International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 |
International Tables for Crystallography (2018). Vol. H, ch. 2.5, p. 142
Section 2.5.4.3.3. Equations for various stress states^{a}Bruker AXS Inc., 5465 E. Cheryl Parkway, Madison, WI 53711, USA |
The general triaxial stress state is not typically measured by X-ray diffraction because of low penetration. For most applications, the stresses in a very thin layer of material on the surface are measured by X-ray diffraction. It is reasonable to assume that the average normal stress in the surface-normal direction is zero within such a thin layer. Therefore, , and the stress tensor has five nonzero components. In some of the literature this stress state is denoted as triaxial. In order to distinguish this from the general triaxial stress state, here we name this stress state as the `biaxial stress state with shear'. In this case, we can obtain the linear equation for the biaxial stress state with shear:where the coefficient and σ_{ph} is the pseudo-hydrostatic stress component introduced by the error in the stress-free d-spacing. In this case, the stresses can be measured without the accurate stress-free d-spacing, since this error is included in σ_{ph}. The value of σ_{ph} is considered as one of the unknowns to be determined by the linear system. With the measured stress-tensor components, the general normal stress (σ_{ϕ}) and shear stress (τ_{ϕ}) at any arbitrary angle ϕ can be given by
Equation (2.5.81) can also be used for other stress states by removing the terms for stress components that are zero. For instance, in the biaxial stress state , so we have
In the 2D stress equations for any stress state with σ_{33} = 0, we can calculate stress with an approximation of d_{o} (or 2θ_{o}). Any error in d_{o} (or 2θ_{o}) will contribute only to a pseudo-hydrostatic term σ_{ph}. The measured stresses are independent of the input d_{o} (or 2θ_{o}) values (He, 2003). If we use to represent the initial input, then the true d_{o} (or 2θ_{o}) can be calculated from σ_{ph} with
Care must be taken that the σ_{ph} value also includes the measurement error. If the purpose of the experiment is to determine the stress-free d-spacing d_{o}, the instrument should be first calibrated with a stress-free standard of a similar material.
References
He, B. B. (2003). Introduction to two-dimensional X-ray diffraction. Powder Diffr. 18, 71–85.Google Scholar