Page 294 - Introduction to Continuum Mechanics

278 Plane Strain

This state of stress is obviously a possible state of stress because it clearly satisfies the equations
of equilibrium in the absence of body forces and the stress components, being linear in $2, & VQ
rise to strain components that are also linear in.*^ so that the compatibility conditions are also
satisfied. Superposing this state of stress to that of part (a), that is, adding Eq. (iic) and Eq.
(iv) we obtain

We note that this is the exact solution for pure bending of the bar with couple vectors parallel
to the direction of 63.
In this example, we have easily obtained, from the plane strain solution where the side faces
x 3 = ± (b/2) of the rectangular bar are prevented from moving normally, the state of stress
for the same rectangular bar where the side faces are traction-free, by simply removing the
component T$$ of the plane strain solution. This is possible for this problem because the T^
obtained in the plane strain solution of part (a) happens to be a linear function of the
coordinates.

Example 5.16.2

Consider the state of stress given by

Show that the most general form of G(x\, KI ) which gives rise to a possible state of stress in
the absence of body force is

Solution. The strain components are