Interfacial metric mechanics: stitching patterns of shape change in active sheets

Geometry, mechanics and actuation of intrinsically curved folds

We combine theory and experiments to explore the kinematics and actuation of intrinsically curved folds (ICFs) in otherwise developable shells. Unlike origami folds, ICFs are not bending isometries of flat sheets, but arise via non-isometric processes (growth/moulding) or by joining sheets along curved boundaries. Experimentally, we implement both, first making joined ICFs from paper, then fabricating flat liquid crystal elastomer (LCE) sheets that morph into ICFs upon heating/swelling via programmed metric changes. Theoretically, an ICF's intrinsic geometry is defined by the geodesic curvatures on either side, κgi. Given these, and a target 3D fold-line, one can construct the entire surface isometrically, and compute the bending energy. This construction shows ICFs are bending mechanisms, with a continuous family of isometries trading fold angle against fold-line curvature. In ICFs with symmetric κgi, straightening the fold-line culminates in a fully-folded flat state that is deployable but weak, while asymmetric ICFs ultimately lock with a mechanically strong finite-angle. When unloaded, freely-hinged ICFs simply adopt the (thickness t independent) isometry that minimizes the bend energy. In contrast, in LCE ICFs a competition between flank and ridge selects a ridge curvature that, unusually, scales as t-1/7. Finally, we demonstrate how multiple ICFs can be combined in one LCE sheet, to create a versatile intrinsically curved gripper that lifts a heavy weight.

Discontinuous Metric Programming in Liquid Crystalline Elastomers

Liquid crystalline elastomers (LCEs) are shape-changing materials that exhibit large deformations in response to applied stimuli. Local control of the orientation of LCEs spatially directs the deformation of these materials to realize a spontaneous shape change in response to stimuli. Prior approaches to shape programming in LCEs utilize patterning techniques that involve the detailed inscription of spatially varying nematic fields to produce sheets. These patterned sheets deform into elaborate geometries with complex Gaussian curvatures. Here, we present an alternative approach to realize shape-morphing in LCEs where spatial patterning of the crosslink density locally regulates the material deformation magnitude on either side of a prescribed interface curve. We also present a simple mathematical model describing the behavior of these materials. Further experiments coupled with the mathematical model demonstrate the control of the sign of Gaussian curvature, which is used in combination with heat transfer effects to design LCEs that self-clean as a result of temperature-dependent actuation properties.

Interfacial metric mechanics: stitching patterns of shape change in active sheets

A flat sheet programmed with a planar pattern of spontaneous shape change will morph into a curved surface. Such metric mechanics is seen in growing biological sheets, and may be engineered in actuating soft matter sheets such as phase-changing liquid crystal elastomers (LCEs), swelling gels and inflating baromorphs. Here, we show how to combine multiple patterns in a sheet by stitching regions of different shape changes together piecewise along interfaces. This approach allows simple patterns to be used as building blocks, and enables the design of multi-material or active/passive sheets. We give a general condition for an interface to be geometrically compatible, and explore its consequences for LCE/LCE, gel/gel and active/passive interfaces. In contraction/elongation systems such as LCEs, we find an infinite set of compatible interfaces between any pair of patterns along which the metric is discontinuous, and a finite number across which the metric is continuous. As an example, we find all possible interfaces between pairs of LCE logarithmic spiral patterns. By contrast, in isotropic systems such as swelling gels, only a finite number of continuous interfaces are available, greatly limiting the potential of stitching. In both continuous and discontinuous cases, we find the stitched interfaces generically carry singular Gaussian curvature, leading to intrinsically curved folds in the actuated surface. We give a general expression for the distribution of this curvature, and a more specialized form for interfaces in LCE patterns. The interfaces thus also have rich geometric and mechanical properties in their own right.