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


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.


Anisotropic elasticity Nonlinear hyperelasticity Finite Elements Deviatoric–volumetric decoupling 



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.


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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

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