Surface energies in a two-dimensional mass-spring model for crystals
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 5, p. 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 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: min y E (n) (y)=nE bulk +nE surface +o(n),n. 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
Keywords: 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},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {5},
     year = {2011},
     pages = {873-899},
     doi = {10.1051/m2an/2010106},
     zbl = {1269.82065},
     mrnumber = {2817548},
     language = {en},
     url = {http://www.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, Volume 45 (2011) no. 5, pp. 873-899. doi : 10.1051/m2an/2010106. http://www.numdam.org/item/M2AN_2011__45_5_873_0/

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