A bridge between the fractional viscoelasticity and time-varying viscosity model: physical interpretation and constitutive modeling

The viscoelastic model involving fractional operators has proved to be a powerful tool to describe the rheological phenomena of viscoelastic materials. It is of great importance to reveal the physical significance of fractional order for the future development in terms of fractional viscoelasticity. In this work, the principle of viscoelasticity equivalence is introduced by comparing the fractional Kelvin model and the time-varying viscosity Kelvin model, in which the time-varying viscosity function in applied to illustrate the physical significance of the fractional order. The results show that the material property undergoes a transformation from Hooke's elasticity to Newtonian viscosity as the order varies from 0 to 1. Moreover, a modified variable-order (VO) fractional model is established where the linear order function is found to be suitable for our model with high accuracy compared to the other three potential functions. Then, the applicability and reasonability of the proposed model is verified by a comparison of uniaxial tension and cyclic loading experiments for tough hydrogels. The physical significance of the fractional order is further investigated in view of the structure evolution and toughening mechanism of hydrogels, and it is found that, as the gel stretches, the weak bond breaks along with the stretch of polymer chains, making the gels exhibit more viscous behavior and produce a greater fractional order. In addition, during the unloading process, the polymer chains relax successively and the reversible metal-coordination bonds will be reformed, enabling the gels incline to harden, which is reflected by the evolution of the variable fractional order.

Automated summary

This study investigates the physical significance of fractional order in viscoelastic materials and reveals the transformation of material properties from elasticity to viscosity as the order varies. The proposed viscoelasticity equivalence principle compares the fractional Kelvin model and the time-varying viscosity Kelvin model to demonstrate the physical meaning of the fractional order. The results show that the structure evolution and toughening mechanism of hydrogels are closely related to the fractional order, with increasing order leading to more viscous behavior.