Calculus of Variations and Geometric Measure Theory

A. Bach - A. Braides - M. Cicalese

Discrete-to-continuum limits of multi-body systems with bulk and surface long-range interactions

created by bach on 30 Jul 2019
modified by braidesa on 30 Dec 2020


Published Paper

Inserted: 30 jul 2019
Last Updated: 30 dec 2020

Journal: SIAM J. Math. Anal.
Volume: 52
Pages: 3600-2665
Year: 2020
Doi: 10.1137/19M1289212


We study the atomistic-to-continuum limit of a class of energy functionals for crystalline materials via $\Gamma$-convergence. We consider energy densities that may depend on interactions between all points of the lattice and we give conditions that ensure compactness and integral-representation of the continuum limit on the space of special functions of bounded variation. This abstract result is complemented by a homogenization theorem, where we provide sufficient conditions on the energy densities under which bulk- and surface contributions decouple in the limit. The results are applied to long-range and multi-body interactions in the setting of weak-membrane energies.

Keywords: Homogenization, $\Gamma$-convergence, discrete-to-continuum, free-discontinuity functionals, multi-body interactions