Page 153 - Introduction to Continuum Mechanics

138 Left Cauchy-Green Deformation Tensor

3.25 Left Cauchy-Green Deformation Tensor

Let

where V is the left stretch tensor. The tensor B is known as the left Cauchy-Green deformation
tensor (also known as the Finger deformation tensor). We note that if there is no deforma-
tion,^ B= I.
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Since F = VR, and R R = I, it is easily verified that

Thus, one can calculate B directly from the deformation gradient F.
Substituting F = RU in Eq. (3.25.2), we obtain the relation between B and C as follows:

We also note that if n is an eigenvector of C with eigenvalue A, then Rn is an eigenvector
of B with the same eigenvalue A.
The components of B have simple geometric meanings which are described below:
T
Consider a material element dX = dSn, where n = R ej, R being the rotation tensor
associated with the deformation gradient F. Then from Eq. (3.23.4), we have

That is

That is

similarly,