Localization of deformation in the central hub of hub-and-spoke kirigami

A recent approach to the design of flexible electronic devices consists of cutting a two-dimensional sheet to form a central hub connected to several tapered `spokes', resembling the hub-and-spoke of a bicycle wheel. When radially compressed, the resulting cut sheet buckles out-of-plane forming a structure whose three-dimensional shape can be chosen by designing the tapering of the spokes. While the deformation of the spokes in this `hub-and-spoke' kirigami are approximately cylindrical (i.e.~zero Gaussian curvature and hence small elastic strain), this is not the case in the central hub. The central hub is deformed radially because of continuity with the spokes but, because of its own circular symmetry, it must develop Gaussian curvature, and hence strain. In this article we quantify this strain, focussing in particular on its magnitude and its location. We find that the strain is localized in a boundary layer near the edge of the hub region, whose size is controlled by the moment applied on it by the deformed spokes. We discuss the implications of our results for avoiding material failure in flexible-electronic devices.