Theoretically, the curvature of a boundary can be harnessed to tune 3D curved shapes based on the classical Gauss-Bonnet theorem in differential geometry 37, which correlates the Gaussian curvature and the geodesic curvature along the boundary (i.e., the projection of boundary curvature). In addition, how to utilize the 3D curved shapes in kirigami sheets for functionalities remains largely unexplored 34, 35, 36. However, it often requires programming intricate cut patterns and arrangements of non-periodic cut units, making the inverse design and optimization for target 3D shapes complicated and challenging 35, 36. The local heterogeneous deformation among non-periodic tessellated cut units induces global out-of-plane buckling of the 2D kirigami sheets, thus, resulting in the formation of different 3D curved shapes 34, 35. Recent work shows that starting from a kirigami sheet or shell, 3D shapes with non-zero Gaussian curvature can be generated by utilizing either forward designs of non-periodic patterns of square cuts/cutouts 34, 36 or inverse designs of tessellation of non-uniform square cuts patterning with irregular polygon cut units 35. There are few studies on the shape shifting from a kirigami sheet to 3D shapes with intrinsic curvature 34, 35, 36. The cuts impart new properties such as auxeticity 9, 11, stretchability 8, 10, 15, 22, 23, 24, conformability 8, multistability 25, and optical chirality 26, which have found broad applications in mechanical metamaterials 11, 15, 27, 28, stretchable devices 8, 10, 23, 29, 30, 3D mechanical self-assembly 31, tunable adhesion 32, and soft machines 17, 18, 33.ĭespite the advance, most studies focus on the local buckling of cut units in a thin sheet patterned with arrays of parallel slits or networked triangular or square cuts etc 8, 9, 10, 11, 12, 13, 15, 17, 18, generating quasi-3D pop-up structures without global curvatures. Starting from a thin sheet with patterned cuts, it can morph into varieties of 2D and pop-up 3D structures via rigid rotation mechanism 20 and/or out-of-plane buckling 21. Compared to continuous thin sheets, the kirigami sheet enables more freedom and flexibility in shape shifting through local or global deformation between cut units 17. Cuts divide the original continuous thin sheets into discretized cut units without sacrificing the global structural integrity. Kirigami, the traditional art of paper cutting, has recently emerged as a new promising approach for creating shape morphing structures and materials 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19. These shape-programmable materials are attractive for broad applications in programmable machines and robots 3, 4, functional biomedical devices 5, and four-dimensional (4D) printing 6, 7. Programmable shape shifting in different materials and structures was realized at all scales utilizing folding, bending, and buckling 2. This study opens a new avenue to encode boundary curvatures for shape-programing materials with potential applications in soft robotics and wearable devices.ĭesigning shape-programming materials from 2D thin sheets to 3D shapes has attracted broad and increasing interest in the past decades due to their novel materials properties imparted by geometrical shapes 1. Leveraging this strategy, we demonstrate its potential applications as a universal and nondestructive gripper for delicate objects, including live fish, raw egg yolk, and a human hair, as well as dynamically conformable heaters for human knees. The strategy largely simplifies the inverse design. Motivated by the Gauss-Bonnet theorem that correlates the geodesic curvature along the boundary with the Gaussian curvature, here, we exploit programming the curvature of cut boundaries rather than the complex cut patterns in kirigami sheets for target 3D curved morphologies through both forward and inverse designs. Existing kirigami designs for target 3D curved shapes rely on intricate cut patterns in thin sheets, making the inverse design challenging. Kirigami, a traditional paper cutting art, offers a promising strategy for 2D-to-3D shape morphing through cut-guided deformation.
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