Localized bulging of an inflated rubber tube with fixed ends.

When a rubber tube with free ends is inflated under volume control, the pressure will first reach a maximum and then decrease monotonically to approach a constant asymptote. The pressure maximum corresponds to the initiation of a localized bulge and is predicted by a bifurcation condition, whereas the asymptote is the Maxwell pressure corresponding to a 'two-phase' propagation state. By contrast, when the tube is first pre-stretched and then has its ends fixed during subsequent inflation, the pressure as a function of the bulge amplitude has both a maximum and a minimum, and the behaviour on the right ascending branch has previously not been fully understood. We show that for all values of the pre-stretch and tube length, the ascending branches all converge to a single curve that is dependent only on the ratio of the tube thickness to the outer radius. This curve represents the Maxwell state to be approached in each case (if Euler buckling or axisymmetric wrinkling does not occur first), but this state is pressure-dependent, in contrast to the free-ends case. We also demonstrate experimentally that localized bulging cannot occur when the pre-stretch is sufficiently large and investigate what strain-energy functions can predict this observed phenomenon. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.