Prandtl Stress Function

The Prandtl stress function is a stress function whose first partial derivatives represent the shear stresses in Saint-Venant torsion. For a homogeneous, isotropic, linearly elastic shaft with shear modulus , twist per unit length , and axis in the -direction, use the normalized scalar function defined by

The dimensional stress potential is . These formulas satisfy the shear equilibrium equation identically (Timoshenko and Goodier 1951, pp. 260-262).

For a bounded, simply connected plane cross section with a boundary consisting of smooth curves, the normalized stress function is the unique solution of the Poisson problem

Here is the two-dimensional Laplacian. The zero Dirichlet boundary conditions fix the additive constant allowed by the traction-free sides. The torsional rigidity is , where

and the torque is .

The same solution has a variational characterization. On the real Sobolev space of functions with zero boundary values, define

The stress function is the unique maximizer, and

Thus , and every admissible trial function gives the lower bound . This is the stress-potential form of the variational formulation given by Lipton (1998, p. 89). It complements the upper bound obtained by minimizing the displacement energy of the warping function.

For a disk , with , the solution and geometric multiplier are

For an ellipse , with , they are

The elliptic solution is given by Timoshenko and Goodier (1951, pp. 263-264).

With the sign convention above, the warping function and normalized stress function satisfy

The normalized stress function used in the torsional rigidity formulas with is .

For a cross section with holes, choose on the outer boundary. On the boundary of the th hole, is an unknown constant . Single-valuedness of the warping function imposes

where is the area of that hole and the unit normal vector points out of the material into the hole. The geometric multiplier becomes

These additional boundary conditions and hole terms are essential (Franců and Rozehnalová-Nováčková 2015, pp. 8-12). For a concentric annulus , with , the solution is still . Its inner boundary value is , and the corrected formula gives .