Study on the performance of variable-order fractional viscoelastic models to the order function parameters

Abstract

In variable-order fractional calculus, the order is allowed to be in a functional form, which provides a novel way to describe the time-dependent physical processes. In this work, the performance of variable-order fractional viscoelastic models to the order function parameters is comprehensively investigated. Firstly, the variable-order fractional constitutive model is deduced from the theory of constant-order fractional viscoelasticity. The effect of order function is studied which shows that the linear order function can effectively describe the viscoelastic properties. Then the variable-order fractional viscoelastic models are developed considering three kinds of viscoelastic behaviors, including stress-strain relationship, creep, and stress relaxation. The sensitivity of the viscoelastic behaviors to the parameters in the linear order function is analyzed in detail. Finally, the effectiveness of the proposed models is validated by comparing them to the existing experimental data of typical viscoelastic behaviors, and the laws of the order function for each case are discussed based on the fractional viscoelastic theories. It is proved that the variable-order fractional models can give an accurate description of the viscoelastic behaviors under various loading conditions, and the change of the fractional order can visualize the evolution of the mechanical properties of the material.