Programming quadric metasurfaces via infinitesimal origami maps of monohedral hexagonal tessellations: Part I

Abstract

The control of the shape of complex metasurfaces is a challenging task often addressed in the literature. This work presents a class of tessellated plates able to deform into surfaces of preprogrammed shape upon activation by any flexural load and that can be controlled by a single actuator. Quadric metasurfaces are obtained from infinitesimal origami maps of monohedral hexagonal tessellations of the plane, that is pavings in which all tiles are congruent to each other. Monohedral tessellated portions can be joined together to obtain more complex shapes, which can be locally synclastic or anticlastic and can have a certain roughness. We broaden previous work by providing a complete characterization of all the three known types of monohedral tessellations composed by irregular hexagons. The proposed two-dimensional structures may have applications in prosthetics, tissue engineering, wearable devices, energy harvesting devices, tunable focus mirrors and adaptive facades. The study is divided in two parts. In Part I, after introducing the discrete kinematics of tessellated plates, it is proved analytically that essentially each type of monohedral hexagonal tessellation possesses only one deformation mode. Afterwards, several numerical examples are provided to demonstrate the variety of achievable surface shapes. In Part II, first the metasurfaces corresponding to assigned tile geometries are given a continuum description, which establishes that the continuous interpolant is always a quadric. Then, experimental results on fabrication, assembly and surface accuracy are reported.

Programming quadric metasurfaces via infinitesimal origami maps of monohedral hexagonal tessellations: Part II

“…, N, and ∇u(x) is a skew-symmetric tensor for almost every x ∈ ∪ N i=1 τ (i) . If the tile undergoes an infinitesimal origami map, the displacement u (1) (x) ∈ R 3 of the point x ∈ τ (1) is given by u (1) (x) = u (1) (x (1) 0 ) + W (1) [x − x (1) 0 ], (2.1) where x (1) 0 is a generic point of the tile, W (1) is a skew-symmetric tensor. If two tiles are connected, we proved that the following condition holds: W (1) − W (2) = ϕ {1,2} e 3 ⊗ n (1,2) , (2.2) where ϕ {i,j} denotes the relative rotation between the tiles τ (i) and τ (i) , and n (i,j) the normal that goes from τ (i) to τ (j) .…”

“…If three tiles are considered, by applying (2.2) three times we have that (1,2) , (2,3) , W (3) − W (1) = ϕ {3,1} e 3 ⊗ n (3,1) .…”

“…In part I, we have proved that ϕ {2,3} = −ϕ {3,1} sin(α (3,1) − α (1,2) ) sin(α (2,3) − α (1,2) ) and ϕ {1,2} = ϕ {3,1} sin(α (3,1) − α (2,3) ) sin(α (2,3) − α (1,2) ) .…”

“…In the companion paper, we have shown that assuming the tessellation to have degree three (every interior vertex of the tessellation is in common to three tiles), and x 1 t (2,3) t (2,3) t (1,2) t (1,2) t (3,1) n (2,3) n (1,2) t (3,1) n (3,1) t (1) t (3) t (2) a (1,2) a (2,3) a (3,1) Figure 1. Geometry of three adjacent tiles.…”

“…We refer the reader to Part I of this report [1] to find the background information, the motivation and the goal of our investigation, together with the related literature survey (table 1).…”