Page 297 - Introduction to Continuum Mechanics

The Elastic Solid 281

state for the beam where the side faces x$ = ±b/2 are traction free. However, if b is very very
small, then it seems reasonable to expect that the application of -Ti$ on these side face alone
will result in a state of stress inside the body which is essentially given by

(Indeed it can be proved that the errors incurred in this equation approach zero with the second
power of b as b approaches zero). Thus, the state of stress obtained in part (b), with 733 taken
to be zero, is the state of stress inside a thin beam under the same external loading as that in
the plane strain case. Such a state of stress is known as the state of plane stress where the stress
matrix given by

The strain field corresponding to the plane stress state is given by

5.17 Plane Strain Problem in Polar Coordinates

In Polar coordinates, the strain components in plane strain problem are, [with
T a = v(Tr+Tw)],

The equations of equilibrium are [see Eqs. (4.8.1)], (noting that there is no z dependence).