# Michell Solution

Michell Solution

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 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 and define a trivial null state of stress and are ignored.