Michell solution

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.

Contents

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]].

φ \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 (ur,uθ) 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 ν is the Poisson's ratio, and μ is the shear modulus.

φ 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. http://plms.oxfordjournals.org/cgi/reprint/s1-31/1/100.pdf. Retrieved 2008-06-25. 
  2. ^ J. R. Barber, 2002, Elasticity: 2nd Edition, Kluwer Academic Publishers.

See also


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Michell — may refer to: Person Anthony Michell (21 June 1870 17 February 1959), an Australian mechanical engineer Bradley Michell (born 14 January 1991), an Australian professional footballer Charles Collier Michell (29 March 1793 –28 March 1851), a… …   Wikipedia

  • Flamant solution — The Flamant solution provides expressions for the stresses and displacements in a linear elastic wedge loaded by point forces at its sharp end. This solution was developed by A. Flamant [A. Flamant. (1892). Sur la répartition des pressions dans… …   Wikipedia

  • John Henry Michell — (26 October 1863 – 3 February 1940) was an Australian mathematician, Professor of Mathematics at the University of Melbourne.Early lifeMichell was the son of John Michell (pronounced Mitchell), a miner, and his wife Grace, née Rowse and was born… …   Wikipedia

  • Linear elasticity — Continuum mechanics …   Wikipedia

  • Black hole — For other uses, see Black hole (disambiguation). Simulated view of a black hole (center) in front of the Large Magellanic Cloud. Note the gravitat …   Wikipedia

  • Historique Des Trous Noirs — Pour les articles homonymes, voir Trou noir (homonymie). Article principal : Trou noir. Cet article traite de la partie historique relative à la découverte et la compréhension des trous noirs. Sommaire 1 XVIIIe siècle  …   Wikipédia en Français

  • Historique de la physique des trous noirs — Historique des trous noirs Pour les articles homonymes, voir Trou noir (homonymie). Article principal : Trou noir. Cet article traite de la partie historique relative à la découverte et la compréhension des trous noirs. Sommaire 1 XVIIIe… …   Wikipédia en Français

  • Historique des trous noirs — Pour les articles homonymes, voir Trou noir (homonymie). Article principal : Trou noir. Cet article traite de la partie historique relative à la découverte et la compréhension des trous noirs. Sommaire 1 XVIIIe siècle : la notion de… …   Wikipédia en Français

  • Stress functions — In linear elasticity, the equations describing the deformation of an elastic body subject only to surface forces on the boundary are (using index notation) the equilibrium equation: where σ is the stress tensor, and the Beltrami Michell… …   Wikipedia

  • GRAVITATION - Gravitation et astrophysique — Si l’on excepte la théorie classique de l’électromagnétisme, introduite bien plus tard par James Clerk Maxwell (1865), aucune théorie physique d’expression aussi simple que la loi du carré inverse de la distance, de Newton (1687), n’a jamais été… …   Encyclopédie Universelle

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”