Energy scaling law for the regular cone
(2016) Journal of Nonlinear Science — Vol. 26, p. 287-314 (2016)
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- Abstract
- We consider a thin elastic sheet in the shape of a disk whose reference metric is that of a singular cone. I.e., 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 L2). 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∗h2|logh|, where the value of C∗ is given explicitly.
- Affiliations
Universität Bonn
Citations
Olbermann, H. (2016). Energy scaling law for the regular cone. Journal of Nonlinear Science, 26, 287-314. https://doi.org/10.1007/s00332-015-9275-4 (Original work published 2016)