Comparative Analysis of Viscoelastic Models Involving Fractional Derivatives of Different Orders

Mathematics Subject Classification: 74D05, 26A33

In this paper, a comparative analysis of the models involving fractional derivatives of di®erent orders is given. Such models of viscoelastic materials are widely used in various problems of mechanics and rheology. Rabotnov's hereditarily elastic rheological model is considered in detail. It is shown that this model is equivalent to the rheological model involving fractional derivatives in the stress and strain with the orders proportional to a certain positive value less than unit. In the scienti¯c literature such a model is referred to as Koeller's model. Inversion of Rabotnov's model developed by himself based on algebra of operators results in similar rheological dependences. Inversion of Koeller's model carried out using Miller's theorem coincides inherently with Rabotnov's inversion procedure.

∗ This paper has been partially supported by the Russian Foundation for Basic Research under Grant No. 05-08-17936.