A homogeneous elliptical shaft may not warp under torsion

When a homogeneous prismatic shaft of an arbitrary cross section is subjected to a twisting moment at its end, an axial displacement, referred to as warping displacement, will be induced in it. The only exception known to date is that the shaft is an isotropic one with circular cross sections. In this work we find that elliptical cross sections may also exhibit zero warping. But now the shafts need to be rectilinearly orthotropic in which the ratio of two associated shear rigidities equals to the square of the aspect ratio of the ellipse. Physically, this means that the elastic orthotropy of the shaft can serve to compensate the geometric deviation from a circular cross-section to an elliptical one. The idea can be further generalized to show that the zero warping property also holds for a number of composite cylinders consisting of an elliptical core or cavity coated with many similarly elliptical layers of different materials.