# Surface energies in a two-dimensional mass-spring model for crystals

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

- Volume: 45, Issue: 5, page 873-899
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topTheil, Florian. "Surface energies in a two-dimensional mass-spring model for crystals." ESAIM: Mathematical Modelling and Numerical Analysis 45.5 (2011): 873-899. <http://eudml.org/doc/197544>.

@article{Theil2011,

abstract = {
We study an atomistic pair potential-energy E(n)(y) that describes
the elastic behavior of two-dimensional crystals with n atoms where
$y \in \{\mathbb R\}^\{2\times n\}$ characterizes the particle positions. The main
focus is the asymptotic analysis of the ground state energy as n
tends to infinity. We show in a suitable scaling regime where the
energy is essentially quadratic that the energy minimum of E(n)
admits an asymptotic expansion involving fractional powers of n:
$\{\rm min\}_y E^\{(n)\}(y) = n \, E_\{\mathrm\{bulk\}\}+ \sqrt\{n\} \, E_\mathrm\{surface\} +o(\sqrt\{n\}), \qquad n \to \infty.$
The bulk energy density Ebulk is given by an explicit
expression involving the interaction potentials. The surface energy
Esurface can be expressed as a surface integral where the
integrand depends only on the surface normal and the interaction
potentials. The evaluation of the integrand involves solving a discrete algebraic Riccati equation. Numerical simulations suggest
that the integrand is a continuous, but nowhere differentiable function of
the surface normal.
},

author = {Theil, Florian},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Continuum mechanics; difference equations; continuum mechanics; two-dimensional crystals},

language = {eng},

month = {2},

number = {5},

pages = {873-899},

publisher = {EDP Sciences},

title = {Surface energies in a two-dimensional mass-spring model for crystals},

url = {http://eudml.org/doc/197544},

volume = {45},

year = {2011},

}

TY - JOUR

AU - Theil, Florian

TI - Surface energies in a two-dimensional mass-spring model for crystals

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2011/2//

PB - EDP Sciences

VL - 45

IS - 5

SP - 873

EP - 899

AB -
We study an atomistic pair potential-energy E(n)(y) that describes
the elastic behavior of two-dimensional crystals with n atoms where
$y \in {\mathbb R}^{2\times n}$ characterizes the particle positions. The main
focus is the asymptotic analysis of the ground state energy as n
tends to infinity. We show in a suitable scaling regime where the
energy is essentially quadratic that the energy minimum of E(n)
admits an asymptotic expansion involving fractional powers of n:
${\rm min}_y E^{(n)}(y) = n \, E_{\mathrm{bulk}}+ \sqrt{n} \, E_\mathrm{surface} +o(\sqrt{n}), \qquad n \to \infty.$
The bulk energy density Ebulk is given by an explicit
expression involving the interaction potentials. The surface energy
Esurface can be expressed as a surface integral where the
integrand depends only on the surface normal and the interaction
potentials. The evaluation of the integrand involves solving a discrete algebraic Riccati equation. Numerical simulations suggest
that the integrand is a continuous, but nowhere differentiable function of
the surface normal.

LA - eng

KW - Continuum mechanics; difference equations; continuum mechanics; two-dimensional crystals

UR - http://eudml.org/doc/197544

ER -

## References

top- R. Alicandro, A. Braides and M. Cicalese, Continuum limits of discrete films with superlinear growth densities. Calc. Var. Par. Diff. Eq.33 (2008) 267–297.
- S. Aubry, The twist map, the extended Frenkel-Kontorova model and the devil's staircase. Physica D7 (1983) 240–258.
- X. Blanc, C. Le Bris and P.L. Lions, From molecular models to continuum mechanics. Arch. Rat. Mech. Anal.164 (2002) 341–381.
- A. Braides and M. Cicalese, Surface energies in nonconvex discrete systems. Math. Models Meth. Appl. Sci.17 (2007) 985–1037.
- A. Braides and A. DeFranchesi, Homogenisation of multiple integrals. Oxford University Press (1998).
- A. Braides and M. Gelli, Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids7 (2002) 41–66.
- A. Braides, M. Solci and E. Vitali, A derivation of linear alastic energies from pair-interaction atomistic systems. Netw. Heterog. Media9 (2007) 551–567.
- J. Cahn, J. Mallet-Paret and E. Van Vleck, Travelling wave solutions for systems of ODEs on a two-dimensional spatial lattice. SIAM J. Appl. Math.59 (1998) 455–493.
- M. Charlotte and L. Truskinovsky, Linear elastic chain with a hyper-pre-stress. J. Mech. Phys. Solids50 (2002) 217–251.
- W. E and P. Ming, Cauchy-Born rule and the stability of crystalline solids: static problems. Arch. Rat. Mech. Anal.183 (2005) 241–297.
- I. Fonseca and S. Müller, A uniqueness proof for the Wulff theorem. Proc. Roy. Soc. Edinburgh Sect. A 119 (1991) 125–136.
- G. Friesecke and F. Theil, Validitity and failure of the Cauchy-Born rule in a two-dimensional mass-spring lattice. J. Nonlinear Sci.12 (2002) 445–478.
- G. Friesecke, R. 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 (2002) 1461–1506.
- D. Gérard-Varet and N. Masmoudi, Homogenization and boundary layer. Preprint available at www.math.nyu.edu/faculty/masmoudi/homog_Varet3.pdf (2010).
- P. Lancaster and L. Rodman, Algebraic Riccati Equations. Oxford University Press (1995).
- J.L. Lions, Some methods in the mathematical analysis of systems and their controls. Science Press, Beijing, Gordon and Breach, New York (1981).
- J.A. Nitsche, On Korn's second inequality. RAIRO Anal. Numér.15 (1981) 237–248.
- C. Radin, The ground state for soft disks. J. Stat. Phys.26 (1981) 367–372.
- B. Schmidt, A derivation of continuum nonlinear plate theory from atomistic models. Multiscale Mod. Sim.5 (2006) 664–694.
- B. Schmidt, On the passage from atomic to continuum theory for thin films. Arch. Rat. Mech. Anal.190 (2008) 1–55.
- B. Schmidt, On the derivation of linear elasticity from atomistic models. Net. Heterog. Media4 (2009) 789–812.
- E. Sonntag, Mathematical Control Theory. Second edition, Springer (1998).
- L. Tartar, The general theory of homogenization. Springer (2010).
- F. Theil, A proof of crystallization in a two dimensions. Comm. Math. Phys.262 (2006) 209–236.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.