Page 293 - Introduction to Continuum Mechanics

The Elastic Solid 277

where

Example 5.16.1

Consider the Airy stress function

(a) Obtain the stresses for the state of plane strain;
(b) If the stresses of part(a) are those inside a rectangular bar bounded by
x\ = 0, jcj = /, jn/2 = ±(h/2) and £3 = ±(6/2), find the surface tractions on the boundaries
(c) If the boundary surfaces *3 = ± (b/2) are traction-free, find the solution.
Solution, (a) From Eq. (5.16.5)

that is,

We note that the surface normal stress on the side faces x$ = ±(£/2)are required to prevent
them from moving in thex^ direction.
(c) In order to obtain the solution for the case where the side facesx^ = ±(b/2) are traction
-free (and therefore have non zero 1/3), it is necessary to remove the normal stresses from these
side faces. Let us consider the following state of stress