Page 514 - Introduction to Continuum Mechanics

498 Special Single Integral Type Nonlinear Constitutive Equations

special cases. In this most general form, only very special flow problems can be solved. A class
of such flows, called the viscometric flow, will be considered in Part C, using this general form
of constitutive equation. However, in the following few sections, we shall first discuss some
special constitutive equations. Some of these constitutive equations have been shown to be
approximations to the general constitutive equation given in Eq. (8.14.2) under certain
conditions (slow flow and/or fading memory). They can also be considered simply as special
fluids. For example, a Newtonian incompressible fluid can be considered either as a special
fluid by itself or as an approximation to the general simple fluid when it has no memory of its
past history of deformation and is under slow flow condition relative to its relaxation time
(which is zero).

8.15 Special Single Integral Type Nonlinear Constitutive Equations

In Section 8.4, we see that the constitutive equation for the linear Maxwell fluids is defined
by

where E is the infinitesimal strain tensor measured with respect to the configuration at time
t. It can be shown that for small deformations, (see Example 8.15.2 below)

Thus, the following two nonlinear viscoelastic fluids represent natural generalizations of the
linear Maxwell fluid in that they reduce to Eq. (8.15.1) under small deformation conditions.

and

where

and/(s) may be given by any one of Eqs. (8.4.9), (8.4.10) and (8.4.11).
We note that since Q(r) is an objective tensor, therefore the constitutive equations defined
by Eqs. (8.15.2) and (8.15.3) are frame indifferent. We note also that even though the fluids
defined by Eqs. (8.15.2) and (8.15.3), with/j =/ 2 have the same behaviors at small deforma-
tion, they are two different nonlinear viscoelastic fluids, behaving differently at large
deformation. Furthermore, if we treat f\(s) and/2(s) as two different memory functions, then