Saint-Venant torsion of exponentially graded micropolar circular and annular bars

In this paper, we consider Saint-Venant torsion of a functionally graded micropolar (Cosserat) beam, where the elastic moduli vary within the plane of the beam cross-section. We state a well-posedness result for the corresponding Neumann boundary-value problem in the weak Sobolev-space setting and show the existence and uniqueness of the solution up to the natural rigid micropolar mode. In view of symmetry, the circular and annular cases reduce to an ordinary differential equation for the radial amplitude. Numerical examples are presented for the stress and couple-stress fields and for the torsional rigidity. The characteristic micropolar length associated with the chosen material parameters is compared with the size of the cross-section, and the classical limit is recovered as a special case.