The Mechanics Difference Between The Outer Torus and Inner Torus

The formulation used by the most of studies on elastic torus are either Reissner 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 paper, we will carry on author's previous work [B.H. Sun, Closed-form solution of axisymmetric slender elastic toroidal shells. J. of Engineering Mechanics, 136 (2010) 1281-1288.], and with the help of our own maple code, 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, and suggest that the analysis of a torus should be done by using the bending theory of a shell, and also reveal that the inner torus is stronger than outer torus due to the property of their Gaussian curvature. One of the most interesting discovery is that the crowns of a torus are the turning point of the Gaussion curvature at ϕ = 0,  π, where the mechanics response of inner and outer torus is almost separated.