Restricted Invariants on the Space of Elasticity Tensors

Forte, Sandra; Vianello, Maurizio

doi:10.1177/1081286505046483

Cited by 4 publications

(3 citation statements)

References 16 publications

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“…Therefore, every SO(3, R)representation V splits into a direct sum of harmonic tensor spaces H n (R 3 ). The space of elasticity tensors admits the following harmonic decomposition which was first obtained by Backus [8] (see also [9,35,36]):…”

Section: Harmonic Decompositionmentioning

confidence: 99%

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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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Section: Harmonic Decompositionmentioning

confidence: 99%

“…Explicit and, due to the aforementioned remark, sometimes different decompositions associated to (4) are provided in [8,66,18,35,36].…”

Section: Harmonic Decompositionmentioning

confidence: 99%

A Minimal Integrity Basis for the Elasticity Tensor

Olive

Kolev

Auffray

2017

Arch Rational Mech Anal

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“…harmonic tensors). The space of elasticity tensors admits the following harmonic decomposition [6,7,19,21]:…”

Section: 2mentioning

confidence: 99%

Invariant-based approach to symmetry class detection

Auffray,

Kolev,

Petitot

2011

Preprint

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In this paper, the problem of the identification of the symmetry class of a given tensor is asked. Contrary to classical approaches which are based on the spectral properties of the linear operator describing the elasticity, our setting is based on the invariants of the irreducible tensors appearing in the harmonic decomposition of the elasticity tensor [19]. To that aim we first introduce a geometrical description of Ela, the space of elasticity tensors. This framework is used to derive invariant-based conditions that characterize symmetry classes. For low order symmetry classes, such conditions are given on a triplet of quadratic forms extracted from the harmonic decomposition of C, meanwhile for higher-order classes conditions are provided in terms of elements of H 4 , the higher irreducible space in the decomposition of Ela. Proceeding in such a way some well known conditions appearing in the Mehrabadi-Cowin theorem for the existence of a symmetry plane [16] are retrieved, and a set of algebraic relations on H 4 characterizing the orthotropic ([D2]), trigonal ([D3]), tetragonal ([D4]), transverse isotropic ([SO(2)]) and cubic ([O]) symmetry classes are provided. Using a genericity assumption on the elasticity tensor under study, an algorithm to identify the symmetry class of a large set of tensors is finally provided. Contents 1.5. Fixed point sets and normalizers 1.6. Linear slices 1.7. Strata dimensions 1.8. Implicit equations for closed strata 2. The isotropic stratification of a n-uple of quadratic forms 2.1. The isotropic stratification of one quadratic form 2.2. The isotropic stratification of n quadratic forms 3. The space of elasticity tensors 3.1. The harmonic decomposition of elasticity tensors 3.2. The isotropic stratification of Ela 3.3. The Cowin-Mehrabadi condition 4. The isotropic stratification of H 4 4.1. Lattice of isotropy 4.2. Global parametrization 4.3. Cubic symmetry ([O]) 4.4. Transversely isotropic symmetry ([O(2)]) 4.5. Trigonal symmetry ([D 3 ]) 4.6. Tetragonal symmetry ([D 4 ]) 4.7. Orthotropic symmetry ([D 2 ]) 4.8. Bifurcation conditions for tensors in H 4

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Harmonic Decomposition of Elastic Constant Tensor and Crystal Symmetry

Dinçkal

2014

Transactions on Engineering Technologies

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Restricted Invariants on the Space of Elasticity Tensors

Cited by 4 publications

References 16 publications

A Minimal Integrity Basis for the Elasticity Tensor

A Minimal Integrity Basis for the Elasticity Tensor

Invariant-based approach to symmetry class detection

Harmonic Decomposition of Elastic Constant Tensor and Crystal Symmetry

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