Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours

We study continuum limits of discrete models for (possibly heterogeneous) nanowires. The lattice energy includes at least nearest and next-to-nearest neighbour interactions: the latter have the role of penalising changes of orientation. In the heterogeneous case, we obtain an estimate on the minimal energy spent to match different equilibria. This gives insight into the nucleation of dislocations in epitaxially grown heterostructured nanowires.

where we compare the minimal energy of heterogeneous defect-free systems and the minimal energy of heterogeneous systems containing dislocations. It turns out that for sufficiently large values of k, the latter are energetically preferred since their energy may grow exactly like k N −1 (see Remark 8). In this respect our result is consistent with the one proven in [14,15] under the non-interpenetration assumption. We recall that the first variational justification of dislocation nucleation in nanowire heterostructures was obtained in [17] in the context of non-linear elasticity. This result was later generalised to a discrete to continuum setting in [14,15] under the non-interpenetration condition, and is here validated without the latter assumption. More recently, variational models for misfit dislocations at semi-coherent interfaces and in elastic thin films have been proposed in [10] and [11] respectively. The paper is organised as follows. In Section 1 we introduce the model. In Section 2 we introduce the minimal costs to bridge different equilibria and study their dependence on the thickness of the nanowire. In Sections 3-5, performing a discrete to continuum limit and a dimension reduction simultaneously, we characterise the Γ-limit of the energy functional for different choices of the topology (Theorems 5.1 and 5.4). All the results are stated in the general case of heterogeneous nanowires. In Section 6 we discuss the effect of boundary conditions on the Γ-limit and briefly study a model including external forces (only in the homogeneous case, for simplicity). In the final part of the paper, Section 7, we compare the model for defect-free nanowires with models including dislocations at the interface, showing that the latter are energetically favoured.
Notation. We recall some basic notions of geometric measure theory for which we refer to [3]. Given a bounded open set Ω ⊂ R N , N ≥ 2, and M ≥ 1, BV (Ω; R M ) denotes the space of functions of bounded variation; i.e., of functions u ∈ L 1 (Ω; R M ) whose distributional gradient Du is a Radon measure on Ω with |Du|(Ω) < +∞, where |Du| is the total variation of Du. If u ∈ BV (Ω; R M ), the symbol ∇u stands for the density of the absolutely continuous part of Du with respect to the Ndimensional Lebesgue measure L N . We denote by J u the jump set of u, by u + and u − the traces of u on J u , and by ν u (x) the measure theoretic inner normal to J u at x, which is defined for H N −1 -a.e. x ∈ J u , where H N −1 is the (N −1)-dimensional Hausdorff measure. A function u ∈ BV (Ω; R M ) is said to be a special function of bounded variation if Du − ∇u L N is concentrated on J u ; in this case one writes u ∈ SBV (Ω; R M ). Given a set E ⊂ Ω, we denote by P (E, Ω) its relative perimeter in Ω and by ∂ * E its reduced boundary. We recall that a partition {E i } i∈N of Ω is called a Caccioppoli partition if i∈N P (E i , Ω) < +∞. Given a rectifiable set K ⊂ Ω, we say that a Caccioppoli partition {E i } i∈N of Ω is subordinated to K if for every i ∈ N the reduced boundary ∂ * E i of E i is contained in K, up to a H N −1 -negligible set.
For N ≥ 2, M N ×N is the set of real N ×N matrices, GL + (N ) is the set of matrices with positive determinant, O(N ) is the set of orthogonal matrices, and SO(N ) is the set of rotations. We denote by I the identity matrix and J the reflection matrix such that Je 1 = −e 1 and Je i = e i for i = 2, . . . , N , where {e i : i = 2, . . . , N } is the canonical basis in R N . The symbol co(X) stands for the convex hull of a set X in M N ×N . Moreover, given N +1 points x 0 , x 1 , . . . , x N ∈ R N , we denote by [x 0 , x 1 , . . . , x N ] the simplex determined by all convex combinations of those points.
Finally, U is the class of subsets of (−L, L) that are disjoint union of a finite number of open intervals.
In the paper, the same letter C denotes various positive constants whose precise value may change from place to place.
1. Setting of the problem. We study the dimension reduction of a discrete model for heterogeneous nanowires. Let L > 0, k ∈ N, Ω kε := (−L, L) × (−kε, kε) N −1 . Up to an affine deformation H ∈ GL + (N ), we can reduce to the case where the lattice is Z N . Thus we consider the discrete thin domain L ε (k) ⊂ R N defined as L ε (k) := εZ N ∩ Ω kε , (1.1) where Ω kε is the union of all hypercubes with vertices in εZ N that have non-empty intersection with Ω kε . In the physically relevant case of N = 3, the set L ε (k) models the crystal structure of a nanowire of length 2L and thickness 2kε, where k is the number of parallel atomic planes. We will nonetheless state all the results for a general N , since their proof does not depend on the dimension. Notice that in definition (1.1) the dependence on k is explicit; this parameter will indeed play a major role in the subsequent analysis. The bonds between the atoms are defined by means of the so-called Kuhn decomposition, which is relevant for modelling some specific Bravais lattices. (See [2, Remark 2.6] for details on the treatment of some lattices in dimension two and three, such as the hexagonal or equilateral triangular, the face-centred cubic, and the body-centred cubic.) First we define a partition T 0 of the unit cube (0, 1) N into N -simplices: we say that T ∈ T 0 if the (N +1)-tuple of its vertices belongs to the set {0, e i1 , e i1 + e i2 , . . . , e i1 + e i2 + · · · + e iN } : where S N is the set of permutations of N elements; see Figure 1. Next, we define T as the periodic extension of T 0 to all of R N . We say that two nodes x, y ∈ Z N are contiguous if there exists a simplex T ∈ T that has both x and y as its vertices. facet) and x 0 and y 0 are opposite vertices. We set and remark that, by periodicity, B 1 and B 2 do not depend on x.
We assume that L ε (k) is composed of two species of atoms, occupying the points contained in the subsets The two species of atoms are characterised by equilibrium distances given by ε and λε, respectively, where λ ∈ (0, 1] is fixed; the case λ ∈ (0, 1) models a heterogeneous nanowire, while the case λ = 1 refers to a homogeneous nanowire. Specifically, the total interaction energy relative to a deformation u : L ε (k) → R N is defined as where p > 1, H ∈ GL + (N ), and the coefficient c(ξ) is equal to some c 1 > 0 for ξ ∈ B 1 and to c 2 > 0 for ξ ∈ B 2 . To simplify the presentation, we restrict our attention to the case of p-harmonic potentials, though our analysis applies, without any significant change, to more general potentials satisfying polynomial growth conditions. More precisely, we may replace for some positive constants C 1 , C 2 . One could consider the case of potentials depending also on x and satisfying suitable periodicity assumptions: this would require a more delicate analysis and would lead to a more complex formula for the Γ-limit. In principle, all the results that we present in the sequel extend to the case when the two components of the nanowire have equilibria of the form H − and H + , where H − , H + ∈ GL + (N ). We have chosen to analyse the case when H + = λH − , since this is particularly meaningful in applications where one has misfit between two crystalline materials with the same lattice structure but different lattice distance at equilibrium (see e.g. [9,13]).
We study the limit behaviour of E 1,λ ε (·, k) as ε → 0 + , thus performing simultaneously a discrete to continuum limit and a dimension reduction to a one-dimensional system. The limit functional was derived in [14,15] by means of Γ-convergence, under the assumption that the admissible deformations fulfil the non-interpenetration condition, namely, that the Jacobian determinant of (the piecewise affine interpolation of) any deformation is strictly positive almost everywhere. The noninterpenetration assumption was used in several parts of the analysis; in particular, it was needed to prove that the limit functional (dependent on k) scales like k N as k → ∞.
The main novelty of the present paper is that we remove the non-interpenetration assumption made in [14,15], allowing for changes of orientations. Furthermore, in the study of the Γ-limit we define a stronger topology that accounts for such changes. In the proof of the new results, only those parts that differ from [14,15] will be shown in details.
In the sequel of the paper we will often consider the rescaled domain 1 ε Ω kε , which converges, as ε → 0 + , to the unbounded strip We define the associated lattice and subsets where Ω k,∞ is the union of all hypercubes with vertices in Z N that have non-empty intersection with Ω k,∞ . For u : We identify every deformation u of the lattice L ε (k) by its piecewise affine interpolation with respect to the triangulation εT . By a slight abuse of notation, such extension is still denoted by u. We can then define the domain of the functional (1.4) as Similarly, for (1.5) we define As customary in dimension reduction problems, we rescale the domain Ω kε to a fixed domain Ω k , independent of ε, by introducing the change of variables z(x) := (x 1 , εx 2 , . . . , εx N ). Accordingly, for each u ∈ A ε (Ω kε ) we defineũ(x) := u(z(x)). Moreover we set Ω k : i.e., z(x) = A ε x. In this way we can recast the functionals (1.4) defined over varying domains into functionals defined on deformations of the fixed domain Ω k . Precisely we set For later use it will be convenient to set the following notation: 2. Definition and properties of minimal energies. We recall that, throughout the paper, I is the identity matrix and J is the reflection matrix such that Je 1 = −e 1 and Je i = e i for i = 2, . . . , N . We will study the Γ-limit of the sequence I 1,λ ε (·, k) as ε → 0 + for every fixed k. For this purpose we introduce the quantity γ(P 1 , P 2 ; k) for P 1 , P 2 ∈ O(N ) ∪ λ O(N ), which represents the minimum cost of a transition from a well to another. Specifically, for each P 1 ∈ O(N ) and P 2 ∈ λ O(N ) we define The next proposition shows that the relevant quantities defined through (2.   γ(λR, λQ; k) = γ(λQ, λR; k) = γ(λI, λJ; k) .
We now prove estimates on the asymptotic behaviour of γ(I, λI) and γ(I, λJ) as k → ∞, which have interesting consequences towards the comparison of this model with those accounting for dislocations in nanowires, see Section 7 below. Indeed, in Theorem 2.2 below we show that for λ = 1 (heterogeneous nanowire) these constants grow faster than k N −1 , while it is known that the corresponding minimum cost for nanowires with dislocations scales like k N −1 (see discussion at the end of Section 7). In contrast, we remark that for λ = 1 one has γ(I, I) = 0 and γ(I, J) ≃ Ck N −1 . An essential tool in the proof of Theorem 2.2 is the following result.
Then there are a subsequence (not relabelled) and a function u ∈ W 1,∞ ((0, 1) N ; Specifically, u is a collection of an at most countable family of rigid deformations, i.e., there exists a Caccioppoli partition {E i } i∈N subordinated to the reduced boundary ∂ * {∇u ∈ SO(N )H}, such that We now prove the main result of this section.
Proof. The upper bound (2.7) is proven by comparing test functions for γ(P 1 , P 2 ; k) with those for γ(P 1 , Note that in the previous inequalities one uses the fact that ∇v ∈ L ∞ and that the energy of the interactions in B 2 can be bounded, using the Mean Value Theorem, by the energy of the interactions in B 1 . For the proof of the lower bound (2.8) we will use Theorem 2.1 in each of the subsets (−1, 0) × (−1, 1) N −1 and (0, 1) × (−1, 1) N −1 . By contradiction, suppose that there exist a sequence k j ր ∞ and a sequence {u j } ⊂ A ∞ (Ω kj ,∞ ) such that . Accordingly, we consider the rescaled lattices (2.10) The above term controls the (piecewise constant) gradient of v j . From (2.9), (2.10), and Theorem 2.1 we deduce that, up to subsequences, ∇v j → ∇v in denotes the reduced boundary of E). Therefore, using a blow-up argument and the fact that v ∈ W 1,∞ ((−1, 1) N ; R N ), we deduce that there exist rank-1 connections between O(N )H and λ O(N )H; see [2,Lemma 3.2]. This implies in particular that λ = 1, which is a contradiction to λ ∈ (0, 1). Hence (2.8) follows. Remark 1. An estimate similar to (2.8) was proven in [14,15] (for a hexagonal lattice in dimension two and a class of three-dimensional lattices) via a different argument, based on the non-interpenetration condition. In fact, in [14,15] a stronger result is proven, namely, that γ(I, λI; k) scales like k N .
The non-interpenetration assumption turns out to be necessary if the energy involves only nearest neighbour interactions; indeed, in such a case, one can exhibit deformations that violate the non-interpenetration condition and for which (2.8) does not hold, see [14,Section 4.2]. Such deformations, which consist of suitable foldings of the lattice, would be energetically expensive (and, in particular, would not provide a counterexample to (2.8)) in the present setting, exactly because of the effect of the interactions across neighbouring cells. It is the latter ones that prevent folding phenomena and allow one to prove (2.8), via Theorem 2.1.

Compactness and lower bound.
Before characterising the Γ-convergence for the rescaled functionals (1.7), we show a compactness theorem for sequences with equibounded energy, as well as bounds from above and from below on those functionals in terms of the changes of orientation in the wire. Such bounds will be used in the proof of the Γ-convergence results, Theorems 5.1 and 5.4.
Essential tools for the compactness and the lower bound are provided by the following rigidity estimates.
The constant C(U ) is invariant under dilation and translation of the domain.
It is convenient to define the energy of a single simplex T with vertices x 0 , . . . , x N , The following lemma provides a lower bound on E cell (u F ; T ) in terms of the distance of F from O(N ). It will be instrumental in using Theorem 3.1.
The next lemma asserts that if in two neighbouring simplices the sign of det ∇u has different sign, then the energy of those two simplices is larger than a positive constant. It will be convenient to define the energetic contribution of the interactions within two neighbouring simplices There exists a positive constant C 0 (depending on H) with the following property: if two neighbouring N -simplices S, T have different orientations in the deformed configuration, i.e., Due to the fact that a minimum energy has to be paid for each change of orientation, see Lemma 3.3, the parts with positive determinant do not mix with those with negative determinant. Hence, passing to the weak* limit we obtain functions taking values in co(SO(N )) ∪ co(O(N )\SO(N )), respectively λ co(SO(N )) ∪ λ co(O(N )\SO(N )). Here, co(X) denotes the convex hull of a set X in M N ×N .
Remark 2. It is well known that co(SO(N )) ∩ co(O(N )\SO(N )) = Ø: indeed, the intersection always contains the zero matrix, here denoted by 0. In dimension N = 2, one can see that In particular, co(SO (2) Henceforth, the symbol U stands for the class of subsets of (−L, L) that are disjoint union of a finite number of open intervals.
Then there exist functionsũ ∈ W 1,∞ (Ω k ; R N ), d 1 , . . . , d N ∈ L ∞ (Ω k ; R N ), and a subsequence (not relabelled) such that  .5), we apply the rigidity estimate (3.1) to the sequence u ε . To this aim, we divide the domain Ω kε into subdomains that are the Cartesian product of intervals (a i , a i + ε), a i ∈ εZ, and the cross-section (−kε, kε) N −1 . We first observe that, by Lemma 3.3 and assumption (3.3), the number of changes of orientation of u ε is uniformly bounded in ε. More precisely, we can find a uniformly bounded number of subdomains (a i , a i + ε) × (−kε, kε) N −1 , i ∈ I ε , #I ε ≤ C, such that if i / ∈ I ε then det ∇u ε has constant sign in (a i , a i + ε) × (−kε, kε) N −1 . In each of these subdomains, we use (3.2) to apply the rigidity estimate (3.1), or its "symmetric" version for O(N )\SO(N ).
Specifically, for each a i with a i < 0 and i / Moreover for i ∈ I ε we set P ε (a i ) = I if a i < 0 and P ε (a i ) = λI if a i ≥ 0. By interpolation one defines a piecewise constant matrix field P ε : . Summing up over i and rescaling the variables, one getŝ where the last inequality of each line follows by applying Lemma 3.2 to each subdomain with i / ∈ I ε and by recalling that each subdomain has volume proportional to ε after rescaling.
We now define the sets and remark that Lemma 3.2, Lemma 3.3, and assumption (3.3) imply that the cardinality of ∂U ε is uniformly bounded. Therefore the sequence {χ Uε } converges, up to subsequences, to χ U strongly in Since we can write where R ε : (−L, L) → SO(N ) is piecewise constant, we deduce that P ε converges, up to subsequences, to some P ∈ L ∞ ((−L, L); M N ×N ) in the weak* topology of L ∞ ((−L, L); M N ×N ). From (3.7) it follows that the weak* limit of ∇ũ ε A −1 ε coincides with P and therefore does not depend on x j for each j = 2, . . . , N . Moreover, inclusion (3.5) follows from the fact that χ Uε P ε converges weakly* to χ U P .
Then, ∇v ε (x) = ∇ũ ε (εx 1 + α ε i , x 2 , . . . , x N )A −1 ε = ∇u ε (εx 1 + α ε i , εx 2 , . . . , εx N ), and, by (3.7), we have 14 ROBERTO ALICANDRO, GIULIANO LAZZARONI AND MARIAPIA PALOMBARÔ (3.10) From (3.10), Theorem 3.1 and the Poincaré inequality, we deduce that there exists a unit interval contained in (− 2σ ε , − σ ε ) such that in the Cartesian product of such interval with the cross-section (−k, k) N −1 , the W 1,p -norm of the difference between v ε and an affine map of the form λQHx + a, with Q ∈ O(N )\SO(N ) and a ∈ R N , is bounded by Cε/σ. By the same argument one can find a unit interval contained in ( σ ε , 2σ ε ) such that in the Cartesian product of such interval with the cross-section (−k, k) N −1 , the W 1,p -norm of the difference between v ε and an affine map of the form λRHx + b, with R ∈ SO(N ) and b ∈ R N , is bounded by Cε/σ. By gluing the function v ε with these maps on such intervals, one can define a functionv ε ∈ A ∞ (Ω k,∞ ) that is a competitor for γ(λJ, λI; k) and such that (cf. (2.2))

only takes into account the interactions between atoms lying in the subset (α
Arguing in a similar way for the other intervals in (3.9) yields (3.6).

Upper bound.
We prove that the bound (3.6) is in fact optimal.  Then there exists a sequence {ũ ε } ⊂ A ε (Ω k ) such that

2)
and lim sup Proof. Using a standard approximation argument we may assume that x 1 → F (x 1 ) is piecewise constant, with values in O(N )H for a.e. x 1 ∈ (−L, 0) and values in λ O(N )H for a.e. x 1 ∈ (0, L). We may also assume that this approximation process does not modify the set U of (4.1). More precisely, there exist m, n ∈ Z, m < 0, n ≥ 0, −L = a m < a m+1 < · · · < a −1 < a 0 = 0 < a 1 < · · · < a n < a n+1 = L, and R i ∈ O(N ) for i = m, . . . , −1, 0, . . . , n such that The following construction is similar to that in [14,Proposition 3.2], so we will show the details only for what concerns the changes of orientation. We introduce a mesoscale {σ ε } such that ε ≪ σ ε ≪ 1 as ε → 0 + . Next we defineũ ε in the sets of the type (a i +σ ε , a i+1 −σ ε ) × (−k, k) N −1 in such a way that its gradient equals R i HA ε if a i+1 ≤ 0 and equals λR i HA ε if a i ≥ 0. This determinesũ ε in those regions, up to some additive constants that will have to be fixed at the end of the construction in order to makeũ ε continuous.
We now complete the definition ofũ ε in the sets of the type (a i −σ ε , a i +σ ε ) × (−k, k) N −1 . Let us first assume i < 0, i.e., a i < 0. Since R i−1 and R i may be in SO(N ) or in O(N )\SO(N ), one can have four cases. If both R i−1 and R i are in SO(N ), it is possible to defineũ ε by interpolating R i−1 and R i so that the cost of the transition has order O(ε/σ ε ), so it gives no contribution to (4.3); we refer to [14] for details. The case R i−1 , R i ∈ O(N )\SO(N ) is completely analogous.
If R i−1 ∈ SO(N ) and R i ∈ O(N )\SO(N ) or viceversa, we defineũ ε in the set (a i −σ ε , a i +σ ε ) × (−k, k) N −1 as a rescaling of a quasiminimiser of (2.1b). More precisely, we fix η > 0 and apply the definition of γ(R i−1 , R i ; k), thus finding M > 0 and v ∈ A ∞ (Ω k,∞ ) such that where we used also Proposition 1. With this at hand, we defineũ ε in the set (a i −σ ε , a i +σ ε ) × (−k, k) N −1 as The constant vector b in the last equation is chosen in such a way thatũ ε is continuous. Since each point of ∂U gives the same contribution γ(I, J; k) to the upper bound, we obtain the first term of (4.3).
The case i > 0, i.e., a i > 0, is treated similarly to i < 0 and gives rise to the second term of (4.3). Finally, for i = 0, i.e., a i = a 0 = 0, we argue as above and defineũ ε by using a rescaling of a quasiminimiser of (2.1a) and applying the definition of γ(R −1 , λR 0 ; k). We then get an interfacial contribution in (4.3) that differs in the two cases 0 ∈ ∂U and 0 / ∈ ∂U .
5. Limit functionals with respect to different topologies. In the next theorem we characterise the Γ-limit of the sequence {I 1,λ ε (·, k)} with respect to the weak* convergence in W 1,∞ (Ω k ; R N ); see [4,8] for an introduction to Γ-convergence. As it can be inferred from the compactness result in Proposition 2, the domain of the Γ-limit turns out to be We show that on such domain the Γ-limit is constant. Hence, the macroscopic description of the model is similar to that of [14,15]; in particular, it does not have memory of the changes of orientation in minimising sequences. In order to keep track of the orientation changes, we need to introduce a stronger topology for the Γ-convergence, as we see in Theorem 5.4.

(5.4)
We assume that u ∈ A 1,λ (k), the other case being trivial. The construction of the recovery sequence depends on the precise value of the minimum in (5.3). Since we do not know such value, we explain how to proceed in the case when γ(k) is any of the two quantities therein.
• If γ(k) = γ(I, λJ; k) we set U := (−L, 0) and construct d 2 , . . . , d N in such a way that Proposition 3 can be now applied to F := (∂ 1 u | d 2 | · · · | d N ), hence providing us with a sequence {ũ ε } ⊂ A ε (Ω k ) satisfying (4.2)-(4.3). In particular we have ∇ũ ε *  Here we picture only a part of the wire containing just one species of atoms, therefore the transition at the interface is not represented. A kink in the profile may be reconstructed by folding the strip, i.e., mixing rotations and rotoreflections (left); or by a gradual transition involving only rotations or only rotoreflections (right). In the limit, the former recovery sequence gives a positive cost, while the latter gives no contribution. If the stronger topology is chosen, the appropriate recovery sequence will depend on the value of the internal variable, which defines the orientation of the wire.
Remark 4. As long as the Γ-convergence is taken with respect to the weak* topology of W 1,∞ (Ω k ; R N ), (5.2) only accounts for the cost of transitions at the interface between the two species of atoms. Indeed, away from the interface it is always possible to construct recovery sequences without mixing rotations and rotoreflections, as done in the proof of the limsup inequality; such transitions have low interaction energy, since γ(I, I) = γ(J, J) = 0, see also Proposition 1. In particular, for λ = 1 the limit functional is trivial, since I 1,1 (u, k) = 0 if u ∈ A 1,1 (k).
Below we show that, if a stronger topology is chosen, the value of the Γ-limit changes. The resulting limit functional depends on an internal variable, D in (5.7), that keeps track of the changes of orientation throughout the thin wire. In fact, different transitions between the energy wells must now be employed according to the value of D; two examples are provided in Figure 2.
We introduce the sequence of functionals defined for u ∈ W 1,∞ (Ω k ; R N ) and D ∈ L ∞ (Ω k ; M N ×N ) bŷ In the next theorem we study the Γ-limit of the sequence {Î 1,λ ε (·, ·, k)} as ε → 0 + with respect to the weak* convergence in W 1,∞ (Ω k ; R N ) × L ∞ (Ω k ; M N ×N ). As a consequence of Proposition 2, the domain of the Γ-limit turns out to bê in Ω + k , where A 1,λ (k) is defined by (5.1). It is convenient to introduce the following definition, where the functional J coincides with the right-hand sides of (3.6) and (4.3).
The last definition will be used to apply Propositions 2 and 3 towards the characterisation of the Γ-limit with respect to the stronger topology. To this end, each pair (u, D) ∈Â 1,λ (k) is associated with a set U realising (5.5). Such U is in general not unique, since co(SO(N )) ∩ co(O(N )\SO(N )) = Ø. Therefore, we choose it to be "optimal", i.e., minimising (5.6). Notice that the minimum in (5.6) is attained since A minimiser needs not be unique as shown in the following example.
Theorem 5.4. The sequence of functionals {Î 1,λ ε (·, ·, k)} Γ-converges, as ε → 0 + , to the functionalÎ with respect to the weak* convergence in Proof. The liminf inequality is obtained by applying Proposition 2 and arguing as in Theorem 5.1. Also the derivation of the limsup inequality is similar to the one performed in Theorem 5.1; let us simply point out that, while in the proof of Theorem 5.1 the matrix field F needed to be reconstructed, here we set F := D and choose U as a minimiser of (5.6). The conclusion follows by applying Proposition 3.
Remark 5. We underline that Theorem 5.4 provides a nontrivial Γ-limit also in the case when λ = 1. Indeed, one hasÎ 1,1 (u, D, k) = γ(I, J; k) H 0 (∂U ∩ (−L, L)) if (u, D) ∈Â 1,1 (k) and U miminises (5.6), where γ(I, J; k) > 0. 6. Boundary conditions and external forces. In the present section we discuss how the previous results extend to the case when the functional (1.4) is complemented by boundary conditions or external forces. Although our considerations apply to the case of general H ∈ GL + (N ) and λ ∈ (0, 1], for simplicity we will focus on the case H = I and λ = 1. We will also test the consistency of the present model with the non-interpenetration condition by looking at minimisers of the Γ-limit when boundary conditions or forces are prescribed. We will see that the continuum limit that keeps track of such constraints is the one provided by the stronger topology (5.7).
Boundary conditions. Let B − , B + ∈ GL + (N ) and suppose that the functional (1.4) is now defined on deformations u ∈ A ε (Ω kε ) that satisfy It is easy to see that while the compactness result of Proposition 2 remains valid, the Γ-limit (5.2) will now contain additional terms corresponding to the minimal energy spent to fix the atoms in the vicinity of the lateral boundaries. However, such extra terms do not depend on the limiting deformations, therefore they do not encode any information about the behaviour of minimising sequences. As far as the stronger topology is concerned, one can see that the limit functional (5.7) will contain the additional quantities γ(B − , P ; k) and γ(P, B + ; k) defined, for P ∈ {I, J}, by where E 1,1 M is as in (2.1b), except that the sum is taken over all atoms contained in the bounded strip (−M, M ) × (−k, k) N −1 . The choice of P = I or P = J depends on whether or not ±L ∈ ∂Ū , whereŪ is a minimiser of (5.6). Precisely, if −L ∈ ∂Ū (resp. L ∈ ∂Ū ), then in (6.2) (resp. (6.3)) we take P = I, otherwise we take P = J. Remark 6. By Proposition 2 and the properties of Γ-convergence, minimisers of (1.4) subjected to (6.1) converge, up to subsequences, to minimisers of (5.7) complemented with the above extra terms. Moreover, if dist(B ± ; SO(N )) is sufficiently small, then such minimisers will not have transitions between co SO(N ) and co O(N )\SO(N ) . This follows from the fact that γ(I, B ± ; k) → 0 as dist(B ± ; SO(N )) → 0 and therefore, as long as γ(I, B + ; k) + γ(B − , I; k) < γ(I, J; k), the optimal transitions will fulfil the non-interpenetration condition. In this respect the quantity γ(I, J; k) can be regarded as an energetic barrier that must be overcome in order to have folding effects.
Theorem 6.1. The following results hold: Then there exists (ũ, D) ∈Â 1,1 (k) and a subsequence (not relabelled) such that (Γ-limit) The sequence of functionals {G ε } Γ-converges, as ε → 0 + , to the functional G(u, D, k) :=Î 1,1 (u, D, k) − F (D, k) , (6.6) with respect to the weak* convergence in As a consequence of the previous theorem and the standard properties of Γconvergence we infer the following result about convergence of minima and minimisers.
Corollary 1. We have that then any cluster point (u, D) of (u ε , D ε ) with respect to the weak* convergence in We now come back to the question of the consistency of the model with the non-interpenetration condition. In this context we cannot expect that minimisers of (6.5) preserve orientation for the whole class of loads defined above. This is clarified in the following remark. Define D := (n 1 | · · · |n N ) if x 1 ∈ (−L, a), and D := (n 1 | · · · |n N −1 | − n N ) if x 1 ∈ (a, L). Note that (x 1 n 1 , D) ∈Â 1,1 (k), and D has a transition point at x 1 = a. Denote byÂ 1,1 0 (k) the subset ofÂ 1,1 (k) of deformations with no transitions; i.e., Therefore, if f 1 , . . . , f N are such that − F (D, k) + γ(I, J; k) < C, then it is energetically preferred to have a transition at a, namely, all minimisers of G are given by (x 1 n 1 + b, D), with b any vector in R N . In contrast, if f N is always positive, then minimisers will not display any transition.

7.
Comparison with models including dislocations. The lattice mismatch in heterostructured materials, corresponding to λ = 1 in the model described in this section, can be relieved by creation of dislocations; i.e., line defects of the crystal structure. We refer to [9,13,19] for an account of the literature on dislocations in nanowires. A model for discrete heterostructured nanowires accounting for dislocations was studied in [14,15] under the assumption that deformations fulfil the non-interpenetration condition. In this paper we have chosen to consider only defect-free configurations in order to both simplify the exposition and to pose emphasis on the difficulties to overcome when the non-interpenetration assumption is removed. In the final part of the paper, we outline the results that can be obtained when dislocations are accounted for.
Following the ideas of [14], in dimension N = 2 we introduce other possible models where the reference configuration represents a lattice with dislocations. More precisely, we fix ρ ∈ [λ, 1] and set  and Ω ε is as in (1.1). For ρ = 1, the number of atomic layers parallel to e 1 is different in the two sublattices (for sufficiently large k); this can be regarded as a system containing dislocations at the interface. In presence of dislocations, the choice of the interactions and of the equilibria strongly depends on the lattice that one intends to model. Therefore, in this section we focus on the simplest situation of hexagonal (or equilateral triangular) Bravais lattices in dimension two and we fix The lattice HL ε (ρ, k) consists of two Bravais hexagonal sublattices with different lattice constants ε and ρε, respectively; see Figure 3. The bonds between nearest and next-to-nearest neighbours are defined first in the lattice HL ε (ρ, k). To this end, one chooses a Delaunay triangulation of HL ε (ρ, k) as defined in [14,Section 1]. Two points x, y of the lattice are said to be nearest neighbours if there is a lattice point z such that the triangle [x, y, z] is an element of the triangulation. Two points x, y are next-to-nearest neighbours if there are z 1 , z 2 such that [x, z 1 , z 2 ] and [y, z 1 , z 2 ] are elements of the triangulation. These definitions coincide with the usual notions of nearest and next-to-nearest neighbours away from the interface. We underline that other choices of interfacial bonds are possible to derive our main results. Indeed, one may start from any triangulation of the lattice satisfying the following properties: the number of nearest neighbours of each point has to be uniformly bounded by a constant independent of ε, while the length of the bonds in HL ε (ρ, k) has to be uniformly bounded by a constant C ε = Cε.
Once the bonds in the lattice HL ε (ρ, k) are defined, we define the bonds of a point x ∈ L ε (ρ, k) as follows: