On the Stability of a Circular Tube Under End Thrust

A circular tube of finite thickness h composed of homogeneous isotropic incompressible material is finitely deformed by end thrust in the direction of its axis so that it changes its length in the ratio λ: 1, and its stability is investigated by considering the equilibrium of a further infinitesimal deformation. A general periodic solution for the displacements in this further deformation is given, but only the special case of radially symmetrical buckling is then considered. The application of the boundary conditions leads to a relation between λ and the frequency of longitudinal rippling v/2π from which the critical λ, determining the onset of instability, is found. Three examples are chosen: (a) thin tube, in which the critical value of λ is derived analytically, (b) tube with radius twice the thickness h, (c) solid cylinder. The last two examples are treated graphically, a neo-Hookean model being assumed. The longitudinal force required to produce instability is given in each case.