Constitutive equations for polymer viscoelasticity derived from hierarchical models in cases of failure of time-temperature superposition

Hierarchical viscoelastic elements whose behaviour is intermediate between linear elasticity and Newtonian viscosity (springpots) have been introduced previously into classical analog models describing linear viscoelastic behaviour. This approach allows a concise description of typical polymer behaviour, including non-exponential relaxation, memory effects and hierarchy of viscoelastic transitions. This approach is extended in the present work to describe complex behaviour, including failure of time-temperature superposition in a semi-crystalline polymer, and the terminal transition from self-similar viscoelasticity to pure flow in a thermoplastic elastomer. 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 exponential 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.