Bifurcation Analysis of a Cylindrical Tube for a Residually-Stressed Material and its Application to Aneurysms in Arterial Tissue

Abstract

An evaluation of the parameters that define a proposed boundary value problem solved with the finite element method as an approach to emulate the bio-mechanical response of arteries at an instability point. Blood vessels are idealized as cylinder tubes with different geometries and boundary conditions that incorporate an invariant based, hyper-elastic and residually stressed material model as a subroutine in a finite element software. A sequential, three-step, mechanical analysis that simultes the in vivo conditioning and physiological actions is applied; the analysis consists on the application of residual stresses, an imposed axial stretch and the application of uniform internal pressure. The models are restricted to obtain instabilities of bending and bulging bifurcation modes, the first is related to mechanical buckling for the mode m=1, while the second one to the propagation of a bulge in the middle section. The average azimuthal and axial stresses are evaluated along the axial direction; uniform distributions are found at the bifurcation point for models in the bending mode, while peaks in the middle section are found for bulging which indicate radial displacements. Stresses in the axial direction tend to increase due to the stretch in that direction, eventually surpassing the ones in the hoop direction. The critical uniform pressure reached at a bifurcation point defines the corresponding onset of bifurcation in terms of the axial stretch. The bending mode is related to low values of axial stretch, but bulging becomes the onset of bifurcation at values above a defined transition value. The transition value increases for more slender cylinder tubes and for lower values of the magnitude of residual stresses. The critical pressure values at their corresponding transition values also tend to decrease for higher magnitudes of residual stresses. A description of the stages of the mechanical analyses are included in terms of the history of configurations for both bifurcation modes. The bulging mode is characterized for having both maximum critical uniform pressure and stresses at the bifurcation point. For certain axial stretches, the bending mode presents the propagation of a secondary lateral bulge at a post-bifurcation stage at a higher critical pressure applied which is related to abdominal aortic aneurysms.