Thermoelasticity and P-wave simulation based on the Cole-Cole model

The classical Zener model of thermoelasticity can be represented by a mechanical (or viscoelastic) model based on two springs and a dashpot, commonly called standard-linear solid, whose parameters depend on the thermal properties and a relaxation time, and yield the isothermal and adiabatic velocities at the low- and high-frequency limits. This model differs from the more general Lord-Shulman theory of thermoelasticity, whose low-frequency velocity is the adiabatic one. These theories are the basis of thermoelastic attenuation in inhomogeneous media, with heterogeneities much smaller than the wavelength, such as Savage's theory of thermoelastic dissipation in a medium with spherical pores. In this case, the shape of the relaxation peak differs from that of the Zener and Lord-Shulman models. In these effective homogeneous media, the anelastic behavior of real materials can better be described by using a stress-strain relation based on fractional derivatives. In particular, wave propagation (dispersion and attenuation) is well described by a Cole-Cole stress-strain equation, as illustrated by the agreement with Savage's theory. We propose a time-domain algorithm based on the Grunwald-Letnikov numerical approximation of the fractional time derivative involved in the time-domain representation of the Cole-Cole model. The spatial derivatives are computed with the Fourier pseudospectral method. We verify the results by comparison with the analytical solution, based on the Green function. The numerical example illustrates wave propagation at an interface separating a porous medium and a purely solid phase.