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Michell solution
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<p>In <a href="/facts/Continuum_mechanics/vI7LpFDk">continuum mechanics</a>, the Michell solution is a general solution to the <a href="/facts/Linear_elasticity/pPLIvs7a">elasticity</a> equations in <a href="/facts/Polar_coordinates/HBko4YoN">polar coordinates</a> ( r , θ {\displaystyle r,\theta } ) developed by <a href="/facts/John_Henry_Michell/mdbkvBLY">John Henry Michell</a> in 1899. The solution is such that the stress components are in the form of a <a href="/facts/Fourier_series/s3fsxPAT">Fourier series</a> in θ {\displaystyle \theta } . </p><p>Michell showed that the general solution can be expressed in terms of an <a href="/facts/Airy_stress_function/DyFgoH3q">Airy stress function</a> 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. </p>