Michell solution

From Wikipedia, the free encyclopedia
Jump to: navigation, search

The Michell solution is a general solution to the elasticity equations in polar coordinates (). The solution is such that the stress components are in the form of a Fourier series in .

Michell[1] 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[edit]

The stress components can be obtained by substituting the Michell solution into the equations for stress in terms of the Airy stress function (in cylindrical coordinates). A table of stress components is shown below.[2]

Displacement components[edit]

Displacements 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.

References[edit]

  1. ^ Michell, J. H. (1899-04-01). "On the direct determination of stress in an elastic solid, with application to the theory of plates" (PDF). Proc. London Math. Soc. 31 (1): 100–124. doi:10.1112/plms/s1-31.1.100. Retrieved 2008-06-25. 
  2. ^ J. R. Barber, 2002, Elasticity: 2nd Edition, Kluwer Academic Publishers.

See also[edit]