Singularities, Structures and Scaling in Deformed Elastic m-Sheets

TL;DR

This paper investigates how elastic energy condenses into singular structures in deformed elastic sheets of various dimensions, revealing different scaling behaviors depending on the embedding space.

Contribution

It provides a numerical analysis of energy condensation in elastic sheets across different dimensions, identifying distinct scaling behaviors and the influence of embedding dimension.

Findings

Identification of cone and ridge scaling behaviors

Differences in energy condensation based on embedding dimension

Numerical methods for finding minimum energy configurations

Abstract

The crumpling of a thin sheet can be understood as the condensation of elastic energy into a network of ridges which meet in vertices. Elastic energy condensation should occur in response to compressive strain in elastic objects of any dimension greater than 1. We study elastic energy condensation numerically in 2-dimensional elastic sheets embedded in spatial dimensions 3 or 4 and 3-dimensional elastic sheets embedded in spatial dimensions 4 and higher. We represent a sheet as a lattice of nodes with an appropriate energy functional to impart stretching and bending rigidity. Minimum energy configurations are found for several different sets of boundary conditions. We observe two distinct behaviors of local energy density fall-off away from singular points, which we identify as cone scaling or ridge scaling. Using this analysis we demonstrate that there are marked differences in the forms of energy condensation depending on the embedding dimension.