Extension of Fractional Calculus Theory in Modeling the Memory Effects of Viscoelastic Polymeric Materials

Abstract

The complex history-dependent behavior exhibited by viscoelastic polymeric materials, known as the memory effect, has long posed a challenge for the accurate prediction of their mechanical properties. Classical linear viscoelasticity theory has inherent theoretical limitations in describing such non-local and cross-scale characteristics. Fractional calculus operators, due to their intrinsic non-locality and hereditary properties, provide a more physically consistent framework for the mathematical description of memory effects. This paper systematically elaborates on the extension of this theory in related modeling. First, it analyzes the physical mechanisms of memory effects and the shortcomings of classical models, establishing the physical significance of fractional operators. Subsequently, a generalized fractional constitutive model is constructed, revealing the cross-scale evolution characteristics of the Mittag-Leffler-type memory kernel, and extending the description to multi-field coupling and non-linear theories. Finally, the strategies for physical identification of model parameters, numerical implementation approaches, and current frontier challenges are discussed. Research indicates that fractional calculus theory can effectively and uniformly describe the memory behavior of polymeric materials from microscopic motion to macroscopic response through continuous relaxation spectrum characterization and flexible memory weight assignment, laying a significant foundation for the development of a new generation of viscoelastic constitutive theories.