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1. Geometric mechanics of ordered and disordered kirigami

   https://doi.org/10.1098/rspa.2022.0822  

   Chaudhary, G.; Niu, L.; Han, Q.; Lewicka, M.; Mahadevan, L. (June 2023, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   The presence of incomplete cuts in a thin planar sheet can dramatically alter its mechanical and geometrical response to loading, as the cuts allow the sheet to deform strongly in the third dimension, most beautifully demonstrated in kirigami art-forms. We use numerical experiments to characterize the geometric mechanics of kirigamized sheets as a function of the number, size and orientation of cuts. We show that the geometry of mechanically loaded sheets can be approximated as a composition of simple developable units: flats, cylinders, cones and compressed Elasticae. This geometric construction yields scaling laws for the mechanical response of the sheet in both the weak and strongly deformed limit. In the ultimately stretched limit, this further leads to a theorem on the nature and form of geodesics in an arbitrary kirigami pattern, consistent with observations and simulations. Finally, we show that by varying the shape and size of the geodesic in a kirigamized sheet, we can control the deployment trajectory of the sheet, and thence its functional properties as an exemplar of a tunable structure that can serve as a robotic gripper, a soft light window or the basis for a physically unclonable device. Overall our study of disordered kirigami sets the stage for controlling the shape and shielding the stresses in thin sheets using cuts. 

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2. Wrapping Liquids, Solids, and Gases in Thin Sheets

   https://doi.org/10.1146/annurev-conmatphys-031218-013533  

   Paulsen, Joseph D. (March 2019, Annual Review of Condensed Matter Physics)

   Many objects in nature and industry are wrapped in a thin sheet to enhance their chemical, mechanical, or optical properties. Similarly, there are a variety of methods for wrapping, from pressing a film onto a hard substrate to inflating a closed membrane, to spontaneously wrapping droplets using capillary forces. Each of these settings raises challenging nonlinear problems involving the geometry and mechanics of a thin sheet, often in the context of resolving a geometric incompatibility between two surfaces. Here, we review recent progress in this area, focusing on highly bendable films that are nonetheless hard to stretch, a class of materials that includes polymer films, metal foils, textiles, and graphene, as well as some biological materials. Significant attention is paid to two recent advances: a novel isometry that arises in the doubly-asymptotic limit of high flexibility and weak tensile forcing, and a simple geometric model for predicting the overall shape of an interfacial film while ignoring small-scale wrinkles, crumples, and folds. 

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3. Kirigami‐Inspired Inflatables with Programmable Shapes

   https://doi.org/10.1002/adma.202001863  

   Jin, Lishuai; Forte, Antonio_Elia; Deng, Bolei; Rafsanjani, Ahmad; Bertoldi, Katia (July 2020, Advanced Materials)

   Abstract 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. 

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4. Wetting and wrapping of a floating droplet by a thin elastic filament

   https://doi.org/10.1039/D0SM01863E  

   Prasath, S Ganga; Marthelot, Joel; Menon, Narayanan; Govindarajan, Rama (February 2021, Soft Matter)

   null (Ed.)

   We study the wetting of a thin elastic filament floating on a fluid surface by a droplet of another, immiscible fluid. This quasi-2D experimental system is the lower-dimensional counterpart of the wetting and wrapping of a droplet by an elastic sheet. The simplicity of this system allows us to study the phenomenology of partial wetting and wrapping of the droplet by measuring angles of contact as a function of the elasticity of the filament, the applied tension and the curvature of the droplet. We find that a purely geometric theory gives a good description of the mechanical equilibria in the system. The estimates of applied tension and tension in the filament obey an elastic version of the Young–Laplace–Dupré relation. However, curvatures close to the contact line are not captured by the geometric theory, possibly because of 3D effects at the contact line. We also find that when a highly-bendable filament completely wraps the droplet, there is continuity of curvature at the droplet-filament interface, leading to seamless wrapping as observed in a 3D droplet. 

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5. A Dual Scale Approach to Modeling Sub-Filter Shear-Induced Instabilities with a Vortex Sheet Method

   Goodrich, A.; Herrmann, M. (May 2021, ILASS-Americas 31st Annual Conference on Liquid Atomization and Spray Systems)

   null (Ed.)

   A method to predict sub-filter shear-induced velocities on a liquid-gas phase interface for use in a dual scale LES model is presented. The method reconstructs the sub-filter velocity field in the vicinity of the interface by introducing a vortex sheet at the interface. The vortex sheet is transported by an unsplit geometric volume and surface area advection scheme with a Piecewise Linear Interface Construction (PLIC) representation of the material interface. At each step and desired location the shear-induced velocities can be calculated by integrating the vortex sheet and other relevant quantities over the liquid-gas interface with the sub-grid velocity reconstruction limited to a small number of cells near the phase interface. The vortex sheet method is tested and compared against prior literature. 

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