Abstract
Origami-inspired folding methods present novel pathways to fabricate three-dimensional (3D) structures from 2D sheets. A key advantage of this approach is that planar printing and patterning processes could be used prior to folding, affording enhanced surface functionality to the folded structures. This is particularly useful for 3D lattices, possessing very large internal surface areas. While folding polyhedral strut-based lattices has already been demonstrated, more complex, curved sheet-based lattices have not yet been folded due to inherent developability constraints of conventional origami. Here, a novel folding strategy is presented to fold flat sheets into topologically complex cellular materials based on triply periodic minimal surfaces (TPMS), which are attractive geometries for many applications. The approach differs from traditional origami by employing material stretching to accommodate non-developability. Our method leverages the inherent hyperbolic symmetries of TPMS to assemble complex 3D structures from a net of self-foldable patches. We also demonstrate that attaching 3D-printed foldable frames to pre-strained elastomer sheets enables self-folding and self-guided minimal surface shape adaption upon release of the pre-strain. This approach effectively bridges the Euclidean nature of origami with the hyperbolic nature of TPMS, offering novel avenues in the 2D-to-3D fabrication paradigm and the design of architected materials with enhanced functionality.
Highlights
Stochastic sheet-based micro-architectures are ubiquitous in engineering and natural materials and appear in the form of foams, sponges, bone tissue, or at the interface of phase-separated materials [1]
Periodic minimal surfaces belong to the realm of hyperbolic geometry and arise from symmetry operations on fundamental patches
The resulting triangular tiling, with angles /2, /4, and /6, is not compatible with the Euclidean plane E2, but is a tiling of the hyperbolic plane H2, as seen in the conformal Poincaré disk model (Fig. 1a). This illustrates the interesting feature that a portion of H2 can be embedded in 3D Euclidean space E3 by wrapping it onto the periodic minimal surface, analogous to embedding E2 in E3 by wrapping it onto a cylinder [24,25]
Summary
Stochastic sheet-based micro-architectures are ubiquitous in engineering and natural materials and appear in the form of foams, sponges, bone tissue, or at the interface of phase-separated materials [1]. Their periodic counterparts, being more tractable to study, have received widespread attention too, especially geometries based on triply periodic minimal surfaces (TPMS). Periodic minimal surfaces form a special class of minimal surfaces that are bicontinuous and periodic in three directions, they extend infinitely and divide space into two continuous, intertwined labyrinths [1]. The unique structure-property relationships offered by TPMS have con-
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