×

Stability of aneurysm solutions in a fluid-filled elastic membrane tube. (English) Zbl 1293.74310

Summary: When a hyperelastic membrane tube is inflated by an internal pressure, a localized bulge will form when the pressure reaches a critical value. As inflation continues the bulge will grow until it reaches a maximum size after which it will then propagate in both directions to form a hat-like profile. The stability of such bulging solutions has recently been studied by neglecting the inertia of the inflating fluid and it was shown that such bulging solutions are unstable under pressure control. In this paper we extend this recent study by assuming that the inflation is by an inviscid fluid whose inertia we take into account in the stability analysis. This reflects more closely the situation of aneurysm formation in human arteries which motivates the current series of studies. It is shown that fluid inertia would significantly reduce the growth rate of the unstable mode and thus it has a strong stabilizing effect.

MSC:

74L15 Biomechanical solid mechanics
74K15 Membranes
74G60 Bifurcation and buckling
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)

Software:

Mathematica
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Chater, E., Hutchinson, J.W.: On the propagation of bulges and buckles. ASME J. Appl. Mech. 51, 269–277 (1984) · doi:10.1115/1.3167611
[2] Kyriakides, S. Chang, Y. C.: The initiation and propagation of a localized instability in an inflated elastic tube. Int. J. Solid Struct. 27, 1085–1111 (1991) · doi:10.1016/0020-7683(91)90113-T
[3] Pamplona, D. C., Goncalves, P. B., Lopes, S. R. X.: Finite deformations of cylindrical membrane under internal pressure. Int. J. Mech. Sci. 48, 683–696 (2006) · doi:10.1016/j.ijmecsci.2005.12.007
[4] Holzapfel, G. A., Gasser, T. C., Ogden, R. W.: A new constitutive framework for arterial wall mechanics and a comparative study of material models. J. Elast. 60, 1–48 (2000) · Zbl 1023.74033 · doi:10.1023/A:1010835316564
[5] Fu, Y. B., Pearce, S. P., Liu, K. K.: Post-bifurcation analysis of a thin-walled hyperelastic tube under inflation. Int. J. Nonlinear Mech. 43, 697–706 (2008) · Zbl 1203.74043 · doi:10.1016/j.ijnonlinmec.2008.03.003
[6] Pearce, S. P., Fu, Y. B.: Characterisation and stability of localised bulging/necking in inflated membrane tubes. IMA J. Appl. Math. 75, 581–602 (2010) · Zbl 1425.74326 · doi:10.1093/imamat/hxq026
[7] Fu, Y. B., Rogerson, G. A., Zhang, Y. T.: Initiation of aneurysms as a mechanical bifurcation phenomenon. Int. J. Non-linear Mech. 47, 179–184 (2012) · doi:10.1016/j.ijnonlinmec.2011.05.001
[8] Fu, Y. B., Xie, Y. X.: Effects of imperfections on localized bulging in inflated membrane tubes. Phil. Trans. R. Soc. A 370, 1896–1911 (2012) · Zbl 1250.74018 · doi:10.1098/rsta.2011.0297
[9] Fu, Y. B., Xie, Y. X.: Stability of localized bulging in inflated membrane tubes under volume control. Int. J. Eng. Sci. 48, 1242–1252 (2010) · Zbl 1231.74042 · doi:10.1016/j.ijengsci.2010.08.007
[10] Haughton, D. M., Ogden, R.W.: Bifurcation of inflated circular cylinders of elastic material under axial loading. I. Membrane theory for thin-walled tubes. J. Mech. Phys. Solids 27, 179–212 (1979) · Zbl 0412.73065 · doi:10.1016/0022-5096(79)90001-2
[11] Ogden, R. W.: Large deformation isotropic elasticity-on the correlation of theory and experiment for incompressible rubber-like solids. Proc. R. Soc. Lond. A326, 565–584 (1972) · Zbl 0257.73034 · doi:10.1098/rspa.1972.0026
[12] Gent, A. N.: A new constitutive relation for rubber. Rubber Chem. Technol. 69, 59–61 (1996) · doi:10.5254/1.3538357
[13] Epstein, M., Johnston, C.: On the exact speed and amplitude of solitary waves in fluid-filled elastic tubes. Proc. R. Soc. Lond. A 457, 1195–1213 (2001) · Zbl 0984.76010 · doi:10.1098/rspa.2000.0715
[14] Demiray, H.: Solitary waves in initially stressed thin elastic tubes. Int. J. Non-Linear Mech. 32, 1165–1176 (1996) · Zbl 0890.73050 · doi:10.1016/S0020-7462(96)00129-1
[15] Pearce, S. P.: Effect of strain-energy function and axial prestretch on the bulges, necks and kinks forming in elastic membrane tubes. Math. Mech. Solids, DOI: 10.1177/1081286511433084
[16] Wolfram, S.: Mathematica: A System for Doing Mathematics by Computer (2nd edn). California: Addison-Wesley (1991) · Zbl 0671.65002
[17] Fu, Y. B., Ilíchev, A.: Solitary waves in fluid-filled elastic tubes: existence, persistence, and the role of axial displacement. IMA J. Appl. Math. 75, 257–268 (2010) · Zbl 1189.35386 · doi:10.1093/imamat/hxq004
[18] Chen, Y. C.: Stability and bifurcation of finite deformations of elastic cylindrical membranes – part I. stability analysis, Int. J. Solids Structures 34, 1735–1749 (1997) · Zbl 0944.74550 · doi:10.1016/S0020-7683(96)00119-9
[19] Alexander, J. C., Sachs, R.: Linear instability of solitary waves of a Boussinesq-type equation: A computer assisted computation. Nonlin. World 2, 471–507 (1995) · Zbl 0833.34046
[20] Pego, R. L., Weinstein, M. I.: Eigenvalues, and instability of solitary waves. Phil. Trans. R. Soc. Lond. A 340, 47–94 (1992) · Zbl 0776.35065 · doi:10.1098/rsta.1992.0055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.