Michell Solution
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The Michell solution is a general solution to the elasticity equations in
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to t ...
( r, \theta \,) developed by J. H. Michell. The solution is such that the stress components are in the form of a
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
in \theta \, . Michell showed that the general solution can be expressed in terms of an Airy stress function of the form : \begin \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^ + B_1'~r~\theta + C_1~r^3 + D_1~r~\ln(r)\right) \cos\theta \\ & + \left(E_1~r + F_1~r^ + F_1'~r~\theta + G_1~r^3 + H_1~r~\ln(r)\right) \sin\theta \\ & + \sum_^ \left(A_n~r^n + B_n~r^ + C_n~r^ + D_n~r^\right)\cos(n\theta) \\ & + \sum_^ \left(E_n~r^n + F_n~r^ + G_n~r^ + H_n~r^\right)\sin(n\theta) \end 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 cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference d ...
). A table of stress components is shown below. J. R. Barber, 2002, ''Elasticity: 2nd Edition'', Kluwer Academic Publishers.


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 3 - 4~\nu & \rm \\ \cfrac & \rm \\ \end where \nu is the
Poisson's ratio In materials science and solid mechanics, Poisson's ratio \nu ( nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Po ...
, and \mu is the
shear modulus In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain: :G \ \stack ...
. Note that a rigid body displacement can be superposed on the Michell solution of the form : \begin u_r &= A~\cos\theta + B~\sin\theta \\ u_\theta &= -A~\sin\theta + B~\cos\theta + C~r\\ \end to obtain an admissible displacement field.


References

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See also

*
Linear elasticity Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mec ...
* Flamant solution *
John Henry Michell John Henry Michell, FRS (26 October 1863 – 3 February 1940) was an Australian mathematician, Professor of Mathematics at the University of Melbourne. Early life Michell was the son of John Michell (pronounced Mitchell), a miner, and his wife ...
Elasticity (physics)