Prandtl Stress Function ($\phi$)

The traction free BC is obviously difficult to satisfy if the cross-section is not a circle or an ellipse.

To simplify matters, we define the Prandtl stress function $\phi(x_1,x_2)\,$ using

${ \sigma_{13} = \phi_{,2} ~~;~~ \sigma_{23} = -\phi_{,1} }$

You can easily check that this definition satisfies equilibrium.

It can easily be shown that the traction-free BCs are satisfied if

${ \frac{d\phi}{ds} = 0 ~~\forall~(x_1,x_2) \in \partial\text{S} }$

where $s$ is a coordinate system that is tangent to the boundary.

If the cross section is simply connected, then the BCs are even simpler:

${ \phi = 0 ~~\forall~(x_1,x_2) \in \partial\text{S} }$

From the compatibility condition, we get a restrictionon $\phi$

${ \nabla^2{\phi} = C ~~\forall~(x_1,x_2) \in \text{S} }$

where $C$ is a constant.

Using relations for stress in terms of the warping function $\psi$, we get

${ \nabla^2{\phi} = -2\mu\alpha ~~\forall~(x_1,x_2) \in \text{S} }$

Therefore, the twist per unit length is

${ \alpha = -\frac{1}{2\mu} \nabla^2{\phi} }$

The applied torque is given by

${ T = -\int_{S} (x_1 \phi_{,1} + x_2 \phi_{,2}) dA \, }$

For a simply connected cylinder,

${ T =2 \int_{S} \phi dA \, }$

The projected shear traction is given by

${\tau = \sqrt{(\phi_{,1})^2+ (\phi_{,2})^2}}$

The projected shear traction at any point on the cross-section is tangent to the contour of constant $\phi\,$ at that point.

The relation between the warping function $\psi\,$ and the Prandtl stress function $\phi\,$ is

${ \psi_{,1} = \frac{1}{\mu\alpha} \phi_{,2} + x2 ~;~~ \psi_{,2} = -\frac{1}{\mu\alpha} \phi_{,1} - x1 }$

Membrane Analogy

The equations

$\nabla^2{\phi} = -2\mu\alpha ~~\forall~(x_1,x_2) \in \text{S}~~;~~~ \phi = 0 ~~\forall~(x_1,x_2) \in \partial\text{S}$

are similar to the equations that govern the displacement of a membrane that is stretched between the boundaries of the cross-sectional curve and loaded by an uniform normal pressure.

This analogy can be useful in estimating the location of the maximum shear stress and the torsional rigidity of a bar.

• The stress function is proportional to the displacement of the membrane from the plane of the cross-section.
• The stiffest cross-sections are those that allow the maximum volume to be developed between the deformed membrane and the plane of the cross-section for a given pressure.
• The shear stress is proportional to the slope of the membrane.