[Paper Review] Post-bifurcation behaviour of elasto-capillary necking and bulging in soft tubes

Dominic Emery, Yibin Fu|arXiv (Cornell University)|Apr 10, 2021

Advanced Materials and Mechanics63 references15 citations

TL;DR

This study investigates the post-bifurcation behavior of elasto-capillary instabilities in soft, axially stretched cylindrical tubes using weakly nonlinear analysis. It reveals that localised necking or bulging emerges sub-critically depending on axial stretch, with a supercritical kink-wave solution at a critical stretch threshold, validated by FEM simulations and applicable to biological and soft robotic systems.

ABSTRACT

Previous linear bifurcation analyses have evidenced that an axially stretched soft cylindrical tube may develop an infinite-wavelength (localised) 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 localised 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 non-linear analysis conducted for a general material model, the initial sub-critical bifurcation solution is found to be localised 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 localisation. 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 (FEM) simulations.

Motivation & Objective

• To understand the initiation and evolution of localised instabilities in soft tubes under axial tension and surface tension.
• To resolve the controversy over whether necking or bulging dominates post-bifurcation behavior in soft tubular structures.
• To characterize the transition from localised solutions to a final 'two-phase' kink-wave state in soft tubes with constrained lateral boundaries.
• To extend linear bifurcation results by performing a weakly nonlinear analysis for general hyperelastic material models.
• To validate theoretical predictions with finite element method (FEM) simulations and interpret the role of loading paths and boundary conditions.

Proposed method

• Conducting a weakly nonlinear near-critical analysis for a general hyperelastic material model under axial stretch and surface tension.
• Deriving amplitude equations for localised solutions using asymptotic expansions in the vicinity of bifurcation points.
• Distinguishing between Case 2 (inner surface fixed, outer free) and Case 3 (outer surface fixed, inner free) boundary conditions to analyze their impact on instability type.
• Using rescaling techniques to reveal universal behavior near the critical axial stretch λmin.
• Applying the nonlinear Schrödinger equation (NLSE) framework to model static solitary wave solutions representing dark (necking) and bright (bulging) solitons.
• Validating theoretical results with FEM simulations from prior work, particularly for Cases 2 and 3.

Experimental results

Research questions

• RQ1 How does the type of localised bifurcation (necking vs. bulging) depend on axial stretch and loading path in soft tubes with surface tension?
• RQ2 What determines whether the initial post-bifurcation solution is sub-critical or super-critical, and under what conditions does a kink-wave solution emerge?
• RQ3 How do different boundary conditions—specifically radial fixation of one lateral surface—affect the onset and nature of localised instabilities?
• RQ4 What is the role of tube thickness (A) in modulating the number and location of bifurcation points, particularly in Case 3?
• RQ5 How does the system evolve from a localised solution to a final two-phase kink-wave state with distinct axial stretches?

Key findings

• For Case 2, localised solutions are sub-critical and correspond to necking when λ < λmin and bulging when λ > λmin, with a supercritical kink-wave solution at λ = λmin.
• In Case 3, when tube thickness A < 0.08567, two additional extrema emerge in the critical surface tension γcr(λ), leading to multiple intervals of necking and bulging depending on λ.
• At λ = λmax (the local maximum of γcr), the bifurcation becomes sub-critical due to a sign flip in the dispersion coefficient, and the solution reduces to a dark solitary wave.
• For fixed λ, the system initially admits a localised necking or bulging solution that 'jumps' to a final two-phase kink-wave state with distinct stretches λL and λR.
• The kink-wave solution is explicitly derived in the vicinity of λmin, confirming a supercritical transition at this critical point.
• FEM simulations support the theoretical prediction that the transition from localised to two-phase behavior is consistent across loading scenarios and boundary conditions.