Michell solution


The Michell solution is a general solution to the elasticity equations in polar coordinates developed by J. H. Michell. The solution is such that the stress components are in the form of a Fourier series in .
Michell showed that the general solution can be expressed in terms of an Airy stress function of the form
The terms and define a trivial null state of stress and are ignored.

Stress components

The stress components can be obtained by substituting the Michell solution into the equations for stress in terms of the Airy stress function. A table of stress components is shown below.

Displacement components

can be obtained from the Michell solution by using the stress-strain and strain-displacement relations. A table of displacement components corresponding the terms in the Airy stress function for the Michell solution is given below. In this table
where is the Poisson's ratio, and is the shear modulus.
Note that a rigid body displacement can be superposed on the Michell solution of the form
to obtain an admissible displacement field.