LATTICE MACROSCOPIC BEHAVIORTalk by Annie Raoult, Laboratoire MAP5, UMR 8145,

Universite Paris Descartes

Covers a series of works since 2006.

D. Caillerie (3S-R, UMR 5521, Grenoble), A. Mourad (Universite libanaise,Beyrouth), A.R., Discrete homogenization in graphene sheet modeling, J.Elast., 2006,N. Meunier (MAP5), O. Pantz (CMAP, UMR 7641), A.R., Square latticeswith three point interactions, ' 2011,

H. Le Dret (JLL, UMR 7598), A.R., Homogenization of hexagonal lattices,

C.R. Acad. Sci. Paris, Ser. I, 2011. + submitted paper.

Related recent results:C. Davini, F. Ongaro, A homogenized model for honeycomb cellularmaterials, J. Elast, 2011R. Alicandro, M. Cicalese, A. Gloria, Integral representation results forenergies defined on stochastic lattices and application to nonlinearelasticity, Arch. Rational Mech. Anal., 2011

Graphene sheet (Artistic impression), Condensed Matter Group, Manchester

,Graphene with strain, Condensed Matter, Manchester

Carbon nanotubes (Iijima, 1991) : rolled graphene, ended bysemi-fullerenes (Smalley, Curl, Kroto, Nobel 1996)

- mechanical properties,- electrical properties depending on their geometry: from insulators tosupra-conductors.

Interatomic distance: r = 0.14 10−9 m

Looking for a macroscopic behavior.

Cauchy-Born rule:

- Restricted version (Cauchy): when submitted to affine imposed boundaryconditions associated with a matrix F and no internal loading, the whole of thelattice deforms accordingly.- Extended version (Born): the above statement is valid for simple lattices. Forcomplex lattices, some nodes may relax.

- Larger range of applicability (E & Ming): within the “elastic range” (before

plasticity or dislocation), for more general boundary conditions, the lattice can

be replaced by an elastic membrane whose internal energy density is given by

the so-called Cauchy-Born density WCB .

WCB : F ∈Mn,2 7→WCB(F ) ∈ R, n = 2, 3.

Recipe (for simple lattices). Take the limit of the mean value overincreasing domains ωR , R → +∞, of the energy due to Fx on ωR .

Confusing: R plays no role. It suffices to compute on a single elementarycell.

Very likely: Cauchy and Born had in mind nd → nd deformations.

First easy case: square lattice with length changes.

Well known method: ε→ 0. Introduce a sequence of lattices with meshsize ε.Internal energy:

I ε(ψ) =kε

2[∑(

|ψ(ε(i + 1, j))− ψ(ε(i , j))| − ε)2

+∑(

|ψ(ε(i , j + 1))− ψ(ε(i , j))| − ε)2

]

for lattice deformations ψ : N ε 7→ Rn.External loads: G ε(ψ) =

∑gε(iε, jε) · ψ(iε, jε)

Dirichlet boundary conditions: part of the boundary

min Jε = I ε − G ε over the lattice deformations

(straightforward existence)

Scalings: kε = k , gε = ε2 g .

We expect an equivalent macroscopic internal (elastic) energy:

I (ψ) =

∫ω

Z(∇ψ(x)

)dx that applies to ψ : ω 7→ Rn, n = 2, 3.

Aim: Determine Z : F ∈Mn×2 7→ R. Notation: F = [f1, f2], f1, f2 ∈ Rn.

Classical trick 1: introduce lattice triangulations, identify ψ : Nε 7→ Rn

with the globally continuous function, still denoted ψ, that is piecewiseaffine and that coincides with ψ at the nodes.This makes a functional space Aε.

Obviously,

∀x1 ∈]iε, (i + 1)ε[, ψ((i + 1)ε, jε)− ψ(iε, jε) = ε ∂1ψ(x1, jε),

∀x2 ∈]jε, (j + 1)ε[, ψ(iε, (j + 1)ε)− ψ(iε, jε) = ε ∂2ψ(iε, x2),

and the partial derivatives are constant per triangle. Therefore,(straightforward)

I ε(ψ) =

∫ω

W (∇ψ) dx , W (F ) = k((|f1| − 1)2 + (|f2| − 1)2

).

To be rigorous. Use Γ-limit arguments in L2(ω;Rn).

Close to obvious by classical trick 2 : extend Jε to L2(ω;Rn) by

Jε(ψ) = +∞ if ψ ∈ L2(ω;Rn) \ Aε.

One easily obtains that Jε Γ-converges to J such that

J(ψ) =

∫ω

QW (∇ψ(x)) dx −∫ω

g(x) · ψ(x) dx , ψ ∈ H1(ω;Rn),

and that QW is explicitely given by

QW (F ) = k(([|f1| − 1]+)2 + ([|f2| − 1]+)2

).

Much more in Alicandro-Cicalese, Braides et al.

Squares with moments or three point interactions

Correct energies in mechanical network modeling should incorporate someresistance to shear. Stillinger-Weber and Tersoff models for atomisticlattice behavior do include angle changes, as well.

Alternate springs (unrealistic) Springs everywhere

Angle changes between deformed bars or bonds have to be incorporated.

Planar deformations: oriented angle in [0, 2π[ , 3d deformations: angle in [0, π].

In all cases, vectors have to be nonzero.

Alternate springs.

Internal energy: Sum over (i , j) of terms

kε(|ψ(Mεi+1,j)− ψ(Mε

i,j)| − ε)2 + kε(|ψ(Mεi,j+1)− ψ(Mε

i,j)| − ε)2

+ K ε

(ψ(Mε

i+1,j)− ψ(Mεi,j)

|ψ(Mεi+1,j)− ψ(Mε

i,j)|·ψ(Mε

i,j+1)− ψ(Mεi,j)

|ψ(Mεi,j+1)− ψ(Mε

i,j)|

)2

+ K ε

(ψ(Mε

i−1,j)− ψ(Mεi,j)

|ψ(Mεi−1,j)− ψ(Mε

i,j)|·ψ(Mε

i,j−1)− ψ(Mεi,j)

|ψ(Mεi,j−1)− ψ(Mε

i,j)|

)2

A first remark : we cannot ignore the nonsuperposition condition.Admissible deformations:

A∗ε = {ψ ∈ C0(ω;R3); ∀T , ψ|T ∈ P1(T ),

∀(i , j), (i ′, j ′)s.t.|i ′ − i |+ |j ′ − j | = 1, ψ(i ′ε, j ′ε) 6= ψ(iε, jε)}.

Integral form of the exact lattice energy: Let us forget the coefficients.

∀ψ ∈ A∗ε , I ε(ψ) =

∫ω

((|∂1ψ| − 1)2 + (|∂2ψ| − 1)2 +

( ∂1ψ

|∂1ψ|· ∂2ψ

|∂2ψ|)2)dx .

This works because three point interactions couple “bars” or “bonds” thatbelong to a single triangle.Equivalently,

I ε(ψ) =

∫ω

W (∇ψ) dx , W (F ) = (|f1| − 1)2 + (|f2| − 1)2 +( f1|f1|· f2|f2|)2.

As above, we extend Jε to L2(ω;Rn) by Jε(ψ) = +∞ if ψ ∈ L2 \ A∗ε .

Let us mention at once some technicalities:- density results involving A∗ε , in order to prove that the Γ-limit is finite onH1(ω;Rn),- (surprisingly?) loading terms,- W is not defined on the whole of Mn×2.We will deal with these points in a more general setting.

First, angular springs everywhere.

We have to deal with the green and red angles. No way ofexpressing them in terms of our piecewise affine functions.

????? Which remedy ?????

Introduce a second triangulation

and a second piecewise affine mapping: ψ piecewise affine on the newtriangulation that coincides with ψ at all nodes.

Integral form of the energy: ∀ψ ∈ A∗ε,

I ε(ψ) =

∫ω

(1

2(|∂1ψ| − 1)2 +

1

2(|∂2ψ| − 1)2 +

( ∂1ψ

|∂1ψ|· ∂2ψ

|∂2ψ|)2

+1

2(|∂1ψ| − 1)2 +

1

2(|∂2ψ| − 1)2 +

( ∂1ψ

|∂1ψ|· ∂2ψ

|∂2ψ|)2)dx .

General three point interactions

Frame indifference for a discrete energy w acting on three points:∀ψ : {M0,M1,M2} 7→ Rn, s.t. ψ(M1) 6= ψ(M0), ψ(M2) 6= ψ(M0),

w(Rψ(M0) + c,Rψ(M1) + c,Rψ(M2) + c) = w(ψ(M0), ψ(M1), ψ(M2))

where R ∈ SO(n), c ∈ Rn. This implies that

∀(x , y , z) ∈ (Rn)3, y 6= x , z 6= x , w(x , y , z) = w(y − x , z − x)

= w(|y − x |, |z − x |, (y − x , z − x).

We consider periodic square lattices such that each point is involved into fourinteractions bringing three points into play. Therefore, we have four three pointenergies:

wa,b with (a, b) = (e1, e2), (e2,−e1), (−e1,−e2), (−e2, e1).

We assume that

w−e1,−e2 = we1,e2 , w−e2,e1 = we2,−e1 , we2,−e1 (f2,−f1) = we1,e2 (f1, f2),

or, equivalently, w−e1,−e2 = we1,e2 , w−e2,e1 = we2,−e1 ,

we2,−e1 (d ′, d , π − θ) = we1,e2 (d , d ′, θ),

and letting w = we1,e2 , we can write on A∗ε

I ε(ψ) = 2

∫ω

w(∇ψ(x)) dx + 2

∫ω

w(∇ψ(x)) dx .

The form

I ε(ψ) = 2

∫ω

w(∇ψ(x)) dx + 2

∫ω

w(∇ψ(x)) dx

is tractable for the asymptotic analysis under the classical assumptions:w : M∗n×2 = (Rn \ {0})× (Rn \ {0}) 7→ R+ satisfies

c1(||F ||p − 1) ≤ w(F ) ≤ c2(||F ||p + 1).

Main trick: Define

∀F ∈Mn×2, W (F ) =

{w(F ) on M∗n×2,

c2(||F ||p + 1) on Mn×2 \M∗n×2.

Jε(ψ) reads as well on A∗ε

Jε(ψ) = 2K (ψ) + 2K (ψ)

where K is defined on W 1,pΓ (ω;Rn) by

∀ψ ∈W 1,pΓ (ω;Rn), K (ψ) =

∫ω

W (∇ψ(x) dx .

And extend Jε to Lp(ω;Rn) by

Jε(ψ) = +∞ if ψ ∈ Lp(ω;Rn) \ A∗ε,

Convergence result. I ε Γ-converges to I 0 defined by

I 0(ψ) = 4

∫ω

QW (∇ψ(x)) dx −∫ω

g(x) · ψ(x) dx , ψ ∈W 1,pΓ (ω;Rn).

Proof :- density results : For any ψ in W 1,p

Γ (ω;Rn), there exists a sequence ψε suchthat ψε ∈ A∗ε ∩ B.C and ψε → ψ in W 1,p(ω;Rn),

- asymptotic weak or strong behavior of ψε induced by the behavior of ψε,

- sequential weak lower semicontinuity of K on W 1,p(ω;R3) where W is (only)Borel, see Dacorogna.

Cauchy-Born rule. See discussions in Ericksen, Friesecke and Theil, E and

Ming ...

The Cauchy-Born rule is valid for the cases we considered (with thequasiconvexification step neglected) with arbitrary loads, and “arbitrary”boundary conditions.

Zero-energy configurations: Choose

we1,e2 (f1, f2) = k1(|f1| − r1)2 + k2(|f2| − r2)2 + K (cos θ)2.

Then, QW (F ) = 0 for any F ∈M∗n×2 such thatvi (Fdiag(1/r1, 1/r2)) ≤ 1, i = 1, 2. In particular, any F = [f1, f2] suchthat |f1| ≤ r1, |f2| ≤ r2 and f1 · f2 = 0. But others with angles not equalto π/2 as well.

Hexagonal lattices(deforming in R3 for definiteness)

Complex lattices with type 1 nodes (black), and type 2 nodes (white),

Let us first describe the global hexagonal network in R2. Let (e1,e2) be anorthonormal basis in R2, and introduce the three vectors

s =√

3e1, t =

√3

2e1 +

32

e2 and p =13(s+ t).

In our description, the network is comprised of two types of nodes: The type 1nodes that occupy points is + jt with (i, j) ∈ Z2, and the type 2 nodes that oc-cupy points is + jt + p, again with (i, j) ∈ Z2, see Figure 1 below. The hexagonalnetwork is thus a complex lattice, a superposition of two simple Bravais latticeswhich are translates of each other, shown with different dashed lines below. Weare following here the standard description of such complex lattices, see [5].

s

tp

Figure 1: •: type 1 nodes, ◦: type 2 nodes

The hexagonal nature of the sheet is not yet apparent. We now assume thatthe internal energy of the sheet only derives from chemical bonds that join type 1nodes to their nearest neighboring type 2 nodes. We model these bonds by bars.There are thus three types of bars: Type 1 bars parallel to s− p, type 2 bars parallelto t − p, and type 3 bars parallel to p, see Figure 2 below. This classification ofbars is only for labeling reasons, all bars are physically equivalent.

3

21

Figure 2: Hexagonal structure and the three different types of bars

2

obtained by superposing two Bravais lattices, as seen below.

Let us first describe the global hexagonal network in R2. Let (e1,e2) be anorthonormal basis in R2, and introduce the three vectors

s =√

3e1, t =

√3

2e1 +

32

e2 and p =13(s+ t).

In our description, the network is comprised of two types of nodes: The type 1nodes that occupy points is + jt with (i, j) ∈ Z2, and the type 2 nodes that oc-cupy points is + jt + p, again with (i, j) ∈ Z2, see Figure 1 below. The hexagonalnetwork is thus a complex lattice, a superposition of two simple Bravais latticeswhich are translates of each other, shown with different dashed lines below. Weare following here the standard description of such complex lattices, see [5].

s

tp

Figure 1: •: type 1 nodes, ◦: type 2 nodes

The hexagonal nature of the sheet is not yet apparent. We now assume thatthe internal energy of the sheet only derives from chemical bonds that join type 1nodes to their nearest neighboring type 2 nodes. We model these bonds by bars.There are thus three types of bars: Type 1 bars parallel to s− p, type 2 bars parallelto t − p, and type 3 bars parallel to p, see Figure 2 below. This classification ofbars is only for labeling reasons, all bars are physically equivalent.

3

21

Figure 2: Hexagonal structure and the three different types of bars

2

Unit cell: Tf full triangle, Te empty triangle.

Realistic domain and boundary conditions

In the present study, we restrict to lattice energies that are due to lengthchanges.

I ε =∑

all bars

k((deformed bar length)− ε

)2.

Deformation description: χε : N ε 7→ R3.

Deformation description: χε : N ε 7→ R3. With χε we associate

- a piecewise affine function ϕε by ϕε(ε(is + jt)) = χε(ε(is + jt)). Thiscorresponds to nodes 1.

- a piecewise constant function γε, deviation between nodes 2 and 1.On full triangles γε = χε(ε(is + jt + p))− χε(ε(is + jt)). On empty triangles,γε = 0.

Deformed bar 1 length:|χε(ε((i + 1)s + jt))− χε(ε(is + jt + p))| = |(ε∂sϕε − γε)(x)| for any x

in the associated full triangle. This allows to rewrite the internal energy as anintegral

I ε(ϕε, γε) =

∫ω

W ε(ε−1x ,Dϕε(x), γε(x))dx

where W ε : R2 × L(R2;R3)× R3 → R is defined by

W ε(y , g , τ) = 2k[(|g(s)− ε−1τ | − 1)2 + (|g(t)− ε−1τ | − 1)2 + (ε−1|τ | − 1)2],

if y ∈ Tf + L, 0 if y ∈ Te + L. No way of getting rid of the oscillation.

Figure: Not forgetting anything

Obtaining the Γ-limit.

Several steps:

- Bounds on minimizers and Γ- lim Jε(ψ, δ) = +∞ if ψ /∈ H1γ0

(ω;R3) or δ 6= 0.

- W0 : R2 × L(R2;R3)→ R, W0(y , g) = infτ∈R3 W ε(y , g , τ).W0 no longer depends on ε. It is still Y -periodic and vanishes for y ∈ Te + L.Potentially analog construction in E & Ming, and in Born.

Expliciting W0 leads to innocent looking problems on triangles. Unsolved so far.

- Auxiliary function: I ε0 (ψ) =∫ωW0(ε−1x ,Dψ(x)) dx if ψ ∈ A(ε).

Then, for all ψ ∈ H1γ0

(ω;R3), (Γ- limε→0 Iε)(ψ, 0) = (Γ- limε→0 I

ε0 )(ψ).

- Finally (see Muller for related arguments) for all ψ ∈ H1Γ (ω;R3),

(Γ- limε→0

I ε0 )(ψ) =

∫ω

Whom(Dψ(x)) dx

where

Whom(g) = infk∈N

{1

k2

(inf

θ∈A(kY )

∫kY

W0(y , g + Dθ(y)) dy)}

.

Proof : adapting the slicing method.

Expected properties are there:

- Whom is frame indifferent (obvious),

- Its material symmetry group contains rotations with angle a multiple of π/3(not obvious).

More on W0: Let A, B, C be three distinct points, R be the radius of thecircumcircle and P its center. Minimize

f (M) = (MA− 1)2 + (MB − 1)2 + (MC − 1)2 either for M ∈ R3 or M ∈ R2.

- if R = 1, then P is the unique minimizer of f , and the min is 0,

- if R < 1, then there are two out-of-plane minimizers, and the min is 0,

- if R > 1, write f ′(M0) = 0, and obtain that any minimizer M0 is in the plane.

More information on M0? on f (M0)?

Next step: incorporate an energy due to angle changes.

X- 10 - 5 0 5 10 15 20

- 1

0

1

2

3

Z

Y 4

5

6

7

18

Figure: Planar. Non affine

Z

- 0.8- 0.6- 0.4- 0.2

0.00.20.40.60.8

- 10

- 5

0

5X

10

15

876520 4321 Y0

Figure: Spatial deformation. Non affine

0

2

Z

- 1.0

- 0.8

- 0.6

- 0.4

- 0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 4

1

62

X38

4Y

105

612

7

8 14

Figure: Zero energy

05/03/11 19:52Figure n°0

Page 1 sur 1file:///Users/ledret/Desktop/Scilab%20nanotrucs/test.svg

Z

-0.8-0.4

0.00.40.8

-30

-20

-10

0

10X

20

30

40

50

60 2520151050 Y

k = 30

Figure: 30 cells