Torsion of an elastic non-homogeneous layer by two circular discs

Abstract

Two circular discs of different radii on the opposite faces of an infinite, non-homogeneous elastic layer, whose rigidity is assumed to vary with two cylindrical coordinates r, z by a power law (% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiabeY7aTjabg2da9iabeY7aTnaaBaaaleaa% cqaHXoqycaGGSaaabeaakiabek7aInaaCaaaleqabaGaeqiXdq3aaW% baaWqabeaacqaHXoqyaaWccaWG6bWaaWbaaWqabeaacqaHYoGyaaaa% aaaa!4CAB!\[\mu = \mu _{\alpha ,} \beta ^{\tau ^\alpha z^\beta } \]), are forced to rotate through two different angles of rotation. The rest of each surface is kept stress free. Using the Hankel integral transform, this problem is shown to lead to two pairs of dual integral equations, the solution of which is governed by two simultaneous Fredholm integral equations of the second kind. The latter may be solved either numerically or by iteration (in the case of sufficiently large values of the layer's thickness compared to the maximum of the radii of the circles and for β=0). The solutions for some particular cases previously investigated are recovered by assigning specific numerical values to physical and geometrical parameters.

Expressions for some quantities of physical importance, such as the torques applied on the two surfaces and the stress intensity factors, are obtained for the two surfaces and some numerical values are displayed graphically.