Analytic description and explicit parametrisation of the equilibrium shapes of elastic rings and tubes under uniform hydrostatic pressure

International Journal of Mechanical Sciences

Volumn 53, Issue 5, 2011, Pages 355-364

  Djondjorov, P A ^a   Vassilev, V M ^a   Mladenov, I M ^b  

^a INSTITUTE OF MECHANICS   (Bulgaria)

^b INSTITUTE OF BIOPHYSICS AND BIOMEDICAL ENGINEERING   (Bulgaria)

Author keywords

Elastic ring (tube); Equilibrium shapes; Hydrostatic pressure; Parametric equations; Similarity law; Tube conductivity

Indexed keywords

ELASTIC RINGS; EQUILIBRIUM SHAPES; PARAMETRIC EQUATION; SIMILARITY LAW; TUBE CONDUCTIVITY;

HYDRAULICS; HYDRODYNAMICS; HYDROSTATIC PRESSURE;

TUBES (COMPONENTS);

EID: 79955059962     PISSN: 00207403     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ijmecsci.2011.02.005     Document Type: Article

References (27)

1. G.F. Carrier On the buckling of elastic rings J Math and Phys 26 1947 94 103
2. I. Tadjbakhsh, and F. Odeh Equilibrium states of elastic rings J Math Anal Appl 18 1967 59 74
3. J. Chaskalovic, and S. Naili Bifurcation theory applied to buckling states of a cylindrical shell ZAMP 46 1995 149 155
4. J.E. Flaherty, J.B. Keller, and S.I. Rubinow Post buckling behaviour of elastic tubes and rings with opposite sides in contact SIAM J Appl Math A 23 1972 446 455
5. Antman SS. Nonlinear problems of elasticity. In: Applied Mathematical Sciences, vol. 107. New York: Springer-Verlag; 1995.
6. C. Pozrikidis Buckling and collapse of open and closed cylindrical shells J Eng Math 42 2002 157 180 (Pubitemid 34623585)
7. M.M. Lévy Mémoire sur un nouveau cas intégrable du problème de l'élastique et l'une de ses applications Journal de Mathematiques Pures et Appliquees série 3X 1884 5 42
8. Halphen GH. La corbe élastique plane sous pression uniforme, Chap. V in Traité des fonctions elliptiques et de leurs applications. Deuxième partie: Applications a la mécanique, a la physique, a la géodésie, a la géométrie et au calcul intégral, Gauthier-Villars et fils, Paris; 1888.
9. A.G. Greenhill The elastic curve under uniform normal pressure Math Ann LII 1889 465 500
10. S.G. Zhang A complete classification of closed shapes for cylindrical vesicles Acta Phys Sin (Overseas Edn) 6 1997 641 655
11. A.J. Adams Postbuckling of circular rings: an analytical solution J Math Phys 49 2008 032902
12. K. Watanabe, and I. Takagi Representation formula for the critical points of the TadjbakhshOdeh functional and its application Japan J Ind Appl Math 25 2008 331 372
13. P.A. Djondjorov, V.M. Vassilev, and I.M. Mladenov Plane curves associated with integrable dynamical systems of the FrenetSeret type Proceedings of the 9th international workshop on complex structures, integrability and vector fields, Sofia 2529 August 2008 2009 World Scientific Publishing Co. Singapore 56 63
14. V.M. Vassilev, P.A. Djondjorov, and I.M. Mladenov On the translationally-invariant solutions of the membrane shape equation I. Mladenov, M. de León, Geometry, integrability and quantization vol. 8 2007 SOFTEX Sofia 312 321
15. V.M. Vassilev, P.A. Djondjorov, and I.M. Mladenov Cylindrical equilibrium shapes of fluid membranes J Phys A: Math Theor 41 2008 435201
16. B.S. Wu, Y.P. Yu, and Z.G. Li Analytical approximations to large post-buckling deformation of elastic rings under uniform hydrostatic pressure Int J Mech Sci 49 2007 661 668 (Pubitemid 46365706)
17. The definitions and properties of the elliptic functions and integrals can be found, e.g., in: Hancock H, editor. Elliptic integrals. New York: Dover; 1958; Jahnke E, Emde F, Lösch F. Tafeln Höherer Funktionen. Stuttgart: Teubner; 1960; Chandrasekharan, K. Elliptic functions. New York: Springer-Verlag; 1985; Lawden D. Elliptic functions and applications. New York: Springer-Verlag; 1989.
18. Y. Fukumoto Stationary configurations of a vortex filament in background flows Proc R Soc Lond A 453 1997 1205 1232
19. G. Arreaga, R. Capovilla, C. Chryssomalakos, and J. Guven Area-constrained planar elastica Phys Rev E 65 2002 031801
20. R. Capovilla, C. Chryssomalakos, and J. Guven Elastica hypoarealis Eur Phys J B 29 2002 163 166 (Pubitemid 135695448)
21. J. Guven Laplace pressure as a surface stress in fluid vesicles J Phys A: Math Gen 39 2006 3771 3785 (Pubitemid 43453834)
22. H.S.M. Coxeter Introduction to geometry 2nd ed 1969 Wiley New York
23. G.A. Korn, and T.M. Korn Mathematical handbook for scientists and engineers: definition, theorems, and formulas for reference and review 1968 McGraw-Hill New York
24. D. DeTurck, H. Gluck, D. Pomerleano, and D.S. Vick The four vertex theorem and its converse Notices Am Math Soc 54 2007 192 207
25. H. Hopf Über die Drehung der Tangenten und Sehnen ebener Kurven Compositio Math 2 1935 50 62
26. Morris CG, editor. Academic press dictionary of science and technology. San Diego: Academic Press Inc.; 1992.
27. M.G. Blyth, and C. Pozrikidis Buckling and collapse of heavy tubes resting on a horizontal or inclined plane Eur J Mech A Solids 21 2002 831 843