Volume 45, Number 5, September-October 2011
|Page(s)||873 - 899|
|Published online||23 February 2011|
Surface energies in a two-dimensional mass-spring model for crystals
Mathematics Institute, University of Warwick,
Coventry, CV4 7AL, UK. firstname.lastname@example.org
Revised: 28 October 2010
We study an atomistic pair potential-energy E(n)(y) that describes the elastic behavior of two-dimensional crystals with n atoms where 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:
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.
Mathematics Subject Classification: 74Q05
Key words: Continuum mechanics / difference equations
© EDP Sciences, SMAI, 2011
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