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Material Tensors and Pseudotensors of Weakly-Textured Polycrystals with Orientation Distribution Function Defined on the Orthogonal Group

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Abstract

A representation theorem on material tensors of weakly-textured polycrystals was recently established by Man and Huang, which delineates quantitatively the effect of crystallographic texture on the material tensor in question. Man and Huang’s theorem is based on the classical assumption in texture analysis that ODFs are defined on the rotation group SO(3), which strictly speaking makes it applicable only to polycrystals with crystallite symmetry defined by one of the 11 proper point groups. In this paper we let ODFs be defined on the orthogonal group O(3) and extend the representation theorem of Man and Huang to cover pseudotensors and polycrystals with crystallite symmetry defined by any of the 21 improper point groups. In SO(3)-based texture analysis, as a result of the inherent limitation imposed by the restricted definition of the ODF, each improper group of crystallite symmetry in question is routinely treated as if it were its peer proper group in the same Laue class. In light of the extended representation theorem, we examine the conditions under which this ad hoc practice will still work as far as effects of texture on material tensors and pseudotensors are concerned and the circumstances under which it won’t.

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Notes

  1. See, e.g., Lee [21] for proof of these preliminaries.

  2. For the definition and notation of Euler angles, we follow the convention adopted by Roe [18]. The angles \((\psi, \theta, \phi)\) are also commonly written as \((\alpha , \beta, \gamma)\) in the physics literature [23, 24].

  3. In what follows we use the word “tensor” in two senses: (i) a multilinear mapping as defined in (5) for some \(r \geq2\); (ii) as in “material tensor”, a multilinear mapping in the preceding sense that further obeys the law of transformation (13).

  4. For brevity, in Eqs. (13) and in (14) we use the same symbol \({\mathcal {T}}_{{ {\boldsymbol {Q}}}}\) for the transformation induced by \({\boldsymbol {Q}}\), irrespective of whether it acts on material tensors or pseudotensors. In Sect. 4 we will also use this symbol to denote the transformation induced by \({\boldsymbol {Q}}\) on ODFs. What the symbol \(\mathcal{T}_{{ {\boldsymbol {Q}}}}\) really means should be clear from the context.

  5. It suffices to consider \(k \leq r\); see the proof of Theorem 3 below.

  6. Man and Huang [5] showed the existence of an orthonormal set of irreducible basis tensors \({\boldsymbol {H}}_{m}^{k, s}\) (\(-k \leq m \leq k\), \(1\leq s \leq n_{k}\) where \(n_{k}\) is the multiplicity of \(\mathcal {D}_{k}\)) which together span \(Z_{c}\), and they developed a procedure to determine explicitly \({\boldsymbol {H}}_{m}^{k, s}\).

  7. Cf. [5] for the existence and a procedure for the construction of such a basis.

  8. \(c^{0,R/L}_{00} = 1/(16\pi^{2})\).

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Acknowledgements

The research reported here was supported in part by a grant (No. DMS-0807543) from the U.S. National Science Foundation.

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Authors and Affiliations

  1. Department of Science and Mathematics, Glenville State College, Glenville, WV, 26351, USA

    Wenwen Du

  2. Department of Mathematics, University of Kentucky, Lexington, KY, 40506-0027, USA

    Chi-Sing Man

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  1. Wenwen Du
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Du, W., Man, CS. Material Tensors and Pseudotensors of Weakly-Textured Polycrystals with Orientation Distribution Function Defined on the Orthogonal Group. J Elast 127, 197–233 (2017). https://doi.org/10.1007/s10659-016-9610-5

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