Version 1
: Received: 8 March 2020 / Approved: 10 March 2020 / Online: 10 March 2020 (03:19:02 CET)
Version 2
: Received: 25 May 2020 / Approved: 26 May 2020 / Online: 26 May 2020 (07:45:00 CEST)
How to cite:
Sun, B. On Symmetrical Deformation of Toroidal Shell with Circular Cross-Section. Preprints2020, 2020030156. https://doi.org/10.20944/preprints202003.0156.v1
Sun, B. On Symmetrical Deformation of Toroidal Shell with Circular Cross-Section. Preprints 2020, 2020030156. https://doi.org/10.20944/preprints202003.0156.v1
Sun, B. On Symmetrical Deformation of Toroidal Shell with Circular Cross-Section. Preprints2020, 2020030156. https://doi.org/10.20944/preprints202003.0156.v1
APA Style
Sun, B. (2020). On Symmetrical Deformation of Toroidal Shell with Circular Cross-Section. Preprints. https://doi.org/10.20944/preprints202003.0156.v1
Chicago/Turabian Style
Sun, B. 2020 "On Symmetrical Deformation of Toroidal Shell with Circular Cross-Section" Preprints. https://doi.org/10.20944/preprints202003.0156.v1
Abstract
By introducing a variable transformation $\xi=\frac{1}{2}(\sin \theta+1)$, the complicated deformation equation of toroidal shell is successfully transferred into a well-known equation, namely Heun's equation of ordinary differential equation, whose exact solution is obtained in terms of Heun's functions. The computation of the problem can be carried out by symbolic software that is able to with the Heun's function, such as Maple. The geometric study of the Gauss curvature shows that the internal portion of the toroidal shell has better bending capacity than the outer portion, which might be useful for the design of metamaterials with toroidal shell cells.
Keywords
toroidal shell; deformation; Gauss curvature; Heun's function; hypergeometric function; Maple
Subject
Physical Sciences, Applied Physics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.