On the Bending Energy of Buckled Edge-Dislocations

TL;DR

This paper investigates the bending energy of elastic membranes with a single edge-dislocation, showing that the minimal energy diverges logarithmically with system size when the membrane buckles to relieve strain.

Contribution

It provides a rigorous proof that the minimum bending energy for strain-free configurations with a dislocation diverges logarithmically, advancing understanding of defect energetics in elastic membranes.

Findings

Minimum bending energy diverges logarithmically with system size.

Buckling allows membranes to relieve in-plane strain.

Rigorous mathematical proof of energy divergence.

Abstract

The study of elastic membranes carrying topological defects has a longstanding history, going back at least to the 1950s. When allowed to buckle in three-dimensional space, membranes with defects can totally relieve their in-plane strain, remaining with a bending energy, whose rigidity modulus is small compared to the stretching modulus. In this paper, we study membranes with a single edge-dislocation. We prove that the minimum bending energy associated with strain-free configurations diverges logarithmically with the size of the system.