Effects of a Change of Poisson’s Ratio Analyzed by Twinned Gradients: Illustrated by Simple Solutions of Boussinesq’s Problem of a Normal Force and Cerruti’s Problem of a Tangential Force Acting on the Surface of a Large Solid, and by a Simple Derivation of the Stresses in a Rotating Thick Disk

Abstract

Abstract Some problems of elasticity have a simple solution for a particular value of Poisson’s ratio. For example, Boussinesq’s problem of a normal force and Cerruti’s problem of a tangential force, acting on the plane surface of a semi-infinite solid, are solved when Poisson’s ratio is 1/2 by referring to Kelvin’s problem of a force at a point in the interior of an infinite solid. For, when Poisson’s ratio is 1/2, the solution of Kelvin’s problem can be stated in terms of one principal stress at each point, acting along the radial line from the point of the load; the other principal stresses are zero; and one half of the total force may be assigned to one half of the infinite solid. For other values of Poisson’s ratio terms must be added in the formulas for the displacements and stresses. The derivations that have been available are somewhat lengthy, especially for Cerruti’s problem. The difficulties are reduced by a simple analytical device, here called “the twinned gradient.” The displacement to be added by the change of Poisson’s ratio is stated as the gradient of a potential except that one of the components is replaced by its twin, an identical component in reversed direction. This device also lends itself to a simplification of the analysis of stresses in a rotating thick disk.