Page 206 - Introduction to Continuum Mechanics

Equations of Motion in Cylindrical and Spherical Coordinates 191

and A is a constant. Verify that the given state of stress is in equilibrium in the absence of body
forces.
Solution. From R = r + z , we obtain

Thus,

Thus, the left hand side of Eq, (4.8. la) becomes, with B r = 0

In other words, the r-equation of equilibrium is satisfied.
Since T& = TQ Z = 0 and TQQ is independent of 0, therefore, with BQ — a@ — 0, the second
equation of equilibrium is also satisfied.
The third equation of equilibrium Eq. (4.8. Ic) with B z — a z = 0 can be similarly verified.
[see Prob. 4.35].

Spherical coordinates

Again, we note that for symmetric stress tensor, T^- T0 r = 0 and7^- 7^=0.