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Register a new userA theorem for the design of deployable kirigami tessellations with different topologies
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The concept of kirigami has been extensively utilized to design deployable structures and reconfigurable metamaterials. Despite heuristic utilization of classical kirigami patterns, the gap between complex kirigami tessellations and systematic design principles still needs to be filled. In this paper, we develop a unified design method for deployable quadrilateral kirigami tessellations perforated on flat sheets with different topologies. This method is based on the parameterization of kirigami patterns formulated as the solution of a linear equation system. The geometric constraints for the deployability of parameterized cutting patterns are given by a unified theorem covering different topologies of the flat sheets. As an application, we employ the design method to achieve desired shapes along the deployment path of kirigami tessellations, while preserving the topological characteristics of the flat sheets. Our approach introduces new perspectives for the topological design of kirigami-inspired structures and metamaterials.
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We present a theorem on the compatibility upon deployment of kirigami tessellations restricted on a spherical surface with patterned cuts forming freeform quadrilateral meshes. We show that the spherical kirigami tessellations have either one or two compatible states, i.e., there are at most two isolated strain-free configurations along the deployment path. The proof of the theorem is based on analyzing the number of roots of the compatibility condition, under which the kirigami pattern allows a piecewise isometric transformation between the undeployed and deployed configurations. As a degenerate case, the theorem further reveals that neutral equilibrium arises for planar quadrilateral kirigami tessellations if and only if the cuts form parallelogram voids. Our study provides new insights into the rational design of morphable structures based on Euclidean and non-Euclidean geometries.
Shape-morphing finds widespread utility, from the deployment of small stents and large solar sails to actuation and propulsion in soft robotics. Origami structures provide a template for shape-morphing, but rules for designing and folding the structures are challenging to integrate into broad and versatile design tools. Here, we develop a sequential two-stage optimization framework to approximate a general surface by a deployable origami structure. The optimization is performed over the space of all possible rigidly and flat-foldable quadrilateral mesh origami. So, the origami structures produced by our framework come with desirable engineering properties: they can be easily manufactured on a flat reference sheet, deployed to their target state by a controlled folding motion, then to a compact folded state in applications involving storage and portability. The attainable surfaces demonstrated include those with modest but diverse curvatures and unprecedented ones with sharp ridges. The framework provides not only a tool to design various deployable and retractable surfaces in engineering and architecture, but also a route to optimizing other properties and functionality.
Kirigami, the art of introducing cuts in thin sheets to enable articulation and deployment, has till recently been the domain of artists. With the realization that these structures form a novel class of mechanical metamaterials, there is increasing interest in using periodic tiling patterns as the basis for the space of designs. Here, we show that aperiodic quasicrystals can also serve as the basis for designing deployable kirigami structures and analyze their geometrical, topological and mechanical properties. Our work explores the interplay between geometry, topology and mechanics for the design of aperiodic kirigami patterns, thereby enriching our understanding of the effectiveness of kirigami cuts in metamaterial design.
Kirigami involves cutting a flat, thin sheet that allows it to morph from a closed, compact configuration into an open deployed structure via coordinated rotations of the internal tiles. By recognizing and generalizing the geometric constraints that enable this art form, we propose a design framework for compact reconfigurable kirigami patterns, which can morph from a closed and compact configuration into a deployed state conforming to any prescribed target shape, and subsequently be contracted into a different closed and compact configuration. We further establish a condition for producing kirigami patterns which are reconfigurable and rigid deployable allowing us to connect the compact states via a zero-energy family of deployed states. All together, our inverse design framework lays out a new path for the creation of shape-morphing material structures.
Kirigami, the art of paper cutting, has become a paradigm for mechanical metamaterials in recent years. The basic building blocks of any kirigami structures are repetitive deployable patterns that derive inspiration from geometric art forms and simple planar tilings. Here we complement these approaches by directly linking kirigami patterns to the symmetry associated with the set of seventeen repeating patterns that fully characterize the space of periodic tilings of the plane. We start by showing how to construct deployable kirigami patterns using any of the wallpaper groups, and then design symmetry-preserving cut patterns to achieve arbitrary size changes via deployment. We further prove that different symmetry changes can be achieved by controlling the shape and connectivity of the tiles and connect these results to the underlying kirigami-based lattice structures. All together, our work provides a systematic approach for creating a broad range of kirigami-based deployable structures with any prescribed size and symmetry properties.
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