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.
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 }, pages = {873--899}, publisher = {EDP-Sciences}, volume = {45}, number = {5}, year = {2011}, doi = {10.1051/m2an/2010106}, mrnumber = {2817548}, zbl = {1269.82065}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2010106/} }
TY - JOUR AU - Theil, Florian TI - Surface energies in a two-dimensional mass-spring model for crystals JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2011 SP - 873 EP - 899 VL - 45 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2010106/ DO - 10.1051/m2an/2010106 LA - en ID - M2AN_2011__45_5_873_0 ER -
%0 Journal Article %A Theil, Florian %T Surface energies in a two-dimensional mass-spring model for crystals %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2011 %P 873-899 %V 45 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2010106/ %R 10.1051/m2an/2010106 %G en %F 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 , 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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