Michell solution Elasticity equation 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. Stress components[edit] 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 )\,} Displacement components[edit] 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. See also[edit] Linear elasticity Flamant solution John Henry Michell References[edit] ^ 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. ^ J. R. Barber, 2002, Elasticity: 2nd Edition, Kluwer Academic Publishers.
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