Page 298 - Introduction to Continuum Mechanics

282 Plane Strain

The third equation is automatically satisfied, because T z$ - T n - 0 and T zz is not a function
of z.

It can be easily verified that the equations of equilibrium Eq. (5.17.2a) are identically
satisfied if

where <p is the Airy stress function. In Section 5.16, we see that in order to satisfy the
compatibility conditions, the Cartesian stress components TU + T^i must satisfy
Eq. (5.16.9), i.e.,

To derive the equivalent expression in cylindrical coordinates, we note that TU + 722 * s tne
first scalar invariant of the stress tensor. Therefore

2
2
2
Also, the Laplacian operator V = (d /drf + d /At§ ) takes the following form in polar
coordinates

Thus, the function <p must satisfy the biharmonic equation

If <p is a function of r only, we have,