Quadratic Invariants of the Elasticity Tensor

Itin, Yakov

doi:10.1007/s10659-016-9569-2

Cited by 6 publications

(8 citation statements)

References 30 publications

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“…In this manner, we can identify the sets of subtensors having similar algebraic and physical properties. This procedure is known to be relevant for fourth-order constitutive tensors in electromagnetism [23], linear elasticity [24][25][26], and gravity [27].…”

Section: Introductionmentioning

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: Introductionmentioning

confidence: 99%

Decomposition of third-order constitutive tensors

Itin

Reches

2021

Mathematics and Mechanics of Solids

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“…In order to deal with the material parameters that are independent of the coordinate system, one looks for invariants of the elasticity tensor , see [5]. As was shown recently [6–9], these invariants can be obtained through the unique irreducible decomposition of the elasticity tensor space into a direct sum of five invariant subspaces. When this decomposition is applied to the Voigt matrix, the corresponding five ( 6 × 6 ) matrices do not show any systematic order.…”

Section: Introductionmentioning

confidence: 99%

“…Our classification turns out to be almost equivalent to that presented in [5], although being based on a different approach. In addition, the problem of a closest symmetry class for a given material (generalized Fedorov problem), see [5, 14], obtains a useful reformulation in terms of the irreducible decomposition [7]. The most intriguing problem is a possibility to design artificial elastic materials with prescribed properties.…”

Section: Introductionmentioning

confidence: 99%

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Irreducible matrix resolution for symmetry classes of elasticity tensors

Itin

2020

Mathematics and Mechanics of Solids

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In linear elasticity, a fourth-order elasticity (stiffness) tensor of 21 independent components completely describes deformation properties elastic constants of a material. The main goal of the current work is to derive a compact matrix representation of the elasticity tensor that correlates with its intrinsic algebraic properties. Such representation can be useful in design of artificial materials. Owing to Voigt, the elasticity tensor is conventionally represented by a (6 × 6) symmetric matrix. In this paper, we construct two alternative matrix representations that conform with the irreducible decomposition of the elasticity tensor. The 3 × 7 matrix representation is in correspondence with the permutation transformations of indices and with the general linear transformation of the basis. An additional representation of the elasticity tensor by two scalars and three 3 × 3 matrices is suitable to describe the irreducible decomposition under the rotation transformations. We present the elasticity tensor of all crystal systems in these compact matrix forms and construct the hierarchy diagrams based on this representation.

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“…The problem of finding out SO(3, R)-invariants of a fourth-order tensor is not new, and has been investigated by many authors (for e.g [47,89,12,88,97,62,53]). Generally, these invariants are computed using traces of tensor products [17,82] and the method relies on some tools developed by Rivlin and others [83,84,76] for a family of second-order tensors.…”

Section: Introductionmentioning

confidence: 99%

A Minimal Integrity Basis for the Elasticity Tensor

Olive

Kolev

Auffray

2017

Arch Rational Mech Anal

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We definitively solve the old problem of finding a minimal integrity basis of polynomial invariants of the fourth-order elasticity tensor C. Decomposing C into its SO(3)-irreducible components we reduce this problem to finding joint invariants of a triplet (a, b, D), where a and b are second-order harmonic tensors, and D is a fourth-order harmonic tensor. Combining theorems of classical invariant theory and formal computations, a minimal integrity basis of 297 polynomial invariants for the elasticity tensor is obtained for the first time.

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Quadratic Invariants of the Elasticity Tensor

Cited by 6 publications

References 30 publications

Decomposition of third-order constitutive tensors

Decomposition of third-order constitutive tensors

Irreducible matrix resolution for symmetry classes of elasticity tensors

A Minimal Integrity Basis for the Elasticity Tensor

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