# Michell solution

In continuum mechanics, the Michell solution is a general solution to the elasticity equations in polar coordinates ($r,\theta$) developed by John Henry Michell in 1899. The solution is such that the stress components are in the form of a Fourier series in $\theta$.
Michell showed that the general solution can be expressed in terms of an Airy stress function of the form ${\begin{aligned}\varphi (r,\theta )&=A_{0}r^{2}+B_{0}r^{2}\ln(r)+C_{0}\ln(r)\\&+\left(I_{0}r^{2}+I_{1}r^{2}\ln(r)+I_{2}\ln(r)+I_{3}\right)\theta \\&+\left(A_{1}r+B_{1}r^{-1}+B_{1}'r\theta +C_{1}r^{3}+D_{1}r\ln(r)\right)\cos \theta \\&+\left(E_{1}r+F_{1}r^{-1}+F_{1}'r\theta +G_{1}r^{3}+H_{1}r\ln(r)\right)\sin \theta \\&+\sum _{n=2}^{\infty }\left(A_{n}r^{n}+B_{n}r^{-n}+C_{n}r^{n+2}+D_{n}r^{-n+2}\right)\cos(n\theta )\\&+\sum _{n=2}^{\infty }\left(E_{n}r^{n}+F_{n}r^{-n}+G_{n}r^{n+2}+H_{n}r^{-n+2}\right)\sin(n\theta )\end{aligned}}$ The terms $A_{1}r\cos \theta$ and $E_{1}r\sin \theta$ 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.
| $\varphi$ | $\sigma _{rr}\,$ | $\sigma _{r\theta }\,$ | $\sigma _{\theta \theta }\,$ |
| $r^{2}\,$ | $2$ | $0$ | $2$ |
| $r^{2}~\ln r$ | $2~\ln r+1$ | $0$ | $2~\ln r+3$ |
| $\ln r\,$ | $r^{-2}\,$ | $0$ | $-r^{-2}\,$ |
| $\theta \,$ | $0$ | $r^{-2}\,$ | $0$ |
| $r^{3}~\cos \theta \,$ | $2~r~\cos \theta \,$ | $2~r~\sin \theta \,$ | $6~r~\cos \theta \,$ |
| $r\theta ~\cos \theta \,$ | $-2~r^{-1}~\sin \theta \,$ | $0$ | $0$ |
| $r~\ln r~\cos \theta \,$ | $r^{-1}~\cos \theta \,$ | $r^{-1}~\sin \theta \,$ | $r^{-1}~\cos \theta \,$ |
| $r^{-1}~\cos \theta \,$ | $-2~r^{-3}~\cos \theta \,$ | $-2~r^{-3}~\sin \theta \,$ | $2~r^{-3}~\cos \theta \,$ |
| $r^{3}~\sin \theta \,$ | $2~r~\sin \theta \,$ | $-2~r~\cos \theta \,$ | $6~r~\sin \theta \,$ |
| $r\theta ~\sin \theta \,$ | $2~r^{-1}~\cos \theta \,$ | $0$ | $0$ |
| $r~\ln r~\sin \theta \,$ | $r^{-1}~\sin \theta \,$ | $-r^{-1}~\cos \theta \,$ | $r^{-1}~\sin \theta \,$ |
| $r^{-1}~\sin \theta \,$ | $-2~r^{-3}~\sin \theta \,$ | $2~r^{-3}~\cos \theta \,$ | $2~r^{-3}~\sin \theta \,$ |
| $r^{n+2}~\cos(n\theta )\,$ | $-(n+1)(n-2)~r^{n}~\cos(n\theta )\,$ | $n(n+1)~r^{n}~\sin(n\theta )\,$ | $(n+1)(n+2)~r^{n}~\cos(n\theta )\,$ |
| $r^{-n+2}~\cos(n\theta )\,$ | $-(n+2)(n-1)~r^{-n}~\cos(n\theta )\,$ | $-n(n-1)~r^{-n}~\sin(n\theta )\,$ | $(n-1)(n-2)~r^{-n}~\cos(n\theta )$ |
| $r^{n}~\cos(n\theta )\,$ | $-n(n-1)~r^{n-2}~\cos(n\theta )\,$ | $n(n-1)~r^{n-2}~\sin(n\theta )\,$ | $n(n-1)~r^{n-2}~\cos(n\theta )\,$ |
| $r^{-n}~\cos(n\theta )\,$ | $-n(n+1)~r^{-n-2}~\cos(n\theta )\,$ | $-n(n+1)~r^{-n-2}~\sin(n\theta )\,$ | $n(n+1)~r^{-n-2}~\cos(n\theta )\,$ |
| $r^{n+2}~\sin(n\theta )\,$ | $-(n+1)(n-2)~r^{n}~\sin(n\theta )\,$ | $-n(n+1)~r^{n}~\cos(n\theta )\,$ | $(n+1)(n+2)~r^{n}~\sin(n\theta )\,$ |
| $r^{-n+2}~\sin(n\theta )\,$ | $-(n+2)(n-1)~r^{-n}~\sin(n\theta )\,$ | $n(n-1)~r^{-n}~\cos(n\theta )\,$ | $(n-1)(n-2)~r^{-n}~\sin(n\theta )\,$ |
| $r^{n}~\sin(n\theta )\,$ | $-n(n-1)~r^{n-2}~\sin(n\theta )\,$ | $-n(n-1)~r^{n-2}~\cos(n\theta )\,$ | $n(n-1)~r^{n-2}~\sin(n\theta )\,$ |
| $r^{-n}~\sin(n\theta )\,$ | $-n(n+1)~r^{-n-2}~\sin(n\theta )\,$ | $n(n+1)~r^{-n-2}~\cos(n\theta )\,$ | $n(n+1)~r^{-n-2}~\sin(n\theta )\,$ |

## Displacement components

Displacements $(u_{r},u_{\theta })$ 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
    $\kappa ={\begin{cases}3-4~\nu &{\rm {for~plane~strain}}\\{\cfrac {3-\nu }{1+\nu }}&{\rm {for~plane~stress}}\\\end{cases}}$
where $\nu$ is the Poisson's ratio, and $\mu$ is the shear modulus.
| $\varphi$ | $2~\mu ~u_{r}\,$ | $2~\mu ~u_{\theta }\,$ |
| $r^{2}\,$ | $(\kappa -1)~r$ | $0$ |
| $r^{2}~\ln r$ | $(\kappa -1)~r~\ln r-r$ | $(\kappa +1)~r~\theta$ |
| $\ln r\,$ | $-r^{-1}\,$ | $0$ |
| $\theta \,$ | $0$ | $-r^{-1}\,$ |
| $r^{3}~\cos \theta \,$ | $(\kappa -2)~r^{2}~\cos \theta \,$ | $(\kappa +2)~r^{2}~\sin \theta \,$ |
| $r\theta ~\cos \theta \,$ | ${\frac {1}{2}}[(\kappa -1)\theta ~\cos \theta +\{1-(\kappa +1)\ln r\}~\sin \theta ]\,$ | $-{\frac {1}{2}}[(\kappa -1)\theta ~\sin \theta +\{1+(\kappa +1)\ln r\}~\cos \theta ]\,$ |
| $r~\ln r~\cos \theta \,$ | ${\frac {1}{2}}[(\kappa +1)\theta ~\sin \theta -\{1-(\kappa -1)\ln r\}~\cos \theta ]\,$ | ${\frac {1}{2}}[(\kappa +1)\theta ~\cos \theta -\{1+(\kappa -1)\ln r\}~\sin \theta ]\,$ |
| $r^{-1}~\cos \theta \,$ | $r^{-2}~\cos \theta \,$ | $r^{-2}~\sin \theta \,$ |
| $r^{3}~\sin \theta \,$ | $(\kappa -2)~r^{2}~\sin \theta \,$ | $-(\kappa +2)~r^{2}~\cos \theta \,$ |
| $r\theta ~\sin \theta \,$ | ${\frac {1}{2}}[(\kappa -1)\theta ~\sin \theta -\{1-(\kappa +1)\ln r\}~\cos \theta ]\,$ | ${\frac {1}{2}}[(\kappa -1)\theta ~\cos \theta -\{1+(\kappa +1)\ln r\}~\sin \theta ]\,$ |
| $r~\ln r~\sin \theta \,$ | $-{\frac {1}{2}}[(\kappa +1)\theta ~\cos \theta +\{1-(\kappa -1)\ln r\}~\sin \theta ]\,$ | ${\frac {1}{2}}[(\kappa +1)\theta ~\sin \theta +\{1+(\kappa -1)\ln r\}~\cos \theta ]\,$ |
| $r^{-1}~\sin \theta \,$ | $r^{-2}~\sin \theta \,$ | $-r^{-2}~\cos \theta \,$ |
| $r^{n+2}~\cos(n\theta )\,$ | $(\kappa -n-1)~r^{n+1}~\cos(n\theta )\,$ | $(\kappa +n+1)~r^{n+1}~\sin(n\theta )\,$ |
| $r^{-n+2}~\cos(n\theta )\,$ | $(\kappa +n-1)~r^{-n+1}~\cos(n\theta )\,$ | $-(\kappa -n+1)~r^{-n+1}~\sin(n\theta )\,$ |
| $r^{n}~\cos(n\theta )\,$ | $-n~r^{n-1}~\cos(n\theta )\,$ | $n~r^{n-1}~\sin(n\theta )\,$ |
| $r^{-n}~\cos(n\theta )\,$ | $n~r^{-n-1}~\cos(n\theta )\,$ | $n(~r^{-n-1}~\sin(n\theta )\,$ |
| $r^{n+2}~\sin(n\theta )\,$ | $(\kappa -n-1)~r^{n+1}~\sin(n\theta )\,$ | $-(\kappa +n+1)~r^{n+1}~\cos(n\theta )\,$ |
| $r^{-n+2}~\sin(n\theta )\,$ | $(\kappa +n-1)~r^{-n+1}~\sin(n\theta )\,$ | $(\kappa -n+1)~r^{-n+1}~\cos(n\theta )\,$ |
| $r^{n}~\sin(n\theta )\,$ | $-n~r^{n-1}~\sin(n\theta )\,$ | $-n~r^{n-1}~\cos(n\theta )\,$ |
| $r^{-n}~\sin(n\theta )\,$ | $n~r^{-n-1}~\sin(n\theta )\,$ | $-n~r^{-n-1}~\cos(n\theta )\,$ |
Note that a rigid body displacement can be superposed on the Michell solution of the form
    ${\begin{aligned}u_{r}&=A~\cos \theta +B~\sin \theta \\u_{\theta }&=-A~\sin \theta +B~\cos \theta +C~r\\\end{aligned}}$
to obtain an admissible displacement field.