Tables for
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 2.5, p. 144

Section Stress calculation

B. B. Hea*

aBruker AXS Inc., 5465 E. Cheryl Parkway, Madison, WI 53711, USA
Correspondence e-mail: Stress calculation

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The final data set after integration and peak evaluation should contain many data points describing the diffraction-ring shape for all collected frames. Each measured data point contains three goniometer angles (ω, ψ, ϕ) and the diffraction-ring position (γ, 2θ). The peak intensity or integrated intensity of the diffraction profile is another value to be determined and may be used in the stress calculation. In most cases the number of data points is more than the number of unknown stress components, so a linear least-squares method can be used to calculate the stresses. In a general least-squares regression, the residual for the ith data point is defined as[{r_i} = {y_i} - {\hat y_i},\eqno(2.5.86)]where [{y_i}] is the observed response value, [{\hat y_i}] is the fitted response value and [{r_i}] is the residual, which is defined as the difference between the observed value and the fitted value. The summed square of residuals is given by[S = \textstyle\sum\limits_{i = 1}^n {r_i^2 = } \textstyle\sum\limits_{i = 1}^n {{{({y_i} - {{\hat y}_i})}^2}}, \eqno(2.5.87)]where n is the number of data points and S is the sum-of-squares error to be minimized in the least-squares regression. For stress calculation, the observed response value is the measured strain at each data point,[{y_i} = \ln \left({{{\sin \theta _0} \over {\sin \theta _i}}} \right),\eqno(2.5.88)]and the fitted response value is given by the fundamental equation as[{\hat y_i} = {p_{11}}{\sigma _{11}} + {p_{12}}{\sigma _{12}} + {p_{22}}{\sigma _{22}} + {p_{13}}{\sigma _{13}} + {p_{23}}{\sigma _{23}} + {p_{33}}{\sigma _{33}} + {p_{\rm ph}}{\sigma _{\rm ph}},\eqno(2.5.89)]where all possible stress components and stress coefficients are listed as a generalized linear equation. Since the response-value function is a linear equation of unknown stress components, the least-squares problem can be solved by a linear least-squares regression. In order to reduce the impact of texture, large grains or weak diffraction on the results of the stress determination, the standard error of profile fitting and the integrated intensity of each profile may be introduced as a weight factor for the least-squares regression (He, 2009[link]).


He, B. B. (2009). Two-dimensional X-ray Diffraction. New York: John Wiley & Sons.Google Scholar

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