Saint-Venant torsion of anisotropic shafts: Theoretical frameworks, extremal bounds and affine transformations

We study the Saint-Venant torsion of anisotropic shafts. Theoretical frameworks for torsion of anisotropic composite shafts are derived in terms of warping function, conjugate function as well as stress potential, parallel to the existing frameworks for torsion of isotropic shafts. We prove an extremal property for the torsional rigidity of anisotropic composite shafts. For homogeneous shafts, an affine coordinate transformation is introduced in the formulation, which demonstrates how the cross-sectional shape of the shaft is deformed (stretching and rotation) under the mapping, and how the warping field and the torsional rigidity of an anisotropic shaft are correlated to those of an isotropic one. We find that a certain class of anisotropic elliptical shafts, simply- or multiply-connected, will not warp under an applied torque. Of all homogeneous shafts with a given cross-sectional area and the same shear rigidity matrix, the torsional rigidity, associated with zero warping displacement, can be proven as extremal upper bounds. Finally, families of anisotropic shafts that are equivalent to isotropic ones, including elliptical and hollow elliptical shafts, and cylindrical shafts with specific cross-sections of parallelogram and triangle shape, are characterized.