Page 512 - Introduction to Continuum Mechanics

496 Transformation law for the Rivlin-Ericksen Tensors under a Change of Frame

Equations (8.12.5),(8.12.6) and (8.12.7) show that the relative right stretch tensor, the relative
right Cauchy-Green deformation tensor Q and its inverse Q~ (the relative Finger tensor)
-1
are objective. On the other hand, \ t E t and B f are nonobjective. We note, this situation is
different from that of the deformation tensor using a fixed reference configuration [See Section
5.31],
From Eq. (8.12.4) and (8.11.11), one can also show that in a change of frame

which shows, as expected that the spin tensor is non-objective.
T
Using Eq. (8.12.11), one can derive for any objective tensor T (i.e., T* = Q(t)TQ (t)) that

is objective, that is, [see Prob. 8.22]

The expression given in (8.12.12) is known as the Jaumann derivative of T which will
be discussed further in a later section.

8.13 Transformation law for the Rivlin-Ericksen Tensors under a Change of
Frame

we obtain

and in fact, for all N,