On Integral Type Constitutive Equation of Viscoelastic Fluids

Abstract

A non-linear constitutive equation is derived through the modification of the memory function in the Lodge equation. It is characterized by the following two assumptions. One is the assumption that the disentanglement rate depends on the time elapsed of entanglement points as well as on the deformation rate of segmental chains. The other is the assumption that the deformation rate is expressed in terms of the time derivative of the first invariant of the relative Finger tensor. Practically, the disentanglement process is assumed to be expresosed by the first order reaction model with the rate coefficient _??_. Here, λ is the relaxation time in the linear region and λ₀ is the characteristic time peculiar to nonlinear region. I_c-1(t; t′) is the first invariant of the relative Finger tensor.“·” denotes the differential with t at constant t′.
The main results obtained from the constitutive equation are as follows.
(1) Relaxation shear modulus at finite strain shows that the strength of the relaxation spectrum depends apparently on shear strain.
(2) Stress relaxation after the cessation of simple shearing flow shows that the strength of relaxation spectrum depends apparently on shear rate.
(3) The experimental results for “double-step” stress relaxation are described more appropriately by the proposed equation than the constitutive equation with “strain dependent relaxation spectrum”.
(4)“Overshoot” of shear stress and normalstress difference are predicted at the beginning of simple shearing flow.
Finally it may be concluded that the proposed constitutive equation describes more appropriately the observed non-linear viscoelastic phenomena than any other theories, and the two assumptions are appropriate.