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

field

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?