Inflation-induced bulge initiation and evolution in graded cylindrical tubes of arbitrary thickness

Abstract

We study the initiation, evolution, and propagation of localized bulging in a pressurized heperelastic tube where the elastic modulus is non-uniform in the radial direction. The primary deformation prior to instability is characterized within the framework of nonlinear elasticity for a general material constitution and a generic modulus gradient. To unravel the influence of modulus gradient on localized bulging, we employ the incompressible Gent model and select three representative modulus gradients, including a linear, an exponential, and a sinusoidal function. In particular, the sinusoidal one is supposed to model an actual artery structure. In addition, two prototypical loading conditions are considered, namely, either the resultant axial force or the axial length can be fixed. Based on an explicit bifurcation condition in terms of the internal pressure and the resultant axial force for localized bulging, an exhaustive theoretical analysis on bulge initiation is carried out and the effect of geometric and material parameters and the modulus gradient on the critical stretch triggering localized bulging is revealed. It turns out that the modulus mismatch, as well as the position of maximum modulus, can dramatically affect the onset of localized bulging. Then we analytically elucidate the influence of modulus gradient on bulge propagation according to Maxwell’s equal area rule and conduct a finite element analysis of bulge evolution. To perform post-bifurcation analysis, a robust finite element model for graded Gent material is established in Abaqus by UHYPER subroutine coding. Interestingly, it is found that a sinusoidally distributed modulus has negligible influence on the critical stretch of bulge initiation, the deformation process of bugle growth, and the maximum size of a bulge. The current analysis not only can be used to qualitatively explain why a healthy human artery evolves into a sandwich structure where the intermediate layer is stiffest but also can provide useful insight into localized instabilities such as necking and beading in graded structures.