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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:msub><a:mi>b</a:mi><a:mi>c</a:mi></a:msub><a:mo>∼</a:mo><a:msub><a:mi>b</a:mi><a:mi>Lamb</a:mi></a:msub><a:mo>/</a:mo><a:msqrt><a:mi>ρ</a:mi></a:msqrt></a:mrow></a:math>, where <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:msub><b:mi>b</b:mi><b:mi>Lamb</b:mi></b:msub><b:mo>∼</b:mo><b:msqrt><b:mrow><b:mi>R</b:mi><b:mi>w</b:mi></b:mrow></b:msqrt></b:mrow></b:math>, with <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>R</c:mi></c:math> being the externally imposed radius of curvature, <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>w</d:mi></d:math> being the plate thickness, and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>ρ</e:mi></e:math> 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. Published by the American Physical Society 2024