Abstract
We provide an extension to previous analysis of the localised beading instability of soft slender tubes under surface tension and axial stretching. The primary questions pondered here are as follows: under what loading conditions, if any, can bifurcation into circumferential buckling modes occur, and do such solutions dominate localisation and periodic axial modes? Three distinct boundary conditions are considered: in case 1 the tube’s curved 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. A linear bifurcation analysis is conducted to determine numerically the existence of circumferential mode solutions. In case 1 we focus on the tensile stress regime given the preference of slender compressed tubes towards Euler buckling over axisymmetric periodic wrinkling. We show that tubes under several loading paths are highly sensitive to circumferential modes; in contrast, localised and periodic axial modes are absent, suggesting that the circumferential buckling is dominant by default. In case 2, circumferential mode solutions are associated with negative surface tension values and thus are physically implausible. Circumferential buckling solutions are shown to exist in case 3 for tensile and compressive axial loads, and we demonstrate for multiple loading scenarios their dominance over localisation and periodic axial modes within specific parameter regimes.
Highlights
Surface tension plays a dominant role in the finite deformation of non-linearly elastic materials below the elasto-capillary length scale s = γ /μ, where γ is the surface tension and μ is the ground state shear modulus [5, 24, 29]
Unlike in case 1, we may extend our linear bifurcation analysis for case 3 to compressive axial loads; this will facilitate an exhaustive investigation into the competition between localisation and periodic axial and circumferential modes when elasto-capillary effects are taken into consideration
For each fixed value of A considered, we find that there exists a fixed compression threshold beyond which circumferential mode solutions are favoured over localisation or periodic axial modes
Summary
In extremely soft materials such as biological tissue or gels, or in solids with a sufficiently high surface area to volume ratio, this length can have an order of magnitude comparable to the microscale or even the milliscale. In such circumstances, it becomes imperative that surface tension is incorporated into the classical continuum framework. Elastocapillary effects have been considered in many investigations of soft materials at finite strains. There have been exhaustive studies into surface instabilities in soft layers under the combined action of surface tension and uni-axial compression [7], equi-biaxial strain [9] and growth [1]. The beading instability of soft cylindrical tubes under axial loading and surface tension has received copious attention given its implication in the axonal degeneration caused by cytoskeletal trauma [16, 21] and neurodegenerative disorders
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