Journal of Elasticity

, Volume 135, Issue 1–2, pp 295–350 | Cite as

The Symmetries of Octupolar Tensors

  • Giuseppe Gaeta
  • Epifanio G. VirgaEmail author


Octupolar tensors are third order, completely symmetric and traceless tensors. Whereas in 2D an octupolar tensor has the same symmetries as an equilateral triangle and can ultimately be identified with a vector in the plane, the symmetries that it enjoys in 3D are quite different, and only exceptionally reduce to those of a regular tetrahedron. By use of the octupolar potential, that is, the cubic form associated on the unit sphere with an octupolar tensor, we shall classify all inequivalent octupolar symmetries. This is a mathematical study which also reviews and incorporates some previous, less systematic attempts.


Order tensors Phase transitions Octupolar tensors Generalized (nonlinear) eigenvalues and eigenvectors 

Mathematics Subject Classification

76A15 15A69 



E.G. Virga acknowledges the kind hospitality of the Oxford Centre for Nonlinear PDE, where part of this work was done while he was visiting the Mathematical Institute at the University of Oxford.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di MilanoMilanoItaly
  2. 2.Dipartimento di MatematicaUniversità di PaviaPaviaItaly

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