Surface energies in a two-dimensional mass-spring model for crystals
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 45 (2011) no. 5, pp. 873-899.

We study an atomistic pair potential-energy E(n)(y) that describes the elastic behavior of two-dimensional crystals with n atoms where $y\in {ℝ}^{2×n}$ 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: ${\mathrm{min}}_{y}{E}^{\left(n\right)}\left(y\right)=n\phantom{\rule{0.166667em}{0ex}}{E}_{\mathrm{bulk}}+\sqrt{n}\phantom{\rule{0.166667em}{0ex}}{E}_{\mathrm{surface}}+o\left(\sqrt{n}\right),\phantom{\rule{2em}{0ex}}n\to \infty .$ 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.

DOI : https://doi.org/10.1051/m2an/2010106
Classification : 74Q05
Mots clés : continuum mechanics, difference equations
@article{M2AN_2011__45_5_873_0,
author = {Theil, Florian},
title = {Surface energies in a two-dimensional mass-spring model for crystals},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {873--899},
publisher = {EDP-Sciences},
volume = {45},
number = {5},
year = {2011},
doi = {10.1051/m2an/2010106},
zbl = {1269.82065},
mrnumber = {2817548},
language = {en},
url = {archive.numdam.org/item/M2AN_2011__45_5_873_0/}
}
Theil, Florian. Surface energies in a two-dimensional mass-spring model for crystals. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 45 (2011) no. 5, pp. 873-899. doi : 10.1051/m2an/2010106. http://archive.numdam.org/item/M2AN_2011__45_5_873_0/

[1] 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. | MR 2429532 | Zbl 1148.49010

[2] S. Aubry, The twist map, the extended Frenkel-Kontorova model and the devil's staircase. Physica D 7 (1983) 240-258. | MR 719055 | Zbl 0559.58013

[3] X. Blanc, C. Le Bris and P.L. Lions, From molecular models to continuum mechanics. Arch. Rat. Mech. Anal. 164 (2002) 341-381. | MR 1933632 | Zbl 1028.74005

[4] A. Braides and M. Cicalese, Surface energies in nonconvex discrete systems. Math. Models Meth. Appl. Sci. 17 (2007) 985-1037. | MR 2337429 | Zbl 1205.82036

[5] A. Braides and A. Defranchesi, Homogenisation of multiple integrals. Oxford University Press (1998). | Zbl 0911.49010

[6] A. Braides and M. Gelli, Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids 7 (2002) 41-66. | MR 1900933 | Zbl 1024.74004

[7] A. Braides, M. Solci and E. Vitali, A derivation of linear alastic energies from pair-interaction atomistic systems. Netw. Heterog. Media 9 (2007) 551-567. | MR 2318845 | Zbl 1183.74017

[8] 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. | MR 1654427 | Zbl 0917.34052

[9] M. Charlotte and L. Truskinovsky, Linear elastic chain with a hyper-pre-stress. J. Mech. Phys. Solids 50 (2002) 217-251. | MR 1892977 | Zbl 1035.74005

[10] W. E and P. Ming, Cauchy-Born rule and the stability of crystalline solids: static problems. Arch. Rat. Mech. Anal. 183 (2005) 241-297. | MR 2278407 | Zbl 1106.74019

[11] I. Fonseca and S. Müller, A uniqueness proof for the Wulff theorem. Proc. Roy. Soc. Edinburgh Sect. A 119 (1991) 125-136. | MR 1130601 | Zbl 0752.49019

[12] 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. | MR 1923388 | Zbl 1084.74501

[13] 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. | MR 1916989 | Zbl 1021.74024

[14] D. Gérard-Varet and N. Masmoudi, Homogenization and boundary layer. Preprint available at www.math.nyu.edu/faculty/masmoudi/homog_Varet3.pdf (2010). | Zbl 1259.35024

[15] P. Lancaster and L. Rodman, Algebraic Riccati Equations. Oxford University Press (1995). | MR 1367089 | Zbl 0836.15005

[16] J.L. Lions, Some methods in the mathematical analysis of systems and their controls. Science Press, Beijing, Gordon and Breach, New York (1981). | MR 664760 | Zbl 0542.93034

[17] J.A. Nitsche, On Korn's second inequality. RAIRO Anal. Numér. 15 (1981) 237-248. | Numdam | MR 631678 | Zbl 0467.35019

[18] C. Radin, The ground state for soft disks. J. Stat. Phys. 26 (1981) 367-372. | MR 643714

[19] B. Schmidt, A derivation of continuum nonlinear plate theory from atomistic models. Multiscale Mod. Sim. 5 (2006) 664-694. | MR 2247767 | Zbl 1117.49018

[20] B. Schmidt, On the passage from atomic to continuum theory for thin films. Arch. Rat. Mech. Anal. 190 (2008) 1-55. | MR 2434899 | Zbl 1156.74028

[21] B. Schmidt, On the derivation of linear elasticity from atomistic models. Net. Heterog. Media 4 (2009) 789-812. | MR 2552170 | Zbl 1183.74020

[22] E. Sonntag, Mathematical Control Theory. Second edition, Springer (1998).

[23] L. Tartar, The general theory of homogenization. Springer (2010). | MR 2582099 | Zbl 1188.35004

[24] F. Theil, A proof of crystallization in a two dimensions. Comm. Math. Phys. 262 (2006) 209-236. | MR 2200888 | Zbl 1113.82016