Distributed order fractional constitutive stress-strain relation in wave propagation modeling

TL;DR

This paper develops a distributed order fractional model for wave propagation in viscoelastic media, establishing fundamental solutions and their properties, linking wave speed to material characteristics, and covering various thermodynamically consistent models.

Contribution

It introduces explicit fundamental solutions for distributed order fractional wave equations and proves their existence and uniqueness across multiple thermodynamically acceptable models.

Findings

Explicit fundamental solutions are derived for distributed order fractional wave equations.

Wave propagation speed is linked to initial material properties.

Existence and uniqueness are established for several classes of fractional models.

Abstract

Distributed order fractional model of viscoelastic body is used in order to describe wave propagation in infinite media. Existence and uniqueness of fundamental solution to the generalized Cauchy problem, corresponding to fractional wave equation, is studied. The explicit form of fundamental solution is calculated, and wave propagation speed, arising from solution's support, is found to be connected with the material properties at initial time instant. Existence and uniqueness of the fundamental solutions to the fractional wave equations corresponding to four thermodynamically acceptable classes of linear fractional constitutive models, as well as to power type distributed order model, are established and explicit forms of the corresponding fundamental solutions are obtained.