Pressure-driven wrinkling of soft inner-lined tubes

Abstract

A simple equation modelling an inextensible elastic lining of an inner-lined tube subject to an imposed pressure difference is derived from a consideration of the idealised elastic properties of the lining and the pressure and soft-substrate forces. Two cases are considered in detail, one with prominent wrinkling and a second one in which wrinkling is absent and only buckling remains. Bifurcation diagrams are computed via numerical continuation for both cases. Wrinkling, buckling, folding, and mixed-mode solutions are found and organised according to system-response measures including tension, in-plane compression, maximum curvature and energy. Approximate wrinkle solutions are constructed using weakly nonlinear theory, in excellent agreement with numerics. Our approach explains how the wavelength of the wrinkles is selected as a function of the parameters in compressed wrinkling systems and shows how localised folds and mixed-mode states form in secondary bifurcations from wrinkled states. Our model aims to capture the wrinkling response of arterial endothelium to blood pressure changes but applies much more broadly.

The θ-formulation of the two-dimensional elastica: buckling and boundary layer theory

“…Equations (2.1)–(2.3) become ∂sn=p−κt−scriptFn,1em∂st=κn−scriptFt,1emB∂sκ=−n,where scriptFt and scriptFn are external axial and shear force per unit length, respectively. Recently, for instance, an analytically solvable model for elastic wrinkling on a nonlinear foundation was introduced, in which scriptFt=0 while scriptFn scales as some power of the local displacement of the elastica [26–28] (see also [29–33] for more experimental and theoretical results on the subject).…”

“…where F t and F n are external axial and shear force per unit length, respectively. Recently, for instance, an analytically solvable model for elastic wrinkling on a nonlinear foundation was introduced, in which F t = 0 while F n scales as some power of the local displacement of the elastica [26][27][28] (see also [29][30][31][32][33] for more experimental and theoretical results on the subject). We may continue to define H as Bκ 2 /2 + t. It now satisfies…”

Universal Wrinkling of Supported Elastic Rings

“…Recent work on a family of simple, yet realistic, models for substrate-supported elastic rings under compression [27] revealed that these models have a special structure that suggests that exact wrinkle solutions can be constructed, and that these may, in turn, be related to the well-known buckled states of the substrate-free case. In this Letter we show that this is indeed the case.…”

“…In order to study the wrinkling of a thin elastic inextensible ring supported by a soft substrate, we consider the following model, which can be derived from the Kirchhoff equations for elastic rods [27,28]:…”

“…The model provides a basis for understanding the nonlinear response of an elastic ring to competing pressure and body forces across a wide range of physical systems. For example, the case n = 2 was used as a simple model able to capture the wrinkle-to-smooth transition that takes place in the endothelium of an artery as the internal blood pressure oscillates [27]. In this case α 2 = (R 1 /λ) 5 , where λ ≡ (B/K) 1/5 is the bending length scale.…”