Isometric immersions, energy minimization and self-similar buckling in non-Euclidean elastic sheets

TL;DR

This paper investigates how hierarchical buckling patterns in non-Euclidean elastic sheets emerge from energy minimization, driven by principal curvature competition and branch-point singularities, even without strain.

Contribution

It reveals that complex buckling patterns arise in zero strain configurations due to curvature competition and branch-point singularities, advancing understanding of elastic sheet morphologies.

Findings

Hierarchical buckling patterns can form without strain.

Principal curvature competition drives pattern formation.

Branch-point singularities are key to complex wrinkling.

Abstract

The edges of torn plastic sheets and growing leaves often display hierarchical buckling patterns. We show that this complex morphology (i) emerges even in zero strain configurations, and (ii) is driven by a competition between the two principal curvatures, rather than between bending and stretching. We identify the key role of branch-point (or "monkey-saddle") singularities in generating complex wrinkling patterns in isometric immersions, and show how they arise naturally from minimizing the elastic energy.