A derivation of the time dependent von Kármán equations from atomistic models

We derive the time-dependent von Kármán 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 Kármán equations in the limit of vanishing interatomic distance $\text{eps}$ and vanishing plate thickness $h$. While the classical plate equations are obtained for $\text{eps}\ll h\ll 1$, we find new plate equations for finitely many layers in the ultrathin case $\text{eps}\sim h$.