Geometry-induced rigidity in elastic torus from circular to oblique elliptic cross-section

B.H Sun

doi:10.1016/j.ijnonlinmec.2021.103754

Abstract

For a given material, different shapes correspond to different rigidities. In this paper, the radii of the oblique elliptic torus are formulated, a nonlinear displacement formulation is presented and numerical simulations are carried out for circular, normal elliptic and oblique tori, respectively. Our investigation shows that both the deformation and the stress response of an elastic torus are sensitive to the radius ratio, and indicate that the analysis of a torus should be done by using the bending theory of shells rather than membrane theory. Numerical study demonstrates that the inner region of the torus is stiffer than the outer region due to the Gauss curvature. The study also shows that an elastic torus deforms in a very specific manner, as the strain and stress concentration in two very narrow regions around the top and bottom crowns. The desired rigidity can be achieved by adjusting the ratio of minor and major radii and the oblique angle.

Highlights

The results indicate that all quantities such as bending moments, surface forces, shear force, and displacement are strongly effected by oblique angle
To verify our formulation, we wrote a computational code in Maple and carried out some numerical simulations
Our investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio and dramatically vary with the meridian angle, the analysis of a torus should be done by the bending theory of a shell rather than membrane theory of shell

Summary

INTRODUCTION

Many natural and man-made objects take the shape of shells to gain better rigidity Both Lazarus et al [1] Vella et al [2] studied egg-like shell’s deformation and geometry-induced rigidity of non-spherical pressurized elastic shells( ellipsoidal and cylindrical). When linear problem of the torus was first studied, high-order and complicated governing equations of a torus under symmetric loads were reduced to lower-order, ordinary differential equation (ODE) by Hans Reissner (1912)[29] when he was a professor at ETH in Switzerland. His colleague at ETH, Meissner (1915) [30] derived a complex-form equations for the shell of revolution.

NONLINEAR DISPLACEMENT FORMULATION OF ELASTIC ELLIPTIC TORUS

A normal elliptic torus

An oblique elliptic torus

CONCLUSIONS