Page 208 - Introduction to Continuum Mechanics

Boundary Condition for the Stress Tensor 193

(c) If the surface traction on the inner surface r = r/ is a uniform pressure /?/ and the outer
surface r = r 0 is free of surface traction, find the constants^ and B.
Solution.

The above results, together with Tj$ = T n = 0, give a value of zero for the left hand side of
Eq. (4.8. la) in the absence of a body force component. Thus, the r-equation of equilibrium is
satisfied. Also, by inspection, one easily sees that Eq. (4.8. Ib) and Eq. (4.8. Ic) are satisfied
when BQ - B z = a$ — a z — 0.
(b) The unit normal vector to the cylindrical surface is n = e,, thus the stress vector is given by

i.e.,

The boundary conditions are

and

Thus,

Eqs. (vii) and (viii) give