Kinematics and Mechanics of Large Elastic Deformations - Continuum Mechanics of Anisotropic Materials - page 303

These equations are then solved for F and F 1 , thus

T and F 1 T

F ð X ; t Þ¼ 1 þ½r o u ð X ; t Þ ð x ; t Þ¼ 1 ½r u ð x ; t Þ :

(11.17)

The Lagrangian strain tensor E and the Eulerian strain tensor e are related to the
displacement gradients by substituting the two equations ( 11.17 ) into the two
equations ( 11.15 ), thus

T T

E ¼ ð 1 = 2 Þf½r o u ð X ; t Þ þ½r o u ð X ; t Þ þ ½r o u ð X ; t Þ½r o u ð X ; t g;

(11.18)

Þ

T T

e ¼ ð 1 = 2 Þf½r uðx; t Þ þ½ruðx; t Þ ½r uðx; t Þ½r uðx; t Þ g: (11.19)

The expanded component forms of ( 11.18 ) and ( 11.19 ) are given by

" #

2
þ

2 2

E II ¼ @

u I 1

2

@ u I @ u II @ u III

@ x I þ

þ ;

@ @ x I @ x I x I

" #

2 2 2

E II II ¼ @

u II 1

2

@ u I @ u II @ u III

x II þ

þ þ ;

@ @ x II @ x II @ x II

" #

;

2 2 2

E III III ¼ @

u III 1

2

@ u I @ u II @ u III

x III þ

þ þ

@ @ x III @ x III @ x III

1
2

@

x II þ @

u I u II

x I þ @

u I

x I @

u II
@ x I þ @

u II
@

@ u II

x II þ @

u III
@

@ u III

@

E III ¼ ;

(11.20)

@ @ @ x I @ x I x II

@

x III þ @

u I u III

@ x I þ @

u I

x I @

u III
@ x I þ @

u II
@

@

x III þ @

u II u III

@

@ u III

1
2

E I III ¼ ;

@ @ x I @ x I @ x III

1
2

@

x III þ @

u II u III

@ x II þ @

u I @

x III þ @

u II u II @

x III þ @

u II u III

@

@ u III

E II III ¼ ;

@ @ x II @ @ x III @ x II @ x III

and

" #

2
þ

2
þ

2

e 11 ¼ @

u 1 1

2

@ u 1 @ u 2 @ u 3

x 1

;

@ @ x 1 @ x 1 @ x 1

" #

2
þ

2
þ

2

e 22 ¼ @

u 2 1

2

@ u 1 @ u 2 @ u 3

x 2

;

@ @ x 2 @ x 2 @ x 2