Page 141 - Introduction to Continuum Mechanics

126 Calculation of the Stretch Tensors From the Deformation Gradient

so that

Since RU = VR = F, therefore, from Eq. (ii), we have

Thus, Rn is an eigenvector of V with eigenvalue A.

3.22 Calculation of the Stretch Tensors From the Deformation Gradient
From a given F, we have F = R U, thus,

That is,

From which the positive definite symmetric tensor U can be calculated as (See Examples
below).

Once U is obtained, R can be obtained from the equation

Since

therefore, [note that U is symmetric],

Thus, from Eq. (3.22.3),

Eq. (iii) states that the tensor R obtained from Eq. (3.22.3) is indeed an orthogonal tensor.
The left stretch tensor V can be obtained from