Attenuation and Shock Waves in Linear Hereditary Viscoelastic Media; Strick–Mainardi, Jeffreys–Lomnitz–Strick and Andrade Creep Compliances

Abstract

Dispersion, attenuation and wavefronts in a class of linear viscoelastic media proposed by Strick and Mainardi (Geophys J R Astr Soc 69:415–429, 1982) and a related class of models due to Lomnitz, Jeffreys and Strick are studied by a new method due to the author. Unlike the previously studied explicit models of relaxation modulus or creep compliance, these two classes support propagation of discontinuities. Due to an extension made by Strick, either of these two classes of models comprise both viscoelastic solids and fluids. We also discuss the Andrade viscoelastic media. The Andrade media do not support discontinuity waves and exhibit the pedestal effect.

Appendix

According to Bernstein’s Theorem, every LICM function $f(t)$ has an integral representation

$$\begin{aligned} f(t) = \int\limits _{[0,\infty [} {\rm e}^{-r \, t}\, \mu ( {\rm d}r) \end{aligned}$$

(62)

where $\mu$ is a positive Radon measure satisfying the inequality

$$\begin{aligned} \int\limits _{[0,\infty [} \frac{\mu ({\rm d}r)}{1 + r} < \infty \end{aligned}$$

Bernstein’s Theorem implies that every Bernstein function $g(t)$ has an integral representation

$$\begin{aligned} g(t) = a + b\, t + \int \limits_{]0,\infty [} ( 1 - {\rm e}^{-r\,t}) \, \nu ( {\rm d}r) \end{aligned}$$

(63)

where $\nu$ is a positive Radon measure satisfying the inequality

$$\begin{aligned} \int\limits _{]0,\infty [} \frac{r\, \nu ( {\rm d}r)}{1 + r} < \infty \end{aligned}$$