The Effect of a Residual Stress on Wave Propagation in a Fluid-Filled Thick Elastic Tube

Year 2019, Volume: 20 Issue: 1, 54 - 69, 01.01.2019

Hakan Erol

https://doi.org/10.18038/aubtda.466203

Abstract

The propagation of harmonic waves in an elastic tubes filled fluid is presented in this study. The tube material is considered to be incompressible, homogeneous, isotropic, initially axially stretched, inflated and thick elastic like human arteries. The viscous fluid is assumed to be incompressible and Newtonian. The differential equations of both materials are obtained in cylindrical coordinates. The analytical solutions of the equations of motion for the fluid, numerical solutions of the equations of motion for the tube have been found. The residual circumferential strain in the unloaded state of artery causes opening angle. The dispersion relation is presented as a function of the axial stretch, opening angle, internal pressure and material parameters. The effects of these parameters are shown and discussed on graphics.

Keywords

Opening angle, constitutive equations, dispersion relation

References

• Womersley JR. 1957. An elastic tube theory of pulse transmission and oscillatory flow in mammalian arteries, WADC. Tech. Rep. TR., 56-614.
• Morgan GW, Kiely, JP. 1954. Wave propagation in a viscous liquid contained in a flexible tube, J. Acoust. Soc. Am. 26, 323–328.
• Mirsky I. 1967. Wave propagation in a viscous fluid contained in an orthotropic elastic tube, Biophys. J. 7, 165–186.
• Atabek HB, Lew HS. 1966. Wave propagation through a viscous incompressible fluid contained in an initially stressed elastic tube, Biophys. J. 6, 481–503.
• Rachev AI. 1980. Effect of transmural pressure and muscular activity on pulse waves in arteries, J. Biomech. Eng. ASME 102, 119–123.
• Demiray H, Antar N. 1996. Effects of initial stresses and wall thickness on wave characteristics in elastic tubes. ZAMM 76, 521–530.
• Cox RH. 1968. Wave propagation through a Newtonian fluid contained within a thick walled viscoelastic tube, Biophys. J. 8, 691–709.
• Kizilova NN. 2006. Pressure wave propagation in liquid-filled tubes of viscoelastic material, Fluid Dynam. 41 434–446.
• Nayfeh AH. 1966. Oscillating two-phase flow through a rigid pipe, AIAA. J. 4 1868–1870.
• Nag SK, Jana RN. 1981. Oscillating two-phase flow in an elastic tube, Acta Mech. 41 121–128.
• Ercengiz A. 2005. Oscillating two-phase flow in a prestressed thick elastic tube, Acta Mech. 179, 169–185.
• Demiray H. 2005. Weakly nonlinear waves in a viscous fluid contained in a viscoelastic tube, European Journal of Mechanics A/ Solids. 24, 337-347.
• Demiray H. 2006. Weakly nonlinear waves in a fluid with variable viscosity contained in a prestressed thin elastic tube. Chaos Solitons & Fractails, 36 196-202.
• Jagielska K, Trzupek D, Lepers M, Pelc A and Zieliński P. 2007. Effect of surrounding tissue on propagation of axisymmetric waves in arteries, Physical Review E 76.
• Chaudhry HR, Bukiet B, Davis A, Ritterj AB and Findley T. J. 1997. Residual stresses in oscillating thoracic arteries reduce circumferential stress gradiants, J. Biomech., 30, 57-62.
• Vaishnav RN, and Vossoughi J. Estimation of residual strains in aortic segments. In: Biomedical Engineering. II. Recent Developments, edited by C. W. Hall. New York: Pergamon, 1983, vol. 2, p.330–333.
• Huang W, Yen RT. 1998. Zero-stress states of human pulmonary arteries and veins, J Appl Physiol 85 867-873.
• Demiray H. 1972. A note on the elasticity of soft biological tissue. J. Biomech. 5, 309–311.
• Baek S, Gleason RL, Rajagopal KR, Humphrey JD, 2007. Theory of small on large: Potential utility in computations of fluid-solid interactions in arteries, Computer Methods in Applied Mechanics and Engineering, 196, 3070-3078.
• Demiray H, Ercengiz A. 1995. Harmonic waves in a prestressed viscoelastic tube filled with a viscous fluid Int. J. Engng Sci. 33, 105-107.
• Eringen AC, Suhubi ES. Elastodynamics, vol. I. New York: Academic Press 1974.
• Atabek HB, Lew HS, 1966. Wave propagation through a viscous incompressible fluid contained in an initially stressed elastic tube, Biophys. J. 6, 481–503.
• [23] Simon BR, Kobayashi AS, Strandness DE, Wiederhielm CA. 1972. Re-evaluation of arterial constitutive laws. Circulation Res. 30, 491–500.
• Demiray H. 1976. Some basic problems in biophysics. Bull. Math. Biol. 38, 701–711.
• Demiray H, Ercengiz A. 1991. Wave propagation in a prestressed elastic tube filled with a viscous fluid, 29, 575-585.
• Bustamante R, Holzapfel GA. 2010. Methods to compute 3D residual stress distributions in hyperelastic tubes with application to arterial walls, International Journal of Engineering Science, 48, 1066-1082.
• Demiray H, Akgün G. 1997. Wave propagation in a viscous fluid contained in a prestressed viscoelastic thin tube, Int. J. Engng Sci. 35 1065-1079.