# Michell solution

The Michell solution is a general solution to the elasticity equations in polar coordinates (${\displaystyle r,\theta \,}$). The solution is such that the stress components are in the form of a Fourier series in ${\displaystyle \theta \,}$.

Michell[1] showed that the general solution can be expressed in terms of an Airy stress function of the form

{\displaystyle {\begin{aligned}\varphi (r,\theta )&=A_{0}~r^{2}+B_{0}~r^{2}~\ln(r)+C_{0}~\ln(r)\\&+\left(I_{0}~r^{2}+I_{1}~r^{2}~\ln(r)+I_{2}~\ln(r)+I_{3}~\right)\theta \\&+\left(A_{1}~r+B_{1}~r^{-1}+B_{1}^{'}~r~\theta +C_{1}~r^{3}+D_{1}~r~\ln(r)\right)\cos \theta \\&+\left(E_{1}~r+F_{1}~r^{-1}+F_{1}^{'}~r~\theta +G_{1}~r^{3}+H_{1}~r~\ln(r)\right)\sin \theta \\&+\sum _{n=2}^{\infty }\left(A_{n}~r^{n}+B_{n}~r^{-n}+C_{n}~r^{n+2}+D_{n}~r^{-n+2}\right)\cos(n\theta )\\&+\sum _{n=2}^{\infty }\left(E_{n}~r^{n}+F_{n}~r^{-n}+G_{n}~r^{n+2}+H_{n}~r^{-n+2}\right)\sin(n\theta )\end{aligned}}}

The terms ${\displaystyle A_{1}~r~\cos \theta \,}$ and ${\displaystyle E_{1}~r~\sin \theta \,}$ 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 (in cylindrical coordinates). A table of stress components is shown below.[2]

${\displaystyle \varphi }$ ${\displaystyle \sigma _{rr}\,}$ ${\displaystyle \sigma _{r\theta }\,}$ ${\displaystyle \sigma _{\theta \theta }\,}$
${\displaystyle r^{2}\,}$ ${\displaystyle 2}$ ${\displaystyle 0}$ ${\displaystyle 2}$
${\displaystyle r^{2}~\ln r}$ ${\displaystyle 2~\ln r+1}$ ${\displaystyle 0}$ ${\displaystyle 2~\ln r+3}$
${\displaystyle \ln r\,}$ ${\displaystyle r^{-2}\,}$ ${\displaystyle 0}$ ${\displaystyle -r^{-2}\,}$
${\displaystyle \theta \,}$ ${\displaystyle 0}$ ${\displaystyle r^{-2}\,}$ ${\displaystyle 0}$
${\displaystyle r^{3}~\cos \theta \,}$ ${\displaystyle 2~r~\cos \theta \,}$ ${\displaystyle 2~r~\sin \theta \,}$ ${\displaystyle 6~r~\cos \theta \,}$
${\displaystyle r\theta ~\cos \theta \,}$ ${\displaystyle -2~r^{-1}~\sin \theta \,}$ ${\displaystyle 0}$ ${\displaystyle 0}$
${\displaystyle r~\ln r~\cos \theta \,}$ ${\displaystyle r^{-1}~\cos \theta \,}$ ${\displaystyle r^{-1}~\sin \theta \,}$ ${\displaystyle r^{-1}~\cos \theta \,}$
${\displaystyle r^{-1}~\cos \theta \,}$ ${\displaystyle -2~r^{-3}~\cos \theta \,}$ ${\displaystyle -2~r^{-3}~\sin \theta \,}$ ${\displaystyle 2~r^{-3}~\cos \theta \,}$
${\displaystyle r^{3}~\sin \theta \,}$ ${\displaystyle 2~r~\sin \theta \,}$ ${\displaystyle -2~r~\cos \theta \,}$ ${\displaystyle 6~r~\sin \theta \,}$
${\displaystyle r\theta ~\sin \theta \,}$ ${\displaystyle 2~r^{-1}~\cos \theta \,}$ ${\displaystyle 0}$ ${\displaystyle 0}$
${\displaystyle r~\ln r~\sin \theta \,}$ ${\displaystyle r^{-1}~\sin \theta \,}$ ${\displaystyle -r^{-1}~\cos \theta \,}$ ${\displaystyle r^{-1}~\sin \theta \,}$
${\displaystyle r^{-1}~\sin \theta \,}$ ${\displaystyle -2~r^{-3}~\sin \theta \,}$ ${\displaystyle 2~r^{-3}~\cos \theta \,}$ ${\displaystyle 2~r^{-3}~\sin \theta \,}$
${\displaystyle r^{n+2}~\cos(n\theta )\,}$ ${\displaystyle -(n+1)(n-2)~r^{n}~\cos(n\theta )\,}$ ${\displaystyle n(n+1)~r^{n}~\sin(n\theta )\,}$ ${\displaystyle (n+1)(n+2)~r^{n}~\cos(n\theta \,)}$
${\displaystyle r^{-n+2}~\cos(n\theta )\,}$ ${\displaystyle -(n+2)(n-1)~r^{-n}~\cos(n\theta )\,}$ ${\displaystyle -n(n-1)~r^{-n}~\sin(n\theta )\,}$ ${\displaystyle (n-1)(n-2)~r^{-n}~\cos(n\theta )}$
${\displaystyle r^{n}~\cos(n\theta )\,}$ ${\displaystyle -n(n-1)~r^{n-2}~\cos(n\theta )\,}$ ${\displaystyle n(n-1)~r^{n-2}~\sin(n\theta )\,}$ ${\displaystyle n(n-1)~r^{n-2}~\cos(n\theta )\,}$
${\displaystyle r^{-n}~\cos(n\theta )\,}$ ${\displaystyle -n(n+1)~r^{-n-2}~\cos(n\theta )\,}$ ${\displaystyle -n(n+1)~r^{-n-2}~\sin(n\theta )\,}$ ${\displaystyle n(n+1)~r^{-n-2}~\cos(n\theta )\,}$
${\displaystyle r^{n+2}~\sin(n\theta )\,}$ ${\displaystyle -(n+1)(n-2)~r^{n}~\sin(n\theta )\,}$ ${\displaystyle -n(n+1)~r^{n}~\cos(n\theta )\,}$ ${\displaystyle (n+1)(n+2)~r^{n}~\sin(n\theta \,}$
${\displaystyle r^{-n+2}~\sin(n\theta )\,}$ ${\displaystyle -(n+2)(n-1)~r^{-n}~\sin(n\theta )\,}$ ${\displaystyle n(n-1)~r^{-n}~\cos(n\theta )\,}$ ${\displaystyle (n-1)(n-2)~r^{-n}~\sin(n\theta )}$
${\displaystyle r^{n}~\sin(n\theta )\,}$ ${\displaystyle -n(n-1)~r^{n-2}~\sin(n\theta )\,}$ ${\displaystyle -n(n-1)~r^{n-2}~\cos(n\theta )\,}$ ${\displaystyle n(n-1)~r^{n-2}~\sin(n\theta )\,}$
${\displaystyle r^{-n}~\sin(n\theta )\,}$ ${\displaystyle -n(n+1)~r^{-n-2}~\sin(n\theta )\,}$ ${\displaystyle n(n+1)~r^{-n-2}~\cos(n\theta )\,}$ ${\displaystyle n(n+1)~r^{-n-2}~\sin(n\theta )\,}$

## Displacement components

Displacements ${\displaystyle (u_{r},u_{\theta })}$ 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

${\displaystyle \kappa ={\begin{cases}3-4~\nu &{\rm {for~plane~strain}}\\{\cfrac {3-\nu }{1+\nu }}&{\rm {for~plane~stress}}\\\end{cases}}}$

where ${\displaystyle \nu }$ is the Poisson's ratio, and ${\displaystyle \mu }$ is the shear modulus.

${\displaystyle \varphi }$ ${\displaystyle 2~\mu ~u_{r}\,}$ ${\displaystyle 2~\mu ~u_{\theta }\,}$
${\displaystyle r^{2}\,}$ ${\displaystyle (\kappa -1)~r}$ ${\displaystyle 0}$
${\displaystyle r^{2}~\ln r}$ ${\displaystyle (\kappa -1)~r~\ln r-r}$ ${\displaystyle (\kappa +1)~r~\theta }$
${\displaystyle \ln r\,}$ ${\displaystyle -r^{-1}\,}$ ${\displaystyle 0}$
${\displaystyle \theta \,}$ ${\displaystyle 0}$ ${\displaystyle -r^{-1}\,}$
${\displaystyle r^{3}~\cos \theta \,}$ ${\displaystyle (\kappa -2)~r^{2}~\cos \theta \,}$ ${\displaystyle (\kappa +2)~r^{2}~\sin \theta \,}$
${\displaystyle r\theta ~\cos \theta \,}$ ${\displaystyle {\frac {1}{2}}[(\kappa -1)\theta ~\cos \theta +\{1-(\kappa +1)\ln r\}~\sin \theta ]\,}$ ${\displaystyle -{\frac {1}{2}}[(\kappa -1)\theta ~\sin \theta +\{1+(\kappa +1)\ln r\}~\cos \theta ]\,}$
${\displaystyle r~\ln r~\cos \theta \,}$ ${\displaystyle {\frac {1}{2}}[(\kappa +1)\theta ~\sin \theta -\{1-(\kappa -1)\ln r\}~\cos \theta ]\,}$ ${\displaystyle {\frac {1}{2}}[(\kappa +1)\theta ~\cos \theta -\{1+(\kappa -1)\ln r\}~\sin \theta ]\,}$
${\displaystyle r^{-1}~\cos \theta \,}$ ${\displaystyle r^{-2}~\cos \theta \,}$ ${\displaystyle r^{-2}~\sin \theta \,}$
${\displaystyle r^{3}~\sin \theta \,}$ ${\displaystyle (\kappa -2)~r^{2}~\sin \theta \,}$ ${\displaystyle -(\kappa +2)~r^{2}~\cos \theta \,}$
${\displaystyle r\theta ~\sin \theta \,}$ ${\displaystyle {\frac {1}{2}}[(\kappa -1)\theta ~\sin \theta -\{1-(\kappa +1)\ln r\}~\cos \theta ]\,}$ ${\displaystyle {\frac {1}{2}}[(\kappa -1)\theta ~\cos \theta -\{1+(\kappa +1)\ln r\}~\sin \theta ]\,}$
${\displaystyle r~\ln r~\sin \theta \,}$ ${\displaystyle -{\frac {1}{2}}[(\kappa +1)\theta ~\cos \theta +\{1-(\kappa -1)\ln r\}~\sin \theta ]\,}$ ${\displaystyle {\frac {1}{2}}[(\kappa +1)\theta ~\sin \theta +\{1+(\kappa -1)\ln r\}~\cos \theta ]\,}$
${\displaystyle r^{-1}~\sin \theta \,}$ ${\displaystyle r^{-2}~\sin \theta \,}$ ${\displaystyle -r^{-2}~\cos \theta \,}$
${\displaystyle r^{n+2}~\cos(n\theta )\,}$ ${\displaystyle (\kappa -n-1)~r^{n+1}~\cos(n\theta )\,}$ ${\displaystyle (\kappa +n+1)~r^{n+1}~\sin(n\theta )\,}$
${\displaystyle r^{-n+2}~\cos(n\theta )\,}$ ${\displaystyle (\kappa +n-1)~r^{-n+1}~\cos(n\theta )\,}$ ${\displaystyle -(\kappa -n+1)~r^{-n+1}~\sin(n\theta )\,}$
${\displaystyle r^{n}~\cos(n\theta )\,}$ ${\displaystyle -n~r^{n-1}~\cos(n\theta )\,}$ ${\displaystyle n~r^{n-1}~\sin(n\theta )\,}$
${\displaystyle r^{-n}~\cos(n\theta )\,}$ ${\displaystyle n~r^{-n-1}~\cos(n\theta )\,}$ ${\displaystyle n(~r^{-n-1}~\sin(n\theta )\,}$
${\displaystyle r^{n+2}~\sin(n\theta )\,}$ ${\displaystyle (\kappa -n-1)~r^{n+1}~\sin(n\theta )\,}$ ${\displaystyle -(\kappa +n+1)~r^{n+1}~\cos(n\theta )\,}$
${\displaystyle r^{-n+2}~\sin(n\theta )\,}$ ${\displaystyle (\kappa +n-1)~r^{-n+1}~\sin(n\theta )\,}$ ${\displaystyle (\kappa -n+1)~r^{-n+1}~\cos(n\theta )\,}$
${\displaystyle r^{n}~\sin(n\theta )\,}$ ${\displaystyle -n~r^{n-1}~\sin(n\theta )\,}$ ${\displaystyle -n~r^{n-1}~\cos(n\theta )\,}$
${\displaystyle r^{-n}~\sin(n\theta )\,}$ ${\displaystyle n~r^{-n-1}~\sin(n\theta )\,}$ ${\displaystyle -n~r^{-n-1}~\cos(n\theta )\,}$

Note that a rigid body displacement can be superposed on the Michell solution of the form

{\displaystyle {\begin{aligned}u_{r}&=A~\cos \theta +B~\sin \theta \\u_{\theta }&=-A~\sin \theta +B~\cos \theta +C~r\\\end{aligned}}}

to obtain an admissible displacement field.

## References

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.