= Torsion stress function for a simply connected bar
{title2=$\Delta\phi=-2\mu\Omega,\quad\phi|_{\partial D}=0,\quad M=2\int_D\phi\,dA$}
For uniform twist $\Omega$ and displacement $(-\Omega yz,\Omega xz,w(x,y))$, <linear elasticity> gives $\tau_{xz}=\mu(w_x-\Omega y)$ and $\tau_{yz}=\mu(w_y+\Omega x)$. Their planar divergence vanishes by axial equilibrium. On a simply connected section there is therefore a stress function $\phi$ with $\tau_{xz}=\phi_y$ and $\tau_{yz}=-\phi_x$.
The mixed derivatives of $w$ imply $\Delta\phi=-2\mu\Omega$. The lateral axial traction is the tangential derivative of $\phi$, so a traction-free connected boundary makes $\phi$ constant; choose it to be zero. Finally the torque is $M=\int_D(x\tau_{yz}-y\tau_{xz})\,dA=-\int_D(x\phi_x+y\phi_y)\,dA=2\int_D\phi\,dA$, since its boundary term vanishes. Thus one scalar <Poisson equation> determines both the shear field and torsional stiffness. Multiply connected sections require additional constants and compatibility conditions on the separate boundary components.
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