# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1701.02155