Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Small Symmetrical Deformation of Thin Torus with Circular Cross-Section

Version 1 : Received: 15 December 2020 / Approved: 17 December 2020 / Online: 17 December 2020 (09:07:55 CET)

How to cite: Sun, B. Small Symmetrical Deformation of Thin Torus with Circular Cross-Section. Preprints 2020, 2020120420 (doi: 10.20944/preprints202012.0420.v1). Sun, B. Small Symmetrical Deformation of Thin Torus with Circular Cross-Section. Preprints 2020, 2020120420 (doi: 10.20944/preprints202012.0420.v1).

## Abstract

By introducing a variable transformation $\xi=\frac{1}{2}(\sin \theta+1)$, a complex-form ordinary differential equation (ODE) for the small symmetrical deformation of an elastic torus is successfully transformed into the well-known Heun's ODE, whose exact solution is obtained in terms of Heun's functions. To overcome the computational difficulties of the complex-form ODE in dealing with boundary conditions, a real-form ODE system is proposed. A general code of numerical solution of the real-form ODE is written by using Maple. Some numerical studies are carried out and verified by finite element analysis. Our investigations show that the mechanics of an elastic torus are sensitive to the radius ratio, and suggest that the analysis of a torus should be done by using the bending theory of a shell. A general Maple code is provided as essential part of this paper.

## Subject Areas

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

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