Energy Scaling Law for the Regular Cone

Abstract

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 ∞ (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.

Energy Scaling Law for a Single Disclination in a Thin Elastic Sheet

“…In the recent work [26], we suggested an ansatz how to prove lower bounds for the free elastic energy of a single disclination without the assumption of radial symmetry. We viewed the elastic sheet in the deformed configuration as an immersed Riemannian manifold, and focused our analysis on intrinsically defined objects, such as the metric and Gauss curvature of the manifold.…”

The shape of low energy configurations of a thin elastic sheet with a single disclination

“…The variational problems considered in these references however are of a very special kind: The constraints on the shape of the elastic sheet are quite restrictive, and the lower bounds use these constraints in an essential way (see [32] for a detailed discussion). This is not the case for our setting, whence our method of proof, which we have developed in [31,32] and which we refine here, is completely different. 3…”

“…Setup and previous work. The present article continues a program [28,31,32] to explore thin elastic sheets with a single disclination from the variational point of view. The free energy that we consider consists of two parts: First, the non-convex membrane energy, that penalizes the difference between the metric that is induced by the deformation and the reference metric, which is the metric of the (singular) cone.…”

“…The idea underlying the recent proofs of ansatz-free lower bounds [31,32] is to control the Gauss curvature (or a linearization thereof) by interpolation between the membrane and the bending term energy. The control over the Gauss curvature allows for a certain control over the Gauss map (or the deformation gradient).…”

On a boundary value problem for conically deformed thin elastic sheets

“…The boundary conditions can be used to show that the Gauss curvature is bounded from below in a certain space "in between" in the sense of interpolation. In the recent paper [23], we show that for the setting of [21,22], it is not necessary to use interpolation, and lower bounds for the bending energy can be obtained by using the control over the membrane energy alone. The present setting with a flat 2 reference metric however defines an interpolation type problem for the Gauss curvature, and we hope that this approach can also yield results for similar variational problems.…”