Page 280 - Introduction to Continuum Mechanics

264 Torsion of a Circular Cylinder

In addition, because the disc is assumed to be rigid, the angle of twist of the left and right shaft
must be equal:

Thus,

From Eqs. (i) and (iii), we then obtain

Example 5.13.3
Consider the angle of twist for a circular cylinder under torsion to be a function of xi and
time t, i.e., 0 = 0 (x\, t).
(a) Determine the differential equation that 6 must satisfy for it to be a possible solution in
the absence of body forces. What are the boundary conditions that 0 must satisfy (b) if the
plane xi = 0 is a fixed end; (c) if the plane jtj = 0 is a free end.
Solution, (a) From the displacements

we find the stress to be

and

The second and third equations of motion give