Dispersion of small amplitute stress waves in pre-stressed elastic, visco-plastic cylindrical bars
The effect of geometrical dispersion on the propagation of longitudinal harmonic stress waves in prestressed elastic, visco-plastic bars is investigated. The solution involves the Hunter and Johnson approximation in which radial and axial displacements and all components of stress tensor are expanded as power series in the radial coordinate. It is shown that the dispersion of the sinusoidal waves is generated by the two simultaneously acting phenomena. These are the purely geometrical dispersion produced by the influence of a free boundary of the bar and the viscous dispersion caused by the viscous properties of material. The obtained solution embraces the combined effect of the phenomena mentioned above. A numerical example is presented in which constants of material characteristic for mild steel were used. The result of the computations provides a plot of phase and group velocities and damping coefficient against the frequency of harmonic waves. The present solution has been compared with the Hunter and Johnson solution for strain-hardening elastic-plastic material and with the exact Pochhammer-Chree solution for elastic rods.