The behaviour of wave propagation in linear thermo-viscoelastic media

We constructed a thermo-viscoelasticity equation based on Lord-Shulman (LS) thermoelasticity with the Kelvin-Voigt (KV) model for viscoelasticity. The plane-wave analysis predicts two compressional waves and a shear wave. These two compressional waves are the fast-P and slow-P diffusion/wave (the T-wave), which have similar characteristics to the fast- and slow-P waves of poroelasticity, respectively. To overcome the nonphysical phenomenon of high-frequency P-waves in the thermo-viscoelastic (KV model), we established the thermo-viscoelasticity equation by combining LS thermoelasticity and the Zener and Cole-Cole model of viscoelasticity. Plane-wave analysis predicts two inflection points on the dispersion and attenuation curves; these are mainly affected by thermal diffusion and viscoelasticity. The dispersion curves of both types of P waves have two-level limit velocities of high frequency, and their attenuation curves also feature two attenuation peaks. Selecting appropriate parameters can cause the two-level limit velocities of high frequency and attenuation peaks to move or overlap. Finally, we consider the experiment data of P-wave velocity varying with frequency of two kinds of sandstone. Indeed, a Cole-Cole fractional model is needed to obtain a good match. These results are helpful for studying the physics of thermo-viscoelasticity and for testing experimental data and numerical algorithms for wave propagation.