Page 270 - Introduction to Continuum Mechanics

254 Simple Extension

5.12 Simple Extension

In this section and the following several sections, we shall present some examples of
elastostatic problems. We begin by considering the problem of simple extension. Again, in
all these problems, we assume small deformations so that there is no need to make a distinction
between the spatial coordinates and the material coordinates in the equations of motion and
in the boundary conditions.
A cylindrical elastic bar of arbitrary cross-section (Fig. 5.6) is under the action of equal and
opposite normal traction o distributed uniformly at its end faces. Its lateral surface is free from
any surface traction and body forces are assumed to be absent.

Fig. 5.6

Intuitively, one expects that the state of stress at any point will depend neither on the length
of the bar nor on its lateral dimension. In other words, the state of stress in the bar is expected
to be the same everywhere. Guided by the boundary conditions that on the planes x\ =* 0 and
x =
i I TU = a ,T2i = T$i = 0 and on the planes x^ = a constant and tangent to the lateral
surface, T^ - T 22 = T-Q - 0, it seems reasonable to assume that for the whole bar

We now proceed to show that this state of stress is indeed the solution to our problem. We
need to show that (i) all the equations of equilibrium are satisfied (ii) all the boundary
conditions are satisfied and (iii) there exists a displacement field which corresponds to the
assumed stress field.
(i) Since the stress components are all constants (either o or zero), it is obvious that in the
absence of body forces, the equations of equilibrium dTij/dXj = 0 are identically satisfied.
(ii)The boundary condition on each of the end faces is obviously satisfied. On the lateral
cylindrical surface,