Exact evaluation of the integral involved in Doi-Edwards constitutive equations
A basic two-dimensional integral, used for establishment of a connection between the stress and the strain tensor, is evaluated exactly and in closed form in terms of elliptic integrals of the first and the second kind. This integral has not been evaluated before. Our result will allow exact analysis of entangled polymeric systems. A generalization of the basic integral is presented in the last section. Introduction. The Doi-Edwards model (Doi and Edwards, 1986), describing dynamics of entangled polymeric systems, was introduced many years ago. Since then, numerous approximating schemes were proposed for the analysis of such systems (Hassager and Hansen, [5]; Marin and Rasmussen, [6], etc.). To the best of my knowledge, nobody has managed so far to compute the basic integral exactly. This is done for the first time here. The problem was brought to my attention by A. Kharlamov. Procedure of evaluation. We need to compute the following integral: