Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 146

The substitution of these representations for w back into the differential
equations above yields the following:

@ f 12 ð

x 2 þ @

x 2 Þ f 21 ð

x 1 ¼ n 13 M 1 x 1

x 1 Þ

; @

f 13 ð

x 3 þ @

x 3 Þ f 31 ð x 1 Þ

x 1 ¼

0 ;

@ @ E 1 I 11 @ @

@ f 31 ð

x 2 þ @

x 1 Þ f 23 ð x 3 Þ M 1 x 3

E 3 I 11 : x 3 ¼ @ @

A study of the equations above shows that certain combinations of terms are
constants, denoted by c 12 , c 13 , and c 32 ,

@ f 12 ð x 2 Þ

Y 12 ¼ @

f 21 ð

x 1 þ n 13 M 1 x 1

x 1 Þ

; @

f 13 ð x 3 Þ

Y 13 ¼ @

f 31 ð x 1 Þ

x 2 ¼ x 3 ¼

;

@ @ E 1 I 11 @ @ x 1

@ f 32 ð x 2 Þ

Y 23 ¼ @

f 23 ð x 3 Þ M 1 x 3

E 1 I 11 ; x 2 ¼ x 3 @ @

thus

Y 12 x 1 þ n 13 M 1 x 1 Y 12 x 2 þ f 12 ð x 2 Þ¼ c 12 ; f 21 ð x 1 Þ¼

2 E 1 I 11 þ

c 21 ;

Y 13 x 1 þ

f 13 ð x 3 Þ¼ Y 13 x 3 þ c 13 ; f 31 ð x 1 Þ¼ c 13 ;

M 1 x 3

Y 23 x 2 þ Y 23 x 3 f 32 ð x 2 Þ¼ c 32 ; f 23 ð x 3 Þ¼

2 E 3 I 11 þ

c 23 :

Substitution of these results into the formulas for w ,

Y 12 x 2 Y 13 x 3 þ

w 1 ð x 2 ; x 3 Þ¼ c 12 þ c 13 ;

x 3 Þ¼ n 13 M 1 x 1

M 1 x 3

Y 12 x 1 Y 23 x 3 þ

w 2 ð x 1 ;

2 E 1 I 11 2 E 3 I 11 þ

c 21 þ c 23 ;

Y 23 x 2 þ Y 13 x 1 þ w 3 ð x 1 ; x 2 Þ¼ c 13 þ c 32 ;

and then into the results of the first integration above, the components of u are
obtained:

u 1 ¼ n 13 M 1 x 2 x 1
E 1 I 11

Y 12 x 2 Y 13 x 3 þ u 1 ;

E 3 n 23 x 2

x 3

E 2 þ n 13 x 1

M 1
2 I 11

Y 23 x 1 Y 23 x 3 þ u 2 ; u 2 ¼ þ

E 1

M 1 x 2 x 3
E 1 I 11 þ

Y 23 x 2 þ Y 13 x 1 þ u 3 ; u 3 ¼

where u 1 ¼ c 32 . The vector u o represents a

superposed rigid object translation as one can see from the fact that u o are the

u 2 ¼ u 3 ¼

c 12 þ c 13 ; c 21 þ c 23 ; c 13 þ