Scalar potential function that can be used to find the stress.
Satisfies equilibrium in the absence of body forces.
Only for two-dimensional problems (plane stress/plane strain).
Airy stress function in rectangular Cartesian coordinates[edit | edit source]
If the coordinate basis is rectangular Cartesian with coordinates denoted by then the Airy stress function is related to the components of the Cauchy stress tensor by
Alternatively, if we write the basis as and the coordinates as , then the Cauchy stress components are related to the Airy stress function by
Airy stress function in polar coordinates[edit | edit source]
In polar basis with co-ordinates , the Airy stress function is related to the components of the Cauchy stress via
Something to think about ...
Do you think the Airy stress function can be extended to three dimensions?
Note that the stress field is independent of material properties in the absence of body forces (or homogeneous body forces).
Therefore, the plane strain and plane stress solutions are the same if the boundary conditions are expressed as traction BCS.
In terms of the Airy stress function
or,
or,
The stress function is biharmonic.
Any polynomial in and of degree less than four is biharmonic.
Stress fields that are derived from an Airy stress function which satisfies the biharmonic equation will satisfy equilibrium and correspond to compatible strain fields.