 Airy stress function

The displayed scalar stress potential automatically satisfies the two in-plane static momentum equations without body force, because the mixed third derivatives cancel. For homogeneous isotropic linear elasticity, strain compatibility further makes $A$ biharmonic. Since stresses depend only on second derivatives, $A$ is unique only up to $ax+by+c$ on a connected domain. Conversely a function with all second derivatives zero is affine, proving that this is exactly the gauge freedom. It does not license adding an arbitrary biharmonic function, which generally changes stress.