Growth of Form in Thin Elastic Structures

TL;DR

This paper investigates the stability of growth in thin elastic structures by modeling it as a quasi-static evolution of a target metric, revealing how local curvature and stress influence stability and elongation.

Contribution

It introduces a model coupling growth laws to local shape properties, providing insights into stability criteria for elastic growth in cylindrical shells.

Findings

Coupling growth to curvature alone causes instability.

Coupling growth to stress can stabilize elongation.

The approach applies to various target geometries.

Abstract

Heterogeneous growth plays an important role in the shape and pattern formation of thin elastic structures ranging from the petals of blooming lilies to the cell walls of growing bacteria. Here we address the stability and regulation of such growth, which we modeled as a quasi-static time evolution of a metric, with fast elastic relaxation of the shape. We consider regulation via coupling of the growth law, defined by the time derivative of the target metric, to purely local properties of the shape, such as the local curvature and stress. For cylindrical shells, motivated by rod-like E. \textit{coli}, we show that coupling to curvature alone is generically linearly unstable and that additionally coupling to stress can lead to stably elongating structures. Our approach can readily be extended to gain insights into the general classes of stable growth laws for different target geometries.