Page 305 - Introduction to Continuum Mechanics
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The Elastic Solid 289
First, we shall demonstrate that the stress field generated from the Airy stress function in
the form of <p = f(r) cos 2 6, can be used to give a stress field which satisfies the boundary
conditions
Then, to this stress field, we will superpose the stress field
which is the solution for a hollow cylinder with a very thick wall (i.e.,6/a-»<»), acted on by a
uniform radial traction — on the outer surface r-b only [see Eqs. (5.18.5) in Example 5.18.1],
In this way, the boundary conditions Eqs. (5.20.2) can be satisfied-
Substituting
into the equation governing the Airy's stress function, Eq. (5.17.6), i.e.,
we obtain that the function f(r) must satisfy the following equation
The general solution for this equation is [see Prob. 5.81]
Thus,
and the corresponding stress components are [see Eqs. (5.17.3)]