Skip to main content
SpringerLink
Log in

Plane Strain Bending of Cylindrical Sectors of Admissible Compressible Hyperelastic Materials

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

Abstract

In the theory of nonlinear elasticity of rubber-like materials, if a homogeneous isotropic compressible material is described by a strain–energy function that is a homogeneous function of the principal stretches, then the equations of equilibrium for axisymmetric deformations reduce to a separable first-order ordinary differential equation. For a particular class of such strain–energy functions, this property is used to obtain a general parametric solution to the equilibrium equation for plane strain bending of cylindrical sectors. Specification of the arbitrary function that appears in such strain–energy functions yields some parametric solutions. In some cases, the parameter can be eliminated to yield closed-form solutions in implicit or explicit form. Other possible forms for the arbitrary constitutive function that are likely to yield such solutions are also indicated.

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. R. Abeyaratne and C.O. Horgan, Initiation of localized plane deformations at a circular cavity in an infinite compressible nonlinearly elastic medium. J. Elasticity 15 (1985) 243–256.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. Aboudi and M.-J. Pindera, High-fidelity micromechanical modelling of continuously reinforced elastic multiphase materials undergoing finite deformations. Math. Mech. Solids 9 (2004) 599–628.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  4. 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  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. M.M. Carroll, On obtaining closed-form solutions for compressible nonlinearly elastic materials. J. Appl. Math. Phys. (ZAMP) 46 (1995) S125–S145.

    Google Scholar 

  7. M.M. Carroll and C.O. Horgan, Finite strain solutions for a compressible elastic solid. Q. Appl. Math. 48 (1990) 767–780.

    MATH  MathSciNet  Google Scholar 

  8. M.M. Carroll and F.J. Rooney, Implications of Shield's inverse deformation theorem for compressible finite elasticity. J. Appl. Math. Phys. (ZAMP) 56 (2005) 1048–1060.

    Article  MATH  MathSciNet  Google Scholar 

  9. D.T. Chung, C.O. Horgan and R. Abeyaratne, The finite deformation of internally pressurized hollow cylinders and spheres for a class of compressible elastic materials. Int. J. Solids Struct. 22 (1986) 1557–1570.

    Article  MATH  Google Scholar 

  10. J.L. Ericksen, Deformations possible in every compressible isotropic perfectly elastic material. J. Math. Phys. 34 (1955) 126–128.

    MathSciNet  Google Scholar 

  11. J.M. Hill, Partial solutions in finite elasticity: I. Plane deformations. J. Appl. Math. Phys. (ZAMP) 24 (1973) 401–408.

    Article  Google Scholar 

  12. J.M. Hill, Partial solutions in finite elasticity. III. Three dimensional deformations. J. Appl. Math. Phys. (ZAMP) 24 (1973) 609–618.

    Article  Google Scholar 

  13. J.M. Hill, Some integrable and exact similarity solutions for plane finite elastic deformations. IMA J. Appl. Math. 44 (1990) 111–126.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. G.A. Holzapfel, T.C. Gasser and R.W. Ogden, A new constitutive framework for arterial wall mechanics and a comparative study of material models. J. Elasticity 61 (2000) 1–48.

    Article  MATH  MathSciNet  Google Scholar 

  16. 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 

  17. 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  MATH  MathSciNet  Google Scholar 

  18. C.O. Horgan and J.G. Murphy, Invariance of the equilibrium equations of finite elasticity for compressible materials. J. Elasticity 77 (2004) 187–200.

    Article  MATH  MathSciNet  Google Scholar 

  19. C.O. Horgan and J.G. Murphy, A Lie group analysis of the axisymmetric equations of finite elastostatics for compressible materials. Math. Mech. Solids 10 (2005) 311–333.

    Article  MATH  MathSciNet  Google Scholar 

  20. J.D. Humphrey, Cardiovascular Solid Mechanics: Cells, Tissues and Organs. Springer, Berlin Heidelberg New York (2002).

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  22. C.B. Sensenig, Non-linear theory for the deformation of pre-stressed circular plates and rings. Comm. Pure Appl. Math. 18 (1965) 147–161.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  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.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Horgan, C.O., Murphy, J.G. Plane Strain Bending of Cylindrical Sectors of Admissible Compressible Hyperelastic Materials. J Elasticity 81, 129–151 (2005). https://doi.org/10.1007/s10659-005-9010-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10659-005-9010-8

Key words

Mathematics subject classifications (2000)

Search

Navigation