On deforming a sector of a circular cylindrical tube into an intact tube: existence, uniqueness, and stability

TL;DR

This paper investigates the deformation of a circular cylindrical tube sector into an intact tube, establishing conditions for existence, uniqueness, and stability of the deformation, with applications to understanding residual stresses in arteries.

Contribution

It provides a theoretical framework linking strain-energy convexity to the existence, uniqueness, and stability of residually-stressed cylindrical tubes, with implications for biomechanics.

Findings

Convexity of strain energy ensures existence and uniqueness.

Bifurcation can occur depending on geometric and physical parameters.

Results are applicable to arterial residual stress analysis.

Abstract

Within the context of finite deformation elasticity theory the problem of deforming an open sector of a thick-walled circular cylindrical tube into a complete circular cylindrical tube is analyzed. The analysis provides a means of estimating the radial and circumferential residual stress present in an intact tube, which is a problem of particular concern in dealing with the mechanical response of arteries. The initial sector is assumed to be unstressed and the stress distribution resulting from the closure of the sector is then calculated in the absence of loads on the cylindrical surfaces. Conditions on the form of the elastic strain-energy function required for existence and uniqueness of the deformed configuration are then examined. Finally, stability of the resulting finite deformation is analyzed using the theory of incremental deformations superimposed on the finite deformation, implemented in terms of the Stroh formulation. The main results are that convexity of the strain energy as a function of a certain deformation variable ensures existence and uniqueness of the residually-stressed intact tube, and that bifurcation can occur in the closing of thick, widely opened sectors, depending on the values of geometrical and physical parameters. The results are illustrated for particular choices of these parameters, based on data available in the biomechanics literature.