Spatial dispersion of elastic waves in a bar characterized by tempered nonlocal elasticity

TL;DR

This paper explores how tempered fractional calculus can model spatial dispersion and attenuation of elastic waves in a nonlocal elastic bar, revealing similarities to time-fractional Zener models and unusual attenuation behaviors.

Contribution

It introduces a tempered fractional calculus framework to model spatial dispersion in nonlocal elasticity, providing numerical analysis of wave behavior in such materials.

Findings

Phase velocity dispersion resembles time-fractional Zener model

Unusual attenuation patterns observed in wave propagation

Numerical methods used to analyze complex dispersion relations

Abstract

We apply the framework of tempered fractional calculus to investigate the spatial dispersion of elastic waves in a one-dimensional elastic bar characterized by range-dependent nonlocal interactions. The measure of the interaction is given by the attenuation kernel present in the constitutive stress-strain relation of the bar, which follows from the Kr\"oner-Eringen's model of nonlocal elasticity. We employ a fractional power-law attenuation kernel and spatially temper it, to make the model physically valid and mathematically consistent. The spatial dispersion relation is derived, but it turns out to be difficult to solve, both analytically and numerically. Consequently, we use numerical techniques to extract the real and imaginary parts of the complex wavenumber for a wide range of frequency values. From the dispersion plots, it is found that the phase velocity dispersion of elastic waves in the tempered nonlocal elastic bar is similar to that from the time-fractional Zener model. Further, we also examine the unusual attenuation pattern obtained for the elastic wave propagation in the bar.