imported>ReyHahn at 18:47, 18 November 2024

2024-11-18T18:47:40Z

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{{Short description|Elasticity equation}}
In [[continuum mechanics]], the '''Michell solution''' is a general solution to the [[Linear elasticity|elasticity]] equations in [[polar coordinates]] (<math> r, \theta </math>) developed by [[John Henry Michell]] in 1899.<ref>{{cite journal |last=Michell |first=J. H. |date=1899-04-01 |title=On the direct determination of stress in an elastic solid, with application to the theory of plates |url=https://zenodo.org/record/1447740 |journal=Proc. London Math. Soc. |volume=31 |issue=1 |pages=100–124 |doi=10.1112/plms/s1-31.1.100}}</ref> The solution is such that the stress components are in the form of a [[Fourier series]] in <math> \theta </math>.

Michell showed that the general solution can be expressed in terms of an [[Airy stress function]] of the form
<math display="block">
\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}
</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.<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 a [[rigid body displacement]] can be superposed 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.

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

== References ==
{{reflist}}

[[Category:Elasticity (physics)]]

Michell solution - Revision history

imported>ReyHahn at 18:47, 18 November 2024