Page 503 - Introduction to Continuum Mechanics

Non-Newtonian Fluids 407

Let

and

We have, (note, Cfat) = I)

The tensor Ai,A2,...A n are known as Rivlin-Ericksen tensors.
We see from the above equation that provided the Taylor series expansion is valid, the
A,/s determine the history of relative deformation. It should be noted however, that not all
histories of relative deformation can be expanded in Taylor series; For example, the stress
relaxation test in which a sudden jump in deformation is imposed on the fluid, has a history of
deformation which is not representable by a Taylor series.

Example 8.9.1
Find the Rivlin-Ericksen tensor for the uni-directional flows of Example 8.8.1.
Solution. We have, from Example 8.8.1

Thus, (see Eq. (8.9.3))