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

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.

arXiv comment6 pages, 6 figures. This article supersedes arXiv:1504.00738

References in corpus (7)

• Elastic theory of unconstrained non-Euclidean plates
• Boundary Layer Analysis of the Ridge Singularity in a Thin Plate
• The von Karman equations for plates with residual strain
• On the Plants Leaves Boundary, "Jupe à Godets" and Conformal Embeddings
• Shape selection in non-Euclidean plates
• The shape of the edge of a leaf
• Defects and boundary layers in non-Euclidean plates

Cited by in corpus (10)

• Geometrically incompatible confinement of solids
• Curved geometries from planar director fields - Solving the two-dimensional inverse problem
• Exact solutions for the wrinkle patterns of confined elastic shells
• Crumples as a generic stress-focusing instability in confined sheets
• Nature's forms are frilly, flexible, and functional
• Distributed branch points and the shape of elastic surfaces with constant negative curvature
• Isometric bending requires local constraints on free edges
• Buckling sheets open a door to understanding self-organization in soft matter
• Continuum mechanics of moving defects in growing bodies
• Weak and Strong Solutions to the Inverse-Square Brachistochrone Problem on Circular and Annular Domains