Nambu-Goldstone modes and diffuse deformations in elastic shells

TL;DR

This paper develops an effective theory for diffuse deformations in thin elastic shells, linking shape changes to Nambu-Goldstone modes and revealing critical thickness effects on deformation persistence.

Contribution

It introduces a novel theoretical framework connecting infinitesimal isometries and low-energy deformations in elastic shells, incorporating stretching and bending energies.

Findings

The theory recovers known results for cylindrical shells.

Identifies two length scales governing deformation relaxation in conical shells.

Discovers a critical thickness where deformations become nearly isometric.

Abstract

I consider the shape of a deformed elastic shell. Using the fact that the lowest-energy, small deformations are along infinitesimal isometries of the shell's mid-surface, I describe a class of weakly-stretching deformations for thin shells based on the Nambu-Goldstone modes associated with those isometries. The main result is an effective theory to describe the diffuse deformations of thin shells that incorporate stretching and bending energies. The theory recovers previous results for the propagation of a "pinch" on a cylinder. A cone, on the other hand, has two length scales governing the persistence of a pinch: one governing the relaxation of the pinch that scales with thickness as a -1/2 power, and one that scales with thickness above which deformations again become isometric. These lengths meet at a critical thickness below which low energy deformations again become nearly isometric.