Page 334 - Introduction to Continuum Mechanics

31S The Elastic Solid

In a change of frame, dx and d\ are related by Eq. (5.32.6), i.e.,

therefore, using Eqs. (i) and (iii), we have

Using Eq. (ii), the above equation becomes

Now, both dX and dX* denote the same material element at the fixed reference time t 0,
therefore, without loss of generality, we can take Q(t 0) = I, so that

Thus,

which is Eq. (5.32.9).

Example 5.32.4
Derive the transformation law for (a) the right Cauchy-Green deformation tensor and (b)
the left Cauchy-Green deformation tensor
Solution.
(a)The right Cauchy-Green tensor C is related to the deformation gradient F by the equation

Thus, from the result of the last example, we have

i.e, in a change of frame

That is, the right Cauchy-Green deformation tensor is not frame-indifferent (or, it is non-ob-
jective ).
(b) The left Cauchy-Green tensor B is related to the deformation gradient F by the equation