An isotropic elastic circular cylinder of length l and radius a hangs under its own
weight, suspended from a horizontal plane x3 = l, where the coordinate x3 is
measured vertically upwards. That is, there are no tractions on the lateral surface
and the free lower end x3 = 0. In these coordinates the body force of gravity,
per unit mass, has coordinates: b1 = 0, b2 = 0, b3 = −g, where g denotes the
acceleration due to gravity. Evaluate the stress components for the displacement
u1 = −Gx1x3, u2 = −Gx2x3, u3 = G{0.5k((x1)^2+(x2)^2) − (k + μ)(l^2 − (x3)^2)},
where k and μ are the Lam´e constants and G is a constant.
Show that the equations of equilibrium are satisfied provided that
G =ng/(2μ(3k + 2μ)),
where n is the density.
Verify that the tractions vanish on the lateral surface and the lower end.
Calculate the traction on the upper end x3 = l and the total force acting on this end.
Why would you expect this value?