Page 143 - Introduction to Continuum Mechanics

128 Right Cauchy-Green Deformation Tensor

T
with respect to the principal axes of ¥ V. After that, one then uses the transformation law
discussed in Chapter 2 to obtain the matrices with respect to the e/ basis. (See Example 3.23.1
below).

Example 3.22.2

and from RjU = R2U , it follows,

(b) Since

thus,

Noting that (R VR') is symmetric, from the result of part (a), we have
R= R

From the decomposition theorem we see that what is responsible for the deformation of a
volume of material in a continuum in general motion is the stretch tensor, either U (the right
2
2
stretch tensor ) or V (the left stretch tensor). Obviously, U and V also characterize the
deformation, as are many other tensors related to them. In the following sections, we discuss
those tensors which have been commonly used to describe finite deformations for a continuum.

3.23 Right Cauchy-Green Deformation Tensor

Let