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X-Ray Diffraction Residual Stress Analysis in Polycrystals

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Theoretical Concepts of X-Ray Nanoscale Analysis

Part of the book series: Springer Series in Materials Science ((SSMATERIALS,volume 183))

Abstract

In the previous chapters, the X-ray analysis has been applied to the samples with the electron density distributed uniformly in a macroscopic volume: the constant value for XRR analysis and three-dimensional periodic function in case of HRXRD analysis.

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Notes

  1. 1.

    The parenthesis near underlined indices mean the symmetrization operation, \(a_{(\underline{i}\underline{j})kl}\equiv \frac{1}{2}(a_{ijkl}+a_{jikl})\). In these notations, the symmetry of stiffness tensor relatively the transposition of indices has a form \(c_{ijkl}=c_{(\underline{i}\underline{j})kl}=c_{ij(\underline{k}\underline{l})}=c_{(\underline{ij}\underline{kl})}\), where the latter equality means the symmetry with respect to the transposition of index pair.

  2. 2.

    To clarify whether the tensor is positive definite, the special representation (7.121) is used (see Sect. 7.4). In this representation, the positive definition corresponds to \(\varvec{C}\,{=}\,(3\kappa _c,2\min (\mu _c,\mu _c'),2\min (\mu _c,\mu _c'))\), and negative definition to \(\varvec{C}\,{=}\,(3\kappa _c,2\max (\mu _c,\mu _c'),2\max (\mu _c,\mu _c'))\), respectively.

  3. 3.

    The statements for Eshelby-Kröner model (7.69) are also valid for Hashin-Strickman model (7.63).

  4. 4.

    A similar situation occurs in Eshelby-Kröner model with the elliptical grains possessing a certain orientation [5].

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Authors and Affiliations

  1. Physics Department, Belarusian State University, Nezavisimosti Av. 4, 200030, Minsk, Belarus

    Andrei Benediktovitch & Ilya Feranchuk

  2. Rigaku Europe SE, Am Hardtwald 11, 76275, Ettlingen, Germany

    Alexander Ulyanenkov

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  1. Andrei Benediktovitch
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Correspondence to Andrei Benediktovitch .

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Benediktovitch, A., Feranchuk, I., Ulyanenkov, A. (2014). X-Ray Diffraction Residual Stress Analysis in Polycrystals. In: Theoretical Concepts of X-Ray Nanoscale Analysis. Springer Series in Materials Science, vol 183. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38177-5_7

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  • DOI: https://doi.org/10.1007/978-3-642-38177-5_7

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