Variational problems in thin elastic structures

TL;DR

This paper derives energy scaling laws for two scenarios of thin elastic structures modeled by 2D Föppl-von Kármán equations, revealing how energy bounds depend on thickness and indentation parameters.

Contribution

It provides new energy scaling laws for singular excess-cone and indented half-sphere structures, with optimal bounds and dependence on geometric parameters.

Findings

Energy bounds scale with thickness parameter h.

Scaling laws depend on indentation depth d.

Optimal bounds are established for both scenarios.

Abstract

For two different scenarios regarding thin elastic structures, described by 2d-F\"oppl-von K\'arm\'an plate models, we obtain energy scaling laws. Firstly, assuming the reference geometry being that of a singular excess-cone, we obtain fairly optimal upper- and lower energy bounds and we highlight how those bounds scale wrt. the thickness-parameter $h.$ Secondly, we consider the half sphere, while being indented by a thin object at the top and perpendicular to its surface. In this situation we provide an energy-scaling law, for radial symmetric admissible maps, this time, depending on $h$ and the indentation depth $d.$