Dynamic measurements in long-memory materials: Fractional calculus evaluation of approach to steady state

Fractional calculus descriptions of polymer viscoelasticity are becoming increasingly popular, as they allow a concise description of non-Debye relaxation and memory of strain history using a minimal parameter set. Use of fractional calculus to this end is frequently restricted to descriptions of dynamic behaviour, for example in dynamic mechanical thermal analysis ( DMTA), where the dependence of the complex modulus on frequency can be expressed algebraically in closed form. However, this approach is only valid in the steady state. The material's approach to the steady state, and the effect of the slowly-decaying transient on DMTA measurements, are addressed here. "Data" are generated by integration of the time-domain fractional integral constitutive equation describing the fractional Zener model, and are analysed by a procedure inspired by a commercial DMTA instrument manual. Results show that the frequency dependence of the decay of the initial transient leads to difficulties in retrieving a single parameter set from the data, demonstrating that a specific procedure is required when evaluating long-memory materials.