Airy stress function

Remarks

If $\varphi$ is of class $C^3$, then $\partial_1 \sigma_{11} + \partial_2 \sigma_{12} = \partial_1 \partial_2^2 \varphi - \partial_2 \partial_1 \partial_2 \varphi = 0$, and likewise $\partial_1 \sigma_{12} + \partial_2 \sigma_{22} = 0$; so the equilibrium equations without body force hold. Conversely, on a simply connected $\Omega$ every symmetric $C^1$ stress field with zero divergence has an Airy stress function, unique up to adding an affine function $a + b x_1 + c x_2$. For a homogeneous isotropic linear elastic material, the compatibility condition becomes the biharmonic equation $\Delta^2 \varphi = 0$.

Example: $\varphi = \tfrac{s}{2} x_2^2$ gives $\sigma_{11} = s$ and $\sigma_{12} = \sigma_{22} = 0$, uniform tension along $x_1$; it is biharmonic.

Non-example: the field $\sigma_{11} = x_1$, $\sigma_{12} = \sigma_{22} = 0$ has no Airy stress function, because $\partial_1 \sigma_{11} + \partial_2 \sigma_{12} = 1 \ne 0$, while every field given by the formulas above with $\varphi \in C^3$ is divergence-free.

In polar coordinates $(r, \theta)$ the formulas read $\sigma_{rr} = \frac{1}{r} \partial_r \varphi + \frac{1}{r^2} \partial_\theta^2 \varphi$, $\sigma_{\theta\theta} = \partial_r^2 \varphi$ and $\sigma_{r\theta} = -\partial_r \big( \frac{1}{r} \partial_\theta \varphi \big)$.