Oscillations and damping in the fractional Maxwell materials

TL;DR

This paper investigates the oscillatory and damping behaviors of fractional Maxwell viscoelastic systems, revealing how fractional relaxation influences system dynamics across different regimes, with applications in biomechanics.

Contribution

It introduces a comprehensive analysis of fractional Maxwell models, detailing their oscillatory responses and damping conditions in various regimes, extending classical oscillator theory to fractional systems.

Findings

Fractional Maxwell models exhibit unique oscillatory behaviors influenced by the fractional parameter.

Critical damping conditions are characterized for fractional systems, relevant to biomechanics.

Long-term and transient responses differ significantly from classical models.

Abstract

This paper examines the oscillatory behaviour of complex viscoelastic systems with power law-like relaxation behaviour. Specifically, we use the fractional Maxwell model, consisting of a spring and fractional dashpot in series, which produces a power-law creep behaviour and a relaxation law following the Mittag-Leffler function. The fractional dashpot is characterised by a parameter beta, continuously moving from the pure viscous behaviour when beta=1 to the purely elastic response when beta=0. In this work, we study the general response function and focus on the oscillatory behaviour of a fractional Maxwell system in four regimes: stress impulse, strain impulse, step stress, and driven oscillations. The solutions are presented in a format analogous to the classical oscillator, showing how the fractional nature of relaxation changes the long-time equilibrium behaviour and the short-time transient solutions. We specifically test the critical damping conditions in the fractional regime, since these have a particular relevance in biomechanics.