Connecting disclinations by ridges

TL;DR

This paper studies the formation of ridges connecting disclinations in thin elastic sheets using a variational approach, establishing energy scaling laws and a novel mathematical estimate under a convexity assumption.

Contribution

It introduces a variational framework for disclinations in elastic sheets and proves the existence of ridges with matching energy bounds, extending the Monge-Ampère measure's properties.

Findings

Minimizers exhibit ridges between disclinations.

Energy scaling law with matching bounds up to logarithmic factors.

Generalization of the Monge-Ampère measure's monotonicity property.

Abstract

We consider a thin elastic sheet with a finite number of disclinations in a variational framework in the F\"oppl-von K\'arm\'an approximation. Under the non-physical assumption that the out-of-plane displacement is a convex function, we prove that minimizers display ridges between the disclinations. We prove the associated energy scaling law with upper and lower bounds that match up to logarithmic factors in the thickness of the sheet. One of the key estimates in the proof that we consider of independent interest is a generalization of the monotonicity property of the Monge-Amp\`ere measure.