Dispersion curves for 3D viscoelastic beams of solid circular cross section with fractional derivatives

The aim is to extend the theory to a viscoelastic beam that satisfies stress-free surface boundary conditions. A viscoelastic material (polyvinyl chloride) was used in the numerical calculation, and the phase and group velocity curves were derived for a viscoelastic beam from the case without damping to the case with damping proportional to the first-order derivative with respect to time. Based on the preliminary data, the phase and group velocity curves were derived for a beam of solid circular cross section. As a result, it was confirmed that, as earlier pointed out for elastic materials, these curves were controlled by the phase velocity inherent to the material. Finally, with the phase velocity and the group velocity of the beam, regularities were derived for the absolute value of the complex velocity on the complex plane.

► A dispersive equation for a viscoelastic 3D beam is given from the derived motion equation.
► The phase velocity increase as the fractional order approaches 0, and decrease as that approaches 1.
► The relation between the absolute value and the real part value of the complex velocity are cleared.