Localised bifurcation in soft cylindrical tubes under axial stretching and surface tension

TL;DR

This paper analyzes the conditions under which localized bulging or necking occurs in soft cylindrical tubes under axial stretch and surface tension, revealing that localization is possible only under certain boundary constraints.

Contribution

It provides analytical bifurcation conditions for localization in hyperelastic tubes and validates them with FEM simulations, highlighting the influence of boundary constraints.

Findings

Localization occurs only when certain boundary conditions are met.

Bifurcation into localized solutions is unattainable under free surface conditions.

Finite Element Method confirms analytical predictions.

Abstract

We investigate localised bulging or necking in an incompressible, hyperelastic cylindrical tube under axial stretching and surface tension. Three cases are considered in which the tube is subjected to different constraints. In case 1 the inner and outer surfaces are traction-free and under surface tension, whilst in cases 2 and 3 the inner and outer surfaces (respectively) are fixed to prevent radial displacement and surface tension. However, each free surface in these latter two cases is still under surface tension. We first state the analytical bifurcation conditions for localisation and then validate them numerically whilst determining whether localisation is preferred over bifurcation into periodic modes. It is shown that bifurcation into a localised solution is unattainable in case 1 but possible and favourable in cases 2 and 3. In contrast, in case 1 any bifurcation must necessarily take the form of a periodic mode with a non-zero wave number. Our results are validated using Finite Element Method (FEM) simulations.