Theoretical and experimental analysis of longitudinal wave propagation in cylindrical viscoelastic rods

Wave propagation in viscoelastic rods is encountered in many applications including studies of impact and fracture under high strain rates and characterization of the dynamic behavior of viscoelastic materials. For viscoelastic materials, both material and geometric dispersion are possible when the diameter of the rod is of the same order as the wavelength. In this work, we simplify the Pochhammer frequency equation for low and intermediate loss viscoelastic materials and formulate corrections for geometric dispersion for both the phase velocity and attenuation. The formulation is then experimentally verified with measurements of the phase velocity and attenuation in commercial polymethylmethacrylate rods that are 12 and 6.4 mm in diameter. Without correcting for geometric dispersion, the usable frequency range for determining the phase velocity and attenuation for the 12 mm rod is about 20 kHz, and about 35 kHz for the 6.4 mm rod. Using the correction procedure developed here, it was possible to accurately determine the phase velocity and attenuation up to frequencies exceeding 55 kHz for the 12 mm rod and 65 kHz for the 6.4 mm rod. These corrections are applicable to many polymers and other viscoelastic materials. From thereon, the viscoelastic properties of the material can be determined over a wide range of frequencies.