Prandtl stress function

Remarks

The equilibrium equation $\partial_x \sigma_{xz} + \partial_y \sigma_{yz} = 0$ holds automatically for any smooth $\phi$. The Poisson equation comes from compatibility, and the boundary condition from the traction-free lateral surface, which makes $\phi$ constant on $\partial \Omega$. Example: for a disc of radius $a$, $\phi = \frac{G\theta}{2}(a^2 - x^2 - y^2)$. Then $\Delta \phi = -2G\theta$, $\phi = 0$ on the circle, $\sigma_{yz} = G\theta x$ and $T = G\theta \cdot \pi a^4/2$. Non-example: $\phi = \frac{G\theta}{2}(a^2 - x^2 - y^2) + 1$ gives the same stresses but is not zero on the boundary, so it violates the normalisation and gives the wrong torque from $2\int \phi$. For a multiply connected cross-section, $\phi$ is constant on each boundary curve, with different constants.