Skip to main content
SpringerLink
Log in

Stability of pre-stressed incompressible hyperelastic cylindrical tubes under axial compression

  • Published:
International Journal of Dynamics and Control Aims and scope Submit manuscript

Abstract

The problems of stability and controllability of the finite deformation are investigated for a pre-stressed hyperelastic cylindrical tube subjected to the axial compression. The tube is assumed to be composed of an incompressible neo-Hookean material. Based on the theory of small deformation superposed on large elastic deformation, the mathematical model is formulated by a system of incremental equilibrium equations and incremental boundary conditions. Moreover, the general solutions describing the finite deformation of the tube are obtained by the form of Bessel functions. Finally, the system of nonlinear equations governing the stability of the tube are given, the criteria of the stability discussed in terms of the corresponding numerical simulations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Mihai LA, Budday S, Holzapfel GA, Kuhl E, Goriely A (2017) A family of hyperelastic models for human brain tissue. J Mech Phys Solids 106:60–79

    Article  MathSciNet  Google Scholar 

  2. Saccomandi G, Vergori L (2019) Generalised Mooney–Rivlin models for brain tissue: a theoretical perspective. Int J Non-linear Mech 109:9–14

    Article  Google Scholar 

  3. Diani J, Tallec PL (2019) A fully equilibrated microsphere model with damage for rubberlike materials. J Mech Phys Solids 124:702–713

    Article  MathSciNet  Google Scholar 

  4. Horgan CO, Murphy JG (2018) Magic angles and fibre stretch in arterial tissue: insights from the linear theory. J Mech Behav Biomed Mater 88:470–477

    Article  Google Scholar 

  5. Zidi M (2000) Combined torsion, circular and axial shearing of a compressible hyperelastic and prestressed tube. J Appl Mech 67:33–40

    Article  Google Scholar 

  6. Zidi M (2001) Effects of a prestress on a reinforced, nonlinearly elastic and compressible tube subjected to combined deformations. Int J Solids Struct 38:4657–4669

    Article  Google Scholar 

  7. Zidi M, Cheref M (2002) Finite deformations of a hyperelastic, compressible and fibre reinforced tube. Eur J Mech Solids 21:971–980

    Article  Google Scholar 

  8. Saravanan U (2011) On large elastic deformation of prestressed right circular annular cylinders. Int J Non-linear Mech 46:96–113

    Article  Google Scholar 

  9. Balbi V, Ciarletta P (2015) Helical buckling of thick-walled, pre-stressed, cylindrical tubes under a finite torsion. Math Mech Solids 20(6):625–642

    Article  MathSciNet  Google Scholar 

  10. Merodio J, Ogden RW (2016) Extension, inflation and torsion of a residually-stressed circular cylindrical tube. Contin Mech Thermodyn 28:157–174

    Article  MathSciNet  Google Scholar 

  11. Li KW, Ogden RW, Holzapfel GA (2018) An exponential constitutive model excluding fibers under compression: application to extension-inflation of a residually stressed carotid artery. Math Mech Solids 23(8):1206–1224

    Article  MathSciNet  Google Scholar 

  12. Sigaeva T, Kolesnikov A, Sudak L (2019) Deformation of a closed hyperelastic helical spring. Int J Non-linear Mech 110:1–8

    Article  Google Scholar 

  13. Green AE, Rivlin RS, Shield RT (1952) General theory of small deformations superposed on finite elastic deformations. Proc R Soc Lond A211:128–154

    MathSciNet  MATH  Google Scholar 

  14. Wilkes EW (1955) On the stability of a circular tube under end thrust. Q J Mech Appl Mech 8:88–100

    Article  MathSciNet  Google Scholar 

  15. Biot MA (1965) Mechanics of incremental deformations. Wiley, New York

    Book  Google Scholar 

  16. Wang SD, Ertepinar A (1972) Stability and vibrations of elastic thick-walled cylindrical and spherical shells subjected to pressure. Int J Non-linear Mech 7:539–555

    Article  Google Scholar 

  17. Haughton DM, Ogden RW (1979) Bifurcation of inflated circular cylinders of elastic material under axial loading—I. Membrane theory for thin-walled tubes. J Mech Phys Solids 27:79–212

    MathSciNet  MATH  Google Scholar 

  18. Haughton DM, Ogden RW (1979) Bifurcation of inflated circular cylinders of elastic material under axial loading—II. Exact theory for thick-walled tubes. J Mech Phys Solids 27:489–512

    Article  MathSciNet  Google Scholar 

  19. Duka ED, England AH, Spencer AJM (1993) Bifurcation of a solid circular elastic cylinder under finite extension and torsion. Acta Mech 98:107–121

    Article  Google Scholar 

  20. Zhu Y, Luo XY, Ogden RW (2008) Asymmetric bifurcations of thick-walled circular cylindrical elastic tubes under axial loading and external pressure. Int J Solids Struct 45:3410–3429

    Article  Google Scholar 

  21. DeBotton G, Bustamante R, Dorfmann A (2013) Axisymmetric bifurcations of thick spherical shells under inflation and compression. Int J Solids Struct 50:403–413

    Article  Google Scholar 

  22. Dai HH, Wang FF, Wang J, Xu J (2015) Pitchfork and octopus bifurcations in a hyperelastic tube subjected to compression: analytical post-bifurcation solutions and imperfection sensitivity. Math Mech Solids 20(1):25–52

    Article  MathSciNet  Google Scholar 

  23. Rivlin RS (1949) Large elastic deformations of isotropic materials. VI. Further results in the theory of torsion, shear and flexure. Philos Trans R Soc 242:173–195

    MathSciNet  MATH  Google Scholar 

  24. Shang XC (2008) The stability of compressed hyperelastic thick-walled cylindrical shells. Adv Solid Mech Relat Res 1:63–68

    Google Scholar 

Download references

Funding

This research was funded by the National Natural Science Foundation of China, Grant Numbers 11902068, 11672069.

Author information

Authors and Affiliations

  1. School of Science, Dalian Minzu University, Dalian, 116600, China

    W. Zhao & X. G. Yuan

  2. Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing, 100124, China

    W. Zhao & W. Zhang

  3. Department of Applied Mechanics, University of Science and Technology Beijing, Beijing, 100083, China

    X. C. Shang

Authors
  1. W. Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. X. G. Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. X. C. Shang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to W. Zhang or X. G. Yuan.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, W., Zhang, W., Yuan, X.G. et al. Stability of pre-stressed incompressible hyperelastic cylindrical tubes under axial compression. Int. J. Dynam. Control 9, 862–871 (2021). https://doi.org/10.1007/s40435-020-00707-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40435-020-00707-x

Keywords

Search

Navigation