Symmetry classes for odd-order tensors

Marc Olive (1) , Nicolas Auffray (2)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 MSME - Laboratoire de Modélisation et Simulation Multi Echelle

We give a complete general answer to the problem, recurrent in continuum mechanics, of determining the number and type of symmetry classes of an odd-order tensor space. This kind of investigation was initiated for the space of elasticity tensors. Since then, this problem has been solved for other kinds of physics such as photoelectricity, piezoelectricity, flexoelectricity, and strain-gradient elasticity. In all the aforementioned papers, the results are obtained after some lengthy computations. In a former contribution we provide general theorems that solve the problem for even-order tensor spaces. In this paper we extend these results to the situation of odd-order tensor spaces. As an illustration of this method, and for the first time, the symmetry classes of all odd-order tensors of Mindlin second strain-gradient elasticity are provided.

Domaines: Mécanique [physics.med-ph]; Théorie des représentations [math.RT]

Marc Olive, Nicolas Auffray. Symmetry classes for odd-order tensors. Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2014, 94 (5), pp.421-447. ⟨10.1002/zamm.201200225⟩. ⟨hal-00832489⟩