Page 511 - Introduction to Continuum Mechanics

Non-Newtonian Fluids 495

then, the tensor A is said to be objective, or frame indifferent (i.e., independent of observers).
From Eq. (8.12.1), we have

and

Now, use the inverse of Eq. (i), we get

Thus,

This is the transformation law for F,(r) under a change of frame. We see that this tensor is not
objective.

This equation can be written:

In the above equation, the tensor inside the first bracket is an orthogonal tensor and the tensor
inside the second bracket is a symmetric tensor. Since the polar decomposition for F* is
unique, therefore, we have

It is a simple matter to show the following transformation laws under a change of frames: