Viscoelastic Behavior Analysis and Application of the Fractional Derivative Maxwell Model

Abstract

A mathematical model of the viscoelastic phenomenon, employing the fractional derivative Maxwell Model (FDMM) is analyzed in order to determine its consistency with thermodynamic principles. In particular, the development of constraints on the parameters of the model guarantees that the FDMM will predict a nonnegative rate of energy dissipation and a nonnegative internal work. The creep compliance and relaxation modulus of the FDMM are obtained in an easier way. It is found that the monotonic non-decreasing creep compliance and monotonic non-increasing relaxation moduli prove a well-behaved viscoelastic phenomenon of the FDMM. The analysis of relaxation modulus indicates that the FDMM represents viscoelastic fluid behavior, with arbitrary fractional derivatives of stress and strain, only if the thermodynamic constraints are satisfied. The steady state sinusoidal response expressions derived in this article are verified using comparisons with experimental force-displacement loops.