Isotropic polynomial invariants of Hall tensor

Liu, Jinjie; Ding, Weiyang; Qi, Liqun; Zou, Wenming

doi:10.1007/s10483-018-2398-9

Cited by 10 publications

(11 citation statements)

References 22 publications

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“…In general, the skew-symmetric tensor κ ijk has nine independent components. The decomposition of this tensor has been studied recently in various publications, for example [36] and the references given therein. We examine how this decomposition follows from the decomposition of the general third-order tensor (in the contravariant version).…”

Section: Glmentioning

confidence: 99%

Decomposition of third-order constitutive tensors

Itin

Reches

2021

Mathematics and Mechanics of Solids

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Third-order tensors are widely used as a mathematical tool for modeling the physical properties of media in solid-state physics. In most cases, they arise as constitutive tensors of proportionality between basic physical quantities. The constitutive tensor can be considered the complete set of physical parameters of a medium. The algebraic features of the constitutive tensor can be used as a tool for proper identification of natural materials, such as crystals, and for designing artificial nanomaterials with prescribed properties. In this paper, we study the algebraic properties of a general asymmetric third-order tensor relative to its invariant decomposition. In correspondence with different groups acting on the basic vector space, we present the hierarchy of different types of tensor decomposition into invariant subtensors. In particular, we discuss the problem of non-uniqueness and reducibility of high-order tensor decomposition. For a general asymmetric third-order tensor, these features are described explicitly. In the case of special tensors with a prescribed symmetry, the decomposition is demonstrated to be irreducible and unique. We present the explicit results for two physically interesting models: the piezoelectric tensor as an example of pair symmetry and the Hall tensor as an example of pair skew-symmetry.

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Section: Glmentioning

confidence: 99%

Decomposition of third-order constitutive tensors

Itin

Reches

2021

Mathematics and Mechanics of Solids

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“…This year, Chen, Liu, Qi, Zheng and Zou [5] showed that eleven isotropic invariants among the Olive-Auffray minimal integrity basis of a third order symmetric tensor form an irreducible function basis of that tensor. Also in this year, a ten invariant minimal integrity basis, which is also an irreducible function basis of the Hall tensor, was presented by Liu, Ding, Qi and Zou [7]. For the piezoelectric tensor, in 2014, Olive [10] gave 495 hemitropic invariants and claimed that these hemitropic invariants form an hemitropic integrity basis.…”

Section: Integrity and Function Bases Of Third Order Tensorsmentioning

confidence: 99%

Irreducible function bases of isotropic invariants of a third order three-dimensional symmetric and traceless tensor

Chen

et al. 2019

Front. Math. China

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Third order three-dimensional symmetric and traceless tensors play an important role in physics and tensor representation theory. A minimal integrity basis of a third order threedimensional symmetric and traceless tensor has four invariants with degrees two, four, six and ten respectively. In this paper, we show that any minimal integrity basis of a third order three-dimensional symmetric and traceless tensor is also an irreducible function basis of that tensor, and there is no polynomial syzygy relation among the four invariants of that basis, i.e., these four invariants are algebraically independent.Key words. minimal integrity basis, irreducible function basis, symmetric and traceless tensor, syzygy. Nomenclature D a third order three-dimensional symmetric and traceless tensor with components D ijk T(m, n) the space of real tensors of order m and dimension n S(m, n) the subspace of symmetric tensors St(m, n) the subspace of symmetric and traceless tensors O(n) the orthogonal group of dimension n SO(n) the special orthogonal group of dimension n Gl(n, R) the general linear group of real matrices m n = m! n!(m−n)! the binomial coefficient for m ≥ n ≥ 0The next theorem claims that there is no syzygy relation among four invariants J 2 , J 4 , J 6 and J 10 , where {J 2 , J 4 , J 6 , J 10 } be an arbitrary minimal integrity basis of D.Theorem 4.1. Let {J 2 , J 4 , J 6 , J 10 } be an arbitrary minimal integrity basis of a third order three-dimensional symmetric and traceless tensor D. Then there is no syzygy relation among four invariants J 2 , J 4 , J 6 and J 10 .

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“…• change the class of its elements: usually polynomial invariants are considered [33,41,25,28,13,21,23], but this is not mandatory; • look for local separating sets instead of global ones: the separating property is then defined, not on the whole vector space, but only on a neighbourhood of a given tensor. In this direction, Bona et al [8] proposed a local parametrization of orbits of generic triclinic elasticity tensors by 18 local algebraic invariants.…”

Section: Introductionmentioning

confidence: 99%

Minimal functional bases for elasticity tensor symmetry classes

Desmorat,

Auffray,

Desmorat

et al. 2021

Preprint

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Functional bases, synonymous with separating sets, are usually formulated for an entire vector space, such as the space Ela of elasticity tensors. We propose here to define functional bases limited to symmetry strata, i.e. sets of tensors of the same symmetry class. We provide such lowcardinal minimal bases for tetragonal, trigonal, cubic or transversely isotropic symmetry strata of the elasticity tensor.

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Isotropic polynomial invariants of Hall tensor

Cited by 10 publications

References 22 publications

Decomposition of third-order constitutive tensors

Decomposition of third-order constitutive tensors

Irreducible function bases of isotropic invariants of a third order three-dimensional symmetric and traceless tensor

Minimal functional bases for elasticity tensor symmetry classes

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