Localized bulging of an inflated rubber tube with fixed ends

Abstract

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’.

A one-dimensional model for axisymmetric deformations of an inflated hyperelastic tube of finite wall thickness

“…6 shows the actual variation of P against ap0q predicted by the 1d model when L is varied and the averaged stretch λ c is fixed or λ c is varied but L is fixed. These results confirm the theoretical prediction of Guo et al (2022) that the right branches of these curves all converge to a master curve that is independent of L or λ c . These curves all terminate at the point where the axial stress near each end of the tube has become compressive enough so that secondary Euler buckling or axisymmetric wrinkling becomes possible.…”

“…It is shown that the bifurcation condition for this case corresponds to the axial force reaching a maximum at a fixed pressure (Fu & Il'ichev, 2015); in other words, as pressure is increased, the critical pressure is the value of pressure at which the axial force reaches a maximum when viewed as a function of the axial stretch. Also, in contrast with the case of fixed axial force where the measured pressure approaches a constant value (the propagation pressure), the measured pressure in the case of fixed ends has an N shape where the right branch approaches a master curve that is independent of the pre-axial-stretch or the tube length (Guo et al, 2022).…”

Localized necking and bulging of finitely deformed residually stressed solid cylinder

“…Drawing upon recent understandings of localized bulging in inflated tubes and localized necking in stretched cylinders, outlined in Fu et al [35], Emery and Fu [36] Wang et al [43], Guo [44], Emery and Fu [61], we now summarize the deformation evolution of the current problem for three distinct loading conditions by virtue of Figures 8 and 9 as follows.…”

“…There, the pressure following bulge initiation decreases to a minimum and then rises again, and the latter is accompanied by bulge propagation. Localized bifurcation of rubber tubes with different end conditions is investigated in Guo et al [44]. In a tube with free ends, bulge initiation occurs at the pressure maximum, followed by a monotonic decrease to approach a constant asymptote.…”

The Ogden model of rubber mechanics: 50 years of impact on nonlinear elasticity

“…Guo et al [21] study the bulging of a rubber tube with fixed ends when it is inflated by internal pressure. Deriving the bifurcation condition for localized bulging is an involved mathematical feat.…”