On the mechanical origin of a doubly curved structure of a cellular meta-plate

Abstract

Transforming a flat physical sheet into a three-dimensional curved surface is a fundamental step for constructing functional shape-shifting surfaces. Among the methodologies relying on the emergent concepts like origami, kirigami and knitted fabric, a traditional cellular solid based meta-plate provides a versatile tool for creating a doubly curved surface by a planar bending actuation. By combining digital fabrication, physical experiment, finite element simulation (FES) and linear elasticity theory, we demonstrate how such a cellular “meta”-plate can morph into a doubly-curved shape. Generalizing the classical Lamb’s theory for the anticlastic effect in a thin elastic plate to our cellular meta-plate, we develop a scaling law for the crossover length below which the doubly-curved surface appears as b_c～b_Lamb/√ρ, where b_Lamb～√Rw, with the radius of curvature R and the plate thickness w, and ρ is the relative density of a given cellular geometry. The prediction is verified by our experiment and FES. The proposed framework is generic, highlighting the fundamental physical aspects of the mechanics of such shape-morphing surfaces, in potential application to a broad of meta-sheets.