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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on January 23, 2008
The Quarterly Journal of Mechanics and Applied Mathematics 2008 61(2):181-203; doi:10.1093/qjmam/hbm027
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Q. Jl Mech. Appl. Math, Vol. 61. No. 2 © The author 2008. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Fourth-order cartesian tensors: old and new facts, notions and applications

Maher Moakher{dagger}

( Laboratory for Mathematical and Numerical Modeling in Engineering Science, National Engineering School at Tunis, ENIT-LAMSIN, B.P. 37, Tunis-Belvédère 1002, Tunisia )

{dagger} < maher.moakher{at}gmail.com>

Received 24 July 2007. Revise 19 November 2007.
  Abstract

Fourth-order tensors can be represented in many different ways. For instance, they can be represented as multilinear maps or multilinear forms. It is also possible to describe a fourth-order tensor in a given vector space by a second-order tensor but in another vector space with higher dimension. Such a representation makes the manipulation of fourth-order tensors similar to that of the more familiar second-order tensors. In this paper, we use these three descriptions to discuss the different symmetries of fourth-order tensors, to present the algebra of the space of fourth-order symmetric tensors and to describe different metrics on this space. Isotropic tensors and orientation tensors are presented using these different representations of fourth-order tensors. Applications to elasticity and high angular resolution diffusion imaging are discussed. Finally, we present a systematic and consistent approach to finding the tensor of an even order that best fits in some sense a tensor of higher even order.


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