Investigation on Fractional and Fractal Derivative Relaxation- Oscillation Models

Abstract

Stress relaxation and damped oscillation of viscoelastic soft materials are characterized with memory effect which is very difficult to describe with the classical integer-order derivative equation models. This study introduced novel relaxation-oscillation models for soft materials by using positive fractional derivative and fractal derivative and investigated their properties against the corresponding fractional derivative model through analytic and numerical solutions. In the relaxation process, the stretched exponential stress relaxation of the fractal derivative model is found to decay fastest, while the slowest decay is with the positive fractional derivative model. In the oscillation process, the fractional and the positive fractional derivative models are observed frequency-dependent dissipative oscillation. We also found that the fractional derivative model is more dissipative than the positive fractional derivative one.