Page 271 - Introduction to Continuum Mechanics

The Elastic Solid 255

and

Thus, the traction-free condition on the whole lateral surface is satisfied.
(Hi) From Hooke's law, the strain components are obtained to be

These strain components are constants, therefore, the equations of compatibility are automat-
ically satisfied. In fact it is easily verified that the following single-valued continuous
displacement field corresponds to the strain field of Eq. (5.12.2)

Thus, we have completed the solution of the problem of simple extension (o>0) or compres-
sion (tf<Q). We note that Eq. (5.12.3) is the unique solution to Eqs. (5.12.2) if rigid body
displacement fields (translation and rotation) are excluded.
If the constant cross-sectional area of the bar is A, the surface traction o on either end face
gives rise to a resultant force of magnitude

passing through the centroid of the area A Thus, in terms of P andy4, the stress components
in the bar are

Since the matrix is diagonal, we know from Chapter 2, that the principal stresses are
P/A ,0,0. Thus, the maximum normal stress is