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Journal of Elasticity
Erschienen in: Journal of Elasticity 1-2/2019

14.01.2019

The Symmetries of Octupolar Tensors

verfasst von: Giuseppe Gaeta, Epifanio G. Virga

Erschienen in: Journal of Elasticity | Ausgabe 1-2/2019

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Abstract

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.
Vorheriger Artikel A Causality Setting for Elasticity Theory
Nächster Artikel Explicit Relaxation of a Two-Well Hadamard Energy

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Fußnoten
1
In the Introduction to [30] (p. IX), we read:
About 25 years ago I started to write notes for a course for seniors and beginning graduate students at Carnegie Institute of Technology (renamed Carnegie-Mellon University in 1968). At first, the course was entitled “Tensor Analysis”. […] The notes were rewritten several times. They were widely distributed and they served as the basis for appendices to the books [9] and [44].
 
2
The superscript \(^{(2)}\) reminds us that this tensor expresses the field induced by polarization as a quadratic function of the external field, whereas the ordinary susceptibility establishes a linear relationship between the two fields.
 
3
More generally, we might consider potentials with contributions up to third order; thus we would have the sum of a scalar part, a vector one, another part described by a second order tensor, and finally the one described by the third order one. Here we focus on this last contribution, as the study of theories with scalar, vector, or second order tensor order parameters is standard (in principle; obviously concrete applications can present endless complications).
 
4
As a general convention, we will denote the potentials in Cartesian coordinates by \(\varPhi \) (with several suffixes) and those in spherical coordinates—which we always consider only for \(r=1\)—by \(\varPsi \) (again with corresponding suffixes).
 
5
It may be worth mentioning that (in particular, if we are satisfied with studying \(\varPhi \) on one hemisphere, which is justified by (3.2)) a third option is present, i.e., setting \(z = \pm \sqrt{1 - x^{2} - y^{2}}\) and considering \(\varPhi \) as a function of \(x\) and \(y\); these take value in the unit disk. This will be used in Sect. 5.3.
 
6
In fact, as pointed out by Walcher [51], this kind of results follow ultimately from the work of Bezout on intersection theory dating back to the 18th century. See his paper [52] for details.
 
7
It should be noted that the “disappearance” of real critical points—w.r.t. the generic situation described by Röhrl’s theorem—is related, at least in our model, to the appearance of a “monkey saddle” [14], i.e., of a critical point with a non-generic index; see below for detail.
 
8
In fact, if \(\mathbf{v}\) is an eigenvector of \(M\) with eigenvalue \(\lambda \), then for any number \(\alpha \neq 0\) also \(\mathbf{w} = \alpha {\mathbf{v}}\) is an eigenvector with the same eigenvalue \(\lambda \).
 
9
In this paper, the adjective “generic” is given the meaning common in algebraic geometry, that is, it designates a property valid away from the roots of a polynomial in parameter space [5].
 
10
This means that we can rule out the possibility to have \(\alpha _{3} = 0\). In fact, even in the case this is a local maximum at height zero, we can always—see Remark 6—choose the North Pole to be an absolute maximum, and this is necessarily positive.
 
11
In order to know the value for the corresponding \(\lambda \), one needs to express the solution in Cartesian coordinates and go back to (5.22); this is due to the fact that our change of coordinates was performed imposing \(r=1\) and thus the constraint term, which represents the dynamical origin of \(\lambda \), is absent in the angular coordinates.
 
12
To compare the expressions worked out in this paper for the Hessian matrix of the octupolar potential with those featuring in [14], the reader should heed that these differ by a scaling factor: the Hessian matrix here is three times the Hessian matrix there.
 
13
It should be noted that in our previous work [14] we have used a slightly different reparametrization, with \(\rho \) instead of \(\rho /2\). This accounts for the differences in many of the forthcoming formulas.
 
14
There are also maps acting on \(\rho \) by changing its sign and leaving the potential invariant; these are not admitted as we have required \(\rho \in [0,2]\).
 
15
Which thus are presumably more complicated than anticipated.
 
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Metadaten
Titel
The Symmetries of Octupolar Tensors
verfasst von
Giuseppe Gaeta
Epifanio G. Virga
Publikationsdatum
14.01.2019
Verlag
Springer Netherlands
Erschienen in
Journal of Elasticity / Ausgabe 1-2/2019
Print ISSN: 0374-3535
Elektronische ISSN: 1573-2681
DOI
https://doi.org/10.1007/s10659-018-09722-8

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