Geometry and Mechanics of Thin Growing Bilayers

TL;DR

This paper develops an analytical model to understand how thin, arbitrarily shaped sheets morph under isotropic in-plane expansion, revealing the influence of shape, slenderness, and natural curvature on their final form.

Contribution

The authors introduce a new measure of slenderness and provide an analytical framework linking shape and growth stimuli to the resulting morphing behavior of thin sheets.

Findings

Mean curvature in the isometric state is three-fourths of the natural curvature.

Shape influences the initial spherical bending and the final isometric form.

The model's scalability allows application across various size scales.

Abstract

We investigate how thin sheets of arbitrary shapes morph under the isotropic in-plane expansion of their top surface, which may represent several stimuli such as nonuniform heating, local swelling and differential growth. Inspired by geometry, an analytical model is presented that rationalizes how the shape of the disk influences morphing, from the initial spherical bending to the final isometric limit. We introduce a new measure of slenderness $\gamma$ that describes a sheet in terms of both thickness and plate shape. We find that the mean curvature of the isometric state is three fourth's the natural curvature, which we verify by numerics and experiments. We finally investigate the emergence of a preferred direction of bending in the isometric state, guided by numerical analyses. The scalability of our model suggests that it is suitable to describe the morphing of sheets spanning several orders of magnitude.