Acta Mechanica

, Volume 227, Issue 3, pp 651–675 | Cite as

Consistent hypo-elastic behavior using the four-dimensional formalism of differential geometry

  • B. Panicaud
  • E. Rouhaud
  • G. Altmeyer
  • M. Wang
  • R. Kerner
  • A. Roos
  • O. Ameline
Original Paper


The covariance principle of the theory of relativity within a four-dimensional framework ensures the validity of any equations and physical relations through any changes of frame of reference, due to the definition of the 4D space–time and the use of 4D tensors, operations and operators. This 4D formalism enables also to clearly distinguish between the covariance principle (i.e., frame-indifference) and the material objectivity principle (i.e., indifference to any rigid body motion superposition). We propose and apply here a method to build a constitutive relation for elastic materials using such a 4D formalism. The present article is specifically devoted in the application of this methodology to construct hypo-elastic materials with the use of the 4D Lie derivative. It enables thus to obtain consistent non-dissipative models equivalent to (hyper)elastic ones.


Constitutive Model Simple Shear Rigid Body Motion Covariant Component Tensor Density 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Wien 2015

Authors and Affiliations

  • B. Panicaud
    • 1
  • E. Rouhaud
    • 1
    • 2
  • G. Altmeyer
    • 1
  • M. Wang
    • 1
  • R. Kerner
    • 2
  • A. Roos
    • 1
  • O. Ameline
    • 1
  1. 1.Institut Charles Delaunay (ICD) - CNRS UMR 6281 Laboratoire des Systèmes Mécaniques et d’Ingénierie Simultanée (LASMIS)Université de Technologie de Troyes (UTT)Troyes cedexFrance
  2. 2.Laboratoire de Physique Théorique de la Matière Condensée (LPTMC)Université Pierre et Marie Curie (UPMC)ParisFrance

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