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{align} \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{align} The terms A_1 r \cos\theta and E_1 r \sin\theta define a trivial null state of stress and are ignored.

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{align} \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{align} The terms A_1 r \cos\theta and E_1 r \sin\theta define a trivial null state of stress and are ignored. \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. Note that a rigid body displacement can be superposed 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.

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