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From Pine Cones to Minimal Surfaces: The Geometry and Mechanics of Morphing Bilayers



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Salsby, Barney 


Ubiquitous to nature, is an inherent ability to program form to fulfill function. In thin biological structures, such as leaves and plants, growth coupled with their slenderness, allows for a wealth of complex geometries to emerge. Underpinning this sophisticated phenomena is a problem of mechanics, in which growth, or more specifically volume changes, impart inhomogeneous strain profiles which frustrate the structure and trigger deformations by way of buckling or wrinkling. Inspired by this, researchers have mathematically and physically characterised these phenomena, enabling 2D sheets to be programmed into 3D shapes and reconfigure between their different stable forms. This would have wide ranging applications in a variety of contexts such as robotics for instance, where smart actuators and mimicking live tissue is needed, or for deployable structures in aeronautics.

A subset of this type of growth is that of bilayers, where the Uniform Curvature model has enabled researchers to successfully investigate problems of buckling and multistability. However, owing to the free edge, this model fails to capture higher order effects pertaining to a boundary layer phenomenon, in which moments must dissipate, and hence geometry vary beyond the quadratic terms near the boundary. Traditionally, researchers have neglected their effect in view of the bulk behaviour. However, the resulting linearly scaled Gauss for the stretching energy does not distinguish between planform geometries, and given recent findings concerning the influence of edge effects on preferred bending direction, the validity of the Uniform Curvature model has been put into question.

By introducing a ’fictitious’ edge moment rotation, the energy function is reduced commensurately with the dissipation that occurs within this boundary layer. These reduction terms are a function of the curvatures and we obtain a system of algebraic equations. By consideration of the neutrally stable shell, we observe the altering in preferred bending configuration and stability properties due to planform geometry effects, which we validate by way of a physical prototype.

By introducing a non-linear scaling of Gauss for the stretching energy, we further investi- gate the cessation of multistability into monostability as aspect ratio is varied. By further coupling this non-linear scaling term with edge effects, we uncover a novel tristable structure and demonstrate how it can be straightforwardly fabricated in a table-top experiment. By use of soft elastomers, we investigate the one- and two-dimensional de-localisation of the boundary layer, noting a minimal surface for opposite-sense prestressing for two-dimensional de-localisation.

This dissertation thus provides an insight into the role edge effects play on bilayers in the context of disparate planform geometries, which we further combine with a non- linear variation of the stretching energy, to see how the coupling of these accounts for the multistable properties and geometry as aspect ratio is varied. Beyond the insights expounded, the approach extends the Uniform Curvature model for the study, design and fabrication of morphing bilayers and their subsequent applications.





Seffen, Keith


Boundary Layer Effects, Growth, Morphing Bilayers, Shell Structures


Doctor of Philosophy (PhD)

Awarding Institution

University of Cambridge