Kinematics and Mechanics of Large Elastic Deformations - Continuum Mechanics of Anisotropic Materials - page 301

Fig. 11.12 The deformation
of a unit square by the U of
Example 11.3.1

11.3.4. Find the square root of the matrix

2
4

5 3 p 0

3
5 :

31

5 3

p

A ¼

21 0

0 0 4

11.3.5. Find the polar decomposition of the tensor F ,

2
4

3
5 :

1 0

010
001

a

F ¼

11.4 The Strain Measures for Large Deformations

The deformation gradient is the basic measure of local deformational and rotational
motion. It maps a small region of the undeformed object into a small region of the
deformed object. If the motion is a pure translation with no rotation, then F

¼ 1 .

If the motion is a rigid object rotation, then F ¼ R where R is an orthogonal matrix,

R T R T

R 1 .

The local state of deformation may be investigated by considering the deforma-
tion of an infinitesimal material filament denoted by d X . In the instantaneous
configuration the same material filament has a position represented by d x . Recalling
the representation (2.2) for a motion, x

¼ R ¼

( X ,t), and the representation (2.16) for
F , it is easy to see that d x and d X are related by d x

¼ x

¼ F d X , or from the polar

decomposition theorem d x ¼ R U d X ¼ V R d X . Thus a pure rotational motion

for which U 1 the length of d X will be preserved but its direction will be

rotated. If the motion includes a deformational component, then the length of d x

¼ V ¼