Hierarchical models for viscoelasticity: Dynamic behaviour in the linear range

Viscoelastic elements derived from hierarchical models are introduced into classical models. Some of the resulting ''modified'' models are shown to yield the fractional differential descriptions of viscoelastic behaviour given previously by Friedrich, and to comply automatically with previously formulated thermodynamic constraints. The reasons underlying the previously observed differences between fractional differential and fractional integral formulations are clarified. The ''modified'' models are shown to be capable of describing experimentally observed dynamic viscoelastic behaviour over a wide frequency range, including multiple transitions, using a limited number of free parameters.