Fractional derivative modeling of time-varying viscosity materials considering initial loading ramp in real experiments

Abstract

The fractional derivative models with time-varying viscosity have been used in characterizing creep or relaxation properties of different viscoelastic material, and many combination models were presented using the Boltzmann superposition principle. However, those models defined as initial ones in this manuscript usually ignored the initial loading ramp, and the ideal-loading condition is commonly assumed as a step function in modeling. The real-loading conditions of tested samples are usually a ramp load followed by constant stress or strain. The difference in loading conditions between the theoretical modeling and experimental procedure strongly influences the models’ rheological property characterization and parameter determination. It is especially the case for the fractional derivative model due to its memory or history-dependent characters, even though the ramp time is short compared with the total experimental time. An application example of the Maxwell model with time-varying viscosity Scott–Blair model (TVSM) shows that the initial loading ramp has a strong influence. To solve this problem, the authors propose modified models of TVSM based on real-loading conditions. The relative errors between initial and modified models are presented. In addition, a history-dependent optimization algorithm for parameter determination is proposed. Three sets of polymer experimental data are employed to suggest that the fitting results of models disregarding initial ramp loads are unreliable. The modified model should be used for characterizing rheological behavior, as this leads to obtaining the best fitting results even for a short experimental time.

Stability of Nonlinear Implicit Differential Equations with Caputo–Katugampola Fractional Derivative

“…With the maturity of the development of fractional calculus theory, fractional differential equations have become research hot-spots for many mathematicians, and appear naturally in various fields such as fluid mechanics, fractals, environmental science, modeling and control theory, signal processing, bioengineering and biomedical science [1][2][3][4]. Due to the nonlocal properties of fractional derivatives, fractional differential equations can better describe complex processes and systems with genetic effects and memory.…”