Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# On Symmetrical Deformation of Toroidal Shell with Circular Cross-Section

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. Preprints 2020, 2020030156 (doi: 10.20944/preprints202003.0156.v1). Sun, B. On Symmetrical Deformation of Toroidal Shell with Circular Cross-Section. Preprints 2020, 2020030156 (doi: 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.

## Subject Areas

toroidal shell; deformation; Gauss curvature; Heun's function; hypergeometric function; Maple

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