Averaging Fourth-Rank Tensors with Weight Functions

Morris, Peter R.

doi:10.1063/1.1657417

Cited by 155 publications

(40 citation statements)

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“…These quantities can therefore be derived once and for all, as has been done by Morris (1969Morris ( , 1970 for orthorhombic (or higher)-orthorhombic symmetry and by Zuo, Xu & Liang (1989) for cubic-triclinic symmetry in a quite general way, and in a more specific form for cubic-orthorhombic symmetry by Bunge (1968). Other symmetries have not been considered.…”

Section: Ljklopqr = ~ Tijklopqr(g')f(o) Dg = ~ Tiiktopq(og"-1)f(g) Dmentioning

confidence: 99%

Elastic properties of polycrystals in the Voigt-Reuss-Hill approximation

Zuo¹,

Humbert²,

Esling³

1992

J Appl Crystallogr

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The problem of calculating polycrystalline elasticity from texture has been treated in the most general way in the framework of the Voigt-Reuss-Hill (VRH) approximation. The analytical solution involves only the even texture coefficients up to fourth order, which can be derived with a limited number of experimental pole figures (e.g. one pole figure for hexagonal polycrystals). The present analysis has been applied to the prediction of the fourth-rank elastic tensors and Young's modulus for a rolled zinc sheet.

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Section: Ljklopqr = ~ Tijklopqr(g')f(o) Dg = ~ Tiiktopq(og"-1)f(g) Dmentioning

confidence: 99%

Elastic properties of polycrystals in the Voigt-Reuss-Hill approximation

Zuo¹,

Humbert²,

Esling³

1992

J Appl Crystallogr

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“…5 The literature is too vast for us to attempt anything that approximates a complete listing of references. Here we are content to cite as examples a sampling of papers in elasticity and acoustoelasticity: orientation averaging in elasticity [10][11][12][13][14], in acoustoelasticity [15,16]; self-consistent methods in elasticity [17][18][19]; perturbation schemes in elasticity [20,21]. Readers who are unfamiliar with quantitative texture analysis should note that for a time (until 1979 when Matthies [22] clarified the cause of "ghost phenomena" in the inversion of X-ray pole figures) texture coefficients W lmn with odd l were generally but erroneously believed to be all equal to zero.…”

Section: For a Measurable Subset A Of M ℘ (A) Is The Probability Thamentioning

confidence: 99%

“…As a linear transformation on V c , Q(n, ω) has three eigenvalues 10 Alternatively, the decomposition formulas can be obtained by other methods [34,41]. [39,42] …”

Section: Decomposition Of a Tensor Into Its Irreducible Partsmentioning

confidence: 99%

“…From the theory of group representations [38], it is well known that decomposition formulas such as (20) and (21) above can be derived by computing the character of the tensor representation in question [33,39,40]. 10 Let χ Zc (ω) be the character of the representation Q → Q ⊗r |Z c , which decomposes into the direct sum given in (19). This decomposition dictates that we have χ Zc = n 0 χ 0 + n 1 χ 1 + · · · + n r χ r .…”

Section: Decomposition Of a Tensor Into Its Irreducible Partsmentioning

confidence: 99%

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A Representation Theorem for Material Tensors of Weakly-Textured Polycrystals and Its Applications in Elasticity

Man

Huang

2010

J Elast

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Material tensors pertaining to polycrystalline aggregates should manifest also the influence of crystallographic texture on the material properties in question. In this paper we make use of tensors which form bases of irreducible representations of the rotation group and prove a representation theorem by which a given material tensor of a weakly-textured polycrystal is expressed as a linear combination of an orthonormal set of irreducible basis tensors, with the components given explicitly in terms of texture coefficients and a set of undetermined material parameters. Once the irreducible basis tensors that appear in the formula are determined, the representation formula, which is valid for all texture and crystal symmetries, will delineate quantitatively the effect of crystallographic texture on the material tensor in question. We present an integral formula and an orthonormalization process which serve as the basis for a procedure to determine explicitly the irreducible basis tensors required in the representation formula. For applications we determine a set of irreducible basis tensors for the elasticity tensor and a set for fourth-order tensors that define constitutive equations in incompressible elasticity and Hill's quadratic yield functions in plasticity. We show that orientation averaging of a tensor can be done easily if we have in hand a set of irreducible basis tensors for the decomposition of the tensor in question. As illustration we derive a formula, which is valid for all texture and crystal symmetries, for the elasticity tensor under the Voigt model.

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“…Relating texture with polycrystal elastic properties can be done by following Roe's idea of spherical expansion [4]. Morris [25] employed Voigt's assumption and expanded both the ODF and the 8th order rotation tensor into spherical harmonics and used the orthogonality of the expansion to eliminate the expansion bases and thereby obtained the polycrystal elastic moduli expressed with single crystal moduli and the ODF coefficients for samples with orthorhombic symmetries. Sayers [26] pushed this idea forward by developing explicit expressions between ODF coefficients and wave velocities in different directions and then applying it to austenitic stainless steel to inversely obtain the coefficients.…”

Section: Introductionmentioning

confidence: 99%