An overview of fractional‐order calculus applied to viscoelasticity

Abstract

Over the past 80 yr, an initially obscure branch of calculus has gained some attention as a compact, mathematically convenient description of linear viscoelasticity. Rieman and Liouville developed the classical definitions of fractional‐order integration and differentiation. These definitions are fading‐memory, linear operators well‐suited for describing relaxation, dispersion, and attenuation effects. Early in this century, several authors suggested ad hoc fractional‐order differentials to describe viscoelastic effects. Scott Blair first suggested the Rieman and Liouville operators in the early 1940s. Several investigators have subsequently echoed Scott Blair’s suggestion. More recently, the fractional‐order calculus viscoelasticity formulation has been expanded to include dynamics and finite‐element formulations. Most recently, the formulation has led to the development of the thermorheologically complex model, a generalization of the thermorheologically simple material model, and the model has been related to molecular bond energy decay and fractals. Other recently developed engineering applications of fractional calculus include corrosion, aeroelasticity, and feedback control.