Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T14:18:33.614Z Has data issue: false hasContentIssue false
  • English
  • Français

Properties and general use of the X-ray elastic factors

Published online by Cambridge University Press:  01 March 2012

Balder Ortner
Balder Ortner
Affiliation:
Erich Schmid Institute of Materials Science, Austrian Academy of Sciences and Institute of Material Physics, University Leoben, Jahnstraße 12, A-8700 Leoben, Austria
Get access Rights & Permissions [Opens in a new window]

Abstract

The equation ε(φ, ψ, hkl)=Fij(φ, ψ, hkl)σij can be directly deduced from Hooke’s law. It is shown that the matrix Fij(φ, ψ, hkl) which is usually called X-ray elastic factors, behaves as a second rank tensor. Since this behaviour is the only criterion for the question of whether or not it is a tensor, the F-matrix must be regarded as a second rank tensor. This allows us to make some statements about the structure of the F-matrix on the basis of Neumann’s principle, to find relationships among F-matrices in different measurement directions, and to apply the methods and strategies for the measurement of a second rank tensor. All this is shown in a few examples. It is further shown that a consistent use of the F-matrix can replace all methods for data evaluation which makes use of linear regressions and, in addition, avoids all difficulties and disadvantages of these methods. One of these disadvantages is that the sin2 ψ-method, as well as its derivatives, is generally not correct least square fits of the measured data. This is also shown in an example. The more complicated cases with stress or constitution gradients in the range of the probed volume or stress measurement after plastic deformation are not discussed.

Keywords

residual stressX-ray elastic factorssin2 ψ-method
Type
X-Ray Diffraction and Related Techniques
Information
Powder Diffraction , Volume 22 , Issue 2 , June 2007 , pp. 103 - 107
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ben-Israel, A. and Greville, T. N. E. (2003). Generalized Inverses, 2nd ed. (Springer-Verlag, Berlin).Google Scholar
Dölle, H. (1979). “The influence of multiaxial stress states, stress gradients and elastic anisotropy on the evaluation of (residual) stresses by X-rays,” J. Appl. Crystallogr.JACGAR10.1107/S0021889879013169 12, 489–501.CrossRefGoogle Scholar
Hauk, V. (1997). Structural and Residual Stress Analysis by Nondestructive Methods (Elsevier, Amsterdam).Google Scholar
Lu, J. (Ed.). (1996). Handbook of Measurement of Residual Stresses (Fairmont Press, Lilburn, Georgia).Google Scholar
Noyan, I. C. and Cohen, J. B. (1987). Residual Stress: Measurement by Diffraction and Interpretation (Springer-Verlag, Berlin).CrossRefGoogle Scholar
Ortner, B. (1989). “On the Selection of Measurement Directions in Second-Rank Tensor (e.g., Elastic Strain) Determination of Single Crystals,” J. Appl. Crystallogr.JACGAR 22, 216–221.CrossRefGoogle Scholar
Ortner, B. (2005). “An Analytic and Generalized Formulation of the sin2 ψ-method,” Z. Metallkd.ZEMTAE 96, 1049–1055.CrossRefGoogle Scholar
Ortner, B. (2006). “Symmetry Properties and Transformation Behaviour of the X-ray Stress Factors,” J. Appl. Crystallogr.JACGAR 39, 401–409.CrossRefGoogle Scholar
Ortner, B. (2007). “Are Data Correctly Fitted in the sin2 ψ- and Similar Methods?,” (to be submitted) Int. J. Mater. Res. (Z. Metallkd.).Google Scholar
Winholtz, R. A. and Cohen, J. B. (1988). “Generalized Least-Squares Determination of Triaxial Stress States by X-ray Diffraction and the Associated Errors,” Aust. J. Phys.AUJPAS 41, 189–199.CrossRefGoogle Scholar