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.
Highlights
The damping of free oscillations is a subject with a long and rich history
This paper examines the oscillatory behavior of complex viscoelastic systems with power law like relaxation behavior
We examine the oscillatory behavior of a viscoelastic element characterized by a fractional Maxwell model [1,2,3]
Summary
The damping of free oscillations is a subject with a long and rich history. there is less work available looking at how the damping of oscillations depends on the viscoelasticity of a system, and in particular, the types of viscoelasticity seen in biological tissues and complex composite materials. The attempts to rationalize a more complex relaxation response often apply a “brute force” fit with multiple relaxation times, the so called generalized Maxwell model Another common relaxation function that fits well with materials with a fractal or hierarchical structure is the power law. It is clear that in a great many situations the MittagLeffler function is as good a fit, if not better, than the powerlaw, asymptotic power law, or a stretched exponential It has the key benefits of a finite value at zero time, having few parameters, and having a compact Laplace transform which is useful for any analytical calculations. We first solve the classical Maxwell model to highlight the differences and similarities
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