Kinematics and Mechanics of Large Elastic Deformations - Continuum Mechanics of Anisotropic Materials - page 298

U

Deformed, but not rotated

Undeformed

F

R

R

V

Deformed and rotated

Rotated, but not deformed

Fig. 11.9 The deformation of a square by F , illustrating the polar decomposition of F

Example Problem 11.3.1
Determine the polar decomposition of the deformation gradient tensor

2
4

3 p 10
020
001

3
5 :

F ¼

Solution : The squares or the right and left stretch tensors are calculated directly
from F , thus

2
4

3 p 0 3

5 ;

2
4

3
5 :

420
240
001

3

3 p 50
001

U 2 F T V 2 F T ¼ F ¼ ¼ F ¼

The square roots of these two tensors are constructed by transforming the tensor
to its principal axes where the eigenvalues are all positive, then the square root of
the tensor is the diagonalized tensor coincident with the principal axes but
containing the square roots of the eigenvalues along the diagonal,

2
4

p A p B 3

5 ;

2
4

3
5 ;

2 A 2 B 0
2 B 2 A 0
001

0

p B 2 A

U ¼ V ¼

þ B 0

0 0 1