Linearly independent sets of isotropic Cartesian tensors of ranks up to eight

Kearsley, E. A.; Fong, Jeffrey T.

doi:10.6028/jres.079b.005

Cited by 43 publications

(51 citation statements)

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“…In fact, 91 independent tensors are needed (Kearsley & Fong 1975). Now, tensors A of components A ijklpqrs are assumed to be symmetric according to indices (i, j), (k, l), (p, q) and (r, s) (called minor symmetries).…”

Section: A Basis For Eighth-order Isotropic Tensorsmentioning

confidence: 99%

Inversion of higher order isotropic tensors with minor symmetries and solution of higher order heterogeneity problems

Monchiet

Bonnet

2010

Proc. R. Soc. A.

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In this paper, the derivation of irreducible bases for a class of isotropic 2nth-order tensors having particular 'minor symmetries' is presented. The methodology used for obtaining these bases consists of extending the concept of deviatoric and spherical parts, commonly used for second-order tensors, to the case of an nth-order tensor. It is shown that these bases are useful for effecting the classical tensorial operations and especially the inversion of a 2nth-order tensor. Finally, the formalism introduced in this study is applied for obtaining the closed-form expression of the strain field within a spherical inclusion embedded in an infinite elastic matrix and subjected to linear or quadratic polynomial remote strain fields.

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Section: A Basis For Eighth-order Isotropic Tensorsmentioning

confidence: 99%

Inversion of higher order isotropic tensors with minor symmetries and solution of higher order heterogeneity problems

Monchiet

Bonnet

2010

Proc. R. Soc. A.

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“…Fundamentally isotropic tensors

{double-struckI}_{2} r

of rank 2 r are linear combinations of Kronecker‐deltas

δ_{i j}^{\otimes normalr}

and permutations thereof [29]. So one fourth order isotropic tensor

I_{4}

I_{i j k l} = δ_{i j} δ_{k l}

.…”

Section: Preliminariesmentioning

confidence: 99%

A micro‐mechanically motivated phenomenological yield function for cubic crystal aggregates

Dyck

Böhlke

2020

Z Angew Math Mech

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A micro-mechanically motivated phenomenological yield function, for polycrystalline cubic metals is presented. In the suggested yield function microstructure is taken into account by the crystallographic orientation distribution function in terms of tensorial Fourier coefficients. The yield function is presented in a polynomial form in powers of the stress state. Known group-theoretic results are used to identify isotropic and anisotropic parts in the yield function, whereby anisotropic parts are characterized by tensorial Fourier coefficients. The form of the presented yield function is inspired by the classic, phenomenological von Mises -Hill yield function first published in 1913. For a specific choice of material parameters, both functions coincide, thus a micro-mechanically motivated generalization of the von Mises -Hill yield function is presented. For the given yield function, two dimensional experimental results are sufficient, to identify a three dimensional anisotropic yield behavior. The work concludes with a treatment of the isotropic special case, i.e. a tension-compression split in yield behavior as well as parameter ranges for convexity and shapes of the yield surface. K E Y W O R D Sanisotropic yield function, crystallographic texture, orientation distribution function, plastic anisotropy, tensorial texture coefficients This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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“…For an independent verification of the form of the isotropic three-dimensional eighth-rank tensor see Ref. 44. The power of representation theory is that one can follow an analogous procedure for shells of higher angular momentum, using a "3" to represent the seven f orbitals, a "4" to represent the nine g orbitals, and so on.…”

Section: Orbital Symmetrymentioning

confidence: 99%

Erratum: Hubbard-like Hamiltonians for interacting electrons in s,p , and d orbitals [Phys. Rev. B 93 , 075101 (2016)]

Coury¹,

Dudarev²,

Foulkes³

et al. 2018

Phys. Rev. B

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Hubbard-like Hamiltonians are widely used to describe on-site Coulomb interactions in magnetic and strongly-correlated solids, but there is much confusion in the literature about the form these Hamiltonians should take for shells of p and d orbitals. This paper derives the most general s, p and d orbital Hubbard-like Hamiltonians consistent with the relevant symmetries, and presents them in ways convenient for practical calculations. We use the full configuration interaction method to study p and d orbital dimers and compare results obtained using the correct Hamiltonian and the collinear and vector Stoner Hamiltonians. The Stoner Hamiltonians can fail to describe properly the nature of the ground state, the time evolution of excited states, and the electronic heat capacity. PACS numbers: 31.15.aq, 31.15.xh, 31.10.+z arXiv:1507.04613v3 [cond-mat.str-el] 12 Nov 2015 C. The multi-orbital model Hamiltonian: p orbital symmetryConsider the case where the local orbitals, α, β, χ and γ appearing in Eq. (4) are real cubic harmonic p orbitals with angular dependence x/r, y/r, and z/r. In this case

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Linearly independent sets of isotropic Cartesian tensors of ranks up to eight

Cited by 43 publications

References 0 publications

Inversion of higher order isotropic tensors with minor symmetries and solution of higher order heterogeneity problems

Inversion of higher order isotropic tensors with minor symmetries and solution of higher order heterogeneity problems

A micro‐mechanically motivated phenomenological yield function for cubic crystal aggregates

Erratum: Hubbard-like Hamiltonians for interacting electrons in s,p , and d orbitals [Phys. Rev. B 93 , 075101 (2016)]

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