A Fractional Derivative Interpretation of Viscoelastic Rubbers. I. Thermodynamically Consistent Fractional-derivative Models for Finite Strain

In this paper, the thermodynamically consistent fractional derivative model is developed for viscoelastic materials such as rubbers under large deformations. As the basis of fractional derivative model, the Maxwell model for finite strain is proposed as a natural extension of the Maxwell model for infinitesimal strain. In the proposed Maxwell model, the stress-strain relationship is defined for both the elastic and viscous components. The viscous stress satisfies the thermodynamic requirement that entropy production is always positive except at zero stress. The two components are combined into the linear evolution equation based on the fact that the stress of the two components is the same. Through the generalized Maxwell model, several types of fractional-derivative stress-strain relations of order less than one have been derived. These relations include a new relation as well as relations already known in the literature. It is shown that one of the relations proposed by Fukunaga and Shimizu (2015, J Comput Nonlin Dyn, vol. 10, 061002) is a special cases of the new relation. For both these fractional derivative models and the generalized Maxwell model under finite strain, the viscous dissipation can be explicitly calculated in thermodynamically consistent manner. The newly obtained model is expected to serve as the powerful tools for understanding the stress behavior of rubberlike materials.