Skip to main content

Advertisement

SpringerLink
Log in

Invariance of the Equilibrium Equations of Finite Elasticity for Compressible Materials

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

Abstract

For homogeneous, isotropic, compressible nonlinearly elastic materials, a wide class of strain-energy density functions are obtained that leave the equations of equilibrium invariant under simple scaling transformations of the material and spatial coordinates. These strain-energy densities are homogeneous functions of the principal stretches. Several illustrative examples of particular strain-energies are provided. For axisymmetric problems, the invariance discussed here ensures that the equations of equilibrium can be solved by quadratures and thus often leads to analytic solutions in parametric or closed-form.

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 J.G. Murphy, A Lie group analysis of the axisymmetric equations of finite elastostatics for compressible materials. Math. Mech. Solids, in press.

  2. C.O. Horgan and J.G. Murphy, Lie group analysis and plane strain bending of cylindrical sectors for compressible nonlinearly elastic materials. IMA J. Appl. Math. 70 (2005) 80–91.

    Article  MathSciNet  Google Scholar 

  3. J.M. Hill, On static similarity deformations for isotropic materials. Quart. Appl. Math. 40 (1982) 287–291.

    Google Scholar 

  4. D. Levi and C. Rogers, Group invariance of a neo-Hookean system: Incorporation of stretch change. J. Elasticity 24 (1990) 295–300.

    Article  Google Scholar 

  5. J.G. Murphy, Equivalent and separable strain-energy functions in compressible finite elasticity. Internat. J. Nonlinear Mech. 40 (2005) 323–329.

    Article  Google Scholar 

  6. J.M. Hill, Cylindrical and spherical inflation in compressible finite elasticity. IMA J. Appl. Math. 50 (1993) 195–202.

    Google Scholar 

  7. D.J. Steigmann, Invariants of the stretch tensors and their application to finite elasticity theory. Math. Mech. Solids 7 (2002) 393–404.

    Google Scholar 

  8. D.J. Steigmann, On isotropic, frame-invariant, polyconvex strain-energy functions. Quart. J. Mech. Appl. Math. 56 (2003) 483–491.

    Article  Google Scholar 

  9. R.G. Bartle, The Elements of Real Analysis. Wiley, New York (1976).

    Google Scholar 

  10. R.T. Shield, Inverse deformation results in finite compressible elasticity. J. Appl. Math. Phys. 18 (1967) 490–500.

    Article  Google Scholar 

  11. M.M. Carroll and F.J. Rooney, Implications of Shield’s inverse deformation theorem for compressible finite elasticity, J. Appl. Math. Phys. (in press).

  12. P.J. Blatz and W.L. Ko, Application of finite elastic theory to the deformation of rubbery materials. Trans. Soc. Rheol. 6 (1962) 223–251.

    Article  CAS  Google Scholar 

  13. C.O. Horgan, Equilibrium solutions for compressible nonlinearly elastic materials. In: Y.B. Fu and R.W. Ogden (eds), Nonlinear Elasticity: Theory and Applications. Cambridge Univ. Press, Cambridge (2001) pp. 135–159.

    Google Scholar 

  14. M.M. Carroll, Finite strain solutions in compressible isotropic elasticity. J. Elasticity 20 (1988) 65–92.

    Article  Google Scholar 

  15. M.M. Carroll, On obtaining closed-form solutions for compressible nonlinearly elastic materials. J. Appl. Math. Phys. 46 (1995) 125–145.

    Google Scholar 

  16. J.G. Murphy, Some new closed-form solutions describing spherical inflation in compressible finite elasticity. IMA J. Appl. Math. 48 (1992) 305–316.

    Google Scholar 

  17. F. John, Plane strain problems for a perfectly elastic material of harmonic type. Comm. Pure Appl. Math. 13 (1960) 239–296.

    Google Scholar 

  18. F. John, Plane elastic waves of finite amplitude: Hadamard materials and harmonic materials. Comm. Pure Appl. Math. 19 (1966) 309–341.

    Google Scholar 

  19. G. Armanni, Sulle deformazioni finite dei solidi elastici isotropi. Il Nuovo Cimento 10(6) (1915) 427–447.

    Google Scholar 

  20. P.K. Currie and M. Hayes, On non-universal finite elastic deformations. In: D.E. Carlson and R.T. Shield (eds), Proceedings of the IUTAM Symposium on Finite Elasticity. Martinus Nijhoff Publishers, The Hague (1981) pp. 143–150.

    Google Scholar 

  21. R.W. Ogden, Nonlinear Elastic Deformations. Ellis Horwood, Chichester (1984). Reprinted by Dover, New York (1997).

    Google Scholar 

  22. P. Podio-Guidugli and G. Tomassetti, Universal deformations for a class of compressible isotropic hyperelastic materials. J. Elasticity 52 (1999) 159–166.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Structural and Solid Mechanics Program, Department of Civil Engineering, University of Virginia, Charlottesville, VA, 22904, USA

    Cornelius O. Horgan

  2. Department of Mechanical Engineering, Dublin City University, Glasnevin, Dublin, 9, Ireland

    Jeremiah G. Murphy

Authors
  1. Cornelius O. Horgan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jeremiah G. Murphy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cornelius O. Horgan.

Additional information

Mathematics Subject Classifications (2000)

74B20, 74G55.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Horgan, C.O., Murphy, J.G. Invariance of the Equilibrium Equations of Finite Elasticity for Compressible Materials. J Elasticity 77, 187–200 (2004). https://doi.org/10.1007/s10659-005-4409-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10659-005-4409-9

Keywords

Search

Navigation