Deterministic and stochastic control of kirigami topology

Topological Metamaterials

The topological properties of an object, associated with an integer called the topological invariant, are global features that cannot change continuously but only through abrupt variations, hence granting them intrinsic robustness. Engineered metamaterials (MMs) can be tailored to support highly nontrivial topological properties of their band structure, relative to their electronic, electromagnetic, acoustic and mechanical response, representing one of the major breakthroughs in physics over the past decade. Here, we review the foundations and the latest advances of topological photonic and phononic MMs, whose nontrivial wave interactions have become of great interest to a broad range of science disciplines, such as classical and quantum chemistry. We first introduce the basic concepts, including the notion of topological charge and geometric phase. We then discuss the topology of natural electronic materials, before reviewing their photonic/phononic topological MM analogues, including 2D topological MMs with and without time-reversal symmetry, Floquet topological insulators, 3D, higher-order, non-Hermitian and nonlinear topological MMs. We also discuss the topological aspects of scattering anomalies, chemical reactions and polaritons. This work aims at connecting the recent advances of topological concepts throughout a broad range of scientific areas and it highlights opportunities offered by topological MMs for the chemistry community and beyond.

Explosive rigidity percolation in kirigami

Controlling the connectivity and rigidity of kirigami, i.e. the process of cutting paper to deploy it into an articulated system, is critical in the manifestations of kirigami in art, science and technology, as it provides the resulting metamaterial with a range of mechanical and geometric properties. Here, we combine deterministic and stochastic approaches for the control of rigidity in kirigami using the power of k choices, an approach borrowed from the statistical mechanics of explosive percolation transitions. We show that several methods for rigidifying a kirigami system by incrementally changing either the connectivity or the rigidity of individual components allow us to control the nature of the explosive transition by a choice of selection rules. Our results suggest simple lessons for the design of mechanical metamaterials.

Engineering by Cuts: How Kirigami Principle Enables Unique Mechanical Properties and Functionalities

Kirigami, the ancient art of paper cutting, has evolved into a design and fabrication framework to engineer multi-functional materials and structures at vastly different scales. By slit cutting with carefully designed geometries, desirable mechanical behaviors-such as accurate shape morphing, tunable auxetics, super-stretchability, buckling, and multistability-can be imparted to otherwise inflexible sheet materials. In addition, the kirigami sheet provides a versatile platform for embedding different electronic and responsive components, opening up avenues for building the next generations of metamaterials, sensors, and soft robotics. These promising potentials of kirigami-based engineering have inspired vigorous research activities over the past few years, generating many academic publications. Therefore, this review aims to provide insights into the recent advance in this vibrant field. In particular, this paper offers the first comprehensive survey of unique mechanical properties induced by kirigami cutting, their underlying physical principles, and their corresponding applications. The synergies between design methodologies, mechanics modeling, advanced fabrication, and material science will continue to mature this promising discipline.

Modular representation and control of floppy networks

Geometric graph models of systems as diverse as proteins, DNA assemblies, architected materials and robot swarms are useful abstract representations of these objects that also unify ways to study their properties and control them in space and time. While much work has been done in the context of characterizing the behaviour of these networks close to critical points associated with bond and rigidity percolation, isostaticity, etc., much less is known about floppy, underconstrained networks that are far more common in nature and technology. Here, we combine geometric rigidity and algebraic sparsity to provide a framework for identifying the zero energy floppy modes via a representation that illuminates the underlying hierarchy and modularity of the network and thence the control of its nestedness and locality. Our framework allows us to demonstrate a range of applications of this approach that include robotic reaching tasks with motion primitives, and predicting the linear and nonlinear response of elastic networks based solely on infinitesimal rigidity and sparsity, which we test using physical experiments. Our approach is thus likely to be of use broadly in dissecting the geometrical properties of floppy networks using algebraic sparsity to optimize their function and performance.

Theorem on the Compatibility of Spherical Kirigami Tessellations

We present a theorem on the compatibility upon deployment of kirigami tessellations restricted on a spherical surface with patterned slits 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 theorem further reveals that the rigid-to-floppy transition from spherical to planar kirigami tessellations is possible if and only if the slits form parallelogram voids along with vanishing Gaussian curvature, which is also confirmed by an energy analysis and simulations. On the application side, we show a design of bistable spherical domelike structure based on the theorem. Our study provides new insights into the rational design of morphable structures based on Euclidean and non-Euclidean geometries.

Wallpaper group kirigami

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 17 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.

Control of connectivity and rigidity in prismatic assemblies

How can we manipulate the topological connectivity of a three-dimensional prismatic assembly to control the number of internal degrees of freedom and the number of connected components in it? To answer this question in a deterministic setting, we use ideas from elementary number theory to provide a hierarchical deterministic protocol for the control of rigidity and connectivity. We then show that it is possible to also use a stochastic protocol to achieve the same results via a percolation transition. Together, these approaches provide scale-independent algorithms for the cutting or gluing of three-dimensional prismatic assemblies to control their overall connectivity and rigidity.

Kirigami-Inspired Inflatables with Programmable Shapes

Kirigami, the Japanese art of paper cutting, has recently enabled the design of stretchable mechanical metamaterials that can be easily realized by embedding arrays of periodic cuts into an elastic sheet. Here, kirigami principles are exploited to design inflatables that can mimic target shapes upon pressurization. The system comprises a kirigami sheet embedded into an unstructured elastomeric membrane. First, it is shown that the inflated shape can be controlled by tuning the geometric parameters of the kirigami pattern. Then, by applying a simple optimization algorithm, the best parameters that enable the kirigami inflatables to transform into a family of target shapes at a given pressure are identified. Furthermore, thanks to the tessellated nature of the kirigami, it is shown that we can selectively manipulate the parameters of the single units to allow the reproduction of features at different scales and ultimately enable a more accurate mimicking of the target.