Kinematics and Mechanics of Large Elastic Deformations - Continuum Mechanics of Anisotropic Materials - page 307

Fig. 11.13 An illustration
of a simple extension, a type
of pure homogeneous
deformation

X III

x 3

X II x 2

X I

x 1

where l III are the principal stretches. The principal stretches represent the

effect of the deformation upon material filaments in the coordinate directions as
they are the ratios of the final length of the filament to the initial length of the
filament. In terms of the principal stretches, the deformation tensors C and c have
the representations

l I , l II ,

2
4

3
5 ;

2
4

3
5 ;

l 2 I 00
0

2

l

I 00
0

2
II

l 2
II

C ¼ c ¼

l 0 0 (11.33)

2
III

l 2
III

00 l 00

and the Lagrangian strain tensor E and the Eulerian strain tensor e ,

2 3

2
I

l 1 0 0

4 5 ;

2
II

2 E ¼ 2 e

0 l 1 0

2
III

0 0 l 1

2
4

3
5 :

1 l 2
I

0 0

1 l 2
II

¼ 0 0 (11.34)

l 2
III

0 0 1

Two special cases of pure homogeneous deformation are of particular interest.
A simple extension is characterized by l I 6¼ l II ¼ l III and is illustrated in

Fig. 11.13 .A uniform dilation is characterized by l ¼ l I ¼ l II ¼ l III and is
illustrated in Fig. 11.14 . In the case of uniform dilation, F ¼ U ¼ V ¼ l1 ,
C ¼ l

2 1 , c ¼ l 2 1 , 2E ¼ 2

l 2 ) 1 . ( l 1) 1 ,2 e ¼ (1