General stress functions in polar coordinates

Michell solution

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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

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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 [from J. R. Barber (2002) [2]].

       
       
       
       
       
       
       
       
       
       
       
       
       
       
       
       
       
       
       
       
       

Displacement components

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Displacements can be obtained from the Michell solution by using the stress-strain and strain-displacement relations. A table of displacement components corresponding the the terms in the Airy stress function for the Michell solution is given below. In this table

 
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     

Note that you can superpose a rigid body displacement on the Michell solution of the form

 

to obtain an admissible displacement field.

References

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  1. J. H. Michell, 1899, On the direct determination of stress in an elastic solid, with application to the theory of plates, Proceedings of the London Mathematical Society, vol. 31, pages 100-124.
  2. J. R. Barber, 2002, Elasticity: 2nd Edition, Kluwer Academic Publishers.
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Introduction to Elasticity