Michell solution

In continuum mechanics, the Michell solution is a general solution to the elasticity equations in polar coordinates ( $r, \theta$ ) developed by John Henry Michell in 1899.{{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}} The solution is such that the stress components are in the form of a Fourier series in $\theta$ .

Michell showed that the general solution can be expressed in terms of an Airy stress function of the form

$\begin{align}
\varphi(r,\theta) &= A_0 r^2 + B_0 r^2 \ln(r) + C_0 \ln(r) \\
& + \left(I_0 r^2 + I_1 r^2 \ln(r) + I_2 \ln(r) + I_3 \right) \theta \\
& + \left(A_1 r + B_1 r^{-1} + B_1' r \theta + C_1 r^3 + D_1 r \ln(r)\right) \cos\theta \\
& + \left(E_1 r + F_1 r^{-1} + F_1' r \theta + G_1 r^3 + H_1 r \ln(r)\right) \sin\theta \\
& + \sum_{n=2}^{\infty} \left(A_n r^n + B_n r^{-n} + C_n r^{n+2} + D_n r^{-n+2}\right) \cos(n\theta) \\
& + \sum_{n=2}^{\infty} \left(E_n r^n + F_n r^{-n} + G_n r^{n+2} + H_n r^{-n+2}\right) \sin(n\theta)
\end{align}$

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

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. J. R. Barber, 2002, Elasticity: 2nd Edition, Kluwer Academic Publishers.

border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; border: 1px #aaa solid; border-collapse: collapse;"
$\varphi$ ! $\sigma_{rr}\,$ ! $\sigma_{r\theta}\,$ ! $\sigma_{\theta\theta}\,$
$r^2\,$ \| $2$ \| $0$ \| $2$
$r^2~\ln r$ \| $2~\ln r + 1$ \| $0$ \| $2~\ln r + 3$
$\ln r\,$ \| $r^{-2}\,$ \| $0$ \| $-r^{-2}\,$
$\theta\,$ \| $0$ \| $r^{-2}\,$ \| $0$
$r^3~\cos\theta \,$ \| $2~r~\cos\theta \,$ \| $2~r~\sin\theta \,$ \| $6~r~\cos\theta \,$
$r\theta~\cos\theta \,$ \| $-2~r^{-1}~\sin\theta \,$ \| $0$ \| $0$
$r~\ln r~\cos\theta \,$ \| $r^{-1}~\cos\theta \,$ \| $r^{-1}~\sin\theta \,$ \| $r^{-1}~\cos\theta \,$
$r^{-1}~\cos\theta \,$ \| $-2~r^{-3}~\cos\theta \,$ \| $-2~r^{-3}~\sin\theta \,$ \| $2~r^{-3}~\cos\theta \,$
$r^3~\sin\theta \,$ \| $2~r~\sin\theta \,$ \| $-2~r~\cos\theta \,$ \| $6~r~\sin\theta \,$
$r\theta~\sin\theta \,$ \| $2~r^{-1}~\cos\theta \,$ \| $0$ \| $0$
$r~\ln r~\sin\theta \,$ \| $r^{-1}~\sin\theta \,$ \| $-r^{-1}~\cos\theta \,$ \| $r^{-1}~\sin\theta \,$
$r^{-1}~\sin\theta \,$ \| $-2~r^{-3}~\sin\theta \,$ \| $2~r^{-3}~\cos\theta \,$ \| $2~r^{-3}~\sin\theta \,$
$r^{n+2}~\cos(n\theta) \,$ \| $-(n+1)(n-2)~r^n~\cos(n\theta) \,$ \| $n(n+1)~r^n~\sin(n\theta) \,$ \| $(n+1)(n+2)~r^n~\cos(n\theta) \,$
$r^{-n+2}~\cos(n\theta) \,$ \| $-(n+2)(n-1)~r^{-n}~\cos(n\theta) \,$ \| $-n(n-1)~r^{-n}~\sin(n\theta)\,$ \| $(n-1)(n-2)~r^{-n}~\cos(n\theta)$
$r^n~\cos(n\theta) \,$ \| $-n(n-1)~r^{n-2}~\cos(n\theta) \,$ \| $n(n-1)~r^{n-2}~\sin(n\theta) \,$ \| $n(n-1)~r^{n-2}~\cos(n\theta) \,$
$r^{-n}~\cos(n\theta) \,$ \| $-n(n+1)~r^{-n-2}~\cos(n\theta) \,$ \| $-n(n+1)~r^{-n-2}~\sin(n\theta) \,$ \| $n(n+1)~r^{-n-2}~\cos(n\theta) \,$
$r^{n+2}~\sin(n\theta) \,$ \| $-(n+1)(n-2)~r^n~\sin(n\theta) \,$ \| $-n(n+1)~r^n~\cos(n\theta) \,$ \| $(n+1)(n+2)~r^n~\sin(n\theta) \,$
$r^{-n+2}~\sin(n\theta) \,$ \| $-(n+2)(n-1)~r^{-n}~\sin(n\theta) \,$ \| $n(n-1)~r^{-n}~\cos(n\theta) \,$ \| $(n-1)(n-2)~r^{-n}~\sin(n\theta) \,$
$r^n~\sin(n\theta) \,$ \| $-n(n-1)~r^{n-2}~\sin(n\theta) \,$ \| $-n(n-1)~r^{n-2}~\cos(n\theta) \,$ \| $n(n-1)~r^{n-2}~\sin(n\theta) \,$
$r^{-n}~\sin(n\theta) \,$ \| $-n(n+1)~r^{-n-2}~\sin(n\theta) \,$ \| $n(n+1)~r^{-n-2}~\cos(n\theta) \,$ \| $n(n+1)~r^{-n-2}~\sin(n\theta) \,$

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\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 \,

| $-2~r^{-1}~\sin\theta \,$

| $0$

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 \,

| $2~r^{-1}~\cos\theta \,$

| $0$

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 $(u_r, u_\theta)$ 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.

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$\varphi$ ! $2~\mu~u_r\,$ ! $2~\mu~u_\theta\,$
$r^2\,$ \| $(\kappa-1)~r$ \| $0$
$r^2~\ln r$ \| $(\kappa-1)~r~\ln r - r$ \| $(\kappa + 1)~r~\theta$
$\ln r\,$ \| $-r^{-1}\,$ \| $0$
$\theta\,$ \| $0$ \| $-r^{-1}\,$
$r^3~\cos\theta \,$ \| $(\kappa-2)~r^2~\cos\theta \,$ \| $(\kappa+2)~r^2~\sin\theta \,$
$r\theta~\cos\theta \,$ \| $\frac{1}{2}[(\kappa-1) \theta~\cos\theta + \{1 - (\kappa+1) \ln r\} ~\sin\theta]\,$ \| $-\frac{1}{2}[(\kappa-1) \theta~\sin\theta + \{1 + (\kappa+1) \ln r\} ~\cos\theta]\,$
$r~\ln r~\cos\theta \,$ \| $\frac{1}{2}[(\kappa+1) \theta~\sin\theta - \{1 - (\kappa-1) \ln r\} ~\cos\theta] \,$ \| $\frac{1}{2}[(\kappa+1) \theta~\cos\theta - \{1 + (\kappa-1) \ln r\} ~\sin\theta] \,$
$r^{-1}~\cos\theta \,$ \| $r^{-2}~\cos\theta \,$ \| $r^{-2}~\sin\theta \,$
$r^3~\sin\theta \,$ \| $(\kappa-2)~r^2~\sin\theta \,$ \| $-(\kappa+2)~r^2~\cos\theta \,$
$r\theta~\sin\theta \,$ \| $\frac{1}{2}[(\kappa-1) \theta~\sin\theta - \{1 - (\kappa+1) \ln r\} ~\cos\theta]\,$ \| $\frac{1}{2}[(\kappa-1) \theta~\cos\theta - \{1 + (\kappa+1) \ln r\} ~\sin\theta]\,$
$r~\ln r~\sin\theta \,$ \| $-\frac{1}{2}[(\kappa+1) \theta~\cos\theta + \{1 - (\kappa-1) \ln r\} ~\sin\theta] \,$ \| $\frac{1}{2}[(\kappa+1) \theta~\sin\theta + \{1 + (\kappa-1) \ln r\} ~\cos\theta] \,$
$r^{-1}~\sin\theta \,$ \| $r^{-2}~\sin\theta \,$ \| $-r^{-2}~\cos\theta \,$
$r^{n+2}~\cos(n\theta) \,$ \| $(\kappa-n-1)~r^{n+1}~\cos(n\theta) \,$ \| $(\kappa+n+1)~r^{n+1}~\sin(n\theta) \,$
$r^{-n+2}~\cos(n\theta) \,$ \| $(\kappa+n-1)~r^{-n+1}~\cos(n\theta) \,$ \| $-(\kappa-n+1)~r^{-n+1}~\sin(n\theta)\,$
$r^n~\cos(n\theta) \,$ \| $-n~r^{n-1}~\cos(n\theta) \,$ \| $n~r^{n-1}~\sin(n\theta) \,$
$r^{-n}~\cos(n\theta) \,$ \| $n~r^{-n-1}~\cos(n\theta) \,$ \| $n(~r^{-n-1}~\sin(n\theta) \,$
$r^{n+2}~\sin(n\theta) \,$ \| $(\kappa-n-1)~r^{n+1}~\sin(n\theta) \,$ \| $-(\kappa+n+1)~r^{n+1}~\cos(n\theta) \,$
$r^{-n+2}~\sin(n\theta) \,$ \| $(\kappa+n-1)~r^{-n+1}~\sin(n\theta) \,$ \| $(\kappa-n+1)~r^{-n+1}~\cos(n\theta)\,$
$r^n~\sin(n\theta) \,$ \| $-n~r^{n-1}~\sin(n\theta) \,$ \| $-n~r^{n-1}~\cos(n\theta) \,$
$r^{-n}~\sin(n\theta) \,$ \| $n~r^{-n-1}~\sin(n\theta) \,$ \| $-n~r^{-n-1}~\cos(n\theta) \,$

border="2" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 0; border: 1px #aaa solid; border-collapse: collapse;"

\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 a rigid body displacement can be superposed 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

Category:Elasticity (physics)

Michell solution

Stress components

Displacement components

See also

References