Page 166 - Introduction to Continuum Mechanics

Kinematics of a Continuum 151

where e 0/ denotes base vectors at the reference position and e,- those at the current position.
Substituting

in the equation d\ = FrfX , we obtain

etc. Thus,

These equations are equivalent to Eqs. (3.30.4).
The matrix

is based on two sets of bases, one at the reference configuration (e or, e^, e oz) and the other
the current configuration (e r ,60 ,e 2 ). The components in this matrix is called the two point
components of the tensor F with respect to ( e r ,e# ,e 2 ) and ( e or , e^, e 02 ).
By using the definition of transpose of a tensor, one can easily establish Eqs. (3.30.5 ) from
Eq. (3.30.4). [see Prob. 3.73]
The components of the left Cauchy-Green tensor, with respect to the basis at the spatial
position x. can be obtained as follows. From the definition B = FF , and by using Eqs. (3.30.4)
and (3.30.5 ) we obtain