Abstract
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.
Similar content being viewed by others
References
Chater, E., Hutchinson, J.W.: On the propagation of bulges and buckles. ASME J. Appl. Mech. 51, 269–277 (1984)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
Gent, A. N.: A new constitutive relation for rubber. Rubber Chem. Technol. 69, 59–61 (1996)
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)
Demiray, H.: Solitary waves in initially stressed thin elastic tubes. Int. J. Non-Linear Mech. 32, 1165–1176 (1996)
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
Wolfram, S.: Mathematica: A System for Doing Mathematics by Computer (2nd edn). California: Addison-Wesley (1991)
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)
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)
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)
Pego, R. L., Weinstein, M. I.: Eigenvalues, and instability of solitary waves. Phil. Trans. R. Soc. Lond. A 340, 47–94 (1992)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Il’ichev, A.T., Fu, Y.B. Stability of aneurysm solutions in a fluid-filled elastic membrane tube. Acta Mech Sin 28, 1209–1218 (2012). https://doi.org/10.1007/s10409-012-0135-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10409-012-0135-2