Energy Scaling Law for the Regular Cone

We consider a thin elastic sheet in the shape of a disk whose reference metric is that of a singular cone. That is, the reference metric is flat away from the center and has a defect there. We define a geometrically fully nonlinear free elastic energy and investigate the scaling behavior of this energy as the thickness h tends to 0. We work with two simplifying assumptions: Firstly, we think of the deformed sheet as an immersed 2-dimensional Riemannian manifold in Euclidean 3-space and assume that the exponential map at the origin (the center of the sheet) supplies a coordinate chart for the whole manifold. Secondly, the energy functional penalizes the difference between the induced metric and the reference metric in \(L^\infty \) (instead of, as is usual, in \(L^2\)). Under these assumptions, we show that the elastic energy per unit thickness of the regular cone in the leading order of h is given by \(C^*h^2|\log h|\), where the value of \(C^*\) is given explicitly.