Loss of ellipticity in the combined helical, axial and radial elastic deformations of a fibre-reinforced circular cylindrical tube

•Extension, inflation, torsion of a tube of transversely isotropic material.•Pressure, axial load and moment formulas.•Admissible directions of transverse isotropy for controllability and universality.•Illustrations for a prototype strain-energy function.

In this paper we consider theoretically the finite deformation of a circular cylindrical tube of a transversely isotropic elastic material, specifically the combined axial stretch, inflation and helical shear deformation, with particular reference to the failure of ellipticity. For a simple form of strain-energy function specific examples involving axial and radial directions of transverse isotropy are then considered, leading to different predictions of the onset of ellipticity failure.