Papers dealing with the generalized Hooke’s law for linearly elastic anisotropic media are reviewed. The papers considered are based on Kelvin’s approach disclosing the structure of the generalized Hooke’s law, which is determined by six eigenmoduli of elasticity and six orthogonal eigenstates.

Anisotropy of elastic properties of materials

In both theoretical and applied mechanics, the modeling of nonlinear constitutive relations of materials is a topic of prime importance. To properly formulate consistent constitutive laws some restrictions need to be impose on tensor functions. To that aim representations theorems for both isotropic and anisotropic functions have been extensively investigated since the middle of the XXth century. Nevertheless, in three-dimensional physical space, most of the results are restricted to sets of tensors up to second-order. The purpose of the present paper is thus to get one step further and to provide an integrity basis for isotropic polynomial functions of a completely symmetric third-order tensor. To explicitly construct this basis, the link that exists between the O(3)-action on harmonic tensors and the SL(2,C)-action on the space of binary forms is exploited. We believe that such an integrity basis may found interesting applications both in continuum mechanics and in other fields of theoretical physics.

/pdf/isotropic-invariants-of-a-completely-symmetric-third-order-104lizhhp1.pdf

Isotropic invariants of a completely symmetric third-order tensor

In this paper, we propose a factorization of a fourth-order harmonic tensor into second-order tensors. We obtain moreover explicit equivariant reconstruction formulas, using second-order covariants, for transverse isotropic and orthotropic harmonic fourth-order tensors, and for trigonal and tetragonal harmonic fourth-order tensors up to a cubic fourth order covariant remainder.

https://hal.archives-ouvertes.fr/hal-01421763/document

Harmonic Factorization and Reconstruction of the Elasticity Tensor

It is known from the theory of group representations that a general tensor can be expressed as a sum of traceless symmetric tensors. In this paper, based on Sylvester9s theorem, it is shown that a general traceless symmetric tensor of any finite order m in three dimensions can be expressed as the traceless symmetric part of tensor product of m unit vectors (called the multipoles) multiplied by a positive scalar. The above two basic structures of tensors allow us to easily give complete and irreducible representations for tensor functions with high–order tensor variables, since those for tensor functions of vectors are well established in the literature. Examples are given for scalar–valued functions of a single fourth–order tensor of the elastic type, and of a number of vectors and second–order tensors. In 1970 Backus gave an alternative proof of Sylvester9s theorem, which shows how to compute the multipoles. Since Backus9s result is not so ‘well known’ to the community of researchers working on continuum mechanics, in the present paper a direct (without using Sylvester9s theorem) and constructive establishment of Maxwell9s multipole representation is provided, which is closer in spirit to a more modern approach to this topic.

Maxwell's multipole representation of traceless symmetric tensors and its application to functions of high-order tensors

Plusieurs lois de comportement mecaniques possedent une formulation tensorielle,
comme c’est par exemple le cas pour l’etude des materiaux elastiques. Dans ce cas in-
tervient un sous-espace de tenseurs d’ordre 4, note Ela et appele espace des tenseurs
d’elasticite. Les questions de classification des materiaux elastiques passent alors par la
necessite de decrire les orbites de l’espace Ela sous l’action du groupe SO(3). Plus ge-
neralement, on est amene a etudier la geometrie d’un espace de tenseurs sur R 3 , via
l’action du groupe O(3).
Cette geometrie est tout d’abord caracterisee par ses differentes classes d’isotropies,
encore appelees classes de symetries. Chaque espace de tenseurs possede en effet un
nombre fini de classes d’isotropies. Nous proposons dans notre travail une methode ori-
ginale et generale pour obtenir les classes d’istropie d’un espace de tenseurs quelconque.
Nous avons ainsi pu obtenir pour la premiere fois les classes d’isotropie d’un espace de
tenseurs d’ordre 8 intervenant en theorie de l’elasticite lineaire du second-gradient de la
deformation.
Dans le cas d’une representation reelle d’un groupe compact, l’algebre des polynomes
invariants separe les orbites, ce qui motive donc la recherche d’une famille generatrice
minimale de polynomes invariants. Celle-ci se fait en exploitant le lien existant entre
les espaces de tenseurs et les espaces de formes binaires et plus precisement la theorie
classique des invariants. On ne fait donc plus intervenir le groupe SO(3) mais le groupe
SL(2, C). Nous avons ainsi repris et re-interprete les approches effectives de cette theo-
rie, notamment developpees par Gordan au XIX e siecle. Cette re-interpretation nous a
permis d’obtenir de nombreux resultats, notamment la determination d’une famille ge-
neratrice minimale d’invariants pour l’elasticite mais aussi pour la piezoelectricte. No-
tons aussi que nous avons pu retrouver d’une facon simple des relations importantes
intervenant en theorie classique des invariants, a savoir les fameuses series de Gordan,
ainsi que des relations plus recentes d’Abdesselam–Chipalkatti sur les transvectants de
formes binaires.

/pdf/geometrie-des-espaces-de-tenseurs-une-approche-effective-1pid7uq2pk.pdf

Géométrie des espaces de tenseurs : une approche effective appliquée à la mécanique des milieux continus

Isotropic invariants of traceless symmetric tensors of orders three and four

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Isotropic invariants of traceless symmetric tensors of orders three and four