On the torsion of a shaft made of polar isotropic or fibre-reinforced linearly elastic material

The fundamental solution of the linear elasticity problem of a non-polar isotropic shaft subjected to an externally applied torsional couple is extended to embrace effects of structural behaviour observed when the shaft is made of either polar isotropic or fibre-reinforced (transversely isotropic) material with fibre-bending stiffness. In the latter case, which is the subject of principal interest, the fibres are assumed parallel to the shaft central axis that is aligned with the axis of the externally applied torque. The shape of the shaft cross-section is essentially considered arbitrary, in the sense that full solution of polar material problem is subjected only to relevant conditions met in its classical, non-polar elasticity version. In dealing with polar isotropy, the obtained solution substantially enhances some initial analytical results that are already available in the literature. In dealing with polar transverse isotropy, the presented models and analytical results are new. It is verified that, in the latter case, the attained solution is exclusively dominated by fibre-twist features of deformation. This verification is underpinned by the observation that, as far as the torsion problem of interest is concerned, the unrestricted version of the theory of fibre-reinforced materials with fibre-bending stiffness provides a reliable relevant model, as well as equally reliable analytical results. However, due to the substantially smaller number of couple-stress elastic moduli involved in the restricted fibre-bending and fibre-splay versions of the theory, neither of the latter versions is able to do the same.