Page 336 - Introduction to Continuum Mechanics

320 The Elastic Solid

Example 5.33.1

Assume that for some elastic medium, the Cauchy stress T is proportional to the right
Cauchy-Green tensor C. Show that this assumption does not result in a frame-indifferent
constitutive equation and is therefore not acceptable.
Solution, The assumption states that,
for the starred frame:

and for the un-starred frame:

where we note that since the same material is considered by the two frames, therefore the
proportional constant must be the same. Now,
T
T * = QTQ [See Eq. (5.33.1)] and C * = C [See Eq. (5.32.11)]
therefore, from Eq. (i)

so that from Eq. (ii) for all Q(t)

The only T for the above equation to be true is T =1. Thus, the law is not acceptable.
More generally, if we assume the Cauchy stress to be a function of the right Cauchy Green
tensor, then for the starred frame T * = f(C *), and for the un-starred frame, T = f(C), where
again, f is the same function for both frames because it is for the same material. In a change
of frame,

That is, again

So that Eq. (i) is not acceptable.

Example 5.33.2

If we assume that the second Piola-Kirchhoff stress tensor T is a function of the right
Cauchy-Green deformation tensor C. Show that it is an acceptable constitutive equation.

Solution. We have, according to the assumption