Page 307 - Introduction to Continuum Mechanics

The Elastic Solid 291

Putting r = a in these equations, we find that

We see therefore, at 0 = — (point m in Fig. 5. 18) and at 0 = — (point n in Fig. 5.18),
Z* £*
TQQ = 3cr. This is the maximum tensile stress which is three times the uniform stress a applied
at the ends of the plate. This is referred to as a stress concentration.

5.21 Hollow Sphere Subjected to Internal and External Pressure
Let the internal and external radii of the hollow sphere be denoted by a/ and a 0 respectively
and let the internal pressure be/?,- and the external pressure bep 0, both pressures are assumed
to be uniform. With respect to the spherical coordinates (r, 6, #>), it is clear that due to the
spherical symmetry of the geometry and the loading that each particle of the elastic sphere will
experience only a radial displacement whose magnitude depends only on r, that is,

substituting Eq. (5.21.1) into the Navier equation of equilibrium in spherical coordinates, Eqs.
(5.6.4) in the absence of body forces, we obtain

where, see Eq. (5.6.3g)

Thus,

The general solution of the above equation is

The stress components corresponding to this displacement field can be obtained from Eqs.
(5.6.3), with e = 3A :