Derivation of a homogenized bending--torsion theory for rods with micro-heterogeneous prestrain

In this paper we investigate rods made of nonlinearly elastic, composite--materials that feature a micro-heterogeneous prestrain that oscillates (locally periodic) on a scale that is small compared to the length of the rod. As a main result we derive a homogenized bending-torsion theory for rods as $\Gamma$-limit from 3D nonlinear elasticity by simultaneous homogenization and dimension reduction under the assumption that the prestrain is of the order of the diameter of the rod. The limit model features a spontaneous curvature-torsion tensor that captures the macroscopic effect of the micro-heterogeneous prestrain. We device a formula that allows to compute the spontaneous curvature-torsion tensor by means of a weighted average of the given prestrain. The weight in the average depends on the geometry of the composite and invokes correctors that are defined with help of boundary value problems for the system of linear elasticity. The definition of the correctors depends on a relative scaling parameter $\gamma$, which monitors the ratio between the diameter of the rod and the period of the composite's microstructure. We observe an interesting size-effect: For the same prestrain a transition from flat minimizers to curved minimizers occurs by just changing the value of $\gamma$. Moreover, in the paper we analytically investigate the microstructure-properties relation in the case of isotropic, layered composites, and consider applications to nematic liquid-crystal-elastomer rods and shape programming.


Introduction
Motivation. Residual stress can have a tremendous effect on the mechanical behavior of slender elastic structures: Equilibrium states of elastic thin films and rods with residual stresses often have a complex shape in equilibrium, and may feature wrinkling and symmetry breaking. Many natural and synthetic materials feature residual stresses due to different physical principles, e.g., growth of soft tissues [6,11], swelling and de-swelling in polymer gels [21], thermo-mechanical coupling in nematic liquid crystal elastomers [47], and thermal expansion in production processes. These mechanisms may be triggered by different stimuli (such as temperature, light, and humidity), and are exploited in the design of active thin structures-elastic structures that are capable to change from an initially flat state into a 3D "programmed" configuration in response to external stimuli, see [22] and [43] for a recent review on shape shifting flat soft matter. Modeling of such structures, requires (next to a description of the stimuli process) a good understanding of the highly nonlinear relation between residual stresses and the geometry of the equilibrium shape. Although intensively studied, no satisfying understanding of this relation has been obtained so far. This is especially the case for composite materials, where material properties and residual stresses feature microstructure-a situation that is relevant for future applications, since "Shape-changing materials offer a powerful tool for the incorporation of sophisticated planar micro-and nano-fabrication techniques in 3D constructs" as pointed out in [43].
Overview of results. In this paper we investigate rods made of nonlinearly elastic, compositematerials that feature a micro-heterogeneous prestrain (or residual stress) that oscillates (locally periodic) on a scale that is small compared to the length of the rod. Our starting point is the energy functional of 3D-nonlinear elasticity with a cylindrical reference domain Ω h = (0, ) × hS ⊂ R 3 : It depends on two small parameters h and ε (describing the thickness of the rod and the period of the composite), and describes prestrain with help of a tensor field A ε,h , see Section 2.1 for the continuum-mechanical interpretation. We suppose W ε to describe a non-degenerate, nonlinear material with stress-free reference state. Moreover, we assume that the amplitude of the prestrain is comparable to the diameter of the rod, i.e., A ε,h = Id +O(h) so that A −1 ε,h ≈ Id +B ε,h . As a main result (see Theorem 2.7) we derive the Γ-limit as (ε, h) ↓ 0 of (1) in the bending regime. In this simultaneous homogenization and dimension reduction limit, we obtain a homogenized bending-torsion theory for rods that features a spontaneous curvature-torsion tensor K eff : (0, ) → Skew (3). It captures the macroscopic effect of the micro-heterogeneous prestrain: where bending and torsion of the rod is described by the isometry u ∈ W 2,2 iso ((0, ), R 3 ) and an attached orthonormal frame R ∈ W 1,2 ((0, ), SO(3)), Re 1 = ∂ 1 u. The elastic moduli of the rod are described by the quadratic form Q hom . It is positive definite on skew symmetric matrices and can computed by a linear relaxation and homogenization formula from L ε -the fourth order elasticity tensor obtained by linearizing W ε at identity. While it is difficult to study energy minimizers of (1) directly, energy minimizers of (2) can easily be obtained by integrating the spontaneous curvature-torsion field K eff . It turns out that K eff depends on L ε (nonlinearly) and on B ε,h (linearly). In addition, we observe that both Q hom and K eff depend on the relative-scaling parameter γ = lim (ε,h)↓0 h ε . Next to the Γ-convergence result, we introduce an effective scheme to evaluate Q hom and K eff , which invokes the definition of suitable correctors that are characterized by corrector equations which essentially come in form of boundary value problems for the system of linear elasticity, see Proposition 3.1. The spontaneous curvature-torsion tensor K eff is than obtained as weighted average of the prescribed prestrain tensor with weights given by the correctors. For isotropic composites with a laterally layered microstructure, we can solve the corrector equations by hand and we obtain explicit formulas for the Q hom and K eff , see Lemma 4.1. We observe a significant qualitative and quantitative dependence of K eff on the relative-scaling parameter γ. In particular, we device an example of a prestrain that yields a transition from a straight minimizer (i.e., K eff ≡ 0) to a curved minimizers (i.e., K eff = 0) by only changing the value of γ, see Section 4.2. Moreover, we briefly discuss applications to nematic liquid crystal elastomers in Section 4.3, shape programming in Section 4.4, and comment on potential applications to rods with varying cross-section in Remark 3.
Survey of the literature. The derivation of mechanical models for rods has a long history. For modeling based on equilibria of forces or conservation of momentum, and derivations via formal asymptotic expansions or based on the assumption of a kinematic ansatz we refer the reader to [4,5,8]. In contrast to these works, we take the perspective of energy minimization, and our result is an ansatz-free derivation that is based on the Γ-convergence methods developed by Friesecke, James & Müller in [15], in particular the geometric rigidity estimate. If we replace in (1) the prestrain tensor A ε,h by the identity matrix, then we recover a standard 3D nonlinear elasticity model without prestrain, i.e., with a stress-free reference configuration. In that case the limit h ↓ 0 with ε > 0 fixed, corresponds to a dimension reduction problem (without homogenization) studied by Mora & Müller in [30] where for the first time a bending-torsion theory for inextensible rods has been derived via Γ-convergence. On the other hand, the limit (h, ε) ↓ 0 corresponds to simultaneous homogenization and dimension reduction and is studied by the second author in [32,33], see also [34,35,18,45] where the same problem for plates is considered. First results that combine dimension reduction in the presence of a prestrain are due to Schmidt: In [40,41] prestrained bending plates are obtained from 3D nonlinear elasticity; see also [27] on the derivation of a model for prestrained von Kármán plates, see also [1] where applications to models for nematic liquid crystal elastomers are studied. Our result can be viewed as a combination of Schmidt's work with [32,33]. We note that a simplified version of our main result is announced in the second author's thesis [32] (together with a rough sketch of the proof). Recently, the derivation of prestrained bilayer rods has been investigated by Kohn & O'Brien [23] and Cicalese, Ruf & Solombrino [9]. In these interesting works not only energy minimizers are studied, but also the convergence of critical points is established and comparision with experiments [42] are discussed. Another interesting direction of active research on related topics are the derivation and analysis of ribbons, e.g., [2,14,13]. In the results discussed so far the prestrain (if present) is assumed to be infinitesimally small. In the last decade, dimension reduction for finite prestrain (yet smoothly varying on a macroscopic scale) has been studied in the framework of non-Euclidean elasticity theory [12], e.g., [24,27,28,7,26] for the derivation of non-Euclidean theories for rods and plates. Rods and shells with nontrivially curved reference configuration lead to similar models when being pulled back to a flat reference configuration (cf. Remark 3 below), e.g., see [39,44,19,20] for shells.
Structure of the paper. We introduce the general framework in Section 2. In particular, we explain the modeling of prestrained composites (based on a multiplicative decomposition of the strain) in Section 2.1. The 3D model and the limiting are described in Sections 2.2 and 2.3. In Section 2.4 we present an abstract definition of the homogenization and averaging formulas that determine Q hom and K eff . In Proposition 3.1 in Section 3 we describe the effective evaluation scheme for these formulas. It is based on the notion of suitable correctors. Eventually, in Section 4 we discuss various applications of the theory to isotropic material for which the correctors, homogenization-, and averaging formulas can be evaluated by hand. All proofs are contained in Section 5.

Notation
• e 1 , e 2 , e 3 denotes the standard basis of R 3 .
• Given a, b ∈ R d we write a⊗b to denote the unique matrix in R d×d given by (a⊗b)c = (b·c)a for all c ∈ R d .
• We write Sym(d), Skew(d), and SO(d) for the space of symmetric, skew-symmetric, and rotation matrices in R d×d . We denote the identity matrix by Id.
• We decompose x = (x 1 ,x) ∈ R 3 into the in-plane component x 1 := x · e 1 and the out-of- • For all x ∈ R 3 we setx(x) := i=2,3 x i e i ∈ R 3 . We tacitly drop the argument and simply writex (instead ofx(x)).

General framework and statement of main results
In this section we state the general framework and our main result.
(W4) W admits a quadratic expansion at Id: where Q : R 3×3 → R is a quadratic form.
• The class Q(α, β) consists of all quadratic forms Q on R 3×3 such that We associate with Q the fourth order tensor L ∈ Lin(R 3×3 , R 3×3 ) defined by the polarization identity LF, G : Stored energy functions of class W(α, β, ρ, r) describe materials that have a stress-free reference state (cf. (W 3)), and that can be linearized at that state (e.g., in the sense of Γ-convergence, see [10,31,16,32]). The elastic moduli of the linearized model are given by the quadratic form Q in condition (W 4), and we have: Lemma 2.7 in [33]). Let W ∈ W(α, β, ρ, r) and denote by Q the quadratic form in (W 4). Then Q ∈ Q(α, β).
We describe prestrained composites with help of a multiplicative decomposition of the strain. To motivate this decomposition, we consider for a moment a composite consisting of two materials. We suppose that each of the materials can be described w.r.t. their individual stress-free reference configurations by stored energy functions W 1 , W 2 ∈ W ∈ W(α, β, ρ, r), respectively. Let Ω = Ω 1∪ Ω 2 ⊂ R 3 denote a common reference configuration of the composite and suppose that material-one (resp. -two) occupies the subdomain Ω 1 (resp. Ω 2 ). We suppose that materialone is stress-free in the reference configuration Ω 1 , and thus the elastic energy coming from material-one is captured by´Ω 1 W 1 (∇u). On the other hand, we suppose that material-two is prestrained in the following sense: If we separate an (infinitesimally small) test-volume U ⊂ Ω 2 from the rest of the body, then it relaxes to a stress-free (energy minimizing) state described by an affine deformation x → Ax where A ∈ R 3×3 is positive definite and independent of U . Thus, Ω := AΩ 2 defines an alternative, stress-free reference state for material-two, and the elastic energy of a deformationũ defined relative to Ω is given by´ Ω W 2 (∇ũ) dx. Since the original deformation u : Ω → R 3 andũ are related by u(x) :=ũ( Ax) (for x ∈ Ω 2 ), we deduce that the energy functional on the level of u associated with material-two is given bŷ Hence, the energy functional for the whole composite takes the form This is corresponds to a multiplicative decomposition F = F el A −1 of the strain. Similar decomposition are used in models for finite strain elasto-plasticity [25] (where A is called the plastic strain tensor and is given by a flow rule), or in biomechanical models for growth and remodeling of tissues and plants, e.g., see [38,17]. If the prestrain is small, then we can simplify decomposition: Suppose that A = R(Id −hB) with R ∈ SO(3), B ∈ R 3×3 , and h > 0. Then for h 1, A can be inverted by the Neumann . Hence, we arrive at an energy functional of the form´Ω W x, ∇u(x)(Id +hB(x)) dx with W (x, ·) ∈ W(α, β, ρ, r) and a tensor B(x) ∈ R 3×3 . The functional describes (up to an error of order smaller than h 2 ) a composite material with heterogeneous prestrain (Id −hB(·)).

The three-dimensional model.
Let S ⊂ R 2 be a Lipschitz domain (open, bounded and connected)-the cross-section of the rod. We may assume without loss of generality that (otherwise we apply a rigid motion). Set ω := (0, ). We denote by Ω h := ω × hS the reference configuration of the rod with thickness h > 0. For our purpose it is convenient to describe the deformation w.r.t. the rescaled reference domain Ω := ω × S, and thus consider for u : Ω → R 3 the scaled deformation gradient, Rescaling (1) and assuming that the prestrain takes the form A ε,h = (Id +hB ε,h ) −1 yields an energy functional of the form I ε,h : L 2 (Ω) → [0, +∞], This parametrized energy functional is the starting point of our derivation. We make the following assumption on the material law: (i) W ε (x, ·) ∈ W(α, β, ρ, r) for almost every x ∈ Ω and for every ε > 0.
We suppose that there exists Q : Ω × R × R 3×3 → R such that (ii) Q(x 1 ,x, y, ·) is a quadratic form that is piecewise continuous in x 1 and periodic in y. More precisely, (a) Q(x, ·) ∈ Q(α, β) for a.e. x ∈ Ω, The fourth order tensor L = L(x 1 ,x, y) associated with Q (cf. Definition 2.1) satisfies Regarding the prestrain, we suppose that B ε,h is locally periodic. Our precise assumption on B ε,h involves the notion of two-scale convergence in a variant for slender domains [32,33] (see [36,3] for the original definition of two-scale convergence). Since this variant of two-scale convergence is sensitive to the relative scaling between h and ε, we introduce a parameter γ ∈ [0, ∞] describing the relative scaling of h and ε.
Remark 1. Note that this notion of two-scale convergence changes if we change the parameter γ. A prototypical example of a strongly two-scale convergent sequence is as follows: Let g ∈ L 2 (Ω; C(Y)), then g h (x) := g(x, x 1 ε(h) ) strongly two-scale converges in L 2 to g. Assumption 2.6 (Prestrain). We suppose that there exists

Limiting model and Γ-convergence.
Under the assumptions above, we can pass to the Γ-limit of I ε,h as (ε, h) = (ε(h), h) → 0. We obtain as a limit a functional defined on the the set A of all deformations of the rod that describe (length-preserving) bending-and twisting-deformations, and an infinitesimal stretch: a ∈ L 2 (ω) .
The Γ-limit is given by I : A → [0, ∞), where Q hom (the homogenized elastic moduli ), K eff (the spontaneous curvature-torsion tensor ), a eff (the spontaneous infinitesimal stretch), and m ≥ 0 (the incompatibility of the prestrain) are quantities that only depend on the linearized material law Q, the prestrain B, the geometry of the cross-section S, and the scale ratio γ; in particular, • m ≥ 0 is a constant given in Definition 2.11 below, • Q hom : ω × Skew(3) × R → R is a positive-definite quadratic form given by the homogenization formula of Definition 2.8 below, • K eff ∈ L 2 (ω; Skew (3)) and a eff ∈ L 2 (ω) are given by the averaging formula of Definition 2.11 below.
Our main result establishes Γ-convergence of I ε(h),h to I: the (scaled) nonlinear strain tensor. Then: Then there exists (u, R, a) ∈ A and a subsequence (not relabeled) such that be a sequence that converges to some (u, R, a) ∈ A in the sense of (10) and (11). Then converging to (u, R, a) in the sense of in the sense of (10) and (11) such that (For the proof see Section 5.1).
Remark 2. Theorem 2.7 also yields a compactness and Γ-convergence result towards a (more conventional) pure bending-torsion model. Indeed, by part (a) of Theorem 2.7 every sequence with equibounded energy satisfies (10) for some rod-deformation (u, R) satisfying (6). Furthermore, by minimizing over a ∈ L 2 (ω) the statements of the parts (b) and (c) in Theorem 2.7 hold with (u, R, a) and I(u, R, a) replaced by (u, R) and I (u, R) := inf a∈L 2 (ω) I(u, R, a) (see Remark 5 below for a more explicit characterization of I ).
Remark 3. Energies of the type (4) naturally emerge in models of rods with varying crosssection. We describe this in the simple set-up of an homogeneous material law that occupies a cylindrical domain with a rapidly oscillating cross-section S ε (x 1 ) ⊂ R 2 : For is sufficiently smooth, and consider the reference domain Then, where For g ε (x 1 ) = εĝ( x 1 ε ) andĝ periodic, the a asymptotic behavior of (12) can be analyzed by the asymptotic behavior of functionals of the form (4) provided γ ∈ (0, ∞].

Homogenization-and averaging formulas.
The definitions of Q hom , K eff , and m rely on the two-scale structure of limiting strains. To motivate the upcoming formulas, we recall a two-scale compactness statement for the nonlinear strain, see [33, Theorem 3.5] (and also Proposition 5.1 below): Suppose (u h ) is a sequence with equibounded energy (cf. (9)) with limit (u, R, a) ∈ A (cf. (10), (11)), then (up to a subsequence) the associated scaled nonlinear strain tensors E h (u h ) weakly two-scale converge in L 2 to a limiting strain E : Ω × Y → R 3×3 sym of the form (3)) is defined as follows: for γ = 0: sym (∂ y Ψ)x + ∂ yφ |∇φ : Note that on the right-hand side in (13) the first and second term are determined by the limiting deformation (u, R, a). Only the third term χ-the only term that involves the fast variable y ∈ Y-depends on the chosen subsequence. We call it the strain corrector. For the following discussion it is convenient to define for (K, a) ∈ Skew(3) × R the affine map and to introduce the two-scale strain space Since R t ∂ 1 R is skew-symmetric (almost surely) for (u, R, a) ∈ A), the limiting strain of (13) satisfies E ∈ L 2 (ω; H γ ).
Formula for Q hom . As in [33] the homogenized quadratic form Q hom is defined by minimizing out the energy contribution coming from χ ∈ H γ rel : Remark 4. We emphasize that the definition of Q hom depends on the small-scale coupling γ via the relaxation space H γ rel . For almost every x 1 ∈ ω, Q hom (x 1 , ·) defines a positive definite quadratic form on Skew(3) × R and the map x 1 → Q(x 1 , K, a) is piecewise continuous for every (K, a) ∈ Skew(3) × R (see Proposition 3.1 below).
Remark 5. As already discussed in Remark 2, a pure bending-torsion model is obtained from I by minimizing out the stretch variable a. This can be made more explicit as follows: where The quadratic form Q hom coincides with the homogenized quadratic form given in [33] where the case without prestrain is studied.
Formulas for K eff and a eff . We first present a "geometric" definition-an alternative "algorithmic" definition that is more practical for numerical investigations is presented in Section 3 below. The geometric definition invokes the following Hilbert-space structure on H := L 2 (S ×Y; Sym (3)): Let L denote the symmetric fourth-order tensor obtained from the quadratic form Q by polarization, and consider for x 1 ∈ ω, Since Q is positive-definite and bounded on symmetric matrices, (·, ·) x 1 defines a scalar product on H. We write · x 1 for the associated norm and note that H γ rel and H γ (see (14) and (16)) are closed, linear subspaces of (H, · x 1 ). We denote by ( We thus have the orthogonal decomposition, A direct consequence is the following observation: In particular, we obtain the following characterization of Q hom : It turns out that any E ∈ (H γ rel ) ⊥x 1 admits a representation via a unique pair (K, a) ∈ Skew(3)× R: defines a linear isomorphism and there exists a constant C = C(α, β, γ, S) such that We denote by P γ,• the unique bounded operator on L 2 (ω; H) defined by the identity and define P γ,• rel and E γ,• analogously. We are now in position to define (K eff (x 1 ), a eff (x 1 )) and m: Definition 2.11 (averaging formula for m and (K eff , a eff )). We set and define (K eff , a eff ) ∈ L 2 (ω; Skew(3) × R) as the unique field such that

Evaluation of the homogenization formulas via BVPs
The definitions of Q hom , K eff and a eff (see Definitions 2.8 and 2.11) are rather abstract. In this section we present a characterization that replaces the "abstract" operator in these definitions by boundary value problems for the system of linear elasticity on the domain S × Y . To benefit from the linearity of the map (K, a) → (K eff , a eff ), we set and note that this defines an orthonormal basis of Skew(3). Moreover, we introduce the maps see (15) for the definition of E. Note that {E (i) : i = 1, . . . , 4} spans the macroscopic strain space. In particular, E (1) corresponds to an infinitesimal stretch (in tangential direction); E (i) (i = 2, 3) corresponds to bending in direction x i , and E (4) corresponds to a twist. We have the following scheme to evaluate the homogenized quantities: Proposition 3.1. For x 1 ∈ ω we define the following objects: (ii) The averaging matrix M(x 1 ) ∈ Sym(4) as the unique matrix with entries (iii) The vector representation of the strain b(x 1 ) ∈ R 4 as the unique vector with entries where B denotes the prestrain tensor of Assumption 2.6. Then: (a) M(x 1 ) is symmetric positive definite and we have 1 (For the proof see Section 5.2).
Remark 6 (Averaging and homogenization). The proposition shows that the spontaneous curvature-torsion tensor K eff and the spontaneous infinitesimal stretch a eff linearly depend on B, and thus, the passage from B to (K eff , a eff ) can be interpreted as a spatial average with a correction that takes the micro heterogeneity of the material, the cross-section S, and the scale ratio γ into account. This is in contrast to the relation between Q and Q hom , which is nonlinear and given by a homogenization formula that has already been obtained in [33] where the case without prestrain is discussed.
Next, we derive boundary value problems (BVP) that allow to compute (27), and to represent the strain correctors χ (i) .
Then the map ι ∞ : defines an isomorphism, and we have (For the proof see Section 5.2).

Examples and explicit formulas for isotropic materials
In this section we restrict our analysis to isotropic materials and the extreme regimes h ε and ε h, i.e., γ ∈ {0, ∞}. In that case the homogenized quantities-the matrix M from Proposition 3.1-can be computed by hand, see Lemma 4.1 below. We further specify the findings of Lemma 4.1 in the case of a bilayer material which was studied in the homogeneous case in [9,23]. We observe a dramatic size effect: We give an explicit example of a prestrain B that produces zero spontaneous bending in the case γ = 0 but non-zero bending in the case γ = ∞. Moreover, we apply Lemma 4.1 to prestrain tensors that originate from models for nematic liquid crystal elastomers and compare the results with the findings of [1,2] in the context of ribbons. Finally, we address shape programming.

Isotropic, laterally periodic composites.
Throughout this section we suppose that the composite is isotropic, periodically oscillating in longitudinal direction, and constant in cross-sectional direction, i.e., we suppose that Q (cf. Assumption 2.3) is of the form with (periodic) Lamé-constants µ, λ ∈ L ∞ (Y) that are (essentially) non-negative, and ess inf y∈R (2µ+ λ) > 0. We recall the definition of some standard moduli for isotropic elastic materials: The formulas for the elastic moduli of the effective model involve the arithmetic and harmonic mean. To shorten notation, for f ∈ L 1 (Y) we set f :=ˆY f (y) dy and f hom := Furthermore, we define the effective moduli and note that ν ∞ = ν and β ∞ = β 0 = β for homogeneous, isotropic materials. Next to the elastic moduli, the homogenized model depends on the geometry of the cross-section S. To capture this effect, we denote by ϕ S ∈ H 1 (S) the unique minimizer to satisfying´S ϕ S = 0. Following [30, Remark 3.5], we refer to the function ϕ S and the parameter τ S as the torsion function and the torsional rigidity.
The following lemma yields an explicit expression for the averaging matrix M of Proposition 3.1 in terms of averages of the Lamé-constants, the torsional rigidity and the torsion function. It can be seen as an extension of the analysis in [30] and [23, Theorem 3] to periodic composites and periodic prestrain.
(iv) The vector k of Proposition 3.1 is given by k i = M −1 ii b i for i = 1, . . . , 4, and it holds where a ∈ R and K = (For the proof see Section 5.3).
Remark 7 (General observations). The qualitative dependency of the spontaneous curvaturetorsion tensor k = (k 1 , k 2 , k 3 , k 4 ) = infinitesimal stretch, curvature, curvature, torsion on the geometry of S, the prestrain B and the material law can be summarized in the following diagram.
In [23, Theorem 3] the statement of Lemma 4.1 is given in the case of a homogeneous material and non-oscillatory prestrain. The values for the induced torsion k 4 in the case γ ∈ {0, ∞} and for the induced stretching and bending k 1 , k 2 , k 3 in the case γ = 0 coincides with the findings of [23] applied to the averaged prestrain B ∈ L 2 (S; R 3×3 ). In the case γ = ∞ the values of k 1 , k 2 , k 3 differ substantially from the homogeneous case. Finally, we note that g ∞ given in (35) satisfies g ∞ = 0. In particular, for homogeneous isotropic materials, i.e. λ and µ are constant, the vector k coincides in the cases γ = 0 and γ = ∞.

Example 2: Nematic rods.
Liquid crystal elastomers are solids made of liquid crystals (rod-like molecules) incorporated into a polymer network. In a nematic phase (at low temperature) the liquid crystals show an orientational order and the material features a coupling between the entropic elasticity of the polymer network and the LC-orientation. The latter leads to a thermo-mechanical coupling that can be used in the design of active thin sheets that show a complex change of shape upon thermo-mechanical (or photo-mechanical) actuation, see [48]. Following [47,37] we describe the elastic energy of a nematic elastomer by the functional where the so-called step-length tensor is given by Above n : Ω → S 2 := {x ∈ R d : |x| = 1} is a director field that describes the local orientation of the liquid crystals, and r is a scalar order parameter. In [7] a non-Euclidean bending plate model is derived via Γ-convergence from (36) under the assumption that the director field n is sufficiently smooth and satisfies additional structural assumptions (in particular it is assumed to be constant in the thickness direction). In [1], the authors derive a plate model from energy (36) with director fields n that are allowed to have large variations across the thickness but with the simplifying assumption that r in (37) is replaced by r h = 1 +rh withr ∈ R, where h denotes the thickness of the plate. Under this assumption, we have Two specific choices for the director field n were studied in [1,2] in detail for the case of plates and ribbons: "Splay bend": In the following, we present the spontaneous curvature-torsion vector k = k(n) for prestrains B(n) defined via (38) with director fields n corresponding to splay bend-and twist configurations. To be precise, set S = (−1, 1) 2 and consider for simplicity the case of an isotropic and homogeneous material law, that is Q (cf. Assumption 2.3) is of the form Q(x, y, G) = Q(G) = 2µ| sym G| 2 + λ(trace G) 2 with µ > 0 and λ ≥ 0.
• (twist). For given ϑ ∈ [0, π), set n (For details on the calculations, we refer to Appendix A.1) Let us now compare the above findings with the results in [1,2]: In [1] the authors derive a 2D-plate model from the energy (36) with ϑ = π 4 . Starting from the resulting plate model a 1D-ribbon model is derived in [2] by cutting out a thin strip from the plate (in a certain angle θ) and perform a dimension reduction limit similar to [13]. The limit model is based on a non-quadratic non strictly-convex function of bending and torsion. Hence, we cannot compare the results directly but at least, up to a non-vanishing prefactor, the preferred bending-torsion (k 2 , k 4 ) derived above lies in the set of preferred bending-torsion given by the model in [2].

Application: Shape programming via isotropic prestrain.
In view of applications it is desirable to recover a given "target" spontaneous curvature-torsion tensor K eff (cf. Definition 2.11) by mixing simple microscopic building blocks that come in the form of parametrized microstructures. Recall that K eff determines (up to an additive constant) the minimizer of the functional (18) (and of (7) up to the infinitesimal stretch). Next, we show that a simple isotropic prestrain suffices to prescribe the bending part of K eff . To simplify the computations, we consider the following specific situation: • The material is isotropic and homogeneous, i.e., we assume that Q (cf. Assumption 2.3) is of the form Q(x, y, G) = Q(G) = 2µ|G| 2 + λ(trace G) 2 with µ > 0 and λ ≥ 0 being fixed from now on, • The cross-section of the rod is circular, i.e. S := {x : |x| ≤ 1}.

Main result -Proof of Theorem 2.7
We first state a compactness and approximation result which is a simple consequence of [33, Proposition 3.2] and [33, Theorem 3.5].
Step 1. Proof of (a). By [33,Proposition 3.2] there exist (u, R) ∈ A and a subsequence (not relabeled) satisfying (10). Moreover, by [33,Theorem 3.5] there exists a ∈ L 2 (ω) and χ ∈ H γ rel such that, up to extracting a further subsequence (not relabeled), we have (By the argument in [33, Proof of Theorem 3.5 (a), Step 4] it a posteriori follows that the rotation field R in (10) and (43) are the same). Hence, it is left to show (11). By two-scale convergence, for every η ∈ L 2 (ω) we havê By (3) and the definition of H γ rel , we have for almost every Combining (44) and (45), we obtain (11).
We recall the following (lower semi-)continuity result with respect to two-scale convergence: Let Q ε and Q be as in Assumption 2.3, and letẼ h be a sequence in L 2 (Ω; R 3×3 ).
Notice that in [33] the statement of Lemma 5.2 is proven, following arguments of [46], under the assumption that x 1 → Q(x 1 ,x, y) is continuous for almost every (x, y) ∈ S × R. Evidently, this extends to the piecewise continuous case considered here. Now, we are in position to prove Theorem 2.7. We follow the argument in [33].
Proof of Theorem 2.7.
Step 1. Compactness. In view of Proposition 5.1 it suffices to show that for every sequence (u h ) ⊂ L 2 (Ω; By the triangle inequality, Now, (41) follows by non-degeneracy of W , cf. (W2), and the equiboundedness of B h in L 2 (Ω).
Step 2. Lower bound. Let (u h ) ⊂ L 2 (Ω; R 3 ) be such that (10) and (11) are valid for some (u, R, a) ∈ A. Without loss of generality, we may assume that By Proposition 5.1 (a), there exists χ ∈ H γ rel such that (42) holds (up to possibly extracting a further subsequence). To shorten notation, we set and Substep 2.1. We claim that The bound follows by a careful Taylor expansion. To that end, set ). Thanks to the non-negativity of W ε(h) , frame-indifference (W1), the quadratic expansion (W4), and minimality at identity (W3), we get where in the last line we used that Since (E h ) and (B h ) are bounded in L 2 and since lim sup h→0 h B h L ∞ (Ω) = 0 (by assumption), we get From E h 2,γ → E, B h 2,γ → B and lim sup h→0 h B h L ∞ (Ω) = 0, and since 1 h → 1 (boundedly in measure), we get and thus (46) follows with help of Lemma 5.2.

Homogenization formulas via
BVPs -Proofs of Lemma 2.10, Proposition 3.1 and Lemma 3.2 Proof of Lemma 2.10. Throughout the proof we denote by C a positive constant that can be chosen only depending on α, β, γ and S. By construction E γ,x 1 is linear and bounded. A standard argument from functional analysis implies that E γ,x 1 is an isomorphism, if E γ,x 1 is surjective and satisfies (23). Surjectivitiy follows from the fact that H γ = range(E γ,x 1 ) + H γ rel , which implies that (H γ rel ) ⊥x 1 ⊂ range(E γ,x 1 ) for every x 1 ∈ ω. The upper bound in (23) is a consequence of (19). We prove the lower bound. Since (H γ rel ) ⊥x 1 ⊂ H γ , for any (K, a) ∈ Skew(3) × R there exists χ K,a ∈ H γ rel such that E γ,x 1 (K, a) = E(K, a) + χ K,a . By the properties of the orthogonal projection, This variational problem has a unique solution, and the associated solution operator T : where · and ·, · denote the standard norm and scalar product in H (i.e. (F, G) := ij˜S×Y F ij G ij ). It is easy to see that E(0, a) is (·, ·)-orthogonal to E(K, 0) and H γ rel . Thus, Moreover, by the short argument in [33, Step 3, Proof of Proposition 2.13], we have This completes the argument for the lower bound in (19).
Step 3. Argument for (b). By the definition of M it suffices to show that the map For every x 1 , x 1 ∈ ω and i = 1, . . . , 4, we have Hence, the piecewise continuity of L(x 1 , ·) ∈ L ∞ (S × R) yields the piecewise continuity of Step 4. Argument for (d).
Proof of Lemma 3.2.
By the definition of (·, ·) x 1 , and thanks to the fact that LF, G = L sym F, sym G , the above equation can be written in the form S×Y L(x 1 ,x, y)(E + D γ ∇φ E ), D γ ∇φ = 0.

Isotropic case -Proof of Lemma 4.1
Proof of Lemma 4.1. We only need to prove (34a)-(34c) (which will done in Step 1 and Step 2 below). The remaining claims then follow from the observation that E(K (j) , 0) + χ (j) , j = 1, 2, 3 and E(0, 1)+χ (4) given by (34a)-(34c) are mutually orthogonal with respect to the inner product (·, ·) x 1 . The latter implies that M(x 1 ) is diagonal and thus a straightforward calculation yields the precise formulas for the entries of M, b and k. For the argument of (34a)-(34c) we first note that χ (i) defined by the variational problem (27) can be equivalently characterized as the minimizer of an associated quadratic-convex energy functional. We exploit this fact in the proof below.
and thus the claim follows.
• Calculations for k 4 , i.e. the preferred twist. Here we have to distinguish between splay bend and twist.
For this, we use an explicit expression of the torsion function ϕ S in case of the square, see Section A.2. We claim that, where c S is given as in (52). Before proving the identity (53), we observe that (53) implies (52). Indeed, B(n = −rc S µ hom cos(2ϑ).

A.2 The torsion function for the square
In this section, in contrast to all other parts of the paper, we use the notation x = (x 1 , x 2 ) ∈ R 2 .
Step 2. Conclusion. In view of Step 1, ϕ Q given as in (57) satisfy − Q ϕ Q = 0 and (in the weak sense) −∆ϕ Q = 0 in Q and where ν denotes the outer normal to ∂Q. Since (61) is the Euler-Lagrange equation for (56) the claim follows.