Oscillations and damping in the fractional Maxwell materials

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.

J. or Rheology 2017

References in corpus (3)

• Power laws in microrheology experiments on living cells: comparative analysis and modelling
• Stress relaxation, dynamics and plasticity of transient polymer networks
• Generalized Elastic Model yields Fractional Langevin Equation

Cited by in corpus (5)

• Fractional viscoelastic models for power-law materials
• Viscoelastic Dynamics in Holography
• Slow-dissipation limit of the harmonic oscillator with general power-law damping
• A Linear Time-Variant Rheological Model for Frictional Aging, Stress Relaxation, and Creep
• Evidence that self-similar microrheology of highly entangled polymeric solutions scales robustly with, and is tunable by, polymer concentration