Homogenization in $BV$ of a model for layered composites in finite crystal plasticity

In this work, we study the effective behavior of a two-dimensional variational model within finite crystal plasticity for high-contrast bilayered composites. Precisely, we consider materials arranged into periodically alternating thin horizontal strips of an elastically rigid component and a softer one with one active slip system. The energies arising from these modeling assumptions are of integral form, featuring linear growth and non-convex differential constraints. We approach this non-standard homogenization problem via Gamma-convergence. A crucial first step in the asymptotic analysis is the characterization of rigidity properties of limits of admissible deformations in the space $BV$ of functions of bounded variation. In particular, we prove that, under suitable assumptions, the two-dimensional body may split horizontally into finitely many pieces, each of which undergoes shear deformation and global rotation. This allows us to identify a potential candidate for the homogenized limit energy, which we show to be a lower bound on the Gamma-limit. In the framework of non-simple materials, we present a complete Gamma-convergence result, including an explicit homogenization formula, for a regularized model with an anisotropic penalization in the layer direction.


Introduction
Metamaterials are artificially engineered composites whose heterogeneities are optimized to improve structural performances. Due to their special mechanical properties, arising as a result of complex microstructures, metamaterials play a key role in industrial applications and are an increasingly active field of research. Two natural questions when dealing with composite materials are how the effective material response is influenced by the geometric distribution of its components, and how the mechanical properties of the components impact the overall macroscopic behavior of the metamaterial.
In what follows, we investigate these questions for a special class of metamaterials with two characteristic features that are of relevance in a number of applications: (i) the material consists of two components arranged in a highly anisotropic way into periodically alternating layers, and (ii) the (elasto)plastic properties of the two components exhibit strong differences, in the sense that one is rigid, while the other one is considerably softer, allowing for large (elasto)plastic deformations. The analysis of variational models for such layered high-contrast materials was initiated in [13]. There, the authors derive a macroscopic description for a two-dimensional model in the context of geometrically nonlinear but rigid elasticity, assuming that the softer component can be deformed along a single active slip system with linear selfhardening.
These results have been extended to general dimensions, to energy densities with p-growth for 1 < p < +∞, and to the case with non-trivial elastic energies, which allows treating very stiff (but not necessarily rigid) layers, see [14,12].
In this paper, we carry the ideas of [13] forward to a model for plastic composites without linear hardening, in the spirit of [18]. This change turns the variational problem in [13], having quadratic growth (cf. also [15,16]), into one with energy densities that grow merely linearly.
The main novelty lies in the fact that the homogenization analysis must be performed in the class BV of functions of bounded variation (see [2]) to account for concentration phenomena. This gives rise to conceptual mathematical difficulties: on the one hand, the standard convolution techniques commonly used for density arguments in BV or SBV cannot be directly applied because they do not preserve the intrinsic constraints of the problem; on the other hand, constraint-preserving approximations in this weaker setting of BV are rather challenging, as one needs to simultaneously regularize the absolutely continuous part of the distributional derivative of the functions and accommodate their jump sets.
To state our results precisely, we first introduce the relevant model with its main modeling hypotheses. Throughout the article, we analyze two versions of the model, namely with and without regularization.
Let e 1 and e 2 be the standard unit vectors in R 2 , and let x = (x 1 , x 2 ) denote a generic point in R 2 . Unless specified otherwise, Ω ⊂ R 2 is an x 1 -connected, bounded domain with Lipschitz boundary, that is, an open set whose slices in the x 1 -direction are ( Assume that Ω is the reference configuration of a body with heterogeneities in the form of periodically alternating thin horizontal layers. To describe the bilayered structure mathematically, consider the periodicity cell Y := [0, 1) 2 , which we subdivide into Y = Y soft ∪ Y rig with Y soft := [0, 1) × [0, λ) for λ ∈ (0, 1) and Y rig := Y \ Y soft . All sets are extended by periodicity to R 2 . The (small) parameter ε > 0 describes the thickness of a pair (one rigid, one softer) of fine layers, and can be viewed as the intrinsic length scale of the system. The collections of all rigid and soft layers in Ω can be expressed as εY rig ∩ Ω and εY soft ∩ Ω, respectively. For an illustration of the geometrical assumptions, see Figure 1. Following the classical theory of elastoplasticity at finite strains (see, e.g., [31] for an overview), we assume that the gradient of any deformation u : Ω → R 2 decomposes into the product of an elastic strain, F el , and a plastic one, F pl . In the literature, different models of finite plasticity have been proposed (see, e.g., [3,22,29,30,37]), as well as alternative descriptions via the theory of structured deformations (see [10,11,24,6] and the references therein). Here, we adopt the classical model by Lee on finite crystal plasticity introduced in [33,35,34], according to which the deformation gradients satisfy ∇u = F el F pl .
The elastoplastic energy of a deformation u ∈ L 1 0 (Ω; R 2 ) := {u ∈ L 1 (Ω; R 2 ) : Ω u dx = 0}, given by represents the internal energy contribution of the system during a single incremental step in a timediscrete variational description. This way of modeling excludes preexistent plastic distortions, and can be considered a reasonable assumption for the first time step of a deformation process. The elastoplastic energy can be complemented with terms modeling the work done by external body or surface forces. The limit behavior of sequences (u ε ) ε of low energy states for (E ε ) ε gives information about the macroscopic material response of the layered composites. In the following, we focus the analysis of this asymptotic behavior on the s = e 1 case, when the slip direction is parallel to the orientation of the layers, cf. also Figure 1. Note that different slip directions can be treated similarly, but the arguments are technically more involved. In fact, for s / ∈ {e 1 , e 2 }, small-scale laminate microstructures on the softer layers need to be taken into account, which requires an extra relaxation step. We refer to [18] for the relaxation mechanism and to [13] for the strategy of how to apply it to layered structures.
An important first step towards identifying the limit behavior of the energies (E ε ) ε (in the sense of Γ-convergence) is the proof of a general statement of asymptotic rigidity for layered structures in the context of functions of bounded variation. The following result characterizes the weak * limits in BV of deformations whose gradients coincide pointwise with rotations on the rigid layers of the material. Note that no additional constraints are imposed on the softer components at this point. Let Ω ⊂ R 2 be an x 1 -connected domain. Assume that (u ε ) ε ⊂ W 1,1 (Ω; R 2 ) is a sequence satisfying ∇u ε ∈ SO(2) a.e. in εY rig ∩ Ω for all ε, (1.8) and that u ε * ⇀ u in BV (Ω; R 2 ) for some u ∈ BV (Ω; R 2 ) as ε → 0. Then, Conversely, any function u ∈ BV (Ω; R 2 ) as in (1.9) can be attained as weak * -limit in BV (Ω; To prove the first part of Theorem 1.1, we adapt the arguments in [13] to the BV -setting. The second assertion follows from a tailored one-dimensional density result in BV , which involves approximating functions that are constant on the rigid layers (see Lemma 3.3 below). Up to minor adaptations, analogous statements hold in higher dimensions. We refer to Remark 3.4 for the specific assumptions on the geometry of the set Ω under which a higher-dimensional counterpart of Theorem 1.1 can be proved.
A natural potential candidate for the limiting behavior of (E ε ) ε in the sense of Γ-convergence (see [8,20] for an introduction, as well as the references therein) is the functional E : L 1 0 (Ω; R 2 ) → [0, ∞], given by where (2)), ψ ∈ BV (a Ω , b Ω ; R 2 ), and det ∇u = 1 a.e. in Ω}. (1.11) We refer to Remark 5.1 for an alternative representation of the functional E. The next theorem states that E provides indeed a lower bound for our homogenization problem.
Theorem 1.2 (Lower bound on the Γ-limit of (E ε ) ε ). Let Ω ⊂ R 2 be an x 1 -connected domain, and let E ε and E be the functionals introduced in (1.7) and (1.10), respectively. Then, every sequence (u ε ) ε ⊂ L 1 0 (Ω; R 2 ) with uniformly bounded energies, sup ε E ε (u ε ) < ∞, has a subsequence that converges weakly * in BV (Ω; R 2 ) to some u ∈ A ∩ L 1 0 (Ω; R 2 ). Additionally, The proof of the first assertion is given in Proposition 4.3. It relies on Theorem 1.1 in combination with a technical argument about the weak continuity properties of Jacobian determinants (see Lemma 4.2). In Section 5, we exhibit two different proofs of (1.12): A first one relying on a Reshetnyak's lower semicontinuity theorem (see, e.g., [2,Theorem 2.38]), and an alternative one exploiting the properties of the admissible layered deformations. The identification of E as the Γ-limit of the sequence (E ε ) ε , though, remains an open problem. Indeed, verifying the optimality of the lower bound in Theorem 1.2 is rather challenging, as it requires to approximate elements of A by means of sequences in A ε at least in the sense of the strict convergence in BV . We refer to Remark 5.2 for a detailed discussion of the main difficulties. Even if the requirement on the convergence of the energies is dropped, recovering the jumps of maps in the effective domain of E under consideration of the non-standard differential inclusions in A ε is by itself another challenging problem. Solving this problem requires delicate geometrical constructions, which are currently not available for all elements in A.
Yet, there are two subclasses of physically relevant deformations in A for which we can find suitable approximations by sequences of admissible layered deformations. The precise statement is given in Theorem 1.3 below.
The first of these two subclasses is A ∩ SBV ∞ (Ω; R 2 ) (we refer to Subsection 2.3 for the definition of the set SBV ∞ ) whose jump sets are given by a union of finitely many lines. Heuristically, this subclass describes deformations that break Ω horizontally into a finite number of pieces, which may get sheared and rotated individually.
The second subclass is In comparison with A, functions in A satisfy two additional constraints, namely the fact that the rotation R is constant and that the jumps of functions in A are parallel to Re 1 . With the notation A , we intend to highlight the second feature. The intuition behind maps in A are non-trivial macroscopic deformations that (up to a global rotation) may make the material break along finite or infinitely many horizontal lines, induce sliding of the pieces relative to each other, and cause horizontal shearing within each individual piece. For an illustration of the two subclasses, see Figure 2.
As a first step towards proving Theorem 1.3, we establish an admissible piecewise affine approximation for limiting deformations with a single jump line (see Lemma 4.5). The construction relies on the characterization of rank-one connections in M e1 proved in [13,Lemma 3.1], with transition lines stretching over the full width of Ω to avoid triple junctions (see Remark 4.6). In Propositions 4.7 and 4.9, we extend the arguments to A ∩ SBV ∞ (Ω; R 2 ) and A , respectively.
Problems in finite crystal plasticity without additional regularizations are generally known to be challenging because of the oscillations of minimizing sequences arising as a byproduct of relaxation mechanisms in the slip systems. This phenomenon is one of the main reasons why a full relaxation theory in finite crystal plasticity is still missing (see [17,Remark 3.2]). In our setting, it hampers the full characterization of weak limits of sequences with uniformly bounded energies. The observation that regularizations can help overcome the above compensated-compactness issue (see also Remark 6.2) motivates the introduction of a penalized version of our problem. After a higher-order penalization of the energy in the layer direction, we obtain the following Γ-convergence result. The attained limit deformations are given by the class A . (1.14) Then, the family (E δ ε ) ε Γ-converges with respect to the strong L 1 -topology to the functional E δ : where ϑ ′ denotes the approximate differential of ϑ (cf. Section 2.2).
The penalization in (1.14) can be viewed in the spirit of non-simple materials [39,40]. Working with stored energy densities that depend on the Hessian of the deformations has proved successful in overcoming lack of compactness in a variety of applications; see, e.g., [5,21,27,36,38]. Very recently, there has been an effort towards weakening higher-order regularizations: It is shown in [7] that the full norm of the Hessian can be replaced by a control of its minors (gradient polyconvexity) in the context of locking materials; for solid-solid phase transitions, an anisotropic second-order penalization is considered in [23]. Along these lines, we introduce the regularized energies in (1.13) that penalize the variation of deformations only in the layer direction. This is enough to deduce that the limiting rotation (as ε → 0) is global and that it determines the direction of the limiting jump. In Section 6, we provide two alternative proofs of this result: A first one relying on Alberti's rank one theorem (see Section 2.1) in combination with the approximation result in Theorem 1.3, and a second one based on separate regularizations of the regular and the singular part of the limiting maps, and inspired by [19,Lemma 3.2]. This paper is organized as follows. In Section 2.1, we collect a few preliminaries, including some background on (special) functions of bounded variation. Section 3 is devoted to the analysis of asymptotic rigidity for layered structures in the setting of BV -functions. A characterization of limits of admissible layered deformations is provided in Section 4. Eventually, Sections 5 and 6 contain the proof of a lower bound for the homogenization problem without regularization (Theorem 1.2) and the full Γ-convergence analysis of the regularized problem (Theorem 1.4), respectively.

Preliminaries
2.1. Notation. In this section, unless mentioned otherwise, Ω is a bounded domain in R N with N ∈ N. Throughout the rest of the paper, we assume mostly that N = 2.
We represent by L N the N -dimensional Lebesgue measure and by H N −1 the (N − 1)-dimensional Hausdorff measure. Whenever we write "a.e. in Ω", we mean "almost everywhere in Ω" with respect to L N ⌊Ω. To simplify the notation, we often omit the expression "a.e. in Ω" in mathematical relations involving Lebesgue measurable functions. Given a Lebesgue measurable set B ⊂ R N , we also use the shorter notation |B| = L N (B) for the Lebesgue measure of B, while the characteristic function of B in For two vectors a, b ∈ R d , a ⊗ b := ab T stands for their tensor product. If a = (a 1 , a 2 ) T ∈ R 2 , we set a ⊥ := (−a 2 , a 1 ) T . We use the standard notation for spaces of vector-valued functions; namely, L p µ (Ω; R d ) with p ∈ [1, ∞] and a positive measure µ for L p -spaces, W 1,p (Ω; R d ) with p ∈ [1, ∞] for Sobolev spaces, C(Ω; R d ) for the space of continuous functions, C ∞ (Ω; R d ) and C ∞ c (Ω; R d ) for the spaces of smooth functions without and with compact support, and C 0,α (Ω; R d ) with α ∈ [0, 1] for Hölder spaces. We denote by C 0 (Ω; R d ) the space of continuous functions that vanish on the boundary of Ω. Moreover, M(Ω; R d ) is the space of finite vector-valued Radon measures. In the case of scalar-valued functions and measures, we omit the codomain; for instance, we write L 1 (Ω) instead of L 1 (Ω; R).
Throughout this manuscript, ε stands for a small (positive) parameter, and is usually thought of as taking values on a positive sequence converging to zero.

2.2.
Functions of bounded variation. We adopt the standard notations for the space BV (Ω; R d ) of vector-valued functions of bounded variation, and refer the reader to [2] for a thorough treatment of this space. Here, we only recall some of its basic properties.
A function u ∈ L 1 (Ω; R d ) is called a function of bounded variation, written u ∈ BV (Ω; R d ), if its distributional derivative Du satisfies Du ∈ M(Ω; R d×N ). The space BV (Ω; R d ) is a Banach space when endowed with the norm u BV (Ω;R d ) := u L 1 (Ω;R d ) + |Du|(Ω), where |Du| ∈ M(Ω) is the total variation of Du.
Let D a u and D s u denote the absolutely continuous and the singular part of the Radon-Nikodym decomposition of Du with respect to L N ⌊Ω, and let D j u and D c u be the jump and Cantor parts of Du. The following chain of equalities holds: where ∇u is the approximate differential of u (that is, the density of D a u), u + and u − are the approximate one-sided limits at the jump points, J u is the jump set of u, and ν u is the normal to J u (cf. [2, Chapter 3]). Following [2, p. 186], we can exploit the polar decomposition of a measure and the fact that all parts of the derivative of u in (2.1) are mutually singular to write Du = g u |Du| with a map g u ∈ L 1 |Du| (Ω; R d×N ) satisfying |g u | = 1 for |Du|-a.e. x ∈ Ω and D a u = g u |D a u|, D s u = g u |D s u|, D j u = g u |D j u|, D c u = g u |D c u|.
In the one-dimensional setting, i.e., for ϕ ∈ BV (a, b; R d ) with Ω = (a, b) ⊂ R N and N = 1, we write ϕ ′ in place of ∇ϕ to denote the approximate differential of ϕ. Accordingly, we use the notation Du = ϕ ′ L 1 + D s ϕ for the decomposition of the distributional derivative of ϕ with respect to the Lebesgue measure.
A function ϕ ∈ BV (a, b; R d ) is called a jump or Cantor function if Dϕ = D j ϕ or Dϕ = D c ϕ, respectively. We denote the sets of all jump and Cantor functions by BV j (a, b; R d ) and BV c (a, b; R d ), respectively. As shown in [2, Corollary 3.33], it is a special property of the one-dimensional setting that (2.4) Throughout this paper, two-dimensional functions of the form (2)) and ψ ∈ BV (a, b; R 2 ), play a fundamental role. Maps u as in (2.5) satisfy u ∈ BV (Ω; R 2 ). Denoting by D 1 u := Du ⊗ e 1 and D 2 u := Du ⊗ e 2 , the first and second columns of Du, respectively, we have for all ζ ∈ C 0 (Ω) that where L 1 ⌊(c, d)⊗D s R T and L 1 ⌊(c, d)⊗D s ψ denote the restrictions to the Borel σ-algebra on Ω = Q of the product measures between L 1 ⌊(c, d) and D s R T and D s ψ, respectively. We observe further that there exists θ ∈ BV (a, b; [−π, π]) such that where the representation of R ′ follows from the chain rule in BV; see, e.g., [2, Theorem 3.96].

Special functions of bounded variation.
A function u ∈ BV (Ω; R d ) is said to be a special function of bounded variation, written u ∈ SBV (Ω; R d ), if the Cantor part of its distributional derivative satisfies D c u = 0. In particular, it holds for every u ∈ SBV (Ω; R d ) that Next, we recall the definition of the space SBV ∞ (Ω; R d ) of special functions of bounded variation with bounded gradient and jump length, which is given by It is shown in [9] that the distributional curl of ∇u for u ∈ SBV ∞ (Ω; R d ) is a measure concentrated on J u .
Finally, we introduce the space which contains piecewise constant one-dimensional functions with values in R d .

2.4.
Geometry of the domain. In this section, we specify our main assumptions on the geometry of Ω, which, as mentioned in the Introduction, will mostly be a bounded Lipschitz domain in R 2 . Let us first recall from [14, Section 3] the definitions of locally one-dimensional and one-dimensional functions.
there exists an open cuboid Q x ⊂ Ω, containing x and with sides parallel to the standard coordinate axes, such that for all y = (y 1 , We say that f is (globally) one-dimensional in the e 2 -direction if (2.9) holds for every y, z ∈ Ω.
Analogous arguments to those in [14,Section 3] show that a function f ∈ BV (Ω; R d ) satisfying D 1 f = 0 is locally one-dimensional in the e 2 -direction. The following geometrical requirement is the counterpart of [14, Definitions 3.6 and 3.7] in our setting.
In what follows, we always assume that the set Ω ⊂ R 2 is an x 1 -connected domain. Under this geometrical assumption, the notions of locally and globally one-dimensional functions in the e 2 -direction coincide. We refer to [14,Section 3] for an extended discussion on the topic, as well as for some explicit geometrical examples.

Asymptotic rigidity of layered structures in BV
In this section, we prove Theorem 1.1, which characterizes the asymptotic behavior of deformations of bilayered materials that correspond to rigid body motions on the stiff layers, but do not experience any further structural constraints on the softer layers. This qualitative result is not just limited to applications in crystal plasticity, but can be useful for a larger class of layered composites where fracture may occur.
We start by introducing some notation. Assume that Ω ⊂ R 2 is an x 1 -connected domain. For ε > 0, let represent the class of layered deformations with rigid components, and let be the associated set of asymptotically attainable deformations.
We aim at proving that B 0 coincides with the set of asymptotically rigid deformations given by cf. (1.1). This identity will be a consequence of Propositions 3.1 and 3.2 below.
where B 0 and B are the sets introduced in (3.2) and (3.3), respectively.
Proof. The proof is inspired by and generalizes ideas from [13, Fix 0 < ε < 1, and let we define a strip, P i ε , by setting From Reshetnyak's theorem, we infer that on each nonempty rigid layer εY rig ∩ P i ε with i ∈ I ε , the gradient ∇u ε is constant and coincides with a rotation We claim that there exist a subsequence of (Σ ε ) ε , which we do not relabel, and a function R ∈ BV (−1, 1; SO(2)) such that To prove (3.5), we first observe that the total variation of the one-dimensional function Σ ε coincides with its pointwise variation, and can be calculated to be Next, we show that the right-hand side of (3.6) is uniformly bounded. By linear interpolation in the The first estimate is a consequence of Jensen's inequality, and optimization over translations yields the second one. To be more precise, the last estimate in (3.7) is based on the observation that for any given a ∈ R 2 \{0}, From (3.6) and (3.7), since (u ε ) ε ⊂ W 1,1 (Ω; R 2 ) as a weakly * converging sequence is uniformly bounded in BV (Ω; R 2 ), and recalling that R The convergence in (3.5) follows now from the weak * relative compactness of bounded sequences in BV (−1, 1; R 2×2 ) (see Section 2.2), together with the fact that strong L 1 -convergence is length and angle preserving. The latter guarantees that the limit function R ∈ BV (−1, 1; R 2×2 ) takes values only in SO (2).
for a.e. x ∈ Ω, which implies that u ∈ B and concludes the proof. To this end, we define auxiliary Consequently, Moreover, for x ∈ Ω, which, together with (3.5), proves that Finally, exploiting (3.10) and (3.11), we conclude that there exists b ∈ BV (Ω; R 2 ) such that b ε → b in L 1 (Ω; R 2 ). In view of the one-dimensional character of the stripes P i ε , we infer that ∂ 1 b = 0. Eventually, identifying b with a function ψ ∈ BV (−1, 1; R 2 ) yields (3.9).
The next lemma states a one-dimensional approximation result of BV -maps by Lipschitz functions that are constant on εI rig , which was an important ingredient in the previous proof. Then, there exists a sequence (w ε ) ε ⊂ W 1,∞ (I; R d ) with the following three properties: (2) and w ∈ BV (I; SO (2)), then each w ε may be taken in W 1,∞ (I; SO (2)).
Proof. Let w ∈ BV (I; R d ). By [2, Theorem 3.9, Remark 3.22], w can be approximated by a sequence of as δ → 0. To obtain property (iii), we will reparametrize v δ so that it is stopped on the set εI rig and accelerated otherwise, and eventually apply a diagonalization argument. We start by introducing for every ε > 0 a Lipschitz function ϕ ε : R → R defined by , where ψ is the 1-periodic function such that ψ(t) = 1 λ if 0 ≤ t ≤ λ, and ψ(t) = 0 if λ < t < 1. By the Riemann-Lebesgue lemma on weak convergence of periodically oscillating sequences, it follows that In particular, ϕ ε converges uniformly to ϕ on every compact set K ⊂ R.
To conclude, we address the issue of constraint-preserving approximations for w ∈ BV (I; SO (2)). In this case, we argue as above, but replace the density argument leading to (3.15) by its analogue for BV functions with values on manifolds, see [28,Theorem 1.2]. This allows us to assume that v δ ∈ C ∞ (Ī; SO(2)), and eventually yields w ε ∈ W 1,∞ (I; SO (2)).
By definition, and accounting for the fact that R takes values in SO (2), the jump set of u ∈ B ∩ SBV (Ω; R 2 ) is related to the jump sets of R and ψ via

Asymptotic behavior of admissible layered deformations
In this section, we prove Theorem 1.3, which characterizes the asymptotic behavior of deformations of bilayered materials that coincide with rigid body rotations on the stiffer layers, and are subject to a single slip constraint on the softer layers. The latter is described with the help of the set M e1 = {F ∈ R 2×2 : det F = 1 and |F e 1 | = 1} = {F ∈ R 2×2 : F = R(I + γe 1 ⊗ e 2 ) with R ∈ SO(2) and γ ∈ R}. In the sequel, according to the context, we will always adopt the most convenient representation.
The following lemma on weak continuity of Jacobian determinants for gradients in W 1,1 (Ω; R 2 ) with suitable additional properties will be instrumental in the proof of the inclusion A 0 ⊂ A.
Proof. The claim in Lemma 4.2 would be an immediate consequence of [26,Theorem 2] if in place of (4.9), we required (adj ∇u ε ) ε ⊂ L 2 (Ω; R 2×2 ), (4.10) which, because of the structure of the adjoint matrix in this two-dimensional setting, is equivalent to ∇u ε ∈ L 2 (Ω; R 2×2 ) for all ε. Even though we are not assuming this here, it is still possible to validate the arguments of [26, Proof of Theorem 2] in our context, as we detail next.
Since | adj ∇u ε | = |∇u ε |, it can be checked that in order to mimic the proof of [26,Theorem 2] with N = 2, we are only left to prove the following: If (ϕ j ) j∈N is a sequence of standard mollifiers and Ω ′ is an arbitrary open set compactly contained in Ω, then (det ∇u ε,j ) j∈N converges to det ∇u ε in L 1 (Ω ′ ) as j → ∞ for all ε, where u ε,j := ϕ j * u ε .
We obtain from the following proposition that weak * limits of sequences in A ε belong to A. Proof. The statement follows from the inclusion A ε ⊂ B ε (see (4.2)) and the identity (4.4) in conjunction with Proposition 3.1 and Lemma 4.2, observing that the condition ∇u ε ∈ M e1 a.e. in Ω guarantees |∂ 1 u ε | = |∇u ε e 1 | = 1 a.e. in Ω, and hence ∂ 1 u ε L ∞ (Ω;R 2 ) = 1 for any ε.
The question whether the set A can be further identified as limiting set for sequences in A ε , namely, whether the equality A 0 = A is true, cannot be answered at this point. However, as stated in Theorem 1.3, the inclusions A 0 ⊃ A ∩ SBV ∞ (Ω; R 2 ) and A 0 ⊃ A hold. Before proving these inclusions, we discuss a further characterization of some special subsets of A.
Note that both J R and J ψ are given by an at most countable union of points in (−1, 1), which implies that J u consists of at most countably many segments parallel to e 1 . It is not possible to conclude that the functions R are piecewise constant according to [ with R ∈ P C(−1, 1; SO(2)) and ψ ∈ SBV ∞ (−1, 1; R 2 ) such that ψ ′ · Re 2 = 0 a.e. in (−1, 1)}.
Here, both J R and J ψ are finite sets of points in (−1, 1), and J u is given by a finite union of segments parallel to e 1 . Alternatively, one can express A ∩ SBV ∞ (Ω; R 2 ) with the help of a Caccioppoli partition of Ω into finitely many horizontal strips; precisely, In the following lemma, we construct an admissible piecewise affine approximation for basic limit deformations in A ∩ SBV ∞ (Ω; R 2 ) with a non-trivial jump along the horizontal line at x 2 = 0. Based on this construction, we will then establish the inclusion A 0 ⊃ A ∩ SBV ∞ (Ω; R 2 ) in Proposition 4.7 below. Let Ω = (0, 1)×(−1, 1), and let u ∈ A ∩ SBV ∞ (Ω; R 2 ) be such that u(x) = R(x 2 )x + ψ(x 2 ) for a.e. x ∈ Ω, where with some R ± ∈ SO(2) and ψ ± ∈ R 2 . Then, there exists a sequence (u ε ) ε ⊂ W 1,1 (Ω; R 2 ) with Ω u ε dx = Ω u dx and u ε ∈ A ε for all ε, and such that u ε * ⇀ u in BV (Ω; R 2 ).
To prove that u ε * ⇀ u in BV (Ω; R 2 ), it suffices to show that (4.16) or, equivalently, in view of (4.12), that for every ϕ ∈ C 0 (Ω; Moreover, using (4.14), a change of variables, and Lebegue's dominated convergence theorem together with the continuity and boundedness of ϕ, we have lim ε→0 (0,1)×(0, ελ 4 x1)  Let u ∈ A ∩ SBV ∞ (Ω; R 2 ) be as in Lemma 4.5, and assume that either R + = ±R − or R + = R − . In these cases, we can simplify the construction of (u ε ) ε in the previous proof. We focus here on stating the counterparts of Figure 3 and (4.14), and omit the detailed calculations, which are very similar to (4.18)- (4.24). Note further that these constructions are not just simpler, but also energetically more favorable, see Remark 5.2 below for more details.
(i) If R + = ±R − , we may replace the construction depicted in Figure 3 (ii) If R is constant, i.e., R + = R − , and ψ + − ψ − is not parallel to Re 1 , the construction in Figure 3 can be replaced by: Figure 5. Alternative construction of V ε if R is constant and ψ + − ψ − is not parallel to Re 1 .
(iii) If R is constant, i.e., R + = R − , and ψ + − ψ − is parallel to Re 1 , then we can use the following construction in place of Figure 3: Figure 6. Alternative construction of V ε if R is constant and ψ + − ψ − is parallel to Re 1 .
Note that in case (i), the slope ρ of the interfaces can attain any value between 0 and 1, while in (ii), ρ is determined by the value of β. In terms of the energies, the construction in case (iii) provides an optimal approximation, which will be detailed in Section 6.
An analogous statement to Proposition 4.7 holds in A .

A lower bound on the homogenized energy
In this section, we present partial results for the homogenization problem for layered composites with rigid components discussed in the Introduction. More precisely, we establish a lower bound estimate on the asymptotic behavior of the sequence of energies (E ε ) ε (see (1.7)), and highlight the main difficulties in the construction of matching upper bounds. Note that the following analysis is restricted to the case s = e 1 .
As a start, we first give alternative representations for the involved energies, which will be useful in the sequel.

Remark 5.1 (Equivalent formulations for E ε and E).
In view of the definition of A ε (see (1.6)), it is straightforward to check that the functional E ε in (1.7) satisfies for u ∈ L 1 0 (Ω; R 2 ). Similarly, according to Proposition 4.1, the functional E from (1.10) can be expressed as for u ∈ L 1 0 (Ω; R 2 ).
We can now provide a bound from below on Γ-lim inf ε→0 E ε and prove Theorem 1.2.
Proof of Theorem 1.2. For clarity, we subdivide the proof into two steps. In the first one, we establish the compactness property. In the second step, we provide two alternative proofs of (1.12). The first proof is based on a Reshetnyak's lower semicontinuity result, while the second version is more elementary, relying on the weak * lower semicontinuity of the total variation of a measure. Either of the arguments highlights a different feature of the representation of A.
Step 2: Lower bound. Let (u ε ) ε ⊂ L 1 0 (Ω; R 2 ) and u ∈ L 1 0 (Ω; R 2 ) be such that u ε → u in L 1 (Ω; R 2 ). We want to show that lim inf To prove (5.1), one may assume without loss of generality that the limit inferior on the right-hand side of (5.1) is actually a limit and that this limit is finite. Then, u ε ∈ A ε and E ε (u ε ) < C for all ε, where C > 0 is a constant independent of ε. Hence, by Step 1, u ε * ⇀ u in BV (Ω; R 2 ) and u ∈ A.
(b) The question whether (5.4) holds for a larger class than A is open at this point. We observe that the gradient-based constructions in Lemma 4.5, Remark 4.6 (i)-(ii), and Proposition 4.7 yield upper bounds on the Γ-lim sup, which, however, do not match the lower bound of Theorem 1.2. This indicates that, in general, a more tailored approach will be necessary.
(c) The upper bounds on the Γ-lim sup of (E ε ) ε resulting from Lemma 4.5, Remark 4.6 (i)-(ii), and Proposition 4.7 can be quantified. As previously mentioned, the constructions in Remark 4.6 (iii) and Proposition 4.9 are even recovery sequences. This is not the case for the general construction in Lemma 4.5 and for those highlighted in Remark 4.6 (i)-(ii). In the following, we suppose that u ∈ A ∩ SBV ∞ (Ω; R 2 ) ∩ L 1 0 (Ω; R 2 ) has a single jump as in the statement of Lemma 4.5; i.e., with R ± ∈ SO(2) and ψ ± ∈ R 2 . Then, which can be estimated from above by For the sequence (u ε ) ε constructed in Lemma 4.5 (and Lemma 4.7), we obtain, recalling (4.13), that Regarding the construction of (u ε ) ε in Remark 4.6 (i), it follows that lim ε→0 E ε (u ε ) = |α| + |β − 1| + 1.

Homogenization of the regularized problem
This section is devoted to the proof of our main Γ-convergence result, Theorem 1.4. We first provide an alternative characterization of the class A of restricted asymptotically admissible deformations introduced in (1.13).
We are now in a position to prove the Γ-convergence of the energies (E ε δ ) ε in (1.14) as ε → 0.
Proof of Theorem 1.4. As before in the proofs of Theorems 1.1 and 1.3, one may assume without loss of generality that Ω = (0, 1) × (−1, 1). We proceed in three steps.
Step 3: Upper bound. Let u ∈ L 1 0 (Ω; R 2 )∩A . We want to show that there is a sequence (u ε ) ⊂ L 1 0 (Ω; R 2 ) such that u ε → u in L 1 (Ω; R 2 ), and be the sequence constructed in the proof of Proposition 4.9, that is, u ε ∈ A ε for every ε with which proves (6.10) and completes the proof of the theorem.
The role of the higher-order regularization in (1.14) is exactly that it helps overcome the issue discussed above. In fact, it guarantees the desired compactness properties for sequences of deformations with equibounded energies.
To conclude, we present an alternative construction for the recovery sequence in Step 3 of the proof of Theorem 1.4.
Alternative proof of Theorem 1.4. As before, we may assume without loss of generality that Ω = (0, 1) × (−1, 1). Moreover, the compactness property and lower bound can be studied exactly as in the proof of Theorem 1.4 above.
Step 1. We assume first that u ∈ L 1 0 (Ω; R 2 ) ∩ A is an SBV -function with a single, constant jump line at x 2 = 0.