Page 513 - Introduction to Continuum Mechanics

Non-Newtonian Fluids 497

Thus, from Eqs. (8.9.2), we have, for all N

We see therefore that all AJV are objective. This is quite to be expected because these tensors
characterize the rate and the higher rates of changes of length of material elements at time /
which are independent of the observers.

8.14 Incompressible Simple Fluid

An incompressible simple fluid is an isotropic ideal material having the following constitu-
tive equation

where * depends on the past histories up to the current time t of the relative deformation
tensor C (. In other words, a simple fluid is defined by

where the index r - — «> to t indicates that the values of the functional H depends on all C r from
Q(x,- oo) to Q(x,f). We note that such a fluid is called "simple" because it depends only on
the history of the relative deformation gradient F f(r) = Vx' tensor (from which Cj(r) is
obtained), and not on the history of the higher gradient of the relative deformation tensor (e.g.,
VW).
Obviously, the functional H in Eq. (8.14.2) is to be the same for all observers (i.e.,
H*=H ). However, it can not be arbitrary, because it must satisfy the frame indifference
requirement. That is, in a change of frame,

Since Q(r) transforms in a change of frame as [see Eq. (8.12.6)]

therefore, the functional H [Cj(x,r);-oo<r<f] must satisfy the condition

We note that Eq. (8.14.5) also states that the fluid defined by Eq. (8.14.2) is an isotropic
fluid.

Any function or functional which obeys the condition given by Eq. (8.14.5) is known as an
isotropic function or isotropic functional.
The relationship between stress and .deformation given by Eq. (8.14.2) for a simple fluid is
completely general. In fact, it includes Newtonian incompressible fluid and Maxwell fluids as