Implementation of Fractional Calculus Using Hierarchical Models: Application to the Terminal Transition of a Complex Polymer

Abstract

Hierarchical viscoelastic elements whose behaviour is intermediate between linear elasticity and Newtonian viscosity (springpots) are incorporated into classical analog models describing linear viscoelastic behaviour. This approach is extended in the present work to describe the terminal transition from self-similar viscoelasticity to pure flow. Tschoegl’s formulation of a finite Gross Marvin ladder model is generalized and applied to other models, and this approach is compared with Friedrich’s method based on application of an exponentiel cutoff to the relaxation function. The method is illustrated using dynamic mechanical measurements on a triblock adhesive in isothermal frequency sweeps. This material displays thermorheological complexity precluding application of time-temperature superposition. Tschoegl’s formulation affords a better description of this material than Friedrich’s approach. The method described here is a useful alternative to time-temperature superposition requiring a limited number of adjustable parameters.