Applications of Fractional Calculus to the Theory of Viscoelasticity

Abstract

The connection between the fractional calculus and the theory of Abel’s integral equation is shown for materials with memory. Expressions for creep and relaxation functions, in terms of the Mittag-Leffler function that depends on the fractional derivative parameter β, are obtained. These creep and relaxation functions allow for significant creep or relaxation to occur over many decade intervals when the memory parameter, β, is in the range of 0.05–0.35. It is shown that the fractional calculus constitutive equation allows for a continuous transition from the solid state to the fluid state when the memory parameter varies from zero to one.