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

Abstract

We consider a geometrically fully nonlinear variational model for thin elastic sheets that contain a single disclination. The free elastic energy contains the thickness h as a small parameter. We give an improvement of a recently proved energy scaling law, removing the next-to leading order terms in the lower bound. Then we prove the convergence of (almost-)minimizers of the free elastic energy towards the shape of a radially symmetric cone, up to Euclidean motions, weakly in the spaces W 2,2 (B1 \Bρ; R 3 ) for every 0 < ρ < 1, as the thickness h is sent to 0.

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.…”

Approximation of the Willmore energy by a discrete geometry model

“…A setting that at first sight seems well adapted to investigate the emergence of ridges would be an elastic sheet featuring two disclinations (metric defects). The case of one disclination has been investigated in [Olb16,Olb17,Olb18], where, aside from an energy scaling law, it has been shown that minimizers converge to a conical configuration as the thickness of the sheet approaches zero. In the case of two disclinations, one expects approximately conical shapes in the neighborhood of each of them, and the formation of a ridge between them.…”

Bending energy of buckled edge dislocations

“…A question of both fundamental and practical importance (e.g., the melting transition in 2D membranes [2,13]) is whether low-energy configurations of buckled dislocations remain finite as R tends to infinity. A natural reference system is that of a disclination, which may also be embedded isometrically in 3D Euclidean space, however, with a bending energy diverging logarithmically with the R; see [14,15] for recent rigorous analyses departing from 3D models. Heuristic arguments have suggested that in dislocations, which are bound pairs of disclinations of opposite signs, the logarithmic contributions may cancel out giving rise to an energy bound independent of R. Numerical simulations were performed in [3], supporting these heuristics.…”