Skip to main content
SpringerLink
Log in

Pure Axial Shear of Isotropic, Incompressible Nonlinearly Elastic Materials with Limiting Chain Extensibility

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

Abstract

The purpose of this research is to investigate the pure axial shear problem for a circular cylindrical tube composed of isotropic hyperelastic incompressible materials with limiting chain extensibility. Two popular models that account for hardening at large deformations are examined. These involve a strain-energy density which depends only on the first invariant of the Cauchy–Green tensor. In the limit as a polymeric chain extensibility tends to infinity, all of these models reduce to the classical neo-Hookean form. The stress fields and axial displacements are characterized for each of these models. Explicit closed-form analytic expressions are obtained. The results are compared with one another and with the predictions of the neo-Hookean model.

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

Similar content being viewed by others

References

  1. C.O. Horgan and G. Saccomandi, Simple torsion of isotropic, hyperelastic, incompressible materials with limiting chain extensibility. J. Elasticity 56 (1999) 159-170.

    Article  MATH  MathSciNet  Google Scholar 

  2. A.N. Gent, A new constitutive relation for rubber. Rubber Chemistry and Technology 69 (1996) 59-61.

    MathSciNet  Google Scholar 

  3. E.M. Arruda and M.C. Boyce, A three dimensional constitutive model for the large deformation stretch behavior of rubber elastic materials. J. Mech. Phys. Solids 41 (1993) 389-412.

    Article  ADS  Google Scholar 

  4. M.C. Boyce, Direct comparison of the Gent and the Arruda—Boyce constitutive models of rubber elasticity. Rubber Chemistry and Technology 69 (1996) 781-785.

    Google Scholar 

  5. H.G. Kilian, Equation of state of real networks. Polymer 22 (1981) 209-217.

    Article  Google Scholar 

  6. B. Erman and J.E. Mark, Structures and Properties of Rubberlike Networks. Oxford University Press, Oxford (1997).

    Google Scholar 

  7. X. Jiang and R.W. Ogden, Some new solutions for the axial shear of a circular cylindrical tube of compressible elastic material. Int. J. of Nonlinear Mechanics 35 (2000) 361-369.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. J.K. Knowles, On finite anti-plane shear for incompressible elastic materials. J. Austral. Math. Soc. B 20 (1977) 1-7.

    MATH  MathSciNet  Google Scholar 

  9. C.O. Horgan, Anti-plane shear deformations in linear and nonlinear solid mechanics. SIAM Review 37 (1995) 53-81.

    Article  MATH  MathSciNet  Google Scholar 

  10. R.S. Rivlin, Large elastic deformations of isotropic materials VI. Further results in the theory of torsion, shear and flexure. Phil. Trans. Roy. Soc. London A 242 (1949) 173-195.

    MATH  MathSciNet  ADS  Google Scholar 

  11. R.W. Ogden, Non-linear Elastic Deformations. Ellis Horwood, Chichester (1984). Reprinted by Dover, New York (1997).

    Google Scholar 

  12. Q. Jiang and J.K. Knowles, A class of compressible elastic materials capable of sustaining finite anti-plane shear. J. Elasticity 25 (1991) 193-201.

    MATH  MathSciNet  Google Scholar 

  13. D.A. Polignone and C.O. Horgan, Axisymmetric finite antiplane shear of compressible nonlinearly elastic circular tubes. Quart. Appl. Math. 50 (1992) 323-341.

    MATH  MathSciNet  Google Scholar 

  14. Q. Jiang and M.F. Beatty, On compressible materials capable of sustaining axisymetric shear deformation. Part 1: Anti-plane shear of isotropic hyperlelastic materials. J. Elasticity 39 (1995) 75-95.

    Article  MATH  MathSciNet  Google Scholar 

  15. M.F. Beatty, Topics in finite elasticity: Hyperelasticity of rubber, elastomers, and biological tissues — with examples. Appl. Mech. Rev. 40 (1987) 1699-1733. (Reprinted with minor modifications as “Introduction to nonlinear elasticity” in M.M. Carroll and M.A. Hayes (eds), Nonlinear Effects in Fluids and Solids, Plenum Press, New York (1996), pp. 16–112.)

    Article  Google Scholar 

  16. R.L. Fosdick and B.G. Kao, Transverse deformations associated with rectilinear shear in elastic solids. J. Elasticity 8 (1978) 117-142.

    Article  MATH  MathSciNet  Google Scholar 

  17. F. Mollica and K.R. Rajagopal, Secondary deformations due to axial shear of the annular region between two eccentrically placed cylinders. J. Elasticity 48 (1997) 103-123.

    Article  MATH  MathSciNet  Google Scholar 

  18. R.L. Fosdick and G. MacSithigh, Helical shear of an elastic, circular tube with a non-convex stored energy. Arch. Rational Mech. Anal. 84 (1983) 31-53.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  19. R.L. Fosdick and Y. Zhang, Coexistent phase mixtures in the anti-plane shear of an elastic tube. Z. Angew. Math. Phys. 65 (1994) 202-244.

    Article  MathSciNet  Google Scholar 

  20. R.L. Fosdick and Y. Zhang, A structured phase transition for the anti-plane shear of an elastic circular tube, Quart. J. Mech. Appl. Math. 48 (1995) 189-210.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Applied Mechanics Program, Department of Civil Engineering, University of Virginia, Charlottesville, VA, 22903, U.S.A. E-mail

    Cornelius O. Horgan

  2. Dipartimento di Ingegneria dell'Innovazione, Università degli Studi di Lecce, 73100, Lecce, Italy. E-mail

    Giuseppe Saccomandi

Authors
  1. Cornelius O. Horgan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giuseppe Saccomandi
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Horgan, C.O., Saccomandi, G. Pure Axial Shear of Isotropic, Incompressible Nonlinearly Elastic Materials with Limiting Chain Extensibility. Journal of Elasticity 57, 307–319 (1999). https://doi.org/10.1023/A:1007639129264

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1007639129264

Search

Navigation