# An atomistic-based F\"oppl-von K\'arm\'an model for graphene

## TL;DR

This paper develops a non-linear continuum F"oppl-von Kármán model for graphene based on atomistic interactions, capturing out-of-plane displacements and lattice effects with derived constitutive coefficients.

## Contribution

It introduces a novel atomistic-to-continuum derivation of a F"oppl-von Kármán model for graphene, accounting for lattice shifts and self-stress effects.

## Key findings

- Derived continuum model depends on macroscopic displacement and lattice shift.
- Constitutive coefficients explicitly linked to atomistic Brenner's REBO potential.
- Formal minimization yields a F"oppl-von Kármán type model for graphene.

## Abstract

We deduce a non-linear continuum model of graphene for the case of finite out-of-plane displacements and small in-plane deformations. On assuming that the lattice interactions are governed by the Brenner's REBO potential of 2nd generation and that self-stress is present, we introduce discrete strain measures accounting for up-to-the-third neighbor interactions. The continuum limit turns out to depend on an average (macroscopic) displacement field and a relative shift displacement of the two Bravais lattices that give rise to the hexagonal periodicity. On minimizing the energy with respect to the shift variable, we formally determine a continuum model of F\"oppl-von K\'arm\'a type, whose constitutive coefficients are given in terms of the atomistic interactions.