Qualitative properties of a continuum theory for thin films

Qualitative properties of a continuum theory for thin films Propriétés qualitatives d’une théorie de continuum

pour des couches minces

Bernd Schmidt

Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany

Received 5 December 2005; received in revised form 24 April 2006; accepted 29 September 2006 Available online 20 December 2006

Abstract

We discuss qualitative aspects of a continuum theory for thin films rigorously derived in [B. Schmidt, On the passage from atomic to continuum theory for thin films, preprint 82/2005, Max-Planck Institut für Mathematik in den Naturwissenschaften, Leipzig].

The stored energy density is examined for convexity properties and limiting behavior under large and small strains. A study of the dependence of the theory on relaxation parameters leads to the result that the scale of convergence used in [B. Schmidt, On the passage from atomic to continuum theory for thin films, preprint 82/2005, Max-Planck Institut für Mathematik in den Naturwissenschaften, Leipzig] is the only scale for which a limiting theory that also accounts for atomic relaxation effects is non-trivial.

©2007 Elsevier Masson SAS. All rights reserved.

Résumé

Nous discutons des aspects qualitatifs d’une théorie de continuum pour des couches minces, dérivée rigoureusement dans [B. Schmidt, On the passage from atomic to continuum theory for thin films, preprint 82/2005, Max-Planck Institut für Mathe- matik in den Naturwissenschaften, Leipzig]. La densité d’énergie emmagasinée est examinée pour des propriétés de convexité et comportement en limite sous des distorsions grandes et petites. Une recherche de la dépendance de la théorie à l’égard des para- mètres de relaxation mène au résultat que l’échelle de la convergence employée dans [B. Schmidt, On the passage from atomic to continuum theory for thin films, preprint 82/2005, Max-Planck Institut für Mathematik in den Naturwissenschaften, Leipzig] est la seule échelle pour laquelle une théorie limite qui inclut également des effets de la relaxation atomique est non-triviale.

©2007 Elsevier Masson SAS. All rights reserved.

Keywords:Thin films; Discrete-to-continuum limits; Effective theories

E-mail address:bschmidt@aero.caltech.edu.

0294-1449/$ – see front matter ©2007 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2006.09.001

1. Introduction

The aim of this paper is to examine qualitative features of a macroscopic theory for thin films that was derived as an effective continuum theory from atomic models in [21]. Deriving thin film limits from three-dimensional elasticity is still an active area of research, see, e.g., [17–19,13,14,16] and, most recently, [15] where a whole hierarchy of different scaling limits is discussed. For the more classical developments see, e.g., [20,8]. On the other hand, by now there are also rigorousΓ-convergence results for the passage from discrete to continuum theory: for suitable pair interaction models, especially in one dimension, see [4–6]; more complicated potentials under additional assumptions as, e.g., the Cauchy–Born rule are considered in [2,3].

In [12], starting from reference configurations Lk=Z³∩ [0, k] × [0, k] × [0, ν−1]

for fixedν∈N, the number of film layers, andk∈N, a limiting continuum theory for the energy of deformations was proposed in the limitk→ ∞taking into account atomistic relaxation effects. In [21], this effective theory was obtained rigorously as a variational limit of the elastic energy functionalE(y^(k))of deformationsy^(k):Lk →R³. This continuum theory was expressed in terms of the gradient of a mapu:[0,1]²→R³andν−1 director fields bⁱ:[0,1]²→R³,i=1, . . . , ν−1:

Theorem 1.1.(Cf.[21])Under suitable assumptions on the energy functionE, and for an appropriate definition of convergence of deformations, there existsϕ:R^3×2×(R³)^ν−1→Rsuch that

E

y^(k)“Γ”

→

[0,1]²

ϕ

∇u, b¹, . . . , b^ν−1

ask→ ∞.

It is worth mentioning that the scheme described in [21] can be applied not only to thin films but to general three- dimensional bodies leading to a stored energy densityϕonly depending on the deformation gradient∇u∈R^3×3, if one assumes sufficiently fast decay of atomic interactions. The main technical difficulty in fact stems from the non- local convergence of relative layer displacements to the family of vector fields(b¹, . . . , b^ν−1)(cf. Definition 2.3). For the qualitative aspects examined in the present paper we will however make use of the ‘thin film structure’ as we allow atoms to explore regions perpendicular to the macroscopic film surface.

In Section 2, after introducing the model, we will recall the precise statements from [21]. Also we will collect some preparatory material that was proved in [21] and will be needed in the sequel.

The following sections are devoted to studying this continuum theory, i.e. the macroscopic energy densityϕqual- itatively. First, cf. Section 3, we examine the dependence ofϕon the relaxation parameterc0(cf. Definition 2.3 and Theorems 2.7, 2.8 and 2.9) and study the limiting casesc₀→ ∞andc₀→0. Moreover, we will see that the physi- cally motivated rate of convergence for which a continuum theory was derived in [21] is the only scale that leads to a non-trivial limiting theory.

In the following two Sections 4 and 5, we derive the limiting behavior under large tensile and compressive strains, and explore the convexity properties and symmetries of the limiting energy functional.

Finally, in Section 6, the scaling behavior of certain systems nearO(2,3), i.e.,(∇u)^T∇u≈Id_R2, is examined.

We still find a non-trivial energy response to compressive strains in this regime. It is, however, weaker than cal- culated without taking into account atomic relaxation effects. In order to prove this result we are led to study the one-dimensional version, an atomic chain, in detail. The results of this paragraph might be of independent interest.

2. The passage from atomic to continuum theory

We give a brief account of the results obtained in [21] on the passage from atomic models to a continuum theory for thin films. For details, motivations of the concepts, and proofs of the results of this section we refer to [21].

Fig. 1.

2.1. The model 2.1.1. Kinematics

We consider a film ofνatomic layers whose reference configuration will be Lk=L∩

Sk× [0, h] ,

whereSk:= [0, k] × [0, k]fork∈N,h:=ν−1 is the height of the film and, for sake of simplicity,L=Z³ (see Fig. 1).

The deformations of this configuration will be denoted by y=y^(k):Lk→R³.

In order fory to be defined not only on the atomic positions, we will assume some interpolation between the atomic positions: for a deformationy:Lk→R³letx¯=x+(1/2,1/2)forx∈ {0, . . . , k−1}²and set

y(x, i)¯ =1 4

z∈Z²,

|z− ¯x|=1/√ 2

y(z, i), i=0, . . . , ν−1.

Now on each of the four triangles with corners(x, i), (z, i), (z¯ , i), wherez, z ∈Z²with|z− ¯x| =1/√

2,|z−z| =1 interpolate linearly to obtainy(x, i)forx∈Sk. Interpolating in between the layers is not so subtle, for definiteness we chooseyto be linear on the segments[(x, i−1), (x, i)]. By this particular choice we guarantee that (local) averages depend only on atomic positions.

Our aim being to study the limitk→ ∞, it is natural to introduce the rescaled functionsy˜defined on the common domainS1× [0, h]:

˜

y^(k)(x):=1

ky^(k)(kx1, kx2, x3).

Considering weak*-limiting points ofy˜as natural variables for a continuum theory, we are led to elementsuof W^1,∞([0,1]²;R³)as limiting deformations. In our regime of thin films of fixed atomic height, we also introduce the quantities

ⁱy˜^(k)(x_p)= ˜y^(k)(x₁, x₂, i)− ˜y^(k)(x₁, x₂,0), i=1, . . . , ν−1,

x_p=(x₁, x₂), to measure the relative shift of the layers of our film. Also these have weak*-limits inL^∞. As in [21] we define:

Definition 2.1.Letu∈W^1,∞(S1;R³)andb=(b¹, . . . , b^ν−1)∈L^∞(S1;(R³)^ν−1). We say that(u,b)is admissible (for givenc₀>0), i.e.(u,b)∈A, if there existsc₁>0 such that

u(x)−u(z)c₁|x−z| ∀x, z∈S1 (1) (minimal strain hypothesis), and there existsb⁰∈L^∞such that

b⁰

∞,bⁱ−b⁰

∞c₀, i=1, . . . , ν−1. (2)

The unrescaled version ofuis denotedU, i.e.U=u. An easy consequence of our interpolation is the following

Lemma 2.2.Supposeuis admissible andy:Lk→R³some deformation withsup_x∈Lk|y(x)−U (xp)|c. Thenyis Lipschitz. For any(rescaled)Lipschitz interpolationy:Sk× [0, h] →R³(y˜:S1× [0, h] →R³)there are constants C₁, C₂, C₃>0such that,

(i) sup_{x∈S1×[0,h]}| ˜y(x)|C2,

(ii) C1|x−z| −C3|y(x)−y(z)|C2|x−z| ∀x, z∈Sk× [0, h].

We next define in what sense we understand deformations to converge to the limiting quantitiesuandb.

Definition 2.3. Let u∈W^1,∞(S1;R³), b∈L^∞(S1;R³). Choose c₀>0 a constant. We say that y^(k)→(u,b) (w.r.t.c₀) if

y˜^(k)−uc₀/k and ∀i: kⁱy˜^(k) b^{∗ i} inL^∞ask→ ∞.

Here and in the sequel we denote byf, respectively ˜fin rescaled variables, f := sup

x∈Lk

f (x), resp. ˜f := sup

x∈Lk

f (x˜ _p/k, x₃).

As detailed in [21], this corresponds to a relaxation scheme where the individual atoms are allowed to move in a region comparable to atomic dimensions.

2.1.2. Energy

The energy of a system ofN atoms at positionsy₁, . . . , y_N∈R³will be a functionE:(R³)^N→Ronly depending on atomic positions. To studyEwe will endow the configuration space(R³)^N with the norm

(y1, . . . , yN)= sup

1iN

|yi|2.

The elastic energy of a deformation y, i.e. the energy of the system (y(x): x ∈Lk) respectively a subsystem M=y(K),K⊂Lk, is denoted

E(y)=E

y(x): x∈Lk

resp. E(M)=E

y(x): x∈K . We normalizeEso thatE(∅)=0.

The main two assumptions onEare firstly the following splitting estimate.

Assumption 2.4.Supposeuis admissible. There exists a functionψ:[0,∞)→Rsuch that

|ψ|M and ψ (r)Mr^−q (3)

whereM, qare constants,M >0,q >3, such that for disjoint setsMandN of atoms we have E(M∪N)−E(M)−E(N)

v∈M, w∈N

ψ

|v−w| ,

whenevery −U_∞C. (The function ψ may depend onC and on u throughc₁ andc₂ wherec₁|x₁−x₂|

|u(x₁)−u(x₂)|c₂|x₁−x₂|.)

Secondly, we need to assume some regularity ofE:

Assumption 2.5.Letube admissible. We assume thatEis locally Lipschitz and in aC-neighborhood ofU ∂

∂y_iE(y) L

whereLmight depend onCand onUthroughc₁,c₂.

Furthermore, we assumeE to be frame indifferent and only depending on the atomic positions, i.e.E remains unchanged after renumbering of atoms and rigid motion of the configurationy(K).

For some results we will have to impose an additional restriction:

Assumption 2.6.Assume thatψ andL of Assumption 2.4 resp. 2.5 depend only onC1 andC3 wherey satisfies

|y(x)−y(z)|C1|x−z| −C3. 2.2. Convergence theorems

SupposeEsatisfies Assumptions 2.4 and 2.5, and a relaxation parameterc0>0 is chosen. The main result of [21]

is the following variational convergence result:

Theorem 2.7.There exists a macroscopic stored energy functionϕsuch that(in the spirit ofΓ-convergence, cf.[10]), (i) ify^(k)→(u,b),(u,b)admissible, then

lim inf

k→∞ E y^(k)

E(u,b),

(ii) and for all admissible(u,b)there exists a sequencey^(k)→(u,b)such that

klim→∞E y^(k)

=E(u,b).

HereE(u,b)is the macroscopic energy E(u,b)=

S1

ϕ

∇u, b¹, . . . , b^ν−1

. (4)

To computeϕby an associated cell problem, set N_k^0,1(A,b)=

y:Lk→R³: y−Ac₀and 1 (k+1)²

x∈Z²∩Sk

ⁱy(x)=bⁱ

. (5)

Theorem 2.8.The macroscopic energy densityϕof Theorem2.9is given by ϕ(A,b)= lim

k→∞ϕ_k(A,b) (6)

where for later use we have introduced the quantities ϕk(A,b)= 1

νk² inf

y∈Nk^0,1(A,b)

E(y). (7)

This limit is uniform on compact subsets ofAhomand depends continuously onA,b.

Here,Ahom⊂R^3×2×(R³)^ν−1, the set ofadmissible(A,b), is defined by Ahom:=

A, b¹, . . . , b^ν−1

: rank(A)=2,∃b⁰∈R³s.t.b⁰, max

1iν−1

bⁱ−b⁰c₀

for matricesA∈R^3×2and vectorsb¹, . . . , b^ν−1.

We also mention the following quantitative version of Theorem 2.7:

Theorem 2.9.Supposel=l(k)is such thatl(k)→0andkl(k)→ ∞ask→ ∞. Let W_k^l(u,b):=

y: ˜y−uc₀/k, kⁱy˜−bⁱ

W^−1,∞l

wherefW^−1,∞:=sup

f·χ: χ∈W₀^1,1, χ_W^1,1

0 = ∇χL¹=1

. Then

klim→∞

1 νk² inf

y∈Wk^l(u,b)

E(y)=

S1

ϕ

∇u(x),b dx.

In fact, Theorems 2.7 and 2.8 also apply to the more general case whereEis of the form E(y)=1

2

i=j

W

|y_i−y_j|

+E₀(y) (8)

whereE0 satisfies the usual assumptions, butW (r)becomes infinitely large as r tends to zero. (In particular, the Lennard–Jones potential is covered by these energy functions.)

Theorem 2.10.For anyr0>0assume thatW is Lipschitz on[r0,∞)and there existsM=M(r0)∈Rsuch that for (a.e.)rr0

W (r)Mr^−q, W(r)Mr^−q+1,

for rr₀. Then Theorem 2.7extends to energy functions of the form (8) where, as in Theorem2.8,ϕ:Ahom→ (−∞,∞]is given by(6), continuous as a function with values inR∪ {∞}.

As another extension we note that the above results also apply to suitable systems of distinguishable particle systems with finite range interaction. Leta >0. To eachx_i∈Lkwe assign a neighborhood

U_xi=

x_j∈L: |x_j−x_i|a

=

x₁ⁱ, . . . , x_rⁱ

a

where the enumeration of elements ofU_xi shall be such thatxⁱ₁=x_i and if(x_i1)₃=(x_i2)₃, then x_jⁱ¹−xi₁=x_jⁱ²−xi₂ forj=1, . . . , ra.

LetS_k^a= [a, k−a]²and suppose the energy of a deformationyis given by Efr(y)=

xi∈L∩(S^a_k×[0,h])

f_xi y

x₂ⁱ

−y x₁ⁱ

, . . . , y xⁱ_r

a

−y x₁ⁱ

+O(k), (9)

wherefx_i:R^3(ra−1)→Rare given functions representing the energy of the interactions between thei-th atom at its positiony(xi)=y(x₁ⁱ)and its neighboring atoms in their positionsy(x₂ⁱ), . . . , y(x_rⁱ_a). (The termO(k)is introduced to compensate for boundary effects, sinceUx_i is not contained inSk× [0, h]for xi in a boundary layer of constant width.) We need the following periodicity assumption: there exist fixedp1, p2∈Nsuch that

f_{(x1+p1,x2,x3)}=f_x=f_{(x1,x2+p2,x3)} (10)

forx=(x₁, x₂, x₃)∈(Z₊)²× {0, . . . , ν−1}.

Proposition 2.11.SupposeE_fris defined as in(9)and(10)holds. Assume that thef_xiare locally Lipschitz. Then the limitϕ_frof Theorem2.8exists and we have

klim→∞

1 νk² inf

y∈Wk^l(u,b)

E_fr(y)=

S1

ϕ_fr

∇u(x),b(x) dx

asl→0andkl→ ∞.

Remark.For such systems we do not need to suppose thatusatisfies a minimal strain hypothesis. Thus,ϕis defined on all ofR^3×2×(R³)^ν−1.

Fig. 2.

2.3. Technical results

We now collect some of the technical results obtained in [21] that will be useful in the following sections.

Consider deformationsy:kΩ× [0, h] →R³forΩ⊂ [0,1]².

Lemma 2.12.Letybe a deformation satisfying| ˜y−u|c/kandK⊂L∩(kΩ× [0, h]). Then there is a constantC (not depending onK)such that ifK=K1∪K2for disjointK1andK2, then

E

y(x): x∈K

−E

y(x): x∈K1C#K2.

SupposeQ= [0, a)²,a1, is partitioned by squaresU₁, . . . , U_r of side-lengthl wherec₀/kla plus some restRwith|R| =O(a·l),l a, as in Fig. 2. (Thenr∼(a/ l)².)

SetM:= {y(x): x∈L∩(kQ× [0, h])},Mi:= {y(x): x∈L∩(kU_i× [0, h])}.

Lemma 2.13.Supposey:kQ× [0, h] →R³satisfies| ˜y−u|c/kfor some admissibleu. Then there existsC >0 such that

E(M)− r i=1

E(Mi) C

ka² l +k²al

.

Remark.In both of the previous lemmas,Cwill only depend onC₁andC₃provided Assumption 2.6 is satisfied.

To measure local spatial averages, we define the measureρ=ρ(k)=

x∈Z²δ_{x/ k}whereδ_{x/ k}is the Dirac measure atx/k. Also set (after extendingbⁱboundedly outsideS1(constantly ifbⁱis constant))

b¯ⁱ(x)= −

x+[−1/2k,1/2k]²

bⁱ(z)dz. (11)

Letb⁰as in (2) be given. For later use we introduce the deformationsv=v^(k), defined by (interpolation of) v(x₁, x₂, i)=

u(x₁, x₂)−¹_kb¯⁰(x₁, x₂) fori=0,

u(x₁, x₂)+¹_k(b¯ⁱ(x₁, x₂)− ¯b⁰(x₁, x₂)) for 1iν−1 (12) for(x1, x2)∈¹_kZ²∩S1. Clearly,v^(k)→(u,b). Its un-rescaled version will be denotedV, i.e.V˜ =v.

Lemma 2.14.Supposey is a deformation with y−Uc₀+δ₁ and

−

[0,1]²

kⁱy˜− ¯bⁱ dρ

δ₂, δ1, δ21. Then there existsy :Lk→R³with

y −Uc0, −

[0,1]²

kⁱy˜ dρ= ¯bⁱ,

and E(y)−E(y)C

δ₁^1/5+δ₂^1/5 k². (This combines Lemmas 3.11 and 3.13 in [21].)

Instead ofbⁱ, it is sometimes more convenient to work with the quantitiesBⁱ defined by choosingb¯⁰minimizing max

1maxiν−1

b¯ⁱ− ¯b⁰,b¯⁰ (c₀) and setting

Bⁱ:= ¯bⁱ⁻¹− ¯b⁰ fori=2, . . . , ν, B¹:= − ¯b⁰. (13)

3. The dependence ofϕon the relaxation scheme

Our notion of convergencey^(k)→(u,b)of atomic deformations to macroscopic variables u,b depends on the constantc₀(cf. Definition 2.3). (To keep track of this dependence, we will sometimes addc₀as an additional sub- script as e.g. inN_k,c^0,1₀,ϕ_k,c0.) Our first task is to analyze this dependence of our continuum theory on the relaxation parameterc₀. It will turn out that we cannot relax sendingc₀ to infinity. This is due to the (physically reasonable) decay assumptions on atomic interactions. Moreover,c0/k will prove to be the only scale which both accounts for atomistic relaxation effects and yields a non-trivial continuum theory. We start by proving the following regularity result.

Proposition 3.1.Fix(A,b)∈Ahom. The mappingc₀→ϕ_c0(A,b)is decreasing and continuous.

Proof. Supposec₀> c₀. By Theorem 2.8,ϕ_c0(A,b)ϕ_c

0(A,b). Conversely, giveny∈N_k,c^0,1₀(A,b), by Lemma 2.14 we find a deformationy ∈N_k,c^0,1

0

(A,b)withE(y)E(y)+C(c₀−c₀)^1/5k² provided(A,b)is admissible forc₀ and|c₀−c₀|1. Thereforeϕ_c

0(A,b)ϕ_c0(A,b)+C(c₀−c₀)^1/5. 2 3.1. The limitc₀→ ∞

SupposeE is an admissible pair potential with purely attractive pair interactionW 0, W ≡0. Considering deformations with larger and larger periodic cells where every atom is mapped to a single point, we see that for all admissibleA,b,

c₀lim→∞ϕ_c0(A,b)= −∞.

In this paragraph we will show that the limitc₀→ ∞in general will be trivial if Assumption 2.6 is satisfied.

Theorem 3.2.SupposeEsatisfies Assumptions2.4, 2.5, and2.6. Defineϕ_∞:=limc₀→∞ϕc₀.(This limit exists point- wise in [−∞,∞)by Proposition 3.1.) Then ϕ_∞(A,b)=ϕ_∞(A,b)for all matrices A, A of rank two and all vectorsb,b ∈(R³)^ν−1.(Every such matrix resp. vector is admissible ifc0is large enough.)

Proof. Suppose first thatA =A. ByV_A,bwe denote the un-rescaled version ofv(cf. (12)) corresponding tou=A andb⁰set to zero. Forbsuch that the projection of eachbⁱonto graph(A)has norm less than 2|A|,

V_A,b(x)−V_A,b(x)=A(x_p−x_p)+b^x3−b^x3 A(x_p−x_p)−4|A| C₁|x−x | −C₃,

C₁, C₃ independent ofb. From Assumption 2.6 and Lemma 2.12 we then find a constantC such that for thoseb, E(V_A,b)Ck². On the other hand, if for two vectorsb1,b2and somei∈ {1, . . . , ν−1},

b^j₂=b^j₁, forj=i, and bⁱ₂=b₁ⁱ +Az, z∈Z²,

Fig. 3.

thenE(V_A,b1)=E(V_A,b2)+O(|z|k). So for allbwe obtain limk→∞ 1

νk²E(V_A,b)C, whenceϕ_∞(A,·)is an upper bounded function onR^3(ν−1)with values in[−∞,∞). Since it is convex (by Proposition 5.3 allϕc₀(A,·)are convex), it must be constant.

For the remaining part it suffices to show that ϕ_∞(A,b)ϕ_∞(A,b).

We proceed similarly as in the proof of Proposition 3.16 of the existence ofϕunder homogeneous conditions in [21].

Fixc0andδ >0. Choosingk0large enough we find by Theorem 2.8y∈N_k^0,1_0,c0(A,b)with 1

νk²₀E(y)ϕc₀(A,b)+δ

2. (14)

We construct a deformationy :Lk→R³,kk₀, by patching together appropriately translated copies ofy: let U₁, . . . , U_sbe translates of[0, k0+1)²as in Fig. 3.

Letz₁, . . . , z_s denote the lower left corners of these sets, setfⁱ=Azⁱ and define y(x₁, x₂, x₃)=y

x₁−zⁱ₁, x₂−zⁱ₂, x₃ +fⁱ forx∈L∩(U_i× [0, h]). Then

y −A = sup

x∈Lk0

y(x)−Axp sup

x∈Lk0

y(x)+ sup

xp∈Sk0

|Axp| =:c0.

Soc₀depends onk₀(andA, A) but is independent ofk. Since

−

[0,1]²

kⁱy˜ −bⁱ

dρ= −

U_j

kⁱy˜ −bⁱ dρ+O

k₀² k

= 1 sk₀²

s j=1

Uj

kⁱy˜ −bⁱ dρ+O

k₀² k

=O k₀²

k

(note|kⁱy˜|2c0), by Lemma 2.14 we find a deformation ˆ

y∈N_k,^0,1_c0(A,b) (15)

such that 1

νk²E(y)− 1 νk²E(y)ˆ

C(c0) k₀²

k 1/5

. (16)

Using Lemma 2.13 and translational invariance, we would now like to split the energy to find that 1

νk²E

y(x): x∈Lk

− 1 νk²₀E

y(x): x∈Lk0C 1

k0+k0

k

. (17)

Fig. 4.

If this is possible, we find that by (17), (15), (16) and (14) forkk₀1 ϕ_k,c0(A,b) 1

νk²E ˆ

y(x): x∈Lk

1

νk₀²E

y(x): x∈Lk0

+δ 2 ϕ_c0(A,b)+δ.

Letting firstk→ ∞, we deduce from Proposition 3.1 ϕ_∞(A,b)ϕc₀(A,b)+δ.

Sinceδwas arbitrary, we finally get sendingc0→ ∞ ϕ_∞(A,b)ϕ_∞(A,b).

It remains to justify the application of Lemma 2.13. The problem is thatc₀depends onk₀. (For nearest neighbor models as discussed in Proposition 2.11, this splitting in (17) will in general not be possible: fory as described above neglecting the bonds between setsy(U_i × [0, h])could result in neglecting an essential part of the energy.) By the remark after Lemma 2.13, however, this will be possible if we can replacey by somey such that stilly −Ac₀ depends only onk₀andy consists of translates ofy(Lk0), but in addition satisfies a far-field minimal strain hypothesis with constantsC1, C3independent ofk0, i.e.

y (x₁)−y (x₂)C₁|x₁−x₂| −C₃. (18) We re-enumerate the squaresU₁, . . . , U_sas depicted in Fig. 4. (r∈Nto be specified later.) Depending onA, A, k (andc₀,c₀) we choose a unit vectore∈R³perpendicular to the graph ofA and numbers 0< a₁<· · ·< a_r2 (to be specified later), and define

y (x₁, x₂, x₃)=y(x₁, x₂, x₃)+a_je ifx∈L∩U_i,j× [0, h], j∈ {1, . . . , r²}.

We will now findC₁, C₃independent ofk₀such that (18) holds. Since still, on each of the setsU_i,j× [0, h],y is a translated copy ofy, we may replacey byy . Applying (17) then finishes the proof.

Ifx1andx2lie in the sameUi,j× [0, h], this is clear from Lemma 2.2 sincey∈N_k^0,1_0,c0(A,b).

Now suppose this is not the case, but still|x1−x2|∞< (r−1)(k0+1). Thenx1∈Ui₁,j₁×[0, h],x2∈Ui₂,j₂×[0, h] withj1=j2. But then

y (x1)−y (x2)|a_j1−a_j2| −y(x1)−y(x2) |a_j1−a_j2| −f^i1,j1−f^i2,j2−y

x₁−(z^i1,j1,0)

−y

x₂−(z^i2,j2,0) |a_j1−a_j2| −f^i1,j1−f^i2,j2−2c0−A

(x₁)_p−z^i1,j1

−A

(x₂)_p−z^i2,j2

|aj₁−aj₂| −Crk0−2c0−Ck0

2rk0 for|a_j1−a_j2|sufficiently large |x₁−x₂|.

So we assume that|a_j1−a_j2|,j₁, j₂∈ {1, . . . , r²}, are large enough to justify the above calculation.

Finally, letx₁∈U_i1,j1× [0, h],x₂∈U_i1,j2× [0, h]and|x₁−x₂|_∞(r−1)(k0+1). Sinceeis perpendicular to the graph ofA andy lies in ac₀-neighborhood of that graph, we find that forrnot too small

y (x₁)−y (x₂)=(a_j1−a_j2)e+y (x₁)−y(x₂)

(a_j1−a_j2)e+A(x₁−x₂)−y(x₁)−Ax₁−y(x₂)−Ax₂ |Ax1−Ax2| −2c0

y(x1)−y (x2)−4c0

f^i1,j1−f^i2,j2−y

x1−z^i2,j1

−y

x2−z^i2,j2−4c0

f^i1,j1−f^i2,j2−2c0−2|A|k0−4c0

cz^i1,j1−z^i2,j2−2c0−2|A|k0−4c0

c

2z^i1,j1−z^i2,j2

c

6|x₁−x₂|,

wherec=min_|x|=1|Ax|. The last but one inequality follows from the fact that fori₁=i₂ c

2z^i1,j1−z^i2,j2c(r−1)k0

4 2c0+2|A|k0+4c0

for|x₁−x₂|_∞> (r−1)k0if we choosersufficiently big.

Settingc₀=c₀+max1jr²|a_je|we furthermore havey −Ac₀. So by possibly enlargingc₀toc₀, we can indeed split the energy to obtain (17), and the proof is finished. 2

For systems that do not satisfy Assumption 2.6,ϕ_∞may be non-trivial (for an example see Proposition 4.5 in [21]).

In Section 5.1 we will prove thatϕ_∞is quasiconvex with respect to the first variable and convex with respect to the second.

3.2. The limitc₀→0

In our definition of convergencey^(k)→(u,b), it does not make sense to consider the limiting case of very restricted relaxation, i.e.c₀→0, unless allbⁱ are zero. Instead of asking ˜y−u in Definition 2.3 to be less thanc₀/k one could demand that

˜y−vc0/k (19)

wherev is as in (12) corresponding to u,bwithb⁰ set to zero. (Condition (2) is not needed for this definition of convergence.) This alternative set-up leads to analogous results in the passage to continuum theory, as shown in [21].

It is not hard to calculate the limit ϕ₀(A,b):= lim

c₀→0ϕ_c0(A,b)

which exists in(−∞,∞]sincec₀→ϕ_c0(A,b)is decreasing.

Proposition 3.3.LetV_A,bbe as in(12)for constant∇u=Aandb. Then ϕ₀(A,b)= lim

k→∞

1 νk²E

V_A,b(x): x∈Lk

.

In particular, the limit on the right-hand side exists(inRunder the usual Assumptions2.4and2.5, in(−∞,∞]for energies of the form(8)).

Proof. Suppose first E is of the form (8) and there arei=j ∈ {0, . . . , ν−1}such that bⁱ∈b^j+AZ². Then, if y−V_A,br,

E(y)k² 4 inf

0<srW (s)−Ck²→ ∞

asr→0. For the remaining cases note thatE(V_A,b)is bounded by Lemma 2.12 and, ify−V_A,br, E(y)−E(V_A,b)Lνk²r.

Therefore, lim sup

k→∞ sup

y∈Nk^0,1(A,b)

1

νk²E(y)− 1

νk²E(VA,b) Lc0. Now lettingc₀→0 proves the claim. 2

Example.For admissible pair potentials (i.e.Wsatisfies the conditions of Theorem 2.10) E_pp(y)=1

2

i=j

W

|y_i−y_j|

, (20)

we get

ϕ₀(A,b)= lim

k→∞

1 2νk²

x,z∈Lk

x=z

WV_A,b(x)−V_A,b(z).

Restricting this sum to those x such that dist(xp, ∂[0, k]²) > l where 1lk yields an error term of order O(kl/k²)=o(1). Then summing over allz∈Z²× {0,1, . . . , ν−1},z=x, instead ofLk\ {x}gives another error term of orderO(l^2−q)=o(1). This sum now being independent ofxp, we obtain

ϕ0(A,b)= 1 2ν

ν−1

i=0

z∈L∩(R²×[0,h]) z=(0,0,i)

WV_A,b(z)−VA,b(0,0, i)

= 1 2ν

ν−1

i,j=0

z_p∈Z² (z_p,j )=(0,0,i)

WAz_p+b^j−bⁱ.

The corresponding macroscopic energy functional is given by E(u,b)=

S1

1 2ν

ν−1

i,j=0

z∈Z² (z,j )=(0,0,i)

W∇u(x)z+b^j(x)−bⁱ(x)dx.

This expression can be seen as a thin-film version with directorsb¹, . . . , b^ν−1of a formula derived in [3].

3.3. Triviality for slowly converging deformations

By our definition of convergence, the effective continuum theory depends on the scalel₁=c₀/kmeasuring the rate of uniform convergence ofy˜^(k)tou. This paragraph serves to prove that in fact only the physically motivated choice l₁(k)=const./kyields non-trivial results. Physically this amounts to the fact that thin film configurations are expected to be only locally energy minimizing: admitting for fracture, i.e., large interatomic distances, the film could locally (3d-) crystallize. Physically reasonable decay assumptions on atomic interactions will then lead to trivial macroscopic limits.

It is easy to see that forl11/kwe reproduce the limit obtained in Proposition 3.3. So suppose nowl1=l1(k) 1/k. (Then allb∈L^∞(S1;(R³)^ν−1)will be admissible.) In analogy toW_k^l (cf. Theorem 2.9) we define

W_k^l1,l2(u,b):=

y: ˜y−ul₁, kⁱy˜−bⁱ

W^−1,∞l₂ .

Theorem 3.4.SupposeE satisfies Assumptions 2.4, 2.5, and 2.6. Assumel1(k),l2(k)satisfy kl1(k), kl2(k)→ ∞.

Then for all admissibleu(cf.(1))and allbthe limit E=E(u,b)= lim

k→∞

1

νk² inf

y∈Wk^l1,l2(u,b)

E(y)

exists in[−∞,∞)and is the same for all(u,b).

Proof. Let

E(u,b):=lim inf

k→∞

1

νk² inf

y∈Wk^l1,l2(u,b)

E(y).

Suppose thatb,b ∈L^∞(S1;(R³)^ν−1)andu,u are admissible. The proof follows along the lines of the proof of Theorem 3.2, we indicate the necessary modifications. Choosing a suitable largek0, we findy∈W_k^l1₀^,l₋²₁(u,b)with

1

ν(k0−1)²E(y)E(u,b)+δ/3

(resp.−1/δforE(u,b)= −∞). In addition to the setsU_i=zⁱ+[0, k0+1)²consider the subsetsU_i=zⁱ+[0, k0)², and constructy similar as in the proof of Theorem 3.2 by

y(x1, x2, x3)=y

x1−zⁱ₁, x2−zⁱ₂, x3

+U zⁱ

onLk∩ Ui× [0, h] ,

where U denotes the unrescaled version of u. On the remaining (2k0+1)ν atoms of U_i × [0, h] we define y appropriately such that

−

U_i/ k

kⁱy˜ dρ^(k)= 1 (k₀+1)²

x_p∈Z²∩U_i

y(xp, i)−y(xp,0)= −

U_i/ k

b¯ⁱdρ^(k).

We may assume that forx, x ∈Lkwithx_p∈U_i\U_i andx_p∈U_i,|y(x)−y(x )||x_p−x_p|and thaty −Uis bounded in terms ofk₀independently ofk.

Considering local spatial averages, we still findyˆ∈W_k^l1,l2(u,b)such that fork₀fixed 1

νk²E(y)− 1 νk²E(y)ˆ

→0 ask→ ∞.

(To prove this, one may choose a scale l3 such that 1/kl3l2 and apply Lemmas 3.13 and 3.14 in [21] resp.

Lemmas 2.2.12 and 2.213 in [22] with constants depending onk₀.)

In order to show that the energy splits, again we possibly have to replacey byy . For the construction ofy we can only guarantee that

y (x₁)−y (x₂)C₁|x₁−x₂| −C₃

withC₁, C₃independent ofk₀andkforx₁andx₂that do not lie in the sameU_i,j∩U_i. But Lemma 2.13 still works in this more general case.rnow might not be a fixed number, but still it only depends onk₀, the same being true for a₁, . . . , a_r2. Also note that for the same reason and by translational invariance

E

y(x): x∈L∩

Ui× [0, h]

−E(y)Ck0. Finally sendingkto infinity gives