depicted in gray, softer components with one active slip system (slip direction s) in white. and ), the space of L 2 -functions with vanishing mean value.By (1.3) and (1.2), one has the following equivalent representations of E ε , ∇u ∈ SO(2) a.e. in εY rig ∩ Ω, (1.6) and E ε (u) = ∞ otherwise in L 2 0 (Ω; R 2 ).It becomes apparent from (1.6) that the functionals E ε are subject to non-convex constraints in the form of partial differential inclusions.Even though E ε matches with an integral expression with quadratic integrand when finite, the constraints render the associated homogenization problem non-standard.In particular, it is not directly accessible to by now classical homogenization methods for variational integrals with quadratic growth as e.g. in [34,4].Due to the non-convexity of the sets M s and SO(2), it does not fall within the scope of works on gradient-constraint problems like [6,7,11], either.
Our main result is the following theorem, which holds under the additional assumption that Ω is simply connected.It amounts to an explicit characterization of the Γ-limit of (E ε ) ε as ε tends to zero (for an introduction to Γ-convergence see e.g.[16,3]), and therefore, provides the desired homogenized model that describes the effective material response in the limit of vanishing layer thickness.
Theorem 1.1 (Homogenization via Γ-convergence).The family (1.7) The pointwise restriction K s,λ for s = (s 1 , s 2 ) and λ ∈ (0, 1) is given by (1.8) Moreover, bounded energy sequences of Recalling the definition of Γ-convergence, Theorem 1.1 can be formulated in terms of these three statements: (Compactness) For ε j → 0 and (u j ) j ⊂ L 2 0 (Ω; R 2 ) with E ε j (u j ) < C for all j ∈ N, there exists a subsequence of (u j ) j (not relabeled) and (1.9) Remark 1.2.In comparison with E ε , the differential constraints in the formulation of E are substantially more restrictive, and cause the limit functional to be essentially one-dimensional.While the gradients of finite-energy deformations for E ε lie pointwise in the set M s , those for E take values in M e 1 , independent of s, and satisfy the additional restriction of a constant rotation.In particular, this implies that ∂ 1 γ = 0, as gradient fields are curl-free.Notice also that the second term in E is non-negative due to the pointwise restriction γ ∈ K s,λ .
Remark 1.3 (Generalizations of Theorem 1.1).a) Except for only minor changes, the quadratic growth in the energies E ε can be replaced by p-growth with p ≥ 2. Calling the modified functionals E p ε , we have that E p = Γ(L p )-lim ε→0 E p ε is characterized by a.e. in Ω, and E p (u) = ∞ otherwise in L p 0 (Ω; R 2 ).For p = 2 this is a reformulation of (1.7).b) In the case s = e 1 , we characterize the Γ-limit of the family (E τ ε ) ε defined in (4.1), which results from (E ε ) ε by adding a linear dissipative term with prefactor τ ≥ 0. For the details see Section 4. We remark that this extension is motivated by [14] and [12].Whereas an explicit relaxation of the model involving the sum of a quadratic and linear expression is (to the best of our knowledge) unsolved, the homogenization result in the special case s = e 1 gets by without microstructure formation and can therefore manage the mixed expression.
In the special cases, where the slip direction is parallel or orthogonal to the layered structure, the result of Theorem 1.1 reflects some basic physical intuition.While for s = e 2 the effective body can only be rotated as a whole, as the rigid layers lead to a complete blocking of the slip system, the slip system is unimpeded if s = e 1 , so that, macroscopically, (up to global rotations) exactly all shear deformations in horizontal direction can be achieved.If the slip direction is inclined, i.e. s / ∈ {e 1 , e 2 }, the pointwise restriction γ ∈ K s,λ implies both that the effective horizontal shearing is only uni-directional (with the relevant direction depending on the orientation of s), which indicates a loss of symmetry, and that its maximum amount is capped.In the limit energy, the factor s 2 1 /λ in front of the quadratic expression in γ corresponds to an effective hardening modulus, recalling that λ ∈ (0, 1) stands for the relative thickness of the softer material layers.For s / ∈ {e 1 , e 2 }, one observes (maybe surprisingly) an additional energy contribution that is linear in γ, which can be interpreted as a dissipative term.
Regarding the proof of Theorem 1.1, we perform the usual three steps for Γ-convergence results by showing compactness and establishing matching upper and lower bounds.
The key to compactness and the lower bound is to capture the macroscopic effects of the bilayered material structure, which lead to an anisotropic reinforcement of the elastoplastic body.In Proposition 2.1, we establish a new type of asymptotic rigidity result, which is not specific to the context of plasticity, but potentially applies to any kind of composite with rigid layers, provided that the macroscopic material response is a priori known to be volume-preserving.The reasoning relies on a well-known result by Reshetnyak (cf.Lemma 2.3), which implies that the stiff layers can only rotate as a whole, on an explicit estimate showing that rotations on neighboring rigid layers are close, and on a suitable one-dimensional compactness argument.As a consequence of Proposition 2.1, the weak limits of finite energy sequences for (E ε ) ε coincide necessarily with globally rotated shear deformations in e 1 -direction.Gradients of the latter have the form R(I + γe 1 ⊗ e 2 ) with a constant rotation R ∈ SO(2) and scalar valued function γ.Note that this result holds for any orientation of the slip system s.
For the upper bound, we construct recovery sequences, meaning sequences of admissible deformations for (E ε ) ε that are energetically optimal in the limit ε → 0. If s = e 1 , the construction is quite intuitive, one simply compensates for the rigid layers by gliding more in the softer components, namely by a factor 1/λ. Analogue constructions for s = e 1 are in general not compatible, which makes this case more involved.After suitable approximation and localization, we may focus on affine limit deformations u with gradient ∇u = F ∈ M e 1 ∩ M qc s , where M qc s denotes the quasiconvex hull of M s , cf. (3.3).The observation that admissible sequences which are affine on all layers do not exist due to a lack of appropriate rank-one connections between M s and SO(2) (see Lemma 3.1 and [10]) motivates to drop the assumption of admissibility at first.Indeed, functions with piecewise constant gradients oscillating between the larger set M qc s and SO(2) yield asymptotically optimal energy values.Finally, to make this construction admissible, we glue fine simple laminates with gradients in M s into the softer layers, ensuring the preservation of the affine boundary values.This approximating laminate construction, as well as the adaption argument for the boundary, is based on work by Conti and Theil [14,12], which uses, in particular, convex integration in the sense of Müller and Šverák [35].
The manuscript is organized as follows.In Section 2, we state and prove the asymptotic rigidity result along with a useful corollary.These are the essential ingredients for proving our main result.We collect some preliminaries on admissible macroscopic deformations in Section 3, including both necessary conditions and relevant construction tools for laminates that are needed for finding recovery sequences.After these preparations, we proceed with the proof of Theorem 1.1, which is subdivided into two sections.Section 4 covers the simpler case s = e 1 in a slightly generalized setting, and Section 5 gives the detailed proofs for s = e 1 .Finally, Section 6 briefly discusses the relation between the limit functional E and (multi)cell formulas.
Notation.The standard unit vectors in R 2 are denoted by e 1 , e 2 , and a ⊥ = (−a 2 , a 1 ) for a = (a 1 , a 2 ) ∈ R 2 .For the tensor product between vectors a, b ∈ R 2 we write a⊗b = ab T ∈ R 2×2 .Further, let |F | = (F F T ) 1/2 be the Frobenius norm of F ∈ R 2×2 .With ⌈t⌉ and ⌊t⌋, let us denote the smallest integer not less and largest integer not greater than t ∈ R, respectively.For a set U ⊂ R 2 , the characteristic function ½ U is given by ½ U (x) = 1 for x ∈ U , and ½ U (x) = 0 if x / ∈ U .When referring to a domain Ω ⊂ R 2 , we mean that Ω is an open, connected, and nonempty set.
In the two-dimensional setting of this paper, the curl operator is defined as follows, curl Notice that we often use generic constants, so that the value of a constant may vary from one line to the other.Moreover, families indexed with ε > 0, may refer to any sequence (ε j ) j with ε j → 0 as j → 0.

Asymptotic rigidity of materials with stiff layers
In this section, we examine the qualitative effect of rigid layers on the macroscopic material response of the composite.The following result provides quite restrictive structural information on volume-preserving effective deformations.
Proposition 2.1 (Asymptotic rigidity for layered materials).Let Ω ⊂ R 2 be a bounded Lipschitz domain.Suppose that the sequence in Ω, and for all ε > 0 with Y rig as defined in (1.1).Then there exists a matrix R ∈ SO (2) (2.1) Furthermore, Remark 2.2.a) Considering the model introduced in Section 1, any weakly converging sequence (u ε ) ε of bounded energy for (E ε ) ε as defined in (1.5) fulfills the requirements of Proposition 2.1.
As a consequence of the weak continuity of the Jacobian determinant (precisely, ⇀ det ∇u in the sense of measures, see e.g.[19] and the references therein), the weak limit function u satisfies the volume constraint det ∇u = 1 a.e. in Ω.
In fact, (2.1) provides a necessary condition for the class of admissible deformations in the effective limit model.It indicates that, macroscopically, (up to a global rotation) only horizontal shear can be achieved.
b) Notice that due to the gradient structure of ∇u in (2.1), the function γ is independent of x 1 in the sense that its distributional derivative ∂ 1 γ vanishes.This follows immediately from 0 The outline of the proof of Proposition 2.1 is as follows.First, we conclude from the well-known rigidity result in Lemma 2.3, applied to the connected components of Ω ∩ εY rig , that each stiff layer can only be rotated as a whole.The resulting rotation matrices are then used to construct a sequence of one-dimensional piecewise constant auxiliary functions for which we establish compactness and from which we obtain structural information on ∇u.More precisely, as a consequence of the explicit estimate in Lemma 2.4, the rotations of neighboring stiff layers are close for small ε, and the auxiliary sequence has bounded variation.By Helly's selection principle one can extract a pointwise converging subsequence whose limit function lies in SO(2) a.e. in Ω, since lengths are preserved in this limit passage.Along with det ∇u = 1, this observation translates into the representation ∇u = R(I + γe 1 ⊗ e 1 ) with R ∈ SO(2) a.e. in Ω.Finally, to prove that R is constant, we exploit essentially the gradient structure of ∇u.
Before giving the detailed arguments, let us briefly state one of the key tools, which, in its classical version, is also known as Liouville's theorem.The first proof in the context of Sobolev maps goes back to Reshetnyak [40], for a quantitative generalization of the result we refer to [23,Theorem 3.1].

Lemma 2.3 (Rigidity for Sobolev functions).
Let Ω ⊂ R 2 be a bounded Lipschitz domain and u ∈ W 1,2 (Ω; R 2 ) with ∇u(x) ∈ SO(2) for a.e.x ∈ Ω.Then u is harmonic and there is a constant rotation R ∈ SO (2) An explicit estimate of the distance between rotations of neighboring stiff layers is given in the following lemma.
Lemma 2.4.Let P = (0, L) × (0, H) with L, H > 0. For i = 1, 2, let w i : P → R 2 be the affine functions defined by in the sense of traces, then has a unique solution.Indeed, by Jensen's inequality, the minimizer v of (2.4) is given by linear interpolation as v we therefore obtain Minimizing this expression with respect to b 1 and b 2 gives (2.3).
Proof of Proposition 2.1.To characterize the limit function u, we will show that the statement holds locally, i.e. on any open cube Q ⊂ Ω with sides parallel to the coordinate axes.Precisely, there exists a rotation To deduce (2.1), it suffices to exhaust Ω with overlapping cubes Q.This way one finds that all R Q coincide, leading to a global rotation R ∈ SO(2).Without loss of generality, let us assume in the following that Q = (0, l) 2 with l > 0. To describe the layered structure of the material, we introduce the notation for the horizontal strips in a larger open cube Q ′ ⊂ Ω that compactly contains Q.The index set selects those strips of thickness ε that are fully contained in Q ′ .Then, by taking ε sufficiently small one has that Q ⊂ i∈Iε P i ε .We subdivide the remaining proof in six steps.
Step 1: Classical rigidity and approximation by piecewise affine functions.Applying Lemma 2.3 to each strip P i ε with i ∈ I ε yields the existence of rotation matrices R i ε ∈ SO(2) and translation and let For each i ∈ I ε , we apply a 1-d version of the Poincaré inequality to derive that with constants c > 0 independent of ε.Summing over all i ∈ I ε gives , and thus, (2.5).
We point out that (σ ε ) ε and (b ε ) ε are uniformly bounded in L ∞ (Q; R 2 ) and L 2 (Q; R 2 ), respectively.The latter follows together with (2.5) and the uniform boundedness of (u ε ) ε in L 2 (Ω; R 2 ).Consequently, there are subsequences of (σ ε ) ε and (b ε ) ε (not relabeled) and func- ), so that, in view of (2.5) and the uniqueness of weak limits, Notice that ∂ 1 b = 0 due to the fact that the functions b ε are independent of x 1 considering the definition of the strips P i ε .
As a continuous, piecewise affine function Π ε is almost everywhere differentiable, and together with (2.9) or (2.8) it holds that In particular, (Π ε ) ε is uniformly bounded in W 1,2 (0, l; R 2×2 ), and therefore (Π ε ) ε admits a weakly converging subsequence with limit Π ∈ W 1,2 (0, l; R 2×2 ).From (2.10), (2.11), and the uniqueness of the limit we infer that Σ = Π ∈ W 1,2 (0, l; SO(2)). (2.12) By constant extension of Σ in x 1 -direction we define a map R on Q, precisely we set Step 4: Establishing ∇u ∈ M e 1 pointwise.The estimate along with (2.10) and the first part of (2.6) leads to From (2.7) and the independence of b of x 1 we then conclude that This shows in particular that R, Σ, σ, and b are independent of the choice of subsequences in (2.10) and (2.6).Moreover, by (2.12) and (2.13) it is immediate to see that σ, b ∈ W 1,2 (Q; R 2 ).Since R ∈ SO(2) pointwise by Step 3, one has that |∇ue 1 | = 1 a.e. in Q.In conjunction with det ∇u = 1 a.e. in Q, we conclude that ∇u ∈ M e 1 a.e. in Q.In view of (2.14), there exists a function γ ∈ L 2 (Q) such that (2.15) Step 5: Proving R constant.Using (2.7) and (2.13), we compute that for a.e.x ∈ Q.Then, along with (2.15) and the independence of R of x 1 , it follows for the distributional derivative of γ that

16)
As curl ∇u = 0 in Q in the sense of distributions, the representation (2.15) entails , where we have used ∂ 1 R = 0 and (2.16).This shows ∂ 2 Re 1 = 0, which implies that R is a constant rotation.
As discussed in Remark 2.2, Proposition 2.1 imposes structural restrictions on the limits of bounded energy sequences for (E ε ) ε .As a consequence, we obtain an asymptotic lower bound energy estimate, which constitutes a first step toward the proof of the liminf-inequality (1.9) for the Γ-convergence result in Theorem 1.1.
Proof.One may assume in the following that u is affine, otherwise the same arguments can be applied to each affine piece of u.

Discussion of admissible deformations
In preparation for the proof of Theorem 1.1, we exploit the specific form of the functionals E ε to identify further properties of the weak limits of bounded energy sequences.Moreover, we provide the basis for the laminate constructions that are the key to obtaining suitable recovery sequences.
3.1.Necessary conditions for admissible macroscopic deformations.Let (u ε ) ε ⊂ W 1,2 (Ω; R 2 ) satisfy ∇u ε ∈ M s a.e. in Ω, and suppose that u ε ⇀ u in W 1,2 (Ω; R 2 ) for some u ∈ W 1,2 (Ω; R 2 ).As the convex set {F ∈ L 2 (Ω; R 2×2 ) : |F s| ≤ 1 a.e. in Ω} is weakly closed in L 2 (Ω; R 2×2 ) and det ∇u ε * ⇀ det ∇u in the sense of measures (cf.Remark 2.2 a)), we know that ∇u ∈ N s a.e. in Ω, ( where According to [14] (see also [13]), the set N s is exactly the quasiconvex hull M qc s of M s .With S = (s|m) = (s|s ⊥ ) ∈ SO(2), another alternative representation of N s is If we assume in addition that (u ε ) ε is a sequence of bounded energy, precisely, E ε (u ε ) < C for all ε, then ∇u ∈ M e 1 pointwise almost everywhere in Ω by the rigidity result in Proposition 2.1 and Remark 2.2 a).Thus, together with (3.1), ∇u ∈ M e 1 ∩ N s a.e. in Ω. (3.3) For s = e 1 the restriction in (3.3) is equivalent to ∇u ∈ M e 1 a.e., while for s = e 1 a straightforward computation shows that with K s,1 as defined in (1.8).In the case s = e 1 , condition (3.3) can be refined even further by exploiting the presence of the rigid layers with their asymptotic volume fraction |Y rig | = 1 − λ.Indeed, from Proposition 2.1 we infer that there are R ∈ SO(2) and γ ∈ L 2 (Ω) such that 3.2.Tools for the construction of admissible deformations.We start by characterizing all rank-one connections in M s , cf. also [10].In particular, in case of i), F − G = (γ − ζ)Rs ⊗ m, while for ii) one has (3.5) Proof.Considering (3.2), it is enough to prove the statement for s = e 1 .Moreover, we may assume without loss of generality that Thus, either Re 1 = e 1 , i.e.R = I, or It was first proven in [14] that N s = M qc s coincides with the rank-one convex hull M rc s , which in particular, means that every N ∈ N s can be expressed as a convex combination of rank-one connected matrices in M s .A specific type of rank-one directions, which turns out optimal for the relaxation of W soft , was discussed by Conti in [12], see also [13].Here we give a different argumentation based on Lemma 3.1.For the first part of (3.7), it is necessary that where we have used (3.5) along with (3.8).Taking squared norms in the above equation imposes a constraint on µ in the form of a quadratic equation, which has two solutions µ 1 ∈ (0, 1/2) and µ 2 ∈ (1/2, 1) with µ 1 + µ 2 = 1.Depending on which of these values is selected for µ, we adjust the rotation Q so that (3.9) holds.It follows from (3.9) and (3.8) that As |N m| −1 ≤ |N s|, the set A N contains exactly two elements, one of which being N m.Finally, we take µ ∈ (0, 1) with corresponding Q such that Gm = N m, which finishes the proof.Let us remark that choosing γ = − |N m| 2 − 1 in (3.8) essentially comes up to switching F and G.
In the case of a non-horizontal slip direction, optimal constructions of admissible deformations cannot be achieved based on rank-one connections in M s .Instead, we employ simple laminates with gradients in SO(2) and N s (and normal e 2 ).The following one-to-one correspondence between γ ∈ K s,λ and R ∈ SO(2), and N ∈ N s with |N e 1 | = 1 is helpful for the explicit constructions.The following two theorems are taken from Conti & Theil [14] and Müller & Šverák [35], respectively.In combination, they allow us modify a simple laminate with gradients in M s in a small part of the domain in such a way that the resulting Lipschitz function takes affine boundary values in N s , while preserving the constraint that gradients lie pointwise in M s , see Corollary 3.7.

Theorem 3.5 ([14, Theorem 4]).
Let Ω ⊂ R 2 be a bounded domain and µ ∈ (0, 1).Suppose that F, G ∈ M s are rank-one connected with F s = Gs and Then for every δ > 0 there are h 0 δ > 0 and Ω δ ⊂ Ω with |Ω \ Ω δ | < δ such that the restriction to Ω δ of any simple laminate between the gradients F and G with weights µ and 1 − µ and period h < h 0 δ can be extended to a finitely piecewise affine function Convex integration methods help to obtain exact solutions to partial differential inclusions.
an in-approximation of M, i.e., the sets U i are open in {F ∈ R 2×2 : det F = 1} and uniformly bounded, U i is contained in the rank-one convex hull of U i+1 for every i ∈ N, and (U i ) i converges to M in the following sense: if Then, for any F ∈ U 1 and any open domain Ω ⊂ R 2 , there exists u ∈ W 1,∞ (Ω; R 2 ) such that ∇u ∈ M a.e. in Ω and u = F x on ∂Ω.
Proof.From Theorem 3.5 we obtain for δ > 0 the desired set Ω δ along with a finitely piecewise affine function v δ : Ω → R 2 that coincides in Ω δ with a simple laminate between the gradients F and G of period h δ < min{δ, h 0 δ }, satisfies ∇v δ ∈ N s a.e. in Ω, and v δ = N x on ∂Ω.In view of (3.7), |∇v δ m| = |N m| a.e. in Ω δ and a.e. in Ω.Finally, the sought function u δ results from a modification of v δ in the (finitely many) domains where ∇v δ / ∈ M s by applying Theorem 3.6 with the in-approximation ( see [14, Proof of Lemma 2] for more details.

Proof of Theorem 1.1 for s = e 1
As indicated in the introduction, in the special case of a horizontal slip direction s = e 1 , we can prove Theorem 1.1 in a slightly more general setting, where W soft has an additional linear term that can be interpreted as a dissipative energy contribution.
More precisely, for a given τ ≥ 0 let us replace W soft with Then, E ε of (1.4) with ε > 0 turns into for u ∈ L 2 0 (Ω; R 2 ), cf.(1.5).In this section, we prove the following generalization of Theorem 1.1 in the case s = e 1 , assuming that Ω is simply connected.
Proof.The proof is divided into three steps.
The main idea for the construction of a recovery sequence is to set γ = 0 in the stiff layers, as the functional E τ ε requires, while compensating with more gliding in the softer layers.Therefore, for ε > 0 we put Let us assume for the moment that the function R(I + γ ε e 1 ⊗ e 2 ) ∈ L 2 (Ω; R 2×2 ) has a potential, meaning that there exists u ε ∈ W 1,2 (Ω; R 2 ) with without loss of generality, we can take u ε ∈ L 2 0 (Ω; R 2 ).By the weak convergence of oscillating periodic functions one has that ½ εY soft ∩Ω * ⇀ λ in L ∞ (Ω), which implies Consequently, it follows in view of Poincaré's inequality that u ε ⇀ u in W 1,2 (Ω; R 2 ), and by compact embedding u ε → u in L 2 (Ω; R 2 ).Regarding the convergence of energies we argue that It remains to prove the existence of u ε ∈ W 1,2 (Ω; R 2 ) such that (4.4) holds.If Ω is a cube Q ⊂ R 2 with sides parallel to the coordinate axes, say Q = (0, l) 2 with l > 0, then γ ε is independent of x 1 in view of Remark 2.2 b) and the orientation of the layers, hence, depending on the context, it can be interpreted as an element in L 2 (Q) or L 2 (0, l).Then, for any a ∈ R 2 , satisfies To construct u ε for a general Ω, we exhaust Ω successively with shifted, overlapping cubes Q, using (4.5) with suitably adjusted translation vectors a.
Step 3: Lower bound.Let (ε j ) j with ε j → 0 as j → ∞, and u j → u in L 2 (Ω; R 2 ).We assume without loss of generality that (u j ) j is a sequence of uniformly bounded energy for (E τ ε j ) j , so that (u j ) j and u satisfy the properties of Step 1.
If u is piecewise affine, the desired liminf inequality then follows directly from Corollary 2.5, as , and from (4.3).To prove the statement for general u, we perform an approximation argument inspired by the proof of Müller's homogenization result in [34,Theorem 1.3].Due to the differential constraints in E τ ε , however, the construction of suitable comparison functions is slightly more involved.Suppose that Ω = Q ⊂ R 2 is a cube with sides parallel to the coordinate axes, otherwise we perform the arguments below on any finite union of disjoint cubes contained in Ω and take the supremum over all these sets, exploiting the fact that the energy density in (4.1) is non-negative.
Accounting for Remark 2.2 b) allows us to find a sequence of one-dimensional simple functions (ζ k ) k (identified with a sequence in L 2 (Q) by constant extension in x 1 -direction) such that . Further, let (v j ) j and (v k,j ) j be the recovery sequences (as constructed in Step 2) for u and w k , respectively.We define observing that z k,j ⇀ w k in W 1,2 (Ω; R 2 ) for all k ∈ N as j → ∞.Moreover, ∇z k,j = ∇u j a.e. in ε j Y rig ∩ Ω, so that in particular, |∇z k,j e 2 | = 1 a.e. in ε j Y rig ∩ Ω for all j, k ∈ N, and for k ∈ N as j → ∞ by (2.2).Considering that w k is piecewise affine, it follows as in the proof of Corollary 2.5 that for all k ∈ N.With (4.7) one obtains for j, k ∈ N, where by construction and the uniform boundedness of (∇u j ) j in L 2 (Ω; R 2×2 ) by (4.2) we infer that for j, k ∈ N. Passing to the limit j → ∞ in the above estimate yields Here we have used (4.8), as well as (4.3).Finally, due to (4.6), taking k → ∞ finishes the proof of the liminf inequality.

Proof of Theorem 1.1 for s = e 1
This section is concerned with the proof of Theorem 1.1 in the case of an inclined or vertical slip direction, that is s = e 1 .Due to the strong restrictions on rank-one connections between SO(2) and M s with normal e 2 (see Remark 3.2), the construction of recovery sequences is more involved than for s = e 1 .
Step 1: Compactness.The proof of compactness is identical with the beginning of Step 1 in Theorem 4.1 for τ = 0, when substituting e 1 with s and e 2 with m.
Step 2: Recovery sequence.Let u ∈ W 1,2 (Ω; R 2 ) ∩ L 2 0 (Ω; R 2 ) such that ∇u = R(I + γe 1 ⊗ e 2 ) with R ∈ SO(2) and γ ∈ L 2 (Ω) be given.The idea of the construction is to specify first a sequence of functions with asymptotically optimal energy that are piecewise affine on the layers and whose gradients lie in N s .Then, to obtain admissible deformations, we approximate these functions in the softer layers with fine simple laminates between gradients in M s , which requires tools from relaxation theory and convex integration as discussed in Section 3.2.
Step 2a: Auxiliary functions for constant γ.Let γ ∈ K s,λ be constant.By Lemma 3.4 we find N ∈ N s such that (5.2) and N e 1 = Re 1 , which guarantees the compatibility for constructing laminates between the gradients R and N with e 2 the normal on the jump lines of the gradient.Precisely, we define for ε > 0 the function v ε ∈ W 1,2 (Ω; R 2 ) with zero mean value characterized by ∇v ε = ∇v 1 (ε −1 q ), where v 1 ∈ W 1,∞ loc (R 2 ; R 2 ) is such that (5.3) Then, by the weak convergence of highly oscillating functions and (5.2), Regarding the energy contribution of the sequence (v ε ) ε it follows that Notice that v ε is not admissible for Step 2b: Admissible recovery sequence for constant γ.Next, we modify the construction of Step 2a in the softer layers to obtain admissible functions, while preserving the energy.This is done by approximation with the simple laminates established in Corollary 3.7, see Figure 2 for illustration.
In view of the independence of γ on x 1 by Remark 2.2 b), we may identify γ ∈ L 2 (Ω) with a one-dimensional simple function with γ i ∈ K s,λ for i = 1, . . ., n and 0 = t 0 < t 1 < . . .< t n = l.Let us denote by N i ∈ N s the matrices corresponding to γ i according to Lemma 3.4 i ).
For i ∈ {1, . . ., n}, let (u i,ε ) ε ⊂ W 1,2 ((0, l) × (t i−1 , t i ); R 2 ) be the recovery sequences corresponding to γ i as constructed in Step 2b.We then define u ε ∈ W 1,2 (Ω; R 2 ) with vanishing mean value by Notice that u ε is well-defined due to the compatibility between ∇u i,ε and R along the jump lines R × εZ.Then, (5.6) leads to and regarding the energy contributions it follows that To generalize the result to a Lipschitz domain Ω, we exhaust Ω successively with shifted, overlapping cubes, performing the necessary adaptions of the glued-in laminate constructions as well as the appropriate translations, cf.Step 2 of Theorem 4.1 for a related argument.
Step 2d: Approximation and diagonalization for general γ.For general γ ∈ L 2 (Ω) with γ ∈ K s,λ a.e. in Ω we use an approximation and diagonalization argument. and From the selection principle by Attouch [1, Corollary 1.16], we infer the existence of a diagonal sequence (u ε ) ε with u ε = w k(ε),ε such that Step 3: Lower bound.Let (ε j ) j with ε j → 0 as j → ∞.Suppose (u j ) j is a bounded energy sequence for (E ε j ) j with u as an immediate consequence of Corollary 2.5.Similarly to Step 3 in the proof of Theorem 4.1, we use approximation to establish the lower bound for general u.Here again, we may restrict ourselves to working with the assumption that Ω is a cube, say Ω = Q = (0, l) 2 for l > 0.
Let (ζ k ) k ⊂ L 2 (0, l) be one-dimensional simple functions (identified with a sequence in L 2 (Ω) by constant extension in x 1 ) of the form (5.9) Moreover, let w k ∈ W 1,2 (Ω; R 2 ) ∩ L 2 0 (Ω; R 2 ) be given by ∇w k = R(I + ζ k e 1 ⊗ e 2 ) for k ∈ N. In the following, we aim at finding sequences (v j ) j , (v k,j ) j ⊂ W 1,2 (Ω; R 2 ) with vanishing mean value such that v j ⇀ u and v k,j ⇀ w k both in W 1,2 (Ω; R 2 ) as j → ∞ (5.10) for all k ∈ N, and ∇v k,j = ∇v j a.e. in ε j Y rig ∩ Ω (5.11) for all j, k ∈ N.Moreover, we seek to have an estimate of the type with c > 0 independent of j, k.
Notice that instead of using recovery sequences for (v j ) j and (v k,j ) j , we will choose the piecewise affine functions obtained from Step 2, when skipping Step 2b (where fine laminates are glued in the softer layers).Indeed, the lack of admissibility does not cause any issues here.The advantage, though, is that due to their simpler structure, these functions are easier to compare in the sense of (5.12).Recall that the full recovery sequences of Step 2d involve regions resulting from convex integration, where the functions are not explicitly known and therefore hard to control.
Precisely, for j, k ∈ N we define v k,j by with N k,i ∈ N s corresponding to ζ k,i in the sense of Lemma 3.4 i ), while (v j ) j results from a diagonalization argument as in Step 2d, i.e. v j = v k(j),j for j ∈ N. Hence, (5.10) and (5.11) are satisfied.Regarding (5.12), we argue that for j, k, K ∈ N, where we have used (3.11).Thus, which yields (5.12).Now we set for j, k ∈ N. Due to (5.11), it holds that |∇z k,j e 2 | = 1 a.e. in ε j Y rig ∩ Ω, as well as for all k ∈ N. Along with (5.12) and (5.9), we finally conclude that in analogy to Step 3 in the proof of Theorem 4.1.
Remark 5.1.It may be more intuitive from the point of view of applications -yet technically more elaborate -to replace the recovery sequence obtained in Step 2b for the affine case by optimal deformations showing "non-stop" simple laminates throughout the softer layers.For this construction, just dispense with the adjustment of the affine boundary conditions along the vertical edges of the unit cell, and instead refine and shift the laminate appropriately to guarantee Y -periodicity.

Comparison with the (multi)cell formula
It is a well-known result in the theory of periodic homogenization of integral functionals with standard growth that the integrand of the effective limit functional is characterized by a multicell formula, or, in the convex case, by a cell formula, see e.g.[34], [31].In this final section, we show that the same is true for the homogenization result in Theorem 1.1, where extended-valued functionals appear.
Recalling W defined in (1.3), we consider the multicell formula Moreover, let us denote by W hom the density of the limit energy E in (1.7), i.e. where This alternative representation of E follows from a straightforward calculation, see (5.1).Before focusing on the relation between W hom , W # , and W cell , we prove the following auxiliary result.As the map det : F → det F is quasiaffine, i.e. det and − det are both quasiconvex (see e.g.[15] for an introduction to generalized notions of convexity), it follows that det F = This finishes the proof.
As indicated above, the multicell and cell formula for W both coincide with the homogenized integrand W hom .Proposition 6.2 (Characterization of the (multi)cell formula).With the definitions above it holds that W hom = W # = W cell .
Proof.Since trivially W # ≤ W cell , it is enough to prove W # ≥ W hom and W hom ≥ W cell .From Lemma 6.1 i ) we infer that W # (F ) = W cell (F ) = ∞ for F / ∈ M e 1 ∩N s ⊃ {F ∈ R 2×2 : W hom (F ) < ∞}, cf.(3.4).Therefore, in the following, we can restrict ourselves to F ∈ M e 1 ∩ N s .

Figure 2 .
Figure 2. Construction of admissible deformations by approximation with fine simple laminates in the softer component (illustrated by ∇z ε in the unit cell Y ).Here, N e 1 = Re 1 , N = µF + (1 − µ)G with F, G ∈ M s and µ ∈ (0, 1) as in Lemma 3.7, and the measure of the boundary region of Y soft is smaller than ε.
for j → ∞ by (5.10) with w k piecewise affine, we argue as in the proof of Corollary 2.5 to derive lim inf j→∞ Ω |∇z k,j m| 2 − 1 dx ≥ E(w k )