Dispersion curves of viscoelastic plane waves and Rayleigh surface wave in high frequency range with fractional derivatives

The moduli of conventional elastic structural materials are extended to one of the viscoelastic materials through a modification whereby the dynamic moduli converge to the static moduli of elasticity as the fractional order approaches zero. By plotting phase velocity curves and group velocity curves of plane waves and Rayleigh surface wave for a viscoelastic material (polyvinyl chloride foam), the influence of the fractional order of viscoelasticity is examined. The phase and group velocity curves in the high frequency range were derived for longitudinal, transverse, and Rayleigh waves inherent to the viscoelastic material. In addition, the equation for the phase velocity was mathematically derived on the complex plane, too, and graphically illustrated. A phenomenon was found that, at the moment when the fractional order of the time derivative reaches an integer value 1, the curve on the complex plane becomes completely different, exhibiting snap-through behavior. We examined the mechanism of the snap-through mathematically. Numerical calculation examples were solved, and good agreement was confirmed between the numerical calculation and the analytical expression mentioned above. From the results of the numerical example, regularities were derived for the absolute value of the complex phase and group velocities on the complex plane.