Analysis of models for viscoelastic wave propagation

TL;DR

This paper compares classical and fractional viscoelastic models for wave propagation, analyzing their stability, deriving time-domain estimates, and illustrating differences through numerical experiments.

Contribution

It develops a comprehensive theoretical framework linking Laplace domain and time-domain analyses for various viscoelastic models, including coupled and fractional versions.

Findings

Stability results for different models in Laplace and time domains

Sharper time-domain estimates using semigroup theory

Numerical experiments showing model differences and parameter effects

Abstract

We consider the problem of waves propagating in a viscoelastic solid. For the material properties of the solid we consider both classical and fractional differentiation in time versions of the Zener, Maxwell, and Voigt models, where the coupling of different models within the same solid are covered as well. Stability of each model is investigated in the Laplace domain, and these are then translated to time-domain estimates. With the use of semigroup theory, some time-domain results are also given which avoid using the Laplace transform and give sharper estimates. We take the time to develop and explain the theory necessary to understand the relation between the equations we solve in the Laplace domain and those in the time-domain which are written using the language of causal tempered distributions. Finally we offer some numerical experiments that highlight some of the differences between the models and how different parameters effect the results.