Static and dynamic responses of a circular cylindrical shell made of hyperelastic arterial material are studied. The material is modeled as a combination of Neo-Hookean and Fung materials. Two types of pressure loads are studied—distributed radial forces and deformation-dependent pressure. The static responses of the shell under these two loads differ essentially at moderate strains, while the behavior is similar for small loads. The principal difference is that the axial displacements are much larger for the shell under distributed radial forces, while for actual pressure the shell is stretched both in circumferential and axial directions. Free and forced vibrations around preloaded configurations are analyzed. In both cases, the nonlinearity of the single-mode (driven mode) response of the preloaded shell is quite weak, but a resonant regime with both driven and companion modes active has been found with more complicated nonlinear dynamics.
Issue Section:
Research Papers
Keywords:
Vibration
References
1.
Lamé
, G.
, 1852
, Leçons sur la théorie mathématique de l'élasticité des corps solides
, Bachelier
, Paris
.2.
Love
, A. E. H.
, 1927
, A Treatise on the Mathematical Theory of Elasticity
, 4th ed., Cambridge University Press
, Cambridge, UK
.3.
Alijani
, F.
, and Amabili
, M.
, 2014
, “Non-Linear Vibrations of Shells: A Literature Review From 2003 to 2013
,” Int. J. Non-Linear Mech.
, 58
, pp. 233
–257
.4.
Lurie
, A. I.
, 2005
, Theory of Elasticity
, Springer-Verlag
, Berlin
.5.
Lopes
, S. R. X.
, Gonçalves
, P. B.
, and Pamplona
, D. C.
, 2007
, “Influence of Initial Geometric Imperfections on the Stability of Thick Cylindrical Shells Under Internal Pressure
,” Commun. Numer. Methods Eng.
, 23
(6
), pp. 577
–597
.6.
Gonçalves
, P. B.
, Pamplona
, D.
, and Lopes
, S. R. X.
, 2008
, “Finite Deformations of an Initially Stressed Cylindrical Shell Under Internal Pressure
,” Int. J. Mech. Sci.
, 50
(1
), pp. 92
–103
.7.
Guo
, Z.
, Wang
, S.
, Li
, L.
, Ji
, H.
, Wang
, Z.
, and Cai
, S.
, 2014
, “Inflation of Stressed Cylindrical Tubes: An Experimental Study
,” Proc. SPIE
, 9234
, p. 92340H
.8.
Haughton
, D. M.
, and Ogden
, R. W.
, 1979
, “Bifurcation of Inflated Circular Cylinders of Elastic Material Under Axial Loading—II: Exact Theory for Thick-Walled Tubes
,” J. Mech. Phys. Solids
, 27
(5–6
), pp. 489
–512
.9.
Akyuz
, U.
, and Ertepinar
, A.
, 1998
, “Stability and Asymmetric Vibrations of Pressurized Compressible Hyperelastic Cylindrical Shells
,” Int. J. Non-Linear Mech.
, 34
(3
), pp. 391
–404
.10.
Chen
, Y.-C.
, and Haughton
, D. M.
, 2003
, “Stability and Bifurcation of Inflation of Elastic Cylinders
,” Proc. R. Soc. London, Ser. A
, 459
(2029
), pp. 137
–156
.11.
Zhu
, Y.
, Luo
, X. Y.
, and Ogden
, R. W.
, 2008
, “Asymmetric Bifurcations of Thick-Walled Circular Cylindrical Elastic Tubes Under Axial Loading and External Pressure
,” Int. J. Solids Struct.
, 45
(11–12
), pp. 3410
–3429
.12.
Zhu
, Y.
, Luo
, X. Y.
, and Ogden
, R. W.
, 2010
, “Nonlinear Axisymmetric Deformations of an Elastic Tube Under External Pressure
,” Eur. J. Mech. A/Solids
, 29
(2
), pp. 216
–229
.13.
Zhu
, Y.
, Luo
, X. Y.
, Wang
, H. M.
, Ogden
, R. W.
, and Berry
, C.
, 2013
, “Three-Dimensional Non-Linear Buckling of Thick-Walled Elastic Tubes Under Pressure
,” Int. J. Non-Linear Mech.
, 48
, pp. 1
–14
.14.
Kozlovsky
, P.
, Zaretsky
, U.
, Jaffa
, A. J.
, and Elad
, D.
, 2014
, “General Tube Law for Collapsible Thin and Thick-Wall Tubes
,” J. Biomech.
, 47
(10
), pp. 2378
–2384
.15.
Knowles
, J. K.
, 1960
, “Large Amplitude Oscillations of a Tube of Incompressible Elastic Material
,” Q. Appl. Math.
, 18
, pp. 71
–77
.16.
Knowles
, J. K.
, 1962
, “On a Class of Oscillations in the Finite Deformation Theory of Elasticity
,” ASME J. Appl. Mech.
, 29
(2
), pp. 283
–286
.17.
Wang
, A. S. D.
, and Ertepinar
, A.
, 1972
, “Stability and Vibrations of Elastic Thick-Walled Cylindrical and Spherical Shells Subjected to Pressure
,” Int. J. Non-Linear Mech.
, 7
(5
), pp. 539
–555
.18.
Calderer
, C.
, 1983
, “The Dynamical Behaviour of Nonlinear Elastic Spherical Shells
,” J. Elasticity
, 13
(1
), pp. 17
–47
.19.
Ren
, J.-S.
, 2008
, “Dynamical Response of Hyper-Elastic Cylindrical Shells Under Periodic Load
,” Appl. Math. Mech.
, 29
(10
), pp. 1319
–1327
.20.
Breslavsky
, I.
, Amabili
, M.
, and Legrand
, M.
, 2014
, “Physically and Geometrically Non-Linear Vibrations of Thin Rectangular Plates
,” Int. J. Non-Linear Mech.
, 58
, pp. 30
–40
.21.
Amabili
, M.
, Karazis
, K.
, Mongrain
, R.
, Païdoussis
, M. P.
, and Cartier
, R.
, 2012
, “A Three-Layer Model for Buckling of a Human Aortic Segment Under Specific Flow-Pressure Conditions
,” Int. J. Numer. Methods Biomed. Eng.
, 28
(5
), pp. 495
–512
.22.
Breslavsky
, I. D.
, Amabili
, M.
, and Legrand
, M.
, 2014
, “Nonlinear Vibrations of Thin Hyperelastic Plates
,” J. Sound Vib.
, 333
(19
), pp. 4668
–4681
.23.
Holzapfel
, G. A.
, 2006
, “Determination of Material Models for Arterial Walls From Uniaxial Extension Tests and Histological Structure
,” J. Theor. Biol.
, 238
(2
), pp. 290
–302
.24.
Amabili
, M.
, and Breslavsky
, I. D.
, 2015
, “Displacement Dependent Pressure Load for Finite Deflection of Shells and Plates
,” Int. J. Non-Linear Mech.
, 77
, pp. 265
–273
.25.
Breslavsky
, I. D.
, Amabili
, M.
, and Legrand
, M.
, 2016, “Axisymmetric Deformations of Circular Rings Made of Linear and Neo-Hookean Materials Under Internal and External Pressure
,” Int. J. Non-Linear Mech.
(to be published).26.
Amabili
, M.
, and Reddy
, J. N.
, 2010
, “A New Non-Linear Higher-Order Shear Deformation Theory for Large-Amplitude Vibrations of Laminated Doubly Curved Shells
,” Int. J. Non-Linear Mech.
, 45
(4
), pp. 409
–418
.27.
Ogden
, R.
, 1997
, Non-Linear Elastic Deformations
, Dover Publications
, New York
.28.
Amabili
, M.
, 2008
, Nonlinear Vibrations and Stability of Shells and Plates
, Cambridge University Press
, New York
.29.
Atkinson
, K. E.
, 1989
, An Introduction to Numerical Analysis
, 2nd ed., Wiley
, New York
.30.
Amabili
, M.
, 2015
, “A New Third-Order Shear Deformation Theory With Non-Linearities in Shear for Static and Dynamic Analysis of Laminated Doubly Curved Shells
,” Compos. Struct.
, 128
, pp. 260
–273
.31.
Leissa
, A. W.
, 1973
, Vibrations of Shells
, National Aeronautics and Space Administration
, Washington, DC
.32.
Parker
, T. S.
, and Chua
, L. O.
, 1989
, Practical Numerical Algorithms for Chaotic Systems
, Springer-Verlag
, New York
.33.
Doedel
, E. J.
, Champneys
, A. R.
, Fairgrieve
, T. F.
, Kuznetsov
, Y. A.
, Sandstede
, B.
, and Wang
, X.
, 1998
, AUTO 97: Continuation and Bifurcation Software for Ordinary Differential Equations (With HomCont)
, Concordia University
, Montreal, QC
, Canada.Copyright © 2016 by ASME
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