View source for 4-Hydroxy-3-methylbut-2-enyl diphosphate reductase

The [[John Henry Michell|Michell]] solution is a general solution to the [[Linear elasticity|elasticity]] equations in [[polar coordinates]] (<math> r, \theta \,</math>).  The solution is such that the stress components are in the form of a [[Fourier series]] in  <math> \theta \, </math>.

Michell<ref>{{cite journal|title=On the direct determination of stress in an elastic solid, with application to the theory of plates|journal=Proc. London Math. Soc.|date=1899-04-01|first=J. H.|last=Michell|coauthors=|volume=31|issue=1|pages=100–124|doi= 10.1112/plms/s1-31.1.100|url=http://plms.oxfordjournals.org/cgi/reprint/s1-31/1/100.pdf|format=|accessdate=2008-06-25 }}</ref> showed that the general solution can be expressed in terms of an [[Airy stress function]] of the form
:<math>
  \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}
 </math>

The terms <math>A_1~r~\cos\theta\,</math> and <math>E_1~r~\sin\theta\,</math> define a trivial null state of stress and are ignored.

== Stress components ==
The [[stress (physics)|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) <ref> J. R. Barber, 2002, ''Elasticity: 2nd Edition'', Kluwer Academic Publishers. </ref>].
{| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; border: 1px #aaa solid; border-collapse: collapse;"
|-
! <math>\varphi</math>
! <math>\sigma_{rr}\, </math>
! <math>\sigma_{r\theta}\, </math>
! <math>\sigma_{\theta\theta}\, </math>
|-
| <math>r^2\,</math>
| <math>2</math>
| <math>0</math>
| <math>2</math>
|-
| <math>r^2~\ln r</math>
| <math>2~\ln r + 1</math>
| <math>0</math>
| <math>2~\ln r + 3</math>
|-
| <math>\ln r\,</math>
| <math>r^{-2}\,</math>
| <math>0</math>
| <math>-r^{-2}\,</math>
|-
| <math>\theta\,</math>
| <math>0</math>
| <math>r^{-2}\,</math>
| <math>0</math>
|-
|-
| <math> r^3~\cos\theta \,</math>
| <math> 2~r~\cos\theta \,</math>
| <math> 2~r~\sin\theta \,</math>
| <math> 6~r~\cos\theta \,</math>
|-
| <math> r\theta~\cos\theta \,</math>
| <math> 2~r^{-1}~\sin\theta \,</math>
| <math> 0 </math>
| <math> 0 </math>
|-
| <math> r~\ln r~\cos\theta \,</math>
| <math> r^{-1}~\cos\theta \,</math>
| <math> r^{-1}~\sin\theta \,</math>
| <math> r^{-1}~\cos\theta \,</math>
|-
| <math> r^{-1}~\cos\theta \,</math>
| <math> -2~r^{-3}~\cos\theta \,</math>
| <math> -2~r^{-3}~\sin\theta \,</math>
| <math> 2~r^{-3}~\cos\theta \,</math>
|-
|-
| <math> r^3~\sin\theta \,</math>
| <math> 2~r~\sin\theta \,</math>
| <math> -2~r~\cos\theta \,</math>
| <math> 6~r~\sin\theta \,</math>
|-
| <math> r\theta~\sin\theta \,</math>
| <math> 2~r^{-1}~\cos\theta \,</math>
| <math> 0 </math>
| <math> 0 </math>
|-
| <math> r~\ln r~\sin\theta \,</math>
| <math> r^{-1}~\sin\theta \,</math>
| <math> -r^{-1}~\cos\theta \,</math>
| <math> r^{-1}~\sin\theta \,</math>
|-
| <math> r^{-1}~\sin\theta \,</math>
| <math> -2~r^{-3}~\sin\theta \,</math>
| <math> 2~r^{-3}~\cos\theta \,</math>
| <math> 2~r^{-3}~\sin\theta \,</math>
|-
|-
| <math> r^{n+2}~\cos(n\theta) \,</math>
| <math> -(n+1)(n-2)~r^n~\cos(n\theta) \,</math>
| <math> n(n+1)~r^n~\sin(n\theta) \,</math>
| <math> (n+1)(n+2)~r^n~\cos(n\theta \,</math>
|-
| <math> r^{-n+2}~\cos(n\theta) \,</math>
| <math> -(n+2)(n-1)~r^{-n}~\cos(n\theta) \,</math>
| <math> -n(n-1)~r^{-n}~\sin(n\theta)\, </math>
| <math> (n-1)(n-2)~r^{-n}~\cos(n\theta) </math>
|-
| <math> r^n~\cos(n\theta) \,</math>
| <math> -n(n-1)~r^{n-2}~\cos(n\theta) \,</math>
| <math> n(n-1)~r^{n-2}~\sin(n\theta) \,</math>
| <math> n(n-1)~r^{n-2}~\cos(n\theta) \,</math>
|-
| <math> r^{-n}~\cos(n\theta) \,</math>
| <math> -n(n+1)~r^{-n-2}~\cos(n\theta) \,</math>
| <math> -n(n+1)~r^{-n-2}~\sin(n\theta) \,</math>
| <math> n(n+1)~r^{-n-2}~\cos(n\theta) \,</math>
|-
|-
| <math> r^{n+2}~\sin(n\theta) \,</math>
| <math> -(n+1)(n-2)~r^n~\sin(n\theta) \,</math>
| <math> -n(n+1)~r^n~\cos(n\theta) \,</math>
| <math> (n+1)(n+2)~r^n~\sin(n\theta \,</math>
|-
| <math> r^{-n+2}~\sin(n\theta) \,</math>
| <math> -(n+2)(n-1)~r^{-n}~\sin(n\theta) \,</math>
| <math> n(n-1)~r^{-n}~\cos(n\theta)\, </math>
| <math> (n-1)(n-2)~r^{-n}~\sin(n\theta) </math>
|-
| <math> r^n~\sin(n\theta) \,</math>
| <math> -n(n-1)~r^{n-2}~\sin(n\theta) \,</math>
| <math> -n(n-1)~r^{n-2}~\cos(n\theta) \,</math>
| <math> n(n-1)~r^{n-2}~\sin(n\theta) \,</math>
|-
| <math> r^{-n}~\sin(n\theta) \,</math>
| <math> -n(n+1)~r^{-n-2}~\sin(n\theta) \,</math>
| <math> n(n+1)~r^{-n-2}~\cos(n\theta) \,</math>
| <math> n(n+1)~r^{-n-2}~\sin(n\theta) \,</math>
|}

== Displacement components ==
[[Displacement field (mechanics)|Displacements]] <math>(u_r, u_\theta)</math> can be obtained from the Michell solution by using the [[linear elasticity|stress-strain]] and [[linear elasticity|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
:<math>
   \kappa = \begin{cases}
            3 - 4~\nu & \rm{for~plane~strain} \\
            \cfrac{3 - \nu}{1 + \nu} & \rm{for~plane~stress} \\
            \end{cases}
 </math>
where <math>\nu</math> is the [[Poisson's ratio]], and <math>\mu</math> is the [[shear modulus]].

{| border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; border: 1px #aaa solid; border-collapse: collapse;"
|-
! <math>\varphi</math>
! <math>2~\mu~u_r\, </math>
! <math>2~\mu~u_\theta\, </math>
|-
| <math>r^2\,</math>
| <math>(\kappa-1)~r</math>
| <math>0</math>
|-
| <math>r^2~\ln r</math>
| <math>(\kappa-1)~r~\ln r - r</math>
| <math>(\kappa + 1)~r~\theta</math>
|-
| <math>\ln r\,</math>
| <math>-r^{-1}\,</math>
| <math>0</math>
|-
| <math>\theta\,</math>
| <math>0</math>
| <math>-r^{-1}\,</math>
|-
|-
| <math> r^3~\cos\theta \,</math>
| <math> (\kappa-2)~r^2~\cos\theta \,</math>
| <math> (\kappa+2)~r^2~\sin\theta \,</math>
|-
| <math> r\theta~\cos\theta \,</math>
| <math> \frac{1}{2}[(\kappa-1) \theta~\cos\theta + \{1 - (\kappa+1) \ln r\} ~\sin\theta]\,</math>
| <math> -\frac{1}{2}[(\kappa-1) \theta~\sin\theta + \{1 + (\kappa+1) \ln r\} ~\cos\theta]\, </math>
|-
| <math> r~\ln r~\cos\theta \,</math>
| <math> \frac{1}{2}[(\kappa+1) \theta~\sin\theta - \{1 - (\kappa-1) \ln r\} ~\cos\theta] \,</math>
| <math> \frac{1}{2}[(\kappa+1) \theta~\cos\theta - \{1 + (\kappa-1) \ln r\} ~\sin\theta] \,</math>
|-
| <math> r^{-1}~\cos\theta \,</math>
| <math> r^{-2}~\cos\theta \,</math>
| <math> r^{-2}~\sin\theta \,</math>
|-
|-
| <math> r^3~\sin\theta \,</math>
| <math> (\kappa-2)~r^2~\sin\theta \,</math>
| <math> -(\kappa+2)~r^2~\cos\theta \,</math>
|-
| <math> r\theta~\sin\theta \,</math>
| <math> \frac{1}{2}[(\kappa-1) \theta~\sin\theta - \{1 - (\kappa+1) \ln r\} ~\cos\theta]\, </math>
| <math> \frac{1}{2}[(\kappa-1) \theta~\cos\theta - \{1 + (\kappa+1) \ln r\} ~\sin\theta]\,</math>
|-
| <math> r~\ln r~\sin\theta \,</math>
| <math> -\frac{1}{2}[(\kappa+1) \theta~\cos\theta + \{1 - (\kappa-1) \ln r\} ~\sin\theta] \,</math>
| <math> \frac{1}{2}[(\kappa+1) \theta~\sin\theta + \{1 + (\kappa-1) \ln r\} ~\cos\theta] \,</math>
|-
| <math> r^{-1}~\sin\theta \,</math>
| <math> r^{-2}~\sin\theta \,</math>
| <math> -r^{-2}~\cos\theta \,</math>
|-
|-
| <math> r^{n+2}~\cos(n\theta) \,</math>
| <math> (\kappa-n-1)~r^{n+1}~\cos(n\theta) \,</math>
| <math> (\kappa+n+1)~r^{n+1}~\sin(n\theta) \,</math>
|-
| <math> r^{-n+2}~\cos(n\theta) \,</math>
| <math> (\kappa+n-1)~r^{-n+1}~\cos(n\theta) \,</math>
| <math> -(\kappa-n+1)~r^{-n+1}~\sin(n\theta)\, </math>
|-
| <math> r^n~\cos(n\theta) \,</math>
| <math> -n~r^{n-1}~\cos(n\theta) \,</math>
| <math> n~r^{n-1}~\sin(n\theta) \,</math>
|-
| <math> r^{-n}~\cos(n\theta) \,</math>
| <math> n~r^{-n-1}~\cos(n\theta) \,</math>
| <math> n(~r^{-n-1}~\sin(n\theta) \,</math>
|-
|-
| <math> r^{n+2}~\sin(n\theta) \,</math>
| <math> (\kappa-n-1)~r^{n+1}~\sin(n\theta) \,</math>
| <math> -(\kappa+n+1)~r^{n+1}~\cos(n\theta) \,</math>
|-
| <math> r^{-n+2}~\sin(n\theta) \,</math>
| <math> (\kappa+n-1)~r^{-n+1}~\sin(n\theta) \,</math>
| <math> (\kappa-n+1)~r^{-n+1}~\cos(n\theta)\, </math>
|-
| <math> r^n~\sin(n\theta) \,</math>
| <math> -n~r^{n-1}~\sin(n\theta) \,</math>
| <math> -n~r^{n-1}~\cos(n\theta) \,</math>
|-
| <math> r^{-n}~\sin(n\theta) \,</math>
| <math> n~r^{-n-1}~\sin(n\theta) \,</math>
| <math> -n~r^{-n-1}~\cos(n\theta) \,</math>
|}

Note that you can superpose a [[rigid body displacement]] on the Michell solution of the form
:<math>
   \begin{align}
   u_r &= A~\cos\theta + B~\sin\theta \\
   u_\theta &= -A~\sin\theta + B~\cos\theta + C~r\\
   \end{align}
 </math>
to obtain an admissible displacement field.

== References ==
{{reflist}}

== See also ==
* [[Linear elasticity]]
* [[Flamant solution]]
* [[John Henry Michell]]

[[Category:Elasticity (physics)]]