Simultaneous Characterization of Relaxation, Creep, Dissipation, and Hysteresis by Fractional-Order Constitutive Models

Jun-Sheng Duan,Yang-Quan Chen,Di-Chen Hu

doi:10.3390/fractalfract5020036

Jun-Sheng Duan, Yang-Quan Chen + Show 1 more

Open Access

https://doi.org/10.3390/fractalfract5020036

Copy DOI

Abstract

We considered relaxation, creep, dissipation, and hysteresis resulting from a six-parameter fractional constitutive model and its particular cases. The storage modulus, loss modulus, and loss factor, as well as their characteristics based on the thermodynamic requirements, were investigated. It was proved that for the fractional Maxwell model, the storage modulus increases monotonically, while the loss modulus has symmetrical peaks for its curve against the logarithmic scale log(ω), and for the fractional Zener model, the storage modulus monotonically increases while the loss modulus and the loss factor have symmetrical peaks for their curves against the logarithmic scale log(ω). The peak values and corresponding stationary points were analytically given. The relaxation modulus and the creep compliance for the six-parameter fractional constitutive model were given in terms of the Mittag–Leffler functions. Finally, the stress–strain hysteresis loops were simulated by making use of the derived creep compliance for the fractional Zener model. These results show that the fractional constitutive models could characterize the relaxation, creep, dissipation, and hysteresis phenomena of viscoelastic bodies, and fractional orders α and β could be used to model real-world physical properties well.

Highlights

Theories and applications of fractional calculus have attracted much attention and acquired rapid developments during the last several decades because fractional calculus is appropriate for representing the memory and hereditary properties of materials and processes.Oldham and Spanier [1] completed the first monograph in 1974, and Ross [2] contributed the first proceedings in 1975
We consider the storage modulus, loss modulus, and loss factor, as well as their characteristics for different fractional constitutive models based on the thermodynamic requirements
For the fractional Zener model, we demonstrate that the storage modulus monotonically increases, while the loss modulus and the loss factor have symmetrical peaks for their curves against the logarithmic scale log(ω )

Summary

Introduction

Theories and applications of fractional calculus have attracted much attention and acquired rapid developments during the last several decades because fractional calculus is appropriate for representing the memory and hereditary properties of materials and processes. In [9,44], this relation was called the Scott-Blair model Macromolecule materials, such as butyl and polybutadiene, are typical viscoelastic bodies whose constitutive relations can be modeled by using fractional derivatives [45]. The purpose of this work is to consider dissipation, creep, relaxation, and hysteresis for a six-parameter fractional constitutive model and its particular cases. For the fractional Zener model, we demonstrate that the storage modulus monotonically increases, while the loss modulus and the loss factor have symmetrical peaks for their curves against the logarithmic scale log(ω ).

Fractional Models and Thermodynamic Requirements

Conclusions

Methods