Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T20:40:52.580Z Has data issue: false hasContentIssue false
  • English
  • Français

Stress determination through diffraction: establishing the link between Kröner and Voigt/Reuss limits

Published online by Cambridge University Press:  08 May 2015

Conal E. Murray ,
Jean L. Jordan-Sweet ,
Stephen W. Bedell  and
E. Todd Ryan
Conal E. Murray*
Affiliation:
IBM T.J. Watson Research Center, Yorktown Heights, New York 10598
Jean L. Jordan-Sweet
Affiliation:
IBM T.J. Watson Research Center, Yorktown Heights, New York 10598
Stephen W. Bedell
Affiliation:
IBM T.J. Watson Research Center, Yorktown Heights, New York 10598
E. Todd Ryan
Affiliation:
GLOBALFOUNDRIES Inc., Albany, New York 12203
*
a) Author to whom correspondence should be addressed. Electronic mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The quantification of stress in polycrystalline materials by diffraction-based methods relies on the proper choice of grain interaction model that links the observed strain to the elastic stress state in the aggregate. X-ray elastic constants (XEC) relate the strain as measured using X-rays to the state of stress in a quasi-isotropic ensemble of grains. However, the corresponding interaction models (e.g., Voigt and Reuss limits) often possess unlikely assumptions as to mechanical response of the individual grains. The Kröner limit, which employs a self-consistent scheme based on the Eshelby inclusion method, is based on a more physical representation of isotropic grain interaction. For polycrystalline aggregates composed of crystals with cubic symmetry, Kröner limit XEC are equal to those calculated from a linear combination of Reuss and Voigt XEC, where the weighting fraction, x Kr, is solely a function of the single-crystal elastic constants and scales with the material's elastic anisotropy. This weighting fraction can also be experimentally determined using a linear, least-squares regression of diffraction data from multiple reflections. Data on metallic thin films reveals that this optimal experimental weighting fraction, x*, can vary significantly from x Kr, as well as that of the Neerfeld limit (x = 0.5).

Keywords

stress determinationpolycrystalline materialsx-ray diffractionstrain anisotropyelasticity
Type
Technical Articles
Information
Powder Diffraction , Volume 30 , Issue 2 , June 2015 , pp. 99 - 103
Check for updates
Copyright
Copyright © International Centre for Diffraction Data 2015 

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

Behnken, H. and Hauk, V. (1986). “Berechnung der Röntgenographischen Elastizitätskonstanten (REK) des Vielkristalls aus den Einkirstalldaten fur beliebige Kristallsymmetrie,” Z. Metallk. 77, 620–626.Google Scholar
Bollenrath, F., Hauk, V. and Müller, E. H. (1967). “Zur Berechnung der vielkristallinen Elastizitätskonstanten aus den Werten der Einkristalle,” Z. Metallk. 58, 76–82.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. 12, 489–501.Google Scholar
Eshelby, J. D. (1957). “The determination of the elastic field of an ellipsoidal inclusion, and related problems,” Proc. R. Soc. Lond. A 241, 376–396.Google Scholar
Hill, R. (1952). “The elastic behaviour of a crystalline aggregate,” Proc. Phys. Soc. A 65, 349–354.CrossRefGoogle Scholar
Kröner, E. (1958). “Berechnung der elastichen Konstanten des Vielkristalls aus den Konstanten des Einkristalls,” Z. Phys. 151, 504–518.Google Scholar
Macherauch, E. (1966). “X-ray stress analysis”. Exp. Mech. 6, 140–153.Google Scholar
Möller, H. and Martin, G. (1939). “Elastiche Anisotropie und rontgenographische Spannungsmessung,” Mitt. Kaiser-Wilhelm-Inst. Eisenforsch. Düsseldorf 21, 261–269.Google Scholar
Murray, C. E. (2013). “Equivalence of Kröner and weighted Voigt–Reuss models for x-ray stress determination,” J. Appl. Phys. 113, 153509.Google Scholar
Murray, C. E., Bedell, S. W. and Ryan, E. T. (2013). “Weighted mechanical models for residual stress determination using x-ray diffraction,” J. Appl. Phys. 114, 033518.Google Scholar
Neerfeld, H. (1942). “Zur Spannungsberechnung aus röntgenographischen Dehnungsmessungen,” Mitt. K.-Wilh.-Inst. Eisenforsch. 24, 61–70.Google Scholar
Noyan, I. C. and Cohen, J. B. (1987). “Residual Stress” (Springer-Verlag, NY).Google Scholar
Reuss, A. (1929). “Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle,” Z. Agnew. Math. Mech. 9, 49–58.Google Scholar
Serruys, W., Van Houtte, P. and Aernoudt, E. (1988). “Why are X-ray measurements different from mechanical residual stress measurements?” Manuf. Technol. 37, 527–530.Google Scholar
Simmons, G. and Wang, H. (1971). “Single Crystal Elastic Constants and Calculated Aggregate Properties” (MIT Press, MA).Google Scholar
Stickforth, J. (1966). “Über den Zusammenhang zwischen röntgenographischer Gitterdehnung und makroskopischen elastischen Spnnungen,” Tech. Mitt. Krupp Forsch. Ber. 24, 89–102.Google Scholar
Tanaka, K., Suzuki, K., Sakaida, Y., Kimachi, H. and Akiniwa, Y. (2000). “Single crystal elastic constants of β-Silicon Nitride determined by X-ray powder diffraction,” Mater. Sci. Res. Int. 6, 249–254.Google Scholar
Van Houtte, P. and De Buyser, L. (1993). “The influence of crystallographic texture on diffraction measurements of residual stress,” Acta Metall. Mater. 41, 323–336.Google Scholar
Voigt, W. (1928). Lehrbuch der Krustallphysik (Teubner, Leipzig).Google Scholar
Wood, R. M. (1962). “The lattice constants of high purity alpha titanium,” Proc. Phys. Soc. 80, 783–786.CrossRefGoogle Scholar