A link between a variable-order fractional Zener model and non-Newtonian time-varying viscosity for viscoelastic material: relaxation time

In this work, by constructing the equivalent viscoelasticity between a fractional Zener model and a time-varying viscosity Zener model, we obtain the time-varying viscosity of creep and relaxation response. The derived time-varying viscosity can well interpret the physical meaning of the evolution of material from Hooke body to Newtonian fluid body. With the relaxation time gradually becoming larger, the viscosity of creep is closer to the viscosity of relaxation. The relaxation modulus and creep compliance of a time-varying viscosity Zener model obtained by inserted viscosity of creep and relaxation approximate that of a fractional Zener model. And the variation of fractional order is accompanied by the transformation of viscoelasticity. Based on the importance of relaxation time, a variable-order function related to relaxation time is proposed, and it can well interpret the physical meaning of evolution of viscoelasticity. Then, it also can reveal a process where the viscosity is consumed to resist deformation in rheology. The viscosity increases with an increase in time and a decrease in the fractional order. Finally, based on the variable-order function, a variable-order fractional viscoelastic Zener model is proposed, and it is in well agreement with experimental data.