Initiation of aneurysms as a mechanical bifurcation phenomenon

Fu, Yibin; Rogerson, G. A.; Zhang, Y.T.

doi:10.1016/j.ijnonlinmec.2011.05.001

Cited by 46 publications

(38 citation statements)

References 36 publications

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“…Since this paper was submitted, it has subsequently been demonstrated by Fu et al [48], using a realistic constitutive model, that the initial onset of aortic abdominal aneurysms can indeed be modelled as a bifurcation phenomenon, and that in the bifurcation interpretation of aneurysm formation, the imperfection sensitivity determined in this paper plays a pivotal role.…”

Section: Resultsmentioning

confidence: 74%

Effects of imperfections on localized bulging in inflated membrane tubes

Xie

2012

Phil. Trans. R. Soc. A.

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The problem of localized bulging in inflated membrane tubes shares the same features with a variety of other localization problems such as formation of kink bands in fibrereinforced composites and layered structures. This type of localization is known to be very sensitive to imperfections, but the precise nature of such sensitivity has not so far been quantified. In this paper, we study effects of localized wall thinning/thickening on the onset of localized bulging in inflated membrane tubes as a prototypical example. It is shown that localized wall thinning may reduce the critical pressure or circumferential stretch by an amount of the order of the square root of maximum wall thickness reduction. As a typical example, a 10 per cent maximum wall thinning may reduce the critical circumferential stretch by 19 per cent. This square root law complements the wellknown Koiter's two-thirds power law for subcritical periodic bifurcations. The relevance of our results to mathematical modelling of aneurysm formation in human arteries is also discussed.

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Section: Resultsmentioning

confidence: 74%

Effects of imperfections on localized bulging in inflated membrane tubes

Xie

2012

Phil. Trans. R. Soc. A.

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“…The major difference between our theory and the prevalent point of view [3,4] is that in the latter there is no bifurcation preceding the remodelling process. Parts (a) and (b) have been accomplished in our earlier papers [5,6,7], although whether bifurcation can take place or not was found to be very sensitive to the arterial models used and there is currently still much uncertainty about the realistic material models for live arteries. In order to establish (c), we have recently examined the effect of fluid inertia on the stability properties of localized bulging solutions [8], following on from an earlier work [9] where fluid inertia was neglected.…”

Section: Introductionmentioning

confidence: 99%

Localized standing waves in a hyperelastic membrane tube and their stabilization by a mean flow

Ильичев

2014

Mathematics and Mechanics of Solids

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We first give a complete analysis of the dispersion relation for traveling waves propagating in a pre-stressed hyperelastic membrane tube containing a uniform flow. We present an exact formula for the so-called pulse wave velocity, and demonstrate that as any pre-stress parameter is increased gradually, localized bulging would always occur before a superimposed small-amplitude traveling wave starts to grow exponentially. We then study the stability of weakly and fully nonlinear localized bulging solutions that may exist in such a fluid-filled hyperelastic membrane tube. Previous studies have shown that such localized standing waves are unstable under pressure control in the absence of a mean-flow, whether the fluid inertia is taken into account or not. Stability of such localized aneurysm-type solutions is desired when aneurysm formation in human arteries is modelled as a bifurcation phenomenon. It is shown that in the near-critical regime axisymmetric perturbations are governed by the Korteweg-de Vries equation, and so the associated (weakly nonlinear) aneurysm solutions are (orbitally) stable with respect to axisymmetric perturbations. Stability of the fully nonlinear aneurysm solutions are studied numerically using the Evans function method. It is found that for each wall-fluid density ratio there exists a critical mean-flow speed above which no axisymmetric unstable modes can be found, which implies that a fully nonlinear aneurysm solution may be completely stabilized by a mean flow.

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“…If this were proved to be the case, stable bulged configurations with even larger amplitude would become possible and their effects on the blood flow would be even more pronounced. However, we appreciate that there is still a lot of uncertainty in the constitutive modeling of arterial walls in vivo and whether bifurcation is possible depends very much on the material models used [8]. Our results should at least be indicative of what might happen if bifurcation is possible when more realistic material models are used.…”

Section: Resultsmentioning

confidence: 75%

“…This study is part of our recent research effort that explores the postulate that initiation of at least some acute aneurysms in human arteries may be modeled as a bifurcation phenomenon [8]. This postulate is motivated by two main results.…”

Section: Introductionmentioning

confidence: 99%

Stability of an inflated hyperelastic membrane tube with localized wall thinning

Ильичев

2014

International Journal of Engineering Science

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It is now well-known that when an infinitely long hyperelastic membrane tube free from any imperfections is inflated, a transcritical-type bifurcation may take place that corresponds to the sudden formation of a localized bulge. When the membrane tube is subjected to localized wall-thinning, the bifurcation curve would "unfold" into the turning-point type with the lower branch corresponding to uniform inflation in the absence of imperfections, and the upper branch to bifurcated states with larger amplitude. In this paper stability of bulged configurations corresponding to both branches is investigated with the use of the spectral method. It is shown that under pressure control and with respect to axi-symmetric perturbations, configurations corresponding to the lower branch are stable but those corresponding to the upper branch are unstable. Stability or instability is established by demonstrating the non-existence or existence of an unstable eigenvalue (an eigenvalue with a positive real part). This is achieved by constructing the Evans function that depends only on the spectral parameter. This function is analytic in the right half of the complex plane where its zeroes correspond to the unstable eigenvalues of the generalized spectral problem governing spectral instability. We show that due to the fact that the skew-symmetric operator J involved in the Hamiltonian formulation of the basic equations is onto, the zeroes of the Evans function can only be located on the real axis of the complex plane. We also comment on the connection between spectral (linear) stability and nonlinear (Lyapunov) stability.

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Initiation of aneurysms as a mechanical bifurcation phenomenon

Cited by 46 publications

References 36 publications

Effects of imperfections on localized bulging in inflated membrane tubes

Effects of imperfections on localized bulging in inflated membrane tubes

Localized standing waves in a hyperelastic membrane tube and their stabilization by a mean flow

Stability of an inflated hyperelastic membrane tube with localized wall thinning

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