  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
localised bulge and is predicted by a bifurcation condition, whereas the
asymptote is the Maxwell pressure corresponding to a "two-phase" propagation
state. In contrast, when the tube is first pre-stretched and then has its ends
fixed during subsequent inflation, the pressure versus 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
pre-stretch and tube length, the ascending branches all converge to a single
curve that is only dependent on the ratio of 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 with 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.
