Self-contact and instabilities in the anisotropic growth of elastic membranes

TL;DR

This study explores the morphological evolution of growing elastic membranes, revealing that beyond a critical growth, skewed e-cone solutions are energetically favored over symmetric ones, with implications for understanding membrane stability.

Contribution

It demonstrates that symmetric e-cone solutions are not always energetically minimal, identifying skewed e-cones as more stable beyond a certain growth threshold, and generalizes findings to various geometries.

Findings

Symmetric e-cones lose stability beyond a critical growth factor.

Skewed e-cone solutions have lower energy and exhibit spiral winding.

Results apply to discs with varying thickness and rings with different radii.

Abstract

We investigate the morphology of thin discs and rings growing in the circumferential direction. Recent analytical results suggest that this growth produces symmetric excess cones (e-cones). We study the stability of such solutions considering self-contact and bending stress. We show that, contrary to what was assumed in previous analytical solutions, beyond a critical growth factor, no symmetric \textit{e}-cone solution is energetically minimal any more. Instead, we obtain skewed e-cone solutions having lower energy, characterized by a skewness angle and repetitive spiral winding with increasing growth. These results are generalized to discs with varying thickness and rings with holes of different radii.