Warping Function

A warping function in Saint-Venant torsion is a harmonic function on a plane cross section that describes axial displacement during uniform twisting of a homogeneous, isotropic, linearly elastic prismatic shaft. If is the twist per unit length, the axial displacement is (Lipton 1998, pp. 86-87). The warping function is determined by the geometry and enters the calculation of torsional rigidity.

Let be a bounded, connected plane domain whose boundary consists of disjoint smooth curves, and let be its outward unit normal vector. The defining boundary-value problem is

where is the two-dimensional Laplacian and . The Neumann boundary conditions express the absence of traction on the lateral surface (Lipton 1998, p. 87). They apply to every boundary component, including those surrounding holes. The divergence theorem gives the compatibility condition

A solution exists and is unique up to an additive constant. The normalization

selects a unique warping function.

The variational characterization uses the functional

The warping function minimizes over the Sobolev space of square-integrable real-valued functions whose first weak partial derivatives are also square integrable. No boundary values are prescribed in this minimization. The minimum is the geometric multiplier in the torsional rigidity , where is the shear modulus (Lipton 1998, p. 89). Its weak formulation is

for every . Expanding the squares and using this identity yields

This proves minimality and shows that all minimizers differ by a constant. In particular, testing with gives

where is the polar area moment of inertia about the origin. Consequently , with equality iff is constant. Any trial function therefore gives the upper bound .

For a disk centered at the origin, the normal derivative is zero and the normalized solution is . Thus

The same solution holds for a concentric annulus , with , giving

These cross sections undergo no axial warping. For the ellipse , with positive semiaxes and , the solution is

and its energy is

These formulas agree with the displacement and torsional rigidity given by Timoshenko and Goodier (1951, pp. 264-265), after dividing their axial displacement by . For , the nonzero level curves of lie on rectangular hyperbolas. The circular limit recovers .

The harmonic displacement warping function differs from the Prandtl stress function used in the Poisson problem formulation of torsional rigidity. For a simply connected domain, define the normalized Prandtl stress function by

Then , and the additive constant can be chosen to impose zero Dirichlet boundary conditions. With this normalization . For a domain with holes, is constant on each boundary component, but these constants need not be equal.