Michell solution
Encyclopedia
The Michell
solution is a general solution to the elasticity
equations in polar coordinates (
). The solution is such that the stress components are in the form of a Fourier series
in
.
Michell showed that the general solution can be expressed in terms of an Airy stress function of the form
The terms
and 
define a trivial null state of stress and are ignored.
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 [from J. R. Barber (2002) ].
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
where
is the Poisson's ratio
, and
is the shear modulus.
Note that you can superpose a rigid body displacement on the Michell solution of the form
to obtain an admissible displacement field.
John Henry Michell
John Henry Michell FRS was an Australian mathematician, Professor of Mathematics at the University of Melbourne.-Early life:...
solution is a general solution to the elasticity
Linear elasticity
Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...
equations in polar coordinates (
). The solution is such that the stress components are in the form of a Fourier seriesFourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
in
.Michell showed that the general solution can be expressed in terms of an Airy stress function of the form
The terms
and define a trivial null state of stress and are ignored.Stress components
The stressStress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...
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 [from J. R. Barber (2002) ].
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Displacement components
DisplacementsDisplacement field (mechanics)
A displacement field is an assignment of displacement vectors for all points in a region or body that is displaced from one state to another. A displacement vector specifies the position of a point or a particle in reference to an origin or to a previous position...
can be obtained from the Michell solution by using the stress-strainLinear elasticity
Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...
and strain-displacement
Linear elasticity
Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...
relations. A table of displacement components corresponding the terms in the Airy stress function for the Michell solution is given below. In this table
where
is the Poisson's ratioPoisson's ratio
Poisson's ratio , named after Siméon Poisson, is the ratio, when a sample object is stretched, of the contraction or transverse strain , to the extension or axial strain ....
, and
is the shear modulus.  |
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Note that you can superpose a rigid body displacement on the Michell solution of the form
to obtain an admissible displacement field.