Elasticity/Prandtl stress function

Prandtl Stress Function φ
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 restriction on $$\phi$$

{ \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.