Post-bifurcation behaviour of elasto-capillary necking and bulging in soft tubes

Abstract

Previous linear bifurcation analyses have evidenced that an axially stretched soft cylindrical tube may develop an infinite-wavelength (localized) instability when one or both of its lateral surfaces are under sufficient surface tension. Phase transition interpretations have also highlighted that the tube admits a final evolved ‘two-phase’ state. How the localized instability initiates and evolves into the final ‘two-phase’ state is still a matter of contention, and this is the focus of the current study. Through a weakly nonlinear analysis conducted for a general material model, the initial sub-critical bifurcation solution is found to be localized bulging or necking depending on whether the axial stretch is greater or less than a certain threshold value. At this threshold value, an exceptionally super-critical kink-wave solution arises in place of localization. A thorough interpretation of the anticipated post-bifurcation behaviour based on our theoretical results is also given, and this is supported by finite-element method simulations.

Localized necking and bulging of finitely deformed residually stressed solid cylinder

“…In the previous section, a detailed analysis of the bifurcation threshold for localized necking/bulging of a residual stressed hyperelastic cylinder under extension has been presented. We found that different loading scenarios share the same bifurcation condition, consistent with the findings reported in Fu et al [35], Emery and Fu [36], Ye et al [41], and Emery and Fu [61].…”

“…Surface tension and residual stress play similar roles when evaluating their influence on generating localized instabilities. Fu et al [35] and Emery and Fu [36, 61] report that for fixed axial length localized bifurcation occurs when d N / d λ = 0 . Therefore, we may assume that the bifurcation condition is again given by equation (91) with the residual stress magnitude ν (or κ ν 2 ) now being the independent loading parameter.…”

“…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.…”

“…An exception is the Blatz–Ko material where strain localization is allowed (see, for example, Dai and Peng [32]). Recently, it has been shown in Fu et al [33], Lestringant and Audoly [34], Fu et al [35], and Emery and Fu [36] that necking localization occurs when the purely mechanical response is coupled with electric actuation or with surface tension.…”

Localized bulging of an inflated rubber tube with fixed ends

“…Recognizing that localized bulging is a 'zero wavenumber' bifurcation phenomenon, recent studies have examined the weakly nonlinear near-critical behaviour [38] and the effects of rotation, fibre-reinforcement, tethering and multi-layering on bulge initiation [39][40][41][42][43]. The methodology developed has also been extended to studying surface tension-induced necking [44][45][46][47]. The commonly used approach of modelling localized bulging/necking as a 'finite wavenumber' bifurcation phenomenon has been appraised in a recent study [48].…”