Surface energies in a two-dimensional mass-spring model for crystals
ESAIM: Mathematical Modelling and Numerical Analysis , 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 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 : 10.1051/m2an/2010106
Classification : 74Q05
Mots clés : continuum mechanics, difference equations
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     title = {Surface energies in a two-dimensional mass-spring model for crystals},
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     pages = {873--899},
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Theil, Florian. Surface energies in a two-dimensional mass-spring model for crystals. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 5, pp. 873-899. doi : 10.1051/m2an/2010106. http://www.numdam.org/articles/10.1051/m2an/2010106/

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