Stability of the Von K\'arm\'an regime for thin plates under Neumann boundary conditions

TL;DR

This paper investigates the stability of the Von Kármán model for thin plates with Neumann boundary conditions under dead loads, extending previous results and identifying force types incompatible with the model.

Contribution

It extends stability analysis of the Von Kármán model to Neumann conditions and modifies stability notions to account for rotational invariance.

Findings

Proves a stability alternative for the Von Kármán model under Neumann conditions.

Shows the model is incompatible with certain force types, favoring the Kirchhoff model in those cases.

Extends previous Dirichlet case results to Neumann boundary conditions.

Abstract

We analyze the stability of the Von K\'arm\'an model for thin plates subject to pure Neumann conditions and to dead loads, with no restriction on their direction. We prove a stability alternative, which extends previous results by Lecumberry and M\"uller in the Dirichlet case. Because of the rotation invariance of the problem, their notions of stability have to be modified and combined with the concept of optimal rotations due to Maor and Mora. Finally, we prove that the Von K\'arm\'an model is not compatible with some specific types of forces. Thus, for such, only the Kirchoff model applies.