Fractional Burgers wave equation on a finite domain

TL;DR

This paper analyzes the dynamic response of a finite viscoelastic rod using Laplace transforms, revealing finite wave speeds and distinct displacement and stress profiles based on Burgers model classes.

Contribution

It introduces an analytical approach for the fractional Burgers wave equation on finite domains, highlighting differences between model classes in wave propagation and response shapes.

Findings

Finite wave propagation speed in second class models.

Classical displacement and stress profiles in first class models.

Distinct response behaviors between Burgers model classes.

Abstract

Dynamic response of the one-dimensional viscoelastic rod of finite length, that has one end fixed and the other subject to prescribed either displacement or stress, is analyzed by the analytical means of Laplace transform, yielding the displacement and stress of an arbitrary rod's point as a convolution of the boundary forcing and solution kernel. Thermodynamically consistent Burgers models are adopted as the constitutive equations describing mechanical properties of the rod. Short-time asymptotics implies the finite wave propagation speed in the case of the second class models, contrary to the case of the first class models. Moreover, Burgers model of the first class yield quite classical shapes of displacement and stress time profiles resulting from the boundary forcing assumed as the Heaviside function, while model of the second class yield responses that resemble to the sequence of excitation and relaxation processes.