Michell solution

In continuum mechanics, the Michell solution is a general solution to the elasticity equations in polar coordinates ( r , θ {\displaystyle r,\theta } ) developed by John Henry Michell in 1899.^[1] The solution is such that the stress components are in the form of a Fourier series in θ {\displaystyle \theta } .

Michell showed that the general solution can be expressed in terms of an Airy stress function of the form φ ( r , θ ) = A 0 r 2 + B 0 r 2 ln ⁡ ( r ) + C 0 ln ⁡ ( r ) + ( I 0 r 2 + I 1 r 2 ln ⁡ ( r ) + I 2 ln ⁡ ( r ) + I 3 ) θ + ( A 1 r + B 1 r − 1 + B 1 ′ r θ + C 1 r 3 + D 1 r ln ⁡ ( r ) ) cos ⁡ θ + ( E 1 r + F 1 r − 1 + F 1 ′ r θ + G 1 r 3 + H 1 r ln ⁡ ( r ) ) sin ⁡ θ + ∑ n = 2 ∞ ( A n r n + B n r − n + C n r n + 2 + D n r − n + 2 ) cos ⁡ ( n θ ) + ∑ n = 2 ∞ ( E n r n + F n r − n + G n r n + 2 + H n r − n + 2 ) sin ⁡ ( n θ ) {\displaystyle {\begin{aligned}\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{aligned}}} The terms A 1 r cos ⁡ θ {\displaystyle A_{1}r\cos \theta } and E 1 r sin ⁡ θ {\displaystyle E_{1}r\sin \theta } define a trivial null state of stress and are ignored.

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.^[2]

φ {\displaystyle \varphi }	σ r r {\displaystyle \sigma _{rr}\,}	σ r θ {\displaystyle \sigma _{r\theta }\,}	σ θ θ {\displaystyle \sigma _{\theta \theta }\,}
r 2 {\displaystyle r^{2}\,}	2 {\displaystyle 2}	0 {\displaystyle 0}	2 {\displaystyle 2}
r 2   ln ⁡ r {\displaystyle r^{2}~\ln r}	2   ln ⁡ r + 1 {\displaystyle 2~\ln r+1}	0 {\displaystyle 0}	2   ln ⁡ r + 3 {\displaystyle 2~\ln r+3}
ln ⁡ r {\displaystyle \ln r\,}	r − 2 {\displaystyle r^{-2}\,}	0 {\displaystyle 0}	− r − 2 {\displaystyle -r^{-2}\,}
θ {\displaystyle \theta \,}	0 {\displaystyle 0}	r − 2 {\displaystyle r^{-2}\,}	0 {\displaystyle 0}
r 3   cos ⁡ θ {\displaystyle r^{3}~\cos \theta \,}	2   r   cos ⁡ θ {\displaystyle 2~r~\cos \theta \,}	2   r   sin ⁡ θ {\displaystyle 2~r~\sin \theta \,}	6   r   cos ⁡ θ {\displaystyle 6~r~\cos \theta \,}
r θ   cos ⁡ θ {\displaystyle r\theta ~\cos \theta \,}	− 2   r − 1   sin ⁡ θ {\displaystyle -2~r^{-1}~\sin \theta \,}	0 {\displaystyle 0}	0 {\displaystyle 0}
r   ln ⁡ r   cos ⁡ θ {\displaystyle r~\ln r~\cos \theta \,}	r − 1   cos ⁡ θ {\displaystyle r^{-1}~\cos \theta \,}	r − 1   sin ⁡ θ {\displaystyle r^{-1}~\sin \theta \,}	r − 1   cos ⁡ θ {\displaystyle r^{-1}~\cos \theta \,}
r − 1   cos ⁡ θ {\displaystyle r^{-1}~\cos \theta \,}	− 2   r − 3   cos ⁡ θ {\displaystyle -2~r^{-3}~\cos \theta \,}	− 2   r − 3   sin ⁡ θ {\displaystyle -2~r^{-3}~\sin \theta \,}	2   r − 3   cos ⁡ θ {\displaystyle 2~r^{-3}~\cos \theta \,}
r 3   sin ⁡ θ {\displaystyle r^{3}~\sin \theta \,}	2   r   sin ⁡ θ {\displaystyle 2~r~\sin \theta \,}	− 2   r   cos ⁡ θ {\displaystyle -2~r~\cos \theta \,}	6   r   sin ⁡ θ {\displaystyle 6~r~\sin \theta \,}
r θ   sin ⁡ θ {\displaystyle r\theta ~\sin \theta \,}	2   r − 1   cos ⁡ θ {\displaystyle 2~r^{-1}~\cos \theta \,}	0 {\displaystyle 0}	0 {\displaystyle 0}
r   ln ⁡ r   sin ⁡ θ {\displaystyle r~\ln r~\sin \theta \,}	r − 1   sin ⁡ θ {\displaystyle r^{-1}~\sin \theta \,}	− r − 1   cos ⁡ θ {\displaystyle -r^{-1}~\cos \theta \,}	r − 1   sin ⁡ θ {\displaystyle r^{-1}~\sin \theta \,}
r − 1   sin ⁡ θ {\displaystyle r^{-1}~\sin \theta \,}	− 2   r − 3   sin ⁡ θ {\displaystyle -2~r^{-3}~\sin \theta \,}	2   r − 3   cos ⁡ θ {\displaystyle 2~r^{-3}~\cos \theta \,}	2   r − 3   sin ⁡ θ {\displaystyle 2~r^{-3}~\sin \theta \,}
r n + 2   cos ⁡ ( n θ ) {\displaystyle r^{n+2}~\cos(n\theta )\,}	− ( n + 1 ) ( n − 2 )   r n   cos ⁡ ( n θ ) {\displaystyle -(n+1)(n-2)~r^{n}~\cos(n\theta )\,}	n ( n + 1 )   r n   sin ⁡ ( n θ ) {\displaystyle n(n+1)~r^{n}~\sin(n\theta )\,}	( n + 1 ) ( n + 2 )   r n   cos ⁡ ( n θ ) {\displaystyle (n+1)(n+2)~r^{n}~\cos(n\theta )\,}
r − n + 2   cos ⁡ ( n θ ) {\displaystyle r^{-n+2}~\cos(n\theta )\,}	− ( n + 2 ) ( n − 1 )   r − n   cos ⁡ ( n θ ) {\displaystyle -(n+2)(n-1)~r^{-n}~\cos(n\theta )\,}	− n ( n − 1 )   r − n   sin ⁡ ( n θ ) {\displaystyle -n(n-1)~r^{-n}~\sin(n\theta )\,}	( n − 1 ) ( n − 2 )   r − n   cos ⁡ ( n θ ) {\displaystyle (n-1)(n-2)~r^{-n}~\cos(n\theta )}
r n   cos ⁡ ( n θ ) {\displaystyle r^{n}~\cos(n\theta )\,}	− n ( n − 1 )   r n − 2   cos ⁡ ( n θ ) {\displaystyle -n(n-1)~r^{n-2}~\cos(n\theta )\,}	n ( n − 1 )   r n − 2   sin ⁡ ( n θ ) {\displaystyle n(n-1)~r^{n-2}~\sin(n\theta )\,}	n ( n − 1 )   r n − 2   cos ⁡ ( n θ ) {\displaystyle n(n-1)~r^{n-2}~\cos(n\theta )\,}
r − n   cos ⁡ ( n θ ) {\displaystyle r^{-n}~\cos(n\theta )\,}	− n ( n + 1 )   r − n − 2   cos ⁡ ( n θ ) {\displaystyle -n(n+1)~r^{-n-2}~\cos(n\theta )\,}	− n ( n + 1 )   r − n − 2   sin ⁡ ( n θ ) {\displaystyle -n(n+1)~r^{-n-2}~\sin(n\theta )\,}	n ( n + 1 )   r − n − 2   cos ⁡ ( n θ ) {\displaystyle n(n+1)~r^{-n-2}~\cos(n\theta )\,}
r n + 2   sin ⁡ ( n θ ) {\displaystyle r^{n+2}~\sin(n\theta )\,}	− ( n + 1 ) ( n − 2 )   r n   sin ⁡ ( n θ ) {\displaystyle -(n+1)(n-2)~r^{n}~\sin(n\theta )\,}	− n ( n + 1 )   r n   cos ⁡ ( n θ ) {\displaystyle -n(n+1)~r^{n}~\cos(n\theta )\,}	( n + 1 ) ( n + 2 )   r n   sin ⁡ ( n θ ) {\displaystyle (n+1)(n+2)~r^{n}~\sin(n\theta )\,}
r − n + 2   sin ⁡ ( n θ ) {\displaystyle r^{-n+2}~\sin(n\theta )\,}	− ( n + 2 ) ( n − 1 )   r − n   sin ⁡ ( n θ ) {\displaystyle -(n+2)(n-1)~r^{-n}~\sin(n\theta )\,}	n ( n − 1 )   r − n   cos ⁡ ( n θ ) {\displaystyle n(n-1)~r^{-n}~\cos(n\theta )\,}	( n − 1 ) ( n − 2 )   r − n   sin ⁡ ( n θ ) {\displaystyle (n-1)(n-2)~r^{-n}~\sin(n\theta )\,}
r n   sin ⁡ ( n θ ) {\displaystyle r^{n}~\sin(n\theta )\,}	− n ( n − 1 )   r n − 2   sin ⁡ ( n θ ) {\displaystyle -n(n-1)~r^{n-2}~\sin(n\theta )\,}	− n ( n − 1 )   r n − 2   cos ⁡ ( n θ ) {\displaystyle -n(n-1)~r^{n-2}~\cos(n\theta )\,}	n ( n − 1 )   r n − 2   sin ⁡ ( n θ ) {\displaystyle n(n-1)~r^{n-2}~\sin(n\theta )\,}
r − n   sin ⁡ ( n θ ) {\displaystyle r^{-n}~\sin(n\theta )\,}	− n ( n + 1 )   r − n − 2   sin ⁡ ( n θ ) {\displaystyle -n(n+1)~r^{-n-2}~\sin(n\theta )\,}	n ( n + 1 )   r − n − 2   cos ⁡ ( n θ ) {\displaystyle n(n+1)~r^{-n-2}~\cos(n\theta )\,}	n ( n + 1 )   r − n − 2   sin ⁡ ( n θ ) {\displaystyle n(n+1)~r^{-n-2}~\sin(n\theta )\,}

Displacements ( u r , u θ ) {\displaystyle (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

κ = { 3 − 4   ν f o r   p l a n e   s t r a i n 3 − ν 1 + ν f o r   p l a n e   s t r e s s {\displaystyle \kappa ={\begin{cases}3-4~\nu &{\rm {for~plane~strain}}\\{\cfrac {3-\nu }{1+\nu }}&{\rm {for~plane~stress}}\\\end{cases}}}

where ν {\displaystyle \nu } is the Poisson's ratio, and μ {\displaystyle \mu } is the shear modulus.

φ {\displaystyle \varphi }	2   μ   u r {\displaystyle 2~\mu ~u_{r}\,}	2   μ   u θ {\displaystyle 2~\mu ~u_{\theta }\,}
r 2 {\displaystyle r^{2}\,}	( κ − 1 )   r {\displaystyle (\kappa -1)~r}	0 {\displaystyle 0}
r 2   ln ⁡ r {\displaystyle r^{2}~\ln r}	( κ − 1 )   r   ln ⁡ r − r {\displaystyle (\kappa -1)~r~\ln r-r}	( κ + 1 )   r   θ {\displaystyle (\kappa +1)~r~\theta }
ln ⁡ r {\displaystyle \ln r\,}	− r − 1 {\displaystyle -r^{-1}\,}	0 {\displaystyle 0}
θ {\displaystyle \theta \,}	0 {\displaystyle 0}	− r − 1 {\displaystyle -r^{-1}\,}
r 3   cos ⁡ θ {\displaystyle r^{3}~\cos \theta \,}	( κ − 2 )   r 2   cos ⁡ θ {\displaystyle (\kappa -2)~r^{2}~\cos \theta \,}	( κ + 2 )   r 2   sin ⁡ θ {\displaystyle (\kappa +2)~r^{2}~\sin \theta \,}
r θ   cos ⁡ θ {\displaystyle r\theta ~\cos \theta \,}	1 2 [ ( κ − 1 ) θ   cos ⁡ θ + { 1 − ( κ + 1 ) ln ⁡ r }   sin ⁡ θ ] {\displaystyle {\frac {1}{2}}[(\kappa -1)\theta ~\cos \theta +\{1-(\kappa +1)\ln r\}~\sin \theta ]\,}	− 1 2 [ ( κ − 1 ) θ   sin ⁡ θ + { 1 + ( κ + 1 ) ln ⁡ r }   cos ⁡ θ ] {\displaystyle -{\frac {1}{2}}[(\kappa -1)\theta ~\sin \theta +\{1+(\kappa +1)\ln r\}~\cos \theta ]\,}
r   ln ⁡ r   cos ⁡ θ {\displaystyle r~\ln r~\cos \theta \,}	1 2 [ ( κ + 1 ) θ   sin ⁡ θ − { 1 − ( κ − 1 ) ln ⁡ r }   cos ⁡ θ ] {\displaystyle {\frac {1}{2}}[(\kappa +1)\theta ~\sin \theta -\{1-(\kappa -1)\ln r\}~\cos \theta ]\,}	1 2 [ ( κ + 1 ) θ   cos ⁡ θ − { 1 + ( κ − 1 ) ln ⁡ r }   sin ⁡ θ ] {\displaystyle {\frac {1}{2}}[(\kappa +1)\theta ~\cos \theta -\{1+(\kappa -1)\ln r\}~\sin \theta ]\,}
r − 1   cos ⁡ θ {\displaystyle r^{-1}~\cos \theta \,}	r − 2   cos ⁡ θ {\displaystyle r^{-2}~\cos \theta \,}	r − 2   sin ⁡ θ {\displaystyle r^{-2}~\sin \theta \,}
r 3   sin ⁡ θ {\displaystyle r^{3}~\sin \theta \,}	( κ − 2 )   r 2   sin ⁡ θ {\displaystyle (\kappa -2)~r^{2}~\sin \theta \,}	− ( κ + 2 )   r 2   cos ⁡ θ {\displaystyle -(\kappa +2)~r^{2}~\cos \theta \,}
r θ   sin ⁡ θ {\displaystyle r\theta ~\sin \theta \,}	1 2 [ ( κ − 1 ) θ   sin ⁡ θ − { 1 − ( κ + 1 ) ln ⁡ r }   cos ⁡ θ ] {\displaystyle {\frac {1}{2}}[(\kappa -1)\theta ~\sin \theta -\{1-(\kappa +1)\ln r\}~\cos \theta ]\,}	1 2 [ ( κ − 1 ) θ   cos ⁡ θ − { 1 + ( κ + 1 ) ln ⁡ r }   sin ⁡ θ ] {\displaystyle {\frac {1}{2}}[(\kappa -1)\theta ~\cos \theta -\{1+(\kappa +1)\ln r\}~\sin \theta ]\,}
r   ln ⁡ r   sin ⁡ θ {\displaystyle r~\ln r~\sin \theta \,}	− 1 2 [ ( κ + 1 ) θ   cos ⁡ θ + { 1 − ( κ − 1 ) ln ⁡ r }   sin ⁡ θ ] {\displaystyle -{\frac {1}{2}}[(\kappa +1)\theta ~\cos \theta +\{1-(\kappa -1)\ln r\}~\sin \theta ]\,}	1 2 [ ( κ + 1 ) θ   sin ⁡ θ + { 1 + ( κ − 1 ) ln ⁡ r }   cos ⁡ θ ] {\displaystyle {\frac {1}{2}}[(\kappa +1)\theta ~\sin \theta +\{1+(\kappa -1)\ln r\}~\cos \theta ]\,}
r − 1   sin ⁡ θ {\displaystyle r^{-1}~\sin \theta \,}	r − 2   sin ⁡ θ {\displaystyle r^{-2}~\sin \theta \,}	− r − 2   cos ⁡ θ {\displaystyle -r^{-2}~\cos \theta \,}
r n + 2   cos ⁡ ( n θ ) {\displaystyle r^{n+2}~\cos(n\theta )\,}	( κ − n − 1 )   r n + 1   cos ⁡ ( n θ ) {\displaystyle (\kappa -n-1)~r^{n+1}~\cos(n\theta )\,}	( κ + n + 1 )   r n + 1   sin ⁡ ( n θ ) {\displaystyle (\kappa +n+1)~r^{n+1}~\sin(n\theta )\,}
r − n + 2   cos ⁡ ( n θ ) {\displaystyle r^{-n+2}~\cos(n\theta )\,}	( κ + n − 1 )   r − n + 1   cos ⁡ ( n θ ) {\displaystyle (\kappa +n-1)~r^{-n+1}~\cos(n\theta )\,}	− ( κ − n + 1 )   r − n + 1   sin ⁡ ( n θ ) {\displaystyle -(\kappa -n+1)~r^{-n+1}~\sin(n\theta )\,}
r n   cos ⁡ ( n θ ) {\displaystyle r^{n}~\cos(n\theta )\,}	− n   r n − 1   cos ⁡ ( n θ ) {\displaystyle -n~r^{n-1}~\cos(n\theta )\,}	n   r n − 1   sin ⁡ ( n θ ) {\displaystyle n~r^{n-1}~\sin(n\theta )\,}
r − n   cos ⁡ ( n θ ) {\displaystyle r^{-n}~\cos(n\theta )\,}	n   r − n − 1   cos ⁡ ( n θ ) {\displaystyle n~r^{-n-1}~\cos(n\theta )\,}	n   r − n − 1   sin ⁡ ( n θ ) {\displaystyle n~r^{-n-1}~\sin(n\theta )\,}
r n + 2   sin ⁡ ( n θ ) {\displaystyle r^{n+2}~\sin(n\theta )\,}	( κ − n − 1 )   r n + 1   sin ⁡ ( n θ ) {\displaystyle (\kappa -n-1)~r^{n+1}~\sin(n\theta )\,}	− ( κ + n + 1 )   r n + 1   cos ⁡ ( n θ ) {\displaystyle -(\kappa +n+1)~r^{n+1}~\cos(n\theta )\,}
r − n + 2   sin ⁡ ( n θ ) {\displaystyle r^{-n+2}~\sin(n\theta )\,}	( κ + n − 1 )   r − n + 1   sin ⁡ ( n θ ) {\displaystyle (\kappa +n-1)~r^{-n+1}~\sin(n\theta )\,}	( κ − n + 1 )   r − n + 1   cos ⁡ ( n θ ) {\displaystyle (\kappa -n+1)~r^{-n+1}~\cos(n\theta )\,}
r n   sin ⁡ ( n θ ) {\displaystyle r^{n}~\sin(n\theta )\,}	− n   r n − 1   sin ⁡ ( n θ ) {\displaystyle -n~r^{n-1}~\sin(n\theta )\,}	− n   r n − 1   cos ⁡ ( n θ ) {\displaystyle -n~r^{n-1}~\cos(n\theta )\,}
r − n   sin ⁡ ( n θ ) {\displaystyle r^{-n}~\sin(n\theta )\,}	n   r − n − 1   sin ⁡ ( n θ ) {\displaystyle n~r^{-n-1}~\sin(n\theta )\,}	− n   r − n − 1   cos ⁡ ( n θ ) {\displaystyle -n~r^{-n-1}~\cos(n\theta )\,}

Note that a rigid body displacement can be superposed on the Michell solution of the form

u r = A   cos ⁡ θ + B   sin ⁡ θ u θ = − A   sin ⁡ θ + B   cos ⁡ θ + C   r {\displaystyle {\begin{aligned}u_{r}&=A~\cos \theta +B~\sin \theta \\u_{\theta }&=-A~\sin \theta +B~\cos \theta +C~r\\\end{aligned}}}

to obtain an admissible displacement field.

• Linear elasticity
• Flamant solution
• John Henry Michell