Analysis of Harmonic Wave Propagation in Fractional Derivative Viscoelastic Media Based on Time-Dependent Modulus of the P-Wave

Abstract

In the present paper, harmonic waves propagating in 3D isotropic viscoelastic media are analyzed using the fractional derivative Scott-Blair model, Kelvin-Voigt model, Maxwell model and standard linear solid model. It is known that only the first and second Lamé constants, or the bulk and shear moduli, appear in Hooke’s law for three-dimensional media, but not Young’s modulus or Poisson’s ratio. This indicates that the bulk and Lamé operators are the most intrinsic operators to express stress in terms of strain when studying wave propagation in 3D viscoelastic media. That is why in the present paper, the emphasis is made on the comprehensive analysis of time-dependent operators for Lamé parameters. In so doing, the fractional derivative models are utilized for defining the time-dependent modulus of the P-wave, which governs the velocity of the longitudinal wave. Asymptotic values of the wave velocities, their coefficients of attenuation and logarithmic decrements have been found for the case of absence of bulk relaxation.