Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 145

E 11 ¼ n 13 M 1 x 2
E 1 I 11

E 22 ¼ n 23 M 1 x 2
E 2 I 11

M 1 x 2
E 3 I 11 ;

; ; E 33 ¼ E 12 ¼ E 13 ¼ E 23 ¼ 0 ;

by integration of the strain-displacement relations (2.49).

Solution : From this strain solution and the strain-displacement relations (2.49), a
system of six first-order partial differential equations for the components of the
displacement vector are obtained:

@ u 1

x 1 ¼ n 13 M 1 x 2

; @

u 2

x 2 ¼ n 23 M 1 x 2

; @

u 3 M 1 x 2

E 3 I 11 ; @

u 1

x 2 þ @

u 2

x 1 ¼ 0 ; @

u 1

x 3 þ @

u 3

x 3 ¼ @ E 1 I 11 @ E 2 I 11 @ @ @ @ @ x 1

; @

u 3

x 2 þ @

u 2

¼ 0

x 3 ¼

0 :

@ @

Integration of the first three of these equations yields

u 2 ¼ n 23 M 1 x 2

u 1 ¼ n 13 M 1 x 2 x 1
E 1 I 11

þ w 1 ð x 2 ; x 3 Þ;

2 E 2 I 11 þ

w 2 ð x 1 ; x 3 Þ; u 3

M 1 x 2 x 3
E 3 I 11 þ

¼ w 3 ð x 1 ; x 2 Þ:

This representation for the components of u can then be substituted into the
second set of three equations above, thus

@ w 1 ð x 2 ; x 3 Þ

þ @

w 2 ð x 1 ; x 3 Þ

¼ n 13 M 1 x 1
E 1 I 11

; @

w 1 ð x 2 ; x 3 Þ

þ @

w 3 ð x 1 ; x 2 Þ

¼ 0

@ x 2 @ x 1 @ x 3 @ x 1

@ w 3 ð x 1 ; x 2 Þ

þ @

w 1 ð x 1 ; x 3 Þ M 1 x 3

E 1 I 11 : x 3 ¼ @ x 2 @

The problem of determining the displacement field u is now the problem of
determining the functions w 1 ( x 2 , x 3 ), w 2 ( x 1 , x 3 ), and w 3 ( x 1 ,x 2 ). Differentiation of
the first equation above with respect to x 3 , the second equation above with respect to
x 2 , and the third equation above with respect to x 1 , one obtains the following:

2 w 1 ð 2 w 2 ð 2 w 1 ð 2 w 3 ð

@ x 2 ;

x 3 ¼ @

x 3 Þ x 1 ; x 3 Þ

; @

x 1 ;

x 3 ¼ @

x 3 Þ x 1 ; x 2 Þ

@ x 2 @ @ x 1 @ x 3 @ x 2 @ @ x 1 @ x 2

2 w 3 ð 2 w 2 ð

@ x 1 ;

x 2 ¼ @

x 2 Þ x 1 ; x 3 Þ

;

@ x 1 @ @ x 1 @ x 3

2 w 1 ð 2 w 2 ð 2 w 3 ð

from which we conclude that @

x 1 ; x 3 Þ

¼ @

x 1 ; x 3 Þ

¼ @

x 1 ; x 2 Þ

0 . It follows that
each component of w must be the sum of two functions, each of a different single
variable, thus.

¼

@ x 2 @ x 3 @ x 1 @ x 3 @ x 1 @ x 2

w 1 ð x 2 ; x 3 Þ¼ f 12 ð x 2 Þþ f 13 ð x 3 Þ; w 2 ð x 1 ; x 3 Þ¼ f 21 ð x 1 Þþ f 23 ð x 3 Þ;

w 3 ð x 1 ; x 2 Þ¼ f 31 ð x 1 Þþ f 32 ð x 2 Þ: