Michell solution

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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[edit]

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[edit]

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[edit]

  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.

See also[edit]