Implementation of the self-consistent Kröner-Eshelby model for the calculation of X-ray elastic constants for any crystal symmetry

Arnold C. Vermeulen, Christopher M. Kube, Nicholas Norberg

Research output: Contribution to journalArticle

Abstract

In this paper, we will report about the implementation of the self-consistent Kröner-Eshelby model for the calculation of X-ray elastic constants for general, triclinic crystal symmetry. With applying appropriate symmetry relations, the point groups of higher crystal symmetries are covered as well. This simplifies the implementation effort to cover the calculations for any crystal symmetry. In the literature, several models can be found to estimate the polycrystalline elastic properties from single crystal elastic constants. In general, this is an intermediate step toward the calculation of the polycrystalline response to different techniques using X-rays, neutrons, or ultrasonic waves. In the case of X-ray residual stress analysis, the final goal is the calculation of X-ray Elastic constants. Contrary to the models of Reuss, Voigt, and Hill, the Kröner-Eshelby model has the benefit that, because of the implementation of the Eshelby inclusion model, it can be expanded to cover more complicated systems that exhibit multiple phases, inclusions or pores and that these can be optionally combined with a polycrystalline matrix that is anisotropic, i.e., contains texture. We will discuss a recent theoretical development where the approaches of calculating bounds of Reuss and Voigt, the tighter bounds of Hashin-Shtrikman and Dederichs-Zeller are brought together in one unifying model that converges to the self-consistent solution of Kröner-Eshelby. For the implementation of the Kröner-Eshelby model the well-known Voigt notation is adopted. The 4-rank tensor operations have been rewritten into 2-rank matrix operations. The practical difficulties of the Voigt notation, as usually concealed in the scientific literature, will be discussed. Last, we will show a practical X-ray example in which the various models are applied and compared.

Original languageEnglish (US)
Pages (from-to)103-109
Number of pages7
JournalPowder Diffraction
Volume34
Issue number2
DOIs
StatePublished - Jun 1 2019

Fingerprint

Crystal symmetry
Elastic constants
elastic properties
X rays
symmetry
crystals
x rays
coding
inclusions
Point groups
stress analysis
Ultrasonic waves
ultrasonic radiation
matrices
Stress analysis
residual stress
Tensors
Residual stresses
Neutrons
textures

All Science Journal Classification (ASJC) codes

  • Radiation
  • Materials Science(all)
  • Instrumentation
  • Condensed Matter Physics

Cite this

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title = "Implementation of the self-consistent Kr{\"o}ner-Eshelby model for the calculation of X-ray elastic constants for any crystal symmetry",
abstract = "In this paper, we will report about the implementation of the self-consistent Kr{\"o}ner-Eshelby model for the calculation of X-ray elastic constants for general, triclinic crystal symmetry. With applying appropriate symmetry relations, the point groups of higher crystal symmetries are covered as well. This simplifies the implementation effort to cover the calculations for any crystal symmetry. In the literature, several models can be found to estimate the polycrystalline elastic properties from single crystal elastic constants. In general, this is an intermediate step toward the calculation of the polycrystalline response to different techniques using X-rays, neutrons, or ultrasonic waves. In the case of X-ray residual stress analysis, the final goal is the calculation of X-ray Elastic constants. Contrary to the models of Reuss, Voigt, and Hill, the Kr{\"o}ner-Eshelby model has the benefit that, because of the implementation of the Eshelby inclusion model, it can be expanded to cover more complicated systems that exhibit multiple phases, inclusions or pores and that these can be optionally combined with a polycrystalline matrix that is anisotropic, i.e., contains texture. We will discuss a recent theoretical development where the approaches of calculating bounds of Reuss and Voigt, the tighter bounds of Hashin-Shtrikman and Dederichs-Zeller are brought together in one unifying model that converges to the self-consistent solution of Kr{\"o}ner-Eshelby. For the implementation of the Kr{\"o}ner-Eshelby model the well-known Voigt notation is adopted. The 4-rank tensor operations have been rewritten into 2-rank matrix operations. The practical difficulties of the Voigt notation, as usually concealed in the scientific literature, will be discussed. Last, we will show a practical X-ray example in which the various models are applied and compared.",
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Implementation of the self-consistent Kröner-Eshelby model for the calculation of X-ray elastic constants for any crystal symmetry. / Vermeulen, Arnold C.; Kube, Christopher M.; Norberg, Nicholas.

In: Powder Diffraction, Vol. 34, No. 2, 01.06.2019, p. 103-109.

Research output: Contribution to journalArticle

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