Advertisement

Computational Mechanics

, Volume 52, Issue 5, pp 1185–1197 | Cite as

On anisotropic elasticity and questions concerning its Finite Element implementation

  • Luigi Vergori
  • Michel Destrade
  • Patrick McGarry
  • Ray W. Ogden
Original Paper

Abstract

We give conditions on the strain–energy function of nonlinear anisotropic hyperelastic materials that ensure compatibility with the classical linear theories of anisotropic elasticity. We uncover the limitations associated with the volumetric–deviatoric separation of the strain–energy used, for example, in many Finite Element (FE) codes in that it does not fully represent the behavior of anisotropic materials in the linear regime. This limitation has important consequences. We show that, in the small deformation regime, a FE code based on the volumetric–deviatoric separation assumption predicts that a sphere made of a compressible anisotropic material deforms into another sphere under hydrostatic pressure loading, instead of the expected ellipsoid. For finite deformations, the commonly adopted assumption that fibres cannot support compression is incorrectly implemented in current FE codes and leads to the unphysical result that under hydrostatic tension a sphere of compressible anisotropic material deforms into a larger sphere.

Keywords

Anisotropic elasticity Nonlinear hyperelasticity Finite Elements Deviatoric–volumetric decoupling 

Notes

Acknowledgments

This work was supported by the Royal Society through an International Joint Project awarded to the second and fourth authors. Finally, the authors are grateful to Jerry Murphy (Dublin City University) for stimulating discussions on the topic. The work of the second author was also partially supported by a ‘Government of Ireland New Foundations Award’ received from the Irish Research Council.

References

  1. 1.
    Musgrave MJP (1970) Crystal acoustics. Holden-Day, San FranciscozbMATHGoogle Scholar
  2. 2.
    Destrade M, Martin PA, Ting TCT (2002) The incompressible limit in linear anisotropic elasticity, with applications to surface waves and elastostatics. J Mech Phys Solids 50:1453–1468MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Royer D, Dieulesaint E (1984) Rayleigh wave velocity and displacement in orthorhombic, tetragonal, hexagonal, and cubic crystals. J Acoust Soc Am 76:1438–1444CrossRefGoogle Scholar
  4. 4.
    Merodio J, Ogden RW (2003) Instabilities and loss of ellipticity in fiber-reinforced compressible non-linearly elastic solids under plane deformation. Int J Solids Struct 40:4707–4727MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Holzapfel GA, Gasser TC, Ogden RW (2000) A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elast 61:1–48MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Spencer AJM (1972) Deformations of fibre-reinforced materials. University Press, OxfordzbMATHGoogle Scholar
  7. 7.
    Ogden RW (1978) Nearly isochoric elastic deformations: application to rubberlike solids. J Mech Phys Solids 26:37–57MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    ABAQUS/standard user’s manual, Ver. 6.10 (2010) Dassault Systèmes Simulia Corporation, PawtucketGoogle Scholar
  9. 9.
    ANSYS 14.0 training manual-fluent (2011) ANSYS, Inc., CanonsburgGoogle Scholar
  10. 10.
    FEBio theory manual, Ver. 1.5 (2012). http://mrl.sci.utah.edu. Accessed 15 May 2013
  11. 11.
    ADINA theory and modeling guide (2005). ADINA R &D, Inc., WatertownGoogle Scholar
  12. 12.
    Federico S (2010) Volumetric-distortional decomposition of deformation and elasticity tensor. Math Mech Solids 15:672–690.Google Scholar
  13. 13.
    Merodio J, Ogden RW (2006) The influence of the invariant \(I_8\) on the stress-deformation and ellipticity characteristics of doubly fiber-reinforced non-linearly elastic solids. Int J Non-Linear Mech 41:556–563Google Scholar
  14. 14.
    Royer D, Dieulesaint E (2000) Elastic waves in solids I. Free and guided propagation. Springer, Berlin Google Scholar
  15. 15.
    Gasser TC, Ogden RW, Holzapfel GA (2006) Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. J R Soc Interface 3:15–35CrossRefGoogle Scholar
  16. 16.
    Ní Annaidh A, Destrade M, Gilchrist MD, Murphy JG (in press) Deficiencies in numerical models of anisotropic nonlinearly elastic materials. Biomech Model Mechanobiol. doi: 10.1007/s10237-012-0442-3.
  17. 17.
    Sansour C (2008) On the physical assumptions underlying the volumetric–isochoric split and the case of anisotropy. Eur J Mech A 27:28–39MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Luigi Vergori
    • 1
  • Michel Destrade
    • 2
    • 3
  • Patrick McGarry
    • 4
  • Ray W. Ogden
    • 5
  1. 1.Marie Curie Fellow of the Istituto Nazionale di Alta Matem- atica, School of Mathematics, Statistics & Applied MathematicsNational University of Ireland GalwayGalwayIreland
  2. 2.School of Mathematics, Statistics & Applied Mathematics, Istituto Nazionale di Alta Matematica National University of Ireland GalwayGalwayIreland
  3. 3.School of Mechanical & Materials EngineeringUniversity College DublinDublin 4Ireland
  4. 4.Department of Mechanical & Biomedical Engineering National University of Ireland GalwayGalwayIreland
  5. 5.School of Mathematics & StatisticsUniversity of GlasgowGlasgowScotland

Personalised recommendations