# Michell solution

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

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

\begin{align} \varphi &= A_0~r^2 + B_0~r^2~\ln(r) + C_0~\ln(r) + D_0~\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{align}

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

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

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

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

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". 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.