Page 291 - Introduction to Continuum Mechanics

The Elastic Solid 275

5.16 Plane Strain

If the deformation of a cylindrical body is such that there is no axial components of the
displacement and that the other components do not depend on the axial coordinate, then the
body is said to be in a state of plane strain. Such a state of strain exists for example in a
cylindrical body whose end faces are prevented from moving axially and whose lateral surface
are acted on by loads that are independent of the axial position and without axial components.
Letting the 63 direction correspond to the cylindrical axis, we have

The strain components corresponding to this displacement field are:

and the nonzero stress components are TU , 7\2, 722, ^33, where

This last equation is obtained from the Hooke's law, Eq. (5.4.8c) and the fact that £"33 = 0 for
the plane strain problem.
Considering a static stress field with no body forces, the equilibrium equations reduce to

Because 733 = T^ (xi, KI ), the third equation is trivially satisfied. It can be easily verified
that for any arbitrary scalar function <p, if we compute the stress components from the following
equations

then the first two equations are automatically satisfied. However, not all stress components
obtained this way are acceptable as a possible solution because the strain components derived
from them may not be compatible; that is, there may not exist displacement components which