Structure of Crumpled Thin Elastic Membranes

Alexander E. Lobkovsky
The James Franck Institute
The University of Chicago
5640 South Ellis Avenue Chicago, Illinois 60637

In this thesis we explore forced crumpling of a thin elastic membrane. As in many problems that involve bending of elastic plates and shells, the limit of the vanishing membrane thickness leads to a boundary layer phenomenon. We argue that the structure of a crumpled sheet in that limit is simple. It consists of a collection of flat facets that are bounded by straight edges that in turn meet at sharp vertices. These edges become infinitely sharp in the small thickness limit. A boundary layer solution in the ridge region determines the details of how the singular limit of a sharp crease is approached. Most of the elastic energy is confined into the ridges. A scaling law allows one to estimate the energy of a ridge given only its length and dihedral angle. Thus, if for a given compression factor, a crumpled sheet can be characterized in terms of the underlying ridge network, on can estimate its elastic energy and therefore its resistance to further compression.