Origin of anticlastic curvature in a cellular metaplate

Generating a three-dimensional curved surface from flat sheets using simple mechanical actuation is a crucial step in developing functional shape-shifting materials. Among the methodologies relying on emergent concepts, the traditional cellular solid-based plate, with its highly porous structures, provides a versatile tool for creating doubly curved surfaces through planar bending actuation. Leveraging a combination of digital fabrication, physical experiments, finite element simulation (FES), and linear elasticity theory, we demonstrate how such a cellular metaplate can exhibit doubly curved shapes. By extending Lamb's classical theory of the anticlastic effect in thin elastic plates to our cellular metaplate, we develop a scaling law for the crossover length below which the doubly curved surface appears as b_{c}∼b_{Lamb}/sqrt[ρ], where b_{Lamb}∼sqrt[Rw], with R being the externally imposed radius of curvature, w being the plate thickness, and ρ being the relative density of a given cellular geometry. The prediction is validated by our experiment and FES. The proposed framework is versatile and emphasizes the fundamental physical principles governing the mechanics of shape-morphing surfaces.