Origami polyhedrons can realize polyhedral transformations and large deployed/folded ratios through the design of creases and slits, demonstrating broad application prospects in metamaterials and deployable structures, such as mechanical, acoustic, and electromagnetic metamaterials, as well as deployable antennas, solar arrays, and capsules. Cubes have the advantages of good symmetry and easy extension, making them highly promising for engineering applications. To meet the requirements of being sealed in the deployed configuration with a high deployed/folded ratio and easy deployment, it is necessary to design rigid and flat-foldable cubes with few slits and few degrees of freedom (DOFs). The existing techniques for designing creases and slits mostly rely on experience or enumeration, making it difficult to achieve polyhedrons that meet the requirements. There is an urgent need to develop systematic techniques for designing creases and slits.
        Recently, Prof. Yan Chen’s team at the School of Mechanical Engineering, Tianjin University, in collaboration with Senior Engineer Ming Li at Aerospace Systems Engineering Shanghai, proposed systematic techniques for designing creases and slits on origami cubes to achieve rigid and flat folding. Additionally, the researchers introduced a method of sealing slits using elastic membranes to construct bistable cubes and achieve customizable peak energy through parametric analysis.
        For rigid and flat-foldable kirigami cubes, the design of creases utilizes the 3D straight skeleton method. There are a total of 31 patterns, as illustrated in Fig. 1. Among these, there are 10 distinct patterns when considering rotational and mirror symmetries.

Fig. 1. The crease patterns of origami cubes designed by the 3D straight skeleton method.

        According to the bellows theorem, any closed polyhedron cannot be folded rigidly, necessitating the introduction of slits on the crease patterns of origami cubes designed by the straight-skeleton method. To achieve the design of origami polyhedrons with few DOFs and slits, the researchers proposed a novel technique for designing slits: Taking cube 2 in Fig. 1 as an example, by converting partial creases into prismatic joints based on the equivalence between prismatic joints and slits, the equivalent mechanism of cube 2 can be obtained (Fig. 2a). By formulating the relationships between the selection of movable prismatic joints and the DOF of the mechanism, the relationships between slit arrangements and DOF are established. Utilizing these relationships, the slit arrangements that meet the desired DOF can be determined directly without enumeration. For cube 2, a one-DOF scheme of movable prismatic joints is shown in Fig. 2b (where the red prismatic joints are movable and the black prismatic joints are fixed). Figure 2c displays a slit (red) arrangement transformed from it. The corresponding origami model is presented in Fig. 2d.

Fig. 2. The technique for arranging slits and its application. (a) The equivalent mechanism transformed from cube 2; (b) a one-DOF scheme of movable prismatic joints (where the red prismatic joints are movable and the black prismatic joints are fixed); (c) the slit (red) arrangement transformed from movable prismatic joints; (d) the corresponding origami model.

        The slits of the kirigami cubes designed using the proposed techniques are closed in both the fully deployed configuration and the fully folded configuration, while open during the folding process. The researchers utilize this characteristic to propose a method for constructing bistable cubes by sealing slits with elastic membranes. Through a hyperelastic model, the relationships between peak energy and geometric parameters, as well as the bistable parameter region, are determined (Fig. 3b). The results indicate that the peak energy of the bistable structure can be customized by modifying the geometric parameters. A set of bistable design parameters was chosen to fabricate a physical prototype, and its bistable characteristics were verified by theory and experiments, as illustrated in Figs. 3c and 3d.

Fig. 3. The geometric parameters of the bistable cube and the results of theoretical analysis and experiments. (a) the geometric parameters; (b) the relationships between peak energy and geometric parameters; (c) the experimental and theoretical force versus folding ratio curves; (d) the experimental and theoretical energy versus folding ratio curves.

        This work was published online on July 28, 2024, in the International Journal of Mechanical Sciences. The corresponding author of the paper is Prof. Yan Chen of Tianjin University, and the first author is Yuehao Zhang, a PhD student at Tianjin University. This research proposes a non-enumerative technique for arranging slits based on the equivalence between prismatic joints and slits, and successfully applies it to the design of rigid and flat-foldable cubes. Additionally, a bistable structure with customizable peak energy is created using a kirigami cube. The results enrich the design technique of kirigami structures, and the proposed bistable cube demonstrates promising engineering applications in the fields of deployable capsules and programmable mechanical metamaterials.

Zhang Y#, Zhang X#, Li M, Chen Y*. The rigid and flat-foldable kirigami cubes. International Journal of Mechanical Sciences, 2024, 282: 109605.
(https://doi.org/10.1016/j.ijmecsci.2024.109605)