Shape programming lines of concentrated Gaussian curvature

TL;DR

This paper explores how patterned liquid crystal elastomer sheets can be designed to form complex curved shapes with sharp ridges, combining theory, experiments, and numerics to understand their shape programming capabilities.

Contribution

It introduces a method to encode Gauss curvature in LCE sheets through specific director patterns, analyzing the formation and deformation of sharp ridges with potential applications in actuation.

Findings

Ridges with V-shaped cross-sections encode Gauss curvature.

Ridges cannot be flattened isometrically but can deform by trading curvature.

Finite thickness blunts ridges, smoothing out curvature effects.

Abstract

Liquid crystal elastomers (LCEs) can undergo large reversible contractions along their nematic director upon heating or illumination. A spatially patterned director within a flat LCE sheet thus encodes a pattern of contraction on heating, which can morph the sheet into a curved shell, akin to how a pattern of growth sculpts a developing organism. Here we consider, theoretically, numerically and experimentally, patterns constructed from regions of radial and circular director, which, in isolation, would form cones and anticones. The resultant surfaces contain curved ridges with sharp V-shaped cross-sections, associated with the boundaries between regions in the patterns. Such ridges may be created in positively and negatively curved variants and, since they bear Gauss curvature (quantified here via the Gauss-Bonnet theorem), they cannot be flattened without energetically prohibitive stretch. Our experiments and numerics highlight that, although such ridges cannot be flattened isometrically, they can deform isometrically by trading the (singular) curvature of the V angle against the (finite) curvature of the ridge line. Furthermore, in finite thickness sheets, the sharp ridges are inevitably non-isometrically blunted to relieve bend, resulting in a modest smearing out of the encoded singular Gauss curvature. We close by discussing the use of such features as actuating linear features, such as probes, tongues and limbs, and highlighting the similarities between these patterns of shape change and those found during the morphogenesis of several biological systems.