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 coordinatesedit
If the coordinate basis is rectangular Cartesian $(\mathbf {e} _{1},~\mathbf {e} _{2})$ with coordinates denoted by $(x_{1},~x_{2})$ then the Airy stress function $(\varphi )$ is related to the components of the Cauchy stress tensor $({\boldsymbol {\sigma }})$ by
Alternatively, if we write the basis as $(\mathbf {e} _{x},\mathbf {e} _{y})$ and the coordinates as $(x,y)\,$, then the Cauchy stress components are related to the Airy stress function by
In polar basis $(\mathbf {e} _{r},\mathbf {e} _{\theta })$ with co-ordinates $(r,\theta )\,$, the Airy stress function is related to the components of the Cauchy stress via
Any polynomial in $x_{1}$ and $x_{2}$ 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.