Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 152

With respect the coordinate system shown the displacements in the 1 and
2 directions are given by

x 2 ;

x 2 þ

d 2
2

A
2 x 1 x 2 þ

A
3

þ 2

A
2

L 2 u 1 ¼ 1 ð 1 þ nÞ

AL 2
2

AL 3
3 :

u 2 ¼ n

A
2 x 1 x 2 þ

A
6 x 1

x 1 þ

In the case of plane stress the strain-stress relations reduce to the following:

1
E ð

1
E ð

E 12 ¼ ð

1 þ nÞ

E

E 11 ¼ T 11 n T 22 Þ; E 22 ¼ T 22 n T 11 Þ; T 12 :

(a) Calculate the strain field in the rectangular region of length L and
width d .
(b) Calculate the stress field in the rectangular region of length L and
width d .
(c) Calculate the values of the stress field at the surfaces x 2 ¼

d
2

of the

rectangular region of length L and width d .
(d) Calculate the stress applied on the surfaces x 2 ¼

d

2 of the rectangular

region of length L and width d .
(e) Calculate the stress applied on the surface x 1 ¼ 0 of the rectangular

region of length L and width d .
(f) Name the equation that was employed in the two previous calculations.

2 u 1
@t 2

2 u 1

6.3.17 Show that the differential equation @ c S @

@x 2 and the initial conditions
that require that the displacement and velocity at time t

¼

¼ 0 to be given by

u 1 ( x 2 , 0) and ( ∂ u 1 / ∂ t )( x 2 , 0) are identically satisfied by the solution

Z x 2 þc S t

1
2 ½

1
2 c S

@ u 1

@

u 1 ð x 2 ; t Þ¼ u 1 ð x 2 þ c S t ; 0 Þþ u 1 ð x 2 c S t ; 0 Þ þ

t ðx;

0 Þ d x:

x 2 c S t