The equivalence of the class of Rivlin-Sawyers equations and a class of stochastic models for polymer stress

This paper establishes the precise relationship between the macroscopic cla ss of factorized Rivlin-Sawyers equations and a class of microscopic-based stochastic models. The former is a well-established and popular class of rh eological models for polymeric fluids, while the latter is a more recently introduced class of rheological models which combines aspects of network an d reptation theory with aspects of continuum mechanic models. It is shown t hat the two models are equivalent in a defined sense under certain unrestri ctive assumptions. The first part of the proof gives the functional relatio nship between the linear viscoelastic memory function of the Rivlin-Sawyers model and the probability density for creation times of random variables i n the stochastic model. The main part of the proof establishes the relation ship between the strain descriptions in each model by showing that the diff erence in corresponding strain expressions can be made arbitrarily small us ing the appropriate weighted norm from spectral approximation theory. (C) 2 001 American Institute of Physics.