The Mechanics Difference Between the Outer Torus and Inner Torus

Abstract

The formulation used by the most of studies on elastic torus are either Reissner’s mixed formulation or Novozhilov’s complex-form one; however, for vibration and some displacement boundary-related problem of torus, those formulations face a great challenge. It is highly demanded to have a displacement-type formulation for torus. In this article, we will carry on the first author’s previous work (Sun, 2010, “Closed-Form Solution of Axisymmetric Slender Elastic Toroidal Shells,” J. Eng. Mech., 136, pp. 1281–1288.), and with the help of our own maple codes, we are able to simulate some typical problems of torus. The numerical results are verified by both finite element analysis and H. Reissner’s formulation. Our investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio. The analysis of a torus must be done by using the bending theory of a shell instead of membrane theory of shells, and also reveal that the inner torus is stronger than outer torus due to their Gaussian curvature. One of the most interesting discovery is that the crowns of a torus, the turning point of the Gaussian curvature at ϕ = 0, π, are the line where the mechanics response of inner and outer torus is almost separated.