Energy scaling law for a single disclination in a thin elastic sheet

TL;DR

This paper establishes optimal energy scaling laws for a single disclination in thin elastic sheets across multiple models, providing rigorous lower bounds that match known upper bounds in the leading order of thickness.

Contribution

It provides the first ansatz-free, optimal lower bounds for the elastic energy of a disclination in thin sheets across three different elasticity models.

Findings

Lower bounds match upper bounds in leading order of h

Results are valid for fully nonlinear, 3D, and Föppl-von Kármán models

Establishes fundamental energy scaling laws for disclinations

Abstract

We consider a single disclination in a thin elastic sheet of thickness $h$. We prove ansatz-free lower bounds for the free elastic energy in three different settings: First, for a geometrically fully non-linear plate model, second, for three-dimensional nonlinear elasticity, and third, for the F\"oppl-von K\'arm\'an plate theory. The lower bounds in the first and third result are optimal in the sense that we find upper bounds that are identical to the respective lower bounds in the leading order of $h$.