Computational bifurcation analysis for hyperelastic residually stressed tubes under combined inflation and extension and aneurysms in arterial tissue

Abstract

In this investigation, the mechanical modelling of a hyperelastic residually stressed thick-walled circular cylindrical tube under inflation and extension is performed from a numerical standpoint. The constitutive relation is derived for residually stressed solids using an invariant-based free energy approach. A three-dimensional residual stress field is introduced in the argument of the free-energy function. This formulation is incorporated into a numerical procedure using the Finite Element Method (FEM), with the use of a nonlinear solution scheme via the modified Riks method in order to capture bifurcation and post-bifurcation of the tube. The FE implementation of the proposed formulation is carried out in the general purpose code ABAQUS by means of user-defined capabilities. A comprehensive analysis of the structural performance of the tube is conducted with special focus on the effect of the residual stresses on the bifurcation behaviour. Results are analysed mainly in the context of aneurysms formation and propagation in arterial wall tissues, although it has other applications such as in the context of venous tortuosity. Bulging and bending modes are obtained, and post-bifurcation is also captured. Results suggest that the onset and location of the bulge along the axis of the tube depend mainly on the axial stretch and on the residual stress field. Computations also show that for sufficiently large values of axial stretch the onset of bifurcation is bulging while for small values of axial stretch the onset of bifurcation is bending. Post-bifurcation behaviour of this latter case gives rise to bulges on one side of the tube, which is an irregular shape that appears in the advancement of abdominal aortic aneurysms (AAA). • Bifurcation and postbifurcation of an inflated and extended residually-stressed circular cylindrical tube are analysed. • Bending and bulging motion are captured depending on the axial stretch of the structure. • The onset of bifurcation and the motion of the bifurcation are exploited in the context of aneurysm formation and propagation. • The combination of bending and bulging is consistent with the development of abdominal aortic aneurysms (AAA).