General stress functions in polar coordinates

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[edit] 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.

[edit] 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) \,

[edit] Displacement components

Displacements can be obtained from the Michell solution by using the stress-strain and strain-displacement relations. A table of displacement components corresponding the 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}
\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.

[edit] References

  1. ↑ J. H. Michell, 1899, On the direct determination of stress in an elastic solid, with application to the theory of plates, Proceedings of the London Mathematical Society, vol. 31, pages 100-124.
  2. ↑ J. R. Barber, 2002, Elasticity: 2nd Edition, Kluwer Academic Publishers.

[edit] Related Content

Introduction to Elasticity

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