ON THE DERIVATION OF LINEAR ELASTICITY FROM ATOMISTIC MODELS. Bernd Schmidt. (Communicated by Antonio DeSimone)

Transcription

1 NETWORKS AND HETEROGENEOUS MEDIA doi: /nhm c American Institute of Mathematical Sciences Volume 4, Number 4, December 009 pp ON THE DERIVATION OF LINEAR ELASTICITY FROM ATOMISTIC MODELS Bernd Schmidt Zentrum Mathemati, Technische Universität München Boltzmannstr. 3, Garching, Germany (Communicated by Antonio DeSimone) Abstract. We derive linear elastic energy functionals from atomistic models as a Γ-limit when the number of atoms tends to infinity, respectively, when the interatomic distances tend to zero. Our approach generalizes a recent result of Braides, Solci and Vitali []. In particular, we study mass spring models with full nearest and next-to-nearest pair interactions. We also consider boundary value problems where a part of the boundary is free. 1. Introduction. The passage from discrete atomic models to continuum theories is an active area of current research in continuum mechanics. For elastic systems one usually refers to the Cauchy-Born rule to obtain macroscopic energy densities from atomistic interaction functionals. The Cauchy-Born rule states that roughly speaing each individual atom follows the macroscopic deformation gradient and in particular does not tae into account fine scale oscillations on the microscopic scale. For a two-dimensional mass spring model, the validity of the Cauchy-Born rule for deformations close to a rigid motion has been proved by Friesece and Theil in [8]. Their result has been generalized to arbitrary dimensions by Conti, Dolzmann, Kirchheim and Müller in [3]. If the deformation gradients are very close to SO(d), the set of orientation preserving rigid motions, then we expect linear elasticity theory to apply. This relation has been made rigorous by Dal Maso, Negri and Percivale who derive the energy functional of linear elasticity as a Γ-limit of nonlinear elasticity for small displacements in [5]. (See also the author s article [10] for a strong convergence result for the associated minimum problems.) Recently it has been noted that one can derive linear elasticity functionals directly from certain atomistic pair potentials: For a special class of pair interaction models Braides, Solci and Vitali prove Γ-convergence of the discrete energy functionals to the energy functional of an associated continuum linear elasticity energy functional (see []). In this set-up one has to deal with two small parameters ε and δ measuring the typical interatomic distance and the local distance of the deformations from the set of rigid motions, respectively. The aim of the present article is to extend these results in three directions. Firstly, we will drop the assumption that atoms are allowed to interact only along 000 Mathematics Subject Classification. 74B05, 74B0, 49J45. Key words and phrases. Atomistic systems, linear elasticity, discrete-to-continuum limits, Γ- convergence. 789

2 790 BERND SCHMIDT special edges of a simplicial decomposition of the lattice unit cell (thus answering positively an open problem posed in []). In particular, our approach will allow us to deal with full next-to-nearest neighbor interactions in mass spring models. (In a D square lattice, e.g., the contributions of both diagonal springs can be considered.) Secondly, by considering more general cell energies, it will be possible to also deal with mass spring systems for which individual pair interactions might not be equilibrated in the reference lattice. In this regime it turns out that one has to carefully choose the two small parameters in the Γ-limit process to obtain the desired result: We will assume that ε δ. This seems to be a reasonable assumption from a physical point of view as ε refers to the ratio of microscopic to macroscopic length scales, while δ is a small macroscopic parameter describing the range of applicability of linear elasticity theory. Note that this would always be satisfied if one first applies the Cauchy-Born rule, i.e., sends ε 0 and then derives the linear limit δ 0. Technically this condition ensures that even for mismatching equilibria of surface and bul energy contributions, the surface terms cannot dominate the bul terms. Finally we will also consider more general non-affine boundary conditions and in particular also investigate the case when a part of the boundary is free. For the sequence of minimizers of the associated boundary value problem we also prove a strong convergence result to the minimizer of the limiting continuum functional. (If Dirichlet boundary conditions are imposed on every surface atom, by a suitable renormalization of the elastic energy, one can in fact drop the assumption ε δ even for incompatible surface and bul energy expressions.) To be more specific, we consider the portion εl of some scaled Bravais lattice L that lies in some fixed domain R d. If E ε (y) denotes the elastic energy of a deformation y : εl R d, our main modeling assumption is that, roughly speaing, E ε be decomposable as a sum of the form E ε (y) = Q W cell ( y Q ) + surface terms, where the sum runs over scaled unit cells Q induced by εl and the discrete gradient y Q consists of all the relative displacements of the corners of Q. Here W cell is of order one (in atomic units) and so we have to consider the scaled quantity ε d E ε (y) so as to arrive at macroscopic energy expressions for small interatomic distances ε. For mass spring systems for which not every individual spring is minimal in the reference configuration, W cell typically assumes a positive mininmum value µ; and one has to renormalize by subtracting a term µ in the energy functional in order to ensure W cell = 0 at the identity. Note, however, that the surface terms, which are also decomposable into unit cell contributions, will in general not be equilibrated at the identity. As the number of surface atoms scales with ε 1 d, our energy functional becomes ε d E ε (y) = ε d Q W cell ( y Q ) + O(ε). In order to derive the functional of linear elasticity theory, we consider deformations y = Id + δu in terms of the small displacement δu and multiply the energy by δ. This way, we finally are led to investigate the functional δ ε d E ε (Id + δu) = δ ε d Q W cell ( (Id + δu) Q ) + O(δ ε).

3 LINEAR ELASTICITY FROM ATOMISTIC MODELS 791 This formula also illustrates why the assumption ε δ will be of importance. Rigidity estimates will now guarantee that in the limit ε, δ 0: 1. Manifestation of the Cauchy-Born rule: The discrete gradient of the atomic displacements reduce to classical gradients, i.e., the microscopic deformation gradient follows the macroscopic deformation gradient.. Linearization of the energy functional: W cell reduces to its Hessian Q cell at the identity. 3. Passage from discrete to continuum theory: The discrete energy functional converges to an integral functional. As a consequence, the discrete energy functionals will be seen to Γ-converge to the common energy functional of linear elasticity, which is derived from W cell by the Cauchy-Bern rule. The paper is organized as follows. In Section we introduce the atomistic models and state the main results on their discrete-to-continuum convergence properties: Theorem.6 states compactness of finite energy sequences and the Γ-convergence to a corresponding continuum limit as an integral functional over a suitable quadratic form of the limiting linear strain. In Theorem.7 we prove strong convergence of minimizers of the discrete problems to the minimizer of the continuum theory. The following Section 3 is devoted to technical preliminaries for the proofs of these results. The main tool is a careful interpolation in between the atomic positions, which will enable us to mae use of the corresponding continuum results. By our general interaction assumptions, however, there is no associated triangulation of the body where the energy can be recovered from an energy density defined on the d d gradients of some linear interpolation. Instead, we will have to consider discrete gradients, i.e., d d matrices that will account for all relative displacements in a typical lattice cell. The ideas in this section are inspired by [9]. As in this paper (also compare [11] for the D case), an important ingredient for our compactness results is (a discrete version of) the geometric rigidity result of Friesece James and Müller, cf. [6]. However, near the free part of the boundary, discrete rigidity may fail. Our main focus in Section 3 will be on how to overcome this difficulty. As a by-product, we state a slightly generalized version of the discrete geometric rigidity result of [9], which allows for a more general shape of the macroscopic region occupied by the atoms. In Section 4 we will then give the proofs of Theorems.6 and.7. The scheme to arrive at a linearized limiting functional follows [5, 10] and also draws ideas from [6]. In fact, the strategy of our proofs is to reduce the problem to the continuum setting investigated in [5, 10], and we refer the reader to these papers rather than re-deriving the results that are needed here. However, as mentioned above and also noticed in [], the discrete nature of the atomistic interaction raises additional difficulties for the simultaneous linearization/discrete-to-continuum limiting process. We exploit techniques developed in [3, 9, 11] to overcome these problems. In the last Section 5 we will give some examples of mass spring models as admissible atomistic interaction functionals and their limiting continuum linear energy functionals. In particular, we will discuss the nearest neighbor interaction in a triangular lattice (recovering the functional derived in []) and the nearest and next-to-nearest neighbor interaction in a square lattice in D. In 3D we will first discuss a general nearest and next-to-nearest neighbor model. By way of example of

4 79 BERND SCHMIDT an equilibrated bcc crystal, we will then also show how our technique can yield continuum theories rigorously even if the basic assumption on the interactions, namely, to be decomposable into individual lattice cell contributions, fails.. The model and main results. Let R d be a bounded Lipschitz domain and L some Bravais-lattice in R d, i.e., L = {λ 1 v λ d v d : λ 1,...,λ d Z} = AZ d for linearly independent vectors v 1,...,v d R d, where A is the matrix (v 1,...,v d ). For notational convenience we will suppose that the v i are labeled such that deta > 0. We assume that the atomic positions in the reference configuration are given by the points of the scaled lattice L ε = εl that lie within. Here ε is a small parameter measuring the interatomic distance and eventually tending to 0. Note that L ε partitions R d into unit cells εa(λ + [0, 1) d ), λ Z d. Figure 1. Reference configuration of an atomic system. We denote the shifted lattice εa(( 1,..., 1 )T + Z d ) consisting of the midpoints of the unit cells by L ε. Accordingly, if x Rd we denote by x = x(x, ε) the element of L ε lying in the same unit cell as x. The ε-cell corresponding to x is defined by Q ε (x) = x + A[ ε, ε )d. A corner of Q ε (x) is an element of x + A{ ε, ε }d. The deformations of our system are mappings y : L ε R d. In order to eep trac of the images of atoms under such deformations we choose a numbering z 1,..., z d of the corners A{ 1, 1 }d of the reference cell A[ 1, 1 )d and view Y (x) = (y 1,..., y d) = (y( x + εz 1 ),..., y( x + εz d)) Z = (z 1,..., z d) as elements of R d d. Let L ε() = { x L ε : Q ε ( x) }, and L ε () = L ε() + ε{z 1,..., z d}. Whenever convenient we will extend the deformations y : L ε R d to a function on L ε (). (The energy functional will of course not be affected by this extension. For the precise definition of the values on the additional sites we refer to the next section.) Our basic assumption is that the energy of a deformation y can be expressed by cell energies W ε : L ε() R d d [0, ] in the form E ε (y) = W ε ( x, Y ( x)), () x L ε () (1)

5 LINEAR ELASTICITY FROM ATOMISTIC MODELS 793 where W ε ( x, ) splits into a bul and a surface part { W cell (ε 1 F), if I ε ( x) = {1,..., d }, W ε ( x, F) = W surface (I ε ( x), ε 1 F), if I ε ( x) {1,..., d }, (3) where I ε ( x) = {i : x + εz i } {1,..., d }. We also assume that the surface terms only depend on the atomic positions of those atoms that lie in L ε in the reference configuration, i.e., W surface (I, ε 1 Y ) depends on the second variable only through (y i ) i I. Note in particular that there are no more than d different surface functions W surface (I, ). The rescaling by ε 1 is due to our choice of measuring the energy contributions of cell deformations in atomic units. Remar. Note that such a decomposition of the energy is possible for suitable mass spring models; e.g., in a two-dimensional square lattice with nearest and next-tonearest neighbor interaction one would assign half of the interaction energy of the nearest neighbor bonds to each of the two adjacent cells. Note also that even in this simple example W surface and W cell could have mismatching equilibria if the ratio of the preferred distance between next-to-nearest neighbors and nearest neighbors is not. See Section 5 for more general examples, also in 3D. Before we describe the properties of these cell energies in more detail, we introduce the discrete gradient of a lattice deformation y : L ε () R d : If x Q ε ( x) with x L ε(), set y(x) := ε 1 (y 1 ȳ,..., y d ȳ), ȳ := 1 d d y i, y i = y( x + εz i ), i = 1,..., d. (So in particular y is a piecewise constant function on with values in R d d.) Also let SO(d) := { R := RZ : R SO(d)} R d d, where Z is the d d matrix (z 1,..., z d) introduced in (1). Our general assumption on the cell energies is the following Assumption.1. (i) Each of the energies W cell, W surface (I, ) : R d d [0, ], I {1,..., d }, is invariant under translations and rotations, i.e. for F R d d, W cell (RF + (c,..., c)) = W cell (F), W surface (I, RF + (c,..., c)) = W surface (I, F) I {1,..., d } for all R SO(d), c R d. (ii) W cell (Y ) is minimal (= 0) if and only if there exists R SO(d) and c R d such that y i = Rz i + c, i = 1,..., d. W surface is bounded in a neighborhood of SO(d). (iii) W cell is C in a neighborhood of SO(d) and the Hessian Qcell = D W cell (Z) at the identity is positive definite on the orthogonal complement of the subspace spanned by infinitesimal translations (x 1,...,x d) (c,...,c) and rotations (x 1,..., x d) (Ax 1,..., Ax d), A T = A. i=1

6 794 BERND SCHMIDT (iv) W cell grows at infinity at least quadratically on the orthogonal complement of the subspace spanned by infinitesimal translations, i.e. lim inf F F V W cell (F) F > 0. Here V = {F R d d : F F d = 0}. Note that for the quadratic form Q cell these assumptions imply that Q cell (v,..., v) = 0 and Q cell (Az 1,...,Az d ) = 0 (4) for all v R d and A R d d with A T = A. Definition.. We say that W surface is compatible with W cell if W surface (I, ) CW cell in a neighborhood of SO(d). In order to study boundary value problems, we consider the Dirichlet boundary, which is a closed subset of of positive H d 1 -measure, and boundary data given by g W 1, (). The space of continuum displacements is H 1 (g,, ), the H 1 -closure of {u W 1, (, R d ) : u(x) = g(x) x }. The definition of the appropriate function space in the discrete setting is subtle, in particular since we do not assume that is smooth or that =. In order to avoid pathologies due to the loss of rigidity near the free boundary, one has to prescribe boundary values in a suitable ε-neighborhood of for the discrete displacements. We introduce the following notation. Definition.3. (i) Fix a point x 0 ( (independent of ε). Then by ε we ) denote the connected component of Q( x) Q ε( x) containing x0. (ii) We write x L ε() and call Q ε ( x) an inner cell if x L ε ε. If x L ε () := L ε () \ L ε (), then Q ε ( x) is called a boundary cell. The set of the corners of all boundary cells, respectively inner cells, is denoted L ε (), respectively L ε (). Note that as ε 0, ε does not depend on the particular ( choice of x 0. For ) sufficiently small ε, ε is the connected component of Q( x) Q ε( x) which contains the bul part of the inner cells. Also note that since is a Lipschitz domain, there exists a constant C = C() such that sup{dist(x, ε ) : x } < Cε. For two points x, x L ε() we denote their lattice geodesic distance, i.e., the length of the shortest polygonal path ( x = x 0, x 1,..., x n = x ) with x i L ε () and x i+1 x i {±v 1,..., ±v d } connecting x and x by dist L ε ()( x, x ). (Also because is a Lipschitz domain, dist L ε ()( x, x ) C x x.) By extension we may furthermore assume that g W 1, (R d, R d ). Definition.4. (i) A boundary cell Q = Q ε ( x), x L ε (), is called a Dirichlet boundary cell ( x L ε () ) if there exists a cell Q = Q ε ( x ), x L ε () such that Q and dist L ε ()(Q, Q ) dist L ε ()(Q, ε ). The set of corners of these cells is denoted L ε (). (ii) Now suppose u : L ε () R d is a lattice mapping. We say that u A ε (g,, ), if u(x) = g(x) for every x L ε (). Remar. This definition of A ε guarantees that even under very wea assumptions on the surface energy W surface, the bul part of the body feels the boundary condition g. Under additional assumptions either on the geometry of (asing, e.g.,

7 LINEAR ELASTICITY FROM ATOMISTIC MODELS 795 to be C 1 ) or on the rigidity properties of W surface (e.g., imposing suitable growth conditions), the notion u A ε can be relaxed in the sense that only those boundary cells that actually contain a piece of are defined to be Dirichlet boundary cells. Finally we have to mae precise in which sense discrete lattice mappings are understood to converge to continuum deformations. To this end, for f L (R d, R m ) define P ε f L by P ε f(x) := f(ξ)dξ, (5) Q ε(x) where Q ε (x) is the lattice cell containing x. It is not hard to see that P ε f f in L as ε 0. Now if f L (, R m ), then we first extend f to a function in L (R d, R m ), again denoted f, and define P ε f as before. Definition.5. Let ε 0 and suppose y : L ε R d, y L (, R d ). We say that y converges to y, i.e., y y, if ε d y (x) P ε y(x) 0. x L ε Note that this definition does not depend on the particular extension that has been chosen for y. We also remar that our notion of convergence is equivalent to asing that suitable interpolations (e.g., piecewise constant or piecewise affine on a triangulation subordinate to the lattice) of y converge to y in L (, R d ). We will derive linear elasticity functionals by studying the functionals I ε,δ : A ε (g,, ) R, u δ ε d E ε (Id + δu) = δ ε d x L ε () W ε ( x, ε(z + δ u( x))) for lattice displacements u : L ε () R d. The main results of this paper are the following: Theorem.6. Suppose W cell and W surface satisfy Assumption.1 and ε, δ are sequences converging to 0 as. In case W cell and W surface are not compatible, assume that also lim ε δ = 0. Set I = I ε,δ. Compactness: If I (u ) is equibounded for a sequence (u ) of lattice displacements, then there exist modifications u A ε (g,, ) with u = u on L ε () L ε () such that for a subsequence (not relabeled) u u for some u H 1 (g,, ). Moreover, the (piecewise constant) discrete gradients u converge to u Z wealy in L () and thus u u Z in L loc ( \ \ ). Γ-convergence: The functionals I Γ-converge to the functional I : H 1 (g,, ) R, u 1 Q cell (e(u) Z). deta (Here e(u) denotes the linear strain e(u) = 1 (( u)t + u).) I.e., (i) Γ-liminf inequality: All sequences (u ), u A ε (g,, ), converging to some u H 1 (g,, ) satisfy the estimate lim inf I (u ) I(u).

8 796 BERND SCHMIDT In fact, this is true under the weaer assumption ε d u (x) P ε u(x) 0. x L ε () L ε () (ii) Existence of recovery sequences: For every u H 1 (g,, ) there is a sequence (u ), u A ε (g,, ), such that u u and lim I (u ) = I(u). By general arguments in the theory of Γ-convergence and the fact that the minimizer of the limiting problem is unique (by Korn s inequality), one can now deduce that, if w is a minimizer of I, then w w (and so w w away from the free boundary ) and w w Z wealy in L loc ( \ \ ), where w is the minimizer of I. In fact, following the approach in [10] we can show that recovery sequences converge even strongly: Theorem.7. Suppose (u ) is a recovery sequence for u. Then u u Z strongly in L loc ( \ \ ). We call (w ), w A ε (g,, ), a sequence of almost minimizers if lim (I (w ) inf{i (v) : v A ε (g,, )}) 0 for. The previous theorem immediately implies strong convergence of w : Corollary.8. If (w ) is a sequence of almost minimizers, then w w Z strongly in L loc ( \ \ ), where w is the unique minimizer of I. Remars. (i) As our definition of convergence can be reformulated in terms of the usual L -convergence of suitable interpolations as remared before, we are indeed proving a Γ-limit result in the usual set-up of convergence in metric spaces (see, e.g., [1, 4] for general introductions to the theory of Γ- convergence). In the next section we will associate carefully chosen modifications u (on the free part of the boundary) and interpolations ũ to lattice displacements u. Our proofs will in particular show that one has wea H 1 -compactness for the interpolations: I (u ) C implies ũ u in H 1 (up to subsequences). For recovery sequences u u and sequences of almost minimizers (w ) we will see that ũ u and w w strongly in H 1. (ii) By standard arguments in the theory of Γ-convergence it is straightforward to include loading terms of the form L ε (u) = l ε (x) u(x), x L ε where l ε l for some l L () in the sense of Definition.5, e.g., l ε (x) = P ε l(x). (iii) In case = we can renormalize the energy by setting Eε ren (y) := E ε (y) W ε ( x, Y ( x)) = W cell ( y( x)). x L ε () x L ε ()\ L ε () Then I ren ε,δ, defined accordingly, will satisfy all the assertions in Theorems.6,.7 and Corollary.8 with limiting functional I, even if W surface is not admissible in the sense of Assumption.1.

9 LINEAR ELASTICITY FROM ATOMISTIC MODELS 797 (iv) Note that indeed F 1 det A Q cell(f Z) is the quadratic form of the energy per unit volume of linear elasticity obtained from the atomistic model by applying the Cauchy-Born rule. Here, the Cauchy-Born rule does not enter as an assumption but rather is a consequence of our analysis: For a given linear macroscopic deformation gradient F, the discrete gradient of a typical unit cell is the discrete gradient F Z induced by F. 3. Discrete deformations. In the following paragraphs we introduce the main technical tools for passing from discrete lattice deformations to continuum objects which are more amenable to the analytical limiting process in the proofs of Section 4. Recall Definitions.3 and.4. In the following we will call two cells Q and Q neighbors if Q Q. The d d matrix assumed by the discrete gradient y on some cell Q will be denoted y Q Modification. To every lattice displacement u A ε (resp., deformation y = Id + δu) we associate a modified displacement u (resp. deformation y ) in the following way. On Dirichlet boundary cells and inner cells in the bul nothing changes, i.e., y (x) = y(x) for x (L ε () ε ) L ε () ; in particular, u(x) = g(x) for x L ε (). On the remaining boundary cells we proceed as follows: Consider the d sublattices L ε,i with L ε,i = z i +L ε and extend successively in the following Steps 1.1+, 1.1-, 1.+, 1.-,..., 1.d+, 1.d-,.1+,.1-,.+,...,.d-, 3.1+,..., where Step i.j± : For every cell Q = Q ε ( x) with x L ε,i such that there exists a cell Q = Q ε ( x±εv j ), i.e. sharing a (d 1)-face with Q, on the corners of which y has been defined already, we extend y to all corners of Q by choosing an extension y such that dist ( y Q, SO(d)) is minimal. Since is assumed to have a Lipschitz boundary, the number of iterations needed to define y on all boundary cells is bounded independently of ε. If y is being extended to the corners of Q in some step as described above, let Q Q be the set of cells on every corner of which y has already been defined in a ( ) previous step. Note that Q Q Q Q is connected by Definition.4. Since has a Lipschitz boundary, we can choose a subset B Q of Q Q containing all neighbors ( ) of Q that lie in Q Q in such a way that Q BQ Q is connected and such that #B Q is bounded independently of Q and ε (see Figure ). The advantage of this particular modification scheme is that the rigidity of the modified deformations on non-dirichlet boundary cells is controlled by the behavior in the bul and on the Dirichlet boundary cells. Lemma 3.1. Suppose y is defined on a boundary cell Q = Q ε ( x), x L ε () \ L ε () as described above. Then there is a constant C such that dist ( y Q, SO(d)) C dist ( y Q, SO(d)). Q B Q We defer the proof of this lemma, which is based on a discrete geometric rigidity estimate, to the end of this section. As a consequence we note the following estimate.

10 798 BERND SCHMIDT Q BQ Figure. B Q (shaded) and Q. Lemma 3.. There exist constants c, C > 0 (independent of ε and y) such that, setting V ε := {x : dist(x, ) cε}, the estimate dist ( y ( x), SO(d)) C dist ( y ( x, SO(d)), x L ε ()\ L ε () x B ε where B ε = { x L ε() L ε() : x / V ε }, is satisfied. Proof. This follows directly from Lemma 3.1 by induction on the extension steps. The versions of Lemma 3.1 and Lemma 3. for displacements u(x) = δ 1 (y(x) x) read as follows. (Also the proof of Lemma 3.3 will be given at the end of this section.) Lemma 3.3. Suppose u is the extension on a non-dirichlet boundary cell Q = Q ε ( x) as described above. Then there is a constant C such that u Q C Q B Q u Q. Lemma 3.4. There exist constants c, C > 0 (independent of ε, δ and u) such that, setting V ε := {x : dist(x, ) cε}, the estimate u ( x) C u ( x), x L ε ()\ L ε () x B ε where B ε = { x L ε() L ε() : x / V ε }, is satisfied for all for u A ε (g,, ). Proof. This follows directly from Lemma 3.3 by induction on the extension steps. 3.. Interpolation. In the sequel it will be convenient to choose a particular interpolation ỹ for a lattice deformation y. We first explain the procedure on a single cell with ε = 1 and y = y. Let y : A { 1, 1 } d = { z i : i { 1,..., d}} R d, Q = A [ 1, 1 ] d. First interpolate linearly on one-dimensional faces of Q, i.e., on those segments [z i, z j ] with z i z j parallel to one of the vectors v n spanning the lattice. Then

11 LINEAR ELASTICITY FROM ATOMISTIC MODELS 799 define a triangulation and interpolation on all two-dimensional faces of Q in the following way: If the face is co{z i1,...,z i4 }, z i = z i1 + v n, z i3 = z i1 + v n + v m, z i4 = z i1 + v m, then set ζ = 1 4 (z i z i4 ), y(ζ) = 1 4 (y(z i 1 )+...+y(z i4 )) and interpolate linearly on each of the triangles co{z ij, z ij+1, ζ}, j = 1,, 3, 4 (mod 4). In general, assuming that we have chosen a simplicial decomposition and corresponding linear interpolation on all (n 1)-dimensional faces, we decompose and interpolate an n-dimensional face F = co{z i1,..., z i n } as follows. Set ζ = 1 n n j=1 z ij, y(ζ) = 1 n n j=1 y(z ij ). Now decompose F by the simplices co{w 1,..., w n, ζ}, where co{w 1,..., w n } is a simplex that belongs to the decomposition of an (n 1)-face already constructed. On these simplices interpolate linearly. For later reference we note that each simplex constructed in this interpolation scheme contains at least one corner z i of Q. One of the advantages of choosing this particular interpolation is that it is not hard to see that if ỹ is the interpolation of y on the unit cell Q, then Q ỹ(x)dx = 1 d d i=1 y(z i ) = ỹ(0). (6) (By induction, F ỹ(x)dx = 1 n n j=1 y(z i j ) on each n-face F = co{z i1,..., z i n } of Q.) Now let u A ε (g,, ) and let u its modification constructed in Paragraph 3.1. We interpolate u on x L ε () Q ε( x) such that ũ H 1 (g,, ): On Dirichlet boundary cells we set ũ(x) = g(x). If, on the other hand, Q ε (x) is neither a neighbor of a Dirichlet boundary cell nor a Dirichlet boundary cell itself, we define ũ(x) = δ 1 (ỹ(x) x) on Q ε (x) by interpolating as described above (with v n, z i replaced by εv n, x + εz i, respectively). Finally, if Q = Q ε ( x) is a cell having neighbors Q j that are Dirichlet boundary cells, we also follow the interpolation scheme described above, but without linearly interpolating on faces of Q that are shared by some Q j : If F = x + ε co{z i1,...,z i n } Q j for some Q j, let ũ(x) = g(x) for x F. If F Q j for all Q ε j, then also ζ = x + j=1 z i j / j Q j. In this case we define ũ(ζ) = 1 n n j=1 ũ( x + εz i j ) as before and interpolate on the simplices co{w 1,...,w n, ζ} linearly on each segment [ζ, w], w co{w 1,...,w n }, where co{w 1,..., w n } is a simplex that belongs to the decomposition of an (n 1)-face already constructed. The following lemma is an easy consequence of our interpolation procedure. n n Lemma 3.5. Suppose u A ε (g,, ) is (the modification of) a lattice displacement interpolated as described above. There exist constants c, C > 0 such that the following conditions are satisfied.

12 800 BERND SCHMIDT (i) If Q is not a Dirichlet boundary cell itself, nor the neighbor of any Dirichlet boundary cell, then c u Q ũ C u Q. Q (ii) If Q is not a Dirichlet boundary cell, but Q is neighboring a Dirichlet boundary cell, then c u Q C ũ C u Q + C. (iii) If Q is a Dirichlet boundary cell, then u Q, ũ C. Q Q Proof. These estimates can be seen along the same lines as the estimates in the following lemma. We refer to the proof of Lemma 3.6. Lemma 3.6. Suppose u A ε (g,, ) and y (x) = x+δu (x) are lattice mappings modified and interpolated as described above. There exist constants c, C > 0 such that the following conditions are satisfied. (i) If Q is not a Dirichlet boundary cell itself, nor the neighbor of any Dirichlet boundary cell, then c dist ( y Q, SO(d)) dist ( ỹ, SO(d)) Q C dist ( y Q, SO(d)). (ii) If Q is not a Dirichlet boundary cell, but Q is neighboring a Dirichlet boundary cell, then c dist ( y Q, SO(d)) Cδ dist ( ỹ, SO(d)) Q C dist ( y Q, SO(d)) + Cδ. (iii) If Q is a Dirichlet boundary cell, then dist ( ỹ, SO(d)) Cδ and dist ( y Q, SO(d)) Cδ. Q Proof. (i) The first inequality is straightforward: Note that, since ỹ is linear on simplices whose volume is comparable to Q, we have dist ( ỹ, SO(d)) L (Q) C Q dist ( ỹ, SO(d)). For the proof of the second inequality assume without loss of generality ε = 1, Q = A[ 1, 1 ]d. Let R SO(d) such that dist( y, SO(d)) = y R. By induction we prove that on each n-dimensional simplex S = co{w 1,...,w n, ζ} of Q constructed in the interpolation procedure we have ( ỹ R)P S C y R, where P S is the projection onto the space span{w w 1,..., w n w 1, ζ w 1 }: Note first that by the remar above equation (6), we may without loss of generality assume that w 1 = z i for some i. Let e be a unit vector in P S (R d ). If e span{w w 1,...,w n w 1 }, then ( ỹ R)e C y R

13 LINEAR ELASTICITY FROM ATOMISTIC MODELS 801 by assumption. Now if e = ζ zi ζ z i, then ( ỹ R)e = y(ζ) y (z i ) R(ζ z i ) ζ z i C sup y (z j ) Rz j (y (z i ) Rz i ) C y R, j and the induction step is proved. The claim now follows by letting n = d. (ii) Suppose y is the function obtained by interpolating y in Q as for cells described in (i). Since g W 1,, it follows that y ỹ L (Q) Cδ. The estimates now follow from (i). (iii) This is clear since y (x) = x + δg(x) on Q Discrete geometric rigidity. An important ingredient for the derivation of linear elasticity from nonlinear energy functionals is a quantitative rigidity estimate for deformations near SO(d) as given in [6], see [5]. The discrete version of this result proved in [9, Theorem 3.3] states that lattice deformations y : L U R d, where U is a union of closed lattice cells such that U is connected, satisfy the following rigidity estimate (in unrescaled variables): For each y there exists R SO(d) such that y( x) R C dist ( y( x), SO(d)), (7) x L U x L U R = R Z. For later use we record here that Assumption.1 implies the following estimate. Lemma 3.7. There is a constant C such that for all F R d d dist (F, SO(d)) CWcell (F). Proof. This is essentially contained in Lemma 3. of [9]. The arguments detailed there show that this estimate is a consequence of the growth assumptions on W cell near SO(d) and imposed in Assumption.1. Proof of Lemma 3.1. By rescaling we may assume that ε = 1. Let η > 0 and suppose dist ( y Q, SO(d)) η. Q B Q By the discrete geometric rigidity result (7) there is a rotation R SO(d) such that Q B Q y Q R Cη. But then y, restricted to L ε B Q, has an extension y to the corners of Q with y Q R Cη. The claim now follows by construction of y. Proof of Lemma 3.3. Let y (x) = x + δu (x) and suppose Q B Q u Q ηδ. Applying Lemma 3.1 and (7), we find a rotation R such that y Q R + Q B Q y Q R Cη (8)

14 80 BERND SCHMIDT holds. But by assumption we also have Q B Q y Q Z = Q B Q δ u Q Cη, whence R Id Cη. The claim now follows from (8). As a by-product we also mention the following discrete geometric rigidity result, which generalizes Theorem 3.3 in [9] in the sense that the constant C only depends on the domain. Theorem 3.8. There exists a constant C > 0, independent of ε, such that for all lattice deformations y : L ε () R d there exists a rotation R SO(d) such that y( x) R C dist ( y( x), SO(d)). x L ε () x L ε () Proof. Apply the modification scheme of Paragraph 3.1 with = and the interpolation procedure of Paragraph 3. to define y and ỹ. By Lemmas 3.1 and 3.6 we then have dist ( ỹ(x), SO(d))dx C SO(d)). x L ε () dist ( y( x), The continuum geometric rigidity result in [6] now provides a rotation R SO(d) such that ỹ(x) R dx C dist ( ỹ(x), SO(d))dx for a constant C = C() and from Lemma 3.5 we thus obtain y ( x) R C dist ( y( x), SO(d)). x L ε () x L ε () To extend this estimate to L ε (), cover the boundary cells by sets D i such that D i = x L Q ε( x), is connected for all i, sup ε Di i diamd i Cε and each D i contains at least one inner cell Q i. The number of possible shapes of the D i is bounded independently of ε, so by applying the rigidity estimate (7) on each of these sets, we obtain rotations R i such that y( x) R i C dist ( y( x), SO(d)). x L ε Di x L ε Di Since in particular R i R ( y Qi R i + y Qi R ) i i C SO(d)), the claim now follows. x L ε () dist ( y( x), 4. Proofs. The proofs of Theorems.6 and.7 are split into the following three subsections. Throughout this section we will suppose that the assumptions of Theorems.6 and.7 are satisfied.

15 LINEAR ELASTICITY FROM ATOMISTIC MODELS Compactness. We first prove the compactness statement in Theorem.6. Let = Q ε ( x), = Q ε ( x). x L ε () x L ε () L ε () Lemma 4.1. Let (u ) be a sequence in A ε (g,, ). (i) If the sequence (I (u )) of energies is bounded, then the sequences ( ũ H 1 ( ) ), ( u L ( ) ) and ( u L ( ) ) are bounded, too. (ii) If the sequence (χ δ dist (Z + δ u, SO(d))) or, equivalently, the sequence (χ δ dist (Id + δ ũ, SO(d))) is equiintegrable on R d, then both (χ u ) and (χ ũ ) are equiintegrable, too. Proof. (i) If I (u ) C, then by Lemmas 3.6, 3. and 3.7 we also have dist (Id + δ ũ, SO(d)) C dist (Z + δ u, SO(d)) + Cδ Cε d W cell (Z + δ u ( x)) + Cδ Cδ. x L ε () (We follow the convention of denoting possibly different constants with the same letter.) Applying the continuum results of [5], in particular Proposition 3.4 and Poincaré s inequality, we find that the ũ are equibounded in H 1 (g,, ). By the construction of u and by Lemmas 3.5 and 3.4 we then find that u L ( ) C u L ( ) C ũ L ( ) C. (ii) First note that Lemma 3.6 implies that (χ δ dist (Id + δ ũ, SO(d))) is equiintegrable if and only if (χ δ dist (Z + δ u, SO(d))) is equiintegrable. This follows from the fact that there is a constant c > 0 such that δ dist (Z + δ u, SO(d)) M on a cell Q can only hold if δ dist (Id+δ ũ, SO(d)) cm on one of the d-simplices on which ũ is interpolated linearly and, vice versa, if δ dist (Id + δ ũ, SO(d)) M, then δ dist (Z + δ u, SO(d)) cm on Q. An analogous argument shows that (χ ũ ) is equiintegrable if and only if (χ u ) is equiintegrable. As a consequence we can refer to the continuum case investigated in Lemma 4. in [10], from which it follows that if (δ dist (Id+δ ũ, SO(d))) is equiintegrable, then so is (χ ũ ). The claim now follows from Lemmas 3.5 and 3.4. Suppose (u ) is a sequence in A ε (g,, ) such that (I (u )) is bounded. By Lemma 4.1 we immediately deduce that for a suitable subsequence (not relabeled) ũ u in H 1 for some u H 1 (g,, ). Passing to a further subsequence we also see from Lemma 4.1 that there exists f L (, R d ) such that χ u χ f in L, and so χ u χ f in L. Noting that any compact subset of \( \ ) is eventually contained in, we see that u f in L loc ( \ ( \ )). To finish the first part of the proof of Theorem.6, it remains to show that f = u Z. Let V. For i {1,..., d } denote by i u and f i the i-th columns of u and f, respectively. Note that i u (x) f i, 1 u (x + ε (z i z 1 )) f 1 in L (V ).

16 804 BERND SCHMIDT For x V compute i u (x) = ε 1 (ũ ( x + ε z i ) ũ ( x)) = ε 1 (ũ ( x + ε (z i z 1 ) + ε z 1 ) ũ ( x + ε (z i z 1 ))) + ε 1 (ũ ( x + ε (z i z 1 )) ũ ( x)) = 1 u ( x + ε (z i z 1 )) + ε 1 A[ ε, ε ) d ũ ( x + ε (z i z 1 ) + ξ) ũ ( x + ξ)dξ = 1 u (x + ε (z i z 1 )) + ε 1 (P ε ũ (x + ε (z i z 1 )) P ε ũ (x)) by construction (see (6) and (5)). Now observe that for a test function ψ Cc (V, R d ) ε 1 (P ε ũ (x + ε (z i z 1 )) P ε ũ (x)) ψ(x)dx V = P ε ũ (x) ψ(x ε (z i z 1 )) ψ(x) dx ε V for ε sufficiently small. Since ũ u in L and P ε 1 by Jensen s inequality, the first term P ε ũ in the integral converges strongly to u. The second term converges to ψ (z i z 1 ) uniformly. Summarizing, we have shown that (f i f 1 )(x)ψ(x)dx = u(x) ψ(x) (z i z 1 )dx V and hence, since V was arbitrary, The claim now follows from d i=1 f i f 1 = u (z i z 1 ). f i = lim d i=1 V i u = 0 = d i=1 u z i. 4.. The Gamma-liminf inequality. Let ε d x L ε () L ε () u (x) P ε u(x) 0 and assume without loss of generality I (u ) C, so that by the compactness properties proved in the previous paragraph we may assume that u u, u u Z. Using Lemma 3.7 and Lemma 3. (set V = V ε ) we can compute I (u ) = δ εd W ε ( x, ε (Z + δ u ( x))) δ εd δ deta + cδ x L ε () W cell (Z + δ u ( x)) x L ε W cell (Z + δ u (x))dx V \V dist (Z + δ u (x), SO(d))dx

17 LINEAR ELASTICITY FROM ATOMISTIC MODELS 805 for some c > 0. By Assumption.1 we can write W cell (Z + F) = 1 Q cell(f) + ω(f) with sup{ ω(f) F : F ρ} 0 as ρ 0. Let (η ) be a sequence of positive numbers and set χ (x) := χ [0,η )( u (x) ). Using Lemma 3.7 once more, we find I (u ) δ deta + cδ χ (x)w cell (Z + δ u (x))dx V (1 χ (x))dist (Z + δ u (x), SO(d))dx 1 = χ (x) ( Q cell ( u (x)) + δ deta ω(δ u (x)) ) dx V + cδ (1 χ (x))dist (Z + δ u (x), SO(d))dx, where the second term inside the first integral can be bounded by χ u ω(δ u ) δ u. Now if η is such that η and η δ 0, then, since χ u is bounded in L ω(δ and χ χ u ) δ converges to zero uniformly as, we deduce that u lim inf I 1 ( (u ) liminf Q cell χv (x)χ (x) deta u (x) ) dx + c liminf δ (1 χ (x))dist (Z + δ u (x), SO(d))dx. Now observe that χ V χ converges to 1 boundedly in measure, so χ V χ u u Z in L (). By lower semicontinuity we obtain lim inf I 1 (u ) Q cell ( u Z)dx deta + c liminf δ (1 χ (x))dist (Z + δ u (x), SO(d))dx. (9) Since the latter term is non-negative and F Q cell (F Z) vanishes on antisymmetric matrices (see (4)), the lower bound in Theorem.6 is proved. The benefit of proving a sharper estimate in (9) than needed to obtain the Γ- liminf inequality is seen in the proof of the following observation. Lemma 4.. For a recovery sequence (u ) the terms χ δ are equiintegrable. dist (Z+δ u, SO(d)) Proof. Suppose this were not the case. Then there exists γ > 0 such that for each m N we find = (m) N such that δ dist (Z + δ u, SO(d)) γ. { u >m} Without loss of generality we may choose (m) such that (1) < () <... and mδ (m) m 1. Now choose η as the inverse of m (m) wherever it is defined,

18 806 BERND SCHMIDT i.e., such that η (m) = m, and note that δ (m) η (m) = mδ (m) m 1 = η 1 Then for the subsequence (u (m) ) we have δ (m) dist (Z + δ (m) u (m), SO(d)) γ. { u (m) >η (m)} (m) Taing the liminf of this expression we find by using (9) that lim inf I (u ) I(u) + cγ. This contradicts the fact that (u ) is a recovery sequence for u. (m). Lemma 4.3. If (u ) is a recovery sequence, then (χ ũ ) and (χ u ) are equiintegrable. Proof. Immediate from Lemma 4.1(ii) and Lemma Recovery sequences. By density it suffices to provide recovery sequences for u W 1, (, R d ) with u = g on. Extending u we may assume that u W 1, (R d, R d ). Recall the definition of P ε from (5) and define lattice displacements u A ε (g,, ) by { P ε u(x), for x L ε \ L ε (), u (x) = (10) g(x), for x L ε () (and interpolate to obtain ũ : Rd as before). Lemma 4.4. Let (u ) be the sequence of functions defined in (10). Then u u in the sense of Definition.5, χ u χ u Z strongly in L and ũ u strongly in H 1 (). Proof. Note first that u L is bounded. This is easily seen on lattice cells not neighboring a Dirichlet boundary cell and on Dirichlet boundary cells it follows from g L. If Q = Q ε ( x) and Q = Q ε ( x ) are cells with Q Q and x L ε (), then by the remar below Definition.3 and by Definition.4(i) there exists a R d with a bounded independently of x, x and ε such that x + ε a. For all pairs of corners ( x + ε z i, x + ε z j ) of Q we have u ( x + ε z i ) u ( x + ε z j ) u ( x + ε z i ) u( x + ε a) + u ( x + ε z j ) u( x + ε a). Since u( x +ε a) = g( x +ε a), the estimate now also follows on Q from u, g W 1,. Since u = P ε u on all inner cells and u L is bounded, we have u (x) P ε u(x) Cε 0 ε d x L ε as, i.e, u u. Let U = P ε u( + ( ε,..., ε )). In order to see that u u Z in L, it suffices to show that u U Z 0, because U Z u Z in L. Choose c > 0 such that V := {x : dist(x, ) > cε } satisfies L ε () V. Since

19 LINEAR ELASTICITY FROM ATOMISTIC MODELS 807 u and U are constant on lattice cells, we have i u 1 u U (z i z 1 ) L ( ) Cε d i u ( x) 1 u ( x) U ( x) (z i z 1 ) + Cε = Cε d C x L ε V x L ε V ( u( x + A[0,ε ε z i + ξ) u( x + ε z 1 + ξ) ) d ) u( x + ξ) (z i z 1 ) dξ u(ξ + ε z i ) u(ξ + ε z 1 ) u(ξ) (z i z 1 ) ε dξ + Cε ε + Cε by Jensen s inequality. This is easily seen to tend to 0 as ε 0 (see, e.g., Lemma A. in [9]). Since d i=1 i u = 0 for all and d i=1 z i = 0, it follows that u u Z. (It is not hard to see that this also implies that ũ u, see, e.g., Lemma A.1 in [9].) We finish the proof of Theorem.6 by showing that u as defined above is a recovery sequence for u. By Lemma 4.4 it remains to prove that limsup I (u ) I(u). The terms W surface (I ε ( x), Z+δ u ( x)) on boundary cells are uniformly bounded. If W surface and W cell are compatible, they are even bounded by Cδ for some C > 0. It follows that I (u ) 1 deta δ W cell(z + δ u (x))dx + Cδ ε, where the error term can be improved to Cε for W surface and W cell compatible. Since F δ W cell(z + δ F) converges uniformly on compacta to 1 Q cell and the error terms Cδ ε (resp. Cε ) converge to zero, we obtain from Lemma lim sup I (u ) Q cell ( u(x) Z)dx. deta Proof of Theorem.7. Let (u ) be a recovery sequence for u. Strong convergence u u Z follows from a convexity argument and equiintegrability of the discrete gradients of recovery sequences proved in Lemma 4.3 (also compare [10]). For V as in the proof of Lemma 4.4 we have I (u ) I(ũ ) 1 deta 1 deta V { u M} (V { u >M}) (\V ) ( δ W cell(z + δ u ) 1 Q cell( ũ Z) 1 Q cell( ũ Z) for all M > 0. Sending first and then M to, since F δ W cell(z + δ F) converges uniformly on compacta to 1 Q cell, we obtain I(u) limsup I(ũ ) liminf I (u ) I(ũ ) 0 (11) )

20 808 BERND SCHMIDT by Lemma 4.4 and equiintegrability of ( ũ ). But Q cell grows quadratically on the subspace perpendicular to infinitesimal rotations and translations. In particular, 1 Q cell(f Z) c F T + F and so c e(ũ ) e(u) 1 deta Q cell ( ũ Z) Q cell ( ũ Z, u Z) + Q cell ( u Z). By Lemma 4.1 ũ converges wealy to u, so (11) implies lim sup c e(ũ ) e(u) = 0. The strong convergence of ũ to u in L () now follows from Korn s inequality. By Lemma 4.3 we now deduce that in fact χ ũ and χ u converge to χ u and χ u Z strongly in L (R d ), respectively. Using again that any compact subset of \( \ ) is eventually contained in and u = u on, we conclude the proof. Proof of Corollary.8. Just note that I has a unique minimizer w by strict convexity of Q cell on symmetric matrices and Korn s inequality. So if (w ) is a sequence of almost minimizers of I, then (w ) is a recovery sequence for w. The claim thus follows from Theorem Examples. In this section we give some examples of atomic interactions to which the results of the previous sections apply. Motivated by the investigations in [8,, 9] we examine mass spring models: lattices of atoms whose energy is given by springs between nearest and next nearest neighbors. We also provide explicit formulas for the (partly well-nown) limiting functionals, which can be derived by elementary calculations The triangular lattice in D. The nearest neighbor interaction E ε (y) = ε ( K y(x1 ) y(x ) 1), ε x 1,x Lε x 1 x =ε ( ) on the triangular lattice L = AZ 1 1 = 0 3 Z in D for deformations that are locally orientation preserving, i.e., whose affine interpolation has nonnegative determinant on each triangle of the induced triangulation, has been analyzed by Braides, Solci and Vitali in []. We include it here for the sae of completeness. Denoting the i-th ( column of F R) 4 by F i and choosing the numbering of z i such that Z = 1 A , the associated cell energy is given by W cell (F) = K 4 ( ( F F 1 1) + ( F 3 F 1) + ( F 4 F 3 1) + ( F 1 F 4 1) + ( F 4 F 1) )

21 LINEAR ELASTICITY FROM ATOMISTIC MODELS 809 on orientation preserving discrete gradients. W surface (compatible with W cell ) is defined appropriately. This is an admissible cell energy, and we have thus re-derived the limiting functional from []. I(u) = 1 deta 3K Q cell (e(u) Z) = 8 e(u) + (tracee(u)) 5.. A mass spring model in D. The following energy functional has been introduced and studied by Friesece and Theil in [8]. Let L = Z. For a deformation y : L ε R let ( y(x1 ) y(x ) E ε (y) = ε + ε x 1,x Lε x 1 x =ε K 1 x 1,x Lε x 1 x = ε K ε a 1 ) ( ) y(x1 ) y(x ) a + χ( y( x)) ε x L ε () with χ = 0 on orientation preserving cell deformations and = otherwise. (We will only need that χ is zero in a neighborhood of SO() and positive ( c > 0) on Ō() \ SO(). Denoting the i-th column F z i of F R 4 by F i, the associated cell energy is given by W cell (F) = 1 K 1 4 ( F i F j a 1 ) + 1 z i z j =1 z i z j = K ( F i F j a ) + χ(f), the surface energy expressions are defied appropriately. We choose units such that K 1 a 1 + K a = K 1 + K, i.e., such that Z is a stationary point of W cell (with µ := W cell (Z) = K 1 (1 a 1 ) + K ( a ) ). In [8] it is shown that indeed for a parameter region (open with respect to the constraint K 1 a 1 + K a = K 1 + K ) W cell grows quadratically on the subspace perpendicular to infinitesimal translations and rotations. Now note that Theorems.6 and.7 apply to the energy functional E ε µ that arises from E ε by replacing W cell with W cell µ. The limiting linear energy functional is given by I(u) = 1 Q cell (e(u) Z), where the quadratic form R sym F Q cell(f Z) is given by F K 1 a 1 F + a K (tracef) + ( a K a 1 K 1 )f 1. Remars. (i) Note that for a 1 1, a the bul and surface contributions are not compatible and µ > 0. Our result can be interpreted as a Γ-development of the functionals u E ε (Id + δ u): Subtracting the zeroth order constant Γ-limit µ, the above calculated Γ-limit is just the Γ-limit of u δ (E ε (Id + δ u) µ ) = δ Eµ ε (Id + δ u) + Cε δ. (See, e.g., [1] for the notion of development by Γ-convergence.)

22 810 BERND SCHMIDT (ii) If a 1 = 1, a =, we have µ = 0. In this case W cell and W surface are compatible A mass spring model in 3D. The following energy functional has been investigated in [9] for a 1 = 1, a =. Let L = Z 3. For a deformation y : L ε R 3 let E ε (y) = ε3 + ε3 x 1,x Lε x 1 x =ε K 1 x 1,x Lε x 1 x = ε ( ) y(x1 ) y(x ) a 1 ε K ( ) y(x1 ) y(x ) a + χ( y( x)). ε x L ε () The non-negative term χ is assumed to be non-zero only for deformations which are not locally orientation preserving, in particular it is zero in a neighborhood of SO(3) and positive ( c > 0) on Ō(3) \ SO(3). The associated cell energy on R 3 8 is given by W cell (F) = z i z j =1 z i z j = K 1 ( F i F j a 1 ) K ( F i F j a ) + χ(f), the surface energy is chosen appropriately. In [9] it is shown that for a 1 = 1, a =, W cell is an admissible cell energy in the sense of Assumption.1 for all K 1, K > 0. Noting that for a 1 1, a the cell energy can be written as a sum of D cell energies over the cube s faces as in the previous example, a necessary condition that W cell be equilibrated on Z is that K1 4 a 1 + K a = K1 4 + K, i.e., K 1a 1 + K a = K 1 + 4K. For given K 1, K, a perturbation argument using that the in-plane displacements on every face are minimized precisely on the unit square by our previous example shows that (up to a constant) W cell still is an admissible energy function for small deviations from the perfectly equilibrated system if K 1 a 1 + K a = K 1 +4K. So again our results of the previous sections apply (after subtracting the zeroth order term µ, µ = K 1 (1 a 1 ) + 3K ( a ) ). The quadratic form R 3 3 sym F Q cell (F Z) is given by ( ( F K a ) ) K F + a K (trace F) + ( a K a 1 K 1 ) (f 1 + f 13 + f 3) bcc crystals. Although the bcc crystal L = K M with K = Z 3 and M = ( 1, 1, 1 )T +Z 3 is a Bravais-lattice (spanned, e.g., by (1, 0, 0) T, (0, 1, 0) T, ( 1, 1, 1 )T ),

23 LINEAR ELASTICITY FROM ATOMISTIC MODELS 811 the nearest and next-nearest neighbor pair interaction functional ( E ε (y) = ε3 K 1 y(x 1 ) y(x ) 3 ε + ε3 x 1,x Lε x 1 x = 3ε x 1,x Lε x 1 x =ε K ( y(x1 ) y(x ) ε ) ) 1 (with orientation preserving condition) cannot be decomposed in the form () as a sum of suitable cell energies over the lattice unit cells. However, since the individual interactions are equilibrated in the reference configuration, we can still apply our results to derive the limiting linear theory: Assume that boundary conditions are prescribed for every boundary atom K ε () and M ε () of εk and of εm, respectively. For F R 3 8, c R 3, let ( W cell (F, c) = 1 ) K 1 3 F i c i 8 z i z j =1 K ( F i F j 1) + χ(f), χ as before. Then W cell (F) := min c R 3 Wcell (F 1 c,...,f 8 c) is an admissible cell energy function and up to boundary terms E ε (y) W cell ( y 1 ( x)) + W cell ( y ( x)), (1) x (Z 3 ) ε () x (Z 3 ) ε () where y 1, y denote the restrictions of y to the sets K ε() and M ε(), respectively (cf. Definition.3). A simple convexity argument shows that, in the limit y ỹ Z, we obtain W cell ( y) = W cell ( y, 0) and the lower bound (1) becomes sharp. As a consequence, the result of the preceding sections also apply here, and we obtain the limiting functional 1 I(u) = Q cell (e(u) Z) = Q cell (e(u) Z) detid with Q cell (F Z) = K F + K 1 (tracef) + (4K 1 K )(f 1 + f 13 + f 3 ) for F R 3 3 symmetric. REFERENCES [1] A. Braides, Γ-convergence for Beginners, Oxford University Press, Oxford, 00. [] A. Braides, M. Solci and E. Vitali, A derivation of linear elastic energies from pair-interaction atomistic systems, Netw. Heterog. Media, (007), [3] S. Conti, G. Dolzmann, B. Kirchheim and S. Müller, Sufficient conditions for the validity of the Cauchy-Born rule close to SO(n), J. Eur. Math. Soc. (JEMS), 8 (006), [4] G. Dal Maso, An Introduction to Γ-convergence, Birhäuser, Boston Basel Berlin, [5] G. Dal Maso, M. Negri and D. Percivale, Linearized elasticity as Γ-limit of finite elasticity, Set-valued Anal., 10 (00), [6] G. Friesece, R. D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., 55 (00),