<abstract xmlns="http://www.w3.org/1999/xhtml"><p>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.</p></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.

Analysis of models for viscoelastic wave propagation

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Analysis of models for viscoelastic wave propagation"}

We consider the problem of waves propagating in a viscoelastic solid. For the material properties of the solid we consider both classical and fractional...

Analysis of models for viscoelastic wave propagation

Published Online: Oct 03, 2018

Page range: 55 - 96

Received: Dec 01, 2017

Accepted: Mar 09, 2018

DOI: https://doi.org/10.21042/AMNS.2018.1.00006

Keywords
Hyperbolic PDE, viscoelasticity, Laplace transforms, semigroups of operators, stability analysis, fractional derivatives

© 2018 Thomas S. Brown et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Analysis of models for viscoelastic wave propagation

Published Online: Oct 03, 2018

Page range: 55 - 96

Received: Dec 01, 2017

Accepted: Mar 09, 2018

DOI: https://doi.org/10.21042/AMNS.2018.1.00006

KeywordsHyperbolic PDE, viscoelasticity, Laplace transforms, semigroups of operators, stability analysis, fractional derivatives

© 2018 Thomas S. Brown et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
Hyperbolic PDE, viscoelasticity, Laplace transforms, semigroups of operators, stability analysis, fractional derivatives