# Propagating instabilities in long collapsible tubes of nonlinear biological material

## TL;DR

This paper studies how elastic tubes in the body can collapse under pressure, focusing on how nonlinear material behavior affects the process of buckling and propagation.

## Contribution

The study introduces a numerical and analytical approach to understand buckling propagation in nonlinear elastic tubes under external pressure.

## Key findings

- Small deviations from linear elasticity can lead to localized collapse followed by propagation at lower pressures.
- 2D and 3D finite element models show similar predictions for collapse and propagation pressures.
- An analytical model successfully explains collapse localization and propagation.

## Abstract

Proper functionality of human body relies on several continuous physical processes, many of which are carried out through biological ducts/tubes. For instance, veins, arteries and airways into the human body are natural conduit systems where blood and air are conveyed. Those elastic tubular components are prone to structural instability (buckling) and eventually collapse under critical conditions of net external pressure, resulting in malfunctioning of main physical processes. In the present work, collapsible elastic tubes are studied from a structural mechanics perspective, examining their resistance to collapse under uniform external pressure, emphasizing on the influence of nonlinear material behavior. The problem is approached numerically using nonlinear finite element models, to analyze tubes with diameter-to-thickness ratio ranging from 9 to 30, considering different nonlinear elastic material properties and focusing on the post-buckling phenomenon of “buckling propagation”. It is demonstrated that small softening deviations from linear elastic behavior may cause a localized collapse pattern followed by its propagation along the tube with a pressure lower than the collapse pressure. Results from two-dimensional (ring) and more rigorous three-dimensional (3D) finite element models are obtained in terms of the collapse pressure value and the propagation pressure value, i.e., the minimum pressure required for a localized buckling pattern to propagate, and the two models provide very similar predictions. A simple analytical model is also employed to explain the phenomenon of collapse localization and its subsequent propagation. In addition, special emphasis is given on the correlation between the 3D results and those from ring analysis in terms of the propagation profile and the energy required for the collapse pattern to advance. Finally, comparison with numerical results from tubes made of elastic–plastic material is performed to elucidate some special features of the propagation phenomenon.