Symmetry breaking in indented elastic cones

TL;DR

This paper analyzes the energy scaling and symmetry breaking in indented elastic cones, revealing three deformation regimes and proving that large indentations lead to non-radially symmetric minimizers.

Contribution

It identifies the energy regimes in an elastic cone model and rigorously proves symmetry breaking for large indentations.

Findings

Three deformation regimes identified with increasing indentation

Logarithmic singularity dominates at small indentation

Large indentations cause symmetry breaking in minimizers

Abstract

Motivated by simulations of carbon nanocones (see Jordan and Crespi, Phys. Rev. Lett., 2004), we consider a variational plate model for an elastic cone under compression in the direction of the cone symmetry axis. Assuming radial symmetry, and modeling the compression by suitable Dirichlet boundary conditions at the center and the boundary of the sheet, we identify the energy scaling law in the von-K\'arm\'an plate model. Specifically, we find that three different regimes arise with increasing indentation $\delta$: initially the energetic cost of the logarithmic singularity dominates, then there is a linear response corresponding to a moderate deformation close to the boundary of the cone, and for larger $\delta$ a localized inversion takes place in the central region. Then we show that for large enough indentations minimizers of the elastic energy cannot be radially symmetric. We do so by an explicit construction that achieves lower elastic energy than the minimum amount possible for radially symmetric deformations.