Page 337 - Introduction to Continuum Mechanics

Constitutive Equation for an Elastic Medium under Large Deformation. 321

and

where we demand that both frames (the unstarred and the starred) have the same function f
for the same material. Now, in a change of frame, the deformation gradient F and the Cauchy
stress tensor T transform in accordance with the following equation:

Thus, the second Piola-Kirchhoff stress tensor transforms as [See Prob.5.98]

Therefore, in a change of frame, the equation

transforms into

which shows that the assumption is acceptable. In fact, it can be shown that Eq. (5.33.5) is the
most general constitutive equation for an anisotropic elastic solid [See Prob. 5.100].

Example 5.32.3

If we assume that the Cauchy stress T is a function of the left Cauchy Green tensor B, is it
an acceptable constitutive law?
Solution. For the starred frame,

and for the un-starred frame,

where we note both frames have the same function f. In a change of frame, (see
Example 5.32.4, Eq. (5.32.13)),

Thus,

That is