Minimal resonances in annular non-Euclidean strips

TL;DR

This paper investigates the shapes of elastic, swollen strips with inhomogeneous growth, combining analytical and numerical methods to classify minimal energy configurations and conditions for zero mean curvature.

Contribution

It introduces a variational approach using conical closed strips to determine minimal energy shapes under inhomogeneous swelling patterns.

Findings

Classification of strip shapes by wrinkles and swelling patterns

Derivation of conditions for zero mean curvature along the center line

Excellent agreement between analytical models and numerical simulations

Abstract

Differential growth processes play a prominent role in shaping leaves and biological tissues. Using both analytical and numerical calculations, we consider the shapes of closed, elastic strips which have been subjected to an inhomogeneous pattern of swelling. The stretching and bending energies of a closed strip are frustrated by compatibility constraints between the curvatures and metric of the strip. To analyze this frustration, we study the class of "conical" closed strips with a prescribed metric tensor on their center line. The resulting strip shapes can be classified according to their number of wrinkles and the prescribed pattern of swelling. We use this class of strips as a variational ansatz to obtain the minimal energy shapes of closed strips and find excellent agreement with the results of a numerical bead-spring model. Within this class of strips, we derive a condition under which a strip can have vanishing mean curvature along the center line.