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

Abstract

Abstract. We consider a single disclination in a thin elastic sheet of thickness h. We prove ansatz-free lower bounds for the free elastic energy in three different settings: First, for a geometrically fully non-linear plate model, second, for three-dimensional nonlinear elasticity, and third, for the Föppl-von Kármán plate theory. The lower bounds in the first and third result are optimal in the sense that we find upper bounds that are identical to the respective lower bounds in the leading order of h.

On the Role of Curvature in the Elastic Energy of Non-Euclidean Thin Bodies

“…[SRS07,ESK09b,ESK09a,ESK11]. Other scalings can be obtained due to external forces [BK14], or singular metrics [Olb17,COT17], but these are further away from the context of this paper.…”

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.…”

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

“…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).…”

“…This information in turn yields lower bounds for the bending energy, using an inequality of Sobolev/isoperimetric type. For the corresponding result from [32] see equation (2) below. This lower bound does not quite achieve the conjectured scaling behavior, in that there exist next-to leading order terms O(h 2 log log 1 h ) which are not present in the upper bound.…”