A derivation of the time dependent von K\'arm\'an equations from atomistic models

TL;DR

This paper rigorously derives the time-dependent von Kármán plate equations from three-dimensional atomistic models, revealing new equations for ultrathin structures where interatomic distance and plate thickness are comparable.

Contribution

It provides a mathematical derivation of von Kármán equations from atomistic models, including new formulations for ultrathin plates with finite layers.

Findings

Derivation of classical von Kármán equations for thin plates

Identification of new plate equations for ultrathin structures

Validation of atomistic-to-continuum limit in dynamic setting

Abstract

We derive the time-dependent von K\'arm\'an plate equations from three dimensional, purely atomistic particle models. In particular, we prove that a thin structure of interacting particles whose dynamics is governed by Newton's laws of motion is effectively described by the von K\'arm\'an equations in the limit of vanishing interatomic distance $\eps$ and vanishing plate thickness $h$. While the classical plate equations are obtained for $\eps \ll h \ll 1$, we find new plate equations for finitely many layers in the ultrathin case $\eps \sim h$.