Wave propagation in anisotropic viscoelasticity

TL;DR

This paper develops a mathematical framework for analyzing wave propagation in anisotropic viscoelastic materials, deriving explicit formulas for Green's functions and applying the theory to specific cases like isotropic and transversely isotropic media.

Contribution

It extends the theory of complete Bernstein functions to matrix-valued functions and applies it to derive explicit Green's function formulas for anisotropic viscoelasticity.

Findings

Explicit Green's function formulas for 3D isotropic viscoelastic media.

Representation of wave propagation using matrix-valued attenuation and dispersion functions.

Application to transversely isotropic media with a constant symmetry axis.

Abstract

We extend the theory of complete Bernstein functions to matrix-valued functions and apply it to analyze Green's function of an anisotropic multi-dimension\-al linear viscoelastic problem. Green's function is given by the superposition of plane waves. Each plane wave is expressed in terms of matrix-valued attenuation and dispersion functions given in terms of a matrix-valued positive semi-definite Radon measure. More explicit formulae are obtained for 3D isotropic viscoelastic Green's functions. As an example of an anisotropic medium the transversely isotropic medium with a constant symmetry axis is considered.