Shape programming lines of concentrated Gaussian curvature

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 grippers. We speculate on similarities between these patterns of shape change and those found during the morphogenesis of several biological systems.

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

“…As a starting point towards understanding the geometric and mechanical properties of these interfaces, we derive analytical formulae for the GC concentrated along them. These analytic results highlight that the interfaces can form folds with positive or negative GC [ 49 ] (or even both), and give a first-order understanding of the surface’s resultant shape, which we further illustrate with simulations.…”

“…Finally, we note that a key premise of this work has been that stitched interfaces must be metric compatible, otherwise the two regions disagree over the length of the interface on actuation, leading to large internal stresses and ultimately, material failure. However, in the experimental literature, one may find examples of both compatible [ 2 , 27 , 49 ] and incompatible [ 9 , 33 , 42 , 57 ] interfaces. The compatible interfaces follow our interfacial metric mechanics framework exactly, and one can see compatible curved interfaces in the actuated configuration.…”

Multi-functional topology optimization ofVictoria cruzianaveins

“…A Victoria cruziana leaf [12,13,18,66,67] has a circular and flat disc, surrounded by an upright leaf of several centimetres in height along the direction e 3 (figure 1c,d). Complex and rugged veins are distributed on the lower surface of the disc (figure 1d), rather than embedded in the disc [8][9][10][11].…”

Shape Morphing of Planar Liquid Crystal Elastomers

“…While the magnitude of this deformation is constant throughout the entire material, the principal shrinking direction (the nematic director field) may vary throughout the sheet. Determining the implicit geometry induced by a particular two-dimensional director field (also know as the forward problem) has been solved [9,10] and amply explored [11][12][13]. Likewise, the inverse problem of determining the LCE director field that will deform into a desired geometry, has been shown [14][15][16][17] to be solvable locally in the form of a system of nonlinear hyperbolic partial differential equations (PDEs).…”

“…Together, Eqs. (10,11) provide us with x (u, v), y (u, v) and θ(u, v), from which one extracts θ(x, y) and can go on to make their PLCE. The algorithm is illustrated in the Fig.…”