Vol. 1, No. 2, 2013

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Symmetry classes for even-order tensors

Marc Olive and Nicolas Auffray

Vol. 1 (2013), No. 2, 177–210

We give a complete general answer to the problem, recurrent in continuum mechanics, of determining of the number and type of symmetry classes of an even-order tensor space. This kind of investigation was initiated for the space of elasticity tensors, and since then different authors have solved this problem for other kinds of physics, such as photoelectricity, piezoelectricity, flexoelectricity, and strain-gradient elasticity. All these problems were treated using the same computational method, which, though effective, has the drawback of not providing general results. Furthermore, its complexity increases with the tensorial order. Here we provide general theorems that directly give the desired results for any even-order constitutive tensor. As an illustration of this method, and for the first time, the symmetry classes of all even-order tensors of Mindlin second strain-gradient elasticity are provided.

anisotropy, symmetry classes, higher-order tensors, generalized continuum theories, strain-gradient elasticity
Mathematical Subject Classification 2010
Primary: 15A72, 20C35, 74B99
Received: 16 May 2012
Revised: 2 October 2012
Accepted: 2 December 2012
Published: 16 April 2013

Communicated by Pierre Seppecher
Marc Olive
Laboratoire d’Analyse, Topologie, Probabilités
Centre de Mathématiques et Informatique
CNRS, UMR 7353
Aix-Marseille Université, Technopôle Château-Gombert
39 rue F. Joliot-Curie
13453 Marseille
Nicolas Auffray
Laboratoire Modélisation et Simulation Multi Echelle
Université Paris-Est
CNRS, UMR 8208
5 boulevard Descartes
77454 Marne-la-Vallée