Programmable shape-shifting materials can take different physical forms to achieve multifunctionality in a dynamic and controllable manner. Although morphing a shape from 2D to 3D via programmed inhomogeneous local deformations has been demonstrated in various ways, the inverse problem-finding how to program a sheet in order for it to take an arbitrary desired 3D shape-is much harder yet critical to realize specific functions. Here, we address this inverse problem in thin liquid crystal elastomer (LCE) sheets, where the shape is preprogrammed by precise and local control of the molecular orientation of the liquid crystal monomers. We show how blueprints for arbitrary surface geometries can be generated using approximate numerical methods and how local extrinsic curvatures can be generated to assist in properly converting these geometries into shapes. Backed by faithfully alignable and rapidly lockable LCE chemistry, we precisely embed our designs in LCE sheets using advanced top-down microfabrication techniques. We thus successfully produce flat sheets that, upon thermal activation, take an arbitrary desired shape, such as a face. The general design principles presented here for creating an arbitrary 3D shape will allow for exploration of unmet needs in flexible electronics, metamaterials, aerospace and medical devices, and more.

A thin sheet of nematic elastomer attains 3D configurations depending on the nematic director field upon heating. In this Letter, we describe the intrinsic geometry of such a sheet and derive an expression for the metric induced by general nematic director fields. Furthermore, we investigate the reverse problem of constructing a director field that induces a specified 2D geometry. We provide an explicit recipe for how to construct any surface of revolution using this method. Finally, we show that by inscribing a director field gradient across the sheet's thickness, one can obtain a nontrivial hyperbolic reference curvature tensor, which together with the prescription of a reference metric allows dictation of actual configurations for a thin sheet of nematic elastomer.

We provide a geometric-mechanical model for calculating equilibrium configurations of chemical systems that self-assemble into chiral ribbon structures. The model is based on incompatible elasticity and uses dimensionless parameters to determine the equilibrium configurations. As such, it provides universal curves for the shape and energy of self-assembled ribbons. We provide quantitative predictions for the twisted-to-helical transition, which was observed experimentally in many systems, and demonstrate it with synthetic ribbons made of responsive gels. In addition, we predict the bi-stability of wide ribbons and also show how geometrical frustration can cause arrest of ribbon widening. Finally, we show that the model's predictions provide explanations for experimental observations in different chemical systems.

A thin elastic sheet lying on a soft substrate develops wrinkled patterns when subject to an external forcing or as a result of geometric incompatibility. Thin sheet elasticity and substrate response equip such wrinkles with a global preferred wrinkle spacing length and with resistance to wrinkle curvature. These features are responsible for the liquid crystalline smectic-like behaviour of such systems at intermediate length scales. This insight allows better understanding of the wrinkling patterns seen in such systems, with which we explain pattern breaking into domains, the properties of domain walls and wrinkle undulation. We compare our predictions with numerical simulations and with experimental observations.

Thin nematic elastomers, composite hydrogels and plant tissues are among many systems that display uniform anisotropic deformation upon external actuation. In these materials, the spatial orientation variation of a local director field induces intricate global shape changes. Despite extensive recent efforts, to date, there is no general solution to the inverse design problem: how to design a director field that deforms exactly into a desired surface geometry upon actuation, or whether such a field exists. In this work, we phrase this inverse problem as a hyperbolic system of differential equations. We prove that the inverse problem is locally integrable, provide an algorithm for its integration, and derive bounds on global solutions. We classify the set of director fields that deform into a given surface, thus paving the way to finding optimized fields.Many fiber-reinforced thin biological tissues [1-4] and synthetic sheets of responsive materials [5-9] deform into their desired shapes by a uniform anisotropic deformation. Upon actuation these effectively 2D materials expand by a constant factor along the fibers and shrink by a different factor along the perpendicular direction. While the length variations along these principal axes are constant across the material, the spatial variation in the direction of the principal axes allows this simple mode of uniform deformation to result in rich and intricate shapes.The fiber orientation is described by the fieldn(r) called the director. Together with the spatially constant shrinkage/expansion factors the director field uniquely defines the two-dimensional geometry that is obtained upon actuation. The actuation can be achieved through changing a variety of ambient conditions: temperature or light in liquid crystal elastomers [10][11][12], humidity for a variety of plants [1][2][3], and immersion in water in fiberreinforces hydrogels [5].Predicting the geometry obtained upon activation as a function of the prescribed director field has been recently resolved [13][14][15][16]. This geometry, captured by the two dimensional Riemannian metric, however, does not uniquely define the obtained surface. A given metric will correspond to a wide and typically continuous family of surfaces. The geometric rigidity that arises from Gaussian curvature sign variations as well as imposed boundary conditions serve to narrow down this wide family. Nonetheless, selecting the desired surface among all embeddings requires some control over the principal curvatures of the surface. Several techniques to partially control the principal curvatures of the thin sheet have been proposed and implemented [5,7,15], yet these techniques are system-specific and depend strongly on the elastic constitutive relations. In what follows we only address the universal problem associated with controlling the two-dimensional Riemannian geometry. The desired surface is an isometric embedding of the obtained solu-tion, however, other embeddings may exist. Selecting among these will be addressed in the future.Recen...